The Junction of PDEs, Financial Mathematics and Probability: Deriving Classical and Generalized Black-Scholes–Merton Formulas

This paper explores the intersection of three foundational areas—partial differential equations, financial mathematics, and probability—by providing a rigorous framework for the classical Black-Scholes–Merton option pricing model and its generalized extensions. For the classical model, a change in variables is employed to transform the Black-Scholes partial differential equation into the linear heat equation. The resulting formulation enables the use of Fourier transform techniques and the fundamental solution (heat kernel) to derive the closed-form Black-Scholes–Merton formula. To extend the classical setting, the interest rate in the discount factor and the stock’s rate of return are modeled using a multifactor Vasicek process, leading to a more sophisticated and realistic option pricing framework. In addition, a complementary derivation based on probabilistic methods, using a change in measure, yields an alternative explicit pricing formula. Numerical experiments based on Monte Carlo simulation show excellent agreement with the closed-form solutions and illustrate notable gains in computational efficiency. The comparative analysis further demonstrates that stochastic interest rates systematically produce lower option prices than the classical constant-rate model, underscoring the importance of accurate interest-rate modeling in practical valuation.