Error Analysis of Finite Difference and Markov Chain Approximations for Option Pricing

Abstract

Mijatovic and Pistorius (Math. Finance, 2013) proposed an efficient Markov chain approximation method for pricing European and barrier options in general one-dimensional Markovian models. However, sharp convergence rates of this method for realistic financial payoffs, which are non-smooth, are rarely available. In this paper, we solve this problem for general one-dimensional diffusion models, which play a fundamental role in financial applications. For such models, the Markov chain approximation method is equivalent to the method of lines using the central difference. Our analysis is based on the spectral representation of the exact solution and the approximate solution. By establishing the convergence rate for the eigenvalues and the eigenfunctions, we obtain sharp convergence rates for the transition density and the price of options with non-smooth payoffs. In particular, we show that for call-/put-type payoffs, convergence is second order, while for digital-type payoffs, convergence is generally only first order. Furthermore, we provide theoretical justification for two well-known smoothing techniques that can restore second-order convergence for digital-type payoffs and explain oscillations observed in the convergence for options with non-smooth payoffs. As an extension, we also establish sharp convergence rates for European options for a rich class of Markovian jump models constructed from diffusions via subordination. The theoretical estimates are confirmed using numerical examples.

Keywords: diffusions, subordination, Markov chain approximation, finite difference, spectral representation, convergence rate, European and barrier options, non-smooth payoffs, smoothing techniques