Terminal Condition for American Put Option In a recent book I read the author mentioned the terminal condition: 

$$\mathop {\lim }\limits_{t \to T} V(S,t) = \max \left\{ {X - S,0} \right\}$$

This is intuitive to understand. 

Then, the author defines: $$\tau  \equiv T - t$$

With this, the terminal condition above can be simplified to:

$$\mathop {\lim }\limits_{\tau  \to 0} V(S,\tau) = 0$$

This is not so intuitive. How can the value of the option be equal to zero in this case?

(note: in the space)   $${\Sigma _1} = \left\{ {(S,\tau )|B(\tau ) \le S &lt;  + \infty ,0 \le \tau  \le T} \right\}$$

Notations: 

$X$ = exercise price 

$S$ = underlying stock price

$T$ = time to maturity

$t$ = time to today

$B(\tau)$ = optimal exercise boundary
I don't think this boundary condition has anything to do with the optimal exercise price and it should therefore hold for both European and American style options. It is simply a terminal condition which allows us to rewrite the PDE in closed-form.

The relationship you show above in which:


  $\mathop {\lim }\limits_{t \to T} V(S,t) = \max \left\{ {X - S,0} \right\}$


$\to$


  $\mathop {\lim }\limits_{\tau  \to 0} V(S,\tau) = 0$


simply means that the extrinsic value of the option (i.e., the value in excess of $X-S&gt; 0, \forall S&lt;X)$ tends towards $0$ as $t \to T$.

Dropping the "$max$" function is simply a restatement of the payoff condition in terms of the Heaviside function. The Heaviside function is essentially equivalent to the maximum function except that it enforces the following boundary at $t=T$:

${\displaystyle V(S,\,\tau)=0\quad \forall \;\;S&lt;X}$,

Though the difference seems subtle, it is important since $V_t$ need not be finite at S = 0, or even defined for that matter, which allows us to more easily perform the substitution of variables required to express the contingent pay-off in terms of the (heat) diffusion equation which is the general solution to the Black-Scholes model.

I hope that the intuition of this boundary condition is clearer from this explanation.