Investments, tenth edition

 Example  21.2 

Calibrating  u  and  d  to Stock Volatility 

  Event  

  Possible 

Paths  

  Probability  

  Final Stock Price  

 3 down 

movements 

  ddd  

 .446 

3

   5  0.089  

59.48  5  100  3  .841 

3

  

 2 down 

and 1 up 

  ddu,   dud,  

 udd  

 3  3  .446 

2

   3  .554  5  0.330  

84.10  5  100  3  1.189  3  .841 

2

  

 1 down 

and 2 up 

  uud,   udu,  

 duu  

 3  3  .446  3  .554 

2

   5  0.411  

118.89  5  100  3  1.189 

2

   3  .841 

 3 up 

movements 

  uuu  

 .554 

3

   5  0.170  

168.09  5  100  3  1.189 

3

  

 Now we can extend Example 21.2 by breaking up the option maturity into ever-shorter 

subintervals. As we do, the stock price distribution becomes increasingly plausible, as we 

demonstrate in Example 21.3.  

 In Example 21.2, we broke up the year into three subperiods. Let’s now look at the cases 

of six and 20 subperiods. 

  Subperiods,   n  

 D t     5      T/n  

      u 5 exp(s"Dt)

   d 5 exp(2s"Dt)

   

  3 

 

.333 

 exp(.173)  5  1.189  

exp( 2 .173)  5  .841 

   6  

.167 

 exp(.123)  5  1.130  

exp( 2 .095)  5  .885 

 20  

.015 

 exp(.067)  5  1.069  

exp( 2 .067)  5  .935 

 Example  21.3 

Increasing the Number of Subperiods 

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  C H A P T E R  

2 1

 Option 

Valuation 

735

Figure 21.5 

  Probability distributions for final stock price. Possible outcomes and associated probabilities. 

In each panel, the stock’s annualized, continuously compounded expected rate of return is 10% and its 

standard deviation is 30%.  Panel A.  Three subintervals. In each subinterval, the stock can increase by 18.9% 

or fall by 15.9%.  Panel B.  Six subintervals. In each subinterval, the stock can increase by 13.0% or fall by 

11.5%.  Panel C.  Twenty subintervals. In each subinterval, the stock can increase by 6.9% or fall by 6.5%. 

.00

.05

.10

.15

.20

.25

.30

.35

.40

25

50

75

100 125 150 175 200 225 250

Final Stock Price

.45

Probability

A

.00

.05

.10

.15

.20

.25

.30

.35

25

50

75

100 125 150 175 200 225 250

Final Stock Price

Probability

B

.00

.05

.10

.15

.20

25

50

75

100 125 150 175 200 225 250

Final Stock Price

C

Probability

We plot the resulting probability distributions in panels B and C of  Figure 21.5 .  

5

  

 Notice that the right tail of the distribution in panel C is noticeably longer than 

the left tail. In fact, as the number of intervals increases, the distribution progressively 

approaches the skewed log-normal (rather than the symmetric normal) distribution. Even 

if the stock price were to decline in  each  subinterval, it can never drop below zero. But 

there is no corresponding upper bound on its potential performance. This  asymmetry 

gives rise to the skewness of the distribution. 

 Eventually, as we divide the option maturity into an ever-greater number of subintervals, 

each node of the event tree would correspond to an infinitesimally small time interval. The 

possible stock price movement within that time interval would be correspondingly small. 

As those many intervals passed, the end-of-period stock price would more and more closely 

  

5

 We adjust the probabilities of up versus down movements using the formula in footnote 4 to make the distribu-

tions in  Figure 21.5  comparable. In each panel,  p  is chosen so that the stock’s expected annualized, continuously 

compounded rate of return is 10%. 

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736

resemble a lognormal distribution.  

6

   Thus the apparent oversimplification of the two-state 

model can be overcome by progressively subdividing any period into many subperiods.

   

 At any node, one still could set up a portfolio that would be perfectly hedged over the 

next tiny time interval. Then, at the end of that interval, on reaching the next node, a new 

hedge ratio could be computed and the portfolio composition could be revised to remain 

  

6

 Actually, more complex considerations enter here. The limit of this process is lognormal only if we assume 

also that stock prices move continuously, by which we mean that over small time intervals only small price 

movements can occur. This rules out rare events such as sudden, extreme price moves in response to dramatic 

information (like a takeover attempt). For a treatment of this type of “jump process,” see John C. Cox and Stephen 

A. Ross, “The Valuation of Options for Alternative Stochastic Processes,”  Journal of Financial Economics   3 

(January–March 1976), pp. 145–66, or Robert C. Merton, “Option Pricing When Underlying Stock Returns Are 

Discontinuous,”  Journal of Financial Economics  3 (January–March 1976), pp. 125–44.