Assume the following:

S = $10

u = 2

d = .5

r = 1.01 per month

X = 15

1-period = 1 month

Here, the initial stock price of 10 can double or halve every period.  Using the tree representation, we get the results shown in Figure 3.7.

Figure 3.7

Tree Representation for Stock

European option values defined on this tree can be computed in a straightforward manner by using the risk-neutral probabilities.  Recall that these probabilities are:

respectively, for an uptick and a downtick.  In this example, these equal 0.34, and 0.66, respectively.

European Call Option Values

The value of a European call option at the uptick node is obtained from the formula:

where Cuu and Cud are the terminal values of the call option (i.e., the values of the call at the two nodes after the first uptick).  A similar procedure is applied at the downtick node, and finally repeated at the initial node. The call option tree is depicted in Figure 3.8.

Figure 3.8

Tree Representation for Call Option

Here, Cuu = 25 = Suu - X = 40 - 15.

American Call Option Values

You have seen (topic 3.3, American Call Option:  Zero-Dividend Case) that if there are no dividends, a call option should not be exercised early.  Since the stock in this example does not pay any dividends, this means that the American option should be worth exactly the same as the European one.

You can verify this using this same example.  In the tree, shown in Figure 3.9, there are two values depicted at each node.  The bottom value is what you get if you exercise the call, and the top value is what you get if you do not exercise the call.

Figure 3.9

Call Value:  Exercised versus Unexercised

Clearly, you would not choose to exercise at any node, except where there are two upticks.

European Put Option Value

You can compute the European put values, strike = 15, at each node either from the risk-neutral valuation principle or from put-call parity.  The tree diagram for put values is shown in Figure 3.10.

Figure 3.10

European Put Option Values

American Put Option Value

Recall (see topic 3.4, American Put Option:  Zero-Dividend Case) that even without dividends, there may be value in early exercise of an American put option.  As a result, you have to check for this possibility at every node.

The tree in Figure 3.11 displays two values at each middle node and three values at the first node.  The top number is the (European) value of the unexercised put option, and the bottom (middle) number is the value of the put option exercised.  The bottom number at the first node is the American put value and online it appears in red:

Figure 3.11

American Put Option Value

You can see that if a downtick occurs in period 1, the stock price equals 5 and the exercised value of the put option is 15 - 5 = 10.  The  unexercised value is the same as the European put option value of 9.85.  As a result, the risk-neutral price at time 0 is now:

In the next topic Two-Period Asian Option Example you can see how to apply the binomial model to the problem of valuing a non-standard option.