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Making the Valuation Model Practical
As we break the year into progressively finer subintervals, the range of possible year-end stock
prices expands. For example, when we increase the number of subperiods to three, the number
of possible stock prices increases to four, as demonstrated in the following stock price tree:
u
3
S
0
u
2
S
0
uS
0
u
2
dS
0
S
0
udS
0
dS
0
ud
2
S
0
d
2
S
0
d
3
S
0
Thus, by allowing for an ever-greater number of subperiods, we can overcome one of the
apparent limitations of the valuation model: that the number of possible end-of-period
stock prices is small.
Notice that extreme events such as u
3
S
0
or d
3
S
0
are relatively rare, as they require either
three consecutive increases or decreases in the three subintervals. More moderate, or mid-
range, results such as u
2
dS
0
can be arrived at by more than one path—any combination of
two price increases and one decrease will result in stock price u
2
dS
0
. There are three of
these paths: uud, udu, duu. In contrast, only one path, uuu, results in a stock price of u
3
S
0
.
Thus midrange values are more likely. As we make the model more realistic and break
up the option maturity into more and more subperiods, the probability distribution for the
final stock price begins to resemble the familiar bell-shaped curve with highly unlikely
extreme outcomes and far more likely midrange outcomes. The probability of each out-
come is given by the binomial probability distribution, and this multiperiod approach to
option pricing is therefore called the binomial model.
But we still need to answer an important practical question. Before the binomial
model can be used to value actual options, we need a way to choose reasonable val-
ues for u and d. The spread between up and down movements in the price of the stock
reflects the volatility of its rate of return, so the choice for u and d should depend on
that volatility. Call s your estimate of the standard deviation of the stock’s continuously
compounded annualized rate of return, and Δ t the length of each subperiod. To make
the standard deviation of the stock in the binomial model match your estimate of s , it
turns out that you can set u 5 exp(s
"Dt) and d 5 exp(2s"Dt).
3
You can see that the
Show that the initial value of the call option in Example 21.1 is $4.434.
a. Confirm that the spread in option values is C
u
2 C
d
5 $6.984.
b. Confirm that the spread in stock values is uS
0
2 dS
0
5 $15.
c. Confirm that the hedge ratio is .4656 shares purchased for each call written.
d. Demonstrate that the value in one period of a portfolio comprised of .4656 shares and one call
written is riskless.
e. Calculate the present value of this payoff.
f. Solve for the option value.
CONCEPT CHECK
21.4
3
Notice that d 5 1/ u. This is the most common, but not the only, way to calibrate the model to empirical
volatility. For alternative methods, see Robert L. McDonald, Derivatives Markets, 3rd ed., Pearson/Addison-
Wesley, Boston: 2013, Ch. 10.
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P A R T V I
Options, Futures, and Other Derivatives
proportional difference between u and d increases with both annualized volatility as well
as the duration of the subperiod. This makes sense, as both higher s and longer holding
periods make future stock prices more uncertain. The following example illustrates how
to use this calibration.
4
Using this probability, the continuously compounded expected rate of return on the stock is .10. In general,
the formula relating the probability of an upward movement with the annual expected rate of return, r, is
p 5
exp(rDt) 2 d
u 2 d
.
Suppose you are using a 3-period model to value a 1-year option on a stock with
volatility (i.e., annualized standard deviation) of s 5 .30. With a time to expiration of
T 5 1 year, and three subperiods, you would calculate Dt 5 T/n 5 1/3, u 5 exp(s
"Dt) 5
exp (.30
"1/3) 5 1.189 and d 5 exp(2s"Dt) 5 exp(2.30"1/3) 5 .841. Given the
probability of an up movement, you could then work out the probability of any final
stock price. For example, suppose the probability that the stock price increases is .554
and the probability that it decreases is .446.
4
Then the probability of stock prices at the
end of the year would be as follows:
We plot this probability distribution in Figure 21.5 , panel A. Notice that the two middle
end-of-period stock prices are, in fact, more likely than either extreme.
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