8.5Binomial Valuation of American Options

The last section derived a formula for valuing European calls.¹⁴This section illustrates

how to use the binomial model to value options that may be prematurely exercised.

These include American puts and American calls on dividend-paying stocks.

¹⁴As

suggested earlier, American calls on stocks that pay no dividends until expiration can be valued

as European calls with the method described in the last section.

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Chapter 8

Options

275

American Puts

The procedure for modeling American option values with the binomial approach is sim-

ilar to that for European options. As with European options, work backward from the

right-hand side of the tree diagram. At each node in the tree diagram, look at the two

future values of the option and use risk-neutral discounting to determine the value of

the option at that node. In contrast to European options, however, this value is only the

value of the option at that node provided that the investor holds on to it one more

period. If the investor exercises the option at the node, and the underlying asset is worth

S at the node, then the value captured is S Kfor a call and K Sfor a put, rather

than the discounted risk-neutral expectation of the values at the two future nodes, as

was the case for the European option.

This suggests a way to value the option, assuming that it will be optimally exer-

cised. Working backward, at each node, compare (1) the value from early exercise

of the option with (2) the value of waiting one more period and achieving one of

two values. The value to be placed at that decision node is the larger of the exer-

cise value and the value of waiting. Example 8.6 illustrates this procedure for an

American put.

Example 8.6:Valuing an American Put

Assume that each period, the nondividend-paying stock of Chiron, a drug company, can

either double or halve in value, that is, u2, d.5.If the initial price of the stock is $20

per share and the risk-free rate is 25 percent per period, what is the value of an American

put expiring two periods from now with a strike price of $27.50?

Answer:Exhibit 8.8 outlines the path of Chiron’s stock and the option.At each node in

the diagram, the risk-neutral probabilities solve

2.5(1 )

1

1.25

Thus, = .5 throughout the problem.

In the final period, at the far right-hand side of the diagram, the stock is worth $80, $20,

or $5.From node U,where the stock price is $40 a share, the price can move to $80 (node

UU) with an associated put value of $0, or it can move to $20 (node UD), which gives a put

value of $7.50.There is no value from early exercise at node Ubecause the $40 stock price

exceeds the $27.50 strike price.This means the put option at node Uis worth the value of

the $0 and $7.50 outcomes, or

0$7.50(1 )

$3.00

1.25

From node D,where the stock price is $10 a share, early exercise leads to a put value

of $17.50.Compare this with the value from not exercising and waiting one more period,

which leads to stock prices of $20 (node UD) and $5 (node DD), and respective put values

of $7.50 and $22.50.The value of these two outcomes is

$7.50$22.50(1 )

$12.00

1.2

This is less than the value from early exercise.Therefore, the optimal exercise policy is to

exercise when the stock price hits $10 at node D.

From the leftmost node, there are two possible subsequent values for the stock.If the

stock price rises to $40 a share (node U), the put, worth $3, should not be exercised early.

If the stock price declines to $10 a share (node D), the put is worth $17.50, achieved by

early exercise.If the market sets a price for the American put, believing it will be optimally

exercised, the market value at this point would be $17.50.To value the put at the leftmost

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

276	Part IIValuing Financial Assets

EXHIBIT8.8

Binomial Tree Diagram forthe Price of Chiron Stock (Above Node) and anAmerican Put Option (Below Node) with K $27.50 and T 2

$80

	UU
.5
	$0
$40

		U
	.5
$20			$20
		?
			.5
			UD
			.5
?		$10	$7.50
	.5
		D

?		$5
	.5
		DD

$22.50

node, weight the $3.00 and $17.50 outcomes by the risk-neutral probabilities and discount

at the risk-free rate.This value is

$3.00$17.50(1 )

$8.20

1.25

Early exercise leads to a $7.50 put at this node, which is inferior.Thus, the value of the put

is $8.20, and it pays to wait another period before possibly exercising.

Note that even with the possibility of early exercise, it is still possible to track the

put option in Example 8.6 with a combination of Chiron stock and a risk-free asset.

However, the tracking portfolio here is different than it is for a European put option

with comparable features. In tracking the American put, the major difference is due

to what happens at node Din Exhibit 8.8, when early exercise is optimal. To track

the option value, do not solve for a portfolio of the stock and the risk-free security

that has a value of $3.00 if an up move occurs and $12.00 if a down move occurs,

but solve for the portfolio that has a value of $3.00 if an up move occurs and $17.50

if a down move occurs. This still amounts to solving two equations with two

unknowns. With the European option, the variation between the up and down state

is $9.00. With the American option in Example 8.6, the variation is $14.50.

Since the variation in the stock price between nodes Uand Dis $30 ($40 $10),

.3 ( $9/$30) of a share perfectly tracks a European option and .483333

(14.50/$30) of a share perfectly tracks an American option. The associated risk-

free investments solve

.3($40)1.25x$3 or x$12 for the European put

.483333($40)1.25x$3 or x$17.867 for the American put

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Chapter 8

Options

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This also implies that the European put is worth .3($20) $12 $6, which is $2.20

less than the American put. One can also compute this $2.20 difference by multiplying

1 (which is .5) by the $5.50 difference between the nonexercise value and actual value

of the American put at node Dand discounting back one period at a 25 percent rate.