Pricing exotic options in a path integral approach

Abstract

In the framework of the Black–Scholes–Merton model of financial derivatives, a path integral approach to option pricing is presented. A general formula to price European path-dependent options on multidimensional assets is obtained and implemented by means of various flexible and efficient algorithms. As an example, we detail the cases of Asian, barrier knock out, reverse cliquet and basket call options, evaluating prices and Greeks. The numerical results are compared with those obtained with other procedures used in quantitative finance and found to be in good agreement. In particular, when pricing at-the-money and out-of-the-money options, the path integral approach exhibits competitive performances.

Acknowledgments

We would like to thank Bernard Lapeyre for suggesting to us that we compare with the Brownian bridge and stratification technique, as well as two anonymous referees for some pertinent and useful remarks concerning the layout and structure of the paper. We acknowledge partial collaboration with Francesca Rossi at the early stage of this work. We wish to thank Carlo Carloni Calame for helpful assistance with software installation. The work of GB is partially supported by STMicroelectronics.

Notes

†For simplicity, we have included the discount factor exp{−rT} in the definition of the payoff.

‡We refer to Clewlow and Strickland (1998) and Glasserman (2003) for a review of standard Monte Carlo variance reduction techniques.

§By definition, p(z′, t′|z, t) is such that the probability for Z(t′) taking a value in the D-dimensional hyper-cube dz′ centred on z′, conditional on Z(t) = z, is p(z′, t′|z, t)dz′.

†Just take as reference the deterministic integration with the rule of thumb of setting at least 6 integration points for each dimension and 1000 MC paths for each end point, i.e. 6 ^D × 10³ total samples.

‡It is important to recall that, if we perform integration on an interval whose spot values are too low, we will have an underestimation of the price.