The Exponent Expansion: An Effective Approximation of Transition Probabilities of Diffusion Processes and Pricing Kernels of Financial Derivatives

Abstract

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. Several examples are presented, and the application of these results to increase the efficiency of numerical approaches to derivative pricing is discussed.

Approximation Methods for Inhomogeneous Geometric Brownian Motion

“…Here and in the following, in order to ascertain the accuracy of the EE we have computed the transition density, and AD prices using numerical solution by means of the standard Crank-Nicholson method; see, e.g., Andersen and Piterbarg (2010), of the PDEs (2.6), and (2.13), respectively. As previously observed for other diffusions (Capriotti (2006)) the exponent expansion is characterized by a remarkably fast convergence by including successive terms of the approximation. As illustrated in Figure 2, similarly to the case of the transition probabilities, the exponent expansion provides a remarkably good, and fast converging approximation of the AD prices for financially sensible parametrizations, and for a sizeable value of the time step T .…”

“…where δ(·) is the Dirac-delta function. The EE, originally introduced in chemical physics by Makri & Miller (1989), was introduced to finance by Capriotti (2006) and applied to the calculation of transition probabilities and AD prices of several diffusion processes, including Hull & White (1990), Cox et al (1985) and Black and Karasinski (1991), demonstrating a remarkable congruence with the exact analytical solutions available for these models. By the EE, transition probabilities and AD prices are obtained as a power series in the expiry date T of the financial claim, which becomes asymptotically exact if an increasing number of terms is included.…”

An Effective Approximation for Zero-Coupon Bonds and Arrow–debreu Prices in the Black–karasinski Model

“…This technique, originally introduced in Chemical Physics by Makri & Miller (1989), was introduced to Finance by Capriotti (2006) and applied to the calculation of transition probabilities and AD prices of several diffusion processes, including Hull & White (1990), Cox et al (1985), and Constant Elasticity of Variance from Cox & Ross (1976), demonstrating a remarkable agreement with the exact analytical solutions available for these models.…”

Approximations of Bond and Swaption Prices in a Black–karasiński Model

“…Prices of zero-coupon bonds were represented by means of a Taylor series expansion with coefficients represented in a closed form, obtained via specific recurrent relations. Another approximation concept, which originated from chemical physics under the name of "the exponent expansion", was introduced to Finance by Capriotti [5] and applied to the calculation of transition probabilities and Arrow-Debreu prices, for several diffusion processes. This approach appeared to provide very accurate approximations and was further pursued by Stehlikova, Capriotti [11] in the context of the BK model.…”