The use of saddlepoint approximations in statistics is a well-established technique for computing the distribution of a random variable whose moment generating function is known. In this paper, we apply the methodology to computing the prices of various European-style options, whose returns processes are not the Brownian motion with drift assumed in the Black-Scholes paradigm. Through a number of examples, we show that the methodology is generally accurate and fast.
"Saddlepoint approximations to option prices." Ann. Appl. Probab. 9 (2) 493 - 503, May 1999. https://doi.org/10.1214/aoap/1029962752