# Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option:

$$\frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0,$$

where

$\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S \frac{\partial V}{\partial S} - r V$ holds for $S > S_f$, where $S_f$ is contact point.

Why does this equation hold for $S > S_f$? Could you give me link for proof?

Another question is: Why do we need high-contact condition?

Update:

Do I correctly understand that for American Put Option, if $S > S_{f}$, there is no sense to exercise at time $t<T$ (because it causes immediate loss: $-V+S-K<0$). So it behaves like European Option, hence $V^{Am}_{P}=V^{E}_{P}$ and it satisfies Black-Scholes Equation.

• For the American option, the solution is given by a Optimal Stopping/Free-boundary Problem. Here you seem to have European vanilla one – Ilya Jun 3 '13 at 8:15
• Hi I suggest this question to be transferred to the Quantitative Finance Stack Exchange forum. Regards – TheBridge Jun 3 '13 at 14:43

The actual problem one solves for American options is an optimal stopping time problem, so the value of the option is

$$V_0 = \max_\tau E_{\tau}\left[e^{-r \tau} (S_\tau-K)^+ \right]$$

where the maximum is taken over all stopping times (exercise strategies $\tau>0$ permissible in the contract).

With a PDE operator such as you have, the instantaneous equality can be expressed in linear complementarity form as

$$\left(\frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V)\right)\cdot \left(V-g\right) = 0$$

where $g$ represents early exercise value.

Note for convenience that (post exercise) the stock itself satisfies the BS PDE trivially.

The payoff when exercising the option is given by:

$$\max(K-S(t),0)$$

now assume there is a $V(S,t)<\max(K-S(t),0)$: there would be the opportunity for arbitrage. We could buy the asset for $S$ and the put option for $V$. Selling the asset for $K$ would lead to a risk free profit of $K-S-V$. Thus the value of the american put option must hold the additional constraint $$V(S,t)\geq \max(K-S(t),0)$$

As long as $V > K-S$ (or $S>S_f$) it is given by the BS PDE, otherwise the price is given by $K-S$. Most (if not all) textbook introductions to financial derivatives include more details on that and derivations of the BS PDE.

The second (more mysterious) constraint is a consequence of "optimal" behavior of the agents. Keywords to find more on that might be optimal stopping problem or game theory of options.