Saddlepoint Methods for Option Pricing

The Fourier transform of the density for the logarithm of the stock price has seen numerous …nancial applications. For a theoretical perspective we cite as examples Du¢ e, Pan and Singleton (2000), and Bakshi and Madan (2000). This transform has become a standard calibration engine following the methods of Carr and Madan (1999) who invoked the Fast Fourier transform for its speed. Direct Fourier inversion has also been used and we cite Heston (1993), Bates (1996) and Scott (1997). Though adequate for near money options, and this su¢ ces for most calibration exercises, the method is known to break down for deep out of the money options where it often gives rise to negative prices. Rogers and Zane (1999) suggest the use of classical saddlepoint methods to compute option prices and employ in particular the Lugannini and Rice (1980) approximation as developed in Daniels (1987) and extensively studied in Jensen (1995). They consider mainly a Gaussian base density but suggest that one may follow Wood, Booth and Butler (1993) for non-Gaussian bases. Working with a Gaussian base these methods are also used by Xiong, Wong and Salopek (2005) for a variety of models with stochastic rates and volatilities. These applications of classical saddlepoint methods are used to compute the probability that the stock is in the money for the risk neutral probability and the reweighted probability when the stock is itself taken as a numeraire. Thus two saddlepoint computations are involved in constructing one call option price. We recognize following Madan, Roynette and Yor (2008) that the call price is itself quite generally a complimentary probability itself. Hence one should be able to apply saddlepoint methods directly in one step to determine the call price. However, even in the classical Black Scholes case, this density is not Gaussian and the use of a Gaussian base is not exact. It turns out that in the Black Scholes case the density re‡ected in call prices as a complementary probability is the density of Gaussian variable less and independent exponential. This leads us to select for a non-Gaussian base the convolution of a normal