分数阶BS方程中自由边界的数值计算

Numerical Calculation of Free Boundaries in the Fractional BS Equations

本文研究一类分数阶美式期权定价模型的数值计算问题。由于分数阶Brown运动能够更好地反映风险资产价格的自相关性和长期记忆性,本文将研究由分数阶Brown运动推导得到的期权价格模型,即分数阶Black-Scholes方程。由于美式期权可以提前执行,该模型的边界条件可归结为分数阶抛物型方程的自由边值问题。自由边界问题的困难之处在于需要同时求出期权的最佳执行边界及分数阶BS方程的数值解,因此本文在美式期权的持有区域使用θ-差分格式预估数值解,再利用边界条件及Taylor公式校正得到收敛的方程数值解和自由边界。

This paper studies the numerical computation of a class of fractional-order American option pricing models. Since the fractional-order Brownian motion can better reflect the autocorrelation and long-term memory of risky asset prices, this paper will study the option price model derived from the fractional-order Brownian motion, i.e., the fractional-order Black-Scholes equation. And since American options can be executed in advance, the boundary conditions of this model can be reduced to the free boundary problem of the fractional order parabolic equation. The difficulty of the free boundary problem is that the optimal execution boundary of the option and the numerical solution of the fractional-order BS equation need to be found simultaneously, so in this paper, the numerical solution is predicted in the holding region of the American option using the θ-difference formats, and then the converged numerical solution of the equation and the free boundary are obtained by using the boundary conditions and the correction of Taylor’s formula.