Option Pricing Under The Lévy Jump Model

This article derives option pricing under three models.Firstly,the first model introduces random interest rates(described by the CIR model)and a jump driven underlying asset model(described by the Lévy process),avoiding the risk of negative interest rates.Due to the existence of many random terms,it is difficult to provide a closed-form pricing formula for European options.Inspired by two articles by Merton(1976)and He and Zhu(2017),we combine the two methods to give a European option pricing formula in the form of series summation by using the forward measure transform and the inverse Fourier transform(to our knowledge,this is the first time to use the combined method).And numerical simulations are used to demonstrate that the series is convergent and the speed of convergence is very fast,with an absolute difference of 10.Subsequently,through the empirical analysis,compared with the classical Black-Scholes model,it shows that our model is more consistent with the actual market.The equilibrium model of interest rates has been extended at every level from the Merton model,Vasicek model,Brennan Schwartz model to the CIR model,including non negative interest rates and mean reversibility.However,according to the data simulated by the pure diffusion model,there is still a gap in fitting real data,especially the jump nature of actual interest rates that cannot be fitted.Therefore,the second model of this paper adds a jump item on the basis of CIR interest rate model to build a CIR interest rate model with Lévy jump.The purpose is to simulate some unexpected situations that may occur in the market,such as the collapse of a systemically important company leading to short-term market imbalances.We provide an European option pricing formula written in the form of an infinite series of Black-Scholes-type terms under double Lévy jumps model,where both the interest rate and underlying price are driven by Lévy process.The series solution converges with a radius of convergence,and it is complemented by some numerical to demonstrate its speed of convergence.The third model in this article provides an option pricing formula for European basket stocks based on the second model.