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Scale invariance and contingent claim pricing

Author

Listed:
  • Jiri Hoogland

    (CWI, Amsterdam)

  • Dimitri Neumann

    (CWI, Amsterdam)

Abstract

Prices of tradables can only be expressed relative to each other at any instant of time. This fundamental fact should therefore also hold for contingent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces a local scaling invariance in the problem of pricing contingent claims. Due to this symmetry we do {\it not\/} require any martingale techniques to arrive at the price of a claim. If the tradables are driven by Brownian motion, we find, in a natural way, that this price satisfies a PDE. Both possess a manifest gauge-invariance. A unique solution can only be given when we impose restrictions on the drifts and volatilities of the tradables, i.e. the underlying market structure. We give some examples of the application of this PDE to the pricing of claims. In the Black- Scholes world we show the equivalence of our formulation with the standard approach. It is stressed that the formulation in terms of tradables leads to a significant conceptual simplification of the pricing-problem.

Suggested Citation

  • Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing," Finance 9907002, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:9907002
    Note: Type of Document - PDF; prepared on NT/Latex; to print on PDF printer; pages: 17 . See also http://www.cwi.nl/~jiri for postscript version version
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    References listed on IDEAS

    as
    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters, in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305, World Scientific Publishing Co. Pte. Ltd..
    3. Robert A. Jarrow & Arkadev Chatterjea, 2019. "The Heath–Jarrow–Morton Libor Model," World Scientific Book Chapters, in: An Introduction to Derivative Securities, Financial Markets, and Risk Management, chapter 25, pages 618-654, World Scientific Publishing Co. Pte. Ltd..
    4. Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing II: Path-dependent contingent claims," Finance 9907003, University Library of Munich, Germany.
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    Citations

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    Cited by:

    1. Carol Alexandra & Leonardo M. Nogueira, 2005. "Optimal Hedging and Scale Inavriance: A Taxonomy of Option Pricing Models," ICMA Centre Discussion Papers in Finance icma-dp2005-10, Henley Business School, University of Reading, revised Nov 2005.
    2. Wanxiao Tang & Jun Zhao & Peibiao Zhao, 2019. "Geometric No-Arbitrage Analysis in the Dynamic Financial Market with Transaction Costs," JRFM, MDPI, vol. 12(1), pages 1-17, February.
    3. Jiri Hoogland & Dimitri Neumann, 2001. "Tradable Schemes," Finance 0105003, University Library of Munich, Germany.
    4. Chih-Chen Hsu & Chung-Gee Lin & Tsung-Jung Kuo, 2020. "Pricing of Arithmetic Asian Options under Stochastic Volatility Dynamics: Overcoming the Risks of High-Frequency Trading," Mathematics, MDPI, vol. 8(12), pages 1-16, December.
    5. Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing II: Path-dependent contingent claims," Finance 9907003, University Library of Munich, Germany.
    6. Jiri Hoogland & Dimitri Neumann, 2001. "Asians and cash dividends: Exploiting symmetries in pricing theory," Finance 0105002, University Library of Munich, Germany.
    7. Boyle, Phelim & Potapchik, Alexander, 2008. "Prices and sensitivities of Asian options: A survey," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 189-211, February.

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    More about this item

    Keywords

    contingent claim pricing; scale-invariance; homogeneity; partial differential equation;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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