Although Brownian motion and normal distribution have been widely used in finance, two puzzles have got much attention recently; namely the leptokurtic feature that empirically the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical abnormality called ``volatility smile' in option pricing. To incorporate these, the double exponential jump diffusion model was proposed before, in which the price of the underlying asset is modeled by two parts, a continuous part driven by Brownian motion, and a jump part with the jump size having a double exponential distribution. The current paper shows that, in addition to the leptokurtic feature and ``volatility smile', the model is simple enough to produce analytical solutions for a variety of option pricing problems, including barrier and perpetual American options, in terms of the Laplace transform and the $Hh$ function. The numerical implementation will also be discussed. As a by product, this also gives a closed form solution for the first passage time of the jump diffusion processes, which might be of independent interest in applied probability.
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