Option Pricing And Statistical Simulation Analysis Under Mixed Fractional Heston-CIR Model

As an emerging product of the contemporary financial derivatives market,option is becoming an indispensable part of the financial market.Option pricing is one of the most popular topics among scholars and investors.In 1973,the emergence of B-S(Black-Scholes)formula provided strong theoretical support for the pricing of financial derivatives such as options and futures,which attracted extensive attention from academic circles.However,the return rate of the classical B-S model is inconsistent with the empirical characteristics of the real return,such as non-normality,non-linearity and non-independence,so scholars mainly improve it from two aspects.On the one hand,fractional Brownian motion model is proposed to describe the long-term dependence of financial assets.On the other hand,in order to describe the characteristics of the underlying assets,such as "peak thick tail",discontinuity and "volatility smile",jump diffusion model,random interest rate and random volatility model model are proposed.Although these models can fit the real financial data well,there are still some differences compared with the real prices.In order to describe the variation of option price better,based on the Heston model,mixed Heston-CIR model with jump and mixed fractional Heston-CIR model are proposed.There are three parts as follows in this article.In the first part,European call option pricing under Heston-CIR mixed model with jump is solved.Based on the classical Heston model,a hybrid stochastic volatility and stochastic interest rate(Heston-CIR)model with jump is constructed by considering the impact of random interest rate and emergency on the price of financial products.Firstly,the stochastic differential equation satisfied by the underlying asset is transformed to the forward measure by measure transformation.Then,the European option price under the model is solved by the fast Fourier transform method.In the second part,American put option pricing under mixed fractional Heston-CIR model is studied.First of all,in order to reflect the volatility smile and the long dependence of the underlying asset,a mixed fractional Heston-CIR model is constructed to describe the underlying asset price based on the fractional market theory.The linear combination of the standard Brownian motion and the fractional Brownian motion is used to replace the Brownian motion.Secondly,the uniqueness and existence of the solutions to the stochastic differential equations satisfied by the underlying asset price and interest rate are proved respectively.The strong convergence of the Euler scheme discretization of the interest rate equation is also proved.In the third part,the results of simulation analysis are given.Firstly,the historical data of the underlying assets are selected for descriptive statistical analysis.Secondly,the real data are compared with the underlying asset price paths under different models.Finally,the least squares Monte Carlo algorithm is used to obtain the prices of American put options with different expiration dates,and the rationality of the proposed model is proved by numerical simulation.