On the Local equivalence of the Black Scholes and the Merton Garman equations

TL;DR

This paper demonstrates that the Black-Scholes and Merton-Garman equations are locally equivalent through gauge symmetry, revealing a functional relation between stock prices and volatility that can enhance market volatility estimation.

Contribution

It introduces the concept of gauge-Hamiltonian to unify Black-Scholes and Merton-Garman equations and explores the implications of volatility as a gauge field.

Findings

Black-Scholes and Merton-Garman equations are locally equivalent.

Gauge character of volatility leads to specific stock-volatility relations.

Extended martingale condition for gauge-Hamiltonian is proposed.

Abstract

It was demonstrated previously that the stochastic volatility emerges as the gauge field necessary for restoring the local symmetry under changes of the prices of the stocks inside the Black-Scholes (BS) equation. When this occurs, then a Merton-Garman-like equation emerges. From the perspective of manifolds, this means that the Black-Scholes equation and the Merton-Garman (MG) one can be considered as locally equivalent. In this scenario, the MG Hamiltonian is a special case of a more general Hamiltonian, here called gauge-Hamiltonian. We then show that the gauge character of the volatility implies some specific functional relation between the prices of the stock and the volatility. The connection between the prices of the stocks and the volatility, is a powerful tool for improving the volatility estimations in the stock market, which is a key ingredient for the investors to make good decisions. Finally, we define an extended version of the martingale condition, defined for the gauge-Hamiltonian.