We will start with the given information and apply it to the two-step binomial model:
Current stock price (S) = $100 \\
Strike price (K) =$100
Risk-free rate (r) = 5% = 0.05
Up-move potential (u) = 1 + 20% = 1.20
Down-move potential (d) = 1 - 20% = 0.80
Time to expiration (n) = 2 periods
First, we need to ensure that there is no arbitrage opportunity. We can check this by verifying whether 0 < d < 1+r < u. We have:
0 < 0.8 < 1.05 < 1.20
This holds true, and we can proceed with the binomial pricing model.
Now let’s construct the binomial tree of stock prices:
Next, let’s calculate the option value at each node using the "backward induction" approach.
At maturity, the payoffs of the call option are:
$$\begin{aligned}
C_{2,0} &= \max\{S_{2,0} - K, 0\} = \max\{64 - 100, 0\} = 0 \\
C_{2,1} &= \max\{S_{2,1} - K, 0\} = \max\{96 - 100, 0\} = 0 \\
C_{2,2} &= \max\{S_{2,2} - K, 0\} = \max\{144 - 100, 0\} = 44\end{aligned}$$
To find the option price at the earlier nodes, we need to calculate the risk-neutral probabilities and then discount the option values. The risk-neutral probabilities are given by:
$$p = \frac{1 + r - d}{u - d} = \frac{1.05 - 0.8}{1.20 - 0.8} = 0.25 \\
1 - p = 1 - 0.25 = 0.75$$
Next, let’s move one step back and calculate the expected discounted option values for nodes S1, 0 and S1, 1:
$$C_{1,0} = \frac{p C_{2,1} + (1-p) C_{2,0}}{1 + r} = \frac{0.25 \cdot 0 + 0.75 \cdot 0}{1.05} = \$0$$
$$C_{1,1} = \frac{p C_{2,2} + (1-p) C_{2,1}}{1 + r} = \frac{0.25 \cdot 44 + 0.75 \cdot 0}{1.05} = \$10.48$$
Now, since it’s an American call option, we need to check for early exercise at node S1, 1. The value of early exercise at that node is max$120 - $100, 0 = $20. Since the early exercise value is higher than the discounted option value, we update the value of C1, 1:
C1, 1 = max C1, 1, S1, 1 − K = max $10.48, $20 = $20
$$\begin{aligned}
C_{1,1} &= \max\{C_{1,1}, S_{1,1} - K\} = \max\{10.48, 20\} = 20\end{aligned}$$
Finally, we move one more step back to find the price of the American call option:
$$C_{0,0} =\frac{p C_{1,1} + (1-p) C_{1,0}}{1 + r} =\frac{0.25 \cdot 20 + 0.75\cdot 0}{1.05} = \$4.76$$
Thus, the price of the American call option is $4.76.