A path integral way to option pricing

Abstract

An efficient computational algorithm to price financial derivatives is presented. It is based on a path integral formulation of the pricing problem. It is shown how the path integral approach can be worked out in order to obtain fast and accurate predictions for the value of a large class of options, including those with path-dependent and early exercise features. As examples, the application of the method to European and American options in the Black-Scholes model is illustrated. A particularly simple and fast semi-analytical approximation for the price of American options is derived. The results of the algorithm are compared with those obtained with the standard procedures known in the literature and found to be in good agreement.

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

“…(2.7), and equating equal powers of ∆t leads in a straightforward way to a decoupled equation for the order zero in ∆t giving 9) and to the following set of recursive differential equations:…”

Hamiltonian and potentials in derivative pricing models: exact results and lattice simulations

“…Related attempts can be found in the literature [3]. We assume a discretization of the time to maturity τ in intervals ǫ = τ /N, with N an arbitrary (large) integer.…”

Review of quantum path integrals in fluctuating markets

“…In general we do not have an explicit expression for the transition probabilities for small time steps, However using a Gaussian transition probability 13,21,22 …”

“…The method becomes useful when one wants to calculate the transition probabilities, which are associated to a given stochastic process. In this framework finite time transition probability can be written as a convolution of short time transition probabilities 11,13,21,22 Once an explicit method is written down to calculate the transitions probabilities it is possible to calculate the expectation values of the quantities of financial interest, given by the form i.e.,…”

“…This formalism can be extended to other models beyond the dynamics of geometric Brownian motion. In Ref., 13 Montagna Et. Al.…”