Explicit implied vols for multifactor local-stochastic vol models

We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, 3/2 stochastic volatility, and SABR local-stochastic volatility.

[1]  The exact smile of certain local volatility models , 2012, 1207.0750.

[2]  Jan Baldeaux,et al.  Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model , 2012, 1203.5903.

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

[4]  P. Henry-Labordère Calibration of Local Stochastic Volatility Models to Market Smiles: A Monte-Carlo Approach , 2009 .

[5]  Option pricing with quadratic volatility: a revisit , 2011, Finance Stochastics.

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

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

[8]  Eric Benhamou,et al.  Time Dependent Heston Model , 2009, SIAM J. Financial Math..

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

[10]  E. Gobet,et al.  Asymptotic and Non Asymptotic Approximations for Option Valuation , 2012 .

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

[12]  Shinzo Watanabe Analysis of Wiener Functionals (Malliavin Calculus) and its Applications to Heat Kernels , 1987 .

[13]  J. Riordan Derivatives of composite functions , 1946 .

[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]  Carol Alexander,et al.  Stochastic Local Volatility , 2008 .

[16]  P. Hagan,et al.  Equivalent Black volatilities , 1999 .

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

[18]  A. Jacquier,et al.  Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model , 2011 .

[19]  Iain J. Clark Foreign Exchange Option Pricing: A Practitioner's Guide , 2011 .

[20]  C. Ewald Local volatility in the Heston model: a Malliavin calculus approach , 2008 .

[21]  Andrea Pascucci,et al.  Analytical Expansions for Parabolic Equations , 2013, SIAM Journal on Applied Mathematics.

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