IDEAS home Printed from https://ideas.repec.org/a/taf/apmtfi/v14y2007i4p347-363.html
   My bibliography  Save this article

Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model

Author

Listed:
  • Fred Espen Benth
  • Martin Groth
  • Rodwell Kufakunesu

Abstract

Following the increasing awareness of the risk from volatility fluctuations, the market for hedging contracts written on realized volatility has surged. Companies looking for means to secure against unexpected accumulation of market activity can find over-the-counter products written on volatility indices. Since the Black and Scholes model require a constant volatility the need to consider other models is obvious. Swaps written on powers of realized volatility in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard are investigated. A key formula is derived for the realized variance able to represent the swap price dynamics in terms of Laplace transforms, which makes fast numerical inversion methods viable. An example using the fast Fourier transform is shown and compared with the approximation proposed by Brockhaus and Long.

Suggested Citation

  • Fred Espen Benth & Martin Groth & Rodwell Kufakunesu, 2007. "Valuing Volatility and Variance Swaps for a Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(4), pages 347-363.
    • Handle: RePEc:taf:apmtfi:v:14:y:2007:i:4:p:347-363
    DOI: 10.1080/13504860601170609
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/13504860601170609
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/13504860601170609?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53, January.
    2. Fred Benth & Thilo Meyer-Brandis, 2005. "The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps," Finance and Stochastics, Springer, vol. 9(4), pages 563-575, October.
    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. Elisa Nicolato & Emmanouil Venardos, 2003. "Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type," Mathematical Finance, Wiley Blackwell, vol. 13(4), pages 445-466, October.
    5. 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.
    6. Peter Carr & Hélyette Geman & Dilip Madan & Marc Yor, 2005. "Pricing options on realized variance," Finance and Stochastics, Springer, vol. 9(4), pages 453-475, October.
    7. Jérôme Detemple & Carlton Osakwe, 2000. "The Valuation of Volatility Options," Review of Finance, European Finance Association, vol. 4(1), pages 21-50.
    8. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
    9. Ole E. Barndorff-Nielsen, 1997. "Processes of normal inverse Gaussian type," Finance and Stochastics, Springer, vol. 2(1), pages 41-68.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Anatoliy Swishchuk, 2013. "Modeling and Pricing of Swaps for Financial and Energy Markets with Stochastic Volatilities," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8660, January.
    2. Finlay, Richard & Seneta, Eugene, 2012. "A Generalized Hyperbolic model for a risky asset with dependence," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2164-2169.
    3. Ole E. Barndorff-Nielsen & Neil Shephard, 2005. "Variation, jumps, market frictions and high frequency data in financial econometrics," OFRC Working Papers Series 2005fe08, Oxford Financial Research Centre.
    4. Hubalek, Friedrich & Sgarra, Carlo, 2009. "On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2137-2157, July.
    5. Nicolas Merener, 2012. "Swap rate variance swaps," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 249-261, May.
    6. Friedrich Hubalek & Petra Posedel, 2011. "Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 11(6), pages 917-932.
    7. Friedrich Hubalek & Carlo Sgarra, 2008. "On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps," Papers 0807.1227, arXiv.org.
    8. Michael Schröder, 2015. "Discrete-Time Approximation of Functionals in Models of Ornstein–Uhlenbeck Type, with Applications to Finance," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 285-313, June.
    9. Griffin, J.E. & Steel, M.F.J., 2006. "Inference with non-Gaussian Ornstein-Uhlenbeck processes for stochastic volatility," Journal of Econometrics, Elsevier, vol. 134(2), pages 605-644, October.
    10. Dotsis, George & Psychoyios, Dimitris & Skiadopoulos, George, 2007. "An empirical comparison of continuous-time models of implied volatility indices," Journal of Banking & Finance, Elsevier, vol. 31(12), pages 3584-3603, December.
    11. Karl Friedrich Hofmann & Thorsten Schulz, 2016. "A General Ornstein–Uhlenbeck Stochastic Volatility Model With Lévy Jumps," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(08), pages 1-23, December.
    12. Giovanni Salvi & Anatoliy V. Swishchuk, 2012. "Modeling and Pricing of Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities," Papers 1205.5565, arXiv.org.
    13. Friedrich Hubalek & Petra Posedel, 2008. "Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models," Papers 0807.3479, arXiv.org.
    14. Hainaut, Donatien & Devolder, Pierre, 2008. "Mortality modelling with Lévy processes," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 409-418, February.
    15. Madan, Dilip B. & Wang, King, 2021. "The structure of financial returns," Finance Research Letters, Elsevier, vol. 40(C).
    16. Carole Bernard & Zhenyu Cui, 2013. "Prices and Asymptotics for Discrete Variance Swaps," Papers 1305.7092, arXiv.org.
    17. Piotr Szczepocki, 2020. "Application of iterated filtering to stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck process," Statistics in Transition New Series, Polish Statistical Association, vol. 21(2), pages 173-187, June.
    18. Ying Jiao & Chunhua Ma & Simone Scotti & Chao Zhou, 2018. "The Alpha-Heston Stochastic Volatility Model," Papers 1812.01914, arXiv.org.
    19. Park, Yang-Ho, 2016. "The effects of asymmetric volatility and jumps on the pricing of VIX derivatives," Journal of Econometrics, Elsevier, vol. 192(1), pages 313-328.
    20. Ole E. Barndorff-Nielsen, 2004. "Power and Bipower Variation with Stochastic Volatility and Jumps," Journal of Financial Econometrics, Oxford University Press, vol. 2(1), pages 1-37.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:taf:apmtfi:v:14:y:2007:i:4:p:347-363. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Longhurst (email available below). General contact details of provider: http://www.tandfonline.com/RAMF20 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.