Efficient Monte Carlo for Discrete Variance Contracts

We develop an efficient Monte Carlo method for the valuation of a financial contract with payoff dependent on discretely realized variance. We assume a general model in which asset returns are random shocks modulated by a stochastic volatility process. Realized variance is the sum of squared daily returns, depending on the sequence of shocks to the asset and the realized path of the volatility process. The price of interest is the expected payoff, represented as a high dimensional integral over the fundamental sources of randomness. We compute it through the combination of deterministic integration over a two dimensional manifold defined by the sum of squared shocks to the asset and the path average of the modulating variance process, followedby exact conditional Monte Carlo sampling. The deterministic integration variables capture most of the variability in realized variance therefore the residual variance in our estimator is much smaller than that in standard Monte Carlo. We derive theoretical results that quantify the variance reduction achieved by the method. We test it for the Hull-White, Heston, and Double Exponential models and show that the algorithm performs significantly better than standard Monte Carlo for realistic computational budgets.