Closed Quantum Black-Scholes: Quantum Drift and the Heisenberg Equation of Motion

Abstract

In this article we model a financial derivative price as an observable on a market state function, with a view to understanding how some of the non-commutative behaviour of the financial market impacts the dynamics. We integrate the Heisenberg Equation of Motion, by using a Riemannian metric, and illustrate how the non-commutative nature of the model introduces quantum interference effects that can act as either a drag or a boost on the resulting return. The ultimate objective is to investigate the nature of quantum drift in the Accardi-Boukas quantum Black-Scholes framework ([1]) which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus ([15]). In particular we aim to differentiate between randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is intrinsic to a quantum system (Heisenberg Equation of Motion).2010 Mathematics Subject Classification. Primary 81S25; Secondary 91B70.