Some Classes Of Stochastic Partial Differential Equations And Financial Derivatives Pricing

This doctoral thesis studies stochastic partial differential equations and financial derivatives pricing. In the first part, based on the existence and uniqueness of the solu-tion, we focus on the asymptotic behavior of solutions:the exponential stability and the second moment stability. In the other part, we are concerned about variance-optimal hedging strategies for several typical kinds of options and the value of European options with counterparty risk (credit risk).Stochastic partial differential equations can capture some phenomena and specific perturbations in Physics and Biology, and has been one of the most active research ar-eas in Probability. For instance, stochastic Burgers equation is an important equation in the fluid motion to describe the interaction of dissipative and non-linear inertial terms. Stochastic Kuramoto-Sivashinsky equation is used to describe the dynamics of elec-tromagnetic field in Physics. In Chapter1, we consider stochastic Burgers equations and Kuramoto-Sivashinsky equations with the noises driven by compensated Poisson random measures. Under some appropriate conditions, we investigate the exponential stability and the second moment stability on the solutions to the equations. In Chapter2, we mainly focus on the long time behavior of the solutions to stochastic wave equa-tions driven by a non-Gaussian Levy process. In Chapter3, we are concerned about a class of stochastic heat equations with first order fractional noises. After establishing the existence and uniqueness of the solution to the equation, we give the regularity of the solution and model the term structure of forward rate with the solutions.Option pricing has always been an active research area for researchers and market practitioners. Black and Scholes investigated the topic on option pricing and provided a quite new method to derive the value of European options. The great idea is to study a self-financing portfolio consisting of the option and the underlying asset and then de-rive option price from the value of the portfolio. However, some empirical tests tell us that the dynamics of the underlying asset should show some features such as stochastic volatility or fat tail. Hence, an increasing number of processes are introduced and used to describe the dynamics of the underlying asset, including stochastic volatility model, jump-diffusion processes and Levy processes. In the classical Black-Scholes model, the market is complete and options can be hedged completely, that is, there exists a riskless portfolio consisting of the underlying asset and the option. But in general L6vy processes model, the market is incomplete and there exists no perfect hedges. There-fore it is quite important to choose a suitable hedging strategy in incomplete markets. In Chapter4and5, we consider a variance-optimal hedging strategy for geometric Asian options and target volatility options under exponential Levy dynamics. Based on the derived Follmer-Schweizer decomposition of the contingent claim in discrete time, we derive the explicit expressions of hedging strategies. Finally, we directly show that the limit of the expressions of target volatility options in discrete time is a Follmer-Schweizer decomposition in continuous-time setting and then get the explicit expressions of hedging strategies in continuous time.In over-the-counter markets, financial institutions and corporations also trade a large number of options and other financial derivatives. Different from organized ex-change, there is no margin in over-the-counter markets and holders of over-the-counter contracts are vulnerable to counterparty credit risk. To model the effect of credit risk when pricing contingent claims, one should choose what triggers credit events. In Chapter6, we provide a pricing model for vulnerable options using jump-diffusion processes and adopt structural approach to describe default risk. Jumps are divided into idiosyncratic component for each asset price and systematic component affecting the prices of all assets. A closed-form valuation formula is derived for vulnerable European options. Numerical analysis illustrates jump effects on the vulnerable option prices.