Scale-invariance and contingent claim pricing

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 contigent claims, i.e. tradable instruments, whose prices depend on the prices of other tradables. We show that this property induces local scale-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 posses a manifest gauge-invariance. A unique solution can only be given when we impose restrictions on the drifts of 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.