Annals of Finance

, Volume 15, Issue 1, pp 59–101 | Cite as

Extreme-strike asymptotics for general Gaussian stochastic volatility models

  • Archil Gulisashvili
  • Frederi ViensEmail author
  • Xin Zhang
Research Article


We consider a stochastic volatility asset price model in which the volatility is the absolute value of a continuous Gaussian process with arbitrary prescribed mean and covariance. By exhibiting a Karhunen–Loève expansion for the integrated variance, and using sharp estimates of the density of a general second-chaos variable, we derive asymptotics for the asset price density for large or small values of the variable, and study the wing behavior of the implied volatility in these models. Our main result provides explicit expressions for the first three terms in the expansion of the implied volatility, based on three basic spectral-type statistics of the Gaussian process: the top eigenvalue of its covariance operator, the multiplicity of this eigenvalue, and the \(L^{2}\) norm of the projection of the mean function on the top eigenspace. Numerical illustrations using the Stein–Stein and fractional Stein–Stein models are presented, including strategies for parameter calibration.


Stochastic volatility Implied volatility Large strike Karhunen–Loève expansion Chi-squared variates 

Mathematics Subject Classification

60G15 91G20 40E05 

JEL Classification

C6 G13 



We are grateful to the associate editor and the anonymous referee for their valuable suggestions and remarks, which significantly contributed to improving the paper.


  1. Abramovitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Applied Mathematics Series, vol. 55. National Bureau of Standards, Gaithersburg (1972)Google Scholar
  2. Adler, R. J.: An introduction to continuity, extrema, and related topics for general Gaussian processes, vol. 12. IMS Lecture Notes Monogr. Ser. (1990)Google Scholar
  3. Alexanderian, A.: A brief note on the Karhunen-Loève expansion, technical note (2015).
  4. Benaim, S., Friz, P.K.: Smile asymptotics, II: models with known moment generating function. J Appl Probab 45, 16–32 (2008)CrossRefGoogle Scholar
  5. Benaim, S., Friz, P.K.: Regular variation and smile asymptotics. Math Finance 19, 1–12 (2009)CrossRefGoogle Scholar
  6. Benaim, S., Friz, P.K., Lee, R.: On Black–Scholes implied volatility at extreme strikes. In: Cont, R. (ed.) Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling, pp. 19–45. Wiley, Hoboken (2009)Google Scholar
  7. Beran, R.: The probabilities of noncentral quadratic forms. Ann Stat 3, 969–974 (1975)CrossRefGoogle Scholar
  8. Chronopoulou, A., Viens, F.: Stochastic volatility models with long-memory in discrete and continuous time. Quant Finance 12(4), 635–649 (2012)CrossRefGoogle Scholar
  9. Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math Finance 8, 291–323 (1998)CrossRefGoogle Scholar
  10. Corlay, S.: Quelques aspects de la quantification optimale, et applications en finance (in English, with French summary). Ph.D. Thesis. Université Pierre et Marie Curie (2011)Google Scholar
  11. Corlay, S.: Properties of the Ornstein-Uhlenbeck bridge. Preprint (2014). Accessed 24 Sept 2018
  12. Corlay, S., Pagès, G.: Functional quantization-based stratified sampling methods. Preprint (2014). Accessed 24 Sept 2018
  13. Deheuvels, P., Martynov, G.: A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables. J Funct Anal 255, 23263–2394 (2008)CrossRefGoogle Scholar
  14. Deuschel, J.-D., Friz, P.K., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility I: theoretical foundations. Commun Pure Appl Math 67, 40–82 (2014a)CrossRefGoogle Scholar
  15. Deuschel, J.-D., Friz, P.K., Jacquier, A., Violante, S.: Marginal density expansions for diffusions and stochastic volatility I: applications. Commun Pure Appl Math 67, 321–350 (2014b)CrossRefGoogle Scholar
  16. Dieker, A. B.: Simulation of fractional Brownian motion. Masters Thesis, Department of Mathematical Sciences, University of Twente, The Netherlands (2004)Google Scholar
  17. Dieker, A.B., Mandjes, M.: On spectral simulation of fractional Brownian motion. Probab Eng Inf Sci 17, 417–434 (2003)CrossRefGoogle Scholar
  18. Gao, K., Lee, R.: Asymptotics of implied volatility to arbitrary order. To appear in Finance Stoch (2015).
  19. Gatheral, J.: A parsimonious arbitrage-free implied volatility parametrization with application to the valuation of the volatility derivatives. In: Global Derivatives and Risk Management, Madrid, May 26 (2004)Google Scholar
  20. Gatheral, J.: The Volatility Surface: A Practitioner’s Guide. Wiley, New York (2006)Google Scholar
  21. Gatheral, J., Jacquier, A.: Arbitrage-free SVI volatility surfaces. Quant Finance 14(1), 59–71 (2014)CrossRefGoogle Scholar
  22. Gulisashvili, A.: Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes. SIAM J Financial Math 1, 609–641 (2010)CrossRefGoogle Scholar
  23. Gulisashvili, A.: Analytically Tractable Stochastic Stock Price Models. Springer, Berlin (2012a)CrossRefGoogle Scholar
  24. Gulisashvili, A.: Asymptotic equivalence in Lee’s moment formulas for the implied volatility, asset price models without moment explosions, and Piterbarg’s conjecture. Int J Theor Appl Finance 15, 1250020 (2012b)CrossRefGoogle Scholar
  25. Gulisashvili, A., Stein, E.M.: Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models. Appl Math Optim 61, 287–315 (2010)CrossRefGoogle Scholar
  26. Gulisashvili, A., Vives, J.: Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models. SIAM J Finanance Math 6, 158–188 (2015)CrossRefGoogle Scholar
  27. Gulisashvili, A., Viens, F., Zhang, X.: Small-time asymptotics for Gaussian self-similar stochastic volatility models. Preprint (2015). arXiv:1505.05256
  28. Hoeffding, W.: On a theorem of V. M. Zolotarev. Theor Probab Appl 9, 89–92 (1964)CrossRefGoogle Scholar
  29. Ibragimov, I.A., Rozanov, Y.A.: Gaussian Random Processes. Springer, New York (1978)CrossRefGoogle Scholar
  30. Lee, R.: The moment formula for implied volatility at extreme strikes. Math Finance 14, 469–480 (2004)CrossRefGoogle Scholar
  31. Nourdin, I., Peccati, G.: Normal Approximations with Malliavin Calculus: from Stein’s Method to Universality. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  32. Renault, E., Touzi, N.: Option hedging and implicit volatilities. Math Finance 6, 279–302 (1996)CrossRefGoogle Scholar
  33. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (2004)Google Scholar
  34. Roper, M.: Arbitrage free implied volatility surfaces (preprint) (2010)Google Scholar
  35. Rosenbaum, M.: Estimation of the volatility persistence in a discretely observed diffusion model. Stoch Process Appl 118(8), 1434–1462 (2008)CrossRefGoogle Scholar
  36. Stein, E., Stein, J.: Stock price distributions with stochastic volatility: an analytic approach. Rev Financ Stud 4, 727–752 (1991)CrossRefGoogle Scholar
  37. Yaglom, A.M.: Correlation Theory of Stationary and Related Random functions, vol. I. Springer, New York (1987)Google Scholar
  38. Zolotarev, V.M.: Concerning a certain probability problem. Theor Probab Appl 6, 201–204 (1961)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsOhio UniversityAthensUSA
  2. 2.Department of Statistics and ProbabilityMichigan State UniversityEast LansingUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations