The Application Of Bayesian Methods In The Black-Scholes Option Price Model

Bayesian statistical methods have the advantage of being able to naturally orient towards pooling in rigorous way information coming from separate sources. In the financial markets, the option price depends on the volatility and asset prices. While in the classical Black-Scholes framework the drift and diffusion parameters are regarded as constants. Even in the later improved model, only one of the random factors, volatility or asset prices, took into account. So in this paper, we first investigated the statistical properties of the Black-Scholes option price under a Bayesian approach. We incorporate randomness both in the price process and in volatility to derive the prior and posterior densities of a European call option. Expressions for the density of the option price conditional on the sample estimate of volatility and on the asset price respectively are also derived. Numerical evaluation found that both conditioning on the asset price and conditioning on the volatility could reduce the variability of the option price. However, the role of asset price is more important than the volatility. Compared to the classic statistical models, Bayesian statistics treat asset price and volatility as random variables and assign to them probability distributions. These factors that affecting option prices and sample information through bayes rule can be full reflected in the posterior density. The loss and utility function were also to be unified in the posterior density. Posterior density for the use of estimate and forecasting can be straightforwardly combined with a loss function to produce optimal estimates of options prices. Thus it can obtain a better estimated effect than the classic statistical models.Because the posterior density can obtain optimal estimated properties of options prices, we introduced a new parameter in its derivation process, the weight attached to the implied volatility estimate; in the further, we built the Black-Scholes option price formula for the forecast density function under the Bayesian framework. It has been suggested that both historical and implied volatilities convey information about future volatility. However, typically in the literature implied and return volatility series are fed separately into models to provide rival forecasts of volatility or options prices. Although there has been a lot of work done in modifying the specification of volatility to make it a stochastic process there has not yet been a model of stochastic volatility that enjoys the popularity of Black-Scholes. In the posterior density, we introduced volatility correlation coefficient. It is a breakthrough for the original method and can better solve the above mentioned-problem. According to the different study, we can take into account all aspects of information and decide to choose the information focus, to improve the accuracy of the forecasting by changing the value of the volatility correlation coefficient. The use of Bayesian methods provided theoretical support to consider joint information, historical and forecast. We merged the backward looking information with the forward looking information on the posterior density modification. Finally, we got the forecast density function which including historical data and implied volatility information.