Perpetual American Put Optimal Stopping and Smooth Pasting

Scenario

An investor holds a perpetual American put option with strike $K$ on an underlying stock following geometric Brownian motion: $dS_t = r S_t dt + \sigma S_t dW_t^{\mathbb{Q}}$ Because maturity is infinite ($T = \infty$), the pricing function $V(S)$ is time-independent.

Your Task

1. Formulate the stationary ordinary differential equation satisfied by $V(S)$ in the continuation region $S > S^*$.
2. State the Value Matching and Smooth Pasting boundary conditions at the unknown exercise threshold $S^*$.
3. Derive the exact closed-form solution for the optimal early exercise boundary $S^*$ and the option price $V(S)$: $S^* = \frac{2r}{2r + \sigma^2} K, \quad V(S) = (K - S^*) \left( \frac{S}{S^*} \right)^{-\frac{2r}{\sigma^2}} \quad (S > S^*)$
4. For K = \100.00, r = 0.05, \sigma = 0.20$, calculate the exercise threshold$S^*$and evaluate$V(S)$at$S = $100.00$.

Constraints

• Perpetual horizon, constant $r > 0, \sigma > 0$.

Examples For $r = 0.05, \sigma = 0.20$, $\alpha = \frac{2(0.05)}{0.04} = 2.5$. S^* = \frac{2.5}{3.5} \times 100 = \71.43$.