A Nonlocal Approach to the Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

Abstract

The Accardi-Boukas quantum Black-Scholes equation ([1]) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus ([14]). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

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

“…• As markets are more and more dominated by algorithms, and execution times fall correspondingly, this effect will become more and more pronounced. The properties of the resulting quantum Black-Scholes equations are further discussed in [12], [13], and [14]. It has also been noted (for example by Haven in [10], and discussed further by Melnyk and Tuluzov in [16]) that the risk in trading financial derivatives cannot be fully hedged in a 'true' quantum model.…”

“…The benefits of this approach are that one can introduce diffusions within noncommutative spaces which mirror many of the real effects of the financial markets, that are difficult to model using 'classical' Ito processes (for example see [12], [13], and [14]). However, we will see that this particular choice of quantization means that the internal dynamics of the quantum system K governed by the Hamiltonian H are lost, and have no impact on the derivative price.…”

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

“…The properties of the resulting quantum Black-Scholes equations are further discussed in [12], [13], and [14]. It has also been noted (for example by Haven in [10], and discussed further by Melnyk and Tuluzov in [16]) that the risk in trading financial derivatives cannot be fully hedged in a 'true' quantum model.…”

Modelling Illiquid Stocks Using Quantum Stochastic Calculus

“…Quantum stochastic calculus was first applied to the problem of derivative pricing, and Mathematical Finance, by Accardi & Boukas in [1], where the authors derive a general form for a Quantum Black-Scholes equation. In [6]- [8], properties of different example models that can be derived using the Quantum Black-Scholes approach are also investigated.…”