Path Dependent Option Pricing: The Path Integral Partial Averaging Method

Abstract

In this paper I develop a new computational method for pricing path dependent options. Using the path integral representation of the option price, I show that in general it is possible to perform analytically a partial averaging over the underlying risk-neutral diffusion process. This result greatly eases the computational burden placed on the subsequent numerical evaluation. For short-medium term options it leads to a general approximation formula that only requires the evaluation of a one dimensional integral. I illustrate the application of the method to Asian options and occupation time derivatives. *

A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: The Exponent Expansion

“…In fact, it is defined by Eq. ( 16) which is decoupled from the recursive system (17). Indeed, it is possible to obtain the same result for the exponent expansion by expressing the transition density as…”

“…3). In Sections 2.3 and 3.3, we illustrate the exponent expansion through the application to the Vasicek, the Cox-Ingersoll-Ross, and the Constant Elasticity of Variance models, and in Section 4 we discuss its application to Monte Carlo and deterministic numerical methods within the path integral framework [14,15,16,17,18,10]. Finally, we draw our conclusions, and we discuss future developments in Section 6.…”

Pricing exotic options in a path integral approach

“…Comparisons with standard Monte Carlo simulations, as well as with the results of other numerical techniques known in the literature, are presented. Related attempts to price options and, in particular, exotic options, using the path integral method can be found in Baaquie (1997), Linetsky (1998), Bennati et al (1999), Matacz (2000a), Rosa-Clot and Taddei (2002), Baaquie et al (2004), Dash (2004) and Lyasoff (2004).…”

“…Comparisons with standard Monte Carlo simulations, as well as with the results of other numerical techniques known in the literature, are presented. Related attempts to price options and, in particular, exotic options, using the path integral method can be found in [10,11,12,13,14,15,16,17].…”

A path integral way to option pricing

“…The aim of this work is to provide a contribution to the problem of efficient option pricing in financial analysis, showing how it is possible to use path integral methods to develop a fast and precise algorithm for the evaluation of option prices. The path integral method, which traces back to the original work of Wiener and Kac in stochastic calculus [7,8] and of Feynman in quantum mechanics [9], is today widely employed in chemistry and physics, and very recently in finance too [10][11][12][13][14], because it gives the possibility of applying powerful analytical and numerical techniques [15]. Following recent studies on the application of the path integral approach to the financial market as appeared in the econophysics literature (see Refs.…”