Abstract
The application of stochastic volatility (SV) models in the option pricing literature usually assumes that the market has sufficient option data to calibrate the model’s risk-neutral parameters. When option data are insufficient or unavailable, market practitioners must estimate the model from the historical returns of the underlying asset and then transform the resulting model into its risk-neutral equivalent. However, the likelihood function of an SV model can only be expressed in a high-dimensional integration, which makes the estimation a highly challenging task. The Bayesian approach has been the classical way to estimate SV models under the data-generating (physical) probability measure, but the transformation from the estimated physical dynamic into its risk-neutral counterpart has not been addressed. Inspired by the generalized autoregressive conditional heteroskedasticity (GARCH) option pricing approach by Duan in 1995, we propose an SV model that enables us to simultaneously and conveniently perform Bayesian inference and transformation into risk-neutral dynamics. Our model relaxes the normality assumption on innovations of both return and volatility processes, and our empirical study shows that the estimated option prices generate realistic implied volatility smile shapes. In addition, the volatility premium is almost flat across strike prices, so adding a few option data to the historical time series of the underlying asset can greatly improve the estimation of option prices.
Highlights
The constant volatility assumption in the original Black–Scholes model has been criticized over the years for its failure to produce the implied volatility smile
Models of [1] and the generalised autoregressive conditional heteroskedasticity (ARCH) (GARCH) models of [2] offer the possibility of capturing many stylized time-varying volatility facts in a time series perspective. They were mainly used in the investigation of economic series, until Duan ([3]) proposed a GARCH option pricing framework that enabled the GARCH model estimated from asset return series to be conveniently transformed into a risk-neutral process for option pricing purposes
We propose an stochastic volatility (SV) option pricing model inspired by Duan’s GARCH option pricing model, and extend our model by allowing return and volatility to have variance gamma (VG) and Student-t innovations, respectively
Summary
The constant volatility assumption in the original Black–Scholes model has been criticized over the years for its failure to produce the implied volatility smile. When option data are unavailable, market practitioners must resort to estimating the model under the physical measure using historical asset returns, and transforming the model into its risk-neutral counterpart for derivative pricing. All of the methods presented above are useful in the estimation of the SV model under the physical measure, but the transformation into the risk-neutral process for option pricing can be a highly non-trivial task. Bates [16] considered the transformation between the two measures and allowed the parameters of the volatility process to change, but he did not set up the locally risk-neutral valuation formula for option pricing with stochastic volatility.
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