Pricing Exchange Options Under Jump-diffusion Model With Double Stochastic Volatility

With the rapid development of financial market and the continuous innovation of financial derivatives,financial products are increasingly abundant and financial market is increasingly active.As an important financial derivative,option is widely used in hedging and risk management,which plays a positive role in enhancing the anti risk ability and maintaining the stability of financial market.Option pricing is one of the core contents in the field of financial mathematics and financial engineering,reasonable option price is very important,which has a great significance to play the market role of option and improve the function of financial market in the complex and changeable financial market environment.Exchange option is an option contract that exchanges one kind of risk asset with another,it is widely used in asset swap and equity investment.With the growth of investment demand and the need of risk management,exchange options are constantly innovating,and many new derivatives such as power exchange options,Asian exchange options and barrier exchange options have been constructed.The pricing of exchange options and their derivatives is studied based on the traditional geometric Brownian motion model,which has some defects in describing the volatility characteristics of the underlying asset price and describing the characteristics of the discontinuous changes of the asset price,so that there are many limitations in practical application.Therefore,the traditional pricing model is appropriately improved,and a more general market model is established,which can capture the uncertainty changes of the financial market better,so that the pricing results are more in line with the real situation of the financial market.In this paper,a jump-diffusion model with double stochastic volatility for underlying asset price is established to study the pricing of discrete Asian exchange options,digital power exchange option and discrete barrier power exchange options,and analyze and discuss the pricing results.Firstly,Two jump diffusion models with compound Poisson process are used to describe the price movements of two kinds of risky assets in the market,and Heston stochastic volatility model is used to describe the change of underlying asset price volatility,and the jump diffusion affine model of stochastic volatility is established.Then,the analytic expression of joint characteristic function is derived by using Feynman-Kac theorem and partial differential integral equation,The pricing formulas of discrete Asian exchange options,digital power exchange option and discrete barrier power exchange options are derived by using Girsanov theorem and Fourier inverse transformation.Finally,the Fast Fourier Transform method and the Monte Carlo simulation are used for numerical calculation,the changes of option prices under different market models are compared,and the sensitivity of option price to some main parameters such as jump risk factors and correlation coefficient of underlying asset prices is analyzed.The results show that Asian exchange options can disperse the market risk better,the exchange options with restrictions can agree on the execution conditions in advance to better control the investment risk,and the exchange options with power structure can better adjust the income leverage and increase the flexibility of options as an investment tool and these options are cheaper than the corresponding common stock exchange options.In addition,the jump risk factors caused by emergencies in the financial market have a significant impact on the prices of these exchange options.Investors should pay attention to the sharp changes in the underlying asset prices caused by emergencies,so as to effectively avoid risks.