Free boundaries & exotic pricing.

Under Black-Scholes dynamics $dS=rSdt+\sigma SdW$, Itô's lemma and delta-hedging give the pricing PDE

For a European payoff you integrate backwards from maturity and you're done. For an American option the holder may exercise at any time, so the value must additionally satisfy $V\ge \text{payoff}$ everywhere - a linear complementarity problem whose solution carves the $(t,S)$ plane into a continuation region and an exercise region, separated by the free boundary S*(t).

The solver below discretises the PDE on a grid (implicit Euler in time, central differences in space, Thomas algorithm for the tridiagonal system) and enforces the early-exercise constraint by projection at each exercise date à la Brennan-Schwartz: solve the European step, then take $V\leftarrow \max (V,\text{payoff})$. Projecting at every time step gives the American price; projecting at four dates gives a Bermudan; never projecting recovers Black-Scholes exactly - a built-in sanity check against the closed form.

The heatmap is the option value; the red curve is the exercise boundary. Three things worth toggling: switch European → Bermudan → American and watch the boundary appear as isolated points, then a continuous curve, while the price at $S=100$ climbs - the early-exercise premium. Raise r and the boundary rises (deeper incentive to take the strike early); raise σ and it falls (optionality is worth waiting for).

The same machinery scaled up in the fixed-income project: Bermudan swaptions priced by deriving the PDE under a short-rate model via Itô, setting the exercise conditions at each date, and solving numerically - after building the zero-coupon curve and validating European swaption prices against Black's formula. The Monte Carlo side of the same toolbox - Euler vs Milstein convergence, Heston pricing, implied-vol smiles - has its own page.