Volatility Swaps and Volatility Options on Discretely Sampled Realized Variance

Abstract

Volatility swaps and variance options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are priced often by a continuously sampled approximation to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under the Heston stochastic volatility model. We first obtain an accurate approximation for the characteristic function of the discretely sampled realized variance. This characteristic function is then applied to derive semi-analytical (up to an inverse Laplace transform) pricing formulae for variance options, volatility swaps and volatility options. We examine with numerical examples the accuracies of the approach in pricing these volatility derivatives. We also test the effect of discrete sampling in pricing volatility derivatives. For realistic contract specifications and model parameters, we find that although continuously sampled variance swaps and options are cheaper than their discretely sampled counterparts, continuously sampled volatility swaps are, however, more expensive than their discretely sampled counterparts.

Keywords: Variance swaps, Variance options, Stochastic volatility, Characteristic function