Volatility Library

Article by: Neil Chriss, William Moroko
Published by: New York University
Date: Jul 1999

“The market for volatility swaps at the time of this writing is dominated by longer dated instruments with maturities in the one to five year range (for an overview of the market, see Mehta (1999)). Consequently, risk management is largely a matter of understanding fluctuations in the mark-to-market value of the swap. Recently a number of articles focusing on the pricing and hedging of volatility swaps (see Carr and Madan (1998), Demeter, Derman, Kamal and Zou (1999)) have appeared. These articles demonstrate that it is possible to hedge the payout risk of a variance swap using a combination of a static position in options and a dynamic stock strategy, but say nothing of mark-to-market risk. This article exclusively studies mark-to-market risk. We classify the types of risks the holder of a volatility swap faces, and argue that some of these risks are modelable and while others depend exclusively on the valuation of out-of-the-money options whose values are not available in the market.”

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Article by: Robert J. Elliott, Tak Kuen Siu, Leunglung Chan
Published by: Applied Mathematical Finance
Date: 16 Jan 2006

“A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous‐time Markov‐modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Heston’s SV model depend on the states of a continuous‐time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market. Both probabilistic and partial differential equation (PDE) approaches are considered for the valuation of volatility derivatives.”

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Article by: Mark Broadie, Ashish Jain
Published by: Columbia Business School
Date: 10 Jan 2008

“This paper studies the pricing and hedging of variance swaps and other volatility derivatives, including volatility swaps and variance options, in the Heston stochastic volatility model. Pricing and hedging results are derived using partial differential equation techniques. We formulate an optimization problem to determine the number of options required to best hedge a variance swap. We propose a method to dynamically hedge volatility derivatives using variance swaps and a finite number of European call and put options.”

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