带有随机利率与随机波动率的Lévy模型期权定价

摘要

研究一类在随机利率与随机波动率作用下的Lévy随机微分方程,令利率与波动率分别为与资产价格相关的函数,在对其进行一些条件限制下,证明方程有合适的解.同时在对Lévy过程中跳部分和方程其他系数的条件限制下,使方程的解满足股票价格的基本要求,从而建立市场模型.这个模型描述的市场是不完备的,利用F(o)llmer-Schweizer最小鞅测度的方法,在一系列等价鞅测度中找到F(o)llmer-Schweizer最小鞅测度,来得到此模型下欧式期权的Black-Scholes定价公式.%A class of stochastic differential equations driven by the Lévy process with stochastic volatility and stochastic interest rates are considered.The interest rate and volatility are related to asset price and it can be proven that suitable solutions with some regular conditions on the interest rate and volatility of these equations are derived.Assuming that some conditions are followed by the jump of the Lévy process and the coefficients,the stock price in the market can be illustrated with the equation.Note that an incomplete market is corresponded to the Lévy model,a Black-Scholes pricing formula for the European call option is constructed by using the F(o)llmer-Schweizer minimal martingale measure.