Option Pricing

Machine learning for option pricing: an empirical investigation of network architectures ArXiv ID: 2307.07657 “View on arXiv” Authors: Unknown Abstract We consider the supervised learning problem of learning the price of an option or the implied volatility given appropriate input data (model parameters) and corresponding output data (option prices or implied volatilities). The majority of articles in this literature considers a (plain) feed forward neural network architecture in order to connect the neurons used for learning the function mapping inputs to outputs. In this article, motivated by methods in image classification and recent advances in machine learning methods for PDEs, we investigate empirically whether and how the choice of network architecture affects the accuracy and training time of a machine learning algorithm. We find that the generalized highway network architecture achieves the best performance, when considering the mean squared error and the training time as criteria, within the considered parameter budgets for the Black-Scholes and Heston option pricing problems. Considering the transformed implied volatility problem, a simplified DGM variant achieves the lowest error among the tested architectures. We also carry out a capacity-normalised comparison for completeness, where all architectures are evaluated with an equal number of parameters. Finally, for the implied volatility problem, we additionally include experiments using real market data. ...

Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure ArXiv ID: 2306.05750 “View on arXiv” Authors: Unknown Abstract The Barndorff-Nielsen and Shephard model is a representative jump-type stochastic volatility model. Still, no method exists to compute option prices numerically for the non-martingale case with infinite active jumps. We develop two simulation methods for such a case under change of measure and conduct some numerical experiments. Keywords: Barndorff-Nielsen and Shephard Model, Stochastic Volatility, Jump Diffusion, Option Pricing, Monte Carlo Simulation, Options ...

Efficient inverse $Z$-transform: sufficient conditions ArXiv ID: 2305.10725 “View on arXiv” Authors: Unknown Abstract We derive several sets of sufficient conditions for applicability of the new efficient numerical realization of the inverse $Z$-transform. For large $n$, the complexity of the new scheme is dozens of times smaller than the complexity of the trapezoid rule. As applications, pricing of European options and single barrier options with discrete monitoring are considered; applications to more general options with barrier-lookback features are outlined. In the case of sectorial transition operators, hence, for symmetric Lévy models, the proof is straightforward. In the case of non-symmetric Lévy models, we construct a non-linear deformation of the dual space, which makes the transition operator sectorial, with an arbitrary small opening angle, and justify the new realization. We impose mild conditions which are satisfied for wide classes of non-symmetric Stieltjes-Lévy processes. ...

MathematicalFinanceIntroduction to Continuous Time Financial Market Models ArXiv ID: ssrn-976593 “View on arXiv” Authors: Unknown Abstract These are my Lecture Notes for a course in Continuous Time Finance which I taught in the Summer term 2003 at the University of Kaiserslautern. I am aware that t Keywords: continuous time finance, stochastic calculus, option pricing, martingales, stochastic differential equations, Derivatives / Quantitative Finance Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 1.0/10 Quadrant: Lab Rats Why: The paper presents dense, advanced mathematics centered on stochastic analysis, stochastic calculus, and derivations of the Black-Scholes model, with no empirical data or backtesting. flowchart TD A["Research Goal: Develop Continuous Time Financial Market Models"] --> B["Methodology: Stochastic Calculus & Martingales"] B --> C["Data: Geometric Brownian Motion SDE Inputs"] C --> D["Computation: Black-Scholes Option Pricing & PDE Solution"] D --> E["Outcome: Valuation of Derivatives & Risk Management Insights"]