Stefan Problem in Finance

The Stefan problem, originally from thermodynamics, describes the evolution of a boundary between two phases of matter, such as ice melting into water. In quantitative finance, it serves as an analogy for modeling the moving boundaries in derivative pricing.

When pricing path-dependent options or options with complex exercise conditions, the interface between different regions of the state space acts like a phase boundary. The position of this boundary is determined by the heat equation, which is mathematically equivalent to the Black-Scholes partial differential equation.

Financial engineers use Stefan problem solvers to track these moving boundaries under changing market conditions. It is a sophisticated tool for understanding how liquidity and volatility create moving frontiers in trading.