On 25 June 2010, two MSc students at AIMS, Nicole Tchamga and Lydienne Matchie, will be defending their MSc theses in Financial Mathematics at 14:00 and 15:30, respectively. The venue is the AIMS Research Centre, 6 Melrose Road, Muizenberg. Enquiries can be made from Raouf Ghomrashni.
In this thesis, we study the martingale method and convex duality approach to the expected utility maximization portfolio problems for terminal wealth in a fixed and finite time horizon. We begin by formulating the superreplication problem as introduced by El Karoui and Quenez in 1995 and we derive the dual characterization of the set of all initial endowments leading to super-hedge a European contingent claim. We then state and provide a detailed proof of the well-known Kramkov-Schachermayer theorem on portfolio optimization and illustrate the theory by several examples in complete and incomplete markets.
In this thesis, higher order numerical methods for weak approximation of solutions of stochastic differential equations (SDEs) are presented. They are motivated by option pricing problems in finance where the price of a given option can be written as the expectation of a functional of a diffusion process. In 2001, Kusuoka constructed a higher order approximation scheme based on Malliavin calculus. The iterated stochastic integrals are replaced by a family of finitely-valued random variables which moments up to a certain fixed order are equivalent to moments of iterated Stratonovich integrals of Brownian motion. Our work is essentially based on the recently developed higher order schemes based on ideas of Kusuoka approximation and Lyons-Victoir ”Cubature on Wiener space” and mostly applied to option pricing. Namely, Ninomiya-Victoir (N-V) and Ninomiya-Ninomiya (N-N) approximation schemes. We apply these algorithms to the pricing of Asian options with stochastic volatility and we also consider the optimal portfolio strategies problem introduced by Fukaya.