Stochastic Models of Implied Volatility Surfaces

type="main" xml:lang="en"> We propose a market–based approach to the modelling of implied volatility, in which the implied volatility surface is directly used as the state variable to describe the joint evolution of market prices of options and their underlying asset. We model the evolution of an implied volatility surface by representing it as a randomly fluctuating surface driven by a finite number of orthogonal random factors. Our approach is based on a Karhunen–Loeve decomposition of the daily variations of implied volatilities obtained from market data on SP500 and DAX options. We illustrate how this approach extends and improves the accuracy of the well–known ‘sticky moneyness’ rule used by option traders for updating implied volatilities. Our approach gives a justification for the use of ‘Vegas’ for measuring volatility risk and provides a decomposition of volatility risk as a sum of independent contributions from empirically identifiable factors. (J.E.L.: G130, C14, C31).

[1]  P. J. Schonbucher.  A market model for stochastic implied volatility , 1999 .

[2]  Wolfgang Härdle,et al.  Common factors governing VDAX movements and the maximum loss , 2002 .

[3]  B. Dumas,et al.  Implied volatility functions: empirical tests , 1996, IEEE Conference on Computational Intelligence for Financial Engineering & Economics.

[4]  Carol Alexander.  Principles of the skew , 2003 .

[5]  R. Hafner,et al.  The Dynamics of DAX Implied Volatilities , 2000 .

[6]  P. Schönbucher.  A market model for stochastic implied volatility , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  Henri Berestycki,et al.  Asymptotics and calibration of local volatility models , 2002 .

[8]  Jeff Fleming,et al.  Implied volatility functions: empirical tests , 1996, IEEE/IAFE 1996 Conference on Computational Intelligence for Financial Engineering (CIFEr).

[9]  A. Kemna,et al.  Analysis of the Term Structure of Implied Volatilities , 1994, Journal of Financial and Quantitative Analysis.

[10]  Allan M. Malz,et al.  Do Implied Volatilities Provide Early Warning of Market Stress? , 2000 .

[11]  R. Sundaram,et al.  Of Smiles and Smirks: A Term Structure Perspective , 1998, Journal of Financial and Quantitative Analysis.

[12]  Riccardo Rebonato,et al.  Volatility and Correlation: In the Pricing of Equity, FX, and Interest-Rate Options , 1999 .

[13]  Emanuel Derman,et al.  STOCHASTIC IMPLIED TREES: ARBITRAGE PRICING WITH STOCHASTIC TERM AND STRIKE STRUCTURE OF VOLATILITY , 1998 .

[14]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[15]  Joshua V. Rosenberg,et al.  Implied volatility functions: a reprise , 1999, Proceedings of the IEEE/IAFE/INFORMS 2000 Conference on Computational Intelligence for Financial Engineering (CIFEr) (Cat. No.00TH8520).

[16]  Rama Cont,et al.  Dynamics of implied volatility surfaces , 2002 .

[17]  R. Tompkins.  Stock index futures markets: stochastic volatility models and smiles , 2001 .

[18]  MARKET MODEL OF STOCHASTIC IMPLIED VOLATILITY WITH APPLICATION TO THE BGM MODEL , 2000 .

[19]  M. Avellaneda,et al.  An E-Arch Model for the Term Structure of Implied Volatility of FX Options , 1997 .

[20]  P. Balland.  Deterministic implied volatility models , 2002 .

[21]  Wolfgang K. Härdle,et al.  The Dynamics of Implied Volatilities: A Common Principal Components Approach , 2003 .

[22]  Olivier Ledoit,et al.  Relative Pricing of Options with Stochastic Volatility , 1998 .

[23]  P. Carr,et al.  Static Hedging of Exotic Options , 1998 .