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

Gongqiu Zhang () and Lingfei Li ()
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Gongqiu Zhang: School of Science and Engineering, The Chinese University of Hong Kong, 518172 Shenzhen, China
Lingfei Li: Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Hong Kong, China

Operations Research, 2019, vol. 67, issue 2, 407-427

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 nonuniform 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 nonuniform grids that ensure 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; nonuniform grids; convergence rate; smooth convergence; extrapolation; spectral representation; nonsmooth payoffs (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (18)

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