First calculate the probability of a down move as: pd = 1 – pu = 1 – 0.6074 = 0.3926
Next calculate the terminal values of the option at expiration for each node of the tree:
Suu = $30 × 1.10 × 1.10 = $36.30, Puu = $0
Sud = $30 × 1.10 × 0.90 = $29.70, Pud = $2.80
Sdu = $30 × 0.9 × 1.10 = $29.70, Pdu = $2.80
Sdd = $30 × 0.9 × 0.9 = $24.30, Pdd = $8.20
Since this is an American option, we need to compare the discounted present value of the option to its intrinsic value after the end of the first 6-month period to see if the option is worth more dead than alive.
Su = $33, which means the intrinsic value of the put in the up node after six months is $0, and the option is worth more alive than dead.
Su = $27, which means the intrinsic value of the put in the down node after six months is ($32.50 – $27) = $5.50. Since $5.50 > $4.817, the option should be exercised because it is worth more dead than alive.
Finally, we can calculate the price of the American option today as:
