Option Early Exercise

(post from old site)

American call

What happens when you exercise?

• you give up the option (whatever that's worth)
• realize the cash flow (call: St - K; put: K-St)

What is that european option worth approximately?

• some upper bound >= option >= some lower bound
• St >= european call >= max{St - Kd(t,T), 0}
• Kd(t, T) >= european put >= max{Kd(t, T) - St, 0}

When NOT to exercise?

If we end up with CF lower than the minimum value that a given option is worth, we definitely do NOT wanna exercise.

Minimum value of an American Call

@maturity T, Call_Eu is worth max{S-K, 0}

@anytime t, Call_Eu + Kd(t, T) = St + Put_Eu

So, Call_Eu = max{St + Put_Eu - Kd(t, T), 0}

>=  max{St - Kd(t,T), 0}

Call_A >= Call_Eu since American offers early exercise.

so, Call_A >= max{St - Kd(t,T), 0}

@time t

if we don't exercise, we hold on to something that's worth at least St - Kd(t,T)

if we do, we realize St - K.

since K > Kd(t,T), Call_A >= max{St - Kd(t,T), 0} > max{St - K, 0}

we'd realize a cash flow max{St - K, 0} that's less than what it's worth at the minimum ie max{St - Kd(t,T), 0} if we exercise at any time t.  That's why we don't ever wanna early exercise a call if there's no dividend.

Note that the above applies on stock options only, not future options.
"Remember that the result about it never being optimal to early exercise an American call option on a non-dividend paying stock only applies to ... stocks. A futures contract is not a stock. In fact, as I said in one of the lecture you can think of a futures contract as being a security that is always worth zero but that it pays a (sometimes negative) "dividend" in every period."

Minimum value of American put?

@maturity T, Put_Eu is worth max{K-ST, 0}

@anytime t, Put_Eu = Call_Eu + Kd(t, T) - St

So, Put_Eu = max{Call_Eu + Kd(t, T) - St, 0}

>=  max{Kd(t, T) - St, 0}

Put_A >= Put_Eu since American offers early exercise.

so, Put_A >= max{Kd(t, T) - St, 0}

@time t

if we don't exercise, we hold on to something that's >= Kd(t, T) - St

if we do, we realize  K - St.

since K > Kd(t,T), we'd realize a cash flow that's greater than what it's worth at the minimum if we exercise at any time t.  It means that we cannot eliminate the potential opportunity to exercise.

Merely beating the minimum does NOT mean it's optimal.  Say, if you own an ATM Put that's yet to expire.  If you exercise it, you give up the option and you get nothing for CF.  That option is yet to expired and is certainly worth something and you do not wanna exercise at that point just because it beats the minimum.

So, when do we wanna early exercise?  TBD...