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

Zhang, Gongqiu; Li, Lingfei

doi:10.1287/opre.2018.1791

Cited by 20 publications

(29 citation statements)

References 29 publications

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“…The rest of the results in the second part follows from Lemma 3, Lemma 5 and Lemma 6 in Zhang and Li (2019).…”

Section: A4 Regime-switching and Stochastic Volatility Modelsmentioning

confidence: 95%

“…Analysis of v n (D, x; y) (see (4.6) and (4.7)): The quantity is essentially the up-and-out barrier option price with L + as the effective upper barrier and payoff function 1 {x=y} . By Zhang and Li (2019), v n (D, x; y) is represented by a discrete eigenfunction expansion based on the eigenvalues and eigenvectors from a matrix eigenvalue problem. Its continuous counterpart also admits an eigenfunction expansion representation where the eigenvalues and eigenfunctions are solutions to a Sturm-Liouville eigenvalue problem on the interval (l, L + ).…”

Section: Outline Of the Proofsmentioning

confidence: 99%

“…One can view the present paper as a generalization of Mijatović and Pistorius (2013) and Zhang and Li (2019) from barrier options to Parisian options. However, this generalization is by no means trivial.…”

Section: Introductionmentioning

confidence: 99%

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A General Approach for Parisian Stopping Times under Markov Processes

Zhang,

2021

Preprint

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We propose a method based on continuous time Markov chain approximation to compute the distribution of Parisian stopping times and price Parisian options under general one-dimensional Markov processes. We prove the convergence of the method under a general setting and obtain sharp estimate of the convergence rate for diffusion models. Our theoretical analysis reveals how to design the grid of the CTMC to achieve faster convergence. Numerical experiments are conducted to demonstrate the accuracy and efficiency of our method for both diffusion and jump models. To show the versality of our approach, we develop extensions for multi-sided Parisian stopping times, the joint distribution of Parisian stopping times and first passage times, Parisian bonds and for more sophisticated models like regime-switching and stochastic volatility models.

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“…The rest of the results in the second part follows from Lemma 3, Lemma 5 and Lemma 6 in Zhang and Li (2019).…”

Section: A4 Regime-switching and Stochastic Volatility Modelsmentioning

confidence: 95%

Section: Outline Of the Proofsmentioning

confidence: 99%

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A General Approach for Parisian Stopping Times under Markov Processes

Zhang,

2021

Preprint

Self Cite

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“…This idea was later extended to price Asian options ([CSK15, KN20]), and realized variance derivatives ( [CKN17]) under stochastic volatility dynamics. A rigorous error analysis for the CTMC approximation and the optimal discretization grid design was considered in [LZ18,ZL19]. Simulation of two-dimensional diffusions was proposed in [CKN21].…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation of Diffusions by Continuous Time Markov Chain

Kirkby¹,

Nguyen²,

Nguyen³

et al. 2021

Preprint

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In this paper we present a novel method for estimating the parameters of a parametric diffusion processes. Our approach is based on a closed-form Maximum Likelihood estimator for an approximating Continuous Time Markov Chain (CTMC) of the diffusion process. Unlike typical time discretization approaches, such as psuedolikelihood approximations with Shoji-Ozaki or Kessler's method, the CTMC approximation introduces no time-discretization error during parameter estimation, and is thus well-suited for typical econometric situations with infrequently sampled data. Due to the structure of the CTMC, we are able to obtain closed-form approximations for the sample likelihood which hold for general univariate diffusions. Comparisons of the statediscretization approach with approximate MLE (time-discretization) and Exact MLE (when applicable) demonstrate favorable performance of the CMTC estimator. Simulated examples are provided in addition to real data experiments with FX rates and constant maturity interest rates.

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“…They simulate the one-dimensional variance process by a CTMC that approximates it and then sample the integrated variance conditioned on the start and end points of the variance process using a Fourier sampler for the CTMC variance model. Finally, there are also various papers on CTMC approximation of one-dimensional Markov processes with applications in finance; see e.g., Mijatović and Pistorius (2013), Cai et al (2015), Eriksson and Pistorius (2015), Cui et al (2018b), Li and Zhang (2018), Zhang and Li (2019b, 2021a,b, 2019aand Zhang et al (2021).…”

Section: Introductionmentioning

confidence: 99%

Simulation of Multidimensional Diffusions with Sticky Boundaries via Markov Chain Approximation

Meier

Zhang

2021

Preprint

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We develop a new simulation method for multidimensional diffusions with sticky boundaries. The challenge comes from simulating the sticky boundary behavior, for which standard methods like the Euler scheme fail. We approximate the sticky diffusion process by a multidimensional continuous time Markov chain (CTMC), for which we can simulate easily. We develop two ways of constructing the CTMC: approximating the infinitesimal generator of the sticky diffusion by finite difference using standard coordinate directions, and matching the local moments using the drift and the eigenvectors of the covariance matrix as transition directions. The first approach does not always guarantee a valid Markov chain whereas the second one can. We show that both construction methods yield a first order simulation scheme, which can capture the sticky behavior and it is free from the curse of dimensionality. We apply our method to two applications: a multidimensional Brownian motion with all dimensions sticky which arises as the limit of a queuing system with exceptional service policy, and a multi-factor short rate model for low interest rate environment in which the stochastic factors are unbounded but the short rate is sticky at zero.

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

Cited by 20 publications

References 29 publications

A General Approach for Parisian Stopping Times under Markov Processes

A General Approach for Parisian Stopping Times under Markov Processes

Maximum Likelihood Estimation of Diffusions by Continuous Time Markov Chain

Simulation of Multidimensional Diffusions with Sticky Boundaries via Markov Chain Approximation

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