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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Fllmer-Schweizer最小鞅测度的方法,在一系列等价鞅测度中找到Fllmer-Schweizer最小鞅测度,来得到此模型下欧式期权的Black-Scholes定价公式.
In this paper, a class of Lévy stochastic differential equations with stochastic interest rates and stochastic volatilities are studied, and the interest rates and volatilities are respectively functions related to asset prices. Under certain conditions, it is proved that the equations have suitable solutions. Under the condition of jumping part of Lévy process and other coefficients of the equation, the solution of the equation is satisfied with the basic requirements of stock price so as to establish the market model. The market described by this model is incomplete and the Föllmer-Schweizer minimum martingale measure , We find the Fllmer-Schweizer minimum martingale measure in a series of equivalent martingale measures to obtain the Black-Scholes pricing formula for European options under this model.