Path integral Monte Carlo method and maximum entropy: a complete solution for the derivative valuation problem

Abstract

Summary form only given. We propose a combination of the path-integral Monte Carlo method and the maximum entropy method as a comprehensive solution for the problem of pricing of derivative securities. The path-integral Monte Carlo approach relies on the probability distribution of the complete histories of the underlying security, from the present time to the contract expiration date. In our present implementation, the Metropolis algorithm is used to sample the probability distribution of histories (paths) of the underlying security. The advantage of the path integral approach is that complete information about the derivative security, including its parameter sensitivities, is obtained in a single simulation. It is also possible to obtain results for multiple values of parameters in a single simulation. The input to the path-integral Monte Carlo method is the assumed propagator for the stochastic process of the underlying. The path integral method is flexible about the input stochastic process and it can be used for both American and European contracts. Derivative valuation can be viewed as a statistical inference procedure about the underlying stochastic process. In its simplest form it reduces to the computation of implied volatility. It is known that the implied volatility matrix may contain significant variations across strike prices and contract maturities. This implies that parametrization of the underlying process via single volatility parameter is inconsistent with market data. Instead, we formulate an approach which allows one to generate a fully consistent estimate of the complete propagator for the underlying.