Scaling Invariance and Contingent Claim Pricing

Centrum voor Wiskunde en Informatica, MAS Working Paper No. R9914

18 Pages Posted: 17 Nov 1999

See all articles by Jiri Hoogland

Jiri Hoogland

Centrum voor Wiskunde en Informatica (CWI)

Dimitri Neumann

Centrum voor Wiskunde en Informatica (CWI)

Date Written: June 14, 1999

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 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.

JEL Classification: G12, G13

Suggested Citation

Hoogland, Jiri and Neumann, Dimitri, Scaling Invariance and Contingent Claim Pricing (June 14, 1999). Centrum voor Wiskunde en Informatica, MAS Working Paper No. R9914. Available at SSRN: https://ssrn.com/abstract=180888 or http://dx.doi.org/10.2139/ssrn.180888

Jiri Hoogland (Contact Author)

Centrum voor Wiskunde en Informatica (CWI) ( email )

P.O.Box 94079
MAS
NL-1090 GB Amsterdam
Netherlands
+31(20)5924102 (Phone)
+31(20)5924199 (Fax)

Dimitri Neumann

Centrum voor Wiskunde en Informatica (CWI) ( email )

P.O.Box 94079
MAS
NL-1090 GB Amsterdam
Netherlands
+31(20)5924102 (Phone)
+31(20)5924199 (Fax)

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