Option Payoff Types and Path Dependence
European, American, Asian, and barrier options differ mainly in when and how the payoff depends on the price path.
A European option can only be exercised at maturity; its payoff depends on the terminal price S_T at a single date.
An American option can be exercised at any time up to maturity, making its value depend on the whole path of prices and early-exercise opportunities.
Asian options pay based on an average price over time; barrier options switch on or off when prices cross trigger levels, both are path-dependent by construction.
Structured power and gas contracts (storage, swing, tolling) are real-world path-dependent options with operational constraints layered on top of financial payoffs.
One simulated price path. Switch payoff type to see what information drives the payoff: terminal vs exercise time vs average vs barrier hit.
European: payoff uses only S_T.
Hold the seed fixed and change parameters. You will see the same realised path reshaped by σ and T. The key lesson is not the model. It is which functional of the path defines the payoff.
Note: the “American” view here is an exercise-choice simulator. It does not compute the optimal stopping policy. It isolates the idea that early exercise makes value depend on the path and decisions along it.