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arXiv:2002.07389

Quantum Implementation of Risk Analysis-relevant Copulas

Janusz Milek

Published Feb 18, 2020• 1 citation
arXivHTMLPDF

Abstract

Modern quantitative risk management relies on an adequate modeling of the tail dependence and a possibly accurate quantification of risk measures, like Value at Risk (VaR), at high confidence levels like 1 in 100 or even 1 in 2000. Quantum computing makes such a quantification quadratically more efficient than the Monte Carlo method; see (Woerner and Egger, 2018) and, for a broader perspective, (Orús et al., 2018). An important element of the risk analysis toolbox is copula, see (Jouanin et al., 2004) regarding financial applications. However, to the best knowledge of the author, no quantum computing implementation for sampling from a risk modeling-relevant copula in explicit form has been published so far. Our focus here is implementation of simple yet powerful copula models, capable of a satisfactory capturing the joint tail behaviour of the modelled risk factors. This paper deals with a few simple copula families, including Multivariate B11 (MB11) copula family, presented in (Milek, 2014). We will show that this copula family is suitable for the risk aggregation as it is exceptionally able to reproduce tail dependence structures; see (Embrechts et al., 2016) for a relevant benchmark as well as necessary and sufficient conditions regarding the ultimate feasible bivariate tail dependence structures. It turns out that such a discretized copula can be expressed using simple constructs present in the quantum computing: binary fraction expansion format, comonotone/independent random variables, controlled gates, and convex combinations, and is therefore suitable for a quantum computer implementation. This paper presents design behind the quantum implementation circuits, numerical and symbolic simulation results, and experimental validation on IBM quantum computer. The paper proposes also a generic method for quantum implementation of any discretized copula.

DOI
10.48550/arxiv.2002.07389

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Modern quantitative risk management relies on an adequate modeling of the tail dependence and a possibly accurate quantification of risk measures, like Value at Risk (VaR), at high confidence levels like 1 in 100 or even 1 in 2000. Quantum computing makes such a quantification quadratically more efficient than the Monte Carlo method; see (Woerner and Egger, 2018) and, for a broader perspective, (Orús et al., 2018). An important element of the risk analysis toolbox is copula, see (Jouanin et al., 2004) regarding financial applications. However, to the best knowledge of the author, no quantum computing implementation for sampling from a risk modeling-relevant copula in explicit form has been published so far. Our focus here is implementation of simple yet powerful copula models, capable of a satisfactory capturing the joint tail behaviour of the modelled risk factors. This paper deals with a few simple copula families, including Multivariate B11 (MB11) copula family, presented in (Milek, 2014). We will show that this copula family is suitable for the risk aggregation as it is exceptionally able to reproduce tail dependence structures; see (Embrechts et al., 2016) for a relevant benchmark as well as necessary and sufficient conditions regarding the ultimate feasible bivariate tail dependence structures. It turns out that such a discretized copula can be expressed using simple constructs present in the quantum computing: binary fraction expansion format, comonotone/independent random variables, controlled gates, and convex combinations, and is therefore suitable for a quantum computer implementation. This paper presents design behind the quantum implementation circuits, numerical and symbolic simulation results, and experimental validation on IBM quantum computer. The paper proposes also a generic method for quantum implementation of any discretized copula.
Quantum Monte Carlo integration (QMCI) provides a quadratic speed-up over its classical counterpart, and its applications have been investigated in various fields, including finance. This paper considers its application to risk aggregation, one of the most important numerical tasks in financial risk management. Risk aggregation combines several risk variables and quantifies the total amount of risk, taking into account the correlation among them. For this task, there exists a useful tool called copula, with which the joint distribution can be generated from marginal distributions with a flexible correlation structure. Classically, the copula-based method utilizes sampling of risk variables. However, this procedure is not directly applicable to the quantum setting, where sampled values are not stored as classical data, and thus no efficient quantum algorithm is known. In this paper, we introduce a quantum algorithm for copula-based risk aggregation that is compatible with QMCI. In our algorithm, we first estimate each marginal distribution as a series of orthogonal functions, where the coefficients can be calculated with QMCI. Then, by plugging the marginal distributions into the copula and obtaining the joint distribution, we estimate risk measures using QMCI again. With this algorithm, nearly quadratic quantum speed-up can be obtained for sufficiently smooth marginal distributions.
Collateralized debt obligation (CDO) has been one of the most commonly used structured financial products and is intensively studied in quantitative finance. By setting the asset pool into different tranches, it effectively works out and redistributes credit risks and returns to meet the risk preferences for different tranche investors. The copula models of various kinds are normally used for pricing CDOs, and the Monte Carlo simulations are required to get their numerical solution. Here we implement two typical CDO models, the single-factor Gaussian copula model and Normal Inverse Gaussian copula model, and by applying the conditional independence approach, we manage to load each model of distribution in quantum circuits. We then apply quantum amplitude estimation as an alternative to Monte Carlo simulation for CDO pricing. We demonstrate the quantum computation results using IBM Qiskit. Our work addresses a useful task in finance instrument pricing, significantly broadening the application scope for quantum computing in finance.
Collateralized debt obligation (CDO) has been one of the most commonly used structured financial products and is intensively studied in quantitative finance. By setting the asset pool into different tranches, it effectively works out and redistributes credit risks and returns to meet the risk preferences for different tranche investors. The copula models of various kinds are normally used for pricing CDOs, and the Monte Carlo simulations are required to get their numerical solution. Here we implement two typical CDO models, the single-factor Gaussian copula model and normal inverse Gaussian copula model, and by applying the conditional independence approach, we manage to load each model of distribution in quantum circuits. We then apply quantum amplitude estimation as an alternative to Monte Carlo simulation for CDO pricing. We demonstrate the quantum computation results using IBM Qiskit. Our work addresses a useful task in finance instrument pricing, significantly broadening the application scope for quantum computing in finance.
Copulas are mathematical tools for modeling joint probability distributions. In the past 60 years they have become an essential analysis tool on classical computers in various fields. The recent finding that copulas can be expressed as maximally entangled quantum states has revealed a promising approach to practical quantum advantages: performing tasks faster, requiring less memory, or, as we show, yielding better predictions. Studying the scalability of this quantum approach as both the precision and the number of modeled variables increase is crucial for its adoption in real-world applications. In this paper, we successfully apply a Quantum Circuit Born Machine (QCBM) based approach to modeling 3- and 4-variable copulas on trapped ion quantum computers. We study the training of QCBMs with different levels of precision and circuit design on a simulator and a state-of-the-art trapped ion quantum computer. We observe decreased training efficacy due to the increased complexity in parameter optimization as the models scale up. To address this challenge, we introduce an annealing-inspired strategy that dramatically improves the training results. In our end-to-end tests, various configurations of the quantum models make a comparable or better prediction in risk aggregation tasks than the standard classical models.
Monte Carlo (MC) simulations are widely used in financial risk management, from estimating value-at-risk (VaR) to pricing over-the-counter derivatives. However, they come at a significant computational cost due to the number of scenarios required for convergence. If a probability distribution is available, Quantum Amplitude Estimation (QAE) algorithms can provide a quadratic speed-up in measuring its properties as compared to their classical counterparts. Recent studies have explored the calculation of common risk measures and the optimisation of QAE algorithms by initialising the input quantum states with pre-computed probability distributions. If such distributions are not available in closed form, however, they need to be generated numerically, and the associated computational cost may limit the quantum advantage. In this paper, we bypass this challenge by incorporating scenario generation – i.e. simulation of the risk factor evolution over time to generate probability distributions – into the quantum computation; we refer to this process as Quantum MC (QMC) simulations. Specifically, we assemble quantum circuits that implement stochastic models for equity (geometric Brownian motion), interest rate (mean-reversion models), and credit (structural, reduced-form, and rating migration credit models) risk factors. We then integrate these models with QAE to provide end-to-end examples for both market and credit risk use cases.
Abstract We present a quantum algorithm that analyzes risk more efficiently than Monte Carlo simulations traditionally used on classical computers. We employ quantum amplitude estimation to price securities and evaluate risk measures such as Value at Risk and Conditional Value at Risk on a gate-based quantum computer. Additionally, we show how to implement this algorithm and how to trade-off the convergence rate of the algorithm and the circuit depth. The shortest possible circuit depth—growing polynomially in the number of qubits representing the uncertainty—leads to a convergence rate of O ( M −2/3 ), where M is the number of samples. This is already faster than classical Monte Carlo simulations which converge at a rate of O ( M −1/2 ). If we allow the circuit depth to grow faster, but still polynomially, the convergence rate quickly approaches the optimum of O ( M −1 ). Thus, for slowly increasing circuit depths our algorithm provides a near quadratic speed-up compared to Monte Carlo methods. We demonstrate our algorithm using two toy models. In the first model we use real hardware, such as the IBM Q Experience, to price a Treasury-bill (T-bill) faced by a possible interest rate increase. In the second model, we simulate our algorithm to illustrate how a quantum computer can determine financial risk for a two-asset portfolio made up of government debt with different maturity dates. Both models confirm the improved convergence rate over Monte Carlo methods. Using simulations, we also evaluate the impact of cross-talk and energy relaxation errors.