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.

Keywords: Black-Scholes model, option pricing, finite differences, analytical solution, derivative pricing

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper involves advanced mathematics like stochastic calculus, Ito’s lemma, and PDE derivations, but lacks any empirical backtesting, real market data, or statistical metrics, focusing instead on theoretical derivations and basic code scripts.

  flowchart TD
    A["Research Goal<br>Overview of Black-Scholes Model"] --> B["Methodology<br>Calculus Derivation & Numerical Setup"]
    B --> C{"Analysis Methods"}
    C --> D["Analytical Solution<br>Variable Separation"]
    C --> E["Numerical Analysis<br>Finite Differences"]
    F["Inputs: Assumptions<br>GBM, Volatility, Strike"] --> B
    D --> G["Key Outcomes<br>Comparison of Accuracy & Efficiency"]
    E --> G
    G --> H["Modern Applications<br>Option Pricing"]