Pricing Of Discrete Barrier Options Based On Wishart Stochastic Volatility Model

Option is an important tool to realize hedging and risk management,and how to price option reasonably is a very important problem.In recent years,the focus of option pricing theory research is mainly in two directions:one is to study the pricing of various exotic options and construct new options to meet the needs of different investors;The second is to improve various pricing models so as to make option pricing more reasonable.The research work of this paper is around these two directions.For financial and insurance companies,to implement a realistic and effective risk minimization strategy,accurate volatility modeling is a crucial step.Therefore,the pricing model adopted in this paper is Wishart volatility model considering various volatility factors.Barrier option is a kind of price path dependent option,whose price is lower than that of ordinary standard option,so it is favored by investors in financial market and widely used in risk management by investors.We know that in the real financial market,the transaction of financial derivatives is often a discrete situation.it has certain market value and research significance for the research of discrete barrier options pricing.In this paper,we discuss the pricing of European barrier options and Asian barrier options in the discrete-time case when the underlying asset price satisfies the Wishart multi-dimensional s-tochastic volatility model(record as WMS V model.The pricing formulas of discrete European bar-rier option and asian barrier options based on discrete geometric average of asset price are derived by using the random analysis methods and mathematical induction methods such as Ito formula,multi-dimensional joint characteristic function of random variable,Fourier inverse transformation and Girsanov measure transformation,and the approximate pricing formula of European barrier options is given by the multi-dimensional discrete fast Fourier transform(FFT).With the help of Monte Carlo simulation,the approximate price solutions of European barrier options and Asian barrier options with discrete arithmetic mean are obtained.Finally,a numerical example is given by using MATLAB and mathematical calculation programming software.The implied volatility is calculated by dichotomy,and the change rule of implied volatility curve under different volatility parameters is analyzed.The numerical results show that multiple stochastic volatility has more influence on option price than single stochastic volatility,and the effect of fitting implied volatil-ity of WMSV model is better than Heston model.The price of new Asian barrier option under WMSV model is lower than that of single European barrier option and single Asian option;In the WMSV model,various stochastic volatility factors have different significant effects on the implied volatility of European discrete barrier options and the price of discrete geometric average Asian barrier options.When the elements in the correlation coefficient R are all positive,the implied volatility curve fluctuates the most,while when they are all negative,the implied volatility curve fluctuates the least;when the sign of the element value in the mean recovery matrix M changes from negative to positive,the implied volatility decreases,and the sign of the element with small speed changes to positive,the trend of the implied volatility changes is large,and when the sign of the element value in M is both negative or positive,the implied volatility curve is relatively stable The validity of FFT algorithm in calculating option price is verified by comparing FFT algorithm and Monte Carlo simulation method.We use Wishart’s multi-dimensional stochastic volatility model to describe the real and com-plex financial market,and use this model to study the pricing problem of barrier options in the discrete case.The results of this paper can provide reference for investors in the financial market when hedging,the results of this paper can be used for reference in the further study of other path dependent options or American options.