arXiv subjects

Will Hicks

Publications and source records attributed to Will Hicks.

12 recordsLinked to original sources

arXiv2026

Market Makers and Risk Aversion: A Hamiltonian Approach to the Excess Volatility Puzzle

In this article we model chaotic dynamics in financial markets by treating the market price, and market makers' inventory, as anharmonic oscillators with a nonlinear coupling. The market makers' risk appetite being the key parameter that determines the degree of chaos in the system. The article demonstrates that whilst external shocks and random noise are important in the treatment of financial time-series, they are not necessary in order to generate unpredictable price changes.

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arXiv2025

Modelling Financial Market Imperfection Using Open Quantum Systems

We start with the idea that open quantum systems can be used to represent financial markets by modelling events from the external environment and their impact on the market price. We show how to characterize distinct orbits of the time evolution, and look at the development of the reduced density matrix, that represents the state of the market, over long time frames. In particular we distinguish between classical and non-classical modes of time evolution. We show that whilst both tend to the same set of maximum entropy states, this occurs faster in classical systems, with a knock on effect on the resulting probability distributions. We demonstrate how non-classical modes of time-evolution can be used to incorporate factors such as illiquid trades and imperfect trading mechanisms, and distinguish between different mechanisms of non-classical time evolution.

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arXiv2024

Modelling Uncertain Volatility Using Quantum Stochastic Calculus: Unitary vs Non-Unitary Time Evolution

In this article we look at stochastic processes with uncertain parameters, and consider different ways in which information is obtained when carrying out observations. For example we focus on the case of a the random evolution of a traded financial asset price with uncertain volatility. The quantum approach presented, allows us to encode different volatility levels in a state acting on a Hilbert space. We consider different means of defining projective measurements in order to track the evolution of a traded market price, and discuss the results of different Monte-Carlo simulations.

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arXiv2024

Information Entropy of the Financial Market: Modelling Random Processes Using Open Quantum Systems

We discuss the role of information entropy on the behaviour of random processes, and how this might take effect in the dynamics of financial market prices. We then go on to show how the Open Quantum Systems approach can be used as a more flexible alternative to classical methods in terms of modelling the entropy gain of a random process. We start by describing an open quantum system that can be used to model the state of a financial market. We then go on to show how to represent an essentially classical diffusion in this framework. Finally, we show how by relaxing certain assumptions, one can generate interesting and essentially non-classical results, which are highlighted through numerical simulations.

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arXiv2023

Modelling Illiquid Stocks Using Quantum Stochastic Calculus

Quantum Stochastic Calculus can be used as a means by which randomness can be introduced to observables acting on a Hilbert space. In this article we show how the mechanisms of Quantum Stochastic Calculus can be used to extend the classical Black-Scholes framework by incorporating a breakdown in the liquidity of a traded asset. This is captured via the widening of the bid offer spread, and the impact on the nature of the resulting probability distribution is modelled in this work.

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arXiv2023

Modelling Illiquid Stocks Using Quantum Stochastic Calculus: Asymptotic Methods

This article investigates the Fokker-Planck equations that arise from the application of quantum stochastic calculus to the modelling of illiquid financial markets, using asymptotic methods. We present a power series solution for quantum stochastic processes with a non-zero conservation process. Whilst the series in question are in general divergent, we show they can be used to approximate solutions for longer time frames, and provide estimates for the relative error on the higher order terms.

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arXiv2021

Wild Randomness, and the application of Hyperbolic Diffusion in Financial Modelling

The application of the Cauchy distribution has often been discussed as a potential model of the financial markets. In particular the way in which single extreme, or "Black Swan", events can impact long term historical moments, is often cited. In this article we show how one can construct Martingale processes, which have marginal distributions that tend to the Cauchy distribution in the large volatility limit. This provides financial justification to approaches discussed by other authors, and highlights an example of how quantum probability can be used to construct non-Gaussian Martingales. We go on to illustrate links with hyperbolic diffusion, and discuss the insight this provides.

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arXiv2020

Pseudo-Hermiticity, Martingale Processes and Non-Arbitrage Pricing

Financial models based on the Wick product, and White Noise formalism have previously been suggested in order to incorporate integrals with respect to fractional Brownian motion. It has also been pointed out that this leads naturally to a quantum mechanical interpretation of the financial market. In this article we pursue this idea further, and in particular show how the framework of quantum probability can be used to construct Martingales, without relying on Brownian integrals. We go on to suggest benefits of doing so, and avenues for future work.

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arXiv2019

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

In this article we model a financial derivative price as an observable on the market state function. We apply geometric techniques to integrating the Heisenberg Equation of Motion. We 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 which involves modelling the financial market as a quantum observable, and introduces randomness through the Hudson-Parthasarathy quantum stochastic calculus. In particular we aim to differentiate randomness that is introduced through external noise (quantum stochastic calculus) and randomness that is fundamental to a quantum system (Heisenberg Equation of Motion).

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arXiv2019

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

The Accardi-Boukas quantum Black-Scholes equation 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. 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.

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arXiv2018

PT Symmetry, Non-Gaussian Path Integrals, and the Quantum Black-Scholes Equation

The Accardi-Boukas quantum Black-Scholes framework, provides a means by which one can apply the Hudson-Parthasarathy quantum stochastic calculus to problems in finance. Solutions to these equations can be modelled using nonlocal diffusion processes, via a Kramers-Moyal expansion, and this provides useful tools to understand their behaviour. In this paper we develop further links between quantum stochastic processes, and nonlocal diffusions, by inverting the question, and showing how certain nonlocal diffusions can be written as quantum stochastic processes. We then go on to show how one can use path integral formalism, and PT symmetric quantum mechanics, to build a non-Gaussian kernel function for the Accardi-Boukas quantum Black-Scholes. Behaviours observed in the real market are a natural model output, rather than something that must be deliberately included.

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arXiv2018

Nonlocal Diffusions and The Quantum Black-Scholes Equation: Modelling the Market Fear Factor

In this paper, we establish a link between quantum stochastic processes, and nonlocal diffusions. We demonstrate how the non-commutative Black-Scholes equation of Accardi & Boukas (Luigi Accardi, Andreas Boukas, 'The Quantum Black-Scholes Equation', Jun 2007, available at arXiv:0706.1300v1) can be written in integral form. This enables the application of the Monte-Carlo methods adapted to McKean stochastic differential equations (H. P. McKean, 'A class of Markov processes associated with nonlinear parabolic equations', Proc. Natl. Acad. Sci. U.S.A., 56(6):1907-1911, 1966) for the simulation of solutions. We show how unitary transformations can be applied to classical Black-Scholes systems to introduce novel quantum effects. These have a simple economic interpretation as a market `fear factor', whereby recent market turbulence causes an increase in volatility going forward, that is not linked to either the local volatility function or an additional stochastic variable. Lastly, we extend this system to 2 variables, and consider Quantum models for bid-offer spread dynamics.

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