Maturity Cycles in Implied Volatility Jean-Pierre Fouque ∗ George Papanicolaou † Ronnie Sircar ‡ Knut Solna § 14 August 2002; revised 15 February 2003 Abstract The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [7]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients. Contents 1 Introduction 2 1.1 Volatility Mean-Reversion ............................ 3 1.2 Fast Mean-Reversion of Volatility ........................ 4 1.3 Goal ........................................ 5 2 Fit of Implied Volatilities to LMMR Formula 6 2.1 Dataset ...................................... 6 2.2 LMMR Fit across Maturities up to one year .................. 6 2.3 Examination of the Breaks ............................ 9 2.3.1 LMMR Fit to Restricted Data ...................... 9 2.4 Fitting LMMR Maturity-by-Maturity ...................... 9 * Department of Mathematics, NC State University, Raleigh NC 27695-8205, fouque@math.ncsu.edu. Work partially supported by NSF grant DMS-0071744. † Department of Mathematics, Stanford University, Stanford CA 94305, papanico@math.stanford.edu. ‡ Department of Operations Research & Financial Engineering, Princeton University, E-Quad, Princeton, NJ 08544, sircar@princeton.edu. Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance. § Department of Mathematics, University of California, Irvine CA 92697, ksolna@math.uci.edu. 1