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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

91G20 60G51 

JEL Classification

C02 G13 

Notes

Acknowledgements

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.

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

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

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