Schr\"odinger group and quantum finance

TL;DR

This paper explores the symmetries of the Black-Scholes equation by linking it to Schr"odinger group symmetries through quantum mechanics analogies, revealing new invariance properties in financial modeling.

Contribution

It demonstrates that the Black-Scholes equation is invariant under the Schr"odinger group by relating financial variables to quantum particle symmetries, introducing a novel algebraic perspective.

Findings

Black-Scholes equation exhibits Schr"odinger group invariance

Financial variables can be mapped to quantum symmetries

A Schr"odinger algebra representation for finance is constructed

Abstract

Using the one dimensional free particle symmetries, the quantum finance symmetries are obtained. Namely, it is shown that Black-Scholes equation is invariant under Schr\"odinger group. In order to do this, the one dimensional free non-relativistic particle and its symmetries are revisited. To get the Black-Scholes equation symmetries, the particle mass is identified as the inverse of square of the volatility. Furthermore, using financial variables, a Schr\"odinger algebra representation is constructed.