Scale-invariance and contingent claim pricing

TL;DR

This paper introduces a scale-invariance principle in contingent claim pricing, deriving a PDE without martingale methods, simplifying the conceptual framework and connecting to classical models like Black-Scholes.

Contribution

It develops a scale-invariance approach to pricing contingent claims, resulting in a PDE formulation that avoids martingale techniques and emphasizes gauge-invariance.

Findings

The PDE formulation aligns with Black-Scholes in standard cases.

Pricing problems exhibit gauge-invariance, simplifying analysis.

Market drifts and volatilities impose restrictions for unique solutions.

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.