European, American, and Bermuda options

In Chapter 10, Options and Futures, we have learnt that for a European option, the option buyer could exercise their right only on maturity dates, while for an American option buyer, they could exercise their right any time before and on maturity dates. Thus, an American option would be more valuable than its counterparty of European option. Bermudan options could be exercised once or several times on a few predetermined dates. Consequently, the price of a Bermudan option should be between a European and an American option with the same features, such as the same maturity dates and the same exercises prices, see the following two inequalities for call options:

Here is an example for a Bermudan option. Assume that a company issues a 10-year bond. After seven years, the company could call back, that is, retire, the bond at the end of each year for the next three years. This callable property is eventually an embedded Bermudan option with exercise dates...