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This paper aims to create a comprehensive framework for both classical and generalized Black-Scholes–Merton option pricing models.

February 12, 2026AppliedMathOpen Access

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

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Exploratory research derives option pricing formulas in finance, highlighting the need for accurate interest-rate modeling.

This paper aims to create a comprehensive framework for both classical and generalized Black-Scholes–Merton option pricing models.
Transformed the Black-Scholes PDE into a linear heat equation using change of variables.
Applied Fourier transform techniques to derive the closed-form Black-Scholes–Merton formula.
Modeled interest rates and stock returns with a multifactor Vasicek process for extended pricing.
Utilized probabilistic methods and change of measure for an alternative pricing formula.
Conducted numerical experiments through Monte Carlo simulations to compare results.
Successfully derived both classical and generalized Black-Scholes–Merton formulas.
Found that stochastic interest rates lead to lower option prices compared to constant-rate models.
Demonstrated that Monte Carlo simulations align closely with closed-form solutions, showing significant computational efficiency.

Meas et al. (2026) studied this question.

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