Tempered stable process, first passage time, and path-dependent option pricing

In this paper, we will discuss an approximation of the characteristic function of the first passage time for a Lévy process using the martingale approach. The characteristic function of the first passage time of the tempered stable process is provided explicitly or by an indirect numerical method. This will be applied to the perpetual American option pricing and the barrier option pricing. For the numerical illustration, we calibrate risk neutral process parameters using S&P 500 index option prices and apply those parameters to find prices of perpetual American option and barrier option.