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

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  • 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.

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  • 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
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    References listed on IDEAS

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    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.
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