Applications of quantum stochastic calculus and open quantum systems, in the modelling of the financial market

Hicks, Will (2024) Applications of quantum stochastic calculus and open quantum systems, in the modelling of the financial market. Post-Doctoral thesis, Memorial University of Newfoundland.

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Abstract

In this thesis we develop methods that enable the study of the phenomenon of noncommutativity in financial markets. In many cases, for example the purchasing of an item from a supermarket, prices behave in an essentially commutative fashion. Whilst the price may increase or decrease over time, when one comes to execute a transaction, the price is generally known in advance, and is not impacted (within reason) by the size of the transaction or what else is going on in the supermarket. Often financial markets do not operate in this simple fashion. The execution price for a transaction cannot always be known in advance, and factors such as the size of the trade, and what other trades happen in the lead up to our transaction, impact the price we will achieve. This motivates us to study financial markets using quantum probability. We start by applying the techniques of quantum stochastic calculus, and building on the quantum Black- Scholes approach of Accardi and Boukas. We then investigate the use of the open quantum systems method. We define an interaction between the financial market and its external environment, before using the singular coupling limit to derive a Markovian approximation to the master equation for the market dynamics. We go onto consider the role of the information entropy in the modelling of the financial market, and consider the relation between the classical and quantum approaches in this regard. We also show how to incorporate financial effects such as trade size, market risk appetite and bid-offer spread, that arguably lie behind noncommutativity, into the modelling approach. In each case, we investigate the impact of introducing noncommutative like behaviour to classical models, using analytical and numerical methods.

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