A put option is an option that allows its owner to sell, rather than buy, an asset at an agreed-upon exercise price \(E\) by a certain expiration time \(T\). This option is valuable to the holder if the asset value falls below \(E\), because then the option holder can buy the asset in the market at the lower price, then sell it at a price of \(E\), thus making a profit. Redo the analysis of Example 1 to give the value of an American put option. Use the same parameter values, except that the initial price of the asset should be taken as \(\$ 11\).