Estimating Heston's and Bates' models parameters using Markov chain Monte Carlo simulation

Heston's model and Bates' model are very important in option pricing. It is mentioned in Mendoza's paper [Bayesian estimation and option mispricing (job market paper). Cambridge, MA: Massachusetts Institute of Technology; 2011] that Mexican Stock Exchange introduced options over its main index (the indice de Precios y Cotizaciones) in 2004 which used Heston's model to price options on days when there was no trading. The estimation of the parameters in both models is not easy. One of the methods is Markov chain Monte Carlo algorithm (MCMC for short). In this paper, we adopt Li, Wells and Yu's MCMC algorithm [A Bayesian analysis of return dynamics with levy jumps. Rev Financ Stud. 2008;21(5):2345-2377]. We provide the necessary derivation utilizing prior distributions since they are otherwise unavailable in the literature. As Li et al. used their model to analyse S&P 500 data from 2 January 1980 to 29 December 2000, we likewise recreate their analysis, this time using data from 1987 to 2012. We would like to involve the financial crisis and analyse how stable the method is while applying to the financial crisis. Unlike Li et al., we find that the estimation is very sensitive to the prior distribution assumption. In addition, we have R-code available by request. We hope to offer tools for people doing empirical research in financial mathematics or quantitative finance.