Pricing Variance, Gamma, and Corridor Swaps Using Multinomial Trees

Abstract

Pricing and hedging real world derivatives requires approximation methods for all but the plainest of plain vanilla cases. The two standard classes of valuation tools are lattices (trees) and Monte Carlo simulation. Lattice methods start at maturity and work backward through the tree to compute the derivative value at the beginning. This works very well as long as the payoff at a given time step depends on the asset price at that date but not on the path the price followed to arrive there. There are many more paths through a tree than there are time steps, so path-dependent problems normally require Monte Carlo and calculations over a very large number of paths to achieve accuracy. This raises major problems for valuing derivatives based on volatility and other higher moments of the returns distribution, because realized values of these statistics do depend on the path and not just the terminal price. In this article, the authors present a new kind of lattice procedure for variance swaps and related contracts that achieves the accuracy of a Monte Carlo approximation in execution time that is very much less—more than two orders of magnitude faster in a basic example. The technique involves a single backward pass through the tree, with efficient calculations of conditional future variance through expiration at each intermediate node being built up along the way. The procedure does not depend on any specific returns process, and examples with several well-known stochastic volatility models show excellent performance on all of them.