带跳多尺度分数布朗运动下欧式期权定价问题

European Option Pricing Problem Involving Multi-Scale Fractional Brownian Motion with Jump

摘要: 为了更合理地描述金融市场里股票等风险资产的“跳跃”、“尖峰厚尾”以及多周期现象，本文通过引入Merton跳跃、分数阶几何布朗运动以及多尺度理论，研究了在标的资产满足带跳多尺度分数阶几何布朗运动的假设下，欧式期权的定价问题。首先，本文证明了多尺度分数阶跳–扩散过程的伊藤公式，利用无套利原理和风险中性原理，得到了欧式期权价格满足的分数阶Black-Scholes方程。另一方面，本文根据分数布朗运动的Girsanov定理，建立了带跳多尺度分数阶布朗运动下风险中性的等价鞅测度，从而利用鞅定价方法得到欧式看涨看跌期权的定价公式及平价公式。最后通过数值模拟证明了该定价模型的科学性。

Abstract: In order to describe more reasonably the “jump”, “spike thick tail” and multi-period phenomenon of stock prices in the financial market, this paper studies the European options pricing problem while the underlying asset satisfies the assumption of jump-diffusion and multi-scale fractional order geometric Brownian motions, by introducing Merton jump, fractional geometric Brownian motion and multi-scale theory respectively. On the one hand, this paper first proves the Ito’s formula for the multi-scale fractional Brownian motion with jumps, then derives the fractional Black-Scholes equation by using the no-arbitrage principle and the risk neutrality principle. On the other hand, based on Girsanov’s theorem of fractional Brownian motion, this paper establishes the risk-neutral equivalent martingale measure for the multi-scale fractional Brownian motion with jumps. And then, this paper obtains the call-put pricing formula and parity formula for European options by using equivalent martingale measure. Finally, numerical simulation proves the scientific nature of the pricing model.