1. Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior.
- Author
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Zhang, Gongqiu and Li, Lingfei
- Subjects
MATHEMATICAL models of pricing ,HEDGING (Finance) ,MARKOV chain Monte Carlo ,APPROXIMATION theory ,ECONOMIC convergence - Abstract
Markov chain approximation is a general method for option pricing and hedging in Markovian models with continuous-state spaces. A key issue for its efficiency is how to design the grid for the Markov chain. In "Analysis of Markov Chain Approximation for Option Pricing and Hedging: Grid Design and Convergence Behavior," Zhang and Li investigated how grid design affects the convergence behavior of barrier and European options in general diffusion models. They obtain sharp estimates for the convergence rate of option price, delta, and gamma and develop two principles that are sufficient and necessary for designing nonuniform grids that can achieve second-order convergence. They further propose a novel class of nonuniform grids that ensure that convergence is not only second order but also, smooth. This allows extrapolation to be applied to achieve even higher convergence rate. Their grids enable the continuous time Markov chain approximation method to price and hedge a large number of options with different strikes fast and accurately. 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. The e-companion is available at https://doi.org/10.1287/opre.2018.1791. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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