A Personal Finance PDE Whose Aggregation Yields the Black–Scholes Equation

In this note, we propose a micro-level stochastic partial differential equation (PDE) to model the evolution of individual personal wealth over time. The model captures multiplicative growth and uncertainty in assets such as savings, equities, and deposits. We demonstrate that, under a mean-field aggregation and central limit scaling over a large population, the collective dynamics converge to the classical Black–Scholes PDE for option pricing. This formulation offers a bottom-up perspective on financial market models, reinterpreting Black–Scholes as the emergent limit of household-level financial behavior.