| Volatility has played a central role in financial markets due to its close relationship with asset pricing and risk management. Among a rich family of volatility models, stochastic volatility models stand out because of their ability to explain many behaviors observed in real markets. On the other hand, fitting stochastic volatility models is widely considered as a difficult task due to the complex hierarchical structure.; Jacquier, Polson and Rossi (1995) is among the first that introduced the Bayesian Markov chain Monte Carlo (MCMC) methodology to calibrate a stochastic volatility model with historical volatility time series. In this work, we extend the Bayesian MCMC technique to a hybrid stochastic volatility model which combines both the historical volatility and the implied volatility. Since no closed-form option pricing formula is available for us to obtain the implied volatility, we resort to MCMC numerical computation of the option prices. With the additional information provided by option data, the hybrid volatility model enables us to yield better results in estimation of the model parameters, and more importantly, it improves the estimation of the volatility time series itself. Although MCMC methods demonstrate tremendous power and flexibility in model calibration, they are computationally very demanding because we have to repeatedly simulate the option prices in each MCMC iteration. Our numerical computation is carried out on both simulated data and the real market data of DJIA, SP500, NASDAQ and MSFT from April 2000 to April 2002. |
