Saddlepoint Approximation Methods for Pricing Derivatives on Discrete Realized Variance

Abstract

We consider the saddlepoint approximation methods for pricing derivatives whose payoffs depend on the discrete realized variance of the underlying price process of a risky asset. Most of the earlier pricing models of variance products and volatility derivatives use the quadratic variation approximation as the continuous limit of the discrete realized variance. However, the corresponding discretization error may become significant for short-maturity derivatives. Under Lévy models and stochastic volatility models with jumps, we manage to obtain the saddlepoint approximation formulas for pricing variance products and volatility derivatives using the small time asymptotic approximation of the Laplace transform of the discrete realized variance. As an alternative approach, we also develop the conditional saddlepoint approximation method based on a given simulated stochastic variance path via Monte Carlo simulation. This analytic-simulation approach reduces the dimensionality of the simulation of the discrete variance derivatives; and in some cases, the simulation procedure of the realized variance can be effectively performed using an appropriate exact simulation method. We examine numerical accuracy and reliability of various types of the saddlepoint approximation techniques when applied to pricing derivatives on discrete realized variance under different types of asset price processes. The limitations of the saddlepoint approximation methods in pricing variance products and volatility derivatives are also discussed.