Quantum Monte Carlo simulations for estimating FOREX markets: a speculative attacks experience

FOREX is the largest and among the most volatile financial markets, making accurate prediction of exchange rates difficult. Traditional econometric and Monte Carlo-based models have had mixed success, particularly in small or irregular samples. The paper addresses the need for more robust forecasting tools that capture volatility and nonlinearity, proposing the application of Auxiliary-Field Quantum Monte Carlo (AFQMC) to enhance estimation accuracy of exchange rate fundamentals and speculative attack models. The study focuses on USD/EUR and USD/JPY over 2013–2021, comparing Markov Chain Monte Carlo (MCMC), Sequential Monte Carlo (SMC), and AFQMC. Motivated by recent crises (e.g., COVID-19) and growing needs for timely risk management, the authors hypothesize that AFQMC improves accuracy, stability, and computational efficiency across sample sizes, especially under stress and small-sample contexts.

Prior research on exchange rate prediction spans econometric methods (e.g., OLS, factor models, VAR, TVP, ECM) and machine learning (SVM, RF, neural networks), often evaluated by standard deviation, RMSE, MAE/MAPE, and accuracy. Classic linear methods often yield overall standard deviations around 0.93–1.84 (worse in small samples: 1.26–2.11). More advanced nonlinear econometrics achieve 0.58–0.83 in large samples and 0.75–1.24 in small samples. Studies such as Giacomini & Rossi (2010), Park & Park (2013), Byrne et al. (2016), Ince et al. (2016), and Hauzenberger & Huber (2019) show limited forecast improvements over random walk and notable instability. Monte Carlo methods (Metropolis-Hastings, SMC) are widely used in finance for valuation, risk, and high-dimensional problems; recommendations include comparing sampling schemes (e.g., Metropolis-Hastings, Gibbs, MCMC). Machine learning applications report accuracies roughly 68–88% (SVM, RF, deep learning), with some hybrid models outperforming traditional ARIMA/ARFIMA in RMSE/MAE. Nonetheless, prediction remains challenging due to volatility and regime changes. The paper positions Quantum Monte Carlo, specifically AFQMC, as a promising approach to improve sampling accuracy and robustness with few observations and irregular distributions, potentially enabling near real-time risk analysis when deployed on quantum hardware.

Models: The study estimates and evaluates multiple exchange rate fundamentals and crisis models: (1) Uncovered Interest Rate Parity (UIRP), relating expected exchange rate changes to interest rate differentials; (2) Purchasing Power Parity (PPP), linking the log exchange rate to relative price levels; (3) BEER, a behavioral equilibrium exchange rate model using CPI, relative non-tradables price, real interest rate, public debt-to-GDP, terms of trade, and net foreign assets; (4) Sticky-Price Monetary (SPM) model with money, output, interest rate, and inflation; (5) Taylor-rule fundamentals linking exchange rates to output gap and inflation alongside lagged exchange rate; (6) Speculative attacks model (Eichengreen et al., 1994) via an Exchange Market Pressure index built from changes in exchange rates, interest rate differentials, and reserves-to-money ratios; and (7) Second-generation speculative attack model (Flood & Marion, 1997) featuring multiple equilibria and policy trade-offs.

Estimation techniques: Three Monte Carlo approaches are compared:

• MCMC (Metropolis-Hastings): construction of a Markov chain with proposal distributions satisfying detailed balance; acceptance probability min{1, π(Y)/π(X)}; convergence monitored by Gelman–Rubin R statistic (0.97–1.03 threshold after burn-in across multiple chains).
• Sequential Monte Carlo (SMC)/particle filtering: represents posterior distributions with weighted particles updated via predict–update–resample steps; importance weights proportional to likelihood; resampling to combat degeneracy; kernel-based approximations for parameter evolution; variance adaptation using covariance terms.
• Auxiliary-Field Quantum Monte Carlo (AFQMC): constrained-path/auxiliary-field formulation projecting to a ground-state solution via random walks in Slater determinant space with importance sampling and constraints to mitigate the sign problem; employs Hubbard–Stratonovich transformation, back-propagation for observables, and walker weighting by overlaps; designed for accurate sampling with algebraic scaling.

Data and setup: Exchange rates USD/EUR and USD/JPY daily close (Yahoo Finance) for 2013–2021. Macroeconomic fundamentals (monthly): short-term interest rates, international reserves, M1, commodity prices, 10-year government bond yields from FRED, Eurostat, World Bank, and Bank of Japan. Models are estimated under designs with different sample sizes (notably N=200 and N=500 observations) to test small- and larger-sample performance. Evaluations use in-sample and out-of-sample classification accuracy (%), RMSE, and MAPE. Robustness checks include Diebold–Mariano (DM) predictive accuracy tests across methods and timing benchmarks. Code implemented in MATLAB R2019b, run on Intel Core i7-6500U processors with 500 computation runs for robustness.

• USD/EUR results: AFQMC consistently outperformed MCMC and SMC across all models. Out-of-sample accuracy with AFQMC ranged from 86.93% to 89.56% for N=200 and from 87.96% to 92.97% for N=500, with lower RMSE and MAPE than alternatives (e.g., UIRP: N=200 out-of-sample accuracy 89.56% with RMSE 0.39 and MAPE 0.23; N=500 92.97%, RMSE 0.26, MAPE 0.15). Across PPP, BEER, SPM, Taylor, speculative attacks (first and second generation), AFQMC delivered the highest accuracies and lowest errors.
• USD/JPY results: AFQMC achieved superior performance: out-of-sample accuracy 90.48–97.37% (N=200) and 90.09–95.62% (N=500), with minimal RMSE/MAPE (e.g., BEER N=200 out-of-sample 95.02% accuracy; SPM N=200 97.37%; Taylor N=200 96.64%).
• Comparative benchmarks: Reported accuracies substantially exceed prior literature using econometric and ML methods, which often range 68–88% and struggle in small samples. AFQMC exhibits better stability with small and irregular samples than MCMC/SMC and conventional regressions.
• Statistical tests: Diebold–Mariano comparisons favored AFQMC; DM statistics were beyond ±1.96 in most pairwise contrasts (MCMC vs SMC, MCMC vs AFQMC, SMC vs AFQMC) for both N=200 and N=500, indicating statistically significant predictive differences in favor of AFQMC.
• Computational efficiency: Monte Carlo approaches required short runtimes; AFQMC generally the fastest among the three across models and samples. Reported elapsed times for AFQMC typically around 0.05–0.13 minutes per estimation context, shorter than or comparable to SMC and faster than MCMC.
• Practical implication: The AFQMC approach provides higher accuracy and lower error metrics for detecting and simulating speculative pressure, offering improved early-warning insights under stress scenarios.

The findings confirm the hypothesis that AFQMC enhances the precision and robustness of exchange rate modeling relative to traditional Monte Carlo and econometric approaches. AFQMC’s improved sampling efficiency and constraint handling translate into higher classification accuracy and lower RMSE/MAPE across diverse fundamentals and speculative attack models, including challenging small-sample settings. This directly addresses the research goal of better capturing volatility and nonlinear dynamics inherent to FOREX, enabling more reliable in- and out-of-sample forecasts. The statistically significant DM test outcomes substantiate the superiority of AFQMC’s predictive performance. Operationally, AFQMC’s shorter runtimes suggest practicality for frequent re-estimation and potential near real-time risk analysis. For practitioners and policymakers, the method improves anticipatory capabilities regarding large exchange rate swings, speculative pressures, and crisis risks, supporting risk management, policy formulation, and trading decisions.

The study introduces AFQMC to the estimation of FOREX fundamentals and speculative attack models and empirically demonstrates its superiority over MCMC and SMC in accuracy, error reduction, stability, and computational efficiency for USD/EUR and USD/JPY (2013–2021). Out-of-sample accuracies with AFQMC reached approximately 88–93% (USD/EUR) and 90–97% (USD/JPY), exceeding benchmarks from prior econometric and machine learning literature. AFQMC’s robustness under small and irregular samples suggests value for stress-scenario simulation and early warning of speculative pressure. The method has meaningful implications for financial institutions and policymakers, enhancing risk assessment, portfolio and derivative valuation, and macro-financial monitoring. Future research could integrate AFQMC-based forecasts with trading strategy simulations and extend applications to other financial econometrics models commonly estimated via Monte Carlo.

A key limitation noted by the authors is the absence of simulations that integrate the Monte Carlo model outputs into concrete trading strategy backtests. Implementing and evaluating AFQMC-driven strategies in realistic market environments would strengthen practical validation. Additionally, while macroeconomic fundamentals were included at monthly frequency alongside daily exchange rates, further exploration of frequency alignment and additional explanatory variables could be considered in future work.