Asymptotic Analysis of Implied Volatility

  • Archil Gulisashvili
Part of the Springer Finance book series (FINANCE)


The implied volatility was first introduced by H.A. Latané and R.J. Rendleman under the name “the implied standard deviation”. Latané and Rendleman studied standard deviations of asset returns, which are implied in actual call option prices when investors price options according to the Black-Scholes model. Chapter 9 mainly is concerned with the asymptotics of the implied volatility at extreme strikes. It presents sharp model-free asymptotic formulas for the implied volatility established by the author and the higher order extensions of these formulas found by K. Gao and R. Lee. The chapter also provides a characterization of implied volatility models free of static arbitrage, and discusses certain symmetries hidden in stochastic asset price models. These symmetries can be used to analyze the asymptotic behavior of the implied volatility at small strikes knowing how the volatility behaves at large strikes.


Asymptotic Formula Implied Volatility Symmetric Model Static Arbitrage Strike Price 
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  1. [CL09]
    Carr, P., Lee, R., Put-call symmetry: extensions and applications, Mathematical Finance 19 (2009), pp. 523–560. MathSciNetzbMATHCrossRefGoogle Scholar
  2. [CL10]
    Carr, P., Lee, R., Implied volatility in stochastic volatility models, in: Encyclopedia of Quantitative Finance, 2010. Google Scholar
  3. [CdF02]
    Cont, R., da Fonseca, J., Dynamics of implied volatility surfaces, Quantitative Finance 2 (2002), pp. 45–60. MathSciNetCrossRefGoogle Scholar
  4. [CGLS09]
    Corquera, J. M., Guillaume, F., Leoni, P., Schoutens, W., Implied Lévy volatility, Quantitative Finance 9 (2009), pp. 383–393. MathSciNetCrossRefGoogle Scholar
  5. [DM10]
    De Marco, S., On probability distributions of diffusions and financial models with non-globally smooth coefficients, Ph.D. Dissertation, Université Paris-Est Marne-la-Vallée and Scuola Normale Superiore di Pisa, 2010. Google Scholar
  6. [DMM10]
    De Marco, S., Martini, C., The term structure of implied volatility in symmetric models with applications to Heston, preprint, 2010; available at
  7. [Dur04]
    Durrleman, V., From implied to spot volatilities, Ph.D. Dissertation, Princeton University, 2004. Google Scholar
  8. [Fen05]
    Fengler, M. R., Semiparametric Modeling of Implied Volatility, Springer, Berlin, 2005. zbMATHGoogle Scholar
  9. [FPS00]
    Fouque, J.-P., Papanicolaou, G., Sircar, R., Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, 2000. zbMATHGoogle Scholar
  10. [Fri10]
    Friz, P., Implied volatility: large strike asymptotics, in: Cont, R. (Ed.), Encyclopedia of Quantitative Finance, pp. 909–913, Wiley, Chichester, 2010. Google Scholar
  11. [GL11]
    Gao, K., Lee, R., Asymptotics of implied volatility to arbitrary order, preprint, 2011; available at
  12. [Gat06]
    Gatheral, J., The Volatility Surface: A Practitioner’s Guide, Wiley, Hoboken, 2006. Google Scholar
  13. [Gul10]
    Gulisashvili, A., Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes, SIAM Journal on Financial Mathematics 1 (2010), pp. 609–641. MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Haf04]
    Hafner, R., Stochastic Implied Volatility: A Factor-Based Model, Springer, Berlin, 2004. zbMATHCrossRefGoogle Scholar
  15. [H-L09]
    Henry-Labordère, P., Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing, Chapman & Hall/CRC, Boca Raton, 2009. zbMATHGoogle Scholar
  16. [IW77]
    Ikeda, N., Watanabe, S., A comparison theorem for solutions of stochastic differential equations and its applications, Osaka Journal of Mathematics 14 (1977), pp. 619–633. MathSciNetzbMATHGoogle Scholar
  17. [LR76]
    Latané, H. A., Rendleman, R. J. Jr., Standard deviations of stock price ratios implied in option prices, Journal of Finance 31 (1976), pp. 369–381. Google Scholar
  18. [Lee01]
    Lee, R., Implied and local volatilities under stochastic volatility, International Journal of Theoretical and Applied Finance 4 (2001), pp. 45–89. MathSciNetzbMATHGoogle Scholar
  19. [Lee04a]
    Lee, R., Implied volatility: statics, dynamics, and probabilistic interpretation, in: Baeza-Yates, R., Glaz, J., Ghyz, H., et al. (Eds.), Recent Advances in Applied Probability, Springer, Berlin, 2004. Google Scholar
  20. [Pro04]
    Protter, P. E., Stochastic Integration and Differential Equations, 2nd ed., Springer, Berlin, 2004. zbMATHGoogle Scholar
  21. [Reb04]
    Rebonato, R., Volatility and Correlation: The Perfect Hedger and the Fox, 2nd ed., Wiley Finance, New York, 2004. Google Scholar
  22. [RT96]
    Renault, E., Touzi, N., Option hedging and implied volatilities in a stochastic volatility model, Mathematical Finance 6 (1996), pp. 279–302. zbMATHCrossRefGoogle Scholar
  23. [Rop09]
    Roper, M., Implied volatility: general properties and asymptotics, Ph.D. thesis, The University of New South Wales, 2009. Google Scholar
  24. [Rop10]
    Roper, M., Arbitrage free implied volatility surfaces, preprint, 2010. Google Scholar
  25. [SP99]
    Sircar, K. R., Papanicolaou, G., Stochastic volatility, smile & asymptotics, Applied Mathematical Finance 6 (1999), pp. 107–145. zbMATHCrossRefGoogle Scholar
  26. [Ski01]
    Skiadopoulos, G., Volatility smile consistent option models: a survey, International Journal of Theoretical and Applied Finance 4 (2001), pp. 403–438. MathSciNetzbMATHCrossRefGoogle Scholar
  27. [SHK99]
    Skiadopoulos, G., Hodges, S., Klewlow, L., The dynamics of the SP500 implied volatility surface, Review of Derivatives Research 3 (1999), pp. 263–282. CrossRefGoogle Scholar
  28. [Teh09a]
    Tehranchi, M. R., Symmetric martingales and symmetric smiles, Stochastic Processes and Their Applications 119 (2009), pp. 3785–3797. MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Archil Gulisashvili
    • 1
  1. 1.Department of MathematicsOhio UniversityAthensUSA

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