Black-Scholes Model: Mathematical Modeling and Continuous/Discrete Simulation

Battle Count: 89

The Black-Scholes model is foundational to quantitative finance and options trading. This paper provides a thorough mathematical derivation and two solution methods (analytical and numerical), which are essential knowledge for any quantitative trader working with derivatives. However, the paper is educational/expository in nature and does not propose novel trading strategies, signal generation, or empirical backtesting. It does not address practical trading considerations such as discrete hedging, transaction costs, or market microstructure. The numerical implementations are basic and not optimized for production trading systems. The relevance is primarily as foundational knowledge rather than a direct trading tool.

The mathematical derivation involves stochastic calculus (Itô's lemma, GBM), PDE theory (parabolic PDE, Cauchy-Euler equation), and numerical analysis (finite differences, matrix systems). The analytical solution via variable separation requires comfort with ODEs and boundary conditions. The numerical solution requires understanding of grid discretization, stability conditions (δ ≤ 0.5 for explicit method), and tridiagonal matrix systems. The provided Python code is straightforward (~145 lines) and uses only numpy and matplotlib, making it accessible for implementation. However, extending to production-grade solvers with adaptive grids, higher-order methods, or multi-dimensional PDEs would significantly increase complexity.

The paper provides complete Python code in Appendix B for both Geometric Brownian Motion simulation and explicit/implicit finite difference implementations of the Black-Scholes PDE solver. All model parameters (r=5%, σ=20%, K=100, T=1 year, S_max=500, N=200, M=2000) are explicitly stated. The derivation is self-contained with all mathematical steps shown. However, no external dataset is used (it is a theoretical/simulation paper), and no quantitative accuracy benchmarks or convergence studies are provided. The code is straightforward and educational rather than production-grade.