Research On Several Option Pricing Models Under Jump Process

The financial derivatives market has achieved rapid development in the past few decades.As a basic financial derivative tool for financial institutions,enterprises and investors to manage risks,allocate assets and implement arbitrage,option are playing an increasingly important role in financial derivatives market transactions.Therefore,it is very important to conduct pricing research on option.Most of the traditional option pricing theories are based on the perfect assumption of the Black Scholes option pricing model(BS model).However,a lot of empirical evidence shows that there are many characteristics of financial asset prices that cannot be explained by BS model,such as spike and thick tail,volatility smile,leverage effect,and clustering characteristics.In order to overcome the defects of the BS model,many scholars have extended the BS model,so that the BS model has been continuously improved and developed.One of the expansion directions is to construct jump diffusion models driven by Brownian motion and jump processes,and it is found that the model with jump process can effectively describe many financial time series characteristics,such as spikes and thick tails,volatility smiles and so on.Therefore,in order to consider the option pricing model that is more in line with the market reality,this thesis studies the option pricing model in the following aspects under the framework of the jump diffusion model:First,this thesis considers that the underlying asset is subject to an option pricing model with Markov regime switching double stochastic volatility,stochastic interest rate and jump.In the proposed model,the stochastic volatility of the underlying asset follows the CIR process,the stochastic interest rate follows the OU process,and the jump follows the compound Poisson process,and the magnitude of the jump obeys a normal distribution.This thesis have obtained the analytical pricing formula of European options by using Fourier Transform.Numerical experiments illustrate that the option prices and the implied volatility curves under different regimes vary clearly,and the effects of regime-switching and jumps on the option price differ.Second,this thesis considers an option pricing model with stochastic intensity,stochastic interest rate,and regime switching stochastic volatility for the underlying asset under the double exponential jump.In the considered model,the stochastic interest rate of the underlying asset and the stochastic intensity of jumps obey the CIR process,and the magnitude of jumps obeys an asymmetric double exponential distribution.In addition,the average level of stochastic volatility is modulated by Markov chains.Through the Fourier transform,this thesis obtain the explicit solution of the European option.Through numerical simulations,the proposed model is capable of portraying the smile of volatility;The option price during the economic recession period is higher than the option price during the economic prosperity period;The price of an option with a regime-witching model is higher than that of a no regime-switching model,and this difference is more obvious in the part of out-ofthe-money option;The jump model produces a greater implied volatility than the no jump model.Compared with the normal jump model,the asymmetric double exponential jump model can better fit the asymmetric implied distribution.Third,this thesis considers the option pricing model of stochastic interest rate and random correlation under the jump diffusion model.It is assumed that the jump of the underlying asset is driven by a self-exciting Hawkes process,the stochastic interest rate obeys the OU process,and the underlying asset and its volatility are randomly correlated.For the underlying asset model constructed,through the Fourier transform,this paper obtain an analytical expression for the price of European options.In the numerical analysis,this paper discuss the impact of self-exciting Hawkes jump and stochastic interest rate on option prices and the shape of the implied volatility surface generated by this model.The results show that the proposed model in this paper can produce flexible pattern of the implied volatility surface,and has the ability to simultaneously fit the implied volatility smile and implied volatility skew commonly observed in the option markets.Fourth,this thesis investigates the option pricing model with stochastic interest rate and counterparty risk under jump diffusion.Assume that the value of the underlying asset with stochastic volatility and the value of the counterparty asset are randomly correlated,and both the value of the underlying asset and the value of the counterparty asset have jumps,and their jumps are driven by the self-exciting Hawkes processes.The semi-analytical expression for the vulnerable European option price under the proposed model is obtained.In the numerical analysis,the option price is calculated by the fast Fourier transform method,and find the option price under the constructed model is higher than the constant correlation model,the constant interest rate model and the Poisson jump model.Moreover,option price decreases with the bankruptcy cost ratio and the default boundary;option price increases with the underlying initial price and the counterparty asset initial value.Fifth,this thesis considers an option pricing model with stock sentiment and option sentiment under jump diffusion.It is assumed that the arrival of the underlying asset jump follows a Poisson process,the magnitude of the jump follows a normal process,and both stock sentiment and option sentiment follow an OU process.An analytical formula for European option pricing is obtained for the constructed model.Using the SSE 50 ETF call option data for empirical research,and comparing with the geometric Brownian motion model,the geometric Brownian motion model with jumps and the Heston model,this thesis found that in the stable sample period,volatile sample period and full sample period,the fitting and prediction errors of the model in the SSE 50 ETF call option price are the smallest,and the fitting and prediction errors in the implied volatility are also the smallest,indicating that the model in this paper in the stable period,non-stationary period,and both stable and non-stationary periods,the fitting and predictive ability of option prices,and the fitting and predictive ability of the implied volatility are better than the comparative model.