The Small-Maturity Smile for Exponential Lévy Models

We derive a small-time expansion for out-of-the-money call options under an exponential Levy model, using the small-time expansion for the distribution function given in Figueroa-Lopez & Houdre (2009), combined with a change of num\'eraire via the Esscher transform. In particular, we quantify find that the effect of a non-zero volatility $\sigma$ of the Gaussian component of the driving L\'{e}vy process is to increase the call price by $1/2\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the L\'evy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility, which sharpens the first order estimate given in Tankov (2010). Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed L\'evy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options.

[1]  P. Carr,et al.  Saddlepoint methods for option pricing , 2009 .

[2]  J. Jacod,et al.  Testing for Jumps in a Discretely Observed Process , 2009, 0903.0226.

[3]  C. Houdré.  Remarks on deviation inequalities for functions of infinitely divisible random vectors , 2002 .

[4]  S. Levendorskii.  American and European Options Near Expiry, Under Markov Processes with Jumps , 2004 .

[5]  S. Levendorskii,et al.  PRICING OF THE AMERICAN PUT UNDER LÉVY PROCESSES , 2004 .

[6]  M. Yor,et al.  Stochastic Volatility for Lévy Processes , 2003 .

[7]  Elton P. Hsu,et al.  ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS , 2009 .

[8]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[9]  José E. Figueroa-López.  Small-time moment asymptotics for Lévy processes , 2008 .

[10]  F. Olver.  Asymptotics and Special Functions , 1974 .

[11]  B. Mandlebrot.  The Variation of Certain Speculative Prices , 1963 .

[12]  Peter Tankov,et al.  Asymptotic results for time-changed Lévy processes sampled at hitting times , 2011 .

[13]  Implied volatility : small time-to-expiry asymptotics in exponential Lévy models , 2010 .

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

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

[16]  Jean Jacod,et al.  Testing for Jumps in a Discretely Observed Process , 2007 .

[17]  Hongzhong Zhang,et al.  Small-time asymptotics for a general local-stochastic volatility model: curvature and the heat kernel expansion , 2013 .

[18]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  J. Rosínski.  Tempering stable processes , 2007 .

[20]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[21]  Jean Jacod,et al.  Estimating the degree of activity of jumps in high frequency data , 2009, 0908.3095.

[22]  Jean Jacod,et al.  Is Brownian motion necessary to model high-frequency data? , 2010, 1011.2635.

[23]  S. Z. Levendorski,et al.  Early exercise boundary and option prices in Levy driven models , 2004 .

[24]  R. Wolpert.  Lévy Processes , 2000 .

[25]  Johannes Muhle-Karbe,et al.  Asymptotic and exact pricing of options on variance , 2013, Finance Stochastics.

[26]  N. Shephard,et al.  Econometrics of Testing for Jumps in Financial Economics Using Bipower Variation , 2005 .

[27]  P. Carr,et al.  What Type of Process Underlies Options? A Simple Robust Test , 2003 .

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

[29]  M. Forde.  Exact pricing and large-time asymptotics for the modified SABR model and the Brownian exponential functional , 2011 .

[30]  Sergei Levendorskii.  American and European options in multi-factor jump-diffusion models, near expiry , 2008, Finance Stochastics.

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

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

[33]  Jos'e E. Figueroa-L'opez,et al.  Small-time expansions for the transition distributions of Lévy processes , 2008, 0809.0849.

[34]  Felix Schlenk,et al.  Proof of Theorem 3 , 2005 .

[35]  O. Barndorff-Nielsen.  Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling , 1997 .

[36]  Approximations for the distributions of bounded variation Lévy processes , 2010 .

[37]  T. Björk.  Arbitrage Theory in Continuous Time , 2019 .

[38]  S. Levendorskii,et al.  Non-Gaussian Merton-Black-Scholes theory , 2002 .

[39]  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.

[40]  Marc Yor,et al.  Pricing options on realized variance , 2005, Finance Stochastics.