Variance Derivatives: Pricing and Convergence

Abstract

We examine the pricing of variance swaps and some generalizations and variants such as self-quantoed variance swaps, gamma swaps, skewness swaps and proportional variance swaps.We consider the pricing of both discretely monitored and continuously monitored versions of these swaps when the dynamics of the log of the underlying stock price are driven by (possibly, multiple) time-changed Levy processes. We derive exact and easily implementable formulae for the prices of these swaps in terms of, essentially, the characteristic function of the log of the stock price and its derivatives. We consider the convergence of the prices of discretely monitored versions of these swaps to the prices of their continuously monitored counterparts as the number N of monitoring times is allowed to tend to infi nity. We signi ficantly extend results in Broadie and Jain (2008a) by showing that the prices of discretely monitored variance swaps and all of the generalizations and variants listed above all converge to the prices of their continuously monitored counterparts as O(1/N). We generalize results in Carr and Lee (2009) by relating the prices of variance swaps and the generalizations and variants listed above to the prices of log-forward-contracts and entropy-forward-contracts. Carr and Lee (2009) show that, under an independence assumption, discretely monitored variance swaps are worth at least as much as their continuously monitored counterparts. We extend this result in two directions, by dropping the independence assumption and by proving analogous results for some of the other swaps listed above.