IDEAS home Printed from https://ideas.repec.org/p/arx/papers/2308.13717.html
   My bibliography  Save this paper

Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing

Author

Listed:
  • Ricardo T. Fernholz
  • Robert Fernholz

Abstract

This paper is a synthesis of the theories of portfolio generating functions and rational option pricing. For a family of n >= 2 assets with prices represented by strictly positive continuous semimartingales, a contingent claim function is a scalable positive C^{2,1} function of the asset prices and time. We extend the theory of portfolio generation to measure the value of portfolios generated by contingent claim functions directly, with no numeraire portfolio. We show that if a contingent claim function satisfies a particular parabolic differential equation, then the value of the portfolio generated by that contingent claim function will replicate the value of the function. This differential equation is a general form of the Black-Scholes equation.

Suggested Citation

  • Ricardo T. Fernholz & Robert Fernholz, 2023. "Portfolios Generated by Contingent Claim Functions, with Applications to Option Pricing," Papers 2308.13717, arXiv.org, revised Jan 2024.
    • Handle: RePEc:arx:papers:2308.13717
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/2308.13717
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Ioannis Karatzas & Johannes Ruf, 2017. "Trading strategies generated by Lyapunov functions," Finance and Stochastics, Springer, vol. 21(3), pages 753-787, July.
    2. Karatzas, Ioannis & Ruf, Johannes, 2017. "Trading strategies generated by Lyapunov functions," LSE Research Online Documents on Economics 69177, London School of Economics and Political Science, LSE Library.
    3. Robert Fernholz, 1999. "Portfolio Generating Functions," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 15, pages 344-367, World Scientific Publishing Co. Pte. Ltd..
    4. 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. Johannes Ruf & Kangjianan Xie, 2018. "Generalised Lyapunov Functions and Functionally Generated Trading Strategies," Papers 1801.07817, arXiv.org.
    2. Ricardo T. Fernholz & Robert Fernholz, 2022. "Permutation-weighted portfolios and the efficiency of commodity futures markets," Annals of Finance, Springer, vol. 18(1), pages 81-108, March.
    3. Ioannis Karatzas & Donghan Kim, 2020. "Trading strategies generated pathwise by functions of market weights," Finance and Stochastics, Springer, vol. 24(2), pages 423-463, April.
    4. Donghan Kim, 2022. "Market-to-book Ratio in Stochastic Portfolio Theory," Papers 2206.03742, arXiv.org.
    5. Erhan Bayraktar & Donghan Kim & Abhishek Tilva, 2023. "Quantifying dimensional change in stochastic portfolio theory," Papers 2303.00858, arXiv.org, revised Apr 2023.
    6. Ting-Kam Leonard Wong, 2017. "On portfolios generated by optimal transport," Papers 1709.03169, arXiv.org, revised Sep 2017.
    7. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.
    8. Ioannis Karatzas & Donghan Kim, 2018. "Trading Strategies Generated Pathwise by Functions of Market Weights," Papers 1809.10123, arXiv.org, revised Mar 2019.
    9. Donghan Kim, 2019. "Open Markets," Papers 1912.13110, arXiv.org.
    10. Ioannis Karatzas & Donghan Kim, 2021. "Open markets," Mathematical Finance, Wiley Blackwell, vol. 31(4), pages 1111-1161, October.
    11. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Prömel, 2023. "Model‐free portfolio theory: A rough path approach," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 709-765, July.
    12. Kangjianan Xie, 2020. "Leakage of rank-dependent functionally generated trading strategies," Annals of Finance, Springer, vol. 16(4), pages 573-591, December.
    13. Fernholz, E. Robert & Karatzas, Ioannis & Ruf, Johannes, 2018. "Volatility and arbitrage," LSE Research Online Documents on Economics 75234, London School of Economics and Political Science, LSE Library.
    14. Donghan Kim, 2023. "Market-to-book ratio in stochastic portfolio theory," Finance and Stochastics, Springer, vol. 27(2), pages 401-434, April.
    15. Ruf, Johannes & Xie, Kangjianan, 2019. "Generalised Lyapunov functions and functionally generated trading strategies," LSE Research Online Documents on Economics 102424, London School of Economics and Political Science, LSE Library.
    16. Johannes Ruf & Kangjianan Xie, 2019. "The impact of proportional transaction costs on systematically generated portfolios," Papers 1904.08925, arXiv.org.
    17. Patrick Mijatovic, 2021. "Beating the Market with Generalized Generating Portfolios," Papers 2101.07084, arXiv.org.
    18. Martin Larsson & Johannes Ruf, 2021. "Relative arbitrage: Sharp time horizons and motion by curvature," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 885-906, July.
    19. Cuchiero, Christa, 2019. "Polynomial processes in stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1829-1872.
    20. Ricardo T. Fernholz & Caleb Stroup, 2018. "Asset Price Distributions and Efficient Markets," Papers 1810.12840, arXiv.org.

    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:2308.13717. 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.