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On the pricing and hedging of volatility derivatives


  • Sam Howison
  • Avraam Rafailidis
  • Henrik Rasmussen


The paper considers the pricing of a range of volatility derivatives, including volatility and variance swaps and swaptions. Under risk-neutral valuation closed-form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility are provided. A general partial differential equation framework for derivatives that have an extra dependence on an average of the volatility is described. Approximate solutions of this equation are given for volatility products written on assets for which the volatility process fluctuates on a timescale that is fast compared with the lifetime of the contracts, analysing both the 'outer' region and, by matched asymptotic expansions, the 'inner' boundary layer near expiry.

Suggested Citation

  • Sam Howison & Avraam Rafailidis & Henrik Rasmussen, 2004. "On the pricing and hedging of volatility derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 11(4), pages 317-346.
  • Handle: RePEc:taf:apmtfi:v:11:y:2004:i:4:p:317-346 DOI: 10.1080/1350486042000254024

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    References listed on IDEAS

    1. Lucas, Andre & Klaassen, Pieter & Spreij, Peter & Straetmans, Stefan, 2001. "An analytic approach to credit risk of large corporate bond and loan portfolios," Journal of Banking & Finance, Elsevier, vol. 25(9), pages 1635-1664, September.
    2. Bangia, Anil & Diebold, Francis X. & Kronimus, Andre & Schagen, Christian & Schuermann, Til, 2002. "Ratings migration and the business cycle, with application to credit portfolio stress testing," Journal of Banking & Finance, Elsevier, vol. 26(2-3), pages 445-474, March.
    3. Gordy, Michael B., 2000. "A comparative anatomy of credit risk models," Journal of Banking & Finance, Elsevier, vol. 24(1-2), pages 119-149, January.
    4. Mark Carey, 1998. "Credit Risk in Private Debt Portfolios," Journal of Finance, American Finance Association, vol. 53(4), pages 1363-1387, August.
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    Cited by:

    1. Ole E. Barndorff-Nielsen & Neil Shephard, 2005. "Variation, jumps, market frictions and high frequency data in financial econometrics," Economics Papers 2005-W16, Economics Group, Nuffield College, University of Oxford.
    2. Piotr Pluciennik, 2010. "Forecasting Financial Processes by Using Diffusion Models," Dynamic Econometric Models, Uniwersytet Mikolaja Kopernika, vol. 10, pages 51-60.
    3. Windcliff, H. & Forsyth, P.A. & Vetzal, K.R., 2006. "Pricing methods and hedging strategies for volatility derivatives," Journal of Banking & Finance, Elsevier, vol. 30(2), pages 409-431, February.
    4. Mengzhe Zhang & Leunglung Chan, 2016. "Pricing volatility swaps in the Heston’s stochastic volatility model with regime switching: A saddlepoint approximation method," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-20, December.
    5. Nicolas Merener, 2012. "Swap rate variance swaps," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 249-261, May.
    6. Lian, Guanghua & Chiarella, Carl & Kalev, Petko S., 2014. "Volatility swaps and volatility options on discretely sampled realized variance," Journal of Economic Dynamics and Control, Elsevier, vol. 47(C), pages 239-262.
    7. Giovanni Salvi & Anatoliy V. Swishchuk, 2012. "Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities," Papers 1205.5565,
    8. Carole Bernard & Zhenyu Cui, 2013. "Prices and Asymptotics for Discrete Variance Swaps," Papers 1305.7092,
    9. Andrew Papanicolaou & Ronnie Sircar, 2014. "A regime-switching Heston model for VIX and S&P 500 implied volatilities," Quantitative Finance, Taylor & Francis Journals, vol. 14(10), pages 1811-1827, October.
    10. repec:eee:apmaco:v:250:y:2015:i:c:p:920-933 is not listed on IDEAS
    11. Leunglung Chan & Eckhard Platen, 2010. "Exact Pricing and Hedging Formulas of Long Dated Variance Swaps under a $3/2$ Volatility Model," Papers 1007.2968,, revised Jan 2011.
    12. repec:eee:ejores:v:262:y:2017:i:1:p:381-400 is not listed on IDEAS

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