Research On Volatility Inference And Option Pricing Based On The Bayesian Hierarchical Prior Model

The Black-Scholes option pricing model has made outstanding contributions to the development of option pricing.But the volatility should be unknown.Therefore,many studies have been explored the model in various ways,such as assuming the volatility is random,or assuming that the stochastic volatility model with jumps.In this paper,we assume that there is a relationship among the volatilities of some assets.Through the Bayesian method,we adopt a hierarchical prior model to describe the volatilities and make inference of them.The results show that the proposed method of estimating volatility and pricing options is more accurate than the traditional method.In this paper,the following work is carried out:Firstly,suppose there are m assets from the same sector.According to the stochastic differential equation obeyed by the asset price process and the It(?) formula,the distribution of their logarithmic rate of return is derived.Hence,the likelihood function is obtained,too.Suppose there is a relationship among the volatilities of the assets,give the volatilities a hierarchical prior under the Bayesian method.Then the joint posterior density function and full condition posterior density function of each unknown parameter are derived.The Metropolis-within-Gibbs algorithm is designed to sample the unknown parameters and make a posterior inference of the unknown parameters in the model.Then the pricing of the European call option is deduced.Secondly,in the empirical analysis,some assets are randomly selected in two different sectors.The volatilities are inferred under the Bayesian method based on the assumption that there is a relationship among them.Then the European call option price is deduced.The results obtained by this method are compared with those calculated by historical volatility,implied volatility,and the volatility inferred through the non-hierarchical Bayesian method,which means there is no relationship among the volatilities of the assets.The comparison results show that it is more accurate to use this method than others when pricing the European call options.Thirdly,a basket option contains multiple underlying assets.It is assumed that there is a specific relationship among the volatilities of the underlying assets in a basket option.Similarly,the volatilities of the underlying assets are inferred under the Bayesian hierarchical model.Then the price of the basket option is deduced.And the second stage in the prior of the Bayesian hierarchical model adopts the non-informative prior.Because it is difficult to obtain the real data of a basket option,we compare the theoretical option prices and the actual option prices of the underlying assets in the basket option.From the comparison results,it shows that this method proposed in the paper is feasible in calculating a basket option.