Pricing Vulnerable European Options In A Mixed Environment Of Geometric And Fractional Brownian Motion

As the continuous opening and development of China's financial market in recent years,the domestic options market has a greater practical significance.With the footsteps of the new year in 2015,the SSE 50 ETF options landed Shanghai Stock Exchange.This means that the initial 24 years,China mainland stock market ushered in the "era of option".After that,some other types of options will continue to enter the financial market,and the transaction scale will continue to increase.It also appears that the OTC market,due to the lack of supervision mechanism,when options traded in the OTC market,the supervision of similar liquidation options short to fulfill its obligations at maturity,the option will lead to multiple square should withstand the market risk and credit risk,and this will inevitably lead to the emergence of default in the transaction process.Until now,the theoretical research on market credit default risk has been relatively mature,but it is relatively less in the field of market pricing for the risk of default,and less the conditions are being taken into account,so it cannot be a good description of the option price changes.Therefore,this paper considers the introduction of the jump diffusion process,stochastic interest rate and mixed Brownian motion.Considering the combination of the simple model and the structural model,the pricing formula is perfected on the basis of the past,and the accuracy of the pricing formula is enhanced.This paper mainly studies the following questions:(1)Because the stock's price in the real market cannot fully comply with the geometric Brown motion,the distribution of financial assets return is characterized by "fat tail",and the stock's price change is not random walk,it shows long-term different correlation and self-similarity in different times.There is a certain gap between these features and the standard Brown movement,and Fractional Brownian Motion's campaign has self-similarity and long-range dependence,more suitable for the characteristics of the financial market.Therefore,in this paper,on the assumption that the stock price follows the geometric and fractional Brown motion,we work out the price of vulnerable options.(2)This paper introduces the jump diffusion process,but the general Girsanov's theorem cannot be applied in this case.Hence,in this paper,we first study the expression formula of the jump process in the new measure after the measure transformation.After the transformation,we can use the Girsanov transform to price the option under the new measure.(3)In the actual financial market,with the increase of OTC,interest rate and default behavior have a strong randomness.So,based on the stochastic interest rate and stochastic default intensity,this paper makes a study on the pricing of the European vulnerable option under the mixed Brown model.In this paper,the explicit solution of the pricing of European vulnerable options is obtained by martingale measure transformation.Through the numerical experiment,the paper compares the pricing formula with the classical Black-Scholes pricing formula,and the results show that the option pricing formula in this paper is more in line with the characteristics of the actual financial market.