Derivative Pricing

Black-Scholes Model, comparison between Analytical Solution and Numerical Analysis ArXiv ID: 2510.27277 “View on arXiv” Authors: Francesco Romaggi Abstract The main purpose of this article is to give a general overview and understanding of the first widely used option-pricing model, the Black-Scholes model. The history and context are presented, with the usefulness and implications in the economics world. A brief review of fundamental calculus concepts is introduced to derive and solve the model. The equation is then resolved using both an analytical (variable separation) and a numerical method (finite differences). Conclusions are drawn in order to understand how Black-Scholes is employed nowadays. At the end a handy appendix (A) is written with some economics notions to ease the reader’s comprehension of the paper; furthermore a second appendix (B) is given with some code scripts, to allow the reader to put in practice some concepts. ...

Toward Black Scholes for Prediction Markets: A Unified Kernel and Market Maker’s Handbook ArXiv ID: 2510.15205 “View on arXiv” Authors: Shaw Dalen Abstract Prediction markets, such as Polymarket, aggregate dispersed information into tradable probabilities, but they still lack a unifying stochastic kernel comparable to the one options gained from Black-Scholes. As these markets scale with institutional participation, exchange integrations, and higher volumes around elections and macro prints, market makers face belief volatility, jump, and cross-event risks without standardized tools for quoting or hedging. We propose such a foundation: a logit jump-diffusion with risk-neutral drift that treats the traded probability p_t as a Q-martingale and exposes belief volatility, jump intensity, and dependence as quotable risk factors. On top, we build a calibration pipeline that filters microstructure noise, separates diffusion from jumps using expectation-maximization, enforces the risk-neutral drift, and yields a stable belief-volatility surface. We then define a coherent derivative layer (variance, correlation, corridor, and first-passage instruments) analogous to volatility and correlation products in option markets. In controlled experiments on synthetic risk-neutral paths and real event data, the model reduces short-horizon belief-variance forecast error relative to diffusion-only and probability-space baselines, supporting both causal calibration and economic interpretability. Conceptually, the logit jump-diffusion kernel supplies an implied-volatility analogue for prediction markets: a tractable, tradable language for quoting, hedging, and transferring belief risk across venues such as Polymarket. ...

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. ...

Isogeometric Analysis for the Pricing of Financial Derivatives with Nonlinear Models: Convertible Bonds and Options ArXiv ID: 2412.08987 “View on arXiv” Authors: Unknown Abstract Computational efficiency is essential for enhancing the accuracy and practicality of pricing complex financial derivatives. In this paper, we discuss Isogeometric Analysis (IGA) for valuing financial derivatives, modeled by two nonlinear Black-Scholes PDEs: the Leland model for European call with transaction costs and the AFV model for convertible bonds with default options. We compare the solutions of IGA with finite difference methods (FDM) and finite element methods (FEM). In particular, very accurate solutions can be numerically calculated on far less mesh (knots) than FDM or FEM, by using non-uniform knots and weighted cubic NURBS, which in turn reduces the computational time significantly. ...

Mathematics of Differential Machine Learning in Derivative Pricing and Hedging ArXiv ID: 2405.01233 “View on arXiv” Authors: Unknown Abstract This article introduces the groundbreaking concept of the financial differential machine learning algorithm through a rigorous mathematical framework. Diverging from existing literature on financial machine learning, the work highlights the profound implications of theoretical assumptions within financial models on the construction of machine learning algorithms. This endeavour is particularly timely as the finance landscape witnesses a surge in interest towards data-driven models for the valuation and hedging of derivative products. Notably, the predictive capabilities of neural networks have garnered substantial attention in both academic research and practical financial applications. The approach offers a unified theoretical foundation that facilitates comprehensive comparisons, both at a theoretical level and in experimental outcomes. Importantly, this theoretical grounding lends substantial weight to the experimental results, affirming the differential machine learning method’s optimality within the prevailing context. By anchoring the insights in rigorous mathematics, the article bridges the gap between abstract financial concepts and practical algorithmic implementations. ...

Deep Joint Learning valuation of Bermudan Swaptions ArXiv ID: 2404.11257 “View on arXiv” Authors: Unknown Abstract This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine learning, Monte Carlo simulation-like training samples and joint learning to come up with an efficient numerical solution. The application of the latter development represents a novelty in the context of computational finance. We also propose a novel design of interdependent neural networks to price early-exercise products, in this case, Bermudan swaptions. The improvements in efficiency and accuracy provided by the here proposed approach is widely illustrated throughout a range of numerical experiments. Moreover, this novel methodology can be extended to the pricing of other financial derivatives. ...

A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations ArXiv ID: 2404.08456 “View on arXiv” Authors: Unknown Abstract In this work, we propose a novel backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs), where the deep neural network (DNN) models are trained not only on the inputs and labels but also the differentials of the corresponding labels. This is motivated by the fact that differential deep learning can provide an efficient approximation of the labels and their derivatives with respect to inputs. The BSDEs are reformulated as differential deep learning problems by using Malliavin calculus. The Malliavin derivatives of solution to a BSDE satisfy themselves another BSDE, resulting thus in a system of BSDEs. Such formulation requires the estimation of the solution, its gradient, and the Hessian matrix, represented by the triple of processes $\left(Y, Z, Γ\right).$ All the integrals within this system are discretized by using the Euler-Maruyama method. Subsequently, DNNs are employed to approximate the triple of these unknown processes. The DNN parameters are backwardly optimized at each time step by minimizing a differential learning type loss function, which is defined as a weighted sum of the dynamics of the discretized BSDE system, with the first term providing the dynamics of the process $Y$ and the other the process $Z$. An error analysis is carried out to show the convergence of the proposed algorithm. Various numerical experiments up to $50$ dimensions are provided to demonstrate the high efficiency. Both theoretically and numerically, it is demonstrated that our proposed scheme is more efficient compared to other contemporary deep learning-based methodologies, especially in the computation of the process $Γ$. ...

Derivative Pricing using Quantum Signal Processing ArXiv ID: 2307.14310 “View on arXiv” Authors: Unknown Abstract Pricing financial derivatives on quantum computers typically includes quantum arithmetic components which contribute heavily to the quantum resources required by the corresponding circuits. In this manuscript, we introduce a method based on Quantum Signal Processing (QSP) to encode financial derivative payoffs directly into quantum amplitudes, alleviating the quantum circuits from the burden of costly quantum arithmetic. Compared to current state-of-the-art approaches in the literature, we find that for derivative contracts of practical interest, the application of QSP significantly reduces the required resources across all metrics considered, most notably the total number of T-gates by $\sim 16$x and the number of logical qubits by $\sim 4$x. Additionally, we estimate that the logical clock rate needed for quantum advantage is also reduced by a factor of $\sim 5$x. Overall, we find that quantum advantage will require $4.7$k logical qubits, and quantum devices that can execute $10^9$ T-gates at a rate of $45$MHz. While in this work we focus specifically on the payoff component of the derivative pricing process where the method we present is most readily applicable, similar techniques can be employed to further reduce the resources in other applications, such as state preparation. ...