An Application of Dirac's Interaction Picture to Option Pricing

TL;DR

This paper applies Dirac's interaction picture from quantum mechanics to derive solutions for the Black-Scholes option pricing model with time-dependent arbitrage bubbles, introducing new analytical methods and symmetry properties.

Contribution

It introduces a novel application of quantum interaction picture to option pricing, providing approximate and exact solutions for perturbed Black-Scholes equations and revealing symmetry invariances.

Findings

Derived approximate solutions using Greeks Delta, Gamma, and Speed.

Constructed exact solutions involving higher-order derivatives.

Identified invariance under interest rate and asset mean interchange.

Abstract

In this paper, the Dirac's quantum mechanical interaction picture is applied to option pricing to obtain a solution of the Black-Scholes equation in the presence of a time-dependent arbitrage bubble. In particular, for the case of a call perturbed by a square bubble, an approximate solution (valid up third order in a perturbation series) is given in terms of the three first Greeks: Delta, Gamma, and Speed. Then an exact solution is constructed in terms of all higher order $S$-derivatives of the Black-Scholes formula. It is also shown that the interacting Black-Scholes equation is invariant under a discrete transformation that interchanges the interest rate with the mean of the underlying asset and vice versa. This implies that the interacting Black-Scholes equation can be written in a 'low energy' and a 'high energy' form, in such a way that the high-interaction limit of the low energy form corresponds to the weak-interaction limit of the high energy form. One can apply a perturbative analysis to the high energy form to study the high-interaction limit of the low energy form.