Pricing Asian options with stochastic volatility

Abstract In this paper, we generalize the recently developed dimension reduction technique of Vecer for pricing arithmetic average Asian options. The assumption of constant volatility in Vecer's method will be relaxed to the case that volatility is randomly fluctuating and is driven by a mean-reverting (or ergodic) process. We then use the fast mean-reverting stochastic volatility asymptotic analysis introduced by Fouque, Papanicolaou and Sircar to derive an approximation to the option price which takes into account the skew of the implied volatility surface. This approximation is obtained by solving a pair of one-dimensional partial differential equations.

[1]  Ronnie Sircar,et al.  Short time-scale in S&P500 volatility , 2003 .

[2]  B. LeBaron,et al.  Stochastic volatility as a simple generator of apparent financial power laws and long memory , 2001 .

[3]  M. Yor,et al.  The Fine Structure of Asset Retums : An Empirical Investigation ' , 2006 .

[4]  Ronnie Sircar,et al.  Multiscale Stochastic Volatility Asymptotics , 2003, Multiscale Model. Simul..

[5]  E. Eberlein,et al.  The Generalized Hyperbolic Model: Financial Derivatives and Risk Measures , 2002 .

[6]  George Papanicolaou,et al.  FROM THE IMPLIED VOLATILITY SKEW TO A ROBUST CORRECTION TO BLACK-SCHOLES AMERICAN OPTION PRICES , 2001 .

[7]  G. Papanicolaou,et al.  MEAN-REVERTING STOCHASTIC VOLATILITY , 2000 .

[8]  L. Rogers,et al.  The value of an Asian option , 1995, Journal of Applied Probability.

[9]  J. Vecer.  A new PDE approach for pricing arithmetic average Asian options , 2001 .

[10]  G. Papanicolaou,et al.  Derivatives in Financial Markets with Stochastic Volatility , 2000 .

[11]  Jan Večeř.  Unified Pricing of Asian Options , 2002 .

[12]  Regina Y. Liu,et al.  A New Control Variate Estimator for an Asian Option , 2004 .

[13]  Local scale invariance and contingent claim pricing , 2001 .

[14]  A. Gallant,et al.  Alternative models for stock price dynamics , 2003 .

[15]  Vadim Linetsky,et al.  Spectral Expansions for Asian (Average Price) Options , 2004, Oper. Res..

[16]  Ronnie Sircar,et al.  Singular Perturbations in Option Pricing , 2003, SIAM J. Appl. Math..

[17]  J. Hoogland,et al.  Asians and Cash Dividends: Exploiting Symmetries in Pricing Theory , 2000, cond-mat/0006133.

[18]  Jan Vecer,et al.  Pricing Asian options in a semimartingale model , 2004 .

[19]  J. Ingersoll.  Theory of Financial Decision Making , 1987 .

[20]  Jiri Hoogland,et al.  Scaling invariance and contingent claim pricing II : Path-dependent contingent claims , 1999 .

[21]  K. Prause.  The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures , 1999 .

[22]  Michael W. Brandt,et al.  Range-Based Estimation of Stochastic Volatility Models , 2001 .