Discretely sampled variance and volatility swaps versus their continuous approximations

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.

[1]  C. Sin.  Complications with stochastic volatility models , 1998, Advances in Applied Probability.

[2]  F. Delbaen,et al.  A general version of the fundamental theorem of asset pricing , 1994 .

[3]  F. Roush.  Introduction to Stochastic Integration , 1994 .

[4]  Johan Tysk,et al.  Bubbles, convexity and the Black–Scholes equation , 2009, 0908.4468.

[5]  A. Cherny,et al.  ON THE MARTINGALE PROPERTY OF TIME-HOMOGENEOUS DIFFUSIONS , 2009 .

[6]  D. Dufresne.  The integrated square-root process , 2001 .

[7]  P. Carr,et al.  A new approach for option pricing under stochastic volatility , 2007 .

[8]  David Hobson,et al.  Comparison results for stochastic volatility models via coupling , 2010, Finance Stochastics.

[9]  Mark Broadie,et al.  The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps , 2008 .

[10]  Peter Carr,et al.  Volatility Derivatives , 2009 .

[11]  David Williams.  Diffusions, Markov Processes and Martingales: Volume 2, Ito Calculus , 2000 .

[12]  Michael Koller.  An Introduction to Stochastic Integration , 2011 .

[13]  David Hobson,et al.  Local martingales, bubbles and option prices , 2005, Finance Stochastics.

[14]  F. Delbaen,et al.  The fundamental theorem of asset pricing for unbounded stochastic processes , 1998 .

[15]  Mark Broadie,et al.  Pricing and Hedging Volatility Derivatives , 2008 .

[16]  Vladimir V. Piterbarg,et al.  Moment explosions in stochastic volatility models , 2005, Finance and Stochastics.

[17]  Christophe Stricker,et al.  Couverture des actifs contingents et prix maximum , 1994 .

[18]  Roger Lee,et al.  Variance swaps on time-changed Lévy processes , 2012, Finance Stochastics.

[19]  Eckhard Platen,et al.  Pricing and hedging of long dated variance swaps under a 3/2 volatility model , 2010, J. Comput. Appl. Math..

[20]  P. Protter,et al.  ASSET PRICE BUBBLES IN INCOMPLETE MARKETS * , 2008 .

[21]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .