Research On Some Problems Of Option Pricing Based On Fractional And Mixed Sub-Fractional Brownian Motion

With the vigorous development of financial derivatives market,people have made a lot of achievements in the pricing of options,one of the basic tools.The classical B-S formula is the most widely used among that.However,the assumptions of BS model are more and more difficult to meet at the same time in the real market.In this paper,we modify and improve the classical BS model.Some problems of option pricing based on fractional and mixed-sub fractional Brownian motion are mainly discussed from three aspects.The first aspect is to improve the standard Brownian motion to fractional Brown-ian motion,modify the random interest rate model in the fractional Brownian motion environment,consider the impact of default risk at the same time,and then deduce the pricing formulae of European call options.A large number of empirical studies have found that the return on assets shows the characteristics of”high peaks&heavy tails”compared with the normal distribution,and the change of asset prices also shows self-similarity and long-term correlation.Therefore,it is not very appropriate to fit asset prices with standard Brownian motion,but fractional Brownian motion can fit these characteristics effectively.Moreover,interest rate is no longer a fixed constant,and its random changes are becoming more and more obvious.In addition,following the more and more complex and changeable of the financial market,default risk has become an influencing factor that can not be ignored in the study of option pricing.Therefore,we modify the hypothesis of standard Brownian motion to fractional Brownian mo-tion,consider the case that the interest rate is a random process instead of a constant,and combine the influencing factor of default risk.Taking the pricing of standard Eu-ropean call option as an example,we use the stochastic differential equation driven by fractional Brownian motion to describe the company's asset value and underlying asset price.At the same time,we also improve the Ho-Lee and Vasicek stochastic in-terest rate models in the fractional Brownian motion environment,and then establish the company's asset value models under the fractional stochastic interest rate.Using the actuarial method of option pricing and the stochastic analysis theory of fractional Brownian motion,the pricing formulae of European call fragile option are deduced and proved,which avoids the shortcomings of most traditional methods,such as relatively complicated research process and lack of obvious financial meaning.The second aspect is to continue to improve the fractional Brownian motion in-to mixed sub-fractional Brownian motion,and study the pricing of geometric average Asian options in the environment of mixed sub-fractional Brownian motion.Asian op-tions have better risk hedging and more market competitiveness.Among the existing research results of Asian option pricing,the improved fractional BS model is still the most widely used.However,fractional Brownian motion is not a semi-martingale,and arbitrage opportunities will appear when it is used to describe the price of financial assets,so we continue to improve it to mixed sub-fractional Brownian motion.When Hurst index>3/4,the mixed sub-fractional Brownian motion is not only a semi-martingale,but also has good properties such as self-similarity,long-term correlation and non-stationary increment,which can ensure that there is no arbitrage opportunity in the market and the characteristics of the stock price process are well described.We assume that the stock price process is a stochastic differential equation driven by the mixed sub-fractional Brownian motion of Hurst index>3/4.The mixed sub-fractional It^o formula is derived through the fractional It^o formula based on the s-tochastic differential equation theory.Then we find the stochastic differential equation satisfied by stock price and its solution is further obtained,so as to deduce the ana-lytical formula of geometric average Asian option pricing formula in the environment of mixed sub-fractional Brownian motion.In addition,combining power option with geometric average Asian option,the analytical formula of power geometric average Asian option pricing formula in mixed sub-fractional Brownian motion environment is derived and proved.We also do Monte Carlo simulation.The results show that under the assumption of mixed sub-fractional Brownian motion,the stock price process is well simulated,and the fitting effect is better than the standard Brownian motion.The third aspect is to continue to modify the assumption that volatility is a con-stant in BS model,improve the threshold generalized autoregressive conditional het-eroscedasticity model to the general threshold generalized autoregressive conditional heteroscedasticity model,so as to effectively model the asymmetrical random volatility of assets.Asset volatility refers to the variance of the return on the underlying asset,and accurately describing volatility is a crucial link for the study of reasonable option pricing.A large number of empirical studies have found that volatility is a random variable rather than a constant in general.And the asymmetry of volatility is a typical fact widely existing in the research of stock options.We modify the zero threshold in the threshold generalized autoregressive conditional heteroscedasticity T-GARCH model to a positive or negative non-zero threshold,and generalize the general threshold generalized autoregressive conditional heteroscedasticity GT-GARCH model,deduce and prove the condition that the model{h_t}has a unique stationary solution,and give the specific formula of the solution.The existence conditions of the stationary solution are further relaxed,and the sufficient conditions for the second-order stationarity and the existence of higher-order moments of the process{?_t}are given.The parameter-s and thresholds of the model are estimated by using the quasi-maximum likelihood estimation method combined with the direct search method.We also do Monte Car-lo simulation for the quasi-maximum likelihood estimation combined with the direct search method.The results show that the method has good performance.Finally,by applying the model to the daily return and volatility series of Shanghai stock exchange with significant leverage effect,we find that the asymmetric model with general thresh-old has better performance than the traditional model with zero threshold.Since the model can more effectively fit the asymmetric volatility characteristics of stock process,it provides a reference for further research on option pricing.