Minimum and Maximum Value of European/American Options
Lower Bound
We know that the value of an option is equal to the sum of its intrinsic value and time value.
Since an option cannot sell below its intrinsic value, its value cannot be negative, Therefore, the lower bound for both American and European options is zero.
Upper Bound
Call Options
A call option provides the option buyer the right to buy the asset. For the option to have value, its price at any time must be lower than the underlying stock price at any time. This is because if the option price were higher than the stock price, it would be cheaper to just buy the asset directly in the spot market. Therefore, the maximum price for an option is equal to the stock price at that time. This applies to both American and European options.
Put Options
A put option provides the option buyer to sell the asset at the strike price. Since an American option can be exercised at any time, its maximum price can be equal to its strike price. 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.
The following table summarizes the upper and lower bounds for these options.
| Minimum Value | Maximum Value | |
| European Call | ct>=0 | ct<=St |
| American Call | Ct>=0 | Ct<=St |
| European Put | pt>=0 | pt<=X/(1+rf)^(T-t) |
| American Put | Pt>=0 | Pt<=X |
A note about notations:
- c represents European call price (Capital C for American call)
- p represents European call price (Capital P for American put)
- S is the Spot price
- X is the Strike price
- T-t is the time remaining to maturity
These are the theoretical limits for the value of the options. However, for pricing purpose, we need to be more precise about the lower bound for the option’s price.