Consider an American put option with the strike price of $32. Suppose the current price for the underlying risky asset is S0 = $31. Suppose further that the annual standard deviation for the return of the underlying asset is constant throughout the lifetime of the option and is estimated to be = 0.1 The risk-free rate in the economy is rf = 4% per year. The option matures 4 years from now. The underlying asset does not pay dividends. Price this option using binomial trees when one period is exactly one year. Note: for the annual standard deviation of the return of the underlying asset to be constant throughout the lifetime of the option, the following must be true:

u = exp(pt) 1 (1) d = exp(pt) 1 (2)

Repeat for the case when one period is six months. State conclusions. Is it ever optimal to exercise the option early? If yes, in which states?