Perpetual American Straddle Dual Free-Boundary Problem

Scenario

A perpetual American straddle on an asset paying continuous dividend yield $q$ gives the holder the right to exercise at any time to receive payoff $|S - K|$. The continuation region is an open interval $(S_L^*, S_H^*)$ bounded by two unknown optimal exercise thresholds $S_L^* < K < S_H^*$.

Your Task

1. Explain why early exercise of the call side requires a positive dividend yield ($q > 0$).
2. Formulate the general solution $V(S) = A S^{\beta_1} + B S^{\beta_2}$ to the stationary Black-Scholes ODE with dividend yield $q$ in the continuation region, where $\beta_1 > 1$ and $\beta_2 < 0$ are the characteristic roots.
3. State the system of four coupled non-linear equations generated by Value Matching and Smooth Pasting at both boundaries $S_L^*$ and $S_H^*$: $V(S_L^*) = K - S_L^*, \quad V'(S_L^*) = -1$ $V(S_H^*) = S_H^* - K, \quad V'(S_H^*) = +1$
4. Implement a numerical root solver to determine $A, B, S_L^*, S_H^*$ for $K = 100, r = 0.05, q = 0.03, \sigma = 0.20$, and verify that $S_L^* < K < S_H^*$.

Constraints

• Constant parameters $r > 0, q > 0, \sigma > 0$.
• Smooth contact at both exercise boundaries.

Examples For $K = 100, r = 0.05, q = 0.03, \sigma = 0.20$, $\beta_1 \approx 1.581, \beta_2 \approx -1.581$. The put threshold is $S_L^* \approx 47.92$, and the call threshold is $S_H^* \approx 285.42$.