Z -Transform and preconditioning techniques for option pricing

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by: [Fusai, Gianluca]On: 15 April 2011Access details: Access Details: [subscription number 936467848]Publisher RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Quantitative FinancePublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713665537

Z-Transform and preconditioning techniques for option pricingGianluca Fusaia; Daniele Marazzinab; Marina Marenac; Michael Ngd

a Department of Economic Science and Quantitative Methods (SEMeQ), Università degli Studi delPiemonte Orientale, Novara, Italy b Department of Mathematics F. Brioschi, Politecnico di Milano,Milano, Italy c Department of Statistics and Applied Mathematics Diego de Castro, Università degliStudi di Torino, Torino, Italy d Department of Mathematics, Hong Kong Baptist University, KowloonTong, Hong Kong

First published on: 15 April 2011

To cite this Article Fusai, Gianluca , Marazzina, Daniele , Marena, Marina and Ng, Michael(2011) 'Z-Transform andpreconditioning techniques for option pricing', Quantitative Finance,, First published on: 15 April 2011 (iFirst)To link to this Article: DOI: 10.1080/14697688.2010.538074URL: http://dx.doi.org/10.1080/14697688.2010.538074

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713665537

http://dx.doi.org/10.1080/14697688.2010.538074

http://www.informaworld.com/terms-and-conditions-of-access.pdf

Quantitative Finance, 2011, 1–14, iFirst

Z-Transform and preconditioning techniques

for option pricing

GIANLUCA FUSAIy, DANIELE MARAZZINA*z, MARINA MARENAxand MICHAEL NG{

yDepartment of Economic Science and Quantitative Methods (SEMeQ), Universita degli Studi del PiemonteOrientale, Novara, Italy

zDepartment of Mathematics F. Brioschi, Politecnico di Milano, Milano, ItalyxDepartment of Statistics and Applied Mathematics Diego de Castro, Universita degli Studi di Torino, Torino,

Italy{Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

(Received 21 January 2009; in final form 3 November 2010)

In the present paper, we convert the usual n-step backward recursion that arises in optionpricing into a set of independent integral equations by using a z-transform approach. In orderto solve these equations, we consider different quadrature procedures that transform theintegral equation into a linear system that we solve by iterative algorithms and we studythe benefits of suitable preconditioning techniques. We show the relevance of our procedure inpricing options (such as plain vanilla, lookback, single and double barrier options) when theunderlying evolves according to an exponential Levy process.

Keywords: Numerical methods for option pricing; Exotic options; Preconditioners; Linearsystems; Toeplitz matrices

1. Introduction

In the present paper we propose a numerical procedurefor pricing different types of derivative contracts (plainvanilla, single and double barrier, lookback options)when the underlying asset evolves according to a genericLevy process. Our method considers the so-called discretemonitoring case that is much more relevant in practicethan the continuous version. No general analyticalsolutions are available (Kyprianou et al. 2005) andone has to resort to numerical methods, which includesemi-analytical approximations using Wiener–Hopf fac-torization, Monte Carlo, PIDE approaches, lattices andquadrature methods, and Fourier and Hilbert transforms.In recent years, several new approaches have beenproposed for pricing discrete barrier options under Levymodels (among others, see Airoldi 2005, Feng andLinetsky 2008, Jackson et al. 2008 and Lord et al. 2008).

Feng and Linetsky (2008, 2009) proposed an efficientmethod based on Hilbert transforms and a discretizationprocedure using the Whittaker cardinal series. The key

idea is to apply backward induction in the Fourier space,

rather than in the state space. Jackson et al. (2008)applied the Fourier transform to the pricing PIDE to

obtain a linear system of easily solvable ordinary differ-

ential equations. A related approach was introduced byLord et al. (2008). Their convolution method is based on

Fourier transforms combined with quadrature rules, thus

extending Eydeland’s algorithm in the Gaussian setting(Eydeland 1994) to the Levy framework. All methods

have a computational complexity of O(Nm log m), whereN is the number of monitoring dates and m is the number

of discretization grid points. Except for the Feng and

Linetsky approach, which has, remarkably, an errordecaying exponentially in m, all other algorithms have a

polynomial order of convergence.A different approach was introduced by Airoldi (2005).

His algorithm is based on the approximation of a genericprobability distribution function (pdf) as a perturbative

series expansion around another simpler pdf (typicallyGaussian) by matching all moments up to a given order.

The convolution between two distributions is thus

reduced to simple arithmetic and computational time, asin the other methods, is linear in the number of

monitoring dates.*Corresponding author. Email: [email protected]

Quantitative FinanceISSN 1469–7688 print/ISSN 1469–7696 online � 2011 Taylor & Francis

http://www.informaworld.comDOI: 10.1080/14697688.2010.538074

Downloaded By: [Fusai, Gianluca] At: 13:44 15 April 2011

In this paper we extend the approach of Fusai et al.(2006) to the Levy framework. Our method is based on az-transform approach that converts the usual backwardprocedure into a set of integral equations. In order tosolve these equations numerically, we consider differentquadrature rules and we exploit the structure of theobtained matrices by implementing suitable precondi-tioners and iterative solvers. Since the linear systemsarising from the integral equations are independent, theimplementation of our numerical procedure can alsoexploit the potentiality of distributed computing (Fusaiet al. 2010).

The idea for solving the pricing problem consists ofrandomizing the option expiry according to a geometricdistribution of parameter q, which is complex in oursetting. The randomized problem transforms the pricingproblem into an integral equation parametrized by q.Provided certain restrictions on the transition density,which we will assume to be of convolution type, anddepending on the nature of the integration domain, theintegral equation may or may not admit analyticalsolutions. For example, in the single barrier case, weobtain an integral equation of Wiener–Hopf type, whichhas been solved in analytical terms in the Gaussian caseby Fusai et al. (2006). Unfortunately, moving outside theGaussian world, simple analytical formulas are no longeravailable, hence the importance of a reliable numericalmethod. In order to de-randomize the problem andobtain the option price, we have to solve Nþ 1 integralequations parametrized by q according to the ruleq ¼ � expð j

ffiffiffiffiffiffiffi�1p

p=N Þ, j¼ 0, . . . ,N, where � is a freeparameter that can be chosen in (0, 1). Therefore, theoption price is obtained as a weighted sum of thesolutions of the integral equations.

When the integral equations are solved employing therectangle quadrature rule, we obtain linear systems withToeplitz matrices. Since the matrix–vector multiplicationcan be computed efficiently via fast Fourier transforms(FFTs), we solve the linear system by an iterative method.In order to speed up the convergence of the method,circulant matrices may be used as preconditioners forsolving the linear system. A circulant matrix is a specialform of Toeplitz matrix where each column is a circularshift of the preceding column. One of the beauties ofcirculant matrices is that they can always be diagonalizedby the discrete Fourier matrices. Many circulant precondi-tioners have been proposed and analysed (see, for instance,Ng (2004) and references therein). One of the mainimportant results of the circulant preconditioning meth-odology is that the complexity of solving a large class ofm�m Toeplitz systems can be reduced to O(m log m)operations, provided that a suitable preconditioner is used.Besides the reduction of the arithmetic complexity, thereare important types of Toeplitz matrices where fast directToeplitz solvers are notoriously unstable, for exampleindefinite and certain non-Hermitian Toeplitz matrices.Therefore, iterative methods are efficient alternatives forsolving these Toeplitz systems (Ng 2004), even if theconvergence speed is a delicate issue and a real challenge,especially in the presence of ill-conditioning and in the

multilevel case (see, e.g., Serra-Capizzano andTyrtyshnikov 2003, Noutsos et al. 2004, referencestherein).

The main drawback of using the rectangle rule is thatthe order of accuracy of the discretized solution dependsonly linearly on the number of quadrature points. Onecan also use higher-order quadrature rules such as thetrapezoidal rule or Simpson’s rule, which have second-and fourth-order accuracy, respectively. In these cases,the discretization matrices are non-Toeplitz matrices.Instead of constructing matrix preconditioners for thesediscretization matrices, we can consider preconditionersfrom the operator point of view. The performance ofthese proposed preconditioners is very good in terms ofaccuracy and computational work.

The computational complexity of our approach isO(Nm log m), where N is the number of monitoring datesand m is the number of sample points in the spacediscretization. Numerical experiments suggest that theerror decays polynomially. We also consider the Eulersummation to reduce the computational cost of ourmethod when the number of monitoring dates is large.By using this algorithm, the computational cost is madeindependent of N.

We stress that, unlikely other approaches, the proposedrandomization method can easily be extended to univar-iate non-Levy models (e.g., the CEV process).Furthermore, within our framework, we can price otherpath-dependent options like Asians. However, in thesecases, we might lose the Toeplitz structure and thereforewe cannot exploit the preconditioners presented in thispaper (see section 5 for details).

The structure of the paper is as follows. In section 2 weintroduce the option pricing problem, we briefly presentthe main studies in the related literature, and we describeour randomization technique and how it transforms thepricing problem into a set of integral equations. Insection 3 we consider different quadrature methods andwe introduce suitable preconditioning techniques. Insection 4 we validate the randomization procedure withnumerical results. Finally, in section 5 we discuss how wecan extend our procedure to other contracts and non-Levy processes.

2. The model

We consider an asset having log-price x. We are interestedin pricing a derivative contract with D equally spacedmonitoring dates, i.e. D is the distance between monitor-ing dates, having payoff �(x) at maturity T. Let v(x, n) bethe price of the option with n monitoring dates left andmaturity nD. In particular, v(x, 0)¼�(x). We are inter-ested in computing v(x,N ), where N is such that T¼ND(see figure 1). If we consider plain vanilla or barrieroptions, the pricing problem can be written in a recursiveway as

vðx, nÞ ¼ e�rDZ

�

Kðx, �;DÞvð�, n� 1Þd�, n ¼ 1, . . . ,N:

ð1Þ

2 G. Fusai et al.


Here, r is the risk-free rate, K(x, �;D) is the transition

density, assumed to be time homogeneous, from x at time

t to � at time tþD, and � is the domain of integration,

��R. A slight modification of (1) can be used to price

lookback options (see section 2.2.4).The formulation in (1) allows for discrete monitoring,

i.e. the updating conditions occur only at prefixed dates,

and requires only the knowledge of the transition density.

This is particularly important because a great discrepancycan be observed between continuously and discretely

monitored prices for exotic options such as barriers and

lookback. Moreover, the convergence of the discrete price

to the continuous price, as the number of monitoring

dates increases, is very slow, of order Oð1=ffiffiffiffiNpÞ in the

Gaussian case (see, for example, Broadie et al. 1999).The numerical implementation of (1) requires a high-

order recursive quadrature. This is discussed, for example,

by Sullivan (2000), Andricopoulos et al. (2003) and Fusai

and Recchioni (2007) with reference to the Gaussianworld, and by Fusai et al. (2009a). Other methods

presented in the literature are based on a perturbative

expansion around a given probability density function by

matching moments of increasing order (Airoldi 2005), fast

Gauss transform (Broadie and Yamamoto 2005), Fourier

transform (Jackson et al. 2008) or Hilbert transform

(Feng and Linetsky 2008, 2009).Concerning the choice of the kernel, we restrict our

attention to Levy processes. In particular, having Levy

process i.i.d. increments, the transition density will have

a convolution form, i.e. with abuse of notation,K(x, �;D)¼K(�� x; D). Levy processes display a number

of palatable features: they are the most direct generaliza-

tion of Brownian motion, they are analytically tractable,

and Levy processes are sufficiently general to include a

wealth of patterns and thus they account for smile and

skew effects in option prices. For a thorough introduction

to Levy processes with applications to finance, see

Schoutens (2003) and Cont and Tankov (2004). A generic

Levy process is fully determined by the characteristic

exponent D(!) of its log-increments ZD¼XtþD�Xt,

which is defined as the logarithm of the characteristic

function

Dð!Þ � lnðEfeffiffiffiffiffi�1p

!ZDgÞ:

Table 1 lists a few parametric Levy processes and their

associated characteristic exponent.The Gaussian model (G) is the benchmark assumption:

the ensuing process is purely diffusive Brownian motion,

which gives rise to the geometric Brownian motion process

for the price of the underlying. The Jump-Diffusion (JD)

model, introduced by Merton (1976), and the Double

Exponential (DE) model, introduced by Kou (2002), are

jump-diffusion processes that account for the presence of

fat tails in the empirical distribution of the underlying. The

CGMY model, introduced by Carr et al. (2002), is a pure

jump process with finite variation that can display both

finite and infinite activity. It is a subordinated Brownian

motion: in other words, it can be interpreted as Brownian

motion subject to a stochastic time change which is related

to the level of activity in the market.For pricing purposes, we need to consider the martin-

gale version of the Levy process. Due to the incomplete-

ness of the market, the choice, except in the case of the G

model, is not unique: there are many equivalent measures

under which the discounted price process is a martingale.

For simplicity, we opt for the so-called mean-adjusted

martingale measure that consists of choosing a constant

mD so that the risk-neutral characteristic function

(Fourier transform) of the transition density isZ þ1�1

Kðx, xþ z;DÞeffiffiffiffiffi�1p

z!dz ¼ e Dð!Þþffiffiffiffiffi�1p

!mDD ¼ e~ Dð!Þ,

v(x,0)=φ(x)

Δ

Δ Δ

. . .

Δ Δ Δ

Time 0 Δ 2Δ (N−1)Δ NΔ=T

v(x,1)=Ex,Δ[v(ξ,0)]

v(x,2)=Ex,2Δ[v(ξ,1)]

v(x,N)=Ex,NΔ[v(ξ,N−1)]

Figure 1. Pricing recursion for options with equally spaced monitoring dates.

Z-Transform and preconditioning techniques for option pricing 3


i.e.

mD ¼ ðr� gÞ � Dð�

ffiffiffiffiffiffiffi�1pÞ

D,

where g denotes the constant dividend payout rate.

2.1. A Z-transform approach

In this section, we present a new approach for solving (1)

based on the z-transform V(x, q) of v(x, n). The

z-transform is defined by the following power series:

Vðx, qÞ :¼ ð1� qÞXþ1n¼0

qnvðx, nÞ: ð2Þ

Since in the contracts considered in this paper the

coefficients v(x, n) in (2) are positive and bounded in n

for any x (for example, for plain-vanilla and barrier call

options we have v(x, n)5ex), the above power series is

convergent in the circle jqj51.y Thus the z-transform is

well defined.A probabilistic interpretation of the above expression is

as follows. Let us consider a coin tossing game, indepen-

dent of the underlying process, where if we obtain a head

for the first time after n tails, n� 0, we receive an option

having n monitoring dates. Here q is the probability of

tails. This maturity randomization technique allows us to

handle a functional of a Levy process taken at an

independent Geometric time rather than at a fixed time.

This approach is similar to the Laplace transform

technique with respect to time, introduced by Geman

and Yor (1993, 1996) for pricing Asian and barrier

options: with continuous monitoring, the option maturity

is assumed to be exponentially distributed and indepen-

dent of the underlying. However, this probabilistic

interpretation (that we call maturity randomization) is

lost when we have to recover v(x, n) given V(x, q) for

different values of q. In fact, the inversion procedure

requires q to be complex, as discussed below. Notice that

the z-transform technique, in general, requires constant

parameters and equally spaced monitoring dates, albeit if

we have constant values in sub-periods (for example,

weekly monitoring in the first year and daily in the second

year) our methodology can still be considered.If we multiply both sides of (1) by (1�q)qn and then we

sum running over n, n� 1, interchange the order of

integration and summation and finally add (1� q)�(x) toboth sides, we obtain the following integral equation:

Vðx, qÞ ¼ q e�rDZ

�

Kð� � x;DÞVð�, qÞd� þ ð1� qÞ�ðxÞ:

ð3Þ

The randomization has transformed the integral recursion(1) into the integral equation (3). Once we have solved theintegral equation, the original function v(x, n) can beobtained by de-randomizing the option maturity. Thisleads to the complex inversion integral choosing acircular integration contour with radius �51 (Abateand Whitt 1992)

vðx, nÞ ¼1

2p�n

Z 2p

0

eVðx, � e ffiffiffiffiffi�1p

uÞe�ffiffiffiffiffi�1p

nudu,

for n¼ 0, 1, . . . . A numerical approximation vH(x, n) ofthe above integral is given by Abate and Whitt (1992) byapplying a trapezoidal rule with step size H¼�/n,

vHðx, nÞ ¼

eVðx, �Þ þ ð�1ÞneVðx,��Þþ 2

Pn�1j¼1 ð�1Þ

j ReðeVðx, � e ffiffiffiffiffi�1p

�j=nÞÞ

( )2n�n

,

ð4Þ

with eVðx, qÞ :¼ Vðx, qÞ=ð1� qÞ and Re(�) denoting the realpart function. In particular, Abate and Whitt (1992)suggest setting �¼ 10��/2N when 10�� accuracy is desired.Their choice is convenient if jv(x, n)j51 in (2), which inthe present case can be satisfied by re-scaling, since v(x, n)are bounded positive coefficients. Numerical experiments,not reported here, show that the choice �¼ 10��/2N workswell if � 2 (6, 10). All the numerical results in this paperwere obtained by setting � ¼ 8 as suggested by Abate andWhitt (1992).

Since in (4) we have to evaluate the alternating series

vHðx,N Þ ¼1

�N

XNj¼0

ð�1Þ jaj ReðeVðx, � e ffiffiffiffiffi�1p

�j=NÞÞ,

with aj suitably defined, we can exploit the Euler summa-tion algorithm. For further details, see O’Cinneide (1997).The idea of this convergence-acceleration technique is toapproximate the above sum by

vHðx,N Þ �1

2me�N

Xme

j¼0

me

j

� �bneþjðx,N Þ, ð5Þ

Table 1. Characteristic exponents of some parametric Levy processes.

Model D(!) Parameter restrictions

G � 12 �

2!2D �40

CGMY CD�ð�YÞððM�ffiffiffiffiffiffiffi�1p

!ÞY �MY þ ðGþffiffiffiffiffiffiffi�1p

!ÞY � GYÞ C, G, M40, Y52

DE � 12 �

2!2Dþ �D ð1�pÞ22þ

ffiffiffiffiffi�1p

!þ

p11�


!� 1

� ��, �, p, 1, 240

JD � 12 �

2!2Dþ �Dðeffiffiffiffiffi�1p

!�ð1=2Þ!2�2 � 1Þ �, �, �40

yIf, for example, we consider plain vanilla or barrier options, lim infn!þ1 v(x, n)�1/n� limn!þ1 e�x/n¼ 1 holds, and thus by theCauchy–Hadamard theorem, the power series in (2) is convergent with radius greater than 1.

4 G. Fusai et al.


where

bkðx,N Þ ¼Xkj¼0

ð�1Þjaj ReðeVðx, � e ffiffiffiffiffi�1p

�j=NÞÞ,

with ne and me suitably chosen. Note that the Euleralgorithm is convenient when neþme5N, so that we needto solve neþmeþ 1 integral equations (3) instead of Nþ 1as in (4). Given that the choice of ne and me can be madeindependent of N, this accelerating technique makes ourmethod very competitive when pricing with a largenumber of monitoring dates is required. In fact, in ournumerical experiments, setting ne¼ 12 and me¼ 10appears to guarantee sufficient accuracy at a low compu-tational cost.

We now discuss how different path-dependent payoffsfit in the structure (3)–(4) and then in the next section wewill illustrate the numerical solution of our transformedproblem.

2.2. Contracts

The aim of this section is to introduce the different classesof options considered in this work: plain vanilla (orEuropean), single and double barrier, and lookbackoptions. Unfortunately, finite-lived American optionscannot be dealt with using our procedure, even thougha somewhat related procedure has been presented byCarr (1998).

2.2.1. Plain vanilla options. If we consider a plainvanilla option, the domain of integration � is infinite,i.e. �¼R in (1). The payoff function of a call option isgiven by �(x)¼ (ex� ek )þ, where k is the logarithm of thestrike. The integral equations to be solved are

Vðx, qÞ ¼ q e�rDZ 1�1

Kð� � x;DÞVð�, qÞd� þ ð1� qÞ�ðxÞ,

and we truncate the infinite domain to [a, b]. The choiceof a and b is made using the Philips and Nelson (1995)bounds.

For this class of options, an analytical solution in termsof the Fourier transform of the option price with respectto the log-strike k is promptly available once we have thecharacteristic function (Carr and Madan 1999). Indeed,we have

F½vkð!Þ ¼

Z þ1�1

effiffiffiffiffi�1p

!kekvðx,N Þdx

¼e�rT ~ NDð!�


þffiffiffiffiffiffiffi�1pÞ

2 þ � !2 þffiffiffiffiffiffiffi�1pð2þ 1Þ!

,

where is a positive constant (dumping factor) such thatthe (þ 1)th moment of the stock price at maturity exists(in our numerical experiments we set � 1.5). A Fourierinversion, performed by the FFT algorithm, yields theoption price

vðx,N Þ ¼e�k

p

Z þ10

e�ffiffiffiffiffi�1p

!kF½vkð!Þd!:

The price of plain vanilla options does not depend on the

number of monitoring dates N; however, this case isparticularly important because we can use it as a

benchmark for our numerical procedure.

2.2.2. Single barrier options. Barrier options are optionsthat die (down-and-out or up-and-out) or start to live(down-and-in or up-and-in) when the underlying asset

hits a preassigned barrier. In this case, � is a semi-infinite

interval. For synthesis, we consider only the case of adown-and-out call option with lower barrier L. The

payoff function is given by

�ðxÞ ¼ ðex � ekÞþ1fx4lg,

where l¼ log(L), 1{x4l}¼ 1 if x4l, 0 otherwise, and�¼ [l,1). The integral equations to be solved are

Vðx, qÞ ¼ q e�rDZ 1l


x 2 ½l,1Þ,

and we truncate the upper infinite domain to [l, b], where

b is chosen using the Chernoff inequality.

2.2.3. Double barrier options. Double barrier optionsare contracts that die (double knock-and-out options) orstart to live (double knock-and-in options) when the

underlying asset hits the boundary. Here we consider the

double knock-and-out call with a lower barrier L and anupper barrier U: thus � is the finite interval [l, u],

l¼ log(L), u¼ log(U). For this class of options the

payoff function is given by

�ðxÞ ¼ ðex � ekÞþ1fl5x5ug,

and the integral equations to be solved are

Vðx, qÞ ¼ q e�rDZ u

l


x 2 ½l, u:

With respect to previous cases, here the only error will be

due to the quadrature rule adopted and will not beaffected by the truncation of the domain.

2.2.4. Lookback options. The settlement of lookbackoptions is based on the minimum or the maximum value

of the underlying asset as registered during the lifetime ofthe option. At maturity, the holder can ‘look back’ and

select the most favorable figure of the underlying as

occurring at the monitoring dates. In the following, weconsider fixed strike lookback put options. Floating strike

options can be priced by symmetry (Eberlein and

Papapantoleon 2005). For this class of contracts theoption’s strike price ek is fixed at purchase and the option

can be exercised at the asset’s lowest price, i.e. if x(n)

denotes the asset log-price at the nth monitoring date and



we define

mðN Þ :¼ minn¼0,...,N

xðnÞ,

then the payoff function is given by

ðek � emðNÞÞþ:

Assuming x(0)4k, the lookback put with fixed strike hasprice

e�rTEP

0,xð0Þ½ðek � emðNÞÞþ ¼ e�rT

Z k

�1

euP0,xð0ÞðmðN Þ uÞdu,

ð6Þ

where P0,x(�) is the cumulative distribution function ofm(N ). In order to price this option, we need to computethe distribution law of m(N ), i.e.

P0,xð0ÞðmðN Þ uÞ ¼ 1� EP

0,xð0Þ½1fxð0Þ4u,xð1Þ4u,...,xðNÞ4ug:

ð7Þ

In order to compute (7), we set

EP

0,xð0Þ½1fxð0Þ4u,xð1Þ4u,...,xðNÞ4ug ¼ hðxð0Þ, 0Þ,

and h(�, �) is defined by the backward recursion

hðz, j� 1Þ ¼ 1fz4ugEP

ð j�1ÞD,z½hð�, j Þ

¼ 1fz4ug

Z þ1�1

Kðz, �;DÞhð�, j Þd�

¼ 1fz4ug

Z þ1�1

Kð� � z;DÞhð�, j Þd�,

j¼N,N� 1, . . . , 1, starting with h(z,N )¼ 1{z4u}.If we define w(z,N� j)¼ h(zþ u, j) and observe that

h(z, j)¼ 0 if z u, we are led to the following recursiverelation:

wðz, jþ 1Þ ¼

Z þ10

Kð� � z;DÞwð�, j Þd�, 05 z5þ1,

ð8Þ

with w(z, 0)¼ 1{z40}. Note that equation (8) is similar to(1) and by randomization we obtain the integral equation

Vðz, qÞ ¼ q

Z þ10

Kð� � z;DÞVð�, qÞd� þ ð1� qÞ1fz40g,

05 z5þ1: ð9Þ

The numerical solution will require the domain to betruncated to [0, b], where b is chosen using the Chernoffinequality.

3. Numerical approach

The aim of this section is to study the numericaldiscretization of (3) and thus of (4). This leads to a setof linear systems that can be solved efficiently consideringpreconditioning techniques. More precisely, in section 3.1we describe how to solve the integral equations usingdifferent quadrature formulas. Then in section 3.2

we study suitable preconditioners. Finally, in section 3.3we present the algorithm.

3.1. Quadrature rules for the numerical solution of theintegral equation

After truncation, if necessary, equation (3) becomes

Vðx, qÞ ¼ q e�rDZ b

a


which we can discretize by applying the compositerectangle (CR) rule with m nodes xi¼ aþ (i� 1)h,i¼ 1, . . . ,m, h¼ (b� a)/m. This leads us to the linearsystem

ðIm � qKmÞVm ¼ (m, ð10Þ

where Im is the identity matrix of size m, Km is the squarematrix with elements Kij¼ e�rDK(xj� xi;D)h, i,j¼ 1, . . . ,m, Vm is the unknown solution vector,Vi¼V(xi, q), i¼ 1, . . . ,m, and (m is the payoff vector,�i¼ (1� q)�(xi), i¼ 1, . . . ,m.

In a similar way we can discretize equation (9). Notethat, since K is a convolution operator and the abscissasare equally spaced, the matrix A¼ Im� qKm is a Toeplitzmatrix, i.e. a matrix in which each descending diagonalfrom left to right is constant. Therefore, we could solvethe linear system using a suitable preconditioner and aniterative solver as suggested by Ng (2004) (see section 3.2).

We can also consider more accurate Newton Cotesquadrature formulas, such as composite trapezoidal (CT)and composite Simpson’s (CS) rules. In this case, thelinear system to be solved becomes

ðIm � qKmDmÞVm ¼ (m, ð11Þ

where the m�m diagonal matrix Dm contains the weightsassociated with each quadrature rule, i.e.

diagðDmÞCT ¼1

2, 1, 1, 1, 1, . . . , 1, 1,

1

2

� �, ð12Þ

diagðDmÞCS ¼1

31, 4, 2, 4, 2, . . . , 2, 4, 1½ : ð13Þ

For further details, see Quarteroni et al. (2007). Note thateven though for these quadratures the abscissas areequally spaced, the matrix A¼ Im� qKmDm is no longer ofToeplitz type. Moreover, we recall that for the CS rule mmust be odd.

3.2. Preconditioners and iterative solvers

A direct solution of (10) and (11) is in general notsuitable. For example, using Gaussian elimination, alinear system can be solved in approximately 2

3m3 oper-

ations. However, we can largely improve on it byexploiting the Toeplitz structure of the matrix Km andusing a suitable preconditioner in both (10) and (11).Preconditioners are useful when using an iterative methodto solve large linear systems. Given the linear systemAx¼ b, a preconditioner P of a matrix A is a matrix such

6 G. Fusai et al.


that P�1A has a smaller condition number than A.A smaller condition number allows one to increase therate of convergence for iterative linear solvers (Quarteroniet al. 2007) and thus, instead of solving the original linearsystem above, one may solve the equivalent precondi-tioned system

P�1Ax ¼ P�1b:

Typically there is a trade-off in the choice of P betweenthe efficiency in the computation of P�1y, for any givenvector y, and the condition number of P

�1A: one,

therefore, chooses the matrix P in an attempt to achievea minimal number of iterations while keeping the com-putation of P�1y as simple as possible.

Solving (11) by a Krylov subspace iterative method(in our numerical experiments we consider the GMRES(Generalized Minimal Residual) method), in each itera-tion we only need Toeplitz matrix–vector multiplication,which means O(m logm) operations (see Ng (2004) fordetails), and diagonal matrix–vector multiplication, i.e. moperations. Thus the cost per iteration is of orderO(m logm) using fast Fourier transforms (FFTs). Themain aim of this section is to construct preconditionersfor (10) and (11) such that the preconditioned matriceshave clustered spectra, i.e. a small condition number, forfast convergence and that only O(m logm) operations arerequired in each iteration of the Krylov subspace iterativemethod even when a high-order quadrature rule isemployed.

Often the most successful preconditioners are thosebased on structural preconditioning, i.e. the precondi-tioner mimics the structure of the original matrix as muchas possible (in the Toeplitz setting, refer to Huckle et al.(2004) for band indefinite Hermitian preconditioners fordense indefinite Hermitian problems, and to Huckle et al.(2005) for band non-Hermitian preconditioners for densenon-Hermitian problems). In our case, we should con-sider Toeplitz times diagonal preconditioners already ininverted form so that the O(m log m) cost is maintained.

Since the matrix Im�Km exhibits clustering at one, asimple proposal could be to consider as preconditioner ininverted form Imþ qKmDm, so that we have to solve thelinear system

ðIm þ qKmDmÞðIm � qKmDmÞVm ¼ ðIm þ qKmDmÞ(m,

and thus the preconditioned coefficient matrix becomes

Im � q2ðKmDmÞ2:

A similar proposal is to considerP�1¼ Imþ qKmDmþ q2(KmDm)

2, so that the precondi-tioned coefficient matrix becomes Im� q3(KmDm)

3.Another possibility is to opt for the preconditioner

developed by Lin et al. (1997) to preconditionIm� qKmDm. The main feature of this preconditioner isthat it is already inverted. Hence, only Toeplitz matrix–vector products (plus some inner products) are required ineach step of the Krylov subspace iterative algorithm. Incontrast, if circulant preconditioners Cm (Ng 2004) areused with high-order quadrature rules, then one has to

invert a matrix of the form Im� qCmDm, which, ingeneral, has no fast inversion formula. However, onecan still remedy the drawback of circulant preconditionersby using the following approach and, thus, avoiding theinversion of Im� qCmDm.

The construction of the preconditioner proposed byLin et al. (1997) is as follows. For circulant precondi-tioners Cm, for Km we note that if Im� qCm is invertible,then its inverse can be expressed in the form

ðIm � qCmÞ�1¼ Im � Pm,

where the eigenvalues of the circulant matrix Pm aregiven by

�q�k1� q�k

, k ¼ 1, 2, . . . ,m,

and �k are the eigenvalues of Cm. Hence, the matrixpreconditioner can be constructed by using the inverse ofcirculant matrices. The preconditioned equation willbecome

ðIm � PmDmÞðIm � qKmDmÞVm ¼ ðIm � PmDmÞ(m, ð14Þ

and its solution using the GMRES iterative method canbe achieved with a small number of iterations (seetable 2). A numerical comparison of the above precondi-tioners in terms of the total number of iterationsnecessary to achieve a given accuracy is provided intable 3.

3.2.1. Construction of the Lin, Ng and Chan

preconditioner. The construction of the Lin, Ng andChan preconditioner (Lin et al. 1997) is detailed below. Inparticular, we remark that it requires only two fastFourier transforms (FFTs) and one inverse fast Fouriertransform (IFFT).

Given a vector r, in order to computeePm ¼ ðIm � PmDmÞr, we consider the following steps.

(1) Embed Km in a circulant matrix C2m (Ng 2004).(2) Compute the eigenvalues � of C2m using FFT.(3) Define Pm as the circulant matrix with eigenvalues�q�/(1� q�).

(4) Embed Dmr in a 2m vector v¼ [Dmr, 0, 0, . . . , 0] (forthe CR rule, Dm is the identity matrix).

(5) Compute w¼PmDmr using FFT and IFFT.(6) Define ePm ¼ r� wð1 : mÞ, where w(1 : m) refers to

the first m elements of w.

3.3. The algorithm

In this section we detail the numerical algorithm.

3.3.1. Plain vanilla and barrier options.

(1) Using the fractional Fourier transform inversionalgorithm, obtain the transition density K(�; D).

(2) Choose the quadrature method and constructthe matrices Km and Dm and the vector (m (seesection 3.1).



(3) For j¼ 0, . . . ,min(N, neþme), set �¼ 8, �¼ 10��/(2N ),qð j Þ ¼ � expð


jp=N Þ, and solve the linearsystems.

(4) Using the min(N, neþme)þ 1 solutions of theintegral equation parametrized by q(j), reconstruct

the option price v(x,N ) using (4) or its approxi-mation (5).

3.3.2. Lookback options.

(1) Truncate the integral in (6) to [, k] and choose aquadrature rule with nodes uj, j¼ 0, . . . , p, todiscretize it.

(2) Using the fractional Fourier transform inver-sion algorithm, obtain the transition density

K(�; D).(3) Choose the quadrature method and construct

the matrices Km and Dm and the vector (m (seesection 3.1).

(4) For j¼ 0, . . . , min(N, neþme), set �¼ 8,�¼ 10��/(2N ), qð j Þ ¼ � expð


jp=N Þ, and solvethe linear systems.

(5) Using the min(N, neþme)þ 1 solutions of theintegral equation parametrized by q( j), reconstructw(z,N ) using (4) or its approximation (5).

(6) Compute P0,x(0)(m(N ) uj)¼ 1� h(x(0), 0)¼ 1�w(x(0)� uj,N ), j¼ 0, . . . , p, and thus approximate(6) with the quadrature rule chosen in step 1.

4. Numerical experiments

In this section we validate our pricing procedure withnumerical experiments. In section 4.1 we describe theperformance of our preconditioning technique. Then insection 4.2 we present numerical results for plain vanillaoptions, for which an analytical solution is available.Finally, in section 4.3 we price barrier and lookbackoptions, comparing our results with Monte Carlosimulations.

In order to make the prices comparable across models,the parameters in the different Levy processes (see table 1)were chosen assuming that the CGMY model, asestimated by Schoutens (2003), is the true one:

C ¼ 0:0244, G ¼ 0:0765, M ¼ 7:5515, Y ¼ 1:2945:

Therefore, we calibrate the other models by minimizingthe square integrated difference between the real part ofthe characteristic functions of the CGMY and the othermodels (for further details, see Belomestny and Reiß(2006)). Thus the calibrated parameters for the JD modelare � ¼ 0:126349, ¼ �0:390078, � ¼ 0:174814 and

Table 2. GMRES iterations and execution time (ETime, in seconds): m¼ 3001.

CR CT CS

N Iterations ETime Iterations ETime Iterations ETime

No preconditioner25 354 1.02 354 1.01 354 0.9850 473 1.17 473 1.20 473 1.15100 656 1.81 656 1.76 656 1.76

Lin, Ng and Chan preconditioner (Lin et al. 1997)25 92 0.44 92 0.43 199 0.7050 92 0.42 92 0.41 256 0.79100 92 0.41 92 0.39 342 1.02

Table 3. GMRES iterations and different preconditioners: m¼ 3001.

No preconditioner Lin, Ng and Chan preconditioner (Lin et al. 1997)

N CR CT CS CR CT CS

25 354 354 354 92 92 19950 473 473 473 92 92 256100 657 657 657 92 92 342

Iþ qKmDm Iþ qKmDmþ q2(KmDm)2

CR CT CS CR CT CS

25 283 283 283 245 245 24550 354 354 354 304 304 304100 473 473 473 397 397 397

8 G. Fusai et al.


� ¼ 0:338796; the calibrated volatility parameter for the Gmodel is � ¼ 0:17801, while the calibrated parameters forthe DE model are � ¼ 0:120381, � ¼ 0:330966,p ¼ 0:20761, 1 ¼ 9:65997 and 2 ¼ 3:13868.

All the numerical experiments were performed inMatlab using an Intel personal computer equipped with6GB of RAM and an Intel core i7 Q720-1600MHzprocessor. The time to maturity is one year (T¼ 1) andthe risk-free rate is 3.67% per year, the dividend yield isset equal to zero, and the stock price S(0)¼ ex(0)¼ 100.

4.1. Preconditioning techniques

In table 2 we price barrier options under the Gaussiandistribution. We report the total number of GMRESiterations (with tolerance 10�11) necessary for the solutionof the min(neþme, N )þ 1 linear systems, considering thepreconditioner proposed by Lin et al. (1997) and the threequadrature formulas (CR, CT and CS). We recall that weset ne¼ 12 and me¼ 10 (see section 2.1).

From these numerical results it is clear that theproposed preconditioning procedure performs well.Note that this preconditioner works better for therectangle and trapezoidal formulas than for the Simpsonformula: this is due to the fact that, for the CR and CTrules, the matrix Dm is the identity matrix or is similar tothe identity matrix (see equation (12)), while this is nottrue when we use the CS rule (see equation (13)).Moreover, table 2 shows that when the preconditionerand the CR and CT rules are considered, the computa-tional cost is constant as we increase the number ofmonitoring dates, while this is not the case withoutpreconditioning. This depends on the good performanceof the preconditioner: in fact, increasing the number ofmonitoring dates, the condition number of the matrix Km

increases, and thus the number of GMRES iterations.However, the good scalability of our preconditionerprevents this behavior.

Finally, numerical experiments (not reported here)show that the number of GMRES iterations is indepen-dent of the matrix dimension m. Moreover, very similarresults are achieved considering different distributionssuch as double exponential, jump diffusion and CGMY.

Note that, since Km is a Toeplitz matrix, it is necessaryto store only its first row and first column, which means2m� 1 elements instead of m2, and each GMRESiteration for the solution of (10) and (11) costs onlyO(m log(m)) operations. In addition, we observe thatlinear systems can be solved independently (indeed theydiffer by the choice of q) and, therefore, our pricingmethod is suitable for parallelization. Thus, we cancombine the preconditioning technique with distributedcomputing, splitting the solution of the independent linearsystems on different nodes or cores. For example, ifm¼ 10,001, N¼ 252 and the composite Simpson’s quad-rature is considered, the execution time, without pre-conditioner, is 12.41 seconds, whilst using thepreconditioner (Lin et al. 1997) it decreases to 6.89

seconds. In addition if we use the multicore technology ofthe Intel i7 CPU (four cores (eight threads)), then thecomputational time decreases to 3.06 seconds (see Fusaiet al. 2010 for further details).

Finally, in table 3 we compare the behavior of thedifferent preconditioners presented in section 3.2. Thecomparison is performed in terms of the number ofiterations necessary to solve the linear systems related tothe discretization of the integral equations of a Europeanoption. The Lin, Ng and Chan preconditioner (Lin et al.1997) performs best. In the following, all the numericalresults are obtained using this preconditioner.

4.2. Plain vanilla options

In this section we consider the pricing problem for plainvanilla options. Since for this kind of option an analyticalsolution v(x,N ) is available, we can compute thepointwise error of our pricing procedure:

jvðx,N Þ � vHh ðx,N Þj, ð15Þ

where x is the log-price of the underlying asset in t¼ 0and

vHh ðx, nÞ ¼

eVhðx, �Þ þ ð�1ÞneVhðx,��Þ

þ 2Pn�1

j¼1 ð�1Þj ReðeVhðx, � e


�j=nÞÞ

( )2n�n

,

with eVh being the numerical approximation to eV.In table 4 we consider the Gaussian and double

exponential distributions and plain vanilla options withstrike K¼ 100, for which we report the root mean squareerror, i.e.

RMSE ¼Xpi¼1

ðvðxi,N Þ � vHh ðxi,N ÞÞ2

p

" #1=2

, ð16Þ

with 0:8 ek ex1 5 � � � 5 exp 1:2 ek. From these numer-ical experiments and others not reported here, it seemsthat m¼ 3001 and m¼ 6001 are sufficient to obtain anexact option price, at least to the first four and fivedecimal digits, respectively, for all the Levy processesconsidered.

Finally, in table 5 we analyse the performance of theEuler summation in terms of accuracy and computationalcost. From this table and table 2 it is clear that the use ofthe Euler algorithm and the Lin, Ng and Chan precondi-tioner (Lin et al. 1997) makes the computational costindependent of the number of monitoring dates when theCR or CT discretization rules are considered, withoutlosing accuracy.

4.3. Path-dependent options

In this section we compare option prices estimatedaccording to our procedure with a 99.75% confidenceinterval (CI) obtained by running 1,000,000 Monte Carlo



simulations. Linear systems were always solved using the

Lin, Ng and Chan preconditioner. In table 6 we consider

single barrier options (strike K¼ 100 and lower barrier

L¼ 90) in the jump diffusion model. The experiments

were performed using the CR and CS quadrature formu-

las. In table 7 we consider double barrier options

(K¼ 100, lower barrier L¼ 90 and upper barrier

U¼ 110) in the DE model. Here we consider only the

composite Simpson’s formula. Finally, in table 8, always

with reference to the DE distribution, we show the

numerical experiments for lookbacky options with

strike K¼ 90.In all these experiments the option prices fall into the

confidence intervals. Moreover, if we consider table 8, we

see that the prices computed with the trapezoidal and

Simpson’s rule are very similar, but a little different from

those computed with the rectangle rule. Since we have to

solve the linear system

ðIm � qKmDmÞVm ¼ (m,

Table 5. DE model: option price computed with/without the Euler summation andconsidering the CT rule. Exact price: 9.552831.

m¼ 2000 m¼ 4000

N Price ETime Price ETime

Euler summation25 9.552844 0.39 9.552821 0.5050 9.552855 0.35 9.552831 0.53100 9.552856 0.35 9.552833 0.54252 9.552858 0.41 9.552834 0.51504 9.552864 0.38 9.552834 0.50756 9.552877 0.36 9.552834 0.531008 9.552897 0.35 9.552835 0.52

No Euler summation25 9.552844 0.37 9.552821 0.5150 9.552855 0.57 9.552831 0.97100 9.552856 1.01 9.552832 1.68252 9.552857 2.37 9.552834 4.00504 9.552863 4.49 9.552833 7.72756 9.552877 6.68 9.552833 11.521008 9.552897 8.86 9.552835 15.27

Table 4. G and DE models: RMSE*1000, computed as in (16).

Gaussian Double exponential

Composite rectangle

N m¼ 1000 m¼ 2000 m¼ 3000 m¼ 1000 m¼ 2000 m¼ 3000

5 0.008867 0.026738 0.008223 0.116497 0.030333 0.00895325 0.091149 0.0024497 0.003184 0.121617 0.012263 0.00757750 0.010117 0.023533 0.011292 0.180895 0.034149 0.012840100 0.096887 0.001785 0.006793 0.114714 0.048279 0.011603

Composite trapezoidal

m¼ 1000 m¼ 2000 m¼ 3000 m¼ 1000 m¼ 2000 m¼ 3000

5 0.043666 0.038549 0.003524 0.175964 0.024876 0.00982025 0.086306 0.011604 0.008423 0.114144 0.015674 0.00778750 0.057375 0.022031 0.011216 0.180842 0.024660 0.009596100 0.093850 0.014785 0.000806 0.215089 0.026430 0.021988

Composite Simpson’s

m¼ 1001 m¼ 2001 m¼ 3001 m¼ 1001 m¼ 2001 m¼ 3001

5 0.130939 0.032701 0.000062 0.141883 0.078788 0.00536925 0.138341 0.026899 0.008306 0.071052 0.028506 0.00509250 0.118575 0.025397 0.007718 0.038403 0.017244 0.008782100 0.072517 0.020823 0.006937 0.126885 0.070940 0.009398

yThe integral in (6) was computed with a Gauss–Legendre quadrature with mþ 1000 nodes.

10 G. Fusai et al.


and Km is a Toeplitz matrix, we have to store only the first

row and column of �qKm and the diagonal matrix Dm,

which means 2m� 1 and m elements, respectively, i.e.

3m� 1 elements. Note that, if we consider the CR rule,

Dm is the identity matrix (see equation (10)), thus it is

necessary to store only 2m� 1 elements. This means that

we can easily increase the linear system dimension, i.e. m,

at a low computational cost. In table 9 we report the

option prices considering the CGMY and DE distribu-

tions using the CR with m45000. For the sake of

comparison, we benchmark these prices using the CT and

CS rules with m¼ 25,001 and m¼ 50,001.Unlike the case of plain vanilla options, where we

obtain similar results considering the different quadrature

rules, we note a faster convergence using the trapezoidal

and Simpson’s formulas than with the rectangle formula.

This is illustrated in tables 6, 8 and 9. In fact, the main

drawback of using the rectangular rule is that the order of

accuracy of the discretized solution depends only linearly

on the number of quadrature points. Thus in order to

Table 6. Single barrier options and JD model: prices.

Composite rectangle

N m¼ 1000 m¼ 2000 m¼ 3000 m¼ 4000 CI

5 9.448103 9.446095 9.445443 9.445110 9.4118–9.478312 9.394821 9.391867 9.390867 9.390366 9.3464–9.413125 9.351163 9.347386 9.346119 9.345484 9.3114–9.378150 9.316035 9.311512 9.309999 9.309242 9.2628–9.3294

100 9.287407 9.282376 9.280699 9.279860 9.2564–9.3232


m¼ 1001 m¼ 2001 m¼ 3001 m¼ 4001

5 9.444151 9.444095 9.444107 9.444113 9.4118–9.478312 9.388772 9.388863 9.388853 9.388857 9.3464–9.413125 9.343608 9.343587 9.343577 9.343580 9.3114–9.378150 9.306933 9.306987 9.306981 9.306977 9.2628–9.3294

100 9.277381 9.277363 9.277357 9.277354 9.2564–9.3232

Table 7. Double barrier options, DE model and CS formula: prices.

N m¼ 1001 m¼ 2001 m¼ 3001 m¼ 4001 CI

5 0.758691 0.759129 0.759275 0.759348 0.7572–0.768812 0.590891 0.591166 0.591257 0.591303 0.5900–0.600325 0.493099 0.493283 0.493345 0.493374 0.4890–0.498250 0.428672 0.428798 0.428840 0.428860 0.4243–0.4329

100 0.383721 0.383808 0.383836 0.383850 0.3827–0.3908

Table 8. Lookback options and DE model: prices.

Composite rectangle

N m¼ 1000 m¼ 2000 m¼ 3000 m¼ 4000 CI

25 4.353317 4.359733 4.361825 4.362913 4.3097–4.377250 4.434074 4.441753 4.444258 4.445550 4.4047–4.4727

100 4.493637 4.502159 4.504982 4.506391 4.4581–4.5263

Composite trapezoidal

m¼ 1000 m¼ 2000 m¼ 3000 m¼ 4000

25 4.369158 4.367491 4.367033 4.366813 4.3097–4.377250 4.451896 4.450504 4.450055 4.449884 4.4047–4.4727

100 4.512827 4.511483 4.511137 4.510984 4.4581–4.5263


m¼ 1001 m¼ 2001 m¼ 3001 m¼ 4001

25 4.367905 4.366962 4.366703 4.366573 4.3097–4.377250 4.450655 4.450039 4.449780 4.449691 4.4047–4.4727

100 4.511466 4.511034 4.510890 4.510818 4.4581–4.5263



obtain reasonable accuracy, a small step-size has to be

used and, hence, the dimension of the resulting matrix

system will be very large. In order to improve the

accuracy, one can use higher-order quadrature rules such

as the trapezoidal rule or Simpson’s rule, which have

second- and fourth-order accuracy, respectively.

However, the execution time for Simpson’s rule is

longer than the others due to the number of GMRES

iterations necessary for the solution of the preconditioned

linear systems (see table 2).Finally, we compare our method with the Hilbert

transform approach of Feng and Linetsky (2008) by

pricing a double barrier option with 252 monitoring dates

considering the parameters introduced by Feng and

Linetsky (2008): S(0)¼K¼ 100, barriers L¼ 80 and

U¼ 120, time to maturity T¼ 1, risk-free interest rate

r¼ 5% and constant dividend rate g¼ 2%. Numerical

experiments show that our method agrees with the prices

reported by Feng and Linetsky (2008), which are given

with 10�8 accuracy. More precisely, under the JD model,y

we obtain

. 10�6 accuracy considering Simpson’s rule with

m¼ 601 or the trapezoidal rule with m¼ 4001;

and. 10�8 accuracy considering Simpson’s rule with

m¼ 2001 or the trapezoidal rule with

m¼ 50,001.

Figure 2 shows the pointwise error on a logarithmic

scale as a function of m as well as the convergence order.

We verify the polynomial convergence of our method

numerically: linear for the rectangle rule, and second and

fourth order for the trapezoidal and Simpson’s rule,

respectively. Similar plots are obtained for all the other

distributions.

Remark 1: In order to analyse the pointwise error of

our numerical procedure, considering (4) and (15), the

following holds:

jvðx,N Þ � vHh ðx,N Þj jvðx,N Þ � vHðx,N Þj

þ jvHðx,N Þ � vHh ðx,N Þj:

Following Abate and Whitt (1992), i.e. setting �¼ 10�4/N,we expect the error term jv(x,N )� vH(x,N )j to benegligible, while the error term jvHðx,N Þ � vHh ðx,N Þjdepends only on the domain truncation and the quadra-ture formula considered. Moreover, in the case of doublebarrier options, there is no domain truncation and theerror depends only on the chosen quadrature formula.

5. Extensions

In this section we briefly detail how the z-transformapproach can deal with other path-dependent options ornon-Levy processes, such as the CEV process. For

Table 9. Lookback options and CR formula: prices.

DE CGMY ETime

m N¼ 50 N¼ 100 N¼ 50 N¼ 100 N¼ 50 N¼ 100

5000 4.44632 4.50723 4.38417 4.42327 0.66 0.6910,000 4.44786 4.50892 4.38588 4.42511 1.11 1.1215,000 4.44836 4.50947 4.38644 4.42571 1.66 1.6320,000 4.44862 4.50975 4.38673 4.42602 2.38 2.3825,000 4.44877 4.50992 4.38691 4.42620 2.96 2.9850,000 4.44908 4.51026 4.38724 4.42657 6.05 6.07100,000 4.44923 4.51042 4.38741 4.42675 14.45 14.3225,000-CT 4.44946 4.51064 4.38765 4.42698 2.99 3.0150,000-CT 4.44942 4.51062 4.38762 4.42695 6.10 6.0925,001-CS 4.44943 4.51062 4.38763 4.42696 6.54 9.3450,001-CS 4.44941 4.51061 4.38761 4.42694 13.73 18.55

102 103 104 10510−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

m

Poi

ntw

ise

erro

r

RectangleTrapezoidalSimpson

1.0061

2.06753.8716

Figure 2. Pricing error on a logarithmic scale for a doublebarrier option in the jump diffusion model. Daily monitoring(N¼ 252).

yThe model parameters are �¼ 0.1, �¼ 3, ¼�0.05 and �¼ 0.086 (see table 1). The benchmark price of the double barrier calloption is 2.07502090 (Feng and Linetsky 2008).

12 G. Fusai et al.


example, it can be shown (see Fusai et al. (2009b) fordetails) that, for Asian floating strike options, the pricingproblem is related to the solution of a set of integralequations

Vðx, qÞ ¼ q

Z þ11

bKðx, �;DÞVð�, qÞd� þ ð1� qÞ 1�x

Nþ 1

� �,

with

bKðx, �;DÞ ¼ e�rDK logx

� � 1

� �;D

� �x

ð� � 1Þ2:

For univariate non-Levy processes, integral equation (3)becomes

Vðx, qÞ ¼ q e�rDZ

�

Kðx, �;DÞVð�, qÞd� þ ð1� qÞ�ðxÞ:

Here, K(x, �; D) is no longer of convolution type, but ourz-transform approach still holds. In this case, the numer-ical approaches presented in the literature review in theintroduction, exploiting the convolution structure, cannotbe used.

In the above two examples, the integral equations losethe convolution behavior. Fast solution methods for non-convolution problems are reported by Fusai et al.(2009b).

6. Conclusion

In this paper we have shown how to price path-dependentoptions in an exponential Levy setting, introducing a newmethod based on the randomization of the option expiry,according to a geometric distribution. The randomizedproblem transforms the pricing problem into a set ofintegral equations. In order to solve these equationsnumerically, we have considered different quadraturerules and we have exploited the use of preconditioningtechniques to improve the performances of our algorithm.Our numerical experiments show that the proposedmethod performs well; Simpson’s formula is the mostaccurate, but the trapezoidal formula shows the besttrade-off between pricing accuracy and execution time.

Note that we can use distributed computing to furtherreduce the computational cost of our method. Finally,exploiting the Euler algorithm, the computational cost ofthe z-transform approach becomes independent of thenumber of monitoring dates.

Acknowledgements

This work was partially supported by the CRTFoundation for the Alfieri Project in TechnologicalInnovation in Finance. We would also like to thank thetwo anonymous referees and Stefano Serra Capizzanoand his research team at the University of Insubria, Italy,for constructive suggestions.

References

Abate, J. and Whitt, W., Numerical inversion of probabilitygenerating functions. Oper. Res. Lett., 1992, 12, 245–251.

Airoldi, M., A moment expansion approach to option pricing.Quant. Finance, 2005, 5(1), 89–104.

Andricopoulos, A.D., Widdicks, M., Duck, P.W. andNewton, D.P., Universal option valuation using quadraturemethods. J. Financial Econ., 2003, 67(3), 447–471.

Belomestny, D. and Reiß, M., Spectral calibration of exponen-tial Levy models. Finance Stochast., 2006, 10, 449–474.

Broadie, M., Glasserman, P. and Kou, S., Connecting discreteand continuous path-dependent options. Finance Stochast.,1999, 3(1), 55–82.

Broadie, M. and Yamamoto, Y., A double-exponential fastGauss transform algorithm for pricing discrete path-dependent options. Oper. Res., 2005, 53(5), 764–779.

Carr, P., Randomization and the American put. Rev. FinancialStud., 1998, 11(3), 597–626.

Carr, P., Geman, H., Madan, D.B. and Yor, M., The finestructure of asset returns: an empirical investigation.J. Business, 2002, 75, 305–332.

Carr, P. and Madan, D.B., Option valuation using thefast Fourier transform. J. Comput. Finance, 1999, 2(4),61–73.

Cont, R. and Tankov, P., Financial Modelling with JumpProcesses, 2004 (Chapman and Hall: London).

Eberlein, E. and Papapantoleon, A., Symmetries and pricingof exotic options in Levy models. In Exotic Option Pricingand Advanced Levy Models, edited by A. Kyprianon,W. Schoutens, and P. Wilmott, pp. 99–128, 2005 (Wiley:Chichester).

Eydeland, A., A fast algorithm for computing integrals infunction spaces: financial applications. Comput. Econ., 1994,7(4), 277–285.

Feng, L. and Linetsky, V., Pricing discretely monitored barrieroptions and defaultable bonds in Levy process models: a fastHilbert transform approach. Math. Finance, 2008, 18(3),337–384.

Feng, L. and Linetsky, V., Computing exponential moments ofthe discrete maximum of a Levy process and lookbackoptions. Finance Stochast., 2009, 13(4), 501–529.

Fusai, G., Abrahams, I.D. and Sgarra, C., An exactanalysis for discrete barrier options. Finance Stochast.,2006, 10, 1–26.

Fusai, G., Longo, G., Marena, M. and Recchioni, M.C., Levyprocesses and option pricing by recursive quadrature.In Economic Dynamics: Theory, Games and EmpiricalStudies, edited by C.W. Hurlington, pp. 31–57, 2009a.(Novapublishers: New York).

Fusai, G., Marazzina, D. and Marena, M., Pricing discretelymonitored Asian options by maturity randomization. SEMeQTechnical Report No. 15, 2009b.

Fusai, G., Marazzina, D. and Marena, M., Option pricing,maturity randomization and distributed computing. ParallelComput., 2010, 36(7), 403–414.

Fusai, G. and Recchioni, M.C., Analysis of quadrature methodsfor pricing discrete barrier options. J. Econ. Dynam. Control,2007, 31(3), 826–860.

Geman, H. and Yor, M., Bessel processes, Asian options, andperpetuities. Math. Finance, 1993, 3(4), 349–375.

Geman, H. and Yor, M., Pricing and hedging double-barrieroptions: a probabilistic approach. Math. Finance, 1996, 6(4),365–378.

Huckle, T., Serra-Capizzano, S. and Tablino Possio, C.,Preconditioning strategies for Hermitian indefinite Toeplitzlinear systems. SIAM J. Sci. Comput., 2004, 25(5), 1633–1654.

Huckle, T., Serra-Capizzano, S. and Tablino Possio, C.,Preconditioning strategies for non Hermitian Toeplitz linearsystem. Numer. Linear Algebra Applic., 2005, 12(2/3),211–220.



Jackson, K., Jaimungal, S. and Surkov, V., Fourier spacetime-stepping for option pricing with Levy models. J. Comput.Finance, 2008, 12(2), 1–29.

Kou, S., A jump diffusion model for option pricing. Mgmt Sci.,2002, 48, 1086–1101.

Kyprianou, A., Schoutens, W. and Wilmott, P., ExoticOption Pricing and Advanced Levy Models, 2005 (Wiley:New York).

Lin, F., Ng, M. and Chan, R., Preconditioners for Wiener–Hopfequations with higher order quadrature rules. SIAM J.Numer. Anal., 1997, 34, 1418–1431.

Lord, R., Fang, F., Bervoets, F. and Oosterlee, C.W., A fast andaccurate FFT-based method for pricing early-exercise optionsunder Levy processes. SIAM J. Sci. Comput., 2008, 30,1678–1705.

Merton, R.C., Option pricing when underlyingstocks are discontinuous. J. Financial Econ., 1976, 3,125–144.

Ng, M., Iterative Methods for Toeplitz Systems, 2004 (OxfordUniversity Press: Oxford).

Noutsos, D., Serra-Capizzano, S. and Vassalos, P., Matrixalgebra preconditioners for multilevel Toeplitz systems do notinsure optimal convergence rate. Theor. Comput. Sci., 2004,315, 557–579.

O’Cinneide, C.A., Euler summation for Fourier series andLaplace transform inversion. Stochast. Models, 1997, 13,315–337.

Philips, T.K. and Nelson, R., The moment bound is tighter thanChernoff’s bound for positive tail probabilities. Am. Statist.,1995, 49, 175–178.

Quarteroni, A., Sacco, R. and Saleri, F., Numerical Mathematics(Springer Series: Texts in Applied Mathematics, Vol. 37),2nd ed., 2007 (Springer: Berlin).

Schoutens, W., Levy Processes in Finance, 2003 (Wiley:New York).

Serra-Capizzano, S. and Tyrtyshnikov, E., How to prove that apreconditioner can not be superlinear. Math. Comput., 2003,72, 1305–1316.

Sullivan, M.A., Pricing discretely monitored barrier options.J. Comput. Finance, 2000, 3(4), 35–52.

14 G. Fusai et al.


Z -Transform and preconditioning techniques for option pricing

Documents

Transcript of Z -Transform and preconditioning techniques for option pricing