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

作　　者：胡静[1] 黄小涛 

机构地区：[1]南京航空航天大学数学学院,江苏 南京

出　　处：《理论数学》2023年第9期2536-2559,共24页Pure Mathematics

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。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.

关 键 词：欧式期权定价方程 分数布朗运动 Merton跳跃 多尺度布朗运动 等价鞅测度 

分 类 号：F83[经济管理—金融学]