4. Saddlepoint Approximation Formulas for Pricing Options

Though the saddlepoint approximations have been commonly used in computing the tail probability of a random variable whose cumulant generating function (cgf) is known, the use of saddlepoint approximations for option pricing is relatively thin in the literature. Rogers and Zane (1999) first introduce the saddlepoint approximation method to price options under the exponential Lévy models. Nice analytic tractability is exhibited since the cgf of log stock price under a Lévy model is known in closed form. By expressing the option price as the difference of two tail expectations under the risk neutral measure and share measure, they apply the Lugannani-Rice formula 2.15 to obtain the corresponding saddlepoint approximation of the European option price.