= Solution
{c}
Use the usual nondegenerate positive-rate case $r>0$, $\sigma>0$, $K>0$. Define the <Black-Scholes equation>
$$
\mathcal LV=V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV,
\qquad g(S)=(K-S)_+.
$$
The American value dominates immediate exercise. Its discounted value is a <supermartingale> under the <risk-neutral measure>, so $\mathcal LV\leq0$, with equality where it is optimal to continue. The <obstacle problem> is
$$
\boxed{V\geq g,\quad\mathcal LV\leq0,\quad
(V-g)\mathcal LV=0,\quad V(T,S)=g(S).}
$$
Equivalently $\max(\mathcal LV,g-V)=0$.
For the put, write the exercise boundary as $S=b(t)$. On $S>b(t)$ solve $\mathcal LV=0$; on $0<S\leq b(t)$ set $V=K-S$. Value matching and <smooth pasting> require
$$
\boxed{V(t,b(t))=K-b(t),\qquad V_S(t,b(t))=-1.}
$$
The remaining conditions are $V(t,0)=K$, $V(t,S)\to0$ as $S\to\infty$, and the terminal <contingent claim payoff>, with $b(T-)=K$ in this positive-rate setting. The boundary is an unknown to be found with the price, which is what makes this a <free boundary problem>.
Back to article page
