Quantum Finance

Quantum theory is used to model secondary financial markets. Contrary to stochastic descriptions, the formalism emphasizes the importance of trading in determining the value of a security. All possible realizations of investors holding securities and cash is taken as the basis of the Hilbert space of market states. The temporal evolution of an isolated market is unitary in this space. Linear operators representing basic financial transactions such as cash transfer and the buying or selling of securities are constructed and simple model Hamiltonians that generate the temporal evolution due to cash flows and the trading of securities are proposed. The Hamiltonian describing financial transactions becomes local when the profit/loss from trading is small compared to the turnover. This approximation may describe a highly liquid and efficient stock market. The lognormal probability distribution for the price of a stock with a variance that is proportional to the elapsed time is reproduced for an equilibrium market. The asymptotic volatility of a stock in this case is related to the long-term probability that it is traded.Comment: Improved 32 page version that is to appear in Physica A. One appendix scrapped, typos corrected, section on conditions for efficient markets extended. References adde

Quantum computational finance: martingale asset pricing for incomplete markets

A derivative is a financial security whose value is a function of underlying traded assets and market outcomes. Pricing a financial derivative involves setting up a market model, finding a martingale (``fair game") probability measure for the model from the given asset prices, and using that probability measure to price the derivative. When the number of underlying assets and/or the number of market outcomes in the model is large, pricing can be computationally demanding. We show that a variety of quantum techniques can be applied to the pricing problem in finance, with a particular focus on incomplete markets. We discuss three different methods that are distinct from previous works: they do not use the quantum algorithms for Monte Carlo estimation and they extract the martingale measure from market variables akin to bootstrapping, a common practice among financial institutions. The first two methods are based on a formulation of the pricing problem into a linear program and are using respectively the quantum zero-sum game algorithm and the quantum simplex algorithm as subroutines. For the last algorithm, we formalize a new market assumption milder than market completeness for which quantum linear systems solvers can be applied with the associated potential for large speedups. As a prototype use case, we conduct numerical experiments in the framework of the Black-Scholes-Merton model.Comment: 31 pages, 6 figure

Quantum Probability Theoretic Asset Return Modeling: A Novel Schr\"odinger-Like Trading Equation and Multimodal Distribution

Quantum theory provides a comprehensive framework for quantifying uncertainty, often applied in quantum finance to explore the stochastic nature of asset returns. This perspective likens returns to microscopic particle motion, governed by quantum probabilities akin to physical laws. However, such approaches presuppose specific microscopic quantum effects in return changes, a premise criticized for lack of guarantee. This paper diverges by asserting that quantum probability is a mathematical extension of classical probability to complex numbers. It isn't exclusively tied to microscopic quantum phenomena, bypassing the need for quantum effects in returns.By directly linking quantum probability's mathematical structure to traders' decisions and market behaviors, it avoids assuming quantum effects for returns and invoking the wave function. The complex phase of quantum probability, capturing transitions between long and short decisions while considering information interaction among traders, offers an inherent advantage over classical probability in characterizing the multimodal distribution of asset returns.Utilizing Fourier decomposition, we derive a Schr\"odinger-like trading equation, where each term explicitly corresponds to implications of market trading. The equation indicates discrete energy levels in financial trading, with returns following a normal distribution at the lowest level. As the market transitions to higher trading levels, a phase shift occurs in the return distribution, leading to multimodality and fat tails. Empirical research on the Chinese stock market supports the existence of energy levels and multimodal distributions derived from this quantum probability asset returns model