Modelling in Computationally Intense Field Of Finance

Prof Dr A S Chaudhuri, Former Fellow, IIAS Shimla

Professor of Electronics and Communication Engineering

Dr B C Roy Engineering College, Durgapur West Bengal

and Dr Ranjan Chaudhuri, Associate Professor (Marketing)

National Institute of Industrial Engineering (NITIE)

Vihar Lake, PO NITIE Mumbai 400087 India

Abstract

The specification of a model almost exclusively involves purely economic considerations. The model may

be used as an aid in economic analysis, policy simulation, or policy optimization, but each case imposes

special demands on the specification. The result of such considerations generally determines the overal1

size of the model, the number of inputs and outputs, and the definition of these variables. In addition, the

outputs of the model are usually decomposed into two types: the endogenous variables which are outputs of

dynamic equations, called behavioral equations and variables which are outputs of non-stochastic

equations, called definitional equations. A choice must be made as to the use of variables in current price

(inflated) or constant price (deflated). The economic specification stage can be summarized as one in which

the following information is determined:

1) The specific purpose of the model, thereby fixing the overall size; and hence, an enumeration of all the

outputs and their type and an enumeration of all the inputs and their type.

2) The output definitions; whether it is explained by a behavioral equation together with all its explanations

(inputs to the equation), or, whether it is determined by a definitional identity.

The second stage is the most challenging of the two. This stage combines the use of a priori economic

information, hypothesis testing techniques, and cross-correlation analysis from the black box approach. In

econometric terminology, the word “structure” denotes a complete definition of the functional or stochastic

relationships between all of the endogenous and exogenous variables. The specific meaning of structure can

be ascertained by examining each equation of the structural form. Before accepting the results of any

estimation, they must be tested for their adequacy. The auto- and cross-correlation functions for the model

residuals constitute an important diagnostic check. The last diagnostic to be employed is perhaps the most

important,, namely, the model’s forecasting performance. After having successfully met the other

2

diagnostic checks a model is not accepted until it has demonstrated its ability to forecast. Forecasts are then

made with each model from the end of its sample period up to the present, using the (historical) observed

inputs over this period. Thus, forecasts are obtained outside of the sample period . Such simulations more

closely approach reality and serve as a good guide in judging the model’s adequacy in forecasting the

unknown future. This gives additional insight into the time-invariance of the model structure. The modeling

procedure described in this paper was designed to incorporate three concepts. First, employment is made of

all available a priori information provided by thus eliminating beforehand the possibility of expending

effort on fruitless searches for non-existent relationships (interconnections), Second, the basic philosophy

of the “black box” approach is then applied allowing the data to decide the exact dynamic structure. Thus,

overly complex (statistically unsubstantiated) structures are automatically eliminated. Third, diagnostics are

continually employed which are designed to both reveal inadequacies and indicate how improvements can

be made.

Introduction

Financial engineering, the most computationally intense field of finance, has only come to be recognized as

a formal profession over the last four or five years. During that time, the International Association of

Financial Engineers (IAFE) has been instrumental in helping to define the profession and in promoting the

creation of financial engineering research programs. Technological sciences recently reveals the revolution

in financial services. For more than a half-century statistics and technical analysis have been the

technologies of choice for financial analysts. However, it was not until the introduction of the Hamiltonian-

Jacobi-Bellman and Black-Scholes differential equation in the mid-70’s that more advanced forms of

mathematics were used in the field of finance. Since that time there has been a tremendous expansion in

the application of mathematics and other engineering technologies to the field of finance. The use of

mathematics has been spurred by the availability of low-cost computers and supercomputers. In addition

there has been an luminous gathering of engineers, mathematicians, and physicists into the finance and

investments industry. Many of these professionals are excited about the opportunities to apply their

quantitative and analytical skills in new ways. And finally there is the explosive growth in the financial

markets in volume, numbers and types of securities offered, and their internationalization under GATT and

3

WTO regimes which has required the intensive use of engineering technologies for development of highly

skillful computerized software.

The future of the continuing convergence of the fields of finance and engineering into a new era of

computational intelligence and its application in the financial services arena has been envisaged. There are

neural nets that emulate the Black-Scholes equation with better out-of-sample performance than the

original equation. Commercially available software using genetic algorithms provide superior results for

portfolio asset allocation. Moreover today’s computational intelligence methodologies can forecast the

inflation rate to 95 % accuracy. It is the mission of the present monograph to be a forum for new

technologies and applications in the intersection of computational intelligence and financial engineering.

The interest in application of the ingredients of computational intelligence to finance has been growing

steadily.

A few years ago, the term "financial engineering" did not exist in common usage; it is only recently that

the various models and algorithms now regarded collectively as financial engineering have coalesced into a

well-defined discipline. Three factors are largely responsible for this mighty achievement. The first is the

simple fact that the financial system is becoming more complex over time. This is an obvious consequence

of general economic growth and development in which the number of market participants, the variety of

financial transactions, and the sums involved also grow. As the financial system becomes more complex,

the benefits of more highly developed financial technology become greater and greater and, ultimately,

indispensable. The second factor is the set of professional developments in the quantitative modelling of

financial markets, e.g., financial technology, pioneered over the past three decades by the giants of

financial professionals: Black, Cox, Lintner, Markowitz, Merton, Modigliani, Miller, Ross, Samuelson,

Scholes, Sharpe, and others. Their contributions laid the remarkably durable foundations on which all of

modern quantitative financial analysis is built. The third factor is an almost parallel set of technological

breakthroughs in computer technology, including hardware, software, and data collection and organization.

Without these breakthroughs, much of the financial technology developed over the past thirty years would

be irrelevant academic observations. Precisely, Professor Merton’s work in the continuous time

mathematics of finance serves as one of the cornerstones on which the profession of financial engineering

is built. Professor Merton redefined modern financial economics definitively, and helped to launch a multi-

4

trillion-dollar industry that is still enjoying double-digit growth. While the methods and techniques range

broadly, from optimal control and filtering to neural networks to nonparametric regression, the topics are

all sharply focused on financial applications. In this way, financial engineering is following a path not

unlike those of the engineering disciplines in their formative stages: applications tend to drive the

technology, yet research and development are characterized by an intellectual entrepreneurialism that cuts

across many different methodologies. No one has illustrated this entrepreneurialism more eloquently than

Harry Markowitz, the father of modern portfolio theory and a winner of the 1990 Nobel Prize in

economics.

In the past, Wall Street and the City in London employed academics to help them develop innovative financial products

and to carry out quantitative modelling work. Besides academics, scientists and engineers were also employed by

financial institutions and trained in-house as financial engineers. In view of this evident need for engineers and

scientists to work and be trained as financial engineers, universities nowadays provide a formal educational route for

acquiring training in financial engineering. Most graduate programmes are being offered by engineering schools and

mathematics departments, though some are being offered by business schools. The understanding and mitigation of risk

are increasingly important aspects of modern finance. Running on the speed and capacity of the Internet and other

information technologies, the world’s financial markets have grown dramatically in size, interconnectedness and

complexity. With more opportunities available, investors are more willing to accept the risks associated with

entrepreneurial ventures to create new financial products and services. But the combination of technological innovation

and globalisation that created the new economy also brings new sources of risk to financial markets.

Financial engineering is defined by the International Association for Financial Engineering (IAFE), a professional body

for financial engineers based in the USA, as: ‘The development and creative application of financial technology to

solve financial problems and exploit financial opportunities. Further, financial engineering involves the design, the

development and the implementation of innovative financial instruments and processes and the formulation of creative

solutions to problems in finance.”

Financial engineering is a multidisciplinary field that combines finance, mathematics and statistics, engineering and

computer science. The related areas in these fields are financial markets, mathematical finance and econophysics and

computational finance. Quantitative and analytical principles and methods are critical in understanding financial

instruments in financial markets today, so that the new discipline of financial engineering has evolved. Financial

engineering requires a composite of skills. For example, the methodology of science and mathematics has been used in

financial engineering areas such as derivative pricing and hedging, portfolio management and risk management.

5

Stochastic calculus helps financial engineers to price exotic options and to hedge the risks. Advances in Monte Carlo

simulation have been applied to risk management. For selecting optimal portfolios, optimisation techniques have been

applied to the asset and liability management problem. The neural networks and genetic algorithms used in engineering

and physical science fields have been applied to forecasting futures prices for trading and investment. A financial

engineer typically works in areas that require a high level of quantitative skills and innovative ideas in structured

finance, derivative trading, risk management, portfolio investments, corporate financing, financial and insurance

products and financial information technology.

The domain knowledge includes financial markets, financial products and processes, price and hedge modelling,

investment technology, risk analysis, computational methods, and data support systems for trading. When institutions

create desired pay-off patterns that manage risk for their clients or use options and futures to hedge the products they

sell, they are engaging in financial engineering.

A distinct professional category called financial engineers is needed by financial institutions to develop new financial

products, to customise and trade them, to monitor risk exposure to books of complex derivatives, to devise hedging

schemes and to search for arbitrage opportunities in the markets. Therefore, there is a need for a new degree

programme for the training of financial engineers. The range of fields in which financial engineers can establish

careers includes risk management, structured financial products, quantitative trading and research.

A wide range of businesses offer career opportunities for financial engineers, including:

• commercial and investment banks

• brokerage and investment firms

• insurance companies

• accounting and consultancy firms

• treasury departments of non-financial corporations

• public institutions such as federal government agencies, state and local governments, municipalities

and international organisations

• software and technology vendors that provide products and services to the financial industry.

Financial engineering is a fairly recent field of study. It combines technical and conceptual advances from various

disciplines to create a broad interdisciplinary body of knowledge, ready to meet the challenges of a rapidly growing

market and an exciting future. John O'Brien, Executive Director of the University of California Berkeley's MFE

programme believes that 'Financial engineering is taking its place, along with traditional engineering disciplines, as a

major driving force in the global economy". A key to the growth of financial engineering has been its maintenance of a

strong interdisciplinary tradition and its appeal across all quantitative fields. The rapid assimilation of new

6

methodology into practice is both a result of and reason for this interdisciplinarity. Interest from the multidisciplinary

fields in engineering science and finance has spurred the growth of degree programmes in financial engineering.

Financial engineering is a fast-growing dynamic field driven by the needs of industry and based on the rich interplay of

and input from many disciplines, including financial economics, probability, numerical methods, operations research,

systems engineering, physics and statistics.

This monograph highlights the future of financial engineering in the making.

A model building procedure will now be outlined. The procedure follows in three stages: 1) Preliminary

Structure Determination, 2) Parameter Estimation, and 3) Diagnostics. These stages are not disjoint, for in

reality the results of any one may alter the results of another and thus require a repeated application of

some or all of the previous stages. Experience has shown that the actual construction of a model involves

repeated cycling through the stages.

The specification of a model almost exclusively involves purely economic considerations. The model may

be used as an aid in economic analysis, policy simulation, or policy optimization, but each case imposes

special demands on the specification. The result of such considerations generally determines the overal1

size of the model, the number of inputs and outputs, and the definition of these variables. In addition, the

outputs of the model are usually decomposed into two types: the endogenous variable which are outputs of

dynamic equations, called behavioral equations and variables which are outputs of non-stochastic

equations, called definitional equations. A choice must be made as to the use of variables in current price

(inflated) or constant price (deflated). The economic specification stage can be summarized as one in which

the following information is determined:

1) The specific purpose of the model, thereby fixing the overall size; and hence, an enumeration of all the

outputs and their type, and, an enumeration of all the inputs and their type.

2) The output definitions; whether it is explained by a behavioral equation together with all its explanations

(inputs to the equation), or, whether it is determined by a definitional identity.

The second stage is the most challenging of the two This stage combines the use of a priori economic

information, hypothesis testing techniques, and cross-correlation analysis from the black box approach. In

econometric terminology, the word “structure” denotes a complete definition of the functional or stochastic

relationships between all of the endogenous and exogenous variables. The specific meaning of structure can

be ascertained by examining each equation of the structural form. Before accepting the results of any

7

estimation, they must be tested for their adequacy. The auto- and cross-correlation functions for the model

residuals constitute an important diagnostic checks. The last diagnostic to be employed is perhaps the most

important,, namely, the model’s forecasting performance. After having successfully met the other

diagnostic checks a model is not accepted until it has demonstrated its ability to forecast. Forecasts are then

made with each model from the end of its sample period up to the present, using the (historical) observed

inputs over this period. Thus, forecasts are obtained outside of the sample period . Such simulations more

closely approach reality and serve as a good guide in judging the model’s adequacy in forecasting the

unknown future. This gives additional insight into the time-invariance of the model structure. The modeling

procedure described in this monograph was designed to incorporate three concepts. First, employment is

made of all available a priori information provided by thus eliminating beforehand the possibility of

expending effort on fruitless searches for non-existent. relationships (interconnections), Second, the basic

philosophy of the “black box” approach is then applied allowing the data to decide the exact dynamic

structure. Thus, overly complex (statistically unsubstantiated) structures are automatically eliminated.

Third, diagnostics are continually employed which are designed to both reveal inadequacies and indicate

how improvements can be made.

Linear Quadratic Tracking Problem to Economic Stabilization

Some of the typical policy problems of recent interest. have been if how and when taxes should be raised or

lowered, whether the money supply should grow at a constant rate or be adjusted in response to economic

conditions, when and how fast the Reserve Bank of India should change credit conditions, and whether

government expenditures should expand or contract,. The objectives of these policies have included such

successes as the minimization of unemployment and the maintenance of full-capacity output, the control of

inflation, a reasonable rate of economic growth, and perhaps more recently, the elimination of poverty

through income redistribution. The control theory has been found useful and often essential to advance the

economic theory and better regulate economies. Application of control engineering viewpoints and control

techniques to treat stabilization of economic systems using optimal control and optimal filtering theories

have attracted many economists and control engineers.

The economic stabilization has been approached as a dual tracking problem in optimal control. The

problem that is defined and solved involves tracking nominal state and nominal policy trajectories, subject

8

to a quadratic cost functional and the constraint of a linear system. This is actually quite general, and will

enable us to penalize for variations in, as well as for the levels of, the state variables and control variables.

The aim of the policy plan will be to make xi , the state vector, track as closely as possible a nominal state-

vector , but subject to ut, the control vector, tracking a nominal control vector In other words, it is desired

that variables such as GNP, investment., and unemployment to follow as closely as possible nominal or

“ideal” time paths throughout the planning period. The nominal time paths for GNP and investment for

example, would probably grow at some steady rate, while that for unemployment to follow as closely as

possible nominal or “ideal” time paths throughout the planning period. The control variables, of course, are

used to make GNP, investment, and unemployment move in the desired direction, but we are not free to

manipulate the control variables in any way whatsoever; they in turn must also stay close to a set of

nominal or ideal time paths. For example, we are not free to increase government spending or the money

supply by 100 percent in one year and decrease them by 200 percent in the next year, etc. Manipulating

policy variables has very real costs associated with it, and these costs must be embodied in the cost

functional.

The system of interest is of the form

with the initial condition

Here x i is the n-dimensional state vector at time i, ui the r-dimensional control vector at time i, and zi an s-

dimensional vector representing, at time i, s exogenous variables which are known for all i but cannot be

controlled by the policy planner. A, B, and C are time-invariant n X n, n X r, and n X s matrices.

Let xi and ui be the nominal state and control vectors that is desired to be tracked. At time i we would like

xi to be close to xi and ui to be close to ui. We assume that xi and ui have been specified for the entire

planning period, i = , 1, … , N . The cost functional is given by

where Q is an n X n positive semi-definite matrix and R is an r X r positive definite matrix. The optimal

control problem is to find a control sequence {u i , i = 0, 1, . . . , N - 1} such that

9

and the cost functional (3) is minimized. It is important to keep in mind the meaning of the cost functional (3). Both Q and R will normally be

diagonal matrices. The elements of Q give the relative costs of deviating from the nominal path of each

state variable, for example, the cost of deviating from nominal GNP relative to the cost of deviating from

nominal unemployment. Some of the elements of Q may be zero. The elements of R give the relative costs

of deviating from the nominal paths of the control variables. For example, we would expect it to be more

costly to manipulate the tax rate than to manipulate the money supply. All of the diagonal elements of R

must be nonzero. This is both meaningful in terms of the economic problem and necessary for a

mathematical solution. Finally, the comparative magnitudes of Q and R give the costs of controlling the

economy relative to the costs of having the economy deviate from its ideal path, that is, the relative costs of

means versus ends.

Solving for the optimal control

The Hamiltonian is

where p i is the vector of costates. The canonical equations for the problem then are

and these are subject to the split boundary conditions

where (10) is a result of the transversality condition. Finally, the minimization of the Hamiltonian is written as

10

yielding

The assumption that p *

i is of the form

Substituting this into ( 12 ) gives

and substituting ( 13 ) and ( 14 ) into ( 7 ) and ( 8) gives

Substituting ( 13 ) into the left hand side of ( 15b ) rearranging terms gives

It is assumed that E is nonsingular and E -1 exists. Then (16a ) can be written as

11

Equating the coefficients of x *

i

Equations ( 24 ) and ( 25 ) provide the boundary conditions for ( 21 ) and ( 22 ) respectively.

12

All of the above steps involve iterative solutions (and only N iterations) that require little more than

multiplying and adding matrices (albeit large matrices, where n might. be on the order of several hundred

for a large econometric model).

13

The economic stabilization policy is approached as a problem in optimal control. The optimal control

problem is defined as a dual discrete-time tracking problem for a linear time-invariant system with a

quadratic cost functional.

It is important to keep in mind the fact that the cost functional accumulates penalties only over a finite time

period. Optimal paths for some variables behave strangely during the last few quarters of the planning

period. Changes in the money supply, for example, affect the short-term interest rate immediately, but at

least two quarters must elapse before there is any impact on residential investment and hence on GNP.

Therefore, if the cost functiona1 does not penalize directly for interest-rate deviations from the nominal, the

optimal quarterly change in the money supply wi11 always be equal to the nominal value. The solution to

this problem is to extend the planning period beyond the time horizon of actual interest. If, for example,

one was interested in formulating an optimal stabilization policy for the next three years, he should extend

the planning period to four or five years to obtain a numerical solution.

Modelling of Stochastic Analysis and Stochastic Control

Stochastic analysis and stochastic control have entered the field of finance at a very rapid, sometimes

explosive, pace in recent years. Powerful techniques from these disciplines have been brought to bear on

almost all aspects of mathematical finance: the study of arbitrage, hedging, pricing, consumption/portfolio

optimization, incomplete and/or constrained markets, equilibrium, differential information, the term-

structure of interest rates, transaction costs and so on. At the same time, the development of sophisticated

analytical and numerical methods, based primarily on partial differential equations and on their numerical

solutions, have helped to increase the relevance of these contributions in the everyday practice of finance.

Stochastic Analysis and Stochastic Control is addressed to a wide audience of control people from

academia and industry. The mathematics of finance and financial engineering is addressed to both

engineers and applied mathematicians from the following areas: stochastic control, stochastic analysis,

applied probability, estimation and identification, systems theory, stability, adaptive control, linear and

nonlinear systems, computational methods, optimization, modelling and control applications. It is important

to bring together leading stochastic control researchers working in a new exciting area of application of

stochastic control theory. It is also important to stress that the methods used in the mathematics of finance

have been successfully used in other fields.

14

Adaptive Control in the mathematics of finance and financial engineering

Lai and Wong [1] start their paper with a brief review of recent developments in the pricing and hedging of

American options. This paper then modifies the basis function approach to adaptive control and

neurodynamic programming and applies it to develop nonparametric pricing formulas for actively traded

American options as well as simulation-based optimization strategies for complex over-the-counter options;

the corresponding optimal stopping problems are prohibitively difficult to solve numerically by standard

backward induction algorithms because of the curse of dimensionality. An important issue in this approach

is the choice of the basis functions, for which some guidelines and underlying theory are provided.

In [2], Ross writes “Despite such gaps (in the theory), when judged by its ability to explain the empirical

data, option pricing theory is the most successful theory not only in finance, but in all of economics.” A call

(put) option gives the holder the right to buy (sell) the underlying asset by a certain date (known as the

“expiration date” or maturity”) at a certain price (known as the “strike price”). European options can be

exercised only on the expiration date, whereas American options can be exercised at any time up to the

expiration date. For European options, closed-form option pricing formulas were derived in the seminal

papers [3] and [4] through a dynamic hedging argument and a no-arbitrage condition. Except for Merton’s

[4] result that American calls written on nondividend-paying stocks reduce to European calls, American

option valuation involves finite-horizon optimal stopping problems and has to be performed numerically.

One of the “gaps” in option pricing theory noted by Ross is that “surprisingly little is known about the

exercise properties” of American options. Another gap is related to “a surprisingly small empirical

literature” which “should increase; options and option pricing theory give us an opportunity to measure

directly the degree of anticipated uncertainty in the markets.” During the fifteen years that have elapsed

since Ross’ review, there have been many advances in option pricing theory and the gaps mentioned by

Ross have been narrowed considerably. However, much still remains to be done. This work modifies the

basis function approach to adaptive control and neuro-dynamic programming and apply it to theoretical and

empirical analysis of American options. For actively traded American options using market data to provide

a sufficiently large training sample for estimating the parameters of a suitably chosen learning network to

price and hedge a new option. The choice of basis functions for the learning network is based on a

decomposition formula for the value function of the optimal stopping problem that gives the price of an

15

American option. This work also considers over-the-counter (OTC) American options. Since trading is

performed “over the counter,” the options data are not available for nonparametric estimation of pricing

formulas. Therefore, the traditional parametric approach is adopted instead. However, the underlying

optimal stopping problem is often prohibitively difficult to solve numerically by backward induction

because of the “curse of dimensionality” when the payoff is associated with multidimensional state

variables. To circumvent this difficulty, the work uses basis functions to implement a simulation-based

optimization scheme. Tsitsiklis and Van Roy [5], [6] have recently introduced this “neuro-dynamic

programming” approach to pricing highly complex American options that involve multidimensional asset

prices, but have not provided a systematic way of choosing the basis functions for this approach. Lai and

Wong consider this problem, review recent developments in simulation-based optimization for American

option valuation and provide some underlying theory. The interplay of leading-edge stochastic control

theory and American option valuation is discussed.

Neural and other learning networks and simulation-based dynamic programming have a burgeoning

literature in stochastic control [7-43]. In this work Lai and Wang apply these ideas to the valuation of both

actively traded and over-the-counter American options. They point out several basic issues that have come

up in the course of applying these ideas. One is the choice of basis functions. Closely related is the balance

between statistical efficiency and computational complexity in the presence of large data sets. Another

algorithm to compute functions via regression and simulation. Although American option valuation only

involves the simplest kind of stochastic control problems, namely, optimal stopping, it already encounters

the long-standing “curse-of-dimensionality,” and how it handles this “curse” should provide useful clues

for the solution of more complicated stochastic control problems.

Simulation in Financial Engineering

Simulation algorithms in the field of financial engineering offers challenges specific to financial simulation

and approaches that researchers have developed to handle them [283-288]. Many problems in financial

engineering require numerical evaluation of an integration. Several merits make simulation popular among

professionals as a methodology for these computations.

First it is simple to apply to many problems. For most derivative securities and financial models, even those

that are complicated and high dimensional, it takes relatively easy work to create a simulation algorithm for

16

pricing the derivatives under the model. Pitfalls in numerical implementation of simulation algorithms are

relatively rare. For the most part a little knowledge and effort go a long way in financial simulations.

The second merit of simulation is its good performances on high-dimensional problems: the rate of

convergence of a Monte-Carlo estimate does not depend on the dimension of the problem. The issues of

dimension and convergence are becoming increasingly important as securities markets and financial risk

management become more sophisticated.

The third attraction of simulation is the confidence interval that it provides for the Monte-Carlo estimate.

This information makes possible of how much computational effort might be needed in order to produce an

estimate of acceptable quality.

Financial engineers frequently apply simulation techniques to derivative securities. The derivatives are

financial instruments whose payoffs derive from the values of other underlying financial variables, such as

prices and interest rates. The example is the European call option, whose payoff is max{ST – K, 0}, where

ST is the price of the stock at time T, and K is the pre-specified amount called the strike price. This option

gives its owner the right to buy the stock at time T for the strike price K: If ST > K, the owner will exercise

this right, and if not, the option expires worthless. If the future payoff of a derivate derives from the

underlying, is there a way to derive the present price of the derivative from the current value of the

underlying? Under some theoretical conditions on the payoff of the derivative, and the possibilities for

trading in the market, the answer is yes. If it is possible to replicate the derivative’s payoff by trading in a

portfolio of securities available on the open market, then the combination of executing this trading strategy

and selling the derivatives has no risk. This is known as hedging the sale of derivatives, and hedging

strategies are of great practical interest in their own right, as well as being of theoretical interest in

justifying no-arbitrage pricing. The pricing theory has this name because it postulates that there are no

arbitrages, which are opportunities to make a positive amount of money with zero risk or cost. Such

opportunities are suppose to disappear, should they exist, because unlimited demand for them would drive

their cost above zero.

The riskless combination of a derivative minus the initial portfolio of its replicating strategy must have

nonpositive cost to avoid arbitrage; assuming the same of the opposite combination, the price of the

derivative must equal the cost of its initial replicating portfolio. A basic theorem of mathematical finance

17

states that this price us the expectation of the derivative’s discounted payoff under an equivalent martingale

measure. This is a probability measure under which discounted asset prices are martingale, and it generally

does not coincide with the original probability measure which models the real world. When discounting is

done with the value of a riskless money market account, the equivalent martingale measure is known as the

risk-neutral measure, because each investor had a neutral attitude towards risk, he would demand the same

return on all risky assets as on a riskless asset.

Given this, pricing a derivative evaluating the expectation of the sum of all its discounted payoff, under a

specified measure. The discounting is crucial and allows for appropriate comparison between cash flows,

whether positive or negative, at different times. Since the probability measure of financial models typically

have density, derivative pricing is evaluating the integral of the product of payoff and probability density

over all possible paths of the underlying.

As an example, consider the European call option under the Black-Scholes model , for which the

distribution of the log stock price ln ST is normal with mean and variance T under a

probability measure P. Here S0 is the initial stock price and and are called respectively the drift and

volatility. Under the risk neutral measure Q, ln ST is normal with mean ln ST + ( /2)T and the same

variance r is the instantaneous interest rate on a riskless money market account. No-arbitrage price of the

European call option is

and are respectively the cumulative distribution and probability density function of the standard normal.

This is the famous Black-Scholes formula.

The standard Mone-Carlo approach to evaluating such expectation is to simulate under the equivalent

martingle measure a state vector which depends on the underlying variables, then evaluate the sample

average of the derivative payoff over all trials. This is an unbiased estimate of the derivative’s price, and

18

when the number of trials n is large, the central limit theorem CLT provides a confidence interval for the

estimate, based on the sample variance of the discounted payoff.

The Monte-Carlo approach is similar for other financial engineering problems, such as finding hedging

strategies and analyzing portfolio return distributions in order to assess the risk of one current portfolio or

select a portfolio with the most attractive combination of reward and risk. All of these rely on the same

basic approach of simulating many trials, each of which is a path of underlying financial variables over a

period of time, computing the values of derivative on this path, and looking at the distribution of these

values.

In some application of simulation, there is no great conceptual difficulty involved in generating simulated

path, other than that of producing pseudo-random number with a digital computer. For instance, when

estimating the steady-state mean of a random variable in a queuing system, the model specifies the

transition rate from any state, and it is not theoretically difficult to sample the next state from the correct

distribution. The situation in financial situation is not so simple. The models of mathematical finance are

usually specified by stochastic differential equation (SDE) under the equivalent martingale measure used

for pricing. Sometimes it is possible to integrate these SDEs and get a tractable expression for the state

vector, but not always.

An example that poses no difficulties is the Black-Scholes Model, which has

19

In this model, the situation is not appreciable more difficult when a path-dependent option whose payoff

depends on the value of the state vector at many times. For instance, a discretely monitored Asian call

option has the payoff max {ST -K, 0} where ST = is the average price. Now the simulation must

generate the entire path St1 , St2, . . . , Stm. Assuming tk = Tk /m kh. The way to simulate the whole path is

to generate m independent standard normal random variables Z1(i) , Z2

(i) , . . . , Zm(i) for the i-th path and set

This provide the correct multivariate distribution for (St1 , St2, . . . , Stm ) and hence the correct distribution

for ST .

Another challenge in path generation is continuous path dependent. While the payoff of the European call

option depends only on the terminal value of the state vector, and the payoff of the discretely monitored

Asian call option depends only on a finite state of observation of the state vector, some derivatives have

payoff that depend on the entire continuous-time path. An example is a down-and-out option that pays off

only if stock price stays above some barriers, or equivalently, if the minimum stock price is above the

barrier. Suppose the stock price obeys the Black-Scholes model. Because

20

Almost surely, the former is not an acceptable substitute for the latter. It is necessary to introduce a new

component Mt = minu [ 0, t ] Su into the state vector; this can be simulated since the joint distribution of

Mt and St is known.

A slightly subtler example occurs in the Hull-White model of stochastic interest rate. The SDE governing

the instantaneous interest rate rt is

Where r is the long term-interest rate, is the strength of mean reversion, and is the interest rate’s

volatility. Integration of these SDE yields the distribution of rt , which is normal. Then the simulated path

rt1 , . . . , rtm is adequate for evaluating payoff that depends only on these interest rates, but not for

evaluating the discount factor DT = ; the discrete approximation h

does not have the right distribution. Instead one must add Dt to the state vector and simulate using its joint

distribution with rt , which is easily computable.

Some financial models feature SDEs that are not easily integrable, as the Black-Scholes and Hull-White

models are. An example is the Cox-Ingersoll-Ross model, in which the SDE is

This models principal advantage over Hull-White is that the instantaneous interest rate must remain

nonnegative. However, there is no useful expression for the distribution of rt given r0 . A simulation of this

model must rely on an approximate discritisation r of the stochastic process r . Because the laws of these

processes are not the same, the Monte-Carlo estimate based on r may be biased for the true price based on

r . This bias is known as discretization error.

The standard error of a Monte-Carlo estimate decreases as 1/ C , where C is the computational budget.

This is not an impressive rate of convergence for a numerical integration method. For simulation to be

competitive for some problem, it is necessary to de3sign an estimator that has less variance than the most

obvious ones. A variance reduction technique is a strategy for producing from one Monte-Carlo estimator

another with lower variance given the same computational budget.

Portfolio Selection in the mathematics of finance and financial engineering

21

Yin and Zhou [44] focus on Markowitz’s Mean-Variance Portfolio Selection [45] in a model incorporating

“regime switching.” The objective of Markowitz’s Nobel Prize winning single-period mean-variance

portfolio selection model is to minimize the risk measured by the variance of the wealth while meeting a

target expected wealth. This model provided a foundation for modern finance theory and stimulated

numerous extensions and applications. Yin and Zhou are concerned with a discrete-time version of the

mean-variance problem under regime switching that is modeled by a Markov chain. The regime switching

can be regarded as a factor process reflecting the overall market environment (e.g., bullish or bearish) as

well as other economic factors. The incorporation of the regime switching makes the model more realistic,

but the resulting system becomes more complex. To reduce the complexity Yin and Zhou derive, using

weak convergence methods, a continuous-time limit system whose solution they have already obtained.

Based on this solution, asymptotic optimal mean-variance portfolio selection policies are obtained.

Markowitz’s Nobel-prize winning mean-variance portfolio selection model (for a single period) [56], [57]

provides a foundation of modern finance theory; it has inspired numerous extensions and applications. The

Markowitz model aims to maximize the terminal wealth, in the mean time to minimize the risk using the

variance as a criterion, which enables investors to seek highest return upon specifying their acceptable risk

level. There have been continuing efforts in extending portfolio selection from the static single period

model to dynamic multiperiod or continuous-time models. However, the research works on dynamic

portfolio selections have been dominated by those of maximizing expected utility functions of the terminal

wealth, which is in spirit different from the original Markowitz’s model. For example, the multiperiod

utility models were investigated in [52]–[55] and [58]–[60], among many others. As for the continuous-

time case, the famous Merton paper [59] studied a utility maximization problem with market factors

modeled as a diffusion process (rather than as a Markov chain). Along another line, the mean-variance

hedging problem was investigated by Duffie and Richardson [51] and Schweizer [63], where an optimal

dynamic strategy was sought to hedge contingent claims in an imperfect market. Optimal hedging policies

[51], [63] were obtained primarily based on the so-called projection theorem. Very recently, using the

stochastic linear-quadratic (LQ) theory developed in [49] and [67], Zhou and Li [70] introduced a

stochastic LQ control framework to study the continuous-time version of the Markowitz’s problem. Within

this framework, they derived closed-form efficient policies (in the Markowitz sense) along with an explicit

22

expression of the efficient frontier. In the aforementioned references, for continuous-time formulations of

mean-variance problems, stochastic differential equations and geometric Brownian motion models were

used. Although such models have been used in a wide variety of situations, they have certain limitations

since all the key parameters, including the interest rate and the stock appreciation/volatility rates, are

assumed to be insensitive to the (very likely) drastic changes in the market. Typically, the underlying

market may have many “modes” or “regimes” that switch among themselves from time to time. The market

mode could reflect the state of the underlying economy, the general mood of investors in the market, and so

on. For example, the market can be roughly divided as “bullish” and “bearish,” while the market

parameters can be quite different in the two modes. One could certainly introduce more intermediate states

between the two extremes. A system, commonly referred to as the regime switching model, can be

formulated as a stochastic differential equation whose coefficients are modulated by a continuous-time

Markov chain. Such a model has been employed in the literature to discuss options; see [47], [48], and [50].

Moreover, an investment-consumption model with regime switching was studied in [68]; an optimal stock

selling rule for a Markov-modulated Black–Scholes model was derived in [69]; a stochastic approximation

approach for the liquidation problem could be found in [65]. [71] treated the continuous-time version of

Markowitz’s mean-variance portfolio selection with regime switching and derived the efficient portfolio

and efficient frontier explicitly. Motivated by the recent developments of mean-variance portfolio selection

and Markov-modulated geometric Brownian motion formulation, a class of discrete-time mean-variance

portfolio selection models were developed and their relationship with the continuous-time counterparts was

revealed. The discrete-time case is as important as the continuous-time one. First, frequently, one needs to

deal with multiperiod, discrete-time Markowitz’s portfolio selection problems directly; see [56] for a recent

account on the topic, in which efficient strategies were derived together with the efficient frontier. In

addition, to simulate a continuous-time model, one often has to use a discretization technique leading to a

discrete-time problem formulation. Yin and Zhou [44] focus one of the main features of the problem to be

tackled is that all the market coefficients are modulated by a discrete-time Markov chain that has a finite

state–space. Owing to the presence of long-term and short-term investors, the movements of a capital

market can be divided into primary movement and secondary movement naturally leading to two-time

scales. Besides, with various economic factors such as trends of the market, interest rates, and business

23

cycles being taken into consideration, the state space of the Markov chain, representing the totality of the

possible market modes, is often large. If simply treated each possible mode as an individual one distinct

from all others, the size of the problem would be huge. A straightforward implementation of numerical

schemes may deem to be infeasible due to the curse of dimensionality. It is thus crucial to find a viable

alternative. To reduce the complexity, it has been observed that the transition rates among different states

could be quite different. In fact, there is certain hierarchy (in terms of the magnitude of the transition rates)

involved. Therefore, it is possible to lump many states at a similar hierarchical level together to form a big

“state.”

With this aggregated Markov chain, the size of the state–space is substantially reduced. Now, to highlight

the different rates of changes, a small parameter has been introduced into the transition matrix, resulting in

a singular perturbation formulation. Based on the recent progress on two-time-scale Markov chains (see

[61] and [66]), the natural connection between the discrete-time problem and its continuous-time limit has

been established. Under simple conditions, it can be shown that suitably interpolated processes converge

weakly to their limits leading to a continuous-time mean-variance portfolio selection problem with regime

switching. The limit mean-variance portfolio selection problem has an optimal solution [71] that can be

obtained in a very simple way under appropriate conditions. Using that solution, policies that are

asymptotically optimal can be designed. Current findings indicate that in lieu of examining the more

complex original problem, one could use the much simplified limit problem as a guide to obtain portfolio

selection policies that are nearly as good as the optimal one from a practical concern. The advantage is that

the complexity is much reduced. Yin and Zhou [44] remark that although the specific mean-variance

portfolio selection is treated in their work, the formulation and techniques can be generally employed as

well in the so-called hybrid control problems that are modulated by a Markov chain for many other

applications. Yin and Zhou [44] establish the natural connection between the discrete-time and continuous-

time models aiming at reducing the complexity of the underlying systems. They construct policies that are

based on optimal control of the limit problem and derives asymptotic optimal strategy via the constructed

controls. The results can be extended by allowing the Markov chain to be nonhomogeneous and/or

including transient states.

The contribution of H. M. Markowitz to economic theory

24

Prior to Markowitz’s portfolio theory [72,73] several economists had pointed to the need for diversification

in investment (“don’t put all your eggs in one basket”). This approach, however, was founded on lay

observation without due analysis and quantification. As is clear this idea without an appropriate scientific

grounding greatly irritated Markowitz and led him to his famous model on the creation of efficient

allocation and portfolio creation. His further enthusiasm in research was to a significant extent influenced

also by practice. Since 1952 he has worked in many well-known companies, which also created for him

the conditions for his further work. In the Fifties and Sixties he worked in Rand Corporation, General

Electric Corporation, Cairmand, Consolidated Analysis Centres Inc. etc. In his work he learnt in particular

optimisation techniques and methods of linear programming. He enriched logistic simulation models and

programming languages through his theoretical knowledge in the field of portfolio theory (at Rand he was a

co-creator of the programming language SIMSCRIPT, which in an updated version is still used today).

Collective investment via fund management companies, investment companies, and pension funds plays a

permanently important role in capital markets. The professional approach of these companies’ managers

does not occur without the use of portfolio creation methods. With the help of these it is possible to

determine in which securities from a selected range it is advantageous to deposit investors funds and

concurrently to also set the optimal ratios in individual assets. The work of H. M. Markowitz, published in

1952 in the article “Portfolio selection” [72] is considered the keystone of modern portfolio theory. The

work states that the price of a share represents the cash flow of future dividends discounted to the net

present value. According to this model an investor is interested in future expected dividend flows and thus

also the expected share price. Markowitz applied this conclusion also to portfolio theory when he stated

that investors are interested in the expected value of a portfolio, where this value may be quantified.

Among those values which investors are interested in are, according to him, risk and return. In creating his

selection model Markowitz works from certain abstractions and presumptions:

• investors have an aversion to risk,

• all investors invest at the same time,

• the basis of investment decisions is expected utility,

• investors make their investment decisions on the basis of expected risk and return, and

• perfect capital markets exist.

25

What does a portfolio in the financial market represent in the meaning of this theory? A portfolio is a set of

various investments, which an investor creates in order to minimize risks connected with investing and also

to find the best possible proportion between risks and returns. Since according to Markowitz’s theory the

investor is risk averse, the investor will create a portfolio with the aim of achieving the largest return for the

minimum risk. In quantifying the yield (return) of a portfolio Markowitz worked at first from determining

the expected yield of one instrument and then from the expected return of the whole portfolio.

The expected yield of the portfolio is a weighted average of the expected individual fields of individual

instruments in the portfolio, where the weightings are the shares of individual investments in the portfolio.

An investor is interested not only in the rate of return but also in the risk. In measuring risk Markowitz at

first works from the risk of one asset and then from the risk of the portfolio. The risk of a portfolio however

is not simply a weighted average of the risks of individual instruments in the portfolio. The degree of risk

of the portfolio is influenced also by other variables, in particular by the mutual relation between the yields

of individual instruments.

Markowitz stated that if an investor invests in a portfolio which perfectly positively correlated yields, then

it does not at all lower his risk, because the yields move in only one direction and the investor in such a

portfolio can suffer significant losses. The ideally compiled portfolio has negatively correlated yields, i.e.

the yields have an inverse movement. To compile such a portfolio however is in practice impossible. Assets

with non-correlated yields create a portfolio in which the yields have no relation to one another. The

benefits of diversification lie in the fact that a more efficient compensation effect of risk and return will be

achieved through an appropriate combination of assets, the correlation of which does not extend to a form

of completely positive correlation. In such cases the standard deviation of the yield of a portfolio is less

than the weighted average of the standard deviations of the assets in the portfolio. Diversification lowers

risk also in the case of a smaller number of securities – first of this risk is lowered quickly, gradually with

an increasing amount of securities, the effectiveness declines. It is thus possible to assess risk in the context

of a portfolio. We cannot judge the effective risk of any security in a way that we will examine it in

isolation. A part of the uncertainty concerning the yields of a security is “de-diversified” as soon as a

security is grouped with others in the portfolio. From Markowitz’s selection model it thus results that if an

investor wants to reduce the overall risk of the portfolio, then he must combine those assets which are not

26

perfectly positively correlated. Markowitz worked from the assumption that in the selection of a portfolio

the investor can select within the framework of the market various combinations of securities with various

yields and risks. In other words he assembled a so called feasible set of all possible combinations of

investment, which an investor is faced with in the market. The typical shape of a feasible set of portfolios

has entered financial theory under the title of an “umbrella shape”, which is depicted in the graph. From the

set of Pareto optimal combinations of expected yields and variances investors will according to Markowitz

select portfolios which: give the maximum expected rate of return at various levels of risk or offer

minimum risk in the case of various levels of expected rates of return. It was shown in a graph.In the graph

it can be seen and that given conditions fulfil the combinations S, in the case of which the investor will

achieve a maximum yield and E with the lowest risk. The set of portfolios fulfilling these two conditions is

known as the efficient set or efficient frontier. This limit depicts the points with the maximum rate of return

for a given level of risk, and which are measured by the standards deviations of the portfolio’s yields. From

the graph it can also be established that the efficient set will be located between points E and S. To this

efficient frontier he also applied indifference curves, which from the aspect of the theory of frontier utility

express the various combinations, in the case of which an investor tries to achieve the same utility. As

Markowitz states, indifference curves have a different slope in the case of a risk-averse investor and that of

a risk seeking investor. The indifference curves of an investor seeking risk have a more moderate slope and

will move closer to point S, where they will also touch the efficient frontier. In his selection model he gives

preference to the risk-averse investor.

Every portfolio manager recognises the value of the innovative approach of H. M. Markowitz in this field.

All his theoretical conclusions have become the basis and springboard for the development of other

theoretical analyses in the field of portfolio theory.

Bielecki and Pliska [74] present an application of risk sensitive, stochastic control theory to the problem of

optimally managing a portfolio of securities. The model features risky assets whose continuous-time

dynamics depend upon one or more economic factors which, in turn, are modelled as Gaussian processes.

The investor’s objective, an infinite horizon criterion, is to maximize the portfolio’s risk-adjusted growth

rate. The results in this paper are important in at least three aspects. First, by using continuous algebraic

Ricatti equations, the authors derive an explicit expression for the optimal trading strategy, thereby

27

obtaining explicit results for a new class of models. Second, by studying some particular examples, they

develop a better economic understanding about the interactions between factor processes and optimal

trading strategies. For instance, with stochastic interest rates it is rigorously shown that the strategy of

100% cash is not necessarily the least risky one. Third, this paper is one of the first to apply stochastic

control methods to fixed income portfolio management.

Beginning with the pioneering work by Merton [75], [76] and continuing through the recent books by

Karatzas and Shreve [77] and Korn [78], some very sophisticated stochastic control methods have been

applied to portfolio management. But most of these applications have been concerned with, at least

implicitly, the management of equities. In spite of an abundance of well-known mathematical interest rate

models, exemplified by the classical models of Duffie and Kan [79], Heath, Jarrow, and Morton [80], and

Asicek [81], one can find very few applications of modern control theory to fixed-income management.

Bielecki and Pliska [74] develop a stochastic control model that will provide a closed form solution to the

problem of long term optimal management of a fund comprised of equities and fixed-income instruments.

An optimization model for dynamic asset management, whether it is a discrete time or a continuous time

model, should recognize and address the following issues:

a) realistic modeling of the dynamics of asset price movements;

b) selection of planning horizon;

c) selection of optimality criterion;

d) computational feasibility.

Bielecki and Pliska [74] adopt a continuous time approach. Although in principle it is possible to obtain

existence results for optimal portfolio policies relative to selected optimality criteria under general model

dynamics such as (1), these dynamics are nevertheless too general for derivation of closed form solutions.

Some structural postulates need to be made about the parameter processes as well as about the driving

noises. One possibility is to model the parameters as Itô processes. Usually it is assumed that the price and

factor processes can be directly observed, but some models postulate only partial observations. Typically,

closed-form solutions to the corresponding optimal portfolio selection problems have only been obtained

after further specification of the functions , and Merton [76] called model an Inter-temporal Capital Asset

Pricing Model. At this point it is important to mention a key aspect of issue a), namely, model

28

completeness versus incompleteness. A non-degenerate model in which the number of sources of

randomness equals the number of underlying financial assets minus one, that is known as a complete

model; otherwise, the model is incomplete. Incomplete models are much more adequate descriptions of

financial markets. For example, if the factors are macroeconomic variables such as interest and

unemployment rates, then implicit with a complete model would be the foolish assumption that one would

know the full histories of the macroeconomic variables simply by observing the asset prices. On the other

hand, incomplete models are much more difficult to analyze, so there is an obvious tradeoff between issues

a) and d). Turning to model design issue b), the selection of the planning horizon, models in the literature

can be classified as: i) models with finite planning horizon, ii) models with random planning horizon (e.g.,

determined by the random time of bankruptcy), and iii) models with infinite time horizon. Each category

serves an appropriate purpose. In Bielecki and Pliska [74] effort is aimed at providing quantitative tools for

fund managers with long term goals of maximizing the growth rates of the portfolios they manage, keeping

the respective risks under control at the same time. In this case no specific finite horizon is appropriate, and

the model of Bielecki and Pliska [74] belongs in category iii) above. It has to be understood that, although

nobody in reality adheres to truly infinite planning horizons, a mathematical infinite horizon model

provides a convenient approximation for “long planning horizon” situations. Selection of the optimality

criterion, i.e., issue c) above, is another discriminant between competing models. The common approach,

and the one highly favored by most economists, is to choose an optimality criterion based upon a so-called

utility functional. For example, the objective might be to maximize expected utility of wealth at the (finite)

planning horizon or to maximize expected utility of consumption over the whole (possibly infinite)

planning interval. It is common practice to evaluate mutual fund by looking at criteria like average return

and volatility measured over historical periods; utility functions are never utilized. For the portfolio

problem Bielecki and Pliska [74] have decided to select the so called risk-sensitive objective criterion, a

criterion to be maximized. As a matter of fact, the model of Bielecki and Pliska [74] provides for direct

computation of optimal portfolio strategies in terms of well understood algebraic Ricatti equations. Their

results are for asset management in general, without regard to the specific application of fixed-income

management. Some recent, related papers are by Canestrelli [82], Canestrelli and Pontini [83], and Kim and

Omberg [84]; they used expected utility criteria. Detemple et al. [85] and Schroder and Skiadas [86]

29

studied somewhat more general models and also used expected utility criteria. Brennan and Schwartz [87]

and Brennan et al. [88] used numerical methods to solve the Hamilton–Jacobi–Bellman partial differential

equation for the optimal trading strategy, but they were hard pressed to derive a solution even though there

were only three factors and a similar number of assets. Explicit results on the application of the model of

Bielecki and Pliska [74] to fixed-income management were recently obtained by Bajeux-Besnainou et al.

[89], Deelstra et al. [90], Liu [91] and Sørensen [92]. Apparently working independently, they studied

some very similar, if not identical, special cases, all maximizing expected utility over a finite planning

horizon. Unfortunately, they all left unresolved some troublesome issues, such as optimal strategies calling

for unbounded positions in some assets. In particular, Deelstra et al. [90] considered a finite time terminal

utility maximization problem with the utility function. They studied a market consisting of two assets, a

stock and a discount bond maturing at a finite time. They also considered a factor process, the spot interest

rate process. They were interested in characterizing a portfolio process that maximizes the utility of the

wealth at time. Thus the planning horizon in their model coincides with the maturity of the underlying

bond, but this causes a singularity of the resulting stochastic control problem at the terminal time . They

tried to overcome this singularity problem by a judicious choice of the class of admissible controls (trading

strategies). Unfortunately, as it appears, their definition of the class of admissible controls is flawed, for it

involves the solution to the associated SDE which is supposed to be not satisfied by the wealth process,

resulting in a circular argument. Bielecki and Pliska [74] are able to incorporate in a rigorous fashion fixed-

income assets having infinite lives by utilizing the concept of rolling-horizon bonds, a concept introduced

by Rutkowski [93]. Rolling horizon bonds can be viewed, roughly, as mutual funds of zero coupon bonds,

all of which mature at about the same fixed distance in the future; these bonds are rolled over in a self

financing manner so this same fixed distance is preserved through the course of time.

Mean-variance hedging, a portfolio choice problem

Bobrovnytska and Schweizer[94] focus on mean-variance hedging, a portfolio choice problem where the

goal is to approximate (with respect to mean squared error) a given payoff by the final wealth from a self-

financing portfolio strategy. Since the wealth dynamics are linear in the chosen portfolio, it is natural to

attack the problem with linear-quadratic stochastic control methods. The novelty of this paper is that it

considers a setting with continuous semi-martingale asset prices in a general filtration which need not be

30

generated by Brownian motions. By exploiting ideas and concepts from mathematical finance, solvability

of the adjoint equations (which are generalized stochastic Riccati equations) can be studied quite precisely.

In the micro-movement of asset price, transaction data are discrete in value, irregularly-spaced in time and

extremely large in size. It has been documented in the finance literature that ignoring the discrete-type of

trading noise results in substantially inflated volatility estimates.

Bobrovnytska and Schweizer[94] show for continuous semimartingales in a general filtration how the

mean-variance hedging problem can be treated as a linear-quadratic stochastic control problem. The adjoint

equations lead to backward stochastic differential equations for the three coefficients of the quadratic value

process, and we give necessary and sufficient conditions for the solvability of these generalized stochastic

Riccati equations. Motivated from mathematical finance, this paper takes a first step toward linear-

quadratic stochastic control in more general than Brownian settings.

Stochastic control methods have a venerable history in the field of financial engineering, and a number of

Nobel Prizes bear ample witness to the fruitfulness of this interaction. One can for instance think of

Merton’s seminal contributions to portfolio optimization and option pricing, among other things. Even

earlier, Harry Markowitz was concerned with mean-variance analysis in financial markets, and this topic

has retained its popularity even after 50 years; see, for instance, the survey [95] which contains more than

200 references. However, in contrast to portfolio optimization based on utility functions, mean-variance

analysis in dynamic inter-temporal frameworks has only recently been linked to stochastic control in a

more systematic way. Bobrovnytska and Schweizer[94] explore this avenue further and to show that it

leads to results and insights in stochastic control even beyond the usual settings. In a given financial

market, the mean-variance hedging problem is to find for a given payoff a best approximation by means of

self-financing trading strategies; the optimality criterion is the expected squared error. In a series of recent

papers, this problem has been formulated and treated as a linear-quadratic (LQ) stochastic control problem

at increasing levels of generality; see for instance [96], [97], [98], [99], [100], [101], or [102] for an

overview and a historical perspective. In the general case where the market coefficients are random

processes, the adjoint equations turn out to lead to a coupled system of backward stochastic differential

equations (BSDEs) for the coefficients of the (quadratic) value functional. This has led to new interest in

and new results on general LQ stochastic control problems, and the mean-variance hedging problem has

31

been treated fairly explicitly by these methods. From the mathematical finance point of view, one drawback

of this approach is that almost all existing papers impose rather restrictive assumptions. To apply general

results from LQ stochastic control, Bobrovnytska and Schweizer[94] work with Itô processes and assume

that all their coefficients are uniformly bounded, which excludes many practically relevant models.

Moreover, the theory of BSDEs is only rarely used beyond the setting of a filtration generated by a

Brownian motion and, thus, strongly relies on a martingale representation theorem. On the other hand, the

mean-variance hedging problem has been solved in much higher generality by martingale and projection

techniques. One can allow continuous semimartingales in general filtrations and only needs an absence-of-

arbitrage condition; see [103] and [104] for recent overviews. The work of Bobrovnytska and

Schweizer[94] is a first step toward a fusion between mathematical finance and LQ stochastic control at

this more general level. For related recent results, see [105]. Bobrovnytska and Schweizer[94] presents the

basic model, explains the mean-variance hedging problem and casts it in the form of an LQ stochastic

control problem. Combining the martingale optimality principle with the natural guess that the value

process of this problem should have a quadratic structure, they then derive a system of BSDEs for the

conjectured coefficients They gives a necessary and sufficient condition for the first of these BSDEs (for

the quadratic coefficient ) to be solvable under the sole assumption that the underlying asset price process

is continuous. One can also show that is the value process of a dual control problem. Apart from continuity

of, Bobrovnytska and Schweizer[94] need that the filtration is continuous and that the variance-optimal

martingale measure satisfies the reverse Hölder inequality. They show how one can explicitly construct a

solution for the mean-variance hedging problem from the solutions of the BSDEs. This is conceptually well

known, but of course technically slightly different than in the usual case of a Brownian filtration.

Unified Bayesian estimation via filtering approach To overcome Brownian filtration., Zeng’s paper [106] presents a unified approach, via filtering, to estimate

stochastic volatility for micro-movement models. The key feature of the models is that they can be

transformed into filtering problems with counting process observations. The Markov chain approximation

method is applied to the filtering equations to construct consistent recursive algorithms, which compute the

joint posterior and the trade-by-trade Bayesian parameter estimates. To illustrate the approach, a micro-

32

movement model built on a jumping stochastic volatility geometric Brownian motion is studied in detail.

Simulation and a real data example are presented.

Stochastic volatility is well documented for asset prices in both macromovement and micromovement

[107]. Macromovement refers to daily, weekly, and monthly closing price behavior while micromovement

refers to transactional (trade-by-trade) price behavior. There is a strong connection as well as striking

distinctions between the macro and micromovements. The strong connection is observed through the

identity of the overall shapes of both, because the macromovement is an equally spaced time series drawn

from the micromovement data. Their striking distinctions are mainly due to financial noise. In

macromovement, the impact of noise is small and is usually neglected. In micromovement, however, the

impact of noise is substantial and noise must be modelled explicitly. If the noise is ignored, then the impact

of noise is transferred to volatility, and the volatility estimates are substantially inflated. This is

documented by [108], [109], and [110] for discrete noise and further in [111] and [112] for discrete plus

other types of noise. Economically, the asset price is distinguished from its intrinsic value and this

distinction is also noise. Noise, as contrasted with information, is well-documented in the market

microstructure literature. Three important types of noise have been identified and extensively studied:

discrete, clustering and nonclustering. First, intraday prices move discretely (tick by tick), resulting in

“discrete noise.” Second, because prices do not distribute evenly on all ticks, but gather more on some ticks

such as integers and halves, “price clustering” is obtained. [113] confirms that this phenomenon is

remarkably persistent through time, across assets, and across market structures. Third, the “nonclustering

noise” includes other unspecified noise, and the outliers in prices are one of the evidence for the existence

of nonclustering noise. In [112], a novel, economically well-grounded and partially observed

micromovement model for asset price is proposed to bridge the gap between the macro and micro

movements caused by noise. The most prominent feature of the proposed model is that it can be formulated

as a filtering problem with counting process observations. This connects the model to the filtering

literature, which has found great success in engineering and networking. Under this framework, the

observables are the whole sample paths of the counting processes, which contain the complete information

of price and trading time. Then, the continuous-time likelihoods and posterior, built upon the sample paths,

not only exist, but also are uniquely characterized by the unnormalized, Duncan–Mortensen–Zakai (DMZ)-

33

like filtering equation, and the normalized, Kushner–Stratonovich (KS) (or Fujisaki–Kallianpur–Kunita)-

like filtering equations respectively. Transaction (or tick) data are discrete in value, irregularly spaced in

time and extremely large in size. Despite recent advances in statistics and econometrics, obtaining

“reliable” parameter estimates for even simple, nonstochastic volatility, micromovement models are

extremely challenging. Contrasted with [114], [112] develops continuous-time Bayes estimation via

filtering with efficient algorithms for the parameter estimation of the micromovement model. That

represents a significant advance in the estimation for micromovement models, also because the continuous-

time likelihoods and posterior are utilized as the foundation for statistical inference. This foundation is

informationally better than those provided by the discrete-time likelihoods and posterior, which merely

make use of a discrete-time subset of the sample paths. In [112], however,only the parameters of a simple

model with GBM as value process is estimated. In this paper, first, a class of stochastic volatility

micromovement models is developed from the macromovement models by incorporating the three types of

noise mentioned. A new, jumping stochastic volatility (JSV) micromovement model, stemming from

geometric Brownian motion (GBM), is proposed and studied (later, it is called the JSV-GBM

micromovement model). Second, a unified approach, Bayes parameter estimation via filtering, is developed

for the mi-cromovement models, especially for estimating stochastic volatility. Stochastic volatility models

are more realistic and more interesting but more difficult to estimate than the simple model with GBM as

value process, where the parameters, the signal of interest, are fixed. In stochastic volatility model,

estimation becomes a “real” filtering problem: the stochastic volatility, the signal of interest, changes over

time and the stock prices are the observations corrupted by discrete types of noise. The JSV-GBM model is

employed to demonstrate the effectiveness of estimating stochastic volatility using Bayes estimation via

filtering. To illustrate the approach, JSV-GBMs consistent recursive algorithm, which approximates the

normalized filtering equation and calculates the joint posterior, is constructed in detail. Simulation results

show that the Bayes estimates for stochastic volatilities are close to their true volatilities, and are able to

capture the movement of volatility. Trade-by-trade volatility estimates for an actual transaction data set are

computed and they confirm that the volatility changes even more dramatically in micromovement.

A unified Bayesian estimation via filtering approach is developed for estimating stochastic volatility for a

class of micromovement models, which capture the impact of noise at the microlevel. The class of models

34

has an important feature in that it can be formulated as a filtering problem with counting process

observations. Under this formulation, the whole sample paths are observable, and the complete tick data

information is used in Bayes parameter estimation via filtering. A consistent recursive algorithm is

developed to compute the Bayes estimates for the parameters in the model, especially, the stochastic

volatility. Simulation studies show that Bayes estimates for time-invariant parameters are consistent, and

Bayes estimates for stochastic volatility are close to their true values and are able to capture the movement

of volatility quickly. The recursive algorithm is fast and feasible for large data sets and it has the recursive

feature allowing quick and easy update. The recursive algorithm is applied to Microsoft’s transaction data

and we obtain Bayes estimates and provide strong affirmative evidence that volatility changes even more

dramatically in trade-by-trade level. The model and its Bayes estimation via filtering equation can be

extended to jump-diffusion process for the value process, and other kinds of noise according to the sample

characteristics of data. The models and the Bayes estimation can be applied to other asset markets such as

exchange rates and commodity prices. It can also apply to assess the quality of security market, and to

compare information flows and noises in different periods and different markets.

Financial markets and corresponding changes in financial assets

Hanson and Westman [115] have observed that following important economic or financial announcements,

there can be large changes in the financial markets and corresponding changes in financial assets. While

these events may be scheduled or unscheduled, the amplitude of the market response may be unpredictable

or random. Often market volatility is modeled by continuous Brownian motion processes, but these are

inadequate for modelling such events and responses. Discontinuous jump processes are needed to model

these important events. Compound Poisson processes are used to model the unscheduled events and hybrid

deterministic-stochastic jump processes are used to model the time of scheduled events and the random

responses to them. The work of Hanson and Westman [115] was motivated by the “important event model”

of Rishel (116), but cast into stochastic differential equation format to facilitate generalization to

constraints, parameter values and computations. Application of the important events model is illustrated by

the optimal portfolio and consumption with the risk-adverse power utility control problem. However, the

usual separable canonical solution is not strictly valid due to added complexity of the jump events,

especially since the times of the scheduled events do not average out of the problem. Fortunately, iterations

35

about the canonical solution result in computationally feasible approximations. This illustrates that the

combined use of stochastic control theory and computations to successfully handle complex jump events.

A large number of continuous-time models of financial markets have been based upon continuous sample

path geometric Brownian motion processes, such as [117], [118], [119],and [120, Chs. 4–6]. However,

Merton [121], [119, Ch. 9], in the original jump diffusion finance model, applied discontinuous sample

path Poisson processes, along with Brownian motion processes, to the problem of pricing options when the

underlying asset returns are discontinuous. Several extensions of the classical diffusion theory of Black and

Scholes [122] were derived by minimizing portfolio variance techniques to jump diffusion models similar

to those techniques used to derive the classic Black and Scholes diffusion formulas. Earlier, Merton [119],

[120, Chs. 5 and 6] treated optimal consumption and investment portfolios with either geometric Brownian

motion or Poisson noise, and illustrated explicit solutions for constant risk-aversion in either the relative or

the absolute forms. Karatzas et al. [123] pointed out that it is necessary to enforce nonnegativity feasibility

conditions on both wealth and consumption, deriving formally explicit solutions from a consumption

investment dynamic programming model with a time-to-bankruptcy horizon, that qualitatively corrects the

Merton’s results. Sethi and Taksar [124] present corrections to certain formulas of Merton’s [119], [120,

Chs. 5 and 6] finite horizon consumption-investment model. Merton [120, Ch. 6] revisited the problem,

correcting his earlier work by adding an absorbing boundary condition at zero wealth and using other

techniques. Rishel [116] introduced a optimal portfolio model for stock prices dependent on quasi-

deterministic scheduled and stochastic unscheduled jump external events based on optimal stochastic

control theory. The jumps can affect both the stock prices directly or indirectly through parameters. The

quasi-deterministic jumps are deterministic only in the timing of the scheduled events, but the jump

responses are random in magnitude. The response to an event can be unpredictable, being based on solid

analysis, prefactored assessments, nuances or other factors external to the event. Rishel’s theoretical paper

is the motivation for this computational application paper. Much additional motivation comes from our

extensive prior research on computational stochastic control models for jump environments, such as

stochastic bioeconomic models with random disasters (see [125] and [126]) and stochastic manufacturing

systems subject to jumps from failures, repairs and other events (see [127]–[129]). Here, our model

formulation is a modification on Rishel’s [116] paper, with heavier reliance on stochastic differential

36

equations, constrained control, more general utility objectives, generalized functions, and random Poisson

measure. Many of the modifications make the model more realistic and computationally feasible. More

realism has been implemented through modifications systematically relying on linear or geometric

stochastic processes, while using control constraints on stock fractions and consumption policies. The

portfolio and consumption optimization problem is formulated and the subsequent partial differential

equation of stochastic dynamic programming is derived from a generalized Itô [130] chain rule. The

computations [122] for the portfolio optimization model have been carried out in MATLAB [131] to

demonstrate the reasonableness of the calculations.

Structure of interest rates with multiple ratings with emphasis on the arbitrage-free feature of the

model Bielecki and Rutkowski [132] provide approach to the Heath–Jarrow–Morton type modeling of defaultable

term structure of interest rates with multiple ratings. Special emphasis is put on the arbitrage-free feature of

the model, as well as on the explicit construction of the conditionally Markov process of credit migrations.

SOME of the basic elements of financial markets are discount and coupon bonds, which represent so-called

fixed income instruments. Coupon bonds can frequently be considered as portfolios of discount bonds with

various maturities. That is why, in many respects, the main object of study with regard to fixed income

instruments are discount bonds. A discount bond is an instrument that promises to pay a specified notional

amount at a specified maturity date, say . Thus, the value of a discount bond at any time is derived as a

function of the notional amount, as well as some other factors. The dependence of the price of discount

bonds on the physical time and on the maturity dates is known as the term structure of interest rates. It

needs to be said though that the term structure of interest rates can be represented in many alternative ways,

besides in terms of prices of discount bonds. If a discount bond pays the promised notional amount at the

maturity date with certainty, then such a bond is called a default free bond. A good example of such bonds

is provided by Treasury bonds. Frequently, a discount bond is not certain to pay the promised notional

amount at the maturity date. If so, such a bond is considered to be prone to default and is known as

defaultable bond. Most of the corporate bonds are defaultable bonds. The dependence of the price of

defaultable discount bonds on the physical time and on the maturity dates is known as the defaultable term

structure of interest rates.This work continues the line of research on reduced-form (or intensity-based)

modeling of defaultable term structure of interest rates originated in [133-135], [136–139], [140–142], and

37

[143]. We do not make here any attempt to classify or scrutinize various models developed in previous

works. For a detailed analysis of these approaches, the interested reader is referred to [144, Chs. 12 and 13].

From the mathematical perspective, the intensity-based modeling of random times hinges on the techniques

of modeling random times developed in the reliability theory. The key concept in this methodology is the

survival probability of a reference instrument or entity, or, more specifically, the hazard rate that represents

the intensity of default. In the most simple version of the intensity-based approach, nothing is assumed

about the factors generating this hazard rate. More sophisticated versions additionally include factor

processes that possibly impact the dynamics of the credit spreads. Important modeling aspects include: the

choice of the underlying probability measure (real-world or risk-neutral), the goal of modeling (risk

management or valuation of derivatives), and the source of intensities. In a typical reduced-form model, the

value of the firm is not included in the model. The specification of intensities is based either on the model’s

calibration to market data or on the estimation based on historical observations. Both in case of credit risk

management and in case of valuation of credit derivatives, the possibility of migrations of underlying credit

name between different rating grades is essential, as it reflects the fundamental feature of the real-life

market of credit risk sensitive instruments (corporate bonds and loans). In practice, credit ratings are the

natural attributes of credit names. Most authors (see, for instance, [140] or [142]) were approaching the

issue of modeling of the credit migrations from the Markovian perspective. In [134], we presented a

general outline of our Heath–Jarrow–Morton (HJM) type reduced-form approach to the modeling of

defaultable term structures that correspond to multiple ratings of corporate bonds. The present work

complements in many ways the previous one, mainly by providing a complete presentation of mathematical

aspects of our model. An important feature of our model is that it indeed is selfconsistent; that is, it is an

arbitrage-free model. In our previous paper, we have already indicated this arbitrage-free property of the

model. In the present work, this important property receives a full justification, since we provide a detailed

description of the enlarged probability space, which, when combined with the dynamics of the

instantaneous forward rates and the dynamics of the migration process, underly the arbitrage-free feature of

the model. It should be acknowledged that our construction can be extended in several directions, and some

of our assumptions can be weakened substantially. For instance, by combining our approach with results of

[146], Eberlein and Õzkan have shown that the model can be extended to the case of a term structure model

38

driven by a Lévy process (as opposed to the case of a standard Brownian motion examined here). The

important issue of the model’s calibration requires further studies Bielecki and Rutkowski [132]. Observe

that, in general, even if all four pieces of data—namely, the maturity date, the transition intensities, the

recovery scheme, and the initial rating—are identical for the two zero coupon bonds, the bonds themselves

may not be identical. In fact, if they are issued by two different entities, the associated migration processes

and are also distinct, in general. More specifically, if we consider the joint migration process , then the

marginal finite-dimensional distributions for and are identical, but in general . If, the credit migration

processes and may be either (conditionally) independent or dependent. In case of independent migration

processes and , no statistical dependence between credit migrations of the two bonds appears. In case of

mutually dependent migration processes, one needs to calibrate the dependence structure (or, more crudely,

the correlation structure) between and. The foregoing remarks are valid if one considers an application of

the general methodology presented in this paper to the valuation and hedging of individual defaultable

bonds—that is, corporate bonds issued by particular institutions, as well as to the valuation and hedging of

related credit derivatives. As an alternative, let us mention that the methodology presented in this paper

may be applied to a totality of alike defaultable bonds—that is, to the totality of bonds for which all four

features listed above coincide. In the latter approach, we identify all such bonds and we substitute them

with a representative bond with an associated representative migration process. This application of our

methodology aims at valuation and hedging of credit derivatives that are tied to the average market value of

corporate bonds of a given credit quality. Thus, the correlation structure between individual bonds is

deliberately disregarded. All that really matters in this interpretation are the marginal statistical properties

of individual corporate bonds, and they are identical for all bonds in a given class. Let us consider two

different defaultable bonds, and let us denote the associated migration processes as and . The respective

default events are and the respective default times are We may study two types of default correlations: the

correlation between random variables and the correlation between random variables and Various

correlation coefficients, such as Pearson’s (or linear) correlation coefficient, may be used to measure the

strength of these correlations. Likewise, we may analyze the correlations between the survival events of the

form: and . Of course, the correlation structure will typically vary depending on whether one uses the risk-

neutral probability or the real world probability .

39

Institutional money management by tracking a given stochastic benchmark

Paolo Dai Pra, Wolfgang J. Runggaldier, and Marco Tolotti [145] consider a problem from institutional

money management, where the objective of the investor/money manager is that of tracking or, better,

outperforming a given stochastic benchmark; the benchmark may be an index process such as the S&P 500

index, it may however also represent other economic quantities such as the value process of a non traded

asset like the inflation or the exchange rate. Typically, it is assumed that the investor may invest in a certain

number of risky assets in addition to a non risky one and let denote the investor’s wealth at time

corresponding to an investment strategy. The benchmark process will be denoted by and we will make the

realistic assumption that it is not perfectly correlated with the investment opportunities so that the investor

cannot completely control his risk (the market is incomplete). This portfolio problem, that is sometimes

also called active portfolio management, has been studied by various authors with particular reference to

[146]. As in [146], we will consider as relevant state variable the ratio of the investor’s wealth to the

benchmark. A natural way to proceed (see again [146]) is then to consider the process up to the exit from a

bounded domain and choose as objective the minimization of the discounted expected loss that penalizes

the deviation of from the constant 1 in the case of “benchmark tracking” and the amount by which falls

below1 in the case when the objective is that of “beating the benchmark.” While the classical criteria such

as the one described above are criteria in the mean, namely they involve expected values of costs/losses, in

this paper we aim at a stronger form of criterion, more precisely that of optimality (see, e.g., [147-150]) that

may in fact be quite appropriate for benchmark tracking/beating. The optimality criteria in use concern an

infinite horizon, which may still make sense in economic/financial applications every time an investor

makes his plans over a long horizon. To keep matters simple in this first attempt to apply a criterion to an

investment problem, we will consider here only symmetric cost functions, i.e., we will only consider the

benchmark tracking problem, thereby penalizing symmetrically both over and undershoots of with respect

to 1. An economically more meaningful asymmetric cost function that penalizes only undershoots/shortfalls

and corresponds to the typical benchmark beating/outperforming can still be dealt with in our approach, but

at the expense of less analytical tractability. More precisely, the aim is to find an investment strategy such

that for the corresponding ratio process we have . For the standard market models, namely those that that

go back to Merton [151], the price processes are geometric Brownian motions (lognormal processes). The

40

dynamics of the process are modified so that it becomes bounded (in a large domain) and still maintains the

main characteristics corresponding to lognormal models. Given the ergodic criterion, this cannot be

accomplished by simply stopping the process upon exit from a given compact set. The drift and diffusion

coefficients, which under certain assumptions can also be interpreted as a random time change and by

which the controlled process is increasingly slowed down as it gets closer and closer to a given boundary.

An interesting aspect that turns out in this context is that the optimal control and the solution of the

Hamilton–Jacobi–Bellman (HJB) equation do not depend on the particular choice of the random time

change. With the thus modified process, we will not only obtain the right ergodic behavior in order to make

the mean-optimality criterion meaningful, but it will furthermore allow us to show that the mean-optimal

control is also optimal. A final methodological aspect of [145] concerns the problem of solving the HJB

equation associated to the given (infinite-horizon) stochastic control problem with the criterion of

optimality in the mean. The traditional way is to guess a possible function which works only in specific

cases. Since a same objective may also be reached by using analytically different cost functions, we will

generalize the problem of solving the HJB by considering a class of possible cost functions and, given the

dynamics of the controlled process.

Kang Boda, Jerzy A. Filar, Yuanlie Lin, and Lieneke Spanjers [152], have studied a problem of optimal

control of ensuring that an adequate capital accumulates sufficiently quickly with sufficiently high

probability. The objective is to develop a tool that could be used to advise nonprofessional investors who

place their retirement benefits in a fund that permits only a limited number of options and offers only

limited opportunity to reallocate the money among these options; say, once a year. It is assumed that an

investor is primarily interested in maximizing the probability of being to afford the fund equal or exceed a

certain specified target amount at that terminal time. Since the mathematical framework in which we model

this problem is that of Markov decision processes (MDPs) (e.g., see [153]) and since a vast majority of

MDPs have objective criteria that depend on one of a number of “expected utility” criteria, it follows

immediately that our problem is essentially different from these classical MDP models. Instead, the

problem belongs to a class of models that are sometimes called “risk-sensitive MDPs.” The latter can,

perhaps, be traced back to [154] and constitutes an area where there has been a fair bit of research activity

in recent years (e.g., see [155], [156], [157], [158], [159], and [160]–[163]). Some of these contributions

41

tried to capture risk in terms of tradeoffs between mean and variance of suitable random variables, some

have followed [164] in considering the expected value of a suitable exponential utility criterion and some

have focused on the so-called “percentile optimality” (e.g, [155], [165], [158], and [162]). Markowitz [166]

pioneered the notion of mean-variance tradeoffs in finance literature and many more sophisticated,

dynamic and stochastic, financial models involving closely related issues have been studied in recent years

(e.g., see [167], [168], and [169]). More precisely, we consider a finite-horizon discounted MDP model in

which the decision-maker, at each stage, needs to decide what percentage of the current retirement fund to

allocate into the limited (small) number of investment options. We assume that both the initial investment

and the target retirement capital are known and that the number of stages is. Now, the first target hitting

time is a random variable whose distribution is specified by the choice of a policy. As mentioned above the

decision-maker’s goal is to find a policy which maximizes. While at first sight, this might appear to be a

very difficult problem it turns out a version of optimality principle can be shown to hold under mild

conditions when we work in an “extended” state space. However, even in the extended state space the new

process is not a Markov process under a general policy. Hence the existence and characterization of optimal

policies cannot be obtained by standard techniques. Instead, the techniques used by Kang Boda, Jerzy A.

Filar, Yuanlie Lin, and Lieneke Spanjers [152] are similar to those developed in [155] which dealt with a

related problem of minimizing the probability that the total discounted wealth is less than a specified target

level. From the preceding optimality principle, structural results about optimal policies can be easily

derived which, in turn, lead to a dynamic-programming type algorithm. It is assumed that the yield of a

given fund in every year in the future is best modeled as a random variable that takes on the past observed

yields from that fund with equal probability. This is a rather simplistic assumption that may not correspond

to reality. To try to alleviate this problem one could consider a model with rolling horizon policies. The

idea of this approach is that an optimal policy is found and the first decision rule is implemented. Then, if

new data are available, a problem with updated parameters and a new time horizon is solved. The first

decision rule from an optimal policy of the latter is then implemented and so on. Rolling horizons have

been used by many researchers (e.g., see [170]).

Appreciation rate of the stock and the volatility of the stock

42

Several authors consider a financial market where the risk free interest rate, the appreciation rate of the

stock and the volatility of the stock depend on an external finite state Markov chain [171]. The authors

investigate the problem of maximizing the expected utility from terminal wealth and solve it by stochastic

control methods for different utility functions. More precisely, they use the Hamilton–Jacobi–Bellman

equation and prove a verification theorem for the Markov-modulated portfolio optimization problem. Due

to explicit solutions it is possible to compare the value function of the problem to one where the financial

market has constant (average) market data. The case of benchmark optimization is also considered.

Stochastic control methods of utility functions for benchmark optimization

Often the financial is incomplete and consists of one bond and one risky asset. The incompleteness of the

market is due to stochastic coefficients appearing in the price process of the risky asset and the bond. More

precisely it is assumed that the interest rate of the bank account, the appreciation rate of the stock and the

volatility of the stock depend on an external continuous-time, finite state Markov chain . The state of the

Markov chain should represent the general market conditions (for a motivation; see, e.g., [173]). Models

with deterministic coefficients are only good for a relative short period of time and cannot respond to

changing conditions. In this Markov-modulated setting, it is desired to solve the classical portfolio

optimization problem where an investor wants to maximize the expected utility from terminal wealth. As

far as the information is concerned, the investor has at the time point of decision observes the stock price

and the market condition. This is due to the fact that in a diffusion price process model the quadratic

variation and thus the volatility can be approximated arbitrarily well by the price process (cf. [174]).

Therefore it is in principle sufficient to solve the optimization problem with complete observation. This is

done using stochastic control methods for a number of different utility functions, namely for logarithmic

utility, and for benchmark optimization. Motivated by [175], there is a growing literature dealing with

portfolio optimization problems under different aspects. Problems with stochastic volatility have for

example been investigated in [176], [177], [178], and [179], among others. Most of these papers assume

that the external process is a diffusion process itself, like in the established volatility model of [180]. To the

best of our knowledge, the first paper to model the volatility as a continuous-time Markov chain is [181].

As we will see this model has the advantage that many portfolio problems can be solved explicitly in

contrast to the diffusion setting (compare, for example, [176] and[179]). Moreover, a diffusion process can

43

be approximated arbitrarily closely by a continuous-time Markov chain (see [182]). Portfolio optimization

with stochastic interest rates are, e.g., treated in [183] and [184]. The authors of [184] consider the Ho-Lee

and the Vasicek model for the interest rate which are both diffusion processes. The solutions we obtain are

found with the help of stochastic control methods. More precisely, by the use of a verification theorem. For

a comprehensive presentation of this theory the reader is referred to [185] or [186], among others. In the

case of deterministic coefficients, this model has been considered in [187] and in a more general context by

[188]. In our setting, we are only partly able to solve the portfolio problem explicitly. A closed form

solution is derived when the discounted stock price process is a martingale. Portfolio optimization with

stochastic market data is more realistic than standard models with constant coefficients. The formulation of

the market condition as a continuous-time Markov chain makes the analysis simpler as in the case of a

driving diffusion. For the utility functions treated here, the maximal portfolio value can be computed as a

solution of a simple linear differential equation. More complicated is the case of benchmark optimization. It

remains open whether a closed form solution can be derived in the general Markov modulated case.

A generalized mean-variance model via optimal investment policy

For an investor to claim his wealth resulted from his multi-period portfolio policy, he has to sustain a

possibility of bankruptcy before reaching the end of an investment horizon. Risk control over bankruptcy is

thus an indispensable ingredient of optimal dynamic portfolio selection. Shu-Shang Zhu, Duan Li, and

Shou-Yang Wang [189] have proposed that a generalized mean-variance model via which an optimal

investment policy can be generated to help investors not only achieve an optimal return in the sense of a

mean-variance tradeoff, but also have a good risk control over bankruptcy. One key difficulty in solving the

proposed generalized mean-variance model is the nonseparability in the associated stochastic control

problem in the sense of dynamic programming. A solution scheme using embedding has been developed by

Shu-Shang Zhu et al [189] to overcome this difficulty and to obtain an analytical optimal portfolio policy.

Optimal dynamic portfolio selection is to redistribute successively in each time period an investor’s current

wealth among a basket of securities in an optimal way in order to maximize a measure of the investor’s

final wealth. The literature of dynamic portfolio selection has been dominated by the results of maximizing

expected utility functions of the terminal wealth [190–192], [193], [194], [195 - 199], [200]. The

Markowitz’s mean-variance model [201] has been recently extended in [202] to a multiperiod setting. The

44

analytical expression of the efficient frontier for the multiperiod portfolio selection is derived. The

continuous-time mean-variance formulation is studied in [204]. The dynamic mean-variance formulation in

[202] and [203] enables an investor to specify a risk level which he can afford when he is seeking to

maximize his expected terminal wealth or to specify an expected terminal wealth he would like to achieve

when he is seeking to minimize the corresponding risk. It is easier and more direct for investors to provide

this kind of subjective information than for them to construct a utility function in terms of the terminal

wealth. The tradeoff information between the expected return and the risk is clearly shown on the efficient

frontier, that is most useful for an investor to decide his investment decision. Performing an optimal

investment policy in accordance with a dynamic portfolio formulation does not eliminate the possibility

that an investor goes to bankruptcy in a volatile financial market before he claims his wealth at the terminal

stage. One key difficulty in solving the proposed generalized mean-variance model of Shu-Shang Zhu et al

[189] is the nonseparability in the associated stochastic control problem in the sense of dynamic

programming. A solution scheme adopting a Lagrangian dual formulation and using embedding is

developed by Shu-Shang Zhu et al [189] and a case study has been presented to gain insights about the

significance of the risk control in dynamic portfolio selection. Due to the volatility of financial markets,

bankruptcy control is an indispensable issue to be addressed in dynamic portfolio selection. In the mean-

variance formulation, reduction of the probability of bankruptcy can be achieved by increasing the

weighting coefficient for the variance. By incorporating a control of the probability of bankruptcy in the

generalized mean-variance formulation the dynamic portfolio selection problem becomes to seek a balance

among three objectives, which could lead to a more satisfactory tradeoff between the probability of

bankruptcy and the expected value of the final wealth. While the traditional stochastic control theory only

concerns a sole objective of minimizing an expected performance measure, variance control occurs

naturally in dynamic portfolio selection problems with a mean-variance formulation and many other

applications. The celebrated dynamic programming is the only universal solution scheme to achieve an

optimality for stochastic control problems. Dynamic programming, however, is only applicable to problems

that satisfy the property of separability and monotonicity. Variance control problems are not directly

solvable by dynamic programming due to its nonseparability. In this respect, variance minimization is a

notorious kind of stochastic control problems. Using an embedding scheme, a feedback optimal portfolio

45

policy can be obtained for variance control problems via parametric dynamic programming method, while

the corresponding optimal condition for the parameter can be derived by examining the relationship

between the primal and the auxiliary problems. The generalized mean-variance formulation proposed in

this note for risk control over bankruptcy in discrete-time dynamic portfolio selection can be extended to

continuous-time dynamic portfolio selection by imposing probability constraints at distinct time instants in

the continuous time horizon.

Risk-Sensitive Portfolio Optimization with Completely and Partially Observed Factors

Stettner [204] considers a market model with discrete time changes of the portfolio in which the prices of

assets depend on some economic factors. The purpose is to minimize a suitable long-horizon risk sensitive

cost functional. The factors may be completely observed, or partially observed, in particular only through

the changes in the asset prices. The form of the cost functional and the existence of both observed and

unobserved factors create a number of technical difficulties. To describe the influence of unobserved

factors a certain family of measure-valued processes is introduced. Using the so-called vanishing discount

approach the existence of the solutions to the long run risk sensitive cost Bellman equation is shown. It

turns out that the optimal portfolio strategy is a function of the observed factors and the above mentioned

measure valued processes.

Option Pricing Models in Financial Engineering

Modern option pricing techniques are often considered among the most mathematically complex of all

applied areas of finance. Financial analysts have reached the point where they are able to calculate, with

alarming accuracy, the value of a stock option. Most of the models and techniques employed by today's

analysts are rooted in a model developed by Fischer Black and Myron Scholes in 1973.

From the moment of its publication in 1973, the Black and Scholes Option Pricing Model has earned a

position among the most widely accepted of all financial models.

The idea of options is certainly not new. Ancient Romans, Grecians, and Phoenicians traded options against

outgoing cargoes from their local seaports. When used in relation to financial instruments, options are

generally defined as a "contract between two parties in which one party has the right but not the obligation

to do something, usually to buy or sell some underlying asset". Having rights without obligations has

46

financial value, so option holders must purchase these rights, making them assets. This asset derives their

value from some other asset, so they are called derivative assets. Call options are contracts giving the

option holder the right to buy something, while put options, conversely entitle the holder to sell something.

Payment for call and put options, takes the form of a flat, up-front sum called a premium. Options can also

be associated with bonds (i.e. convertible bonds and callable bonds), where payment occurs in installments

over the entire life of the bond, but this paper is only concerned with traditional put and call options.

Modern option pricing techniques, with roots in stochastic calculus, are often considered among the most

mathematically complex of all applied areas of finance. These modern techniques derive their impetus from

a formal history dating back to 1877, when Charles Castelli wrote a book entitled The Theory of Options in

Stocks and Shares. Castelli's book introduced the public to the hedging and speculation aspects of options,

but lacked any monumental theoretical base. Twenty three years later, Louis Bachelier offered the earliest

known analytical valuation for options in his mathematics dissertation "Th‚orie de la Sp‚culation" at the

Sorbonne. He was on the right track, but he used a process to generate share price that allowed both

negative security prices and option prices that exceeded the price of the underlying asset. Bachelier's work

interested a professor at MIT named Paul Samuelson, who in 1955, wrote an unpublished paper entitled

"Brownian Motion in the Stock Market". During that same year, Richard Kruizenga, one of Samuelson's

students, cited Bachelier's work in his dissertation entitled "Put and Call Options: A Theoretical and Market

Analysis". In 1962, another dissertation, this time by A. James Boness, focused on options. In his work,

entitled "A Theory and Measurement of Stock Option Value", Boness developed a pricing model that made

a significant theoretical jump from that of his predecessors. More significantly, his work served as a

precursor to that of Fischer Black and Myron Scholes, who in 1973 introduced their landmark option

pricing model.

The Black and Scholes Option Pricing Model didn't appear overnight, in fact, Fisher Black started out

working to create a valuation model for stock warrants. This work involved calculating a derivative to

measure how the discount rate of a warrant varies with time and stock price. The result of this calculation

held a striking resemblance to a well-known heat transfer equation. Soon after this discovery, Myron

Scholes joined Black and the result of their work is a startlingly accurate option pricing model. Black and

47

Scholes can't take all credit for their work, in fact their model is actually an improved version of a previous

model developed by A. James Boness in his Ph.D. dissertation at the University of Chicago. Black and

Scholes' improvements on the Boness model come in the form of a proof that the risk-free interest rate is

the correct discount factor, and with the absence of assumptions regarding investor's risk preferences.

In order to understand the model itself, we divide it into two parts. The first part, SN(d1), derives the

expected benefit from acquiring a stock outright. This is found by multiplying stock price [S] by the change

in the call premium with respect to a change in the underlying stock price [N(d1)]. The second part of the

model, Ke (-rt) N(d2), gives the present value of paying the exercise price on the expiration day. The fair

market value of the call option is then calculated by taking the difference between these two parts.

Assumptions of the Black and Scholes Model:

1) The stock pays no dividends during the option's life

Most companies pay dividends to their share holders, so this might seem a serious limitation to the model

considering the observation that higher dividend yields elicit lower call premiums. A common way of

48

adjusting the model for this situation is to subtract the discounted value of a future dividend from the stock

price.

2) European exercise terms are used

European exercise terms dictate that the option can only be exercised on the expiration date. American

exercise term allow the option to be exercised at any time during the life of the option, making american

options more valuable due to their greater flexibility. This limitation is not a major concern because very

few calls are ever exercised before the last few days of their life. This is true because when you exercise a

call early, you forfeit the remaining time value on the call and collect the intrinsic value. Towards the end

of the life of a call, the remaining time value is very small, but the intrinsic value is the same.

3) Markets are efficient

This assumption suggests that people cannot consistently predict the direction of the market or an

individual stock. The market operates continuously with share prices following a continuous Itô process.

To understand what a continuous Itô process is, you must first know that a Markov process is "one where

the observation in time period t depends only on the preceding observation." An Itô process is simply a

Markov process in continuous time. If you were to draw a continuous process you would do so without

picking the pen up from the piece of paper.

4) No commissions are charged

Usually market participants do have to pay a commission to buy or sell options. Even floor traders pay

some kind of fee, but it is usually very small. The fees that Individual investor's pay is more substantial and

can often distort the output of the model.

5) Interest rates remain constant and known

The Black and Scholes model uses the risk-free rate to represent this constant and known rate. In reality

there is no such thing as the risk-free rate, but the discount rate on U.S. Government Treasury Bills with 30

days left until maturity is usually used to represent it. During periods of rapidly changing interest rates,

these 30 day rates are often subject to change, thereby violating one of the assumptions of the model.

49

6) Returns are lognormally distributed

This assumption suggests, returns on the underlying stock are normally distributed, which is reasonable for

most assets that offer options.

After the Black and Scholes Model:

Since 1973, the original Black and Scholes Option Pricing Model has been the subject of much attention.

Many financial scholars have expanded upon the original work. In 1973, Robert Merton relaxed the

assumption of no dividends. In 1976, Jonathan Ingerson went one step further and relaxed the the

assumption of no taxes or transaction costs. In 1976, Merton responded by removing the restriction of

constant interest rates. The results of all of this attention, that originated in the autumn of 1969, are

alarmingly accurate valuation models for stock options.

Stochastic Control in Economic Theory

The control theory, in general, and stochastic control theory, in particular, are useful and sometimes even

essential to advance the state of economic theory and better regulate economies. Probabilistic and statistical

methods have been applied to economic problems such as development and planning of production and

inventory, growth models, portfolio selections, and to other aspects of theory of firms and so forth for some

time. These applications are, however, primarily to static or stationary economic situations

Application of control engineering viewpoints and control techniques to economic problems are of more

recent origin. Stochastic control theory has been applied to macroeconomic dynamic systems to answer

various policy questions in stabilization, optimal growth, planning, and others. Control of stochastic linear

dynamic macroeconomic models to minimize some quadratic social cost function is, perhaps, the best

known example of applications of stochastic control theory to macroeconomic systems The policymakers at

various government agencies and central banks are beginning to be seriously interested in modeling

national economics as large scale control systems and using econometric and control theoretic models and

techniques to evaluate alternative economic policies.

It may be emphasized that the aspect of decision making with imperfect information as the thread common

to both the stochastic control theory and economic problems. In assessing real or potential impacts of

stochastic control theory in economics, it is convenient to follow a conventional dichotomy of economics

50

into macroeconomics and microeconomics for expositional purposes. Briefly speaking, control theory has

been developed in our endeavor to guide or modify, to our advantage, time paths of objects in which we are

interested. Stochastic control theory is needed to take account of random variables or stochastic processes

which may be involved in determining, for example, the time paths of the objects of study and control. It is

not enough for policymakers to choose economic policies on a day-to-day or month-to-month basis, since

current policy decisions have impact not only in the present but also over some time in the future. This is

because the object of control, the national economy, is dynamic. The economic system must then be

modeled as a dynamic system and not just as a static system. The same can be said of basis economic

decision making units such as households and firms, which are the objects of study of microeconomic

theory, even though their decision making activities may not always be modeled as dynamic processes.

Besides the rather obvious applications of stochastic control theory to macroeconomic models such as

national economies there are applications to microeconomic situations. For example, economists

have recently constructed various models of market organization with imperfect information. Models of

search behavior, such as by the unemployed for jobs and by firms for workers, are nowadays being

modeled as ones with imperfect information. Many economic agents participate and interact directly or

indirectly with each other in markets. Their decision processes must then be modeled as dynamic decision

making processes under uncertainty or imperfect information. In models of markets or search behavior with

imperfect information, learning or formations of expectation regarding future behavior of economic

variables such as prices play important roles in explaining behavior of economic agents.

There are economic situations which illustrate the advantages of state-space representation of dynamic

models and on topics in which controllability or observability properties of dynamic systems play essential

roles. The concept of observability can be used to establish uniqueness of equilibrium solution in a quantity

adjustment (microeconomic) model.

Recursive dynamic least square instrument variable algorithm

The recursive dynamic least square instrument variable algorithm introduces the concept of the dynamics

of financial engineering based on innovative reasoning which precedes by forming an expectation and

verifying it by proper use of informational data set composed of the available knowledge and intuitive

observation. It deals with an application of the cybernetic method of recursive dynamic least square

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instrument variable algorithm with on line parameter tracking adaptability for on line modelling of short

term national market index movement with a time slot of 1-day having interacting variables such as market

price indices of market-dominating fundamentals of industrial production. The investigation [252] unfolds

the traditions of controls and systems in estimation, identification and exploitation of structures to develop

efficient algorithms providing opportunities of significant research in financial engineering. The work

presented here formalizes a specific dynamic situation, namely the construction of a finite dimensional

process for daily movement of national market index. It has been clearly demonstrated with observed data

that the flexibility of the algorithms is remarkably broad. Indeed, it is possible to choose free variables in

such a way that the entire formal modelling process can be interpreted as a linear quadratic Gaussian

problem.

The system scientists traditionally study systems which involve continuous variables and have dynamics

which can be described by either differential or difference equations. There are many situations, in which

such models are not appropriate as the following examples will illustrate. Consider the daily movement of

all India market index. To identify the process a series of decisions tests in the form of correlation are to be

executed. The state of the system, which changes only at the discrete instant of time instead of

continuously, consists of an assortment of discrete variables. For such an important class of systems there is

a dearth of elegant and succinct identification techniques. Once the identification has been done the system

behavior can be obtained. An initial step in this direction has been exemplified in the situation one has a

discrete set of interpretation of a sequence of observed data, which form the basis for efficient and

relational assessment of system identification in order to provide the operational framework in which many

types of knowledge and information can be incorporated into the system. Development of mathematical

description of a system is often undertaken to predict performance and responses. National market is a

complex system. The market indices yield meaningful measures of systematic market development.

Returns to an index reflect averages of investor holdings and provide a performance benchmark for the

asset class included in the index. Comparison of investment returns to those of an index with the same

target weights has become the most widely accepted criterion of investment performance. The financial

news papers such as The Wall Street Journal and the London Financial Times report global, regional, local,

and industry indices on a broad range of portfolios [263],[264],[265],[266] and [267]. National market

52

indices, such as BSE Sensex, S&P 500 and London’s FTSE provide performance benchmarks for equity

investments in these national markets. Regional indices track stock returns in broad geographic regions

such as Americas, Europe, Nordic or Asia Pacific. [268].

The recursive technique is one in which an estimate is updated on receipt of fresh information. Beck [253],

[254], [255], Ljung [256], [257], [258], [259] and [260]] have given a good coverage of recursive

identification methods.

The input-output relationship of financial engineering fundamentals on national market brings in

spontaneous emergence of order of optimum complexity from the initial featureless states.

It can be clearly demonstrated that the flexibility provided by the recursive algorithms are remarkably

broad. It is evident from the mathematical description of the all India share price index that all new

financial information is quickly understood by the market state variables and the information itself becomes

immediately incorporated in the model.

Money market has been found to exhibit certain universal characteristics in which a large number of agents

interact [261], [262]. It is obvious that movements in money market are immediate and unbiased reflection

of incoming news about future earning prospects. In the present money market model there are two groups

of players: the first group ‘fundamentalists’ follows the premise of efficient market operations, in that, they

allow the price index to follow the fundamentals. The other group, which may be called noise group do not

affect the share price and are automatically filtered out in course operations of the intelligent algorithms

that have been used. A distinguishing feature of the present approach as compared to other simulation

models [269], [270] is that we have adopted a statistical formalization [271] where fundamentals react with

price index.

The fuzzy set theory in decision making in financial management

The fuzzy set theory provides a guide to and techniques for forecasting, decision making, conclusions, and

evaluations in an environment involving uncertainty, vagueness, and impression in business, finance,

management, and socio-economic sciences [273],[274], [275], [276], [277], [278]. It encompasses

applications in case studies including stock market strategy. The fuzzy membership function µ (unit share

price) for low , medium and high and the corresponding Mumbai Stock Exchange Sensitive Index (BSE

SENSEX), µ (BSE SENSEX ) for low , medium and high are described. It has been shown that the unit

53

share price in a dynamically stable market moves along with the sensitive index [272]. The application of

fuzzy control algorithms for market management may appear to be a promising domain for further

investigation.

This study [272] is devoted to the analysis of data, an endeavour that exploits the concepts, constructs, and

mechanisms of fuzzy set theory. Data analysis involves searching for stable, meaningful, easily

interpretable patterns in databases [281], [282]. Data analysis is an immensely heterogeneous research area

that embraces techniques and ideas that stem from probability and statistics, fuzzy sets. data visualization,

databases. And so forth. In spite of such a profound diversity, the focal point is constant: to reveal patterns

that are not only meaningful but also easily comprehensible. A fuzzy set can be regarded as an elastic

constraint imposed on the elements from a universe of discourse. By admitting a certain form of elasticity

when defining concepts and introducing various notions that are encountered in every day life.

Conceptually, fuzzy sets help alleviate problem with the classification of elements of boundary nature by

allowing for a notion of membership to a category. This requires a mathematical framework to handle

common usages of terms that are not Boolean in character, but rather are indistinct, vague, and fuzzy.

Rather than allow for an element to be either in or not in a particular set, each element can be assigned a

degree of membership in the set, often scaled over the range [0,1]. A membership of zero indicates that the

element is not a member of the fuzzy set. A membership of one indicates that the element definitely

belongs to the fuzzy set. Intermediate values correspond to lesser degrees of membership.

The fuzzy set theory provides a guide to and techniques for forecasting, decision making, conclusions, and

evaluations in an environment involving uncertainty, vagueness, and impression in business, finance,

management, and socio-economic sciences [279], [280]. Traditional modeling techniques do not capture

the nature of complex systems especially when humans are involved. Fuzzy logic provides effective tools

for dealing with such systems. It encompasses applications in case studies including Time Forecasting for

Project Management, New Product Pricing, Client Financial Risk Tolerance Policy, Deviation and Potential

Problem Analysis, Inventory Control Model, Stock Market Strategy.

Fuzzy systems, including fuzzy logic and fuzzy set theory, provide a rich and meaningful addition to

standard logic. The mathematics generated by these theories is consistent, and fuzzy logic may be a

generalization of classic logic. The applications which may be generated from or adapted to fuzzy logic are

54

wide-ranging, and provide the opportunity for modeling of conditions which are inherently imprecisely

defined, despite the concerns of classical logicians. Many systems may be modeled, simulated, and even

replicated with the help of fuzzy systems, not the least of which is human reasoning itself. Large data bases

on stock market indices and equity prices are available to a researcher to analyze and to apply decision

algorithms. Financial processes are becoming more complex which leads to the increased use of

computational intelligence (CI) [272] to process the mass of data produced in this field. CI-based

approaches of fuzzy algorithms are being applied to describe the linguistic interpretation of the rules in a

fuzzy system. The transparency of CI combined with good function approximation capabilities are the main

justification for applying fuzzy systems in finance and business.

Group Method of Data Handling

The group method of data handling (GMDH) is a self-contained exposition of a cybernetic approach to

develop mathematical model of an appealing environment with the behavioral equations and latent

variables as the important supporting characters. The framework presented incorporates problems of

representations, questions of parameterisation and procedure for identification algorithm for obtaining

models from observed data. The exposition deals with an application of the cybernetic method to develop

polynomials of optimum complexity with multi-layer group method of data handling algorithm of technical

cybernetics through an observation-conjecture-modelling-validation cycle. One of the goals of technical

cybernetics has been to capture the major elements of a dynamical process under the umbrella of a formal

mathematical synthesis. The work presented by Ranjan Chaudhuri [237] formalises a specific dynamic

situation, namely the construction of a finite dimensional process for daily movement of national market

index. It has been clearly demonstrated with observed data that the flexibility of the algorithms is

remarkably broad. Indeed, it is possible to choose free variables in such a way that the entire formal

modelling process can be interpreted as a loop shaping problem, where the loops are the layers in the multi-

layer selection process, and the loop-breaking takes place at the optimum layer through a certain specific

choice of variables. It has been observed that deep-lying feedback paths exist in national market operation.

To give mathematical description of daily national index movement as a function of a set of exogenous

variables interrelated with one another through deep-lying feedback paths is a complex process. Theories

based on differential or difference equations are not adequate to describe the process. In view of this

55

difficulty, the method of modelling applied here uses a technique of self-organisation. The GMDH is found

to simulate adequately the input-output relationship of the complex process of daily market index

movement as a function of industry-dependent state variables. Money market is one of the most exciting

and sobering parts of economics. It is marked by bubbles in which speculative prices are driven up far

beyond their intrinsic values. Speculative bubbles always produce crashes and often lead to economic fear.

Market is a tool where country's resources are allocated. By proper utilisation of resources a nation can

command its economy decisively. Modelling with interacting market parameters increases our appreciation

of macroeconomic analysis. This explores the exciting world where principles of cybernetics work on the

theories of economics. The Group Method of Data Handling Algorithms (GMDH) describes modeling,

forecasting, decision support and pattern recognition of complex systems [205] – [208] and [209] – [217] .

There are processes for which it is needed to know their future or to analyze inter-relations. Economy,

climate, finance, ecology, medicine, manufacturing and military systems [218] – [230] are areas where the

GMDH method has been successfully applied. This self-organizing method is based on sorting-out of

gradually complicated models and their evaluation by external criterion on data sample. It was developed

for forecasting, extrapolation of multivariate processes, knowledge discovery and data mining, decision

making by "what-if" scenario, diagnostics and pattern recognition. Linear or non-linear, probabilistic

models or clusterizations are selected by minimal value of an external criterion. Ivakhnenko’s [229], [230]

multi-layer group method of data handling is heuristic method of self-organisation of different partial

models. This method involves the generation and comparison of all possible combinations of input output

and to select the best possible ones according to the criterion of integral square error [212].

In multi-layer group method of data handling algorithms, polynomials are used as the basic means of

investigation of complex dynamical systems. The polynomials of prediction are regression equation which

connect the current values of output with the current and/or past values of input variables. Regression

analysis allows to evaluate the coefficients of the polynomial by criterion of minimum mean square error.

Then the polynomials are treated in the same manner as are seeds n the agricultural selection, an unique

mathematical concept propagated and established by Academician A.G. Ivakhnenko and his co-workers of

the Institute of Cybernetics, Kiev, Ukraine.

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Volterra series [231] introduced to non-linear system analysis by Wiener [232] , learning filter of Gabor,

Wilby and Woodcock [233] and the perception of Rosenblatt [234] have provided the conceptual basis for

multilayer GMDH. Astrom and Eykhoff [235] pointed out that problems may arise with the use of

Volterra series or high degree polynomial to approximate non-linear functions because of the fact that there

are many coefficients to estimate, many data are needed and the computation with the resulting large

matrices may be prohibitive. Ivakhnenko’s multi-layer GMDH algorithms are free of these problems. He

models the input output relationships of complex process using multi-layer network structure of

Rosenblatt’s perception type, who designed the model of brain’s perception.

We describe GMDH-type polynomial networks. where xi is a i-th input variable, y is an output The

GMDH-type networks are the multi-layered ones. In the second and next layers r, the size Lr of the

population defined by the number F. The generation and selection of the layers are again performed.

The new layers are created while the criterion value is decreased. In Fig. we depicted an example of

the polynomial network consisting of 3 layers. The GMDH algorithm grew for m = 5 inputs and F = 4.

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An example of polynomial network This network is described by a set of the following polynomials:

where g1, …, g6 are the transfer function of the neurons. With the help of multi-layer GMDH algorithms Ivakhnenko obtained the polynomial description of

British economy for prediction and control. The GMDH is a computer aided self-organization of

spontaneous emergence of order of optimum complexity from the initial featureless states. One of the goals

of the theory of cybernetics has been to capture major elements of a dynamical process under the umbrella

of a formal mathematical synthesis. Analysis has been done to find relationship between different

parameters of stock market indices of different stock exchanges. But the present investigation is indeed

unique in its own nature. The work presented in here formalizes a specific dynamic situation, namely the

construction of a mathematical description for a finite dimension process of the daily all India share price

index. It has been clearly demonstrated that the flexibility provided by the multi-layer group method of

data handling algorithm is remarkably broad. In fact, it is possible to choose free variables in such a way

that the entire formal modelling process can be reinterpreted not as a least square error minimization

problem but as a “loop shaping” problem where the loops are the layers in a multi-layer selection process

and the loop-breaking takes place at the optimal layer via certain specific choice of variables. Thus it may

be justifiably asserted that LQG/GMDH is a practical comparison of LQG/LTR [236] . It is evident from

the mathematical description of the all India share price index that all new financial information is quickly

58

understood by the market state variables and the information itself becomes immediately incorporated in

the model. The operating principles of an efficient market holds that market prices contain all available

information. The price movement in organized sector follows a definite pattern. Thus it can be safely

concluded that Indian market is in an efficient self-monitoring equilibrium state.

marketing applications motivate a specialized model and its adaptive control [238]. Each of r control

variables is set in each of a sequence of time periods. The process being controlled has a response (profit)

function that is the sum of a constant plus linear and quadratic forms in the control variables. The

coefficients of the quadratic form are assumed to be known constants, those of the linear form to change

with time as autoregressive process. Information about the changing coefficients is detected by performing

experiment on a sub-portion of the process being controlled. Provision is made for adding further

information from unspecified sources. Bayesian methods update the distributions of the unknown

coefficients. Dynamic programming determines the values of the control variables and experimental design

parameters to maximize the sum of discounted future profits. The probabilistic assumptions of the model

are chosen so that all distributions are normal with known variances and, for the most part, zero

covariances between variables. Partly as a result of this, optimal control turns out to involve rather simple

exponential smoothing rules.

Optimal Adaptive Control: A Multivariate Model for Marketing Applications

Marketing offers rich possibilities for adaptive control applications with its many decision variables,

reasonably clear objective functions, and dynamic environments. Measurement techniques, though

imprecise by engineering standards, offer actionable information and, by their very imprecision, challenge

the development of new theory. On the other hand, marketing is relatively lean on good descriptive models

of the processes over which control is sought and the implementation of model-based systems of substantial

complication poses formidable organizational, educational, and managerial obstacles. However, the high

importance placed on effective marketing by all companies ensures that new techniques, if truly productive,

are likely to be adopted in time. To give an example of a marketing application, consider a manufacturer

setting an advertising budget for a consumer product. He is never precisely certain how sales will respond

to advertising because response is difficult to measure and changes with time. The changes arise from

various disturbing influences; for example, competitors may introduce new products, alter prices, run

59

promotions, and generally stir the market up in ways that affect advertising response. In addition, the

advertising messages themselves may grow stale or consumer tastes may drift away from the product. Yet

the measurement of sales response to advertising is not impossible, only imprecise. Such measurements can

also form the basis of adaptive control of advertising spending. This possibility has been proposed and

analyzed by Little [239]. In at least one company the proposal has become a reality and an adaptive system

has been used to set annual advertising budgets for a product line. The model and adaptive system of [239],

however, control a single variable, whereas most marketing situations are bursting with manipulable

quantities. Therefore, a multivariate model and control system with wider applicability are sought. A

number of authors, for example, Tse and Athans [240], Deshpanda et al. [241], and Upadhyay and Lainiotis

[242] have presented approaches to optimal adaptive control of stochastic systems. Relative to their work,

the work of Little [238] presents model and control process having a rather specific structure motivated by

the class of problems addressed. As a result, a quite exact analysis is possible and, in addition, the required

calculations are simple. If the measurements yield useful data in their own right and if a meaningful

criterion function can be worked out, then an adaptive system in some more or less recognizable form has a

chance of being sold to management as a guide to setting the control variables. If then the implemented

system recommends definite changes in the control variables and these changes seem correct and not trivial

or quixotic, the system may stay in place and even grow in scope.

Macroeconomic Modeling with Modern Control Techniques

Macroeconomic Modeling [243] in the least complex structural form makes it endowable for the

application of modern control techniques. The method employed is a blend of the “black box’’ approach

where no knowledge of the inner mechanism is assumed and the classical approach of econometrics where

economic theory is used to determine structure. The technique is illustrated in detail by means of a

numerical example consisting of a four-equation sector of a larger model in which a full treatment of error

is given. This serves to reveal to engineers the difficulties of econometrics, demonstrating the many pit-

falls and acute problems that prevent a straightforward application of the methods of control theory. The

emphasis throughout has been on the development of a methodology together with the implementing of

supporting computational algorithms in the form of program packages. The objective is to make a

determined effort to establish to what extent control theory can help resolve what is acknowledged to be a

60

difficult control problem. Such an attempt is fraught with dangers; of criticism from economists on grounds

of lack of realism and from control engineers because the economy is not necessarily a causal mechanism

on which assumption all control theory is founded. The main difficulty lies in the extreme complexity of

the actual economic system. Indeed, on the contrary, a preoccupation with the “black box” approach can

lead to palpable nonsense. Some of the results given are incomplete in that they require further work from

the viewpoint of economic analysis

The modeling procedure described in [243] was designed to incorporate four concept.s. First, employment

is made of all available a priori information provided by economic theory thus limiting beforehand the

possibility of expending effort on fruitless searches for nonexistent relationships (interconnections).

Second, the basic philosophy of the “black box” approach is then applied allowing the data to decide the

exact dynamic structure. Thus, overly complex (statistically unsubstantiated) structures are automatically

eliminated. Third, diagnostics are continually employed which are designed to both reveal inadequacies and

indicate how improvements can be made. Finally, experienced judgment is always solicited whenever the

results of one stage must be used in a decision affecting the input to a succeeding stage. The overall

outcome of this modeling procedure is a model with the fewest required interconnections , and the least

possible order dynamic operators explaining the behavior of the existing interactions. Using a sector of a

small control model of the United Kingdom economy, the difficult task of structure determination has been

illustrated. In particular, it. was shown how a model structure determined from a univariate (single

equation) analysis can be extensively altered in the framework of a simultaneous model, requiring a re-

specification of the joint structure. However, repeated cycling through the estimation and diagnostic stages

allowed the determination of a reasonably well defined model with no more than 11 parameters, after an

economic re-specification to delete import prices from the endogenous variables. A final analysis of the

fitted model was required in order to determine its ability to forecast ahead of the sample period. Such

forecasting revealed a certain deficiency in the model with respect to the over-dominant role played by the

estimated means of the outputs, i.e., better, or more, explanatory variables are required. It is the considered

opinion that any attempt to investigate the applicability of control theory to the problems of

macroeconomic policy optimization needs to be an interdisciplinary effort. For the equations of the model

to have significant meaning in economic terms the importance of sympathetic collaboration with

61

professional economists cannot be over emphasised. On questions of basic relationships their assistance is

essential- a purely black-box approach cannot in this case be made to work. Most control engineers and

theorists are not sufficiently cognizant of the special economic issues involved. Conversely, most,

economist, or econometricians do not. fully understand the generality and unified approach to dynamic

systems afforded by control theory. Both areas have a considerable amount, of mutual interest and much

can be learned from the other.

The Parallel Computing Environment for Financial Engineering Applications

The parallel computing environment [ 283] is an object-oriented C++ library that uses abstractions to

simplify parallel programming for financial engineering applications. The message passing interface

ensures portability and performance over a wide range of parallel cluster and symmetric multiprocessing

machines.

Parallel computing has emerged as a cost-effective means of dealing with computationally intensive

financial and scientific problems. To effectively utilize this technology, developers need software that

reduces the complexity of the process as well as tools to support integration of parallel and desktop

machines. The parallel environment is a C++ library that facilitates development of large scale parallel

applications, particularly financial engineering applications.

Parallel computing provides domain-specific object-oriented libraries for solving partial/stochastic

differential equations using the finite-difference method and Monte Carlo simulation. These libraries factor

out the common operations required for FD and MC computations so that in most cases the user need only

provide the code required for the specific application. The architecture consists of three layers:

• The data abstraction and transportation core layer provides the transportation drivers that facilitate

communications among parallel processes, desktop machines, and databases via the message passing

interface (MPI), Extensible Markup Language (XML), base64 encoding, and open database connectivity

(ODBC). This layer simplifies the code associated with transferring complex data among different types of

processes and machines.

• The parallel application layer provides the domain-specific FD and MC libraries.

• The remote execution layer provides an interface between the parallel computation and the desktop

machine. This layer introduces remote parallel objects, which can encapsulate applications built using the

62

domain-specific libraries. Users can manipulate the handles of these parallel objects on a desktop machine

to access applications residing on the parallel platform. The core layer uses standard MPI primitives

Monte Carlo simulation is a numerical technique for solving problems that stochastic models describe by

generating numerous samples, commonly known as paths. Computation speed is a major barrier to

deploying MC simulations in many large and real-time applications. The MC library in parallel computing

environment facilitates the parallelization of MC applications, freeing users from dealing with these issues

while maintaining extensibility. The implementation works on heterogeneous clusters and uses dynamic

load balancing.

The MC library provides the Simulation abstract base class and related classes that encapsulate a parallel

simulation. To use the library, the user simply codes the computation of a single sample via an abstract

method. The MC library can then handle most other aspects associated with the parallelization and load

balancing of the MC simulation in a manner transparent to the programmer.

The MC library parallelizes the simulation by controlling each co-process to run sample generation, with

intermittent communications to schedule, coordinate, and load-balance the remaining simulation. The

Simulation class run method performs the parallel MC simulation in a series of computing and

synchronization steps.

Conclusion

Economic models are outcome of analysis and synthesis of financial engineering. Economic models are

employed for a variety of purposes; forecasting is an important application, control of the economy is

another. So in constructing an economic model it is important to know what the model is to be used for and

what the economist wants from it. The validity of the. model will require it to be readily admitted to an

economic analysis. Thus in setting up the equations it is necessary to express them in structural form with

which economists are familiar. There are some features of the economic environment that distinguishes this

particular field from others. The very imprecision of economic theory is exasperating; there is no

counterpart to the exactitude of the laws governing the behavior of electrical circuits, for example. The

nearest approach in economic theory would be a statement, such as “voltage affects or is affected by

current.” The nearest to an economic law is the statement that “demand” equals “supply;” rather as

Kirchhoff’s law tells us that, the sum of currents at a node is zero. There is even doubt about the

63

assumption of causality; not only doubt as to what is “cause” and what “affect” but also due to the fact that

action often precedes the action itself due to “smart operators”. A further outstanding feature is the extreme

brevity of the records available. Occasionally a sharp change, such as for example a devaluation, provides

the system with a jolt but owing to the complex interactions of the system its effects are soon smudged out.

The point. is quite a significant one since there are good reasons to believe that the economic system is one

whose underlying parameters are not constant, but evolving with time. Thus, in dealing with a system of

this sort it would be very important to get the maximum information from the recent past of the record.

Another distinctive feature of economic systems is the very high noise level. This arises from the way in

which highly aggregated national statistics are accumulated; a further hazard is the habit of occasionally

redefining the base of such statistics so destroying the continuity of the run and leading to a difficult

smoothing problem in handling such adulterated data. With the high noise levels disturbances do propagate

themselves through the system. The interactions between equations are very pronounced indeed and

simultaneous estimation of parameters of the complete set of equations is essential. It is in this sense that

the modelling of the economy offers new possibilities to the systems expert and represents a new challenge

to his ingenuity.

The investigation of the applicability of system theory of GMDH [205] –[210] to the problems of

macroeconomic policy is an interdisciplinary effort. For the equations of the model to have significant

meaning in economic terms the importance of sympathetic collaboration with professional economists

cannot be over emphasized. On questions of basic relationships their assistance is essential. A purely black-

box approach cannot in this case be made to work. Most system theorists are not sufficiently cognizant of

the special economic issues involved. Conversely, most, economist, or econometricians do not. fully

understand the generality and unified approach to dynamic systems afforded by the system theory of

GMDH. Both areas have a considerable amount, of mutual interest and much can be learned from the other.

We have offered some speculations about the future of interdisciplinary payoffs involving system theorists

and economists working together. Our speculations, although influenced by our own work we hope that

contrary view will help clarify the possible evolution of groups composed of system theorists and

economists working together. This work , it is hoped will have at their disposal a mathematical tool, the

group method of data handling algorithm that, when used together with econometric models, could

64

substantially advance the science of economic and financial management. The paper has also showed that

the GMDH approach has been extremely useful, easy, and computationally cheap for running tradeoff

studies that lead to alternate economic (monetary and fiscal) policies. Macroeconomics is concerned with

the cont.ro1 of economic aggregates. The system dynamics, as a whole, are described by sets of high-order

equations. The mat.hematica1 models are obtained using a combination of economic theory and regression

analysis. The developed models have tried to exploit the power of state variable representations. Once the

models have been identified by GMDH methods the time-varying parameters can be estimated using fresh

information in GMDH algorithms.

This section introduces some of the ideas and methods in the identification and estimation of simultaneous

regression equation systems in econometrics. After pointing out the special features of econometric

systems, it defines the problem of identification and presents several methods for estimating the parameters

in such systems. Hopefully this investigation will be useful to research workers in related fields of

economics and who are interested in the estimation of dynamic econometric systems.

It is important. to estimate the parameters of the structural equations. The main reason is that. economic

hypotheses such are formulated in the form of structural equations. If there is any change in economic

institutions, technological relations, or behavioral patterns of the economic agents as described by the

structural equations, one is able to assess its impact only by modifying the st.ructura1 equation affected.

Economic hypotheses arc mainly qualitative in character. They help specify the important variables which

should appear in each structural equation and frequently also the signs of their coefficients but not the

magnitudes. Limited historical data are employed for the estimation of all the unknown parameters in the

system. Before studying methods for estimating the unknown parameters it is necessary to impose

restrictions on the parameters to insure their identifiability. A set of structural parameters is said to be

identifiable if there exists no other set which will give rise to the same structural equation. If there exist two

sets of values for the structural parameters from which the same equation is deduced, the structural

parameters are unidentifiable. In this case, no consistent estimator for the set of parameters exists. The

GMDH algorithms of the methods of self organization of mathematical models based on multiple normal

regressions give most reliable estimates of the unknown parameters pertaining to the struct.ura1 equation

subject to the identifiability restrictions The nonlinear structural equations as proposed for GDP prediction

65

has suggested an application of the method of instrumental variables to certain type of nonlinear equations.

The methods just mentioned for nonlinear systems have not been tried extensively. and much more

experimentation is required to study their computational problems and their sampling properties.

Economists are probably dealing with large systems. The number of unknown parameters to be estimated is

larger, and the data on which estimates can be derived are limited. Mention should also be made of the

recent interest among econometricians in the estimation of time-varying coefficients in regression models,

a topic well-treated in GMDH algorithms. Having investigated some the methods for identification and

estimation of system theory for using in econometric it is hoped that scholars in this inter-disciplinary field

will find this work interesting and appealing.

GMDH formulates the system equations, state and space constraints, and a criterion functional for an

example for a problem in economic growth, and discusses some interpretation of the underlying economic

structure. Several examples are presented to illustrate particular features of Estimation problems in

economics and to more general work in mathematical system theory of economics involving group method

of algorithms. New developments in the theory of economic growth raise a number of issues of interest to

system theorists. This paper suggests a framework which may be helpful in studying economic growth

models and gives reference to mathematica1 discussions of the principles underlying some of the

economic problems to which system theory can usefully be applied. The description one might take of the

state of an economic system is a record, at, the specified instant, of its aggregates of variables along with a

record of flows and transaction between various agents or groups within the economy. The most recent

growth models have tended to deal with one primary factor, gross domestic product whose growth is

influenced by a number of distinct variables. The state is specified by a finite-dimensional vector whose

components represent economic aggregates. In discussions of economic growth it is usually assumed that

ultimate concern attaches to the rate of increase of GDP. The performance index for an economy can be

taken to be a function of the state so that the criterion functional really depends only on the time paths for

these variables and the initial state. It may be a more satisfactory portfolio theory into the analysis would

entail treating a growth model with important stochastic components; this topic is an open research

problem. The present investigation devises computational methods that improve the analysis of economic

phenomena for helping to improve analysis of such economic indicators of economic growth as GDP. The

66

models may become valuable tools for researchers and financial analysts. The models are directed at an

important aspect of economic movements over time, so-called non-stationary data, such as a long-lasting

effect from a temporary disturbance in economic growth patterns.

References

[1] Tze Leung Lai and Samuel Po-Shing Wong, “Valuation of American Options via Basis Functions”,

IEEE Transactions on Automatic Control, Vol. 49, No. 3, March 2004 [2] S. A. Ross, “Finance,” in The New Palgrave: A Dictionary of Economics, J. Eatwell, M. Milgate, and P. Newman, Eds. New York, NY: Stockton Press, 1987, vol. 2, pp. 322–336. [3] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Polit. Econ., vol. 81, pp. 637–659, 1973. [4] R. C. Merton, “Theory of rational option pricing,” Bell J. Econo.Manage. Sci., vol. 4, pp. 141–181, 1973. [5] J. N. Tsitsiklis and B. Van Roy, “Optimal stopping of Markov processes: Hilbert space theory, approximation algorithms, and an application to pricing high-dimensional financial derivatives,” IEEE

Trans. Automat.Contr., vol. 44, pp. 1840–1851, Oct. 1999. [6] , “Regression methods for pricing complex American-style options,”IEEE Trans. Neural Networks, vol. 12, pp. 694–703, July 2001. [7] F. AitSahlia and T. L. Lai, “A canonical optimal stopping problem for American options and its numerical solution,” J. Computa. Finance, vol. 3, pp. 33–52, 1999. [8] M. J. Brennan and E. S. Schwartz, “The valuation of the American put option,” J. Finance, vol. 32, pp. 449–462, 1977. [9] Apostolos-Paul N. Refenes, A. Neil Burgess, and Yves Bentz . “Neural Networks in Financial Engineering: A Study in Methodology”, IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 8, NO. 6, NOVEMBER 1997, p.1222-1268

[10] G. R. Courtadon, “A more accurate finite difference approximation for the valuation of options,” J.

Finan. Quant. Anal., vol. 17, pp. 697–703, 1982. [11] L. Wu and Y. K. Kwok, “A front-fixing finite difference method for the valuation of American options,” J. Finan. Eng., vol. 6, pp. 83–97, 1997. [12] J. Cox, S. Ross, and M. Rubinstein, “Option pricing: A simplified approach,” J. Finan. Econ., vol. 7, pp. 229–263, 1979. [13] S. D. Jacka, “Optimal stopping and the American put,” Math. Finance, vol. 1, pp. 1–14, 1991. [14] P. Carr, R. Jarrow, and R. Myneni, “Alternative characterizations of American put options,” Math.

Finance, vol. 2, pp. 87–106, 1992. [15] J. Huang, M. G. Subrahmanyam, and G. G. Yu, “Pricing and hedging American options: A recursive integration method,” Rev. Finan. Stud., vol. 9, pp. 277–300, 1996. [16] N. Ju, “Pricing an American option by approximating its early exercise boundary as a multipiece

67

exponential function,” Rev. Finan. Stud., vol. 11, pp. 627–646, 1998. [17] F. AitSahlia and T. L. Lai, “Exercise boundaries and efficient approximations to American option prices and hedge parameters,” J. Computat. Finance, vol. 4, pp. 85–103, 2001. [18] J. C. Hull, Options, Futures & Other Financial Derivatives, 4th ed. Upper Saddle River, NJ: Prentice- Hall, 2000. [19] I. Karatzas, “On the pricing of the American option,” Appl. Math.Optim., vol. 17, pp. 37–60, 1988. [20] P. Boyle and S. H. Lau, “Bumping up against the barrier with the binomial method,” J. Derivatives, vol. 2, pp. 6–14, 1994. [21] F. AitSahlia, L. Imhof, and T. L. Lai, “Fast and accurate valuation of American barrier options,” J

Computat. Finance, vol. 7, pp. 129–145, 2003. [22] T. L. Lai and T. W. Lim, “Exercise regions and efficient valuation of American lookback options,” Math. Finance, vol. 14, pp. 249–269, 2002. [23] J. M. Hutchinson, A. W. Lo, and T. Poggio, “A nonparameric approach to pricing and hedging derivative securities via learning networks,” J Finance, vol. 49, pp. 851–889, 1994. [24] M. Broadie, J. Detemple, E. Ghysels, and O. Torres, “Nonparametric estimation of American options’ exercise boundaries and call prices,” J.Econ. Dyna. Control, vol. 24, pp. 1829–1857, 2000. [25] M. Broadie and J. Detemple, “American option valuation: New bounds, approximations, and a comparison of existing methods,” Rev. Finan.Stud., vol. 9, pp. 1211–1250, 1996. [26] K. J. Hunt, D. Sbarbaro, R. Zbikowski, and P. J. Gawthrop, “Neural networks for control systems—A survey,” Automatica, vol. 28, pp. 1083–1112, 1992. [27] R. M. Sanner and J. J. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural

Networks, vol. 3, pp. 837–863, Nov. 1992. [28] T. J. Hastie and R. J. Tibshirani, Generalized Additive Models. London, U.K.: Chapman & Hall, 1990. [29] G. Wahba, Spline Models for Observational Data. Philadelphia, PA: SIAM, 1990. [30] A. R. Barron, “Universal approximation bounds for superpositions of a sigmoid function,” IEEE

Trans. Inform. Theory, vol. 39, pp. 930–945,May 1993. [31] J. H. Friedman and B.W. Silverman, “Flexible parsimonious smoothing and additive modeling,” Technometrics, vol. 31, pp. 3–21, 1989. [32] H. J. Kushner and P. G. Dupius, Numerical Methods for Stochastic Control Problems in Continuous

Time. New York: Springer-Verlag, 1992. [33] J. Barraquand and D. Martineau, “Numerical valuation of high dimensional multivariate American securities,” J. Finan. Quant. Anal., vol. 30, pp. 383–405, 1995. [34] S. B. Raymar and M. J. Zwecher, “Monte Carlo estimation of American call options on the maximum of several stocks,” J. Derivatives, vol. 5, pp. 7–23, 1997. [35] M. Broadie and P. Glasserman, “Pricing American-style securities using simulation,” J. Econ. Dyna.

Control, vol. 21, pp. 1323–1352, 1997.

68

[36] D. P. Bertsekas and J. N. Tsitsiklis, Neuro-Dynamic Programming. Belmont, MA: Athena Scientific, 1996. [37] F. A. Longstaff and E. S. Schwartz, “Valuing American options by simulation: A simple least-squares approach,” Rev. Finan. Stud., vol. 14, pp. 113–147, 2001. [38] M. Haugh and L. Kogan, “Pricing American options: A duality approach,” Oper. Res., 2004 [39] L. C. G. Rogers, “Monte Carlo valuation of American options,” Math.Finance, vol. 12, pp. 271–286, 2002. [40] M. H. A. Davis and I. Karatzas, “A deterministic approach to optimal stopping, with applications,” in Probability, Statistics and Optimization , F. P. Kelly, Ed. New York, NY: Wiley, 1994, pp. 455–466. [41] T. L. Lai and Q. Zhang, “Pricing Complex American Options by Simulation and Regression,” Dept. Statistics, Stanford University, Technical Rep., 2003. [42] P. Billingsley, Convergence of Probability Measures. New York: Wiley, 1968. [43] E. Clément, D. Lamberton, and P. Protter, “An analysis of a least squares regression method for American option pricing,” Finance Stoch., vol. 6, pp. 449–471, 2002. [44] C. J. Stone, “The use of polynomial splines and their tensor products in multivariate function estimation,” Ann. Statist., vol. 22, pp. 118–184, 1994. [45] G. Yin, Fellow and Xun Yu Zhou, “ Markowitz’s Mean-Variance Portfolio Selection With Regime Switching: From Discrete-Time Models to Their Continuous-Time Limits”, IEEE Transactions on Automatic control, vol. 49, No. 3, March 2004,p.349-360 [46] Harry M. Markowitz, “Foundations of Portfolio Theory” Nobel Lecture, December 7, 1990 [47] G. Barone-Adesi and R. Whaley, “Efficient analytic approximation of American option values,” J.

Finance, vol. 42, pp. 301–320, 1987. [48] J. Buffington and R. J. Elliott, “American options with regime switching,” Int. J. Theoret. Appl.

Finance, vol. 5, pp. 497–514, 2002. [49] S. Chen, X. Li, and X. Y. Zhou, “Stochastic linear quadratic regulators with indefinite control weight costs,” in SIAM J. Control Optim., vol. 36, 1998, pp. 1685–1702. [50] G. B. Di Masi, Y. M. Kabanov, and W. J. Runggaldier, “Mean variance hedging of options on stocks with Markov volatility,” Theory Probab.Appl., vol. 39, pp. 173–181, 1994. [51] D. Duffie and H. Richardson, “Mean-variance hedging in continuous time,” Ann. Appl. Probab., vol. 1, pp. 1–15, 1991. [52] E. J. Elton and M. J. Gruber, Finance as a Dynamic Process. Upper Saddle River, NJ: Prentice-Hall, 1975. [53] J. C. Francis, Investments: Analysis and Management. New York: Mc-Graw-Hill, 1976. [54] R. R. Grauer and N. H. Hakansson, “On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies,” Manage. Sci., vol. 39, pp. 856–871, 1993.

69

[55] N. H. Hakansson, “Multi-period mean-variance analysis: Toward a general theory of portfolio choice,” J. Finance, vol. 26, pp. 857–884, 1971. [56] D. Li and W. L. Ng, “Optimal dynamic portfolio selection: Multi-period mean-variance formulation,” Math. Finance, vol. 10, pp. 387–406, 2000. [57] H. Markowitz, “Portfolio selection,” J. Finance, vol. 7, pp. 77–91, 1952. [58] , Portfolio Selection: Efficient Diversification of Investment. New York: Wiley, 1959. [59] R. C. Merton, “An intertemporal capital asset pricing model,” Econometrica, vol. 41, pp. 867–888, 1973. [60] J. Mossin, “Optimal multiperiod portfolio policies,” J. Business, vol. 41, pp. 215–229, 1968. [61] S. R. Pliska, Introduction to Mathematical Finance: Discrete TimeModels. Oxford, U.K.: Blackwell, 1997. [62] P. A. Samuelson, “Lifetime portfolio selection by dynamic stochastic programming,” Rev. Econ.

Statist., vol. 51, pp. 239–246, 1969. [63] M. Schweizer, “Approximation pricing and the variance-optimal martingale measure,” Ann. Probab., vol. 24, pp. 206–236, 1996. [64] G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications:A Singular Perturbation

Approach. New York: Springer-Verlag, 1998. [65] G. Yin, R. H. Liu, and Q. Zhang, “Recursive algorithms for stock Liquidation: A stochastic optimization approach,” SIAM J. Optim., vol. 13, pp. 240–263, 2002. [66] G. Yin, Q. Zhang, and G. Badowski, “Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states,” Ann.Appl. Probab., vol. 10, pp. 549–572, 2000. [67] J. Yong and X. Y. Zhou, Stochastic Controls: Hamiltonian Systems andHJB Equations. New York: Springer-Verlag, 1999. [68] T. Zariphopoulou, “Investment-consumption models with transactions costs and Markov-chain parameters,” SIAM J. Control Optim., vol. 30, pp. 613–636, 1992. [69] Q. Zhang, “Stock trading: An optimal selling rule,” SIAM J. ControlOptim., vol. 40, pp. 64–87, 2001. [70] X. Y. Zhou and D. Li, “Continuous-Time mean-variance portfolio selection: A stochastic LQ framework,” Appl. Math. Optim., vol. 42, pp. 19–33, 2000. [71] X. Y. Zhou and G. Yin, “Markowitz’s mean-variance portfolio selection with regime switching: A continuous-time model,” SIAM J. ControlOptim., vol. 42, pp. 1466–1482, 2003. [72]. A.-P. N. Refenes, “Methods for optimal network design,” in A.-P. N. Refenes, Ed., Neural Networks

in the Capital Markets. Chichester, U.K.: Wiley, 1995, pp. 33–54.

[73]. H. M. Markowitz, “The utility of wealth”, JPE, 60, April 1952. [74] Tomasz R. Bielecki and Stanley R. Pliska, “Risk-Sensitive ICAPM With Application to Fixed-Income Management”, IEEE Transactions on Automatic Control, VOL. 49, NO. 3, March 2004, p. 420-432.

70

[75] R. C. Merton, “Optimum consumption and portfolio rules in a continuous time model,” J. Econ.

Theory, vol. 3, pp. 373–413, 1971. [76] , “An intertemporal capital asset pricing model,” Econometrica, vol. 41, pp. 866–887, 1973. [77] , Methods of Mathematical Finance. New York: Springer-Verlag, 1998. [78] R. Korn, Optimal Portfolios. Stochastic Models for Optimal Investment and Risk Management in

Continuous Time. Singapore: World Scientific, 1997. [79] D. Duffie and R. Kan, “A yield-factor model of interest rates,” Math.Finance, vol. 6, pp. 379–406, 1996. [80] D. Heath, R. Jarrow, and A. Morton, “Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation,” Econometrica, vol. 60, pp. 77–105, 1992. [81] O. A. Vasicek, “An equilibrium characterization of the term structure,”J. Financial Econ., vol. 5, pp. 177–188, 1977 [82] E. Canestrelli, “Applications of multidimensional stochastic processes to financial investments,” Rend.

Studi Econ. Quant., pp. 54–72, 1999. [83] E. Canestrelli and S. Pontini, “Inquiries on the applications of multidimensional stochastic processes to financial investments,” Econ. Complex., vol. 2, pp. 44–62, 1999. [84] T. S. Kim and E. Omberg, “Dynamic nonmyopic portfolio behavior,” Rev. Finan. Stud., vol. 9, pp. 141–161, 1996. [85] J. B. Detemple, R. Garcia, and M. Rindisbacher, “AMonte Carlo method for optimal portfolios,” J.

Finance, vol. 58, pp. 401–446, 2003. [86] M. Schroder and C. Skiadas, “Optimal consumption and portfolio selection with stochastic differential utility,” J. Econ. Theory, vol. 86, pp. 68–126, 2002. [87] M. J. Brennan and E. S. Schwartz, “The use of treasury bill futures in strategic asset allocation programs,” in World-Wide Asset and LiabilityModeling, J. M. Mulvey and W. T. Ziemba, Eds. Cambridge, U.K.: Cambridge Univ. Press, 1996. [88] M. J. Brennan, E. S. Schwartz, and R. Lagnado, “Strategic asset allocation,” J. Econ. Dyna. Control, vol. 21, pp. 1377–1403, 1997. [89] I. Bajeux-Besnainou, J. V. Jordan, and R. Portait, “Dynamic allocation of stocks, bonds, and cash for long horizons,” J. Business, vol. 76, pp. 263–288, 2003. [90] G. Deelstra, M. Grasselli, and P.-F. Koehl, “Optimal investment strategies in a CIR framework,” J.

Appl. Probab., vol. 37, pp. 936–946, 2000. [91] J. Liu, “Portfolio selection in stochastic environments,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1999. [92] C. Sørensen, “Dynamic asset allocation and fixed-income management,” J. Financial Quant. Anal., vol. 34, pp. 513–531, 2002. [93] M. Rutkowski, “Self-financing trading strategies for sliding, rolling-horizon, and consol bonds,” Math.

Finance, vol. 9, pp. 361–385, 1999.

71

[94] Olga Bobrovnytska and Martin Schweizer, “Mean-Variance Hedging and Stochastic Control:Beyond the Brownian Setting”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004, p.396-408. [95] M. C. Steinbach, “Markowitz revisited: Mean-variance models in financial portfolio analysis,” SIAM

Rev., vol. 43, pp. 31–85, 2001. [96] M. Kohlmann and X. Y. Zhou, “Relationship between backward stochastic differential equations and stochastic controls: A linear-quadratic approach,” SIAM J. Control Optim., vol. 38, pp. 1392–1407, 2000. [97] A. E. B. Lim and X. Y. Zhou, “LQ control and mean-variance portfolio selections: The stochastic parameter case,” in Mathematical Finance. Workshop of the Mathematical Finance Research

Project,M. Kohlmann and S. Tang, Eds. Boston, MA: Birkhäuser, 2001, pp. 261–270. [98] M. Kohlmann and S. Tang, “Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the meanvariance hedging,” Stoch. Processes Applicat., vol. 97, pp. 255–288, 2002. [99] Y. Hu and X. Y. Zhou, “Indefinite stochastic Riccati equations,” SIAM J. Control Optim., vol. 42, pp. 123–137, 2003. [100] , “Minimization of risk and linear quadratic optimal control theory,” SIAM J. Control Optim., vol. 42, pp. 1118–1142, 2003. [101] S. Tang, “General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations,” SIAM J. Control

Optim., vol. 42, pp. 53–75, 2003. [102] , “Multidimensional backward stochastic Riccati equations and applications,” SIAM J. Control

Optim., vol. 41, pp. 1696–1721, 2003. [103] H. Pham, “On quadratic hedging in continuous time,” Math. Meth. Oper.Res., vol. 51, pp. 315–339, 2000. [104] , “A guided tour through quadratic hedging approaches,” in Option Pricing, Interest Rates and Risk

Management, E. Jouini, J. Cvitanic´, and M. Musiela, Eds. Cambridge, U.K.: Cambridge Univ. Press, 2001, pp. 538–574. [105] , “Backward stochastic PDE and imperfect hedging,” Int. J. Theore. Applied Finance, vol. 6, pp. 663–692, 2003. [106] Yong Zeng, “Estimating Stochastic Volatility via Filtering for the Micromovement of Asset Prices”, IEEE Transactions on Automatic Control, vol.. 49, no. 3, March 2004, p.338-348. [107] T. Bollerslev, R. Chou, and K. Kroner, “ARCH modeling in finance: A review of the theory and empirical evidencs,” J. Econometrics, vol. 52, pp. 5–59, 1992. [108] G. Gottlieb and A. Kalay, “Implications of the discreteness of observed stock prices,” J. Finance, vol. 40, pp. 135–153, 1985. [109] C. Ball, “Estimation bias induced by discrete security prices,” J. Finance, vol. 43, pp. 841–865, 1988. [110] D. C. Cho and E. W. Frees, “Estimating the volatility of discrete stock prices,” J. Finance, vol. 43, pp. 451–466, 1988.

72

[111] L. Harris, “Estimation of stock price variances and serial covariances from discrete observations,” J.

Finan. Quantitat. Anal., vol. 25, pp.291–306, 1990. [112] Y. Zeng, “A partially observed model for micromovement of asset prices with Bayes estimation via filtering,” Math. Finance, vol. 13, pp. 411–444, 2003. [113] L. Harris, “Stock price clustering and discreteness,” Rev. Finan. Stud.,vol. 4, pp. 389–415, 1991. [114] P. Del Moral, J. Jacod, and P. Protter, “The Monte-Carlo method for filtering with discrete time observations,” Probab. Theory Related Fields, vol. 120, pp. 346–368, 2001. [115] Floyd B. Hanson, and John J. Westman, “Optimal Portfolio and Consumption Policies Subject to Rishel’s Important Jump Events Model:Computational Methods”, ”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004 [116] R. Rishel, “Modeling and portfolio optimization for stock prices dependent on external events,” in Proc. 38th IEEE Conf. Decision Control,1999, pp. 2788–2793 [117] F. Black and M. Scholes, “The pricing of options and corporate liabilities,” J. Polit. Econ., vol. 81, pp. 637–659, 1973. [118] H. J. Kushner, Stochastic Stability and Control. New York: Academic, 1967. [12] R. C. Merton, “Lifetime portfolio selection under uncertainty: The continuous-time case,” Rev. Econ. Statist., vol. 51, pp. 247–257, 1969. [119] Mertin , “Optimum consumption and portfolio rules in a continuous-time model,” J. Econ. Theory, vol. 3, pp. 373–413, 1971. [120] , Continuous-Time Finance. Cambridge, MA: Basil Blackwell, 1990. [121] , “Option pricing when underlying stock returns are discontinuous,”J. Finan. Econ., vol. 3, pp. 125– 144, 1976. [122] Weekly interest Rrates: Jan 1, 1994 to most recent, Federal Reserve System. (2000). [Online]. Available: http://www.frbchi.org/econinfo/finance/int-rates/rates/cw.prn [123] I. Karatzas, J. P. Lehoczky, S. P. Sethi, and S. E. Shreve, “Explicit solution of a general consumption/investment problem,” Math. Oper. Res., vol. 11, pp. 261–294, 1986. [124] S. P. Sethi and M. Taksar, “A note on Merton’s optimum consumption and portfolio rules in a continuous-time model,” J. Econ. Theory, vol. 46, pp. 395–401, 1988. 125] F. B. Hanson and D. Ryan, “Optimal harvesting with both population and price dynamics,” Math.

Biosci., vol. 148, pp. 129–146, 1998. [126] F. B. Hanson and H. C. Tuckwell, “Population growth with randomly distributed jumps,” J. Math.

Biol., vol. 36, pp. 169–187, 1997. [127] J. J. Westman and F. B. Hanson, “The LQGP problem: A manufacturing application,” in Proc. 1997

Amer. Control Conf., vol. 1, 1997, pp. 566–570. [128] , “State dependent jump models in optimal control,” in Proc. 38

th IEEE Conf. Decision Control,

1999, pp. 2378–2383.

73

[129] , “Nonlinear state dynamics: Computational methods and manufacturing example,” Int. J. Control, vol. 73, pp. 464–480, 2000. [130] K. Itô, “On stochastic differential equations,” Mem. Amer. Math. Soc., no. 4, pp. 1–51, 1972 [131] C. Moler et al., Using MATLAB, Version 5. Natick, MA: Mathworks, 1999. [132] Tomasz R. Bielecki and Marek Rutkowski, “Modeling of the Defaultable Term Structure:Conditionally Markov Approach”,

[133] A. Arvanitis, J. Gregory, and J.-P. Laurent, “Building models for credit spreads,” J. Derivatives, vol. 6, no. 3, pp. 27–43, 1999. [134] T. Bielecki and M. Rutkowski, “Multiple ratings model of defaultable term structure,” Math.

Finance, vol. 10, pp. 125–139, 2000a. [135] , “HJM with multiples,” Risk, vol. 13, no. 4, pp. 95–97, 2000b. [145] T. Bielecki and M. Rutkowski, Credit Risk: Modeling, Valuation, and Hedging. New York: Springer- Verlag, 2002. [136] D. Duffie, “Forward rate curves with default risk,” in Working Paper. Stanford, CA: Stanford Univ., 1994 . [137] , “Defaultable term structure models with fractional recovery of par,” in Working Paper. Stanford, CA: Stanford Univ., 1998. [138] D. Duffie and K. Singleton, “Ratings-based term structures of credit spreads,” Stanford Univ., Stanford, CA, 1998. [139] , “Modeling term structures of defaultable bonds,” Rev. Finan. Stud., vol. 12, pp. 687–720, 1999. [140] R. A. Jarrow, D. Lando, and S. M. Turnbull, “A Markov model for the term structure of credit risk spreads,” Rev. Finan. Stud., vol. 10, pp. 481–523, 1997. [141] R. A. Jarrow and S. M. Turnbull, “Pricing derivatives on financial securities subject to credit risk,” J.

Finance, vol. 50, pp. 53–85, 1995. [142] D. Lando, “On Cox processes and credit-risky securities,” Rev. Derivatives Res., vol. 2, pp. 99–120, 1998. [143] P. J. Schönbucher, “Term structure modeling of defaultable bonds,” Rev. Derivatives Res., vol. 2, pp. 161–192, 1998. [144] E. Eberlein and S. Raible, “Term structure models driven by general Lévy processes,” Math.

Finance, vol. 9, pp. 31–53, 1999. [145] Paolo Dai Pra, Wolfgang J. Runggaldier, and Marco Tolotti, “Pathwise Optimality for Benchmark Tracking’, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004 [146] S. Browne, “Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark,” Finance Stoch., vol. 3, pp. 275–294, 1999. [147] È. L. Presman, “Optimality almost surely and in probability for a stochastic linear-quadratic regulator,” Theory Probab. Applicat., vol. 42, pp. 531–535, 1997.

74

[148] È. L. Presman, V. I. Rotar, and M. Taksar, “Optimality in probability and almost surely. The general scheme and a linear quadratic regulator problem,” Stoch. Stoch. Rep., vol. 43, pp. 127–137, 1993. [149] V. I. Rotar, “Connectivity property and optimality almost surely and in probability,” in Proc. New

Trends in Probability and Statistics, Utrecht, The Netherlands, 1991, pp. 528–539. [150] P. D. Pra, G. B. Di Masi, and B. Trivellato, “Pathwise optimality in stochastic control,” SIAM J.

Control Optim., vol. 39, no. 5, pp. 1540–1557, 2000. [151] R. Merton, “Optimum consumption and portfolio rules in a continuous time model,” J. Econ. Theory, vol. 3, pp. 373–413, 1971. [152] Kang Boda, Jerzy A. Filar, Yuanlie Lin, and Lieneke Spanjers, “Stochastic Target Hitting Time and the Problem of Early Retirement”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004. [153] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming. New York, NY: Wiley, 1994. [154] R. A. Howard and J. E. Matheson, “Risk-sensitive Markov decision processes,” Manage. Sci., vol. 18, pp. 356–369, 1972. [155] C. Wu and Y. Lin, “Minimizing risk models in Markov decision processes with policies depending on target values,” J. Math. Anal. Appl.vol. 231, pp. 47–67, 1999. [156] D. J. White, “Mean, variance, and probabilistic criteria in finite Markov decision processes: A review,” J. Optim. Theory Appl., vol. 56, pp. 1–29, 1988. [157] J. A. Filar, L. C. M. Kallenberg, and H.-M. Lee, “Variance-penalized Markov decision processes,” Math. Oper. Res., vol. 14, pp. 147–161, 1989. [158] J. A. Filar, D. Krass, and K. W. Ross, “Percentile performance criteria for limiting average Markov decision processes,” IEEE Trans. Automat.Contr., vol. 40, pp. 2–10, Jan. 1995. [159] M. Bouakiz andY.Kebir, “Target-level criterion in Markov decision processes,” J. Optim. Theory

Appl., vol. 86, pp. 1–15, 1995. [160] R. Cavazos-Cadena and E. Fernández-Gaucherand, “Controlled Markov chains with risk-sensitive criteria: Average cost, optimality equations, and optimal solutions,” Math. Meth. Oper. Res., vol. 49, pp. 299–324, 1999. [161] M. J. Sobel, “The variance of discounted MDPs,” J. Appl. Probab., vol.19, pp. 794–802, 1982. [162] S. X. Yu, Y. Lin, and P. Yan, “Optimization models for the first arrival target distribution function in discrete time,” J. Math. Anal. Appl., vol. 225, pp. 193–223, 1998. [163] S. C. Jaquette, “A utility criterion for Markov decision processes,” Manage. Sci., vol. 23, pp. 43–49, 1976. [164] P. Whittle, Risk-Sensitive Optimal Control. New York: Wiley, 1990. [165] , “Minimizing a threshold probability in discounted Markov decision processes,” J. Math. Anal.

Applicat., vol. 173, pp. 634–646, 1993. [166] H. Markowitz, Portfolio Selection: Efficient Diversification of Investment. New York: Wiley, 1959.

75

[167] A. D. Roy, “Safety first and the holiday of assets,” Econometrica, vol. 220, pp. 431–449, 1996. [168] R. C. Merton, Continuous-Time Finance. Oxford, U.K.: Basic Blackwell, 1990. [169] S. Roy, “Theory of dynamic portfolio for survival under uncertainty,” Math. Social Sci., vol. 30, pp. 171–194, 1995. [170] R. E. Wildeman, R. Dekker, and A. C. J. M. Smit, “A dynamic policy for grouping maintenance activities,” Eur. J. Oper. Res., vol. 99, pp. 530–551, 1997. [171] Bensoussan, “Remarks on the Pricing of Contingent Claims Under Constraints, ”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004. [172] Nicole Bäuerle andUlrich Rieder, “Portfolio Optimization With Markov-Modulated Stock Prices and Interest Rates”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004. [173] Q. Zhang, “Stock trading: An optimal selling rule,” SIAM J. ControlOptim., vol. 40, pp. 64–87, 2001. [174] R. Frey andW . J. Runggaldier, “A nonlinear filtering approach to volatility estimation with a view towardhigh frequency data,” Int. J. Theoret. Appl. Finance, vol. 4, pp. 199–210, 2001. [175] R. C. Merton, “Optimum consumption andportfolio rules in a continuous-time model,” J. Econ.

Theo., pp. 373–413, 1971. [176] W. H. Fleming andD. Hern´ndez-Hern´ndez, “An optimal consumption model with stochastic volatility,” Finance Stoch., vol. 7, pp. 245–262. [177] T. Zariphopoulou, “Optimal investment andconsumption models with nonlinear stock dynamics,” Math. Meth. Oper. Res., vol. 50, pp. 271–296, 1999. [178] , “A solution approach to valuation with unhedgeable risks,” FinanceStoch., vol. 5, pp. 61–82, 2001. [179] J. P. Fouque, G. Papanicolaou, andK. R. Sircar, Derivatives in FinancialMarkest with Stochastic

Volatility. Cambridge, U.K.: Cambridge Univ.Press, 2000. [180] S. Heston, “A closed-form solution for options with stochastic volatility with applications to bondand currency options,” Rev. Financial Stud., vol. 6, pp. 327–343, 1993. [181] G. B. Di Masi, Y. M. Kabanov, andW. J. Runggaldier, “Mean-variance hedging of options on stocks with Markov volatilities,” Theory Probab.Appl., vol. 39, pp. 211–222, 1994. [182] H. J. Kushner andP. G. Dupuis, Numerical Methods for Stochastic ControlProblems in Continuous

Time. New York: Springer-Verlag, 1992. [183] A. Lioui andP . Poncet, “On optimal portfolio choice under stochastic interest rates,” J. Econ. Dyna.

Control, vol. 25, pp. 1841–1865, 2001. [184] R. Korn andH. Kraft, “A stochastic control approach to portfolio problems with stochastic interest rates,” SIAM J. Control Optim., vol. 40, pp. 1250–1269, 2001. [185] W. H. Fleming andR. W. Rishel, Deterministic and Stochastic OptimalControl. New York: Springer- Verlag, 1975. [186] W. H. Fleming andH. M. Soner, Controlled Markov Processes and ViscositySolutions. New York: Springer-Verlag, 1993.

76

[187] S. Browne, “Reaching goals by deadline: Digital options and continuous-time active portfolio management,” Adv. Appl. Probab., vol. 31, pp. 551–577, 1999. [188] H. Föllmer andP . Leukert, “Quantile hedging,” Finance Stoch., vol. 3, pp. 251–273, 1999. [189] Shu-Shang Zhu, Duan Li, and Shou-Yang Wang, “Risk Control Over Bankruptcy in Dynamic Portfolio Selection:A Generalized Mean-Variance Formulation”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004. [190] B. Dumas and E. Lucinao, “An exact solution to a dynamic portfolio choice problem under transaction costs,” J. Finance, vol. 46, pp. 577–595, 1991. [191] E. J. Elton and M. J. Gruber, “On the optimality of some multiperiod portfolio selection criteria,” J.

Business, vol. 47, pp. 231–243, 1974. [192] E. F. Fama, “Multiperiod consumption-investment decisions,” Amer.Econ. Rev., vol. 60, pp. 163– 174, 1970. [193] R. R. Grauer and N. H. Hakansson, “On the use of mean-variance and quadratic approximations in implementing dynamic investment strategies: A comparison of returns and investment policies,” Manage. Sci., vol. 39, pp. 856–871, 1993. [194] N. H. Hakansson, “On optimal myopic portfolio policies, with and without serial correlation of yields,” J. Business, vol. 44, pp. 324–334, 1971. [195] R. C. Merton, “Lifetime portfolio selection under uncertainty: The continuous-time case,” Rev.

Econ. Statist., vol. 51, pp. 247–257, 1969. [196] , Continuous-Time Finance. Cambridge, MA: Basil Blackwell, 1990. [197] J. Mossin, “Optimal multiperiod portfolio policies,” J. Business, vol. 41, pp. 215–229, 1968. [198] R. Östermark, “Vector forecasting and dynamic portfolio selection: Empirical efficiency of recursive multiperiod strategies,” Eur. J. Oper. Res., vol. 55, pp. 46–56, 1991. [199] S. R. Pliska, Introduction to Mathematical Finance. Cambridge, MA: Basil Blackwell, 1997. [200] P. A. Samuelson, “Lifetime portfolio selection by dynamic stochastic programming,” Rev. Econ.

Statist., vol. 50, pp. 239–246, 1969. [201] H. M. Markowitz, Portfolio Selection: Efficient Diversification of Investment. New York: Wiley, 1959. [202] D. Li and W. L. Ng, “Optimal dynamic portfolio selection: Multiperiod mean variance formulation,” Math. Finance, vol. 10, pp. 387–406, 2000. [203] X. Y. Zhou and D. Li, “Continuous time mean-variance portfolio selection: A stochastic LQ framework,” Appl. Math. Optim., vol. 42, pp. 19–33, 2000. [204] Lukasz Stettner, “Risk-Sensitive Portfolio Optimization With Completely and Partially Observed Factors”, IEEE Transactions on Automatic Control, Vol. 49, NOo. 3, March 2004. [205] Iakhnenko,A.G., and Osipenko,V.V. Algorithms of Transformation of Probability Characteristics into Deterministic Forecast. Sov. J. of Automation and Information Sciences, 1982, vol.15, no.2, pp.7-15.

77

[206] Farlow,S.J.,(ed.) Self-organising Methods in Modeling (Statistics: Textbooks and Monographs, vol.54), Marcel Dekker Inc., New York and Basel, 1984. [207] Spashko V.S. and Kostenko Ju.V. GMDH Algorithm for Two-Level Modeling of Multivariate Cyclic Processes, Sov. J. of Automation and Information Sciences, 1987, vol.20, No. 4. [208] Stepashko V.S. Asymptotic Properties of External Criteria for Model Selection, Sov. J. of Automation and Information Sciences, 1988, vol.21, no.6, pp.84-92.

[209] Sepashko V.S. GMDH Algorithms as Basis of Modeling Process Automation after Experimental Data, Sov. J. of Automation and Information Sciences, vol.21, no.4, 1988, pp.43- 53.

[210] Caudhuri A.S., "Observation , Conjecture and Experiment with Group Method of Data Handling Algorithms ", International Journal of System Analysis, Modeling and Simulation , vol. 6 , no. 11/12 , 1989 , p.867-906, Spinger Verlog, Berlin.

[211] J. Murata and S.Sagara, “Self-organisation of models for Power System load forecasting", International Conference on System Science, Warsaw, Poland, September 19 - 22, –1989.

[212] Belogurov,V.P. “A criterion of model suitability for forecasting quantitative processes”. Sov.

J. of Automation and Information Sciences, 1990, vol.23, no.3, p.21-25 [213] Ivakhnenko,A.G. “An Inductive Sorting Method for the Forecasting of Multidimensional Random Processes and Events with the Help of Analogues Forecast Complexing”. Pattern

Recognition and Image Analysis, 1991, vol.1, no.1, pp.99-108. [214] Zholnarsky,A.A. “Agglomerative Cluster Analysis Procedures for Multidimensional Objects: A Test for Convergence”. Pattern Recognition and Image Analysis, 1992, vol.25, no.4, pp.389-390. [215] Stepashko V.S. “ Structural Identification of Predictive Models under Conditions of a

Planned Experiment”, Sov. J. of Automation and Information Sciences, 1992, vol.25, no.1, pp.24-32.

[216] Ivakhnenko,A.G., Ivakhnenko,G.A., and Müller,J.A. “ Self-Organisation of Optimum

Physical Clustering of the Data Sample for Weakened Description and Forecasting of Fuzzy Objects”. Pattern Recognition and Image Analysis, 1993, vol.3, no.4, pp.415-421.

[217] Madala,H.R. and Ivakhnenko,A.G. “Inductive Learning Algorithms for Complex Systems Modeling”. CRC Press Inc., Boca Raton, 1994. [218] Ivakhnenko,A.G., Ivakhnenko,G.A. and Müller,J.A. “Self-Organisation of Neuronets with

Active Neurons”. Pattern Recognition and Image Analysis, 1994, vol.4, no.2, pp.177-188. [219] Ivakhnenko,G.A.“ Self-Organisation of Neuronet with Active Neurons for Effects of Nuclear

Test Explosions Forecastings”. System Analysis Modeling Simulation (SAMS), vol.20, 1995, pp.107-116.

[220] Ivakhnenko, A.G., Müller, J.-A.“ Present state and new problems of further GMDH Development”. SAMS, vol.20, no.1-2, 1995, pp.3-16. [221] Müller, J.-A., Lemke,F. “ Self-Organising modelling and decision support in economics”. Proc. of IMACS'95, Gordon and Breach Publ. 1995, pp.135-138.

[222] Lemke, F. “ Self Organize! - software tool for modelling and prediction of complex systems”.

78

SAMS, vol.20, no.1-2, 1995, pp.17-28. [223] Ivakhnenko, A.G., Müller, J.-A. “ Self-organisation of nets of active neurons”. SAMS, vol.20,

no.1-2, 1995, pp.93-106.

[224] Hild, Ch.R., Bozdogan H. “ The Use of Information-Based Model Selection Criteria in the GMDH Algorithms”. SAMS, vol.20, no.1-2, 1995, pp.29-50.

[225] Lange T. “ A New Principle of Creation of Selection Criteria for Automatic Model Selection

With Uncertainty of Data”. SAMS, vol.20, no.1-2, 1995, pp.79-92. [226] Wang Xiufeng, L.B. “ Self-Organizing Methods Modelling With "Concentrated" Variables”.

SAMS, vol.20, no.1-2, 1995, pp.51-58. [227] Müller, J.-A. “ Analysis and prediction of ecological systems”. SAMS, vol.21, 1996. [228] Ivakhnenko,A.G. and Ivakhnenko,G.A. “ Simplified Linear Programming Algorithm as

Basic Tool for Open-Loop Control”. System Analysis Modeling Simulation (SAMS), 1996, vol.22, pp.177-184.

[229] Iakhnenko A.G. “Heuristic self-organisation of problems of Engineering Cybernetics”,

Automatica, vol. 6, pp. 207-219, 1970.

[230] Ivakhnko A.G., “Polynomial theory of complex systems”, Transactions of Systems, Man and Cybernetics of Institution of Electrical and Electronic Engineering, IEEE, vol. SMC 1, NO. 4, October 1971

[231] Volterra V., “Theory of functional and integral and integro-differential equations”, London, Blackie, 1930.

[232] Wener N., “Non linear problems in random theory”, New York, Willey, 1958. [233] Gabor D., Wilby W.P.L. and WoodCock R., “A Universal Non Linear Filter Predictor And Simulator which Optimises itself by a Learning Process”, Proceedings of the Institution of Electrical Engineers, London, IEE Proc., Vol. 108, Part B, pp. 422 – 433, July 1961. [234] Rosenlblatt F., “Principles of Neurodynamics”, Spartan Books, Washington, 1962.

[235] Atrom K.J. and Eykhoff P., “System Identification , A survey”, Automatica , Vol.7, pp.123–162, March, 1971.

[236] Stein and M. Athans, “The LQG/LTR Procedure for Multivariable Feedback Control Design”,

IEEE Transactions on Automatic Control, Vol. AC –32, No. 2, February 1987, p. 105 –114 [237] R. Chaudhuri, "Identification of the Interactions of Industry-dependent State Variables on

Movement of National Market Index," Journal of the Institution of Engineers (India), Interdisciplinary Division, November 2000.

[238] John D. C. Little, “Optimal Adaptive Control: A Multivariate Model for Marketing Applications”, IEEE Transactions on Automatic Control, Vol. AC-22, No. 2, April 1977, p. 187-195. [239] J. D. C. Little, “A model of adaptive control of promotional spending,” Oper. Res., vol. 14, pp. 1075-1097, Nov. 1966. [240] E. Tse and M. Athans, “Adaptive stochastic control for a class of linear systems,” IEEE Trans.

Automat. Contr., vol. AC-17, pp. 38-52, Feb. 1972.

79

[241] J. G. Deshpande et al., “Adaptive control of linear stochastic separation theorem,” Int. J. Conrr.,

vol. 15, pp. 877-888, 1972. [242] T. N. Upadhyay and D. G. Lainiotis, “Joint adaptive plant and systems,” Automatica, vol. 9, pp. 107-1 15, Jan. 1973. [243] Wall, K.and Westcott,J. “Macroeconomic modeling for control”, Automatic Control, IEEE Transactions on , Volume: 19 Issue: 6 , Dec 1974 , Page(s): 862 -873 [244] Athans, M.; Kendrick, D., “Control theory and economics: A survey, forecast, and speculations”, Automatic Control, IEEE Transactions on , Volume: 19 Issue: 5 , Oct 1974, page(s): 518 -524 [245] Chow, G., “ Identification and estimation in econometric systems: A survey”, Automatic Control, IEEE Transactions on , Volume: 19 Issue: 6 , Dec 1974, Page(s): 855 –862. [246] H. A. Simon, “Dynamic programnling under uncertainty with a quadratic criterion function,” Ecunometriea, vol. 24, Jan. 1956. [247] H. Theil, “A note on certainty equivalence in dynamic plan- ning,” Econontetricu, vol. 25, Apr. 1957. [248] C. C. Holt, F. Modigliani, J. F. Muth, and H. A. Simon, “ Planning Production, Inventories and Work Force”, Engleaood, Prentice-Hall, 1960. [249] A. R. Dobell and Y. C. Ho, “Optimal investment policy: An example of a control problem in economic theory,’’ IEEE Trans. Automat. Contr., vol. AC-12, pp. 4-14, Feb. 1967.

[250] R. S. Pindyck, “An application of the linear quadratic tracking problem to economic stabilization policy,” IEEE Trans. Automat. Contr., vol. AC-17, pp. 287-300, June 1972.

[251] P. A. Samuelson, “Lifetime portfolio selection by dynamic stochastic programming,” Rev. Econonlics Stat., vol. LI, no. 3, 1969

[252] R. Chaudhuri (2002) , Input-Output Relationship of Industrial Production Fundamentals on National Market”, International Journal of AMSE, France, Modelling D-2002, Vol. 23 nº 1

[253] Beck M.B. (1979), “System identification, estimation and forecasting of water quality Part 1 : Theory” Working paper, WP-79-31, International Institute for Applied Systems Analysis, A-2361 Laxenberg, Austria.

[254] Beck M.B. (1977), “Real time control of water quality and quantity”, International Institute of Applied Systems Analysis, Luxembourg, Austria. [255] Beck M.B. (1977), “The identification and adaptive prediction of Urban Sewer Flows”, International Journal of Control, vol. 25, No. 3, pp. 425-440. [256] Ljung L and Soderstrom T(1983), “Theory and practice of recursive identification”, MIT Press, Boston. [257] Ljung L.(1977) , “Analysis of recursive stochastic algorithms”, IEEE transaction on Automatic Control, vol. AC-22, No. 4. [258] Ljung L.(1978),“Convergence analysis of parameter identification methods”,, IEEE Trans.,vol.AC- 23,No. 5.

80

[259] Ljung L.(1977), “A positive real transfer functions and the convergence of some recursive schemes”, IEEE transaction on Automatic Control, vol. AC-22.

[260] Ljung L. (1979), “Asymptotic Behaviour of Extended Kalman Filter as a Parameter Estimator for Linear Systems”, IEEE Trans. Vol. AC-24, No. 1, pp. 36-50. [261] Lux T. (1996) , “ Long term stochastic dependence in financial prices : evidence from the German stock market” , Appl. Econ. Lett. vol 3, p.701-706. [262] Lux T. (1997), “Time variation of second moments from a noise trader/infection model”, J. Econ. Dyn. Control , vol 22, p.1-38. [263] Lux T. (1998), “ The socio-economic dynamics of speculative market : interacting agents,chaos, and the fat tails of return distribution” , J. Econ. Behav. Organizat. , vol 33 , p. 143- 165. [264] Lux T. and Marchesi M.(1999), “ Scaling and criticality in a stochastic multi-agent model of a financial b market”, Nature, vol 397, p. 498-500.

[265] Philippe Jorion and William N. Goetzmann (!999), “Global Stock Markets in the Twentieth Century,” Journal of Finance, vol.53, June 1999. [266] Nai-Fu Chen, Richard Roll and Stephen A. Ross (1986), “Economic Forces and the Stock Market,” Journal of Business, vol 59, July 1986, pp. 383-404. [267] Eugene F. Fama and Kenneth R. French (1992), “The Cross-Section of Expected Stock Returns”, Journal of Finance, vol 47, June 1992, pp. 427-465.

[268] Narasimhan Jagadeesh and Sheridan Titman (1993), “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency”, Journal of Finance, vol 48, March 1993, pp. 65-91. [269] Campbell R. Harvey (1998), ”Time-varying Conditional Covariences in Test of Asset Pricing Models”, Journal of Financial Economics, vol. 24, no. 2, October 1989, p. 289- 318.

[270] Levy M., Levy H. and Solommon S.(1995); “Microscopic simulation of the stock market : the effect of microscopic diversity”, J. Phys. I (France), vol 5 , p. 1087-1107.

[271] Liu Y., Cizeau P., Meyer M., Peng C. K. and Stanley H. E. (1997), “Correlations in economic time series”, Physica A, vol 245, p. 437-440. [272]. Chaudhuri, R, “ Decision Making in Stock Market Management”, Journal of the Institution of Engineers(India), vol 85, May 2004, IE( I ) Journal-ID, pages 33-37. [273]. Zadeh, L.A,"Fuzzy sets", Information and Control, Vol. 8, 1965, pp. 338-353 [274]. Zadeh, L.A, "Fuzzy Algorithms", Information and Control, Vol. 12, 1968 ,pp. 338-353. [275]. Gougen, J.A," The logic of inexact concepts", Synthese, Vol. 19, 1969, pp 325-373. [276]. Zadeh L.A, "Outline of a New Approach to the Analysis of Complex Systems and Decision

Processes", IEEE Transaction on Systems, Man and Cybernetics, Vol. SMC-3, No.1. 1973, pp. 28-34. [277]. F. Wenstop, "Deductive verbal models of organizations," Int. Jrnl. of Man-Mach. Stud., Vol. 8, 1976, pp. 293-311. [278]. Zadeh, L.A,"PRUF - A meaning representation language for natural languages", International Journal of Man-Machine Studies, Vol. 10, 1978, pp. 395-460.

81

[279]. G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications , Prentice-Hall, New Jersey, 1995

[280]. George Bojadziev (Simon Fraser University) & Maria Bojadziev, Fuzzy Logic for Business, Finance, and Management, July 1997 ,ISBN 981-02-2894-5. [281]. Kaoru Hirota and Witold Pedrycz, “ Fuzzy Computing for Data Mining”, Proceedings of the IEEE , vol. 87. No. 9, September 1999.

[282]. W. Pedrycz and F. Gomide, “ An Introduction to Fuzzy Sets : Analysis and Design”, Cambridge, MA: MIT Press. 1998. [283] Monk-Ping Leong et al. CPE: A Parallel Library for Financial Engineering Applications, IEEE Computer October 2005, vol. 38 no. 10 , pp. 70-77