Bessel势空间中一类偏积分微分方程的解及其在期权定价模型中的应用
On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models
摘要：
本文主要研究一类非局部非线性偏积分微分方程（PIDE）解的定性性质。利用抽象半线性抛物型方程的理论，证明了Bessel位势空间中解的存在唯一性。我们的目的是推广已知的存在性结果的广泛类的Lévy措施，包括与一个强奇异核。作为一个应用，我们考虑一类PIDE的金融数学中产生的。经典的线性Black-Scholes模型依赖于几个限制性假设，如市场的流动性和完备性。放松完全市场假设，假设标的股票价格过程为Levy随机过程动力学，我们得到了一个期权定价模型的PIDE。研究标的资产服从带跳Levy随机过程的看涨期权和看跌期权的定价模型。证明了Levy随机过程下美式期权定价中线性互补性问题的惩罚PIDE解的存在唯一性.我们还提出了数值结果和比较期权价格的各种利维随机过程建模的基础资产动态。
英文摘要：
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black-Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Levy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Levy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Levy stochastic processes. We also present numerical results and comparison of option prices for various Levy stochastic processes modelling underlying asset dynamics.
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