马尔可夫调制的双分数布朗运动模型下亚式期权定价

Asian Option Pricing in Double Fractional Brownian Model with Markovian Switching

针对一种新的增量随机过程——马尔可夫调制的双分数布朗运动,基于可靠性数学思想,利用测度变换技巧将实际概率测度变换成等价鞅测度,研究了在此模型下连续时间的固定价格亚式期权定价问题;通过亚式期权所满足的概率密度转移函数,将经典的测度变换方法与拟鞅相结合,并推广到受双分数布朗运动驱动的B-S市场环境中,利用风险中性定价方法分别得到具有固定执行价格的几何平均亚式看涨和看跌期权的定价公式;双分数布朗运动不具有独立性和平稳增量性,更符合显示情形,且与基于分数布朗运动的期权定价公式进行比较分析,可知分数布朗运动只是双分数布朗运动的一种特殊情形,可基于双分数布朗运动对分数布朗运动的亚式期权期权定价模型进行推广。

Aiming at a new incremental stochastic process in double-fractional Brownian motion with Markovian switching,based on the idea of reliability mathematics,the real probability measure is transformed into the equivalent martingale measure by using the measure transformation technique,and the pricing problem of fixedprice Asian option with continuous time under this model is studied. By using the probability density transfer function satisfied by Asian options,the classical measure transformation method is combined with quasi-martingale method,which is extended to the B-S market driven by double-fractional Brownian motion. The pricing formulas of geometrically average Asian call and put options with fixed execution price are obtained by using risk-neutral pricing method. The double fractional Brown motion is independent and stable incrementally,which is more consistent with the display situation. Compared with the option pricing formula based on fractional Brownian motion,fractional Brownian motion is only a special case of double-fractional Brownian motion,which can be extended to Asian option pricing model based on double-fractional Brownian motion.