Finance and Stochastics

, Volume 17, Issue 1, pp 107–133 | Cite as

Asymptotic and exact pricing of options on variance

  • Martin Keller-Ressel
  • Johannes Muhle-Karbe


We consider the pricing of derivatives written on the discretely sampled realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuous-time limit, the quadratic variation of the underlying log-price. Here, we characterize the small-time limits of options on both objects. We find that the difference between them strongly depends on whether or not the stock price process has jumps. Subsequently, we propose two new methods to evaluate the prices of options on the discretely sampled realized variance. One of the methods is approximative; it is based on correcting prices of options on quadratic variation by our asymptotic results. The other method is exact; it uses a novel randomization approach and applies Fourier–Laplace techniques. We compare the methods and illustrate our results by some numerical examples.


Realized variance Quadratic variation Option pricing Small-time asymptotics Fourier–Laplace methods 

Mathematics Subject Classification

91G20 60G51 

JEL Classification

C02 G13 



Financial support by the National Centre of Competence in Research ‘Financial Valuation and Risk Management’ (NCCR FINRISK), Project D1 (Mathematical Methods in Financial Risk Management) is gratefully acknowledged. The NCCR FINRISK is a research instrument of the Swiss National Science Foundation.

We thank Richard Vierthauer for valuable discussions on the regularity of Laplace transforms and Marcel Nutz for comments on an earlier version. We are also very grateful to two anonymous referees and an anonymous Associate Editor, whose helpful comments significantly improved the present article.


  1. 1.
    Abate, J., Whitt, W.: Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36–43 (1995) zbMATHCrossRefGoogle Scholar
  2. 2.
    Barndorff-Nielsen, O.E., Shephard, N.: Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. B 64, 253–280 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bates, D.S.: Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Financ. Stud. 9, 69–107 (1996) CrossRefGoogle Scholar
  4. 4.
    Broadie, M., Jain, A.: The effect of jumps and discrete sampling on volatility and variance swaps. Int. J. Theor. Appl. Finance 11, 761–797 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bühler, H.: Volatility markets—consistent modeling, hedging and practical implementation. PhD thesis, TU Berlin (2006). Available at
  6. 6.
    Carr, P., Lee, R.: Robust replication of volatility derivatives. NYU Courant Institute Mathematics in Finance working paper 2008-3 (2008) Google Scholar
  7. 7.
    Carr, P., Madan, D.: Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73 (1999) Google Scholar
  8. 8.
    Carr, P., Geman, H., Madan, D., Yor, M.: Pricing options on realized variance. Finance Stoch. 9, 453–475 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Davies, B., Martin, B.: Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gatheral, J.: The Volatility Surface. Wiley, New York (2006) Google Scholar
  11. 11.
    Jacod, J.: Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Mathematics, vol. 714. Springer, Berlin (1979) zbMATHGoogle Scholar
  12. 12.
    Jacod, J.: Asymptotic properties of power variations of Lévy processes. ESAIM Probab. Stat. 11, 173–196 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Jacod, J.: Asymptotic properties of realized power variations and related functionals of semimartingales. Stoch. Process. Appl. 118, 517–559 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) zbMATHGoogle Scholar
  15. 15.
    Kallsen, J., Muhle-Karbe, J., Voß, M.: Pricing options on variance in affine stochastic volatility models. Math. Finance 21, 627–641 (2009) Google Scholar
  16. 16.
    Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance 21, 73–98 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Lee, R.: Realized volatility options. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 1500–1504. Wiley, Chichester (2010) Google Scholar
  18. 18.
    Muhle-Karbe, J., Nutz, M.: Small-time asymptotics of option prices and first absolute moments. J. Appl. Probab. 48(4), 1003–1020 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Neuberger, A.: Volatility trading (1992). Working paper, London Business School Google Scholar
  20. 20.
    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, New York (2005) Google Scholar
  21. 21.
    Raible, S.: Lévy Processes in Finance: Theory, Numerics, and Empirical Facts (2000). Dissertation, Universität Freiburg i. Br. Available at
  22. 22.
    Rudin, W.: Real and Complex Analysis, 1st edn. McGraw-Hill, New York (1966) zbMATHGoogle Scholar
  23. 23.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  24. 24.
    Sepp, A.: Analytical pricing of double-barrier options under a double-exponential jump diffusion process: applications of Laplace transform. Int. J. Theor. Appl. Finance 7, 151–175 (2003) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Sepp, A.: Pricing options on realized variance in Heston model with jumps in returns and volatility. J. Comput. Finance 11, 33–70 (2008) Google Scholar
  26. 26.
    Sepp, A.: Pricing options on realized variance in the Heston model with jumps in returns and volatility II: an approximate distribution of the discrete variance. J. Comput. Finance (2011, to appear). Preprint. Available at
  27. 27.
    Stoyan, D.: Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institut für MathematikTU BerlinBerlinGermany
  2. 2.Departement MathematikETH ZürichZürichSwitzerland

Personalised recommendations