Some Classes Of Stochastic Partial Differential Equations And Financial Derivatives Pricing  Posted on:20150226  Degree:Doctor  Type:Dissertation  Country:China  Candidate:X C Wang  Full Text:PDF  GTID:1109330467465574  Subject:Probability theory and mathematical statistics  Abstract/Summary:  PDF Full Text Request  This doctoral thesis studies stochastic partial differential equations and financial derivatives pricing. In the first part, based on the existence and uniqueness of the solution, we focus on the asymptotic behavior of solutions:the exponential stability and the second moment stability. In the other part, we are concerned about varianceoptimal 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 areas in Probability. For instance, stochastic Burgers equation is an important equation in the fluid motion to describe the interaction of dissipative and nonlinear inertial terms. Stochastic KuramotoSivashinsky equation is used to describe the dynamics of electromagnetic field in Physics. In Chapter1, we consider stochastic Burgers equations and KuramotoSivashinsky 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 equations driven by a nonGaussian 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 selffinancing portfolio consisting of the option and the underlying asset and then derive 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, jumpdiffusion processes and Levy processes. In the classical BlackScholes 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. Therefore it is quite important to choose a suitable hedging strategy in incomplete markets. In Chapter4and5, we consider a varianceoptimal hedging strategy for geometric Asian options and target volatility options under exponential Levy dynamics. Based on the derived FollmerSchweizer 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 FollmerSchweizer decomposition in continuoustime setting and then get the explicit expressions of hedging strategies in continuous time.In overthecounter markets, financial institutions and corporations also trade a large number of options and other financial derivatives. Different from organized exchange, there is no margin in overthecounter markets and holders of overthecounter 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 jumpdiffusion 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 closedform valuation formula is derived for vulnerable European options. Numerical analysis illustrates jump effects on the vulnerable option prices.  Keywords/Search Tags:  Stochastic Partial Differential Equations, Poisson Random Measures, BurgersEquation, KuramotoSivashinsky Equation, Exponential Stability, Second MomentStability, Regularity, Option Pricing, Levy Processes, Asian Options, Target Volatility Options  PDF Full Text Request  Related items 
 
