Partial Differential Equation Modeling

Partial differential equation modeling is the use of calculus-based equations to describe how financial variables change over space and time. In derivatives, PDEs are used to model the evolution of option prices based on underlying price, time, volatility, and interest rates.

The most famous example is the Black-Scholes-Merton PDE, which assumes a continuous market and geometric Brownian motion. Modern models extend these to include stochastic volatility, jumps in price, and transaction costs.

These models provide the theoretical foundation for pricing, hedging, and risk assessment. While computationally intensive, they offer the most rigorous framework for understanding the behavior of complex financial instruments in diverse market environments.