The maximum value of a call or put option could be any value between zero and the difference between the underlying price and exercise price. By establishing lower bounds, we are able to tighten the range so that at expiration, the minimum value of a call and a put is zero.

• The maximum value of a call option is: max (0, underlying price - exercise price).
• The maximum value of a put option is max (0, exercise price - underlying price).
 Remember that the maximum value of a call option is the greater of zero or, its underlying price minus the exercise price. On the other hand, the maximum value of a put option is the greater of zero, or its exercise price minus the underlying price.

The maximum and minimum values for American calls are explained by the following formula:

 Formula 15.7

Notice that the lower bounds for an American call option are the same as the lower bounds for a European call option.

Now, for an American put option, the lower bounds are slightly different:

Formula 15.8

European Calls
Because European options can only be exercised on a specific date (unlike American options, which can be exercised at any time before or at expiration), we need to perform extra manipulations to determine the lower bound. We will skip these manipulations here because it is unlikely that you will be asked to know this on your upcoming exam. The formula is as follows:

Formula 15.9

Where: co = current call value, S0 = current price of underlying asset, X = strike price, r = risk-free rate of interest, and T = time to expiration (# days/365)

European Puts
The lower bounds for European puts is are follows:

Formula 15.10

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