Derivatives

Efficient Importance Sampling under Heston Model: Short Maturity and Deep Out-of-the-Money Options ArXiv ID: 2511.19826 “View on arXiv” Authors: Yun-Feng Tu, Chuan-Hsiang Han Abstract This paper investigates asymptotically optimal importance sampling (IS) schemes for pricing European call options under the Heston stochastic volatility model. We focus on two distinct rare-event regimes where standard Monte Carlo methods suffer from significant variance deterioration: the limit as maturity approaches zero and the limit as the strike price tends to infinity. Leveraging the large deviation principle (LDP), we design a state-dependent change of measure derived from the asymptotic behavior of the log-price cumulant generating functions. In the short-maturity regime, we rigorously prove that our proposed IS drift, inspired by the variational characterization of the rate function, achieves logarithmic efficiency (asymptotic optimality) by minimizing the decay rate of the second moment of the estimator. In the deep OTM regime, we introduce a novel slow mean-reversion scaling for the variance process, where the mean-reversion speed scales as the inverse square of the small-noise parameter (defined as the reciprocal of the log-moneyness). We establish that under this specific scaling, the variance process contributes non-trivially to the large deviation rate function, requiring a specialized Riccati analysis to verify optimality. Numerical experiments demonstrate that the proposed method yields substantial variance reduction–characterized by factors exceeding several orders of magnitude–compared to standard estimators in both asymptotic regimes. ...

Data-driven Feynman-Kac Discovery with Applications to Prediction and Data Generation ArXiv ID: 2511.08606 “View on arXiv” Authors: Qi Feng, Guang Lin, Purav Matlia, Denny Serdarevic Abstract In this paper, we propose a novel data-driven framework for discovering probabilistic laws underlying the Feynman-Kac formula. Specifically, we introduce the first stochastic SINDy method formulated under the risk-neutral probability measure to recover the backward stochastic differential equation (BSDE) from a single pair of stock and option trajectories. Unlike existing approaches to identifying stochastic differential equations-which typically require ergodicity-our framework leverages the risk-neutral measure, thereby eliminating the ergodicity assumption and enabling BSDE recovery from limited financial time series data. Using this algorithm, we are able not only to make forward-looking predictions but also to generate new synthetic data paths consistent with the underlying probabilistic law. ...

Numerical valuation of European options under two-asset infinite-activity exponential Lévy models ArXiv ID: 2511.02700 “View on arXiv” Authors: Massimiliano Moda, Karel J. in ’t Hout, Michèle Vanmaele, Fred Espen Benth Abstract We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation. ...

Quantum Machine Learning methods for Fourier-based distribution estimation with application in option pricing ArXiv ID: 2510.19494 “View on arXiv” Authors: Fernando Alonso, Álvaro Leitao, Carlos Vázquez Abstract The ongoing progress in quantum technologies has fueled a sustained exploration of their potential applications across various domains. One particularly promising field is quantitative finance, where a central challenge is the pricing of financial derivatives-traditionally addressed through Monte Carlo integration techniques. In this work, we introduce two hybrid classical-quantum methods to address the option pricing problem. These approaches rely on reconstructing Fourier series representations of statistical distributions from the outputs of Quantum Machine Learning (QML) models based on Parametrized Quantum Circuits (PQCs). We analyze the impact of data size and PQC dimensionality on performance. Quantum Accelerated Monte Carlo (QAMC) is employed as a benchmark to quantitatively assess the proposed models in terms of computational cost and accuracy in the extraction of Fourier coefficients. Through the numerical experiments, we show that the proposed methods achieve remarkable accuracy, becoming a competitive quantum alternative for derivatives valuation. ...

An Efficient Calibration Framework for Volatility Derivatives under Rough Volatility with Jumps ArXiv ID: 2510.19126 “View on arXiv” Authors: Keyuan Wu, Tenghan Zhong, Yuxuan Ouyang Abstract We present a fast and robust calibration method for stochastic volatility models that admit Fourier-analytic transform-based pricing via characteristic functions. The design is structure-preserving: we keep the original pricing transform and (i) split the pricing formula into data-independent inte- grals and a market-dependent remainder; (ii) precompute those data-independent integrals with GPU acceleration; and (iii) approximate only the remaining, market-dependent pricing map with a small neural network. We instantiate the workflow on a rough volatility model with tempered-stable jumps tailored to power-type volatility derivatives and calibrate it to VIX options with a global-to-local search. We verify that a pure-jump rough volatility model adequately captures the VIX dynamics, consistent with prior empirical findings, and demonstrate that our calibration method achieves high accuracy and speed. ...

Tail-Safe Hedging: Explainable Risk-Sensitive Reinforcement Learning with a White-Box CBF–QP Safety Layer in Arbitrage-Free Markets ArXiv ID: 2510.04555 “View on arXiv” Authors: Jian’an Zhang Abstract We introduce Tail-Safe, a deployability-oriented framework for derivatives hedging that unifies distributional, risk-sensitive reinforcement learning with a white-box control-barrier-function (CBF) quadratic-program (QP) safety layer tailored to financial constraints. The learning component combines an IQN-based distributional critic with a CVaR objective (IQN–CVaR–PPO) and a Tail-Coverage Controller that regulates quantile sampling through temperature tilting and tail boosting to stabilize small-$α$ estimation. The safety component enforces discrete-time CBF inequalities together with domain-specific constraints – ellipsoidal no-trade bands, box and rate limits, and a sign-consistency gate – solved as a convex QP whose telemetry (active sets, tightness, rate utilization, gate scores, slack, and solver status) forms an auditable trail for governance. We provide guarantees of robust forward invariance of the safe set under bounded model mismatch, a minimal-deviation projection interpretation of the QP, a KL-to-DRO upper bound linking per-state KL regularization to worst-case CVaR, concentration and sample-complexity results for the temperature-tilted CVaR estimator, and a CVaR trust-region improvement inequality under KL limits, together with feasibility persistence under expiry-aware tightening. Empirically, in arbitrage-free, microstructure-aware synthetic markets (SSVI $\to$ Dupire $\to$ VIX with ABIDES/MockLOB execution), Tail-Safe improves left-tail risk without degrading central performance and yields zero hard-constraint violations whenever the QP is feasible with zero slack. Telemetry is mapped to governance dashboards and incident workflows to support explainability and auditability. Limitations include reliance on synthetic data and simplified execution to isolate methodological contributions. ...

Volatility Calibration via Automatic Local Regression ArXiv ID: 2509.16334 “View on arXiv” Authors: Ruozhong Yang, Hao Qin, Charlie Che, Liming Feng Abstract Managing exotic derivatives requires accurate mark-to-market pricing and stable Greeks for reliable hedging. The Local Volatility (LV) model distinguishes itself from other pricing models by its ability to match observable market prices across all strikes and maturities with high accuracy. However, LV calibration is fundamentally ill-posed: finite market observables must determine a continuously-defined surface with infinite local volatility parameters. This ill-posed nature often causes spiky LV surfaces that are particularly problematic for finite-difference-based valuation, and induces high-frequency oscillations in solutions, thus leading to unstable Greeks. To address this challenge, we propose a pre-calibration smoothing method that can be integrated seamlessly into any LV calibration workflow. Our method pre-processes market observables using local regression that automatically minimizes asymptotic conditional mean squared error to generate denoised inputs for subsequent LV calibration. Numerical experiments demonstrate that the proposed pre-calibration smoothing yields significantly smoother LV surfaces and greatly improves Greek stability for exotic options with negligible additional computational cost, while preserving the LV model’s ability to fit market observables with high fidelity. ...

Fast and explicit European option pricing under tempered stable processes ArXiv ID: 2510.01211 “View on arXiv” Authors: Gaetano Agazzotti, Jean-Philippe Aguilar Abstract We provide series expansions for the tempered stable densities and for the price of European-style contracts in the exponential Lévy model driven by the tempered stable process. These formulas recover several popular option pricing models, and become particularly simple in some specific cases such as bilateral Gamma process and one-sided TS process. When compared to traditional Fourier pricing, our method has the advantage of being hyperparameter free. We also provide a detailed numerical analysis and show that our technique is competitive with state-of-the-art pricing methods. ...

Enhanced indexation using both equity assets and index options ArXiv ID: 2508.21192 “View on arXiv” Authors: Cristiano Arbex Valle, John E Beasley Abstract In this paper we consider how we can include index options in enhanced indexation. We present the concept of an \enquote{“option strategy”} which enables us to treat options as an artificial asset. An option strategy for a known set of options is a specified set of rules which detail how these options are to be traded (i.e.bought, rolled over, sold) depending upon market conditions. We consider option strategies in the context of enhanced indexation, but we discuss how they have much wider applicability in terms of portfolio optimisation. We use an enhanced indexation approach based on second-order stochastic dominance. We consider index options for the S&P500, using a dataset of daily stock prices over the period 2017-2025 that has been manually adjusted to account for survivorship bias. This dataset is made publicly available for use by future researchers. Our computational results indicate that introducing option strategies in an enhanced indexation setting offers clear benefits in terms of improved out-of-sample performance. This applies whether we use equities or an exchange-traded fund as part of the enhanced indexation portfolio. ...

Risk-Neutral Pricing of Random-Expiry Options Using Trinomial Trees ArXiv ID: 2508.17014 “View on arXiv” Authors: Sebastien Bossu, Michael Grabchak Abstract Random-expiry options are nontraditional derivative contracts that may expire early based on a random event. We develop a methodology for pricing these options using a trinomial tree, where the middle path is interpreted as early expiry. We establish that this approach is free of arbitrage, derive its continuous-time limit, and show how it may be implemented numerically in an efficient manner. ...