Weak Convergence Methods for Approximation of the Evaluation of Path-Dependent Functionals

Abstract

In many applications, one needs to evaluate a path-dependent objective functional $V$ associated with a continuous-time stochastic process $X$. Due to the nonlinearity and possible lack of Markovian property, more often than not, $V$ cannot be evaluated analytically, and only Monte Carlo simulation or numerical approximation is possible. In addition, such calculations often require the handling of stopping times, the usual dynamic programming approach may fall apart, and the continuity of the functional becomes the main issue. Denoting by $h$ the stepsize of the approximation sequence, this work develops a numerical scheme so that an approximating sequence of path-dependent functionals $V^h$ converges to $V$. By a natural division of labors, the main task is divided into two parts. Given a sequence $X^h$ that converges weakly to $X$, the first part provides sufficient conditions for the convergence of the sequence of path-dependent functionals $V^h$ to $V$. The second part constructs a sequence of approximations $X^h$ using Markov chain approximation methods and demonstrates the weak convergence of $X^h$ to $X$, when $X$ is the solution of a stochastic differential equation. As a demonstration, combining the results of the two parts above, approximation of option pricing for the discrete-monitoring-barrier option underlying stochastic volatility model is provided. Different from the existing literature, the weak convergence analysis is carried out by using the Skorohod topology together with the continuous mapping theorem. The advantage of this approach is that the functional under study may be a function of stopping times, projection of the underlying diffusion on a sequence of random times, and/or maximum/minimum of the underlying diffusion.