Derivatives

‘P’ Versus ‘Q’: Differences and Commonalities between the Two Areas of QuantitativeFinance ArXiv ID: ssrn-1717163 “View on arXiv” Authors: Unknown Abstract There exist two separate branches of finance that require advanced quantitative techniques: the “Q” area of derivatives pricing, whose task is to &quo Keywords: Quantitative Finance, Derivatives Pricing, Stochastic Calculus, Fixed Income, Derivatives Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The paper delves deep into stochastic calculus, PDEs, and advanced stochastic processes (e.g., Ornstein-Uhlenbeck, Heston model), indicating high mathematical complexity. However, it is purely theoretical/conceptual with no data, code, backtests, or implementation details, resulting in very low empirical rigor. flowchart TD A["Research Question<br>Differences & Commonalities<br>between P & Q Finance"] --> B["Methodology<br>Literature Review & Comparative Analysis"] B --> C["Key Inputs<br>Stochastic Calculus Models &<br>Derivatives Pricing Frameworks"] C --> D{"Computational Process<br>Analysis of Methodologies"} D --> E["P Area<br>Pricing & Risk Management<br>(Stochastic Control, Calibration)"] D --> F["Q Area<br>Derivatives Pricing & Hedging<br>(Risk-Neutral Valuation)"] E & F --> G["Outcomes<br>Unified Quantitative Framework<br>Distinct Methodologies &<br>Common Mathematical Foundations"]

Phynance ArXiv ID: ssrn-2433826 “View on arXiv” Authors: Unknown Abstract These are the lecture notes for an advanced Ph.D. level course I taught in Spring ‘02 at the C.N. Yang Institute for Theoretical Physics at Stony Brook. The cou Keywords: Stochastic Processes, Financial Mathematics, Brownian Motion, Derivatives Pricing, Derivatives Complexity vs Empirical Score Math Complexity: 9.0/10 Empirical Rigor: 2.0/10 Quadrant: Lab Rats Why: The paper is a PhD-level lecture on advanced stochastic calculus and derivative pricing, heavily featuring formal mathematical derivations and physics-inspired path integral methods, but contains no empirical data, backtests, or implementation details. flowchart TD A["Research Goal: Model Derivatives Pricing via Stochastic Processes"] --> B["Key Methodology: Applied Brownian Motion & Itô Calculus"] B --> C["Data/Inputs: Financial Market Parameters & Hypothetical Models"] C --> D["Computational Process: Solving Stochastic Differential Equations"] D --> E["Outcome: Analytical Derivatives Pricing Frameworks"]

The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process ArXiv ID: 2601.09074 “View on arXiv” Authors: L. J. Espinosa González, Erick Treviño Aguilar Abstract In this paper we study the Fourier estimator of Malliavin and Mancino for the spot volatility. We establish the convergence of the trigonometric polynomial to the volatility’s path in a setting that includes the following aspects. First, the volatility is required to satisfy a mild integrability condition, but otherwise allowed to be unbounded. Second, the price process is assumed to have cadlag paths, not necessarily continuous. We obtain convergence rates for the probability of a bad approximation in estimated coefficients, with a speed that allow to obtain an almost sure convergence and not just in probability in the estimated reconstruction of the volatility’s path. This is a new result even in the setting of continuous paths. We prove that a rescaled trigonometric polynomial approximate the quadratic jump process. ...

Quantum computing for multidimensional option pricing: End-to-end pipeline ArXiv ID: 2601.04049 “View on arXiv” Authors: Julien Hok, Álvaro Leitao Abstract This work introduces an end-to-end framework for multi-asset option pricing that combines market-consistent risk-neutral density recovery with quantum-accelerated numerical integration. We first calibrate arbitrage-free marginal distributions from European option quotes using the Normal Inverse Gaussian (NIG) model, leveraging its analytical tractability and ability to capture skewness and fat tails. Marginals are coupled via a Gaussian copula to construct joint distributions. To address the computational bottleneck of the high-dimensional integration required to solve the option pricing formula, we employ Quantum Accelerated Monte Carlo (QAMC) techniques based on Quantum Amplitude Estimation (QAE), achieving quadratic convergence improvements over classical Monte Carlo (CMC) methods. Theoretical results establish accuracy bounds and query complexity for both marginal density estimation (via cosine-series expansions) and multidimensional pricing. Empirical tests on liquid equity entities (Credit Agricole, AXA, Michelin) confirm high calibration accuracy and demonstrate that QAMC requires 10-100 times fewer queries than classical methods for comparable precision. This study provides a practical route to integrate arbitrage-aware modelling with quantum computing, highlighting implications for scalability and future extensions to complex derivatives. ...

Boundary error control for numerical solution of BSDEs by the convolution-FFT method ArXiv ID: 2512.24714 “View on arXiv” Authors: Xiang Gao, Cody Hyndman Abstract We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method. ...

Utility Maximisation with Model-independent Constraints ArXiv ID: 2512.24371 “View on arXiv” Authors: Alexander M. G. Cox, Daniel Hernandez-Hernandez Abstract We consider an agent who has access to a financial market, including derivative contracts, who looks to maximise her utility. Whilst the agent looks to maximise utility over one probability measure, or class of probability measures, she must also ensure that the mark-to-market value of her portfolio remains above a given threshold. When the mark-to-market value is based on a more pessimistic valuation method, such as model-independent bounds, we recover a novel optimisation problem for the agent where the agents investment problem must satisfy a pathwise constraint. For complete markets, the expression of the optimal terminal wealth is given, using the max-plus decomposition for supermartingales. Moreover, for the Black-Scholes-Merton model the explicit form of the process involved in such decomposition is obtained, and we are able to investigate numerically optimal portfolios in the presence of options which are mispriced according to the agent’s beliefs. ...

CapOptix: An Options-Framework for Capacity Market Pricing ArXiv ID: 2512.12871 “View on arXiv” Authors: Millend Roy, Agostino Capponi, Vladimir Pyltsov, Yinbo Hu, Vijay Modi Abstract Electricity markets are under increasing pressure to maintain reliability amidst rising renewable penetration, demand variability, and occasional price shocks. Traditional capacity market designs often fall short in addressing this by relying on expected-value metrics of energy unserved, which overlook risk exposure in such systems. In this work, we present CapOptix, a capacity pricing framework that interprets capacity commitments as reliability options, i.e., financial derivatives of wholesale electricity prices. CapOptix characterizes the capacity premia charged by accounting for structural price shifts modeled by the Markov Regime Switching Process. We apply the framework to historical price data from multiple electricity markets and compare the resulting premium ranges with existing capacity remuneration mechanisms. ...

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. ...

Signature approach for pricing and hedging path-dependent options with frictions ArXiv ID: 2511.23295 “View on arXiv” Authors: Eduardo Abi Jaber, Donatien Hainaut, Edouard Motte Abstract We introduce a novel signature approach for pricing and hedging path-dependent options with instantaneous and permanent market impact under a mean-quadratic variation criterion. Leveraging the expressive power of signatures, we recast an inherently nonlinear and non-Markovian stochastic control problem into a tractable form, yielding hedging strategies in (possibly infinite) linear feedback form in the time-augmented signature of the control variables, with coefficients characterized by non-standard infinite-dimensional Riccati equations on the extended tensor algebra. Numerical experiments demonstrate the effectiveness of these signature-based strategies for pricing and hedging general path-dependent payoffs in the presence of frictions. In particular, market impact naturally smooths optimal trading strategies, making low-truncated signature approximations highly accurate and robust in frictional markets, contrary to the frictionless case. ...

Constrained deep learning for pricing and hedging european options in incomplete markets ArXiv ID: 2511.20837 “View on arXiv” Authors: Nicolas Baradel Abstract In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets. ...