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.
For an at-the-money option or an out-of-the-money option, the lower bound is zero. For example, for an out-of-the-money call option strike price (X) is higher than the spot price (X). So, if someone purchased this option, he will still be better off by buying the option from the spot market.
For an in-the-money option, the lower bound will be the difference between the strike price (X), and the spot price (S). For in-the-money call options, X is lower than S, and if the price was lower than S-X, the a person can buy the call option and immediately exercise it to make a profit. Apart from this we also need to consider the present value of the exercise price.
We are not getting into the derivation of the minimum and maximum values, however it is important to know what they are. The following table summarizes the lower and upper price bounds for American and European options.
| Minimum Value | Maximum Value | |
| European Call | ct>= Max(0,S-X/(1+rf)^T-t) | ct<=St |
| American Call | Ct>= Max(0,S-X/(1+rf)^T-t) | Ct<=St |
| European Put | pt>= Max(0, X/(1+rf)^T-t – S) | pt<=X/(1+rf)^(T-t) |
| American Put | Pt >= Max(0,X-S) | Pt<=X |
Let’s take a few examples to understand these formulas.
Example 1
Calculate the minimum and maximum price of 6-month American and European calls with a strike price of 90 and currently trading at 96. Assume a risk-free rate of 6%.
American Call
Minimum: Max(0,S-X/(1+rf)^T-t) = Max (0,96-90/(1.06)^0.5) = 8.58
Maximum: Ct<=St = 96
European Call
Minimum: Max(0,S-X/(1+rf)^T-t) = Max (0,96-90/(1.06)^0.5) = 8.58
Maximum: Ct<=St = 96
Example 2
Calculate the minimum and maximum price of 6-month American and European puts with a strike price of 95 and currently trading at 90. Assume a risk-free rate of 6%.
American Put
Minimum: Max(0,X-S) = Max (0,95-90) = 5
Maximum: X = 95
European Put
Minimum: Max(0, X/(1+rf)^T-t – S) = Max (0,95/(1.06)^0.5 – 90) = 2.27
Maximum: X/(1+rf)^(T-t) = 95/(1.06)^0.5 = 92.27
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