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Local Scale Invariance And Contingent Claim Pricing

Author

Listed:
  • J. K. HOOGLAND

    (Centrum voor Wiskun de en Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands)

  • C. D. D. NEUMANN

    (Centrum voor Wiskun de en Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands)

Abstract

Prices of tradables can only be expressed relative to one another 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 scale invariance in the problem of pricing contingent claims. Due to this symmetry we donotrequire 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

  • J. K. Hoogland & C. D. D. Neumann, 2001. "Local Scale Invariance And Contingent Claim Pricing," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-21.
    • Handle: RePEc:wsi:ijtafx:v:04:y:2001:i:01:n:s0219024901000857
    DOI: 10.1142/S0219024901000857
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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. Martin Gremm, 2016. "Global Gauge Symmetries, Risk-Free Portfolios, and the Risk-Free Rate," Papers 1605.03551, arXiv.org.
    3. Andrew Ming-Long Wang & Yu-Hong Liu & Yi-Long Hsiao, 2009. "Barrier option pricing: a hybrid method approach," Quantitative Finance, Taylor & Francis Journals, vol. 9(3), pages 341-352.
    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. Carol Alexander & Leonardo M. Nogueira, 2006. "Hedging Options with Scale-Invariant Models," ICMA Centre Discussion Papers in Finance icma-dp2006-03, Henley Business School, University of Reading.
    6. Vellekoop, M.H. & Vd Kamp, A.A. & Post, B.A., 2006. "Pricing and hedging guaranteed returns on mix funds," Insurance: Mathematics and Economics, Elsevier, vol. 38(3), pages 585-598, June.
    7. Alexander, Carol & Nogueira, Leonardo M., 2007. "Model-free hedge ratios and scale-invariant models," Journal of Banking & Finance, Elsevier, vol. 31(6), pages 1839-1861, June.
    8. Rei[ss], Oliver & Schoenmakers, John & Schweizer, Martin, 2007. "From structural assumptions to a link between assets and interest rates," Journal of Economic Dynamics and Control, Elsevier, vol. 31(2), pages 593-612, February.
    9. 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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