Cubature methods and applications to option pricing

[摘要] ENGLISH ABSTRACT: In this thesis, higher order numerical methods for weak approximation of solutionsof stochastic differential equations (SDEs) are presented. They aremotivated by option pricing problems in finance where the price of a givenoption can be written as the expectation of a functional of a diffusion process.Numerical methods of order at most one have been the most used so far andhigher order methods have been difficult to perform because of the unknowndensity of iterated integrals of the d-dimensional Brownian motion present inthe stochastic Taylor expansion. In 2001, Kusuoka constructed a higher orderapproximation scheme based on Malliavin calculus. The iterated stochasticintegrals are replaced by a family of finitely-valued random variables whosemoments up to a certain fixed order are equivalent to moments of iteratedStratonovich integrals of Brownian motion. This method has been shown tooutperform the traditional Euler-Maruyama method. In 2004, this methodwas refined by Lyons and Victoir into Cubature on Wiener space. Lyons andVictoir extended the classical cubature method for approximating integralsin finite dimension to approximating integrals in infinite dimensional Wienerspace. Since then, many authors have intensively applied these ideas and thetopic is today an active domain of research. Our work is essentially based onthe recently developed higher order schemes based on ideas of the Kusuokaapproximation and Lyons-Victoir 'Cubature on Wiener space and mostly appliedto option pricing. These are the Ninomiya-Victoir (N-V) and Ninomiya-Ninomiya (N-N) approximation schemes. It should be stressed here that manyother applications of these schemes have been developed among which is theAlfonsi scheme for the CIR process and the decomposition method presentedby Kohatsu and Tanaka for jump driven SDEs.After sketching the main ideas of numerical approximation methods inChapter 1 , we start Chapter 2 by setting up some essential terminologiesand definitions. A discussion on the stochastic Taylor expansion based oniterated Stratonovich integrals is presented, we close this chapter by illustratingthis expansion with the Euler-Maruyama approximation scheme. Chapter 3contains the main ideas of Kusuoka approximation scheme, we concentrate onthe implementation of the algorithm. This scheme is applied to the pricing ofan Asian call option and numerical results are presented. We start Chapter 4by taking a look at the classical cubature formulas after which we propose in a simple way the general ideas of 'Cubature on Wiener space also known asthe Lyons-Victoir approximation scheme. This is an extension of the classicalcubature method. The aim of this scheme is to construct cubature formulas forapproximating integrals defined on Wiener space and consequently, to develophigher order numerical schemes. It is based on the stochastic Stratonovichexpansion and can be viewed as an extension of the Kusuoka scheme. Applyingthe ideas of the Kusuoka and Lyons-Victoir approximation schemes, Ninomiya-Victoir and Ninomiya-Ninomiya developed new numerical schemes of order 2,where they transformed the problem of solving SDE into a problem of solvingordinary differential equations (ODEs). In Chapter 5 , we begin by a generalpresentation of the N-V algorithm. We then apply this algorithm to the pricingof an Asian call option and we also consider the optimal portfolio strategiesproblem introduced by Fukaya. The implementation and numerical simulationof the algorithm for these problems are performed. We find that the N-Valgorithm performs significantly faster than the traditional Euler-Maruyamamethod. Finally, the N-N approximation method is introduced. The ideabehind this scheme is to construct an ODE-valued random variable whoseaverage approximates the solution of a given SDE. The Runge-Kutta methodfor ODEs is then applied to the ODE drawn from the random variable anda linear operator is constructed. We derive the general expression for theconstructed operator and apply the algorithm to the pricing of an Asian calloption under the Heston volatility model.

[发布机构] Stellenbosch University