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Scale invariance and contingent claim pricing

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Author Info

  • 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.

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Bibliographic Info

Paper provided by EconWPA in its series Finance with number 9907002.

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Length: 17 pages
Date of creation: 16 Jul 1999
Date of revision:
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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Related research

Keywords: contingent claim pricing; scale-invariance; homogeneity; partial differential equation;

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References

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  1. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
  2. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
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Citations

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Cited by:
  1. Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing II: Path-dependent contingent claims," Finance 9907003, EconWPA.
  2. 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, Reading University, revised Nov 2005.
  3. Jiri Hoogland & Dimitri Neumann, 2001. "Asians and cash dividends: Exploiting symmetries in pricing theory," Finance 0105002, EconWPA.
  4. Jiri Hoogland & Dimitri Neumann, 2001. "Tradable Schemes," Finance 0105003, EconWPA.

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