Monte Carlo Simulation

Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing ArXiv ID: 2601.00815 “View on arXiv” Authors: Mara Kalicanin Dimitrov, Marko Dimitrov, Anatoliy Malyarenko, Ying Ni Abstract Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options. ...

Efficient Calibration in the rough Bergomi model by Wasserstein distance ArXiv ID: 2512.00448 “View on arXiv” Authors: Changqing Teng, Guanglian Li Abstract Despite the empirical success in modeling volatility of the rough Bergomi (rBergomi) model, it suffers from pricing and calibration difficulties stemming from its non-Markovian structure. To address this, we propose a comprehensive computational framework that enhances both simulation and calibration. First, we develop a modified Sum-of-Exponentials (mSOE) Monte Carlo scheme which hybridizes an exact simulation of the singular kernel near the origin with a multi-factor approximation for the remainder. This method achieves high accuracy, particularly for out-of-the-money options, with an $\mathcal{“O”}(n)$ computational cost. Second, based on this efficient pricing engine, we then propose a distribution-matching calibration scheme by using Wasserstein distance as the optimization objective. This leverages a minimax formulation against Lipschitz payoffs, which effectively distributes pricing errors and improving robustness. Our numerical results confirm the mSOE scheme’s convergence and demonstrate that the calibration algorithm reliably identifies model parameters and generalizes well to path-dependent options, which offers a powerful and generic tool for practical model fitting. ...

Sharpening Shapley Allocation: from Basel 2.5 to FRTB ArXiv ID: 2511.12391 “View on arXiv” Authors: Marco Scaringi, Marco Bianchetti Abstract Risk allocation, the decomposition of a portfolio-wide risk measure into component contributions, is a fundamental problem in financial risk management due to the non-additive nature of risk measures, the layered organizational structures of financial institutions, and the range of possible allocation strategies characterized by different rationales and properties. In this work, we conduct a systematic review of the major risk allocation strategies typically used in finance, comparing their theoretical properties, practical advantages, and limitations. To this scope we set up a specific testing framework, including both simplified settings, designed to highlight basic intrinsic behaviours, and realistic financial portfolios under different risk regulations, i.e. Basel 2.5 and FRTB. Furthermore, we develop and test novel practical solutions to manage the issue of negative risk allocations and of multi-level risk allocation in the layered organizational structure of financial institutions, while preserving the additivity property. Finally, we devote particular attention to the computational aspects of risk allocation. Our results show that, in this context, the Shapley allocation strategy offers the best compromise between simplicity, mathematical properties, risk representation and computational cost. The latter is still acceptable even in the challenging case of many business units, provided that an efficient Monte Carlo simulation is employed, which offers excellent scaling and convergence properties. While our empirical applications focus on market risk, our methodological framework is fully general and applicable to other financial context such as valuation risk, liquidity risk, credit risk, and counterparty credit risk. ...

Learning the Exact SABR Model ArXiv ID: 2510.10343 “View on arXiv” Authors: Giorgia Rensi, Pietro Rossi, Marco Bianchetti Abstract The SABR model is a cornerstone of interest rate volatility modeling, but its practical application relies heavily on the analytical approximation by Hagan et al., whose accuracy deteriorates for high volatility, long maturities, and out-of-the-money options, admitting arbitrage. While machine learning approaches have been proposed to overcome these limitations, they have often been limited by simplified SABR dynamics or a lack of systematic validation against the full spectrum of market conditions. We develop a novel SABR DNN, a specialized Artificial Deep Neural Network (DNN) architecture that learns the true SABR stochastic dynamics using an unprecedented large training dataset (more than 200 million points) of interest rate Cap/Floor volatility surfaces, including very long maturities (30Y) and extreme strikes consistently with market quotations. Our dataset is obtained via high-precision unbiased Monte Carlo simulation of a special scaled shifted-SABR stochastic dynamics, which allows dimensional reduction without any loss of generality. Our SABR DNN provides arbitrage-free calibration of real market volatility surfaces and Cap/Floor prices for any maturity and strike with negligible computational effort and without retraining across business dates. Our results fully address the gaps in the previous machine learning SABR literature in a systematic and self-consistent way, and can be extended to cover any interest rate European options in different rate tenors and currencies, thus establishing a comprehensive functional SABR framework that can be adopted for daily trading and risk management activities. ...

Holdout cross-validation for large non-Gaussian covariance matrix estimation using Weingarten calculus ArXiv ID: 2509.13923 “View on arXiv” Authors: Lamia Lamrani, Benoît Collins, Jean-Philippe Bouchaud Abstract Cross-validation is one of the most widely used methods for model selection and evaluation; its efficiency for large covariance matrix estimation appears robust in practice, but little is known about the theoretical behavior of its error. In this paper, we derive the expected Frobenius error of the holdout method, a particular cross-validation procedure that involves a single train and test split, for a generic rotationally invariant multiplicative noise model, therefore extending previous results to non-Gaussian data distributions. Our approach involves using the Weingarten calculus and the Ledoit-Péché formula to derive the oracle eigenvalues in the high-dimensional limit. When the population covariance matrix follows an inverse Wishart distribution, we approximate the expected holdout error, first with a linear shrinkage, then with a quadratic shrinkage to approximate the oracle eigenvalues. Under the linear approximation, we find that the optimal train-test split ratio is proportional to the square root of the matrix dimension. Then we compute Monte Carlo simulations of the holdout error for different distributions of the norm of the noise, such as the Gaussian, Student, and Laplace distributions and observe that the quadratic approximation yields a substantial improvement, especially around the optimal train-test split ratio. We also observe that a higher fourth-order moment of the Euclidean norm of the noise vector sharpens the holdout error curve near the optimal split and lowers the ideal train-test ratio, making the choice of the train-test ratio more important when performing the holdout method. ...

Asymptotic universal moment matching properties of normal distributions ArXiv ID: 2508.03790 “View on arXiv” Authors: Xuan Liu Abstract Moment matching is an easy-to-implement and usually effective method to reduce variance of Monte Carlo simulation estimates. On the other hand, there is no guarantee that moment matching will always reduce simulation variance for general integration problems at least asymptotically, i.e. when the number of samples is large. We study the characterization of conditions on a given underlying distribution $X$ under which asymptotic variance reduction is guaranteed for a general integration problem $\mathbb{“E”}[“f(X)”]$ when moment matching techniques are applied. We show that a sufficient and necessary condition for such asymptotic variance reduction property is $X$ being a normal distribution. Moreover, when $X$ is a normal distribution, formulae for efficient estimation of simulation variance for (first and second order) moment matching Monte Carlo are obtained. These formulae allow estimations of simulation variance as by-products of the simulation process, in a way similar to variance estimations for plain Monte Carlo. Moreover, we propose non-linear moment matching schemes for any given continuous distribution such that asymptotic variance reduction is guaranteed. ...

Machine Learning Based Stress Testing Framework for Indian Financial Market Portfolios ArXiv ID: 2507.02011 “View on arXiv” Authors: Vidya Sagar G, Shifat Ali, Siddhartha P. Chakrabarty Abstract This paper presents a machine learning driven framework for sectoral stress testing in the Indian financial market, focusing on financial services, information technology, energy, consumer goods, and pharmaceuticals. Initially, we address the limitations observed in conventional stress testing through dimensionality reduction and latent factor modeling via Principal Component Analysis and Autoencoders. Building on this, we extend the methodology using Variational Autoencoders, which introduces a probabilistic structure to the latent space. This enables Monte Carlo-based scenario generation, allowing for more nuanced, distribution-aware simulation of stressed market conditions. The proposed framework captures complex non-linear dependencies and supports risk estimation through Value-at-Risk and Expected Shortfall. Together, these pipelines demonstrate the potential of Machine Learning approaches to improve the flexibility, robustness, and realism of financial stress testing. ...

A Sinusoidal Hull-White Model for Interest Rate Dynamics: Capturing Long-Term Periodicity in U.S. Treasury Yields ArXiv ID: 2506.06317 “View on arXiv” Authors: Amit Kumar Jha Abstract This study is motivated by empirical observations of periodic fluctuations in interest rates, notably long-term economic cycles spanning decades, which the conventional Hull-White short-rate model fails to adequately capture. To address this limitation, we propose an extension that incorporates a sinusoidal, time-varying mean reversion speed, allowing the model to reflect cyclic interest rate dynamics more effectively. The model is calibrated using a comprehensive dataset of daily U.S. Treasury yield curves obtained from the Federal Reserve Economic Data (FRED) database, covering the period from January 1990 to December 2022. The dataset includes tenors of 1, 2, 3, 5, 7, 10, 20, and 30 years, with the most recent yields ranging from 1.22% (1-year) to 2.36% (30-year). Calibration is performed using the Nelder-Mead optimization algorithm, and Monte Carlo simulations with 200 paths and a time step of 0.05 years. The resulting 30-year zero-coupon bond price under the proposed model is 0.43, compared to 0.47 under the standard Hull-White model. This corresponds to root mean squared errors of 0.12% and 0.14%, respectively, indicating a noticeable improvement in fit, particularly for longer maturities. These results highlight the model’s enhanced capability to capture long-term yield dynamics and suggest significant implications for bond pricing, interest rate risk management, and the valuation of interest rate derivatives. The findings also open avenues for further research into stochastic periodicity and alternative interest rate modeling frameworks. ...

Agent-based Liquidity Risk Modelling for Financial Markets ArXiv ID: 2505.15296 “View on arXiv” Authors: Perukrishnen Vytelingum, Rory Baggott, Namid Stillman, Jianfei Zhang, Dingqiu Zhu, Tao Chen, Justin Lyon Abstract In this paper, we describe a novel agent-based approach for modelling the transaction cost of buying or selling an asset in financial markets, e.g., to liquidate a large position as a result of a margin call to meet financial obligations. The simple act of buying or selling in the market causes a price impact and there is a cost described as liquidity risk. For example, when selling a large order, there is market slippage – each successive trade will execute at the same or worse price. When the market adjusts to the new information revealed by the execution of such a large order, we observe in the data a permanent price impact that can be attributed to the change in the fundamental value as market participants reassess the value of the asset. In our ABM model, we introduce a novel mechanism where traders assume orderflow is informed and each trade reveals some information about the value of the asset, and traders update their belief of the fundamental value for every trade. The result is emergent, realistic price impact without oversimplifying the problem as most stylised models do, but within a realistic framework that models the exchange with its protocols, its limit orderbook and its auction mechanism and that can calculate the transaction cost of any execution strategy without limitation. Our stochastic ABM model calculates the costs and uncertainties of buying and selling in a market by running Monte-Carlo simulations, for a better understanding of liquidity risk and can be used to optimise for optimal execution under liquidity risk. We demonstrate its practical application in the real world by calculating the liquidity risk for the Hang-Seng Futures Index. ...

DeFi Liquidation Risk Modeling Using Geometric Brownian Motion ArXiv ID: 2505.08100 “View on arXiv” Authors: Timofei Belenko, Georgii Vosorov Abstract In this paper, we propose an analytical method to compute the collateral liquidation probability in decentralized finance (DeFi) stablecoin single-collateral lending. Our approach models the collateral exchange rate as a zero-drift geometric Brownian motion, and derives the probability of it crossing the liquidation threshold. Unlike most existing methods that rely on computationally intensive simulations such as Monte Carlo, our formula provides a lightweight, exact solution. This advancement offers a more efficient alternative for risk assessment in DeFi platforms. ...