具有一般跳过程的期权无差异效用价值过程的定价模型

Exponential utility indifference value process in a general jump model based on random measures

本文采用指数效用最大化的方法研究了期权的动态无差异效用价值过程Ct(H;α).考虑股票价格过程为具有基于随机测度的一般跳的半鞅模型,且期权的无差异效用价值过程的Doob-Meyer分解的鞅部分的GKW(Galtchouk-Kunita-Watanabe)分解满足Jacod鞅表示定理.利用无差异效用价值过程在最小熵测度和最优投资策略下为鞅的事实构建了一个倒向随机微分方程.通过概率测度变换将方程的鞅部分和生成元转化为BMO(bounded mean oscillation)鞅,证明了该方程的解的唯一性.并将方程的生成元分成[?A=0]和[?A≠0],证明了最优投资策略存在.从而给出期权无差异效用价值过程的倒向随机微分方程的表达形式.

This paper deals with the dynamic exponential utility indifferent value process of a contingent claim.We regard the risky asset pricing process as a semimartingale with general jumps based on a random measure,where the GKW（Galtchouk-Kunita-Watanabe） decomposition of the martingale part of the exponential utility indifferent value process＇ s Doob-Meyer decomposition satisfies the Jacod decomposition. A backward stochastic differential equation（BSDE） is established from the fact that the exponential utility indifferent value process is a martingale for the optimal investment strategy under the minimal entropy measure. The uniqueness of the solution to the equation is proved by converting the BSDE to bounded mean oscillation（BMO） martingales under a new probability measure. The division of the equation＇s generator into [△A = 0] and [△A≠0] proves the existence of an optimal investment strategy. As a result,the exponential utility indifference value process of the contingent claim is the solution of the BSDE.