Asymptotic and Non Asymptotic Approximations for Option Valuation

We give a broad overview of approximation methods to derive analytical formulas for accurate and quick evaluation of option prices. We compare different approaches, from the theoretical point of view regarding the tools they require, and also from the numerical point of view regarding their performances. In the case of local volatility models with general time-dependency, we derive new formulas using the local volatility function at the mid-point between strike and spot: in general, our approximations outperform previous ones by Hagan and Henry-Labordere. We also provide approximations of the option delta.

[1]  H. Weitzner,et al.  Perturbation Methods in Applied Mathematics , 1969 .

[2]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[3]  R. Azencott Densité des diffusions en temps petit: développements asymptotiques , 1984 .

[4]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[5]  P. Souganidis,et al.  Asymptotic Series and the Methods of Vanishing Viscosity , 1985 .

[6]  Louis O. Scott Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application , 1987, Journal of Financial and Quantitative Analysis.

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

[8]  Mark Schroder Computing the Constant Elasticity of Variance Option Pricing Formula , 1989 .

[9]  JAPAN STATISTICAL SOCIETY , 1991 .

[10]  N. Yoshida ASYMPTOTIC EXPANSION FOR STATISTICS RELATED TO SMALL DIFFUSIONS , 1992 .

[11]  N. Yoshida Asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe , 1992 .

[12]  Bruno Dupire Pricing with a Smile , 1994 .

[13]  Min Qian,et al.  Stochastic flows of diffeomorphisms , 1995 .

[14]  D. Nualart The Malliavin Calculus and Related Topics , 1995 .

[15]  N. Touzi,et al.  Small noise expansion and importance sampling , 1997 .

[16]  P. Carr,et al.  Option valuation using the fast Fourier transform , 1999 .

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

[18]  Alan L. Lewis Option Valuation under Stochastic Volatility , 2000 .

[19]  G. Papanicolaou,et al.  Derivatives in Financial Markets with Stochastic Volatility , 2000 .

[20]  Akihiko Takahashi,et al.  The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims , 2001 .

[21]  M. Musiela,et al.  Martingale Methods in Financial Modelling , 2002 .

[22]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[23]  D. Duffie,et al.  Affine Processes and Application in Finance , 2002 .

[24]  Paul Glasserman,et al.  Monte Carlo Methods in Financial Engineering , 2003 .

[25]  Ronnie Sircar,et al.  Singular Perturbations in Option Pricing , 2003, SIAM J. Appl. Math..

[26]  Jim Gatheral,et al.  A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives , 2004 .

[27]  Masayuki Uchida,et al.  Asymptotic Expansion for Small Diffusions Applied to Option Pricing , 2004 .

[28]  Ronnie Sircar,et al.  Maturity cycles in implied volatility , 2004, Finance Stochastics.

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

[30]  O. Pironneau,et al.  Computational Methods for Option Pricing (Frontiers in Applied Mathematics) (Frontiers in Applied Mathematics 30) , 2005 .

[31]  Roger Lee,et al.  Implied Volatility: Statics, Dynamics, and Probabilistic Interpretation , 2005 .

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

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

[34]  W. Schachermayer,et al.  HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK–MERTON–SCHOLES? , 2007, 0711.1272.

[35]  Elisa Alòs,et al.  Malliavin differentiability of the Heston volatility and applications to option pricing , 2008, Advances in Applied Probability.

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

[37]  Smart expansion and fast calibration for jump diffusions , 2007, Finance Stochastics.

[38]  Analytical Formulas for Local Volatility Model with Stochastic Rates , 2009 .

[39]  Torben G. Andersen,et al.  Stochastic volatility , 2003 .

[40]  M. Tehranchi Asymptotics of Implied Volatility far from Maturity , 2009, Journal of Applied Probability.

[41]  Antoine Jacquier,et al.  Convergence of Heston to SVI , 2010, 1002.3633.

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

[43]  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..

[44]  New Approximations in Local Volatility Models , 2010 .

[45]  Eric Benhamou,et al.  Expansion formulas for European options in a local volatility model , 2010 .

[46]  Michael Tehranchi,et al.  Can the implied volatility surface move by parallel shifts? , 2010, Finance Stochastics.

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

[48]  A. Jacquier,et al.  Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models , 2011, 1108.3998.

[49]  Antoine Jacquier,et al.  The large-maturity smile for the Heston model , 2011, Finance Stochastics.

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

[51]  Emmanuel Gobet,et al.  Weak approximation of averaged diffusion processes , 2014 .