Numerical pricing of options using high-order compact finite difference schemes

作者:

Highlights:

摘要

We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.

论文关键词:35A35,35A40,65N99,European options,American options,High-order compact scheme,Grid stretching,Front fixing

论文评审过程:Received 29 August 2006, Revised 11 January 2007, Available online 21 February 2007.

论文官网地址:https://doi.org/10.1016/j.cam.2007.01.035