An Integral Equation Method with High-Order Collocation Implementations for Pricing American Put Options

Abstract

The aim of this paper is to solve a free boundary problem arising in pricing American put options. It is known that the free boundary (optimal exercise boundary) satisfies a "nonstandard" Volterra integral equation. This Volterra integral equation is resolved by a high-order collocation method based on graded meshes. With the computed free boundary, a Black-Scholes equation for pricing the American put options is solved by a moving mesh method. Numerical examples are provided to confirm the efficiency of the approach.

High-Accuracy Integral Equation Approach for Pricing American Options with Stochastic Volatility

“…Now we develop a collocation method (Note 3) to solve the integral equation (9). As proved for the case of constant Volatility (see the summary in Ma et al (2010)), the solution of the integral equation is singular in general. So using uniform meshes cannot get an optimal accuracy in the numerical solution of the integral equation (5).…”

“…From the EEP representation, the optimal exercise boundary satisfies an integral equation whose analytical and numerical solutions are difficult to find. To this end, Ma et al (2010) construct a high-order collocation method on non-uniform meshes for solving the integral equation arising in the EEP representation of American put options under lognormal process. In this paper, we extend the collocation methods to solution of the integral equations for the general diffusion process with stochastic volatility.…”

Projection and Contraction Method for the Valuation of American Options

“…Numerical methods for American options have attracted increasing interest, and are mainly of two types -viz. the Monte Carlo method [6,19,22] and the partial differential equation (PDE) method [1,9,10,14,20,23,27]. The Monte Carlo method has a high computational cost due to its slow convergence, and in this article we pursue the famous Black-Scholes PDE approach, which is widely regarded as one of most effective [7,11,15].…”

“…In their seminal contribution, Cox et al [8] introduced the binomial method to price American options, and its convergence was proven by Amin & Khanna [2]. The binomial method is essentially a difference method, and inspired a variety of finite difference schemes for American option pricing [9,20,26]. Refs.…”

Spectral Method for the Black-Scholes Model of American Options Valuation

“…About the first challenge, Cox [15,23] has proved that the optimal exercise boundary satisfied a nonlinear Volterra integral equation, and Ma et al [29] have solved the same nonlinear Volterra integral equation by a high-order collocation method and presented numerical results. We shall follow the idea of [29] to deal with the first difficulty.…”