Pricing American options with the Runge-Kutta-Legendre finite difference scheme

TL;DR

This paper introduces the Runge-Kutta-Legendre finite difference scheme for pricing American options, demonstrating its stability and convergence advantages over traditional methods across various models.

Contribution

It presents a novel Runge-Kutta-Legendre scheme with an added polynomial shift, and evaluates its performance in complex option pricing models.

Findings

The scheme shows favorable stability properties.

It achieves comparable or better convergence rates.

It outperforms traditional schemes like Crank-Nicolson in stability.

Abstract

This paper presents the Runge-Kutta-Legendre finite difference scheme, allowing for an additional shift in its polynomial representation. A short presentation of the stability region, comparatively to the Runge-Kutta-Chebyshev scheme follows. We then explore the problem of pricing American options with the Runge-Kutta-Legendre scheme under the one factor Black-Scholes and the two factor Heston stochastic volatility models, as well as the pricing of butterfly spread and digital options under the uncertain volatility model, where a Hamilton-Jacobi-Bellman partial differential equation needs to be solved. We explore the order of convergence in these problems, as well as the option greeks stability, compared to the literature and popular schemes such as Crank-Nicolson, with Rannacher time-stepping.