Exact Pricing of Discretely-Sampled Variance Derivatives

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Journal Article
Journal of Business Management and Applied Economics, 2013, 2 (4), pp. 1 - 24
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We examine the pricing of variance swaps and some generalisations 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 L ́evy processes. Using the method of Hong (2004), exact and easily implementable formulae for the prices of these swaps are given in terms of, essentially, the characteristic function of the log of the stock price and its derivatives. We generalise results in Carr and Lee (2009) by relating the prices of variance swaps and the generalisations 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. Here, this result is extended in two directions, dropping the independence assumption and proving analogous results for some of the other swaps listed above.
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