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Discretely sampled variance and volatility swaps versus their continuous approximations

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  • Robert Jarrow

    ()

  • Younes Kchia

    ()

  • Martin Larsson

    ()

  • Philip Protter

    ()

Abstract

Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given that the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values. Copyright Springer-Verlag 2013

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File URL: http://hdl.handle.net/10.1007/s00780-012-0183-2
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Bibliographic Info

Article provided by Springer in its journal Finance and Stochastics.

Volume (Year): 17 (2013)
Issue (Month): 2 (April)
Pages: 305-324

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Handle: RePEc:spr:finsto:v:17:y:2013:i:2:p:305-324

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Related research

Keywords: Variance swaps; Volatility swaps; NFLVR; Semimartingales; 60G35; 60G44; C65; C69; G12;

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  1. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
  2. Mark Broadie & Ashish Jain, 2008. "The Effect Of Jumps And Discrete Sampling On Volatility And Variance Swaps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(08), pages 761-797.
  3. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
  4. Peter Carr & Roger Lee & Liuren Wu, 2012. "Variance swaps on time-changed Lévy processes," Finance and Stochastics, Springer, vol. 16(2), pages 335-355, April.
  5. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
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Cited by:
  1. David Hobson & Martin Klimmek, 2011. "Model independent hedging strategies for variance swaps," Papers 1104.4010, arXiv.org, revised May 2011.
  2. Carol Alexander & Johannes Rauch, 2014. "Discretisation-Invariant Swaps," Papers 1404.1351, arXiv.org.

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