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Early exercise of an American put depends on a tradeoff between insurance value

When there are no dividends, the early exercise of an American put depends on a tradeoff

between insurance value (which comes from volatility) and time value (a function

of interest rates). Thus, for example, for a given level of volatility, early exercise of the

put becomes more likely if interest rates are higher. This question provides a numerical

illustration

Consider a two-period binomial model with u = 1.10 and d = 0.90. Suppose the initial

stock price is 100, and we are looking to price a two-period American put option with

a strike of K = 95.

(a) First, consider a “low” interest rate of R = 1.02. Show that early exercise of the

American put is never optimal in this case.

(b) Nowconsider a “high” interest rate of R = 1.05. Showthat it nowbecomes optimal

to exercise the put early in some circumstances. What is the early exercise premium

in this case?

Sample Answer

Part (a)

To determine whether early exercise of the American put is optimal under the “low” interest rate of R = 1.02, we need to compare the option’s intrinsic value at each node to the expected discounted value of exercising the option early and investing the proceeds at the risk-free rate.

At the node where St = 110:

Intrinsic value = 10 – 95 = 5

Expected discounted value of exercising early:

(5) / (1.02)^2 = 4.78

Since the intrinsic value is greater than the expected discounted value of exercising early, it is not optimal to exercise the put early at this node.

Full Answer Section

At the node where St = 90:

Intrinsic value = 5

Expected discounted value of exercising early:

(5) / (1.02)^2 = 4.78

Again, the intrinsic value is greater than the expected discounted value of exercising early, so it is not optimal to exercise the put early at this node.

Therefore, under the “low” interest rate of R = 1.02, early exercise of the American put is never optimal.

Part (b)

Under the “high” interest rate of R = 1.05, the expected discounted value of exercising early becomes more attractive, potentially favoring early exercise at some nodes.

At the node where St = 110:

Intrinsic value = 10 – 95 = 5

Expected discounted value of exercising early:

(5) / (1.05)^2 = 4.50

Since the intrinsic value is still greater than the expected discounted value of exercising early, it is not optimal to exercise the put early at this node.

At the node where St = 90:

Intrinsic value = 5

Expected discounted value of exercising early:

(5) / (1.05)^2 = 4.39

In this case, the intrinsic value is very close to the expected discounted value of exercising early. However, since the intrinsic value is still slightly higher, it is not optimal to exercise the put early at this node.

However, at the final node where St = 99:

Intrinsic value = 1

Expected discounted value of exercising early:

(1) / (1.05)^2 = 0.926

Here, the intrinsic value is lower than the expected discounted value of exercising early. Therefore, it is optimal to exercise the put early at this node to maximize the option’s value.

The early exercise premium can be calculated by subtracting the value of the American put without early exercise from the value of the American put with early exercise.

Value of American put without early exercise:

Max(95 – St, 0)

At the final node where St = 99, the value of the put without early exercise is 0.

Value of American put with early exercise:

Max(95 – St, St / (1.05)^2)

At the final node where St = 99, the value of the put with early exercise is 0.926.

Therefore, the early exercise premium is 0.926 – 0 = 0.926.

In conclusion, early exercise of the American put becomes optimal under the “high” interest rate of R = 1.05 at the final node where St = 99. The early exercise premium in this case is 0.926.

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