Boundary Conditions for Options
For anyone who knows the payoff values for call and put options, they are also aware of minimum and maximum values for call and put options. However those values change according to the type of option, European or American.
For American options, you do not need to wait till the end of contract to exercise your option. On the other hand, you have to wait to the end of contract period in order to exercise your option. That makes a difference in the maximum and minimum value for European and American options.
1. Maximum and Minimum Values
The minimum value for any option is zero.
No option can sell less than zero.
Maximum value of a call is the current value of underlying
Call option gives you the right to buy an underlying asset as a fixed price. It would not make sense to pay more for the right to buy the underlying than the value of the underlying itself.
Maximum value of a European Put is the present value of the exercise price
Put option gives you the right to sell the underlying at a fixed price. If the underlying is $10 and you got the Put option to exercise at $20 than you can sell it at $20. If you have the right at $30 you can sell it at $30. So the maximum value for a put option is the exercise price. But for European put options you have to wait till the end. Thus, the exercise price has to bring back to today’s value, which is the Present Value of Exercise price.
But the maximum and minimum values can be narrowed.
For American options which are exercisable immediately, the lower bound of an American option price is the current intrinsic value
C>= Max ( 0, S-X)
P >=Max ( 0, X-S)
Same applies to European Call options, but remember that the European option has to wait to the end, thus the exercise price is the PV of the X.
So, the lower bound of European Call option is either zero or the underlying price minus the present value of the exercise price, whichever is greater.
Same applies to put options
the lower bound of European Put option is the greater of either zero or the present value of the exercise price minus the underlying price.
But there is one interesting point here, European Call is the the greater than the American Call option at the exercise day which is non-sense. Thus the upper bound of American Call option is actually the pay-off for the European Call option
American Call >= Max [0, S- X/(1+r)^T]
European Call >= Max [0, S-X/(1+r)^T]
for the Put options things change
American Put >= Max [0, X-S]
European Put >= Max [0, X/(1+r)^T-S]