Research On Option Pricing With Stock Liquidity-adjustment And Bayesian Empirical Method

Option pricing theory and its application research have always been the hot topic in the financial field,and have received extensive attention and in-depth research.The BlackScholes option pricing model lays the foundation for option pricing theory,and is widely used as a standard pricing tool in financial investment and risk management,but there exist also some shortcomings.In order to better fit the actual market conditions,many scholars have carried out different extended studies to improve the pricing accuracy of option pricing models.Many empirical studies have shown that market liquidity is an important factor in financial asset pricing and risk management.However,the traditional option pricing models are usually built on a frictionless and perfectly liquid market,ignoring the effect of market liquidity on option pricing.At present,although some scholars have studied the option pricing with liquidity-adjustment,the related research progress is still slow,and it is still necessary to carry out a systematic and in-depth research on the option pricing in imperfectly liquid market.In an incomplete market,therefore,this paper considers the pricing of options on imperfectly liquid stocks,and studies the effects of stock liquidity on European options and Exchange options.Furthermore,the effects of stock liquidity on the option hedging strategy and the risk management are studied.Based on the theory of asset pricing,financial economics,mathematical finance and stochastic analysis,the option pricing models with stock liquidity-adjustment are derived by Esscher measure transformation and the change of numeraire.In addition,considering the influence of parameter uncertainty on option pricing and risk management,this paper proposes a Bayesian approach to perform statistical inference on the liquidity-adjusted option pricing models,and empirically analyzes the theoretical models and numerical algorithms based on market data.The main work of this paper is summarized as follows:First,this paper proposes a Bayesian approach to perform statistical inference on European option pricing model with stock liquidity-adjustment and the empirical research.Under the liquidity-adjusted Black-Scholes model,the stock price dynamics are derived under the risk-neutral measure by Esscher transformation.Combining the likelihood function and the prior information from existing literatures,the posterior densities of model parameters are derived by Bayesian formula.Then,based on the liquidity-adjusted pricing formula of European call option,the posterior density for the option price is derived by using nonlinear transformation.The posterior inferences on model parameters and the option price are performed based on the Metropolis-within-Gibbs sampling algorithm.Finally,an empirical application to S&P 500 index option is illustrated based on the theoretical model and numerical algorithm.The empirical results show that the Bayesian statistical method performs better than classical statistical method in parameter estimations and option pricing,especially for near-the-money options and shorter term options.Second,this paper proposes an Exchange option pricing model with stock liquidityadjustment,and extends the research on the issue of multi-asset option pricing in imperfectly liquid market.The dynamics of underlying stocks prices are described by the liquidityadjusted Black-Scholes model and the correlation of the stocks prices are considered.Then,the dynamics of the stocks prices are derived under the risk-neutral measure by Esscher transformation.And the pricing formula of Exchange option with stock liquidity-adjustment is derived by Esscher measure transformation and the change of numeraire.Furthermore,combining the MCMC numerical algorithm,a Bayesian statistical approach is proposed to estimate the model parameters and perform posterior inference on the option price.Finally,the effect of stock liquidity on the Exchange option price is analyzed through numerical experiments.Comparing with Black-Scholes model,the numerical results show that the effects of stock liquidity on Exchange option price and the comparative statics are significant.This thesis provides the theoretical model and empirical methods for studying the effect of stock liquidity on the pricing of Exchange options.It extends the research on single-asset option pricing with liquidity to the research on multi-asset option pricing with liquidity,and provides a general method for the research on multi-asset option pricing with liquidity.Third,this paper considers the effect of stock liquidity on the option hedging strategy and proposes a theoretical model and empirical method for hedging the Value-at-Risk(Va R)of imperfectly liquid stock by investing in the put option with liquidity-adjustment,and extends the research on hedging the risk exposure to stock price based on put options by minimizing the Va R.In an incomplete and imperfectly liquid market,a closed-form pricing formula of European put option with stock liquidity-adjustment is obtained by Esscher measure transformation and martingale pricing principle.Then,under the limited hedging cost,an expression for the minimal Va R of the portfolio is derived by hedging the risk exposure to stock price based on the put option with liquidity-adjustment.Thus,an optimal hedging strategy which minimizes the Va R is deduced by determining an optimal strike price and the hedging ratio for the put option.Finally,in a numerical experiment,the random walk chain Metropolis-Hastings algorithm is implemented to simulate the posterior distributions of the model parameters.Moreover,the posterior inferences are performed for the model parameters,option price,optimal strike price and the minimal Va R.The numerical results show that the risk hedging strategy with liquidity-adjustment differs from the hedging strategy based on Black-Scholes model.The effect of the stock liquidity on risk hedging strategy is significant.Fourth,this paper extends the application of Bayesian statistical method in option pricing and risk management.In view of the flexibility and advantages of Bayesian statistical methods in model inference,this paper proposes a Bayesian approach to study the issues of option pricing with liquidity-adjustment.To the best of our knowledge,it is the first time to study the liquidity-adjusted option pricing model under Bayesian statistical method.Moreover,under Bayesian statistical method,this thesis extends the research on hedging the risk exposure to stock price based on put options,and it provides a new perspective to evaluate the Va R of the portfolio based on the posterior distribution,which enriches the calculation method of Va R.Comparing with traditional method,the proposed method allows for the influence of prior information and parameter uncertainty on option pricing and risk hedging strategy.Unlike existing literatures usually providing only a point estimation,this paper provides more information about the optimal strike price and the minimal Va R from a probabilistic perspective.These results are useful for financial institutions and investors with different risk preferences to make better decisions.