Black-Scholes Model

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. ...

Model-Free Deep Hedging with Transaction Costs and Light Data Requirements ArXiv ID: 2505.22836 “View on arXiv” Authors: Pierre Brugière, Gabriel Turinici Abstract Option pricing theory, such as the Black and Scholes (1973) model, provides an explicit solution to construct a strategy that perfectly hedges an option in a continuous-time setting. In practice, however, trading occurs in discrete time and often involves transaction costs, making the direct application of continuous-time solutions potentially suboptimal. Previous studies, such as those by Buehler et al. (2018), Buehler et al. (2019) and Cao et al. (2019), have shown that deep learning or reinforcement learning can be used to derive better hedging strategies than those based on continuous-time models. However, these approaches typically rely on a large number of trajectories (of the order of $10^5$ or $10^6$) to train the model. In this work, we show that using as few as 256 trajectories is sufficient to train a neural network that significantly outperforms, in the Geometric Brownian Motion framework, both the classical Black & Scholes formula and the Leland model, which is arguably one of the most effective explicit alternatives for incorporating transaction costs. The ability to train neural networks with such a small number of trajectories suggests the potential for more practical and simple implementation on real-time financial series. ...

Deep Learning vs. Black-Scholes: Option Pricing Performance on Brazilian Petrobras Stocks ArXiv ID: 2504.20088 “View on arXiv” Authors: Joao Felipe Gueiros, Hemanth Chandravamsi, Steven H. Frankel Abstract This paper explores the use of deep residual networks for pricing European options on Petrobras, one of the world’s largest oil and gas producers, and compares its performance with the Black-Scholes (BS) model. Using eight years of historical data from B3 (Brazilian Stock Exchange) collected via web scraping, a deep learning model was trained using a custom built hybrid loss function that incorporates market data and analytical pricing. The data for training and testing were drawn between the period spanning November 2016 to January 2025, using an 80-20 train-test split. The test set consisted of data from the final three months: November, December, and January 2025. The deep residual network model achieved a 64.3% reduction in the mean absolute error for the 3-19 BRL (Brazilian Real) range when compared to the Black-Scholes model on the test set. Furthermore, unlike the Black-Scholes solution, which tends to decrease its accuracy for longer periods of time, the deep learning model performed accurately for longer expiration periods. These findings highlight the potential of deep learning in financial modeling, with future work focusing on specialized models for different price ranges. ...

Deep Gamma Hedging ArXiv ID: 2409.13567 “View on arXiv” Authors: Unknown Abstract We train neural networks to learn optimal replication strategies for an option when two replicating instruments are available, namely the underlying and a hedging option. If the price of the hedging option matches that of the Black–Scholes model then we find the network will successfully learn the Black-Scholes gamma hedging strategy, even if the dynamics of the underlying do not match the Black–Scholes model, so long as we choose a loss function that rewards coping with model uncertainty. Our results suggest that the reason gamma hedging is used in practice is to account for model uncertainty rather than to reduce the impact of transaction costs. ...

Stochastic Expansion for the Pricing of Asian and Basket Options ArXiv ID: 2402.17684 “View on arXiv” Authors: Unknown Abstract We present closed analytical approximations for the pricing of basket options, also applicable to Asian options with discrete averaging under the Black-Scholes model with time-dependent parameters. The formulae are obtained by using a stochastic Taylor expansion around a log-normal proxy model and are found to be highly accurate for Asian options in practice as well as for vanilla options with discrete dividends. ...

Option pricing for Barndorff-Nielsen and Shephard model by supervised deep learning ArXiv ID: 2402.00445 “View on arXiv” Authors: Unknown Abstract This paper aims to develop a supervised deep-learning scheme to compute call option prices for the Barndorff-Nielsen and Shephard model with a non-martingale asset price process having infinite active jumps. In our deep learning scheme, teaching data is generated through the Monte Carlo method developed by Arai and Imai (2024). Moreover, the BNS model includes many variables, which makes the deep learning accuracy worse. Therefore, we will create another input variable using the Black-Scholes formula. As a result, the accuracy is improved dramatically. ...

Instabilities of explicit finite difference schemes with ghost points on the diffusion equation ArXiv ID: 2308.04629 “View on arXiv” Authors: Unknown Abstract Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models. ...