We use a path integral approach for solving the stochastic equations underlying the financial modeling of derivative security pricing. The path integral methodology is well known in the field of quantum mechanics and defines expectation values as functional integrals or sums over histories (paths) of the quantum or stochastic dynamical system. Such integrals are a limit of the sequence of finite-dimensional multiple integrals obtained discretizing the time interval under consideration, which corresponds to the time to expiration in the case of financial derivatives. In essence, a continuous stochastic process is specified by its probability density functional rather than its evolution law (a stochastic differential equation, SDE). The Feynman-Kac formula guarantees the equivalence of the two formulations.

Well known Monte Carlo (MC) and Quasi Monte Carlo (QMC) numerical methods for the calculation of conditional expectation values on stochastic processes are normally seen as devices for generating averages over appropriate random paths. In the framework of the path integral approach they are needed to generate appropriate discretized multivariate probability density functions. While MC and QMC are essentially the only existing numerical methods for high dimensional problems (corresponding to derivatives with many underlying securities), they suffer from slow convergence properties. In the case of low dimensional problems many techniques are available, either based on integrating by finite differencies the partial differential equation (PDE) corresponding to the SDE at hand or on a drastically simplified assumption on the transition probabilities (i.e. binomial methods).

We use an alternative method for the numerical computation of the path integral formulation of the stochastic financial problem which is elegant, easy to extend for path-dependent and American options, amenable to a parallel implementation and with good convergence and stability properties. The Green Function Deterministic Numerical Method (GFDNM) relies upon approximating the transition probabilities for the discretized time steps, and computing the integrals by standard numerical quadrature over a discretized grid. In fact, the transition probability represents the Green function of the PDE corresponding to the SDE underlying the evolution of the financial security. The conditional expectation values are simply products of the payoff vector by a certain number of transition matrices.

The actual implementation of GFDNM does not require storing full matrices since they can be computed analytically step by step. Linear arrays must be manipulated instead, their size being equal to the number of grid points over the random variables. In contrast with PDE-based numerical formulations, American-style options are easily included in the algorithm by introducing testing conditions after each vector-matrix product; this is possible since such products proceed backward in time.

We present some performance results of a parallel, distributed memory implementation of GFDNM for pricing some real-life financial derivatives contingent upon one to three underlying variables. Our parallel implementation distributes the above-mentioned linear array between the available processors and then proceeds to vector-matrix products which need all-to-all interprocess communication pattern. While the parallel code is highly portable on any parallel platform or LAN network where MPI is available, the adopted approach benefits from a tightly-interconnected, high-speed and low-latency dedicated network, such as the SuperCluster architecture from QSW.

Keywords: Path Integral, Stochastic Equations, Financial Derivatives, Green function, Parallel Computing

Date received: March 1, 1999