Exercise: Visualising the early-exercise boundary

The optimal-stopping rule for an American put is: exercise when $S \le S^*(t)$ for some boundary

S^*(t)

. This is a curve in

(t, S)

-space.

Tasks

Modify the binomial tree to record the early-exercise boundary at each time step. (At each time $n$ , find the largest $S$ at which $V^{exercise} \ge V^{hold}$ .)
Plot $S^*(t)$ vs $t \in [0, 1]$ for the American put $K = 100, r = 0.05, \sigma = 0.2, T = 1$ .
Boundary at $t = T$ . What is $S^*(T^-)$ ? Compare with the perpetual put boundary (for $T = \infty$ ).
Effect of $\sigma$ . Repeat for $\sigma \in \{0.1, 0.2, 0.4\}$ . How does the boundary shift?
Effect of $r$ . Repeat for $r \in \{0.01, 0.05, 0.10\}$ . Higher rates push the boundary closer to $K$ — why?