Comparison of numerical solutions of option pricing under two mixed Black–Scholes models

Enhancing the Black–Scholes model with other financial models is a widely used approach to improve its accuracy and adaptability to real market conditions. This enhancement is typically achieved by replacing the fixed parameters of the traditional Black–Scholes model with stochastic variables, allowing for greater flexibility in capturing market dynamics. However, this modification leads to nonlinear partial differential equations (PDEs), which require advanced mathematical techniques for analysis and solution. This study extends the Black–Scholes framework by incorporating two stochastic interest rate models, resulting in nonlinear PDEs that better reflect real-world financial complexities. We systematically analyze and compare the numerical solutions of these nonlinear PDEs using two distinct computational approaches, evaluating their effectiveness and convergence properties. Furthermore, to assess the practical applicability of our models, we conduct a numerical case study using real market data. For each stochastic model, we implement both solution approaches to determine how closely the computed option prices align with actual market prices. This comparative analysis provides insights into the strengths and limitations of each method and highlights the impact of stochastic interest rate modeling on option pricing accuracy.