You need to use a two-step binomial model and consider the possibility of early exercise. First calculate the stock price tree. You have S0=20, so the first step results in either SU=20(1.2)=24 or SD=20(0.8)=16 at the end of year one. At the end of the second year the possible outcomes are SUU=24(1.2)=28.80, SUD= SDU=24(0.8)=19.20, or SDD=16(0.8)=12.80. The PV of the expected payoff in the up node is e-0.05[0.00(0.65)+4.80(0.35)]=$1.60. The payoff from early exercise in the up node is max{24-24, 0}=0. Since the PV of the expected payoff exceeds the payoff from early exercise, early exercise in the up node is not optimal. In the down node the PV of the expected payoff is e-0.05[4.80(0.65)+11.20(0.35)]=$6.70. The payoff from early exercise in the down node is max{24-16, 0} = $8.00. So early exercise is optimal in the down node. The value of the option can now be calculated as the PV of the expected payoffs at the end of the first year, or as e-0.05[1.60(0.65)+8.00(0.35)]=$3.65.
If the option is exercised early at the initial node it is worth $4 (=$24 - 20). This value is greater than $3.65, thus, the option should be exercised early at Node 0 and will be worth $4.
