Discretely sampled variance and volatility swaps versus their continuous approximations

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• 作者：Robert Jarrow (1) (2)
  Younes Kchia (3)
  Martin Larsson (4)
  Philip Protter (5)
  
• 关键词：Variance swaps ; Volatility swaps ; NFLVR ; Semimartingales ; 60G35 ; 60G44 ; C65 ; C69 ; G12
• 刊名：Finance and Stochastics
• 出版年：2013
• 出版时间：April 2013
• 年：2013
• 卷：17
• 期：2
• 页码：305-324
• 全文大小：636KB
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• 作者单位：Robert Jarrow (1) (2)
  Younes Kchia (3)
  Martin Larsson (4)
  Philip Protter (5)
  
  1. Johnson Graduate School of Management, Cornell University, Ithaca, NY, 14853, USA
  2. Kamakura Corporation, 2222 Kalakaua Ave., Honolulu, HI, 96815, USA
  3. Centre de Math茅matiques Appliqu茅es, Ecole Polytechnique, Paris, France
  4. School of Operations Research, Cornell University, Ithaca, NY, 14853, USA
  5. Statistics Department, Columbia University, New York, NY, 10027, USA
  
• ISSN：1432-1122

文摘

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.