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arXiv:quant-ph/0110046

Quantum-like approach to financial risk: quantum anthropic principle

Edward W. Piotrowski, Sładkowski, J.

Published Oct 8, 2001• 35 citations
arXivHTMLPDF

Abstract

We continue the analysis of quantum-like description of market phenomena and economics. We show that it is possible to define a risk inclination operator acting in some Hilbert space that has a lot of common with quantum description of the harmonic oscillator. The approach has roots in the recently developed quantum game theory and quantum computing. A quantum anthropic principle is formulated

DOI
10.48550/arxiv.quant-ph/0110046

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We develop a comprehensive theoretical framework for quantum-enhanced risk modeling in financial systems, establishing mathematical foundations for representing and computing risk factors using quantum states and operations. The theory begins by formulating portfolio risk as quantum observables, where correlations between assets are naturally encoded in entangled quantum states, enabling exponentially compact representations of high-dimensional covariance structures. We derive theoretical bounds on the accuracy of quantum risk estimates and prove convergence theorems for quantum algorithms that compute value-at-risk, conditional value-atrisk, and higher-moment risk measures. Our framework incorporates counterdiabatic evolution techniques to navigate complex risk landscapes efficiently, establishing conditions under which quantum annealing approaches converge to optimal portfolio configurations with provable performance guarantees. We analyze the computational complexity of quantum versus classical risk calculation, identifying scenarios where quantum advantage emerges based on portfolio size, correlation structure complexity, and required solution precision. The theory extends to dynamic risk assessment under changing market conditions, proving that quantum algorithms can adapt to evolving risk profiles with polynomial overhead. We establish fundamental limits on quantum speedups for risk computation and identify problem structures that maximize quantum advantage. This theoretical foundation bridges quantum computing and quantitative finance, offering rigorous principles for designing quantum algorithms that address computational bottlenecks in modern risk management.
Abstract Finance stands as the most quantitatively advanced branch of the social sciences, yet it is also the most humbling. The mean-variance efficient frontier, the CAPM, arbitrage pricing relations, the Black–Scholes–Merton option formula and the efficient market hypothesis possess unparalleled formal elegance, yet they suffered a broad empirical refutation in the global financial crisis of 2008. Standard models understate tail risk, misinterpret the comovement of assets that are supposed to diversify risk, and presuppose liquidity and rationality — two features that vanish precisely when they are most needed. Behavioural finance has documented an extensive catalogue of cognitive biases, yet it remains a patchwork of isolated effects without a unified calculus of probability. This monograph is the first systematic work to establish quantum finance. It argues that the mathematical formalism of quantum mechanics supplies this missing unified structure: financial states are state vectors in Hilbert space; beliefs and valuations coexist as superposed amplitudes; trading and information disclosure act as measurements that collapse states in accordance with the Born rule; assets and markets are entangled with one another; and path interference generates fat tails and herding, which are inadmissible under classical additive probability. The book comprises six parts and thirty chapters in total. Part I lays the philosophical and formal foundations. Starting from the paradigm crisis, it proceeds through quantum probability and quantum stochastic processes to construct the analytical framework built on three core pillars of quantum finance. Part II develops quantum asset pricing, covering the quantum CAPM, quantum APT, quantum option pricing and the term structure of interest rates. Part III advances quantum risk management, including quantum risk measures, Value-at-Risk (VaR) and expected shortfall, stress testing, credit risk and phase transitions in systemic risk. Part IV addresses quantum portfolios and asset allocation, ranging from the quantum meanvariance frontier, strategic and tactical asset allocation, to factor investing and hedge fund strategies. Part V turns to market microstructure and behavioural finance, examining quantum order flows, marketmaker liquidity, quantum prospect theory and market sentiment. Part VI looks ahead to frontier avenues: quantum computational optimization, quantum machine learning, blockchains, postquantum cryptography and quantuminformed financial regulation. This book rigorously recovers the classical CAPM, efficient market hypothesis and Gaussian risk models as special cases under decoherence limits. It is intended for graduate students and faculty in finance and financial engineering, as well as researchers and practitioners working on asset pricing, risk management, portfolio allocation and financial regulation.
We analyze complexity of financial (and general economic) processes by comparing classical and quantum-like models for randomness. Our analysis implies that it might be that a quantum-like probabilistic description is more natural for financial market than the classical one. A part of our analysis is devoted to study the possibility of application of the quantum probabilistic model to agents of financial market. We show that, although the direct quantum (physical) reduction (based on using the scales of quantum mechanics) is meaningless, one may apply so called quantum-like models. In our approach quantum-like probabilistic behaviour is a consequence of contextualy of statistical data in finances (and economics in general). However, our hypothesis on "quantumness" of financial data should be tested experimentally (as opposed to the conventional description based on the noncontextual classical probabilistic approach). We present a new statistical test based on a generalization of the well known in quantum physics Bell's inequality.
This chapter explores a new paradigmatic thinking in financial market behavior, namely, quantum-like framework (QLF), which is a philosophical, mathematical and information-theoretic extension of quantum science formulation to financial market behaviors. Complexity science has already enriched financial market understanding based on network and chaos theories; however, QLF can further enrich such understanding based on a concrete framework of radical uncertainty and ambiguity, which is analogically inspired by quantum physics (rather than any physicalist or reductionist approach). The chapter also provides insights for open quantum system theory-based analysis of financial market interactions, quantum game theory applications and quantum computation applications in financial markets for future.