Download Using The Lognormal Random Variable to Model Stock Prices

Using The Lognormal Random
Variable to Model Stock Prices
Using The Lognormal Random Variable to Model Stock Prices
In reality, stock prices can assume any non-negative value. In a binomial tree, of
course, only a finite number of stock prices are possible. The Lognormal or Geometric
Brownian Motion random variable is often used to model the evolution of stock prices.
The Lognormal model for asset value (or stock price) assumes that in a small time t
the stock price changes by an amount that is normally distributed with
S tan dard Deviation  S t
Mean = St
Here S = current stock price.
 may be thought of as the instantaneous rate of return on the stock. By the way, this
model leads to really "jumpy" changes in stock prices (like real life). This is because
during a small period of time the standard deviation of the stock's movement will greatly
exceed the mean of a stock's movement. This follows because for small t, t will be
much larger than t. This is consistent with reality.
For example, per day Microsoft has in the recent past had mean growth per day = .40/252
= .0016 and  per day = 1 / 252 * (.47)  .03. (note .03>.016!). σ is roughly speaking, a
measure of the percentage volatility in the annual return of a stock. As we will see later,
Microsoft has a volatility of 47%, AOL 65%, and Amazon.COM in 1999 had a volatility
of 120%! The volatility of a stock or project is crucial to a real options calculation.
We use the following equation to simulate future stock prices in EXCEL.
St  S0e(  .5
2
) t  t NORMSINV ( RAND())
. (1)
Here S0 = Today's stock price, μ = instantaneous rate of return on stock, and σ = Annual
volatility of stock (standard deviation of stock price changes over a small unit of time on
an annualized basis). The =NORMSINV(RAND()) generates a sample from a standard
normal (mean 0 and sigma 1) random variable.
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Remarks
 Taking the logarithm of both sides of (1) shows us that we could simulate
Ln( St )using the fact that
Ln (St )  Ln(S 0 )  (  .5 2 )t   t NORMSINV ( RAND()) (2)
This shows than Ln (St) is normally distributed. This explains the name lognormal
distribution.

If our stock is growing instantaneously at rate , then why do (1) and (2) seem to
indicate that we are growing at a rate of  - .52? The reason for this is that over
time increased volatility reduces the mean growth rate of a stock. To see this
consider two stocks:
 Stock 1 always yields 10% a year.
 Stock 2 doubles during half of all years and loses 50% of its value during half of
all years.
On average Stock 1 grows by 10% per year and Stock 2 grows by 25% per year. Yet over
time Stock 2 will certainly end up lower than Stock 1 due to its large volatility or
variability.
Simulating the Lognormal Random Variable
In file lognormal.xls we simulate for 5 years quarterly Microsoft stock prices
under the assumption that  = .30 and  = .38 and that today’s stock price is $100. See
Figure 1.
Figure 1
Copyright All Rights Reserved, February 2000, Wayne L. Winston
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C
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2
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D
E
mu
sigma
Date
Today
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
3.25
3.5
3.75
4
4.25
4.5
4.75
5
F
G
0.3
0.38
Price
100
78.24848
87.4427
83.55709
124.0535
144.4007
177.1342
188.4546
233.7532
354.9943
440.8483
414.2939
590.0955
542.4397
711.7793
720.6463
437.7635
400.2455
398.2597
292.9255
356.3406
RN
0.05584
0.612166
0.294955
0.962476
0.691332
0.78101
0.510497
0.797858
0.971244
0.799619
0.265424
0.940843
0.228761
0.870807
0.407269
0.001732
0.220259
0.372245
0.027651
0.767818
Step 1: In F7:F26 we generate random numbers used in (1) to create each period’s stosk
price. We copy from F7 to F8:F26 the formula
=RAND().
Step 2: In E7:E26 we generate the Microsoft stock price for each of the next 20 quarters
using (1). We copy from E7 to E8:E26 the formula
=E6*EXP((mu-0.5*sigma^2)*0.25+sigma*SQRT(0.25)*NORMSINV(F7)).
Step 3: To generate, say, 10000 realizations of Microsoft’s stock price in 5 years we use
a data table. Enter the numbers 1-10000 in J10:J10009. In cell K9 record the simulated
stock price in 5 years with the formula
=E26.
Now select the table range J9:K10009 and select the DATA Table option. Select any
blank cell as your Column Input cell and you will see 1000 different realizations of
Microsoft’s stock price. You may have to hit F9 to get the table to recalculate. Also
checking TOOLS OPTIONS CALCULATIONS AUTOMATIC EXCEPT FOR TABLES
will ensure that data table will only recalculate when you hit F9.
Copyright All Rights Reserved, February 2000, Wayne L. Winston
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Step 4: After entering a .05 and .95 in cells L2 and L3 we can obtain the 5th percentile
and 95th percentile of Microsoft’s stock prices in 5 years.. Just copy from K2 to K2:K3
the formula
= PERCENTILE($K$10:$K$10009,L2).
As shown in Figure 2, there is (approximately) a 5% chance MSFT stock in 5 years will
be worth $71.07 or less and a 5% chance that the MSFT price in five years is $1252.92 or
more. Thus there is a lot of variability in where MSFT will end up. Don’t forget that we
actually do not know , so there is actually more volatility than our calculations indicate.
In cells L4 and L5 we compute the mean and standard deviation of the MSFT 5
year price with the formulas
= AVERAGE(K10:K10009) and
=STDEV(K10:K10009).
Thus we find our best estimate of the average MSFT price in 5 years to equal $459 and
our best estimate of the standard deviation is $459.
Copyright All Rights Reserved, February 2000, Wayne L. Winston
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Figure 2
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J
K
L
%ile
5th percentile
95th percentile
mean
sigma
1
2
3
4
5
6
7
8
9
80.55697
1305.684
450.7337
459.3412
0.05
0.95
491.4252
244.9976
474.1399
122.667
386.8273
428.0139
109.0213
44.58376
92.68346
593.1868
Copyright All Rights Reserved, February 2000, Wayne L. Winston