Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior

38 Pages Posted: 26 Jul 2017

See all articles by Lingfei Li

Lingfei Li

The Chinese University of Hong Kong

Gongqiu Zhang

Economics and Management School, Wuhan University

Date Written: July 23, 2017

Abstract

Continuous time Markov chain (CTMC) approximation is an intuitive and powerful method for pricing options in general Markovian models. This paper analyzes how grid design affects the convergence behavior of barrier and European options in general diffusion models. Using the spectral method, we obtain sharp estimates for the convergence rate of option price for non-uniform grids. We propose to calculate an option’s delta and gamma by taking central difference of option prices on the grid. For this simple method, we prove that, surprisingly, delta and gamma converge at the same rate as option price does. Our analysis allows us to develop principles that are sufficient and necessary for designing nonuniform grids that can achieve second order convergence for option price, delta and gamma. Based on these principles, we propose a novel class of non-uniform grids, which ensures that convergence is not only second order, but also smooth. This further allows extrapolation to be applied to achieve even higher convergence rate. Our grids enable the CTMC approximation method to price and hedge a large number of options with different strikes fast and accurately. Applicability of our results to jump models is discussed through numerical examples.

Keywords: Diffusions, Jumps, Markov Chain, Non-Uniform Grids, Convergence Rate, Smooth Convergence, Extrapolation, Spectral Representation, Non-Smooth Payoffs

JEL Classification: G13

Suggested Citation

Li, Lingfei and Zhang, Gongqiu, Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior (July 23, 2017). Available at SSRN: https://ssrn.com/abstract=3007318 or http://dx.doi.org/10.2139/ssrn.3007318

Lingfei Li (Contact Author)

The Chinese University of Hong Kong ( email )

Shatin, New Territories
Hong Kong

Gongqiu Zhang

Economics and Management School, Wuhan University ( email )

Luojiashan
Wuhan
Wuhan, Hubei NA
China

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