TY - JOUR
T1 - CTMC integral equation method for American options under stochastic local volatility models
AU - Ma, Jingtang
AU - Yang, Wensheng
AU - Cui, Zhenyu
N1 - Publisher Copyright:
© 2021 Elsevier B.V.
PY - 2021/7
Y1 - 2021/7
N2 - In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the 4/2 model and the α−hypergeometric models. They illustrate the high accuracy and efficiency of the method.
AB - In this paper, a continuous-time Markov chain (CTMC) approach is proposed to solve the problem of American option pricing under stochastic local volatility (SLV) models. The early exercise premium (EEP) representation of the value function, which contains the corresponding European option term and the EEP term, is in general not available in closed-form. We propose to use the CTMC to approximate the underlying asset, and derive explicit closed-form expressions for both the European option term and the EEP term, so that the integral equation characterizing the early exercise surface can be explicitly expressed through characteristics of the CTMC. The integral equations are then solved by the iteration method and the early exercise surface can be computed, and semi-explicit expressions for the values and Greeks of American options are derived. We denote the new method as the CTMC integral equation method, and establish both the theoretical convergence and the precise convergence order. Numerical examples are given for the classical Black-Scholes model and the general stochastic (local) volatility models, such as the stochastic alpha beta rho (SABR) model, the Heston model, the 4/2 model and the α−hypergeometric models. They illustrate the high accuracy and efficiency of the method.
KW - American option pricing
KW - Continuous-time Markov chains
KW - Early exercise premium
KW - Integral equation
KW - Stochastic local volatility models
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U2 - 10.1016/j.jedc.2021.104145
DO - 10.1016/j.jedc.2021.104145
M3 - Article
AN - SCOPUS:85107652124
SN - 0165-1889
VL - 128
JO - Journal of Economic Dynamics and Control
JF - Journal of Economic Dynamics and Control
M1 - 104145
ER -