FGCS

ELSEVIER Future Generation Computer Systems 13 (1997/98) 269-278

tmlTURE

QENERATION OOMPUTER

OYSTEMS

Application of HPC to a portfolio choice problem

Marc Breitler a, StCphane Hegi b, Jean-Daniel Reymond a,*, Nils S. Tuchschmid b,l a Laboratoire de Mkanique des Fluides, DGM, Ecole Polytechnique Ftfdtfrale de Lmsanne, Ecublens CH-1015 Lausanne, Switzerland

b Institut de Gestion Bancaire et Financikre, HEC, Universite’de Lmsanne, CH-1015 Luusanne, Switzerland

Abstract

In this paper, we study a portfolio choice problem when the investment opportunity set is not constant over time. More precisely, the expected returns of financial assets are assumed to be time-varying and driven by four state variables. This allocation problem leads to a set of complex non-linear partial differential equations whose numerical solution requires powerful computing resources. The numerical method for solving this type of convection-diffusion equations in this framework is described. The algorithm is implemented on a vector computer and with another version using the single program multiple data (SPMD) approach on a dedicated massively parallel processing (MPP) system. The efficiency of the method is evaluated on both types of architecture. Numerical results for real market conditions are presented using a wide range of parameter values to explore the validity domain of the algorithm. Copyright 0 1998 Elsevier Science B.V.

Keywords: Dynamic portfolio allocation; Optimal portfolio choice; MPP; MPI; Convection-diffusion equation; Hyperbolic and

parabolic problems

1. Introduction

The use of engineering methods to solve financial

problems is not recent, but today the links between

these two fields are numerous and important. For ex-

ample, strong analogies between turbulence and finan-

cial markets have been found (see [5]), and the partial

differential equations found in many financial prob-

lems show similar characteristics to some mechani-

cal engineering systems (see [14]). Furthermore, the

amount of data produced by numerical simulation dur-

ing risk analysis for instance, may be comparable to

those of a computational fluid dynamics (CFD) prob-

lem. Therefore, powerful supercomputers (like paral-

*Corresponding author. Tel.: (41-21) 693 35 63; fax: (41-21)

693 36 46; e-mail: Jean-Daniel.Reymond@.epfl.ch.

‘Tel.: (41-21) (41-21) 692 34 67; fax: 692 34 35; e-mail:

Nils.Tuchschmid@hec.unil.ch.

lel scalable processors) are more and more used by

financial institutions and business firms.

The main objective of this research project is to

solve numerically an important theoretical problem

in finance: how an investor should optimally and dy-

namically allocate his/her wealth over a given time

during which he/she faces a set of stochastic invest-

ment opportunities. Indeed, this problem is known to

be described by complex non-linear systems of equa-

tions under constraints for which there are no tech-

niques for seeking the analytical solutions. Since CFD

problems are commonly solved with numerical tech-

niques implemented on high performance computing

(HPC) platforms, the interest in exploiting such pro-

cessing technology capacity is natural. In addition,

this research project allows to extend the accumulated

experience in solving non-linear convection_difSusion

equations and in using high speed vector or parallel

0167-739X/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights reserved

PII SO 167-739X(98)00029-0

210 M. Breitler et al./Future Generation Computer Systems 13 (1997/98) 269-278

computing. From a financial or economic viewpoint,

the theoretical framework has been known for a long

time (e.g. [9]). However, as previously mentioned,

there are no analytical tools for finding a solution once

the set of investment opportunities is not constant and

is assumed to evolve randomly over time. Therefore,

few papers have tried to analyse and to assess the

“true” performance of this optimal allocation prob-

lem compared to standard asset allocation criteria even

though one may cite numerous papers implementing

numerical methods to solve stochastic optimal control

problems (see [4] or [6]). In this respect, the recent re-

search conducted by Brennan et al. [l] is an exception.

It is also worthwhile mentioning that the impact of

this research is not only “practical” in the sense that it

may demonstrate the efficiency of parallel computers

and allow to obtain results interesting to numerous in-

vestors in need of an “every-day” asset allocation strat-

egy which could explicitly take into account the impact

of their investment time horizon. Indeed, it should also

provide many theoretical extensions. For instance, the

study of partial differential equations is an important

part of the proposed solution. Its numerical implemen-

tation involves choices for an efficient method, the de-

termination of suitable boundary conditions and the

treatment of the constraints. For that matter, it clearly

offers an opportunity to enlarge the application frame

of the numerical algorithm theory. From a financial

point of view, the numerical results can be compared

with other “ad hoc” allocation criteria and it will be

possible to extend the analysis to other financial allo-

cation problems as when introducing derivative instru-

ments in the investment process and analysing their

effect on the economic performance of long term in-

vestment strategy.

The structure of this paper is as follows: in Section 2

we briefly present the financial model which is

then developed in Section 3. In Section 4, after the

description of the numerical scheme, we provide the

results of various asset allocation strategies. In partic-

ular, we compare the long term and short term invest-

ment strategies and thus demonstrate the impact of

investors’ time horizon. We then present in Section 5

the timing performance of the serial and parallel

versions of the code. Section 6 concludes this study.

2. The financial model

It is assumed that the economy is driven by four

state variables, namely, the short term domestic inter-

est rate (r), the short term foreign interest rate (R), the

domestic dividend yield (8) and the foreign dividend

yield (A). It is furthermore assumed that these state

variables and the foreign exchange rate (e) follow a

joint stochastic process of the form:

dr = pr(‘, t)dt + a,(‘, t)dz,,

dR = P,Y(., t)dt + a~(., t)dzR,

ds=~CLg(.,t)dt+as(.,t)dzs,

dA = pA(‘, t) dt + u.d(‘, t) dzA,

de = &e(., t)dt + a,(., t)dze],

where dz(.) are the increments of Wiener processes.

In each country, there is a short term bond that yields

instantaneous sure returns, r(., t) and R(., t), respec-

tively. The agent can also invest in domestic (SD) and

foreign (SF) stocks, the stochastic processes of which

are driven by the four state variables. Based on this

set of assumptions, the stochastic process followed by

the investor’s wealth (W) - his/her budget constraint - is equal to

$.y[uD($y+wF(g

+KD(%)+M(T)], (1)

where COD and WF are the weights (or fraction of

investor’s wealth) invested in domestic and foreign

stocks, respectively, while KD and KF are the weights

of the domestic and foreign cash markets (SF and

BR are expressed in domestic currency). Since, by

definition, the sum of weights is 1, it finally yields

an equation with only three weight variables (21~ =

WF + KF):

dW - = bD(@ - r) + WFkY, + asFe - R) W

+ Z)F(R+k,--r)+rldt

+ WDffS,, dzs, + WFaSF dzsF + VFC~ dze.

(2)

h4. Breitler et al./Future Generation Computer Systems 13 (1997/98) 269-278 211

Notice that it is assumed that the expected returns

(CL(.)), the variances and covariances (a(.)) of the fi-

nancial assets are parameters known to the investor

at each period of time. In the sequel these param-

eters are estimated using weekly observations from

1976 to 1995. Notice also that even though the risk

estimation problem is known to be important, our

main focus in this study is the validation of the nu-

merical procedure. Therefore, we do not analyse in

great details the sensitivity of our results to parameter

changes.

3. The optimisation problem

The investor’s optimisation problem (see [lo]) is the

maximisation of the investor’s wealth expected utility

between today, or time to, and T, the maturity date.

t is defined as the time period to the horizon, that is,

T - to. In this framework, one supposes that investors

have power utility functions; namely they maximise

the expected utility E,,(u(W~)}, with

u(w,) = Ez, Y

where y is a parameter depending on the investor’s

relative risk aversion (v < 1).

To solve this problem, we apply a standard dynamic

programming method. More formally, let us define the

indirect utility function

J(W, 6, r, A, R, r) ._ .- lWnm~WIWtWVT), T)]l.

Hence, using Bellman’s principle yields

(4)

(5)

which, in turn, gives the system of equations provided

below, Eqs. (6) and (7). Note that this system does

not depend on the state variable W. Indeed, it can be

shown that the power utility function leads to an opti-

mal demand which is independent of investor’s wealth

(e.g. [lo]). As a matter of fact, the indirect value func-

tion, J(.), can be separated into two functions, that

is, u (function of the wealth) times a scalar H (func-

tion of the time and the four state variables). This

system of second order equations - Eq. (6) coupled

with the system (7) - has to be solved for the vector

field of the control variables (WD, WF, LJF)~ and the

scalar field H. The differential equation is a non-linear

convection-diffusion equation with a source term pro-

portional to H. This is a situation similar to the flow

simulation equations in fluid mechanics [7]. Scalar and

vector functions are strongly coupled by the system

of equations and the coupling of the scalar and vector

functions is therefore a crucial step of the numerical

treatment (linearisation).

Eq. (5) leads to

+ wFaS~?i + ales )

(

1 +Y&

7 + WDuSDr

+ wFas,r + woe, >

(

1 +YHA ypA + ~D~SDA

+ ~FC’SFA + WC'eA )

(

1 +YHR ;PR + @Da&R

+ ~F~SFR + WhR >

+ :(Hssai + H,,aT

+ HAda; + Hma;)

-k (Hsrasr •k H8daSd i- H8RasR

+ HARaAR+H,Ra,R+HrAa,a) (6)

212 M. Breitler et al/Future Generation Computer Systems 13 (1997/98) 269-278

and 4. Implementation and results

OsD SF

as,,

1 at

=(v ;: ’ 0

where

al = -_(psD - KS Hr

‘) - --SD6 - -wDr H H

HA HR - -USDA - -~SDR,

H H H8

a2 = -bsF + OsFe - RI - HasFs

Hr HA HR - -~SF~ - -~SFA - --a&R,

H H H HS Hr

a3 = -(R + Pe - r) - ~aes - zaer

HA HR - --o,A - --a,R.

H H

(7)

The matrix of the second order term coefficients is

positive definite, hence the equation is parabolic (in

t for the space variables r, R, 6 and A ) like the time

dependent formulation of the Navier-Stokes equation

in the field of fluid mechanics. This is an important

property for the numerical implementation because the

parabolic term makes it possible to stabilise the ap-

proximation of the hyperbolic part. However, the time

step must be small compared to the space discretisa-

tion in order to provide enough damping. In general,

the parabolic damping of a numerical method may be

less than that of the differential equation. The theoret-

ical parabolic decay rate is not preserved by the nu-

merical approximation and this is exacerbated when

the equation is non-linear. In addition, if the coeffi-

cients of the second order term are small (compared

to the coefficients of the first order terms) this prob-

lem becomes more severe. The algorithm described

in the sequel has converged with the different sets of

estimate parameters for the sample period considered.

For the space discretisation (discretisation of the

state variables) the use of a finite difference method

on a uniform mesh has been chosen. The scheme is

centred in space (for the first and second order deriva-

tives). In order to be used in a real investment strategy,

the discretisation must provide an accuracy of at least

half a per cent in the short term interest rate directions

and every quarter of a per cent in the dividend yield

directions. This implies a large number of degrees of

freedom.

The algorithm is tested with two types of estimated

parameters (either constant over the entire sample pe-

riod or estimated on a set of defined former periods

and used “out-of-sample”). With some sets of param-

eters, the numerical procedure has difficulties to con-

verge over the whole time period which appears to be

due to a boundary phenomenon. Hence, the influence

of the boundary conditions has been carefully exam-

ined. However, tests achieved with different types of

boundary conditions (first and second order boundary

conditions) lead to identical results and therefore the

problem of convergence is not of this type of origin,

There is another reason: in some cases the hyperbolic

character of the equation increases near the bound-

ary and the centred scheme is inappropriate. In the

second stage of the developments, an upwind scheme

has also been implemented in order to cope with this

situation.

Tests have also been performed with a range of

values for the parameter y because the stability do-

main of the algorithm has to be wide enough to in-

clude all the different types of investors. The con-

verged solutions were obtained with a y = + l/2 (low

risk averse agent) up to a y = -20 (high risk averse

investor).

These computations have demonstrated the feasi-

bility of the method. The numerical results give the

opportunity to discuss in great details the impact

of investors’ time horizons on the optimal weights

and therefore the economic performance of their

strategies. Moreover, the analysis shows that the

method can be extended to a system with more state

variables.

M. Breitler et al. /Future Generation Computer Systems 13 (1997/98) 269-278 213

E 8 3 I 2 $ E $ 8 rz 8 z 9 9 z z z c ‘” $ 0) z z z

a) b) Fig. 1. Proportion differential between: (a) long term and short term investors; (b) long term and mean-variance investors.

To illustrate the interest and the practicability of the

numerical approach, some results are provided below

for a set of unconstrained policies. They are obtained

for three different strategies using the set of expected

return, variance and correlation parameters estimated

over the entire sample period 1976-95. For the first

strategy, it is assumed that the investor has a 20-year

time horizon, that is, with a maturity at the end of

1995. For the second strategy, the investor is supposed

to have a 3-month time horizon and this short term

allocation strategy is then repeated, quarter after quar-

ter, during 20 years. Finally, the third strategy is the

one pursued over time by a standard mean-variance

investor who therefore does not have any hedging de-

mand, that is, whose the first and second derivatives

of H with respect to the state variables are forced to

be zero. Notice that the latter strategy would indeed

be the optimal one if, on one side, the prices of risky

assets were assumed to follow geometric Brownian

motions and if, on the other side, the domestic short

term interest rate was supposed to be constant over

time.

These three different allocation strategies allow

us to compare the effect of investors’ time horizons

within a portfolio problem. Indeed, the mean-variance

strategy represents the case of a “myopic” investor

who applies a standard tactical asset allocation tool

over short periods of time and who repeats the pro-

cess every quarter. For the 3-month and 20-year

strategies, investors also have the same reallocation

frequency. In other words, it is assumed that new

proportions are computed every quarter and that the

portfolio composition is changed accordingly. There-

fore, as time elapses and since the maturity is fixed,

it is clear that the “20-year” investment strategy

will draw closer to the 3-month one and that, three

months before maturity, both strategies will lead to

identical investment decisions. However, as opposed

to the mean-variance case, these last two strategies

are indeed dynamic and imply the search of optimal

allocation paths.

In Fig. (1) we present the differences between the

“optimal” fractions of wealth invested in the domestic

stock market by the 20-year and the 3-month investors

(Fig. l(a)) and by the 20-year and the mean-variance

investors (Fig. l(b)). Not surprisingly, the results of

Fig. l(a) show us that the long term investment strat-

egy leads to “overweight” the allocation in stocks com-

pared to the short term ones. Notice that, as expected,

these differences tend to zero as we are close to matu-

rity. This graph also shows that the decision-making of

the 3-month and the mean-variance investors are not

so different, since the standard tactical asset alloca-

tion procedure is by construction a short term strategy.

More interesting is the fact that the proportion differ-

entials are far from being negligible. For instance, the

difference between the proportions invested in the do-

mestic stock market by the 20-year and the 3-month

investors amounts for 5% on average with a maximum

close to 14%. If both the domestic and the foreign

stock markets are taken into account, the average dif-

ferential increases up to 22%. These first results raise

274 M. Breitler et al./Future Generation Computer Systems 13 (1997/98) 269-278

0

a o--Q long-teml investor 5 Q-El long-term investor Q-03 months investor r(Hc 3 month8 compounded

---- exponentiel function 6 4

Y m4

$3

2

2 1

Fig. 2. Standardised certainty equivalence.

two points. First, the effect of time in the investor’s al-

location strategy is important. Second, a corrolary to

the previous remark, any “ad hoc” or exogenous solu-

tion to incorporate the impact of time horizon will be

unsatisfactory.

In order to further analyse the impact of time, it is

useful to look at the evolution of the “standardised cer-

tainty equivalents (SCE)“, reported in Fig. 2. Indeed,

following Brennan et al. [l], the choice of the power

utility function allows to easily interpret the meaning

of the scalar H. The certainty equivalent is defined as

the amount of wealth for which investors will be in-

different between investing his/her current wealth op-

timally or receiving this amount for sure at maturity,

that is,

(CW Y

= yfz(r, R, 6, A, t>

+ $f = H(r, R, 6, A, T)“~ = SCE. (9)

Hence, H to the power of one over y possesses a use-

ful economic or financial interpretation. Indeed, the

higher the “value” of time, the greater the SCE should

be. Fig. 2(a) displays the evolution of this metric. Not

surprisingly, the results clearly show that a long term

investor values the opportunity to invest over time, that

is, the fact to benefit from future investment prospects

whereas, by construction, a short term investor is

unable to do so. In accordance with the financial

interpretation of this metric, the same graph also in-

dicates that the 20-year SCE decreases over time.

This evolution is somewhat accelerated around the

end of 198 1, a period corresponding to a substantial

reduction in the levels of interest rates and therefore,

for the investor’s viewpoint, to a reduction in his/her

future investment prospects.

In Fig. 2(b), “the 3-month compounded SCE” is

plotted. The latter is measured as the SCE of the 3-

month investor to the power of the number of re-

maining reallocation periods (four times the number

of years remaining to the maturity date because four

reallocations are achieved per year). It therefore rep-

resents the SCE of the short term investor as if the lat-

ter could have achieved the same certainty equivalent

quarter after quarter assuming a maturity at the end

of 1995. By doing so, the short term and long term

investors’ SCE can be compared since both metrics

are given on the same time horizon. Clearly, changes

in the investment opportunity set have a greater im-

pact on the short term investor’s expected utility. The

“compounded SCE” appears to be quite volatile while

the one of the long term investor, albeit decreasing, is

rather smooth. In other words, the proper inclusion of

time horizon allows the evolution of investors’ wealth

to be less erratic as theoretically predicted.

Finally, as far as the returns of these strategies are

concerned, Table 1 summarises the main results ob-

tained by investing according to the “optimal” weights

provided by the numerical solutions and the mean-

variance criterion. More precisely, the first positions

Table 1

M. Breitler et al./Future Generation Computer Systems 13 (1997/98) 269-278 275

Mean instantaneous returns quarterly computed and standard deviations

20-year

strategy 3-month strategy

Mean-var. strategy

Domestic stocks

Eq.-weighted strategy

Mean return 3.57% 3.28% Standard deviation 4.89% 4.46%

are taken at the beginning of 1976. They are closed

three months later and the returns are computed. Then,

the new quarterly proportions are used to reinvest and

the returns are measured three months later. The oper-

ation is repeated over time up to 1996. Finally, based

on these series of effective returns, the means and stan-

dard deviations are computed. Table 1 compares these

results with the ones obtained by two “standard’ in-

vestment strategies. In the first one, it is assumed that

agents invest all their wealth in the domestic stock

market. In the second one, the return is computed for

a simple equally weighted strategy, that is, wealth is

equally allocated across the four different categories

of assets and the weights are kept constant over time.

Notice that this last strategy offers a natural point of

comparison. First, it is easy to implement. Second, it

is aimed at limiting the exposure on a given market.

Finally, it can be seen as an “active” contrarian strat-

egy since it leads to sell assets which have performed

well and to buy the ones which have performed poorly.

As can be seen from Table 1, the 20-year, the 3-month

and the mean-variance strategies clearly dominate the

two other ones. The differences are of economic sig-

nificance. In parallel, we observe that the long term

investor’s portfolio return slightly outperforms its 3-

month counterpart which, in turn, outperforms the

standard mean-variance strategy. However, these last

results have to be taken with caution. First, long term

and short term strategies are “optimal” by definition

so that simple comparison in terms of raw returns is

not relevant. Second, even though mean returns are

obtained on the basis of a series of quarterly reallo-

cations computed over 20 years, they still represent a

single result from which no statistical inference can be

drawn. However, these first results are rather stimulat-

ing. First, the numerical procedure allows us to solve

the investors’ portfolio problem and, therefore, to im-

plement true dynamic allocation strategies. Second, all

3.20% 1.70% 1.42% 4.48% 9.16% 5.22%

the myopic strategies repeated over time are subop-

timal, that is, they are dominated by the ones which

correctly take into account the effect of investors’

time horizons. Indeed, the long term investment strat-

egy outperforms the 3-month strategy repeated over

20 years, while the latter outperforms the “myopic”

mean-variance strategy.

5. Numerical performance

To assess the “true” advantage of a long term in-

vestment strategy, repeated simulations are required.

However, computations are costly in time and mem-

ory. Indeed, the computation of a single investor’s op-

timal strategy can take on a vector CRAY computer

up to two and a half hours of CPU time. Due to the

non-linearity of the equations, the convergence may

indeed require very small time steps which can lead to

computation of eight billion unknowns. An example

is given in Fig. 3 illustrating the coupling of the vector

and the scalar fields that are solutions of the system

of equations. For the allocation of Fig. 3(b) 16000

time steps are necessary while the discretisation is of

20 nodes in each space direction. Since MPP com-

putations have considerable potential for the numeri-

cal simulation of complex three-dimensional problems

[2,12] and in order to maximise computation speed of

each simulation, the code is designed to mn efficiently

on a massively parallel computing engine. The latter

will then enable us to achieve a thorough analysis of

any dynamic allocation strategy since it requires re-

peated computations.

Parallelism is achieved by dividing the computa-

tional variable space into a number of sub-domains.

In each sub-domain the model equation is solved us-

ing the same code on the different processors. With

the SPMD programming style, the core of the parallel

216 M. Breitler et al./Future Generation Computer Systems 13 (1997DS) 269-278

Fig. 3. (a) Domain of three variables (the fourth variable is fixed) with the vector and the scalar field for a given time (snap-shot of a 20-year simulation); (b) Allocation for the period 1987-1989 in relation with the evolution of one variable (the foreign dividend yield).

:- . .

. .

. .

._.m.......-.-. s: . .:.

mm.

. . . . . . . ..w-SW. . . . . II . . . . . .._____. ._ . . . . . . . . \ 1

. .

. .

. .

. . . . . . . . . ..-......-

S2

v, v2

Fig. 4. Zone of data transfer.

implementation is fundamentally identical to the serial

version of the code.

Communication between processors is necessary to

exchange data at the interface of neighbouring sub-

domains. Data are exchanged using the message pass-

ing library MPI. Communication overheads have been

minimised by avoiding explicit syncbronisation bar-

rier. The synchronisation is performed by a couple of

processors. Processor Pt is initialised with the nodes

of the domain Sr (sketch in Fig. 4) and processor P2

with the nodes of the domain S2. Processor PI stores

in a vector u1 the values of the column that will also

be needed by P2 at the next time step. Then, P1 re-

ceives the vector v2 sent by 9 and proceeds to the

next time step.

Computations have been made on a Cray T3D sys-

tem but the same code has also been implemented on

a network of HP workstations and on a SGI Origin

2000 system. For the Cray T3D with 64 processors

computation rates are 20 times faster than on a vec-

tor Cray Y-MP M94. These results with up to 64 pro-

cessors demonstrate the scalability of the algorithm.

Fig. 5(a) shows the CPU time needed for a full allo-

cation solution with each architecture. Both parallel

machines have been employed with a number of pro-

cessors smaller than the total number of nodes. In fact,

it has been found that using all the nodes of an Origin

2000 reduces the performances considerably. How-

ever, for the given problem size the method is scalable

(Fig. 5(b)) because the developed algorithm requires

particularly limited communication traffic. This linear

speedup lets us expect that the method can be easily

extended to larger problems. If the number of vari-

ables and the dimensions are greater, it will also be

possible to use more computing nodes without drastic

degradation in performance.

6. Conclusion

An efficient method for solving a portfolio choice

problem has been implemented on different architec-

tures which shows that the problem is numerically

M. Breitler et al./Future Generation Computer Systems 13 (1997198) 269-278

66636 t M T3D

m-4 Origin 2000 1

2046

1024 -

512 -

256

*CrayYMP

126 II number of processors

4

32 -

16 (5 < P 6-

4

- T3D ,

4 6 16 32 64 number of pfocassMs

b)

211

Fig. 5. T3D and Origin 2000 performance timings compared with one vector computation on a Cray Y-MP M94.

consistent. The results are satisfactory and the method

provides not only a qualitative estimation but also

quantitative results with practical implications about

asset allocation. It allows us to explicitly take into ac-

count and to measure the effect of the time horizon on

the investor’s optimal decision making.

This study has contributed to enlarge the experience

for solving non-linear convection-diffusion equations

in multi-dimensional domains. The proposed proce-

dure successfully controls non-linearity and coupling

of the scalar and vector fields.

The parallel version of the code is easily portable on

any workstation cluster. Consequently, since the for-

mulation of the model has no limitation for the number

of state variables, a wide range of allocation strategies

can now be considered. The CPU power accessible

with the parallel version can deal with substantially

larger problems.

Acknowledgements

This project was supported by the interdisciplinary

funding program: “Fonds de Recherche UNIL-EPFL”.

Recent results have also been achieved in collaboration

with the Training Center for Investment Professionals

Inc. (TCIP-CFPI-AZEK).

References

[II

VI

[31

[41

[51

I61

[71

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Marc Breitler is a Research Engineer at the LMF. He graduated in Mathe- matics from the Ecole Polytechnique Fed&ale de Lausanne (EPFL) in 1993 (research domain: numerical simula- tion of a combustion problem). From 1993 to 1994 he worked in a medi- cal institute and was responsible for an image analyser system. Since then he specialised in numerical methods for financial applications.

StCphane Hegi is currently a Ph.D. student and a Research Assistant at the University of Lausanne. After a B.A. in Management Science, he obtained in 1995 his Master degree in Banking and Finance (MBF) from the Univer- sity of Lausanne. His Master’s thesis was entitled Valuation and Hedging of Interest Rate Term Structure Contin- gent Claims for fhe Swiss Market.

Jean-Daniel Reymond is a Research Scientist responsible for projects at the LMF since 1994. He graduated in Mathematics from the University of Geneva in 1989 (research domain: nu- merical analysis and implicit method for differential equations). From 1989 to 1993, he is a full time Research Scientist in direct collaboration with industries at the Institut de Machines Hydrauliques et de Mtcanique des Fluides of the Ecole Polytechnique

Fed&ale de Lausanne. His primary interest is in CAD engineer- ing, grid generation for complex three-dimensional geometries and applications for high-performance parallel system.

Nils Tuchschmid is an Associate Professor of Finance at the Ecole des HEC, Lausanne University. He received his Licence, Diploma and Ph.D. in Economics from the Uni- versity of Geneva. He was a Visiting Scholar at the Wharton School of the University of Pennsylvania and at the Finance Department of the University of Georgia. His main research interests are portfolio management and asset allocation, performance measurement,

domestic and international asset pricing models.