Derivatives Pricing

‘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"]

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. ...

How to choose my stochastic volatility parameters? A review ArXiv ID: 2512.19821 “View on arXiv” Authors: Fabien Le Floc’h Abstract Based on the existing literature, this article presents the different ways of choosing the parameters of stochastic volatility models in general, in the context of pricing financial derivative contracts. This includes the use of stochastic volatility inside stochastic local volatility models. Keywords: Stochastic Volatility, Local Volatility, Derivatives Pricing, Parameter Estimation, Volatility Modeling, Equity Derivatives ...

One model to solve them all: 2BSDE families via neural operators ArXiv ID: 2511.01125 “View on arXiv” Authors: Takashi Furuya, Anastasis Kratsios, Dylan Possamaï, Bogdan Raonić Abstract We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov–Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees. ...

SABR-Informed Multitask Gaussian Process: A Synthetic-to-Real Framework for Implied Volatility Surface Construction ArXiv ID: 2506.22888 “View on arXiv” Authors: Jirong Zhuang, Xuan Wu Abstract Constructing the Implied Volatility Surface (IVS) is a challenging task in quantitative finance due to the complexity of real markets and the sparsity of market data. Structural models like Stochastic Alpha Beta Rho (SABR) model offer interpretability and theoretical consistency but lack flexibility, while purely data-driven methods such as Gaussian Process regression can struggle with sparse data. We introduce SABR-Informed Multi-Task Gaussian Process (SABR-MTGP), treating IVS construction as a multi-task learning problem. Our method uses a dense synthetic dataset from a calibrated SABR model as a source task to inform the construction based on sparse market data (the target task). The MTGP framework captures task correlation and transfers structural information adaptively, improving predictions particularly in data-scarce regions. Experiments using Heston-generated ground truth data under various market conditions show that SABR-MTGP outperforms both standard Gaussian process regression and SABR across different maturities. Furthermore, an application to real SPX market data demonstrates the method’s practical applicability and its ability to produce stable and realistic surfaces. This confirms our method balances structural guidance from SABR with the flexibility needed for market data. ...

Axes that matter: PCA with a difference ArXiv ID: 2503.06707 “View on arXiv” Authors: Unknown Abstract We extend the scope of differential machine learning and introduce a new breed of supervised principal component analysis to reduce dimensionality of Derivatives problems. Applications include the specification and calibration of pricing models, the identification of regression features in least-square Monte-Carlo, and the pre-processing of simulated datasets for (differential) machine learning. Keywords: differential machine learning, principal component analysis, derivatives pricing, least-square Monte-Carlo, dimensionality reduction ...

Convergence of a Deep BSDE solver with jumps ArXiv ID: 2501.09727 “View on arXiv” Authors: Unknown Abstract We study the error arising in the numerical approximation of FBSDEs and related PIDEs by means of a deep learning-based method. Our results focus on decoupled FBSDEs with jumps and extend the seminal work of HAn and Long (2020) analyzing the numerical error of the deep BSDE solver proposed in E et al. (2017). We provide a priori and a posteriori error estimates for the finite and infinite activity case. ...

Finite Element Method for HJB in Option Pricing with Stock Borrowing Fees ArXiv ID: 2501.02327 “View on arXiv” Authors: Unknown Abstract In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging process. In this study, we apply a finite element approach with a non-uniform mesh for the task of option pricing with stock borrowing fees, leading to an HJB equation that bypasses analytical optimal control in favor of direct PDE discretization. The time integration employs the theta-scheme, with initial modifications following Rannacher`s procedure. A Newton-type algorithm is applied to address the penalty-like term at each time step. Numerical experiments are conducted, demonstrating consistency with a benchmark problem and showing a strong match. The CPU time needed to reach the desired results favors P2-FEM over FDM and linear P1-FEM, with P2-FEM displaying superior convergence. This paper presents an efficient alternative framework for the HJB problem and contributes to the literature by introducing a finite element method (FEM)-based solution for HJB applications in mathematical finance. ...

AD-HOC: A C++ Expression Template package for high-order derivatives backpropagation ArXiv ID: 2412.05300 “View on arXiv” Authors: Unknown Abstract This document presents a new C++ Automatic Differentiation (AD) tool, AD-HOC (Automatic Differentiation for High-Order Calculations). This tool aims to have the following features: -Calculation of user specified derivatives of arbitrary order -To be able to run with similar speeds as handwritten code -All derivatives calculations are computed in a single backpropagation tree pass -No source code generation is used, relying heavily on the C++ compiler to statically build the computation tree before runtime -A simple interface -The ability to be used \textit{“in conjunction”} with other established, general-purpose dynamic AD tools -Header-only library, with no external dependencies -Open source, with a business-friendly license ...