Research On Asian Options Pricing Based On Generalized Mixed Fractional Brownian Motion

As the complexity of the financial market increases,the traditional standard European option and American option are difficult to meet the special needs of investors.To meet the growing investment needs of investors,financial institutions have designed many new options with flexible trading methods,among which Asian option is the typical one.The price of Asian options depends not only on the price of the underlying asset on the maturity date but also on the change of the underlying asset price throughout the validity period.It is a strong path-dependent option,which can avoid internal transactions like artificially hyping the price.Because the price of Asian option is generally cheaper than that of standard European option.it has gradually become one of the most active options of the trading market for financial derivatives.Since the appearance of Asian options,the pricing research of Asian options has become one of the hot spot in the field of finance,and its pricing theory has developed mature.At the beginning of the birth of Asian options,most of the researches on its pricing were based on the traditional B-S model.However,there is a certain deviation between the asset change process described by standard geometric Brownian motion and the actual operating results.The peak and thick tail effect and the long-range dependence of assets cannot be described by geometric Brownian motion.Scholars have gradually introduced fractional Brownian motion and mixed fractional Brownian motion with great achievements.In this thesis,fractional Brownian motion is further extended to Generalized Mixed Fractional Brownian Motion(GMFBM).Using the GMFBM model,the pricing of Asian options,Asian power options,and Asian rainbow options have been studied.GMFBM is a stochastic process composed of linear combinations of several fractional Brownian motions.Through the setting of the Hurst index,GMFBM can better reflect the peak and thick tail phenomenon,and long-range memory,and long-range aperiodic statistical dependence of assets in the change of assets.Besides,it can truly realize no arbitrage in the financial market.On the other hand,through the combination of coefficient and Hurst index,generalized mixed fractional Brownian motion can degenerate into standard Brownian motion,fractional Brownian motion,and mixed fractional Brownian motion.The option pricing model based on GMFBM can unify the previous results to a certain extent,which has a certain theoretical significance and practical value.The uncertainties in the real financial market include not only randomness but also fuzziness.Introducing fuzziness into option pricing is one of the research hot spots at present.After getting the option pricing formula based on the GMFBM model,this thesis further focuses on option price in a fuzzy environment,and get the fuzzy pricing formulas of European option and Asian power options based on GMFBM model in fuzzy environment.Based on the Asian option pricing formula in the fuzzy environment,the fuzzy interval of option price determined by the fuzzy pricing formula can be obtained through numerical calculation,which has a certain guiding significance for investors in investment decision.The main research contents of this thesis are as follows:⑴ The pricing formulas of Asian options and Asian power options based on GMFBM are established.In terms of the fractional dimension It? formula and the principle of arbitrage-free,the partial differential equation satisfied by the price of Asian options is acquired.Furthermore,the analytical solution of Asian options and Asian power options price are obtained under fixed strike price and floating strike price respectively,and the corresponding put-call parity formulas are gained.⑵ The pricing formula of the Asian rainbow options based on the GMFBM model is obtained.Considering that the dynamic changes of N underlying assets accord with the generalized mixed fractional Brownian motion,the partial differential equation satisfied by the Asian minimum call rainbow option is obtained by using the multidimensional fractional It? theorem and the principle of arbitragefree.Because the coefficient of the equation is time-dependent,it can not be solved directly through the solution of a conventional partial differential equation.Based on the Stulz’s solution,the equation is transformed into an equation with a coefficient independent of time by using variable transformation.In the end,the analytical solution of the equation is received.⑶ Fuzziness widely exists in the real world.Only single randomness fails to accurately describe the operation law of assets.Based on the option pricing formula obtained from the GMFBM model,this thesis fuzzifies the parameters in the pricing formula by using interval 1 trapezoidal fuzzy number,and get the fuzzy pricing interval of the option price.⑷ Interval 1 fuzzy sets have complete algorithms,but some information will be missing.This thesis further discusses the real option pricing based on interval type 2 trapezoidal fuzzy numbers.Taking the investment project of medical equipment as an example,given the irreversibility of project investment,it is necessary to evaluate whether to invest in advance.