IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2208.09898.html
   My bibliography  Save this paper

Fair pricing and hedging under small perturbations of the num\'eraire on a finite probability space

Author

Listed:
  • William Busching
  • Delphine Hintz
  • Oleksii Mostovyi
  • Alexey Pozdnyakov

Abstract

We consider the problem of fair pricing and hedging under small perturbations of the num\'eraire. We show that for replicable claims, the change of num\'eraire affects neither the fair price nor the hedging strategy. For non-replicable claims, we demonstrate that is not the case. By reformulating the key stochastic control problem in a more tractable form, we show that both the fair price and optimal strategy are stable with respect to small perturbations of the num\'eraire. Further, our approach allows for explicit asymptotic formulas describing the fair price and hedging strategy's leading order correction terms. Mathematically, our results constitute stability and asymptotic analysis of a stochastic control problem under certain perturbations of the integrator of the controlled process, where constraints make this problem hard to analyze.

Suggested Citation

  • William Busching & Delphine Hintz & Oleksii Mostovyi & Alexey Pozdnyakov, 2022. "Fair pricing and hedging under small perturbations of the num\'eraire on a finite probability space," Papers 2208.09898, arXiv.org.
    • Handle: RePEc:arx:papers:2208.09898
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2208.09898
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Mostovyi, Oleksii, 2020. "Asymptotic analysis of the expected utility maximization problem with respect to perturbations of the numéraire," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4444-4469.
    2. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    3. Föllmer, H. & Schweizer, M., 1989. "Hedging by Sequential Regression: an Introduction to the Mathematics of Option Trading," ASTIN Bulletin, Cambridge University Press, vol. 19(S1), pages 29-42, November.
    4. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
    5. Sarah Boese & Tracy Cui & Samuel Johnston & Gianmarco Molino & Oleksii Mostovyi, 2020. "Stability and asymptotic analysis of the F\"ollmer-Schweizer decomposition on a finite probability space," Papers 2002.03286, arXiv.org, revised Jun 2020.
    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. Battulga Gankhuu, 2022. "Augmented Dynamic Gordon Growth Model," Papers 2201.06012, arXiv.org, revised Aug 2023.
    2. Ammann, Manuel & Kind, Axel & Wilde, Christian, 2003. "Are convertible bonds underpriced? An analysis of the French market," Journal of Banking & Finance, Elsevier, vol. 27(4), pages 635-653, April.
    3. Jérôme Detemple, 1999. "American Options: Symmetry Properties," CIRANO Working Papers 99s-45, CIRANO.
    4. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    5. Daniel Oda, 2013. "Introducing Liquidity Risk in the Contingent-Claim Analysis for the Banks," Working Papers Central Bank of Chile 681, Central Bank of Chile.
    6. Jamshidian, Farshid, 2008. "Numeraire Invariance and application to Option Pricing and Hedging," MPRA Paper 7167, University Library of Munich, Germany.
    7. Sven Rady, 1997. "Option pricing in the presence of natural boundaries and a quadratic diffusion term (*)," Finance and Stochastics, Springer, vol. 1(4), pages 331-344.
    8. Lok Man Michel Tong & Gianluca Marcato, 2018. "Modelling Competitive Mortgage Termination Option Strategies: Default vs Restructuring and Prepayment vs Defeasance," ERES eres2018_300, European Real Estate Society (ERES).
    9. Dominique Guegan & Jing Zhang, 2009. "Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market," PSE-Ecole d'économie de Paris (Postprint) halshs-00368336, HAL.
    10. Martzoukos, Spiros H., 2001. "The option on n assets with exchange rate and exercise price risk," Journal of Multinational Financial Management, Elsevier, vol. 11(1), pages 1-15, February.
    11. Battulga Gankhuu, 2021. "Equity--Linked Life Insurances on Maximum of Several Assets," Papers 2111.04038, arXiv.org, revised Aug 2023.
    12. Dominique Guegan & Jing Zang, 2009. "Pricing bivariate option under GARCH-GH model with dynamic copula: application for Chinese market," The European Journal of Finance, Taylor & Francis Journals, vol. 15(7-8), pages 777-795.
    13. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    14. Martins, José & Marques, Rui Cunha & Cruz, Carlos Oliveira, 2014. "Maximizing the value for money of PPP arrangements through flexibility: An application to airports," Journal of Air Transport Management, Elsevier, vol. 39(C), pages 72-80.
    15. Pennacchi, George G., 2005. "Risk-based capital standards, deposit insurance, and procyclicality," Journal of Financial Intermediation, Elsevier, vol. 14(4), pages 432-465, October.
    16. Ascione, Giacomo & Mehrdoust, Farshid & Orlando, Giuseppe & Samimi, Oldouz, 2023. "Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    17. Jamshidian, Farshid, 2007. "Exchange Options," MPRA Paper 4471, University Library of Munich, Germany, revised 14 Aug 2007.
    18. Marcel Philipp Müller & Sebastian Stöckl & Steffen Zimmermann & Bernd Heinrich, 2016. "Decision Support for IT Investment Projects," Business & Information Systems Engineering: The International Journal of WIRTSCHAFTSINFORMATIK, Springer;Gesellschaft für Informatik e.V. (GI), vol. 58(6), pages 381-396, December.
    19. Rombouts, Jeroen V.K. & Stentoft, Lars, 2011. "Multivariate option pricing with time varying volatility and correlations," Journal of Banking & Finance, Elsevier, vol. 35(9), pages 2267-2281, September.
    20. Sebehela, Tumellano, 2015. "Rationally financing an acquisition," Journal of Economics and Business, Elsevier, vol. 81(C), pages 1-20.

    More about this item

    Statistics

    Access and download statistics

    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:arx:papers:2208.09898. 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: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    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.