IDEAS home Printed from https://ideas.repec.org/a/taf/apmtfi/v2y1995i2p117-133.html
   My bibliography  Save this article

Uncertain volatility and the risk-free synthesis of derivatives

Author

Listed:
  • T. J. Lyons

Abstract

To price contingent claims in a multidimensional frictionless security market it is sufficient that the volatility of the security process is a known function of price and time. In this note we introduce optimal and risk-free strategies for intermediaries in such markets to meet their obligations when the volatility is unknown, and is only assumed to lie in some convex region depending on the prices of the underlying securities and time. Our approach is underpinned by the theory of totally non-linear parabolic partial differential equations (Krylov and Safanov, 1979; Wang, 1992) and the non-stochastic approach to Ito's formation first introduced by Follmer (1981a,b). In these more general conditions of unknown volatility, the optimal risk-free trading strategy will, necessarily, produce an unpredictable surplus over the minimum assets required at any time to meet the liabilities. This surplus, which could be released to the intermediary or to the client, is not required to meet the contingent claim. One sees that the effect of unknown volatility is the creation of a 'with profits' policy, where a premium is paid at the beginning, the contingent claim is collected at the terminal time, but that in addition an unpredictable surplus available as well. The risk-free initial premium required to meet the contingent claim is given by the solution to the Dirichlet problem for a totally non-linear parabolic equation of the Pucci-Bellman type. The existence of a risk-free strategy starting with this minimum sum is dependent upon theorems ensuring the regularity of the solution and upon a non-probabilistic understanding of Ito's change of variable formulae. To illustrate the ideas we give a very simple example of a one-dimensional barrier option where the maximum Black-Scholes price of the option over different fixed values for the volatility lying in an interval always underestimates the risk-free 'price' under the assumption that the volatility can vary within the same interval. This paper puts together rather standard mathematical ideas. However, the author hopes that the overall result is more than the sum of its parts. The ability to hedge under conditions of uncertain volatility seems to be of considerable practical importance. In addition it would be interesting if these ideas explained some features in the design of existing contracts.

Suggested Citation

  • T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    • Handle: RePEc:taf:apmtfi:v:2:y:1995:i:2:p:117-133
    DOI: 10.1080/13504869500000007
    as

    Download full text from publisher

    File URL: http://www.tandfonline.com/doi/abs/10.1080/13504869500000007
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1080/13504869500000007?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. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    2. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    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. John Armstrong & Claudio Bellani & Damiano Brigo & Thomas Cass, 2021. "Option pricing models without probability: a rough paths approach," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1494-1521, October.
    2. Marcelo F. Perillo, 2021. "Valuación de Títulos de Deuda Indexados al Comportamiento de un Índice Accionario: Un Modelo sin Riesgo de Crédito," CEMA Working Papers: Serie Documentos de Trabajo. 784, Universidad del CEMA.
    3. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    4. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    5. Karl Friedrich Mina & Gerald H. L. Cheang & Carl Chiarella, 2015. "Approximate Hedging Of Options Under Jump-Diffusion Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(04), pages 1-26.
    6. Anlong Li, 1992. "Binomial approximation in financial models: computational simplicity and convergence," Working Papers (Old Series) 9201, Federal Reserve Bank of Cleveland.
    7. Timothy Johnson, 2015. "Reciprocity as a Foundation of Financial Economics," Journal of Business Ethics, Springer, vol. 131(1), pages 43-67, September.
    8. Jamshidian, Farshid, 2008. "Numeraire Invariance and application to Option Pricing and Hedging," MPRA Paper 7167, University Library of Munich, Germany.
    9. Björn Lutz, 2010. "Pricing of Derivatives on Mean-Reverting Assets," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-642-02909-7, October.
    10. Fergusson, Kevin, 2020. "Less-Expensive Valuation And Reserving Of Long-Dated Variable Annuities When Interest Rates And Mortality Rates Are Stochastic," ASTIN Bulletin, Cambridge University Press, vol. 50(2), pages 381-417, May.
    11. Keppo, Jussi & Moscarini, Giuseppe & Smith, Lones, 2008. "The demand for information: More heat than light," Journal of Economic Theory, Elsevier, vol. 138(1), pages 21-50, January.
    12. Robert Elliott & Tak Siu, 2015. "Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(2), pages 133-149, May.
    13. Jian Guo & Saizhuo Wang & Lionel M. Ni & Heung-Yeung Shum, 2022. "Quant 4.0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence," Papers 2301.04020, arXiv.org.
    14. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    15. A. Fiori Maccioni, 2011. "The risk neutral valuation paradox," Working Paper CRENoS 201112, Centre for North South Economic Research, University of Cagliari and Sassari, Sardinia.
    16. Bossaerts, P. & Ghysels, E. & Gourieroux, C., 1996. "Arbitrage-Based Pricing when Volatility is Stochastic," Cahiers de recherche 9615, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
    17. Sebastian Jaimungal, 2022. "Reinforcement learning and stochastic optimisation," Finance and Stochastics, Springer, vol. 26(1), pages 103-129, January.
    18. Gregory C. Chow, 2003. "Duplicating Contingent Claims by the Lagrange Method," Finance 0306004, University Library of Munich, Germany.
    19. repec:uts:finphd:40 is not listed on IDEAS
    20. Zhu, Ke & Ling, Shiqing, 2015. "Model-based pricing for financial derivatives," Journal of Econometrics, Elsevier, vol. 187(2), pages 447-457.
    21. Geman, Hélyette, 2005. "From measure changes to time changes in asset pricing," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2701-2722, November.

    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:2:y:1995:i:2:p:117-133. 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.