**CFA Level 1 - Derivatives**

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: c _{o} = current call value, S_{0} = 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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