American Options
American Style Options
Our discussion so far has been limited to European style options (i.e., options that can only be exercised at maturity). American style options allow for exercise any point until maturity of the option. While the early exercise feature is not valuable for call options on non dividend paying stocks, deep in the money put options could benefit from early exercise.
When an investor exercises an option early, she captures only the intrinsic value of the option and forgoes the time value. While the intrinsic value can be invested at the risk free rate, the interest so earned is less than the time value in most cases. For a deep in the money put option, the upside is limited (because the stock price cannot fall below zero).
In such cases, the interest on intrinsic value can exceed the option’s time value.
Consider a stock currently trading at $50. The periodically compounded interest rate is 3%. Suppose that U = 1.3 and D = 0.80.
Calculate the value of a two period European style put option on the stock that has an exercise price of $50. Also, determine if early exercise would make economic sense.
Solution
The following tree shows put option value at each of the nodes in the binomial tree.
The value of the option for the down node at t = 1 is calculated as:
The value of the option at time = 0 is calculated as:
For the down move at t = 1, the exercise value of the put option is $10, calculated as or . Clearly, for this node, early exercise results in higher value ($10 as opposed to $9.44).
Had the option in the previous example been an American style put option, the value would be $5.24 as shown in Figure 29.2.
Figure 29.2: Valuing an American Style Put Option
The value of the put option at time 0 can be calculated as the present value of the expected value of the option at time t = 1.
American style call options on dividend paying stocks can be evaluated similarly: determine at each node whether the exercise value is greater than the intrinsic value and, if so, use that higher value. For dividend paying stocks, the stock price falls when the stock goes ex dividend, and it may make sense to exercise the call option before such a decline in price.