The Small-Maturity Heston Forward Smile

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Ito diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.

[1]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[2]  Leif Andersen,et al.  Asymptotics for Exponential Levy Processes and Their Volatility Smile: Survey and New Results , 2012, 1206.6787.

[3]  J. Vives,et al.  On the Short-Time Behavior of the Implied Volatility for Jump-Diffusion Models With Stochastic Volatility , 2006 .

[4]  Roger Lee.  Option Pricing by Transform Methods: Extensions, Unification, and Error Control , 2004 .

[5]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[6]  L. Bergomi.  Smile Dynamics IV , 2009 .

[7]  Y. Lecam.  Implicit Diffusions , 2009 .

[8]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[9]  Ludger Kaup,et al.  Holomorphic Functions of Several Variables , 1983 .

[10]  Kun Gao,et al.  Asymptotics of implied volatility to arbitrary order , 2011, Finance Stochastics.

[11]  P. Friz,et al.  REGULAR VARIATION AND SMILE ASYMPTOTICS , 2006, math/0603146.

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

[13]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[14]  Archil Gulisashvili,et al.  Analytically Tractable Stochastic Stock Price Models , 2012 .

[15]  Paul Glasserman,et al.  Forward and Future Implied Volatility , 2010 .

[16]  Tübinger Diskussionsbeiträge.  Stochastic volatility with an Ornstein-Uhlenbeck process: An extension , 2014 .

[17]  Akihiko Takahashi,et al.  Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems , 2004 .

[18]  Asymptotics of Implied Volatility to Arbitrary Order , 2011 .

[19]  Peter Tankov,et al.  A New Look at Short‐Term Implied Volatility in Asset Price Models with Jumps , 2012 .

[20]  Marcel Nutz,et al.  Small-Time Asymptotics of Option Prices and First Absolute Moments , 2010, Journal of Applied Probability.

[21]  Applying stochastic volatility models for pricing and hedging derivatives , 2002 .

[22]  Antoine Jacquier,et al.  Marginal Density Expansions for Diffusions and Stochastic Volatility II: Applications , 2014 .

[23]  Antoine Jacquier,et al.  Marginal Density Expansions for Diffusions and Stochastic Volatility I: Theoretical Foundations , 2011, 1111.2462.

[24]  Svante Janson.  PROBABILITY ASYMPTOTICS: NOTES ON NOTATION , 2009 .

[25]  R. C. Merton,et al.  Theory of Rational Option Pricing , 2015, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[26]  J. Joseph,et al.  Fourier transforms , 2012 .

[27]  Yasufumi Osajima.  The Asymptotic Expansion Formula of Implied Volatility for Dynamic SABR Model and FX Hybrid Model , 2007 .

[28]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[29]  P. Tankov.  Pricing and Hedging in Exponential Lévy Models: Review of Recent Results , 2011 .

[30]  David Williams,et al.  Probability with Martingales , 1991, Cambridge mathematical textbooks.

[31]  Roger Lee.  THE MOMENT FORMULA FOR IMPLIED VOLATILITY AT EXTREME STRIKES , 2004 .

[32]  Archil Gulisashvili,et al.  Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes , 2009, SIAM J. Financial Math..

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

[34]  M. Keller-Ressel,et al.  MOMENT EXPLOSIONS AND LONG‐TERM BEHAVIOR OF AFFINE STOCHASTIC VOLATILITY MODELS , 2008, 0802.1823.

[35]  H. Berestycki,et al.  Computing the implied volatility in stochastic volatility models , 2004 .

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

[37]  Jim Gatheral.  The Volatility Surface: A Practitioner's Guide , 2006 .

[38]  P. Henry-Labordère.  Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing , 2008 .

[39]  Antoine Jacquier,et al.  The Small-Time Smile and Term Structure of Implied Volatility under the Heston Model , 2012, SIAM J. Financial Math..

[40]  G. Papanicolaou,et al.  Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives , 2011 .

[41]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[42]  Antoine Jacquier,et al.  Asymptotics of Forward Implied Volatility , 2012, SIAM J. Financial Math..

[43]  Forward-Start Options in Stochastic Volatility Model , 2003 .

[44]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[45]  B. Bercu,et al.  Sharp Large Deviations for the Ornstein--Uhlenbeck Process , 2002 .

[46]  E. Eberlein,et al.  Sato processes and the valuation of structured products , 2007 .

[47]  A. Jacquier,et al.  SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL , 2009 .

[48]  C. Houdr'e,et al.  Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps , 2010, 1009.4211.

[49]  Peter Friz,et al.  SMILE ASYMPTOTICS II: MODELS WITH KNOWN MOMENT GENERATING FUNCTIONS , 2006 .

[50]  Antoine Jacquier,et al.  Asymptotic formulae for implied volatility in the Heston model , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.