Volatility Swaps and Volatility Options on Discretely Sampled Realized Variance

Volatility swaps and volatility options are financial products written on discretely sampled realized variance. Actively traded in over-the-counter markets, these products are often priced by continuously sampled approximations to simplify the computations. This paper presents an analytical approach to efficiently and accurately price discretely sampled volatility derivatives, under a general 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 price discrete volatility derivatives through either semi-analytical pricing formulae (up to an inverse Fourier transform) or an efficient Fourier-cosine series method. Numerical experiments show that our approximation is more accurate in comparison to the approximations in the literature. We remark that although discretely sampled variance swaps and options are usually more expensive than their continuously sampled counterparts, discretely sampled volatility swaps are more prone to be cheaper than the continuously sampled counterparts. An analysis is then provided to explain why this is the case in general for realistic contract specifications and reasonable model parameters.