R19.P1.T3.FIN_PRODS_HULL_Ch10_Put_Option_Max&Min_Values

[gargi.adhikari]

In reference to R19.P1.T3.FIN_PRODS_HULL_Ch10_Put_Option_Max & Min Values:-
I am having some trouble with the Max and Min Values of American & European Put Options.

Since an American option can be exercised at any time,
Max Value of an American PUT, P <= K .

However, for a European option, since it cannot be exercised before the expiry date, its maximum value= will be equal to the present value of the strike price.
That is, Max Value of an European PUT <= Ke^-rt

So using the above logic,
Min Value of an American CALL should be, C >= Max [ ( S-K) , 0 ]-->Since an American option can be exercised at any time

Min Value of an European CALL should be , C >= Max [ ( S-K. e^-rt) , 0 ]-->Since an European Option cannot be exercised before the expiry date, the Strike Price has to be discounted to reflect the Present Value of the strike price.

But for Call Options, I see just one formula C >= Max [ ( S-K. e^-rt) , 0 ] --both for the American as well as the European Call Option.

Any insights on this would be much appreciated.

 

[David Harper CFA FRM]

Hi @gargi.adhikari I think the reason is that, given an American option must be worth more than its European counterpart (because it offers everything the European offers plus it offers the option to exercise early, so we can think of an American option as a European option plus an early exercise option!)....

• In the case of the put, the value of (K-S) > K*exp(-rT) - S, because K > K*exp(-rT). However,
• In the case of the call, the value of (S - K) is not > S - K*exp(-rT). Instead, the american (S-K) < the european S-K*exp(-rT). As this is merely the determination of the lower bound, it does not make sense to assert the American call minimum value is less than the European call minimum value.

From my perspective, this is really a matter of time value favoring the call option and somewhat working against the put. If for some reason we assumed a negative risk-fee rate, then the situation would be reversed (!); i.e., there would only be one minimum value, including for the american call, of S-K*exp(-rT), because it would (in that scenario) be greater than (S-K). Why is time value favoring the call? Because the strike price is discounted to the present. These minimum value bounds are very narrow (limited) metrics, they omit the volatility the is a large determinant of value.

FWIW, an intuition around the call is to model out the implicit assumption that the minimum value assumes the stock price only grows at the risk free rate. In that case, the future S = S*exp(rT) and is exercised for future gain of S*exp(rT)-K, such that the PV of the future gain = [S*exp(rT)-K]*exp(-rT) which equals S*K*exp(-rT). I hope that's interesting!

 

[gargi.adhikari]

@David Harper CFA FRM Thanks so much for this insight ! Just had my AHA Moment !