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The graph representing the value of the put therefore

merges into the put's intrinsic value, X - S, for a sufficiently small value of

S which is shown as point A in the graph. When volatility and time to

expiration increase, the value of the put moves in the direction indicated by

the arrows.

In other words, according to Cox and Rubinstein, there must always be some

critical value, S`(z), for every time instant z between time t and time T, at

which the investor will exercise the put option if that critical value, S(z),

falls to or below this value (this is when the investor thinks it is the optimal

decision to follow)...

More importantly, this critical value, S`(z) will depend

on the time left to expiry which therefore also implies that S`(z) is actually a

function of the time to expiry. This function is referred to, according to

Walker, as the Optimum Exercise Boundary (OEB).

However in order to be able to value an American put option, we need to solve

for the put valuation foundation and then optimum exercise boundary at the same

time...

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