A Quantum Finance Model

Abstract

Both academic research and practical application of mathematical finance have been extremely fruitful since the seminal work of Black-Scholes-Merton in the early 1970s. In this framework, the prices of financial assets are modeled as stochastic processes in probability spaces inside which the machinery of stochastic calculus is a powerful tool. The fundamental asset pricing theorem states that the absence of arbitrage opportunities in a market is equivalent to the existence of a probability measure, equivalent to the objective probability, under which the discounted prices of the assets become local martingales. This linkage between finance on the one hand and the probability theory on the other is the key to the success of mathematical finance. In this note, we show that it is possible to extend the classical probability model to a quantum probability model. The classical stochastic calculus is replaced by its quantum counterpart on the Boson Fock space. In particular, we show that the fundamental asset pricing theorem remains valid in this non-commutative setting. As an application, prices of quantum European options are obtained.