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Journal of Stochastic Analysis

MODELLING UNCERTAIN VOLATILITY USING QUANTUM STOCHASTIC CALCULUS: UNITARY VS NON-UNITARY TIME EVOLUTION

Will Hicks

Published Sep 1, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.31390/josa.5.3.02

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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.
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.
Measurement is a key element of quantum theory where only one possible outcome comes to pass, while other ones disappear. Similar phenomena have been observed in psychology, language processing, and economics. Mathematical descriptions that employ elements of quantum theory have successfully modeled effects observable outside of subatomic physics. Price and value are important concepts in finance. We distinguish between price as something fixed in a transaction and publicly available, while value is an agent’s private assessment. Measurement and updates of both may happen in asynchronous timelines and the frequency of measurements might impact change. When considering derivative markets, usually the focus is on price dynamics given by a stochastic process – market liquidity is often assumed to be infinite. A more detailed consideration of liquidity in relation to price dynamics is desirable and this work outlines possible angles from quantum-inspired models. We discuss how market liquidity and volatility relate to each other in the residential real estate and stock market. We investigate quantum walks, quantum models of psychological decision making by individuals and groups, and the role that the framing of a question plays, all with the goal of an improved understanding of the link between observed price volatility and liquidity.
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.
Abstract In this work we study the unitary time-evolutions of quantum systems defined on infinite-dimensional separable time-dependent Hilbert spaces. Two possible cases are considered: a quantum system defined on a stochastic interval and another one defined on a Hilbert space with stochastic integration measure (stochastic time-dependent scalar product). The formulations of the two problems and a comparison with the general theory of open quantum systems are discussed. Possible physical applications of the situations considered are analyzed.