The Double Exponential Jump Diffusion Process The Optimal Stopping Problem,

Since F.Black,M.Scholes and R.Merton made a major breakthrough in the pricing of financial derivatives,rapid progress has been gained in the theory and application of mathematical finance.With the deepness of study in financial practice,especially,from the serious impact concerning recent rare financial events and many problems of financial reform,etc.,the Black-Scholes model based on the Brownian motion is found to be not appropriate to capture the law of modern financial market.In 1976,Merton firstly established a jump-diffusion model where the jump risks are unsymstematic and the jump magnitude of the log of the asset price is assumed to be a normal distribution,and consider option pricing of European option.Hereafter Merton'work,many research achievements have been gained.However,despite the success of the Black-Scholes and Merton's model,recent empirical works indicate the inability of such two models to caputure the true features of asset fluctuating,and suggest:(1) the jump risks can not be ignored,and may represent non-asymmetric leptokurtic features and "implied volatility smile".In recent decades,many research modified Black-Scholes formula by explaining its two limitation,but the common problems is that it is difficult to obtain analytical solutions to option pricing under these models.At the same time,these models did not properly reflect the high peak and non- symmetrical features,especially the high peak feature.In 2000,Kou proposed double exponential jump-diffusion model,which is very simple.It consists of two parts,a continuous part driven by a geometric Brownian motion,and a jump part,with the logarithm of jump sizes having a double exponential distribution and the jump times corresponding to the event times of a Poisson process.The most important feature of the model is that it can generate a highly skewed and leptokurtic distribution.In addition,the model can lead to analytical solutions to the pricing for European options and exotic options.Accordingly,it is reasonable to do some research under this model,which is of great practical and theoretical significance.In this paper,making use of the theory of probalility,stachastic analysis and optimal stopping,we give the analytical solutions for a class of optimal stopping problems under this model,which offer good guidance for people to make dicisions to some extent,especially for choosing the best time to exercise American options.The proofs associated with them are also presented in this paper.