Part $2.$

The current price of a stock is $$45.$ Calculate the value of an American put option on the stock

using a two$-$step binomial tree given the following information.

The strike price of the option, $K=$$$47,$ each time step is one year, the risk$-$free interest rate,

$r=10\%,u=1\mathrm{.}1,$ and $d=0\mathrm{.}91.$

Part $3.$

Given that $u=e^{\sqrt[\phantom{2}]{T}},$ calculate the implied volatility for the American put option in Part $2.$