Lower and Upper Bounds for Options

For those of you with the Kaplan Schweser 2012 notes, it may help to refer to Figure 5 on p211 of the Fixed Inc/Derivatives book for my question.

I’m looking for guidance on the below:

We discount the exercise price (X) by the Risk Free Rate for the minimum value of a European & American Call and for a European Put (assuming it’s >0), and for the maximum value of a European Put.

The Exercise Price (X) of an American Put is not discounted (minimum value or Maximum). This makes sense as it’s immediately exercisable when it is in the money.

What I would like is a bit more info on why we need to discount X at the risk free rate on an American Call.


What is the definition of “Exercise Price”?

Exercise price is the strike price on the option, i.e. the price at which it can be exercised.


the question was to get you thinking about when it is exercised…

and hence the reason to “discount” it.

Oops (it’s pretty late over here).

American Call is in-the-money when the exercise price is below the stock price.

American Put is in-the-money when the stock price falls below the exercise price.

Is it because if we didn’t discount the American Call, the European Call would have a greater value?

as St - XPV > St - X


X = Exercise price

XPV = PV of X

St = Equals stock at time t

Also as:

as St - XPV > St - X

If the American Option was in-the-money, given this condition, selling the option in the secondary market would be worth more than exercising it?


Watch this video. It covers everything in this reading.


the reason why you have to discount an american call ( for minimum value) is because if you dont discount it, it can be potentially lower than a european call. an american call cannot be lower than a european call.

  • american option can be excercised anytime while european only at the end = amercan call must have more value for this extra feature.

  • since all valuations are based on European calls who’s value of X must be discounted (excercised at the end) ==> you have to discount the call of an american value too so it can be worth AT LEAST the value of a European call (if you don’t discount, american call will be worth less) which is not possible