Pricing Path-Dependent Payoff Using Quadrature on Wiener Space

Solution to stochastic differential equation that defines the dynamics of asset prices is interpreted as a function of underlying Brownian motion when permissible, and the expectation values of various functionals of the solution – therefore functionals of Brownian motion (Wiener functionals) – are estimated from numerical approximations to high-dimensional Gaussian integrals using quadrature. The simple method is exemplified for a few standard asset price dynamics used in practice for structured and contingent claim pricing, such as local volatility, Heston stochastic volatility, SABR, and LIBOR market model (LMM/BGM). The quadrature in these concrete implementations is performed in m-dimensional space where m is between 10 and 30 depending on the specific model or a payoff structure. The methodology appears to be a broadly viable alternative to Monte Carlo for short- to medium-dated financial products, with potential uses outside those examined in this paper