LOGNORMAL MODEL FOR STOCK PRICES

MICHAEL J. SHARPE
MATHEMATICS DEPARTMENT, UCSD

1. I
What follows is a simple but important model that will be the basis for a later study of stock prices as a
geometric Brownian motion. Let S0 denote the price of some stock at time t = 0. We then follow the stock
price at regular time intervals t = 1, t = 2, . . . , t = n. Let St denote the stock price at time t . For example, we
might start time running at the close of trading Monday, March 29, 2004, and let the unit of time be a trading
day, so that t = 1 corresponds to the closing price Tuesday, March 30, and t = 5 corresponds to the price at
the closing price Monday, April 5. The model we shall use for the (random) evolution of the the price process
S0 , S1 , . . . , Sn is that for 1 ≤ k ≤ n, Sk = Sk −1 Xk , where the Xk are strictly positive and IID—i.e., independent,
identically distributed. We shall return to this model after the next section, where we set down some reminders
about normal and related distributions.

2. P   N  L D

First of all, a random variable Z is called standard normal (or N (0, 1 ), for short), if its density function fZ (z )
−z 2 /2 Rz
is given by the standard normal density function φ(z ) :== e√2π . The function 8(z ) := −∞ φ(u ) d u denotes the
distribution function of a standard normal variable, so an equivalent condition is that the distribution function
(also called the cdf ) of Z satisfies FZ (z ) = P (Z ≤ z ) = 8(z ). You should recall that
Z ∞
(2.1) φ(z ) d z = 1 i.e., φ is a probability density
−∞
Z ∞
(2.2) z φ(z ) d z = 0 with mean 0
−∞
Z ∞
(2.3) z 2 φ(z ) d z = 1 and second moment 1.
−∞

In particular, if Z is N (0, 1 ), then the mean of Z , E (Z ) = 0 and the second moment of Z , E (Z 2 ) = 1.

In particular, the variance V (Z ) = E (Z 2 ) − (E (Z ))2 = 1. Recall that standard deviation is the square root
of variance, so Z has standard deviation 1. More generally, a random variable V has a normal distribution
with mean µ and standard deviation σ > 0 provided Z := (V − µ)/σ is standard normal. We write for short
V ∼ N (µ, σ 2 ). It’s easy to check that in this case, E (V ) = µ and Var(V ) = σ 2 . There are three essential facts
you should remember when working with normal variates.
Theorem 2.4. Let V1 , . . . , Vk be independent, with each V j ∼ N (µ j , σ 2j ). Then
V1 + · · · + Vk ∼ N (µ1 + · · · + µk , σ12 + · · · + σk2 ).
Theorem 2.5. (Central Limit Theorem:) If a random variable V may be expressed a sum of independent variables,
each of small variance, then the distribution of V is approximately normal.
This statement of the CLT is very loose, but a mathematically correct version involves more than you are
assumed to know for this course. The final point to remember is a few special cases, assuming V ∼ N (µ, σ 2 ).
(2.6) P (|V − µ| ≤ σ) ≈ 0.68; P (|V − µ| ≤ 2σ) ≈ 0.95; P (|V − µ| ≤ 3σ) ≈ 399/400.
1
2 MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD

We’ll say that a random variable X = exp(σ Z + µ), where Z ∼ N (0, 1 ), is lognormal(µ,σ 2 ). Note that the
parameters µ and σ are the mean and standard deviation respectively of log X . Of course, σ Z + µ ∼ N (µ, σ 2 ),
by definition. The parameter µ affects the scale by the factor exp(µ), and we’ll see below that the parameter σ
affects the shape of the density in an essential way.

Proposition 2.7. Let X be lognormal(µ,σ 2 ). Then the distribution function FX and the density function fX of X
are given by
(2.8)
 log x − µ   log x − µ 
FX (x ) = P (X ≤ x ) = P (log X ≤ log x ) = P (σ Z + µ ≤ log x ) = P Z ≤ =8 , x > 0.
σ σ

 
log x −µ
d φ σ
(2.9) fX (x ) = FX ( x ) = , x > 0.
dx σx

These permit us to work out a formulas for the moments of X . First of all, for any positive integer k,
 
log x −µ
Z ∞ Z ∞ xk φ σ
E (X k ) = x k fX (x ) d x = dx
0 0 σx

hence after making the substitution x = exp(σ z + mu ), so that d x = σ exp(σ z + µ), we find
Z ∞ Z ∞
1 2
− z2 +k σ z +k µ 1 1 2 + k σ 2 +k µ k 2 σ2
(2.10) k
E (X ) = √ e dz = √ e − 2 (z −k σ) 2 dz = e 2 +k µ .
2π −∞ 2π −∞

R∞
(We completed the square in the exponent, then used the fact that by a trivial substitution, −∞
φ(z − a ) d z = 1.)
In particular, setting k = 1 and k = 2 give

σ2 2 +2µ 2 +2µ 2
E ( X 2 ) = e 2σ V (X ) = E (X 2 ) − (E (X ))2 = e σ eσ − 1 .


(2.11) E (X ) = e 2 +µ ; ;

The median of X (which continues to be assumed lognormal(µ,σ 2 )) is that x such that FX (x ) = 1/2. By (2.8),
log x −µ log x −µ
this is the same as requiring 8( σ ) = 1/2, hence that σ = 0, and so log x = µ, or x = e µ . That is,

(2.12) X has median e µ .

The two theorems above for normal variates have obvious counterparts for lognormal variates. We’ll state them
somewhat informally as:

Theorem 2.13. A product of independent lognormal variates is also lognormal with respective parameters µ = µj
P

and σ 2 = σ 2j .
P

Theorem 2.14. A random variable which is a product of a large number of independent factors, each close to 1, is
approximately lognormal.

Here is a sampling of lognormal densities with µ = 0 and σ varying over {.25, .5, .75, 1.00, 1.25, 1.50}.
 LOGNORMAL MODEL FOR STOCK PRICES 3

Some lognormal densities

1.5

1.25

0.75

0.5

0.25

1 2 3 4

The smaller σ values correspond to the rightmost peaks, and one sees that for smaller σ , the density is close
to the normal shape. If you think about modeling men’s heights, the first thing one thinks about is modeling
with a normal distribution. One might also consider modeling with a lognormal, and if we take the unit of
measurement to be 70 inches (the average height of men), then the standard deviation will be quite small, in
those units, and we’ll find little difference between those particular normal and lognormal densities.

3. L P M

We continue now with the model described in the introduction: Sk = Sk −1 Xk . The first natural question
here is which specific distributions should be allowed for the Xk . Let’s suppose we follow stock prices not just at
the close of trading, but at all possible t ≥ 0, where the unit of t is trading days, so that, for example, t = 1.3
corresponds to .3 of the way through the trading hours of Wednesday, March 31. Note that SS01 = SS.15 SS.05 , and
S1 S .5
under the time homogeneity postulated above, one should suppose that S .5 and S0 are IID. Continuing in this
way, we see that for any positive integer m, setting h = 1/ m,
S1 Smh S (m−1)h Sh
= ...
S0 S (m−1)h S (m−2)h S0
Skh
where the factors S (k −1 )h are IID. Consequently, taking logarithms, we find
S  m  S 
1 kh
X
log(X1 ) = log = log
S0 S (k −1)h
k −=1
so that for arbitrarily large m, log X1 may be represented as the sum of m IID random variables. In view of
the Central Limit Theorem, under mild additional conditions—for example, if log X1 has finite variance, then
log X1 must have a normal distribution. Therefore, it is reasonable to hypothesize that the Xk are lognormal,
and we may write Xk = exp(σ Zk + µ), where the Zk are IID standard normal.
The first issue is the estimation of the parameters µ and σ from data. The thing you need to recall is that if you
have a sample of n IID normal variates Y1 , . . . , Yn with unknown meanPµ and unknown standard deviation σ ,
(Y −Ȳ )2
then the sample mean Ȳ := Y1 +···+n
Yn
is an unbiased estimator of µ and nk−1 is an unbiased estimator of σ 2 .
If we denote by Y¯2 the mean value of the Yk2 , it is elementary algebra to verify that
(Yk − Ȳ )2
P
n
Y¯2 − Ȳ 2 .


(3.1) =
n−1 n−1
4 MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD

When n is large, the factor n /(n − 1 ) is close to 1, and may be ignored. Be aware that when Excel computes the
variance (VAR) of a list of numbers y1 through yn , it uses this formula.
So, if we have a sample of stock prices S0 through Sn , we compute the n ratios X1 := SS10 through Xn := SSn−n 1
and then set Yk := log Xk . (In the financial literature, Rk := SkS−kS−k1−1 = Xk − 1 is called the return for the k th
day. In practice, Xk is quite close to 1 most of the time, and so Yk is mostly close to 0. For this reason, since
log(1 + z ) is close to z when z is small, Yk is mostly very close to the return Rk .) Apply the estimators described
above to estimate µ by Ȳ and σ 2 by formula (3.1). This kind of calculation can be conveniently handled by an
Excel spreadsheet, or a computer algebra system such as MathematicaT M . Stock price data is available online,
for example at http://biz.yahoo.com. Spreadsheet files of stock price histories may be downloaded from
that site in CSV (comma separated value) format, which may be imported from Excel or MathematicaT M .
The parameters µ and σ arising from this stock price model are called the drift and volatility respectively. The
idea is that stocks price movement is governed by a deterministic exponential growth rate µ, though subject to
random fluctation whose magnitude is governed by σ . The following picture of Qualcomm stock (QCOM) over
roughly the last nine months is shown in the following picture, along with the deterministic growth rate S0e k µ .
You might at this point check out the last page of this handout, where I’ve graphed the result of 10 simulations
starting at the same initial price, but using independent lognormal multipliers with the same drift and volatility
as this data.

65

60

55

50

45

40

50 100 150 200

The graph below shows a plot of the values log X j versus time j, along with a horizontal red line at their mean
µ and horizontal green lines at levels µ ± 2σ . Note that of the 199 points in the plot, only 7 are outside these
levels. This is not far from the roughly 5% of outliers you would expect, based on the normal frequencies.
 LOGNORMAL MODEL FOR STOCK PRICES 5

0.1

0.08

0.06

0.04

0.02

50 100 150 200

-0.02

-0.04

The empirical cdf of the log Xk is pictured next (in red) compared with a normal cdf having the estimated µ
and σ .

0.8

0.6

0.4

0.2

-0.05 -0.025 0.025 0.05 0.075 0.1

The following is only for those who already know about such matters. To test whether the log Xk are normal,
one computes the maximal difference D between the empirical cdf and the normal cdf with the estimated µ
√
and σ using the n = 199 data. Then nD has a known approximate distribution under the null hypothesis, and
approximately,
√ √ √
(3.2) P ( nD > 1.22 ) = .1, P ( nD > 1.36 ) = .05, P ( nD > 1.63 ) = .01.
√
In our case, the observed value of nD is about 1.06, which is not sufficient to reject the null hypothesis at any
reasonable level.
We emphasize that the justifications given here are quite crude. In particular, the hypothesized independence
of the day to day returns is difficult to reconcile with the well known herd mentality of stock investors. There is
an extensive literature on models for stock prices that are much more sophisticated, though of course less easy
6 MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD

to work with. From the point of view of the mathematical modeler, a mathematically simple model that yields
approximately correct insights is certainly worthwhile, though one should always keep its limitations in mind.
Finally, here is the simulated stock price based on the same initial price as the earlier graph of QCOM, but
using independent lognormal multipliers with the same drift and volatility as the QCOM data.

90 90
80 80
70 70
60 60
50 50
50 100 150 200 50 100 150 200

90 90
80 80
70 70
60 60
50 50
50 100 150 200 50 100 150 200

90 90
80 80
70 70
60 60
50 50
50 100 150 200 50 100 150 200

90 90
80 80
70 70
60 60
50 50
50 100 150 200 50 100 150 200

90 90
80 80
70 70
60 60
50 50
50 100 150 200 50 100 150 200

4. A       

This time, we take the postulates from the beginning of the last section, and enhance them a bit so that the
time parameter t may take any positive real value. It is in this application supposed to be a clock that ticks only
 LOGNORMAL MODEL FOR STOCK PRICES 7

during stock trading hours. We suppose St represents the dollar value of a particular stock at time t . Here are
the spruced up hypotheses.
Definition 4.1. We shall say that the stock price process St (> 0 ) follows a lognormal model provided the following
conditions hold.
(4.2) For all s , t ≥ 0, the random variable St +s /St has a distribution depending only on s, not on t . (Loosely
speaking, a stock should have the same chance of going up 10% in the next hour, no matter what time we
start at.)
(4.3) For any n, if we consider the process St at the times 0 < t1 < · · · < tn , the ratios St1 /S0 , St2 /St1 through
Stn /Stn−1 are mutually independent. (Loosely speaking, a prediction of the stock price percentage increase from
time tn−1 to time tn should not be influenced by knowledge of the actual percentage increases during any
preceding periods.)
Let’s examine the consequences of this definition. First of all, arguing just as in the preceding section, the
distribution of St /S 0 is necessarily lognormal with some parameters (µt , σt2 ). Let us define µ := µ1 and σ 2 := σ12 ,
so that S1 /S0 is lognormal (µ, σ 2 ). Now, for any integer m > 0
S 1 S 1/ m S 2/ m Sm / m
= ...
S0 S 0 S 1/ m S (m−1)/ m
and by the first hypothesis in the definition, all the random variables on the right side have the same distribu-
tion, namely lognormal (µ1/ m , σ12/ m . Moreover, by the second condition in the definition, they are mutually
independent. As a product of independent lognormals is also lognormal, and the parameters add, we have
µ = m µ1/ m ; σ 2 = m σ12/ m .
Consequently,
1 1 2
µ1/ m = µ; σ12/ m = σ .
m m
Similarly, we may write
S k / m S 1/ m S 2/ m Sk /m
= ...
S0 S 0 S 1/ m S (k −1)/ m
and deduce by the same reasoning that
k k 2
µk / m = µ; σk2/ m = σ .
m m
Writing this another way, we have proved that
µt = t µ; σt2 = t σ 2 for t of the form k / m .
As the integers k , m > 0 are completely arbitrary, it follows (with some mild but unspecified assumption) that
in fact
(4.4) µt = t µ; σt2 = t σ 2 for t > 0.
As a consequence, we have shown:
Proposition 4.5. If St satisfies Definition 4.1, then (a) S1 /S0 lognormal with some parameters (µ, σ 2 ); (b) St +s /St
is then lognormal (s µ, s σ 2 ).
We further analyze this process by studying its natural logarithm Vt := log St /S0 . The new process Vt clearly
has the following properties:
(4.6) V0 = log(S0 /S 0 ) = 1;
(4.7) Vt ∼ N (t µ, t σ 2 );
(4.8) Vt +s − Vt = log St +s /St ∼ N (s µ, s σ 2 ), and so has the same distribution as Vs ;


(4.9) If 0 < t1 < · · · < tn , then Vt1 , Vt2 − Vt1 through Vtn − Vtn−1 are independent.
8 MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD

We make one further algebraic simplification, setting Bt := Vt −σ t µ , so that the process Bt has the following


properties:
(4.10) B0 = 1;
(4.11) Bt +s − Bt ∼ N (0, s ), and so has the same distribution as Bs ;
(4.12) If 0 < t1 < · · · < tn , then Bt1 , Bt2 − Bt1 through Btn − Btn−1 are independent.
A process Bt , defined for t ≥ 0, satisfing these conditions is called a standard Brownian motion. It may be
proved (quite tricky proof, though) that the process may also be assumed to have continuous sample paths. That
is, we may assume that for every ω, t → Bt (ω) is continuous.
There is a substantial literature available based on this mathematical model of Brownian motion, and the
methods developed to study it are fundamental to the study of mathematical finance at a more advanced level.
Now let’s go back and write
Vt = t µ + σBt .
That is, the process Vt has a uniform drift component t µ and a scaled Brownian component σBt . Finally, we
express St in terms of Bt by
St


= exp Vt = exp t µ + σBt .
S0
It is a consequence of this representation that for any t , s ≥ 0, we have
St +s

(4.13) = exp s µ + σ(Bt +s − Bt ) .
St

5. A B–S 
We now have all the tools available to perform a simple calculation that will prove to solve one of the option
pricing problems to be studied later. We assume that the stock price process St satisfies (4.13), with given drift
µ and volatility σ . Fix x > 0, T > 0 and let s = S 0 denote the price at time 0. We shall prove that
2 log(s /x ) + T µ p

log(s /x ) + T µ


(5.1) E (ST − x )+ = se T (µ+σ /2) 8 √ + Tσ −x 8 √ .
Tσ Tσ
We have E (ST − x )+ = E h (ST /S 0 ), where h ( y ) := (sy√− x )+ , which vanishes for y < x /s and takes the value
(sy − x ) for y ≥ x /s. In view of (4.13), writing BT = T Z where Z ∼ N (0, 1 ), we have ST /S 0 = exp(T µ +
√
σ T Z ), so that

p
E (ST − x )+ = E h (ST /S0 ) = E (exp(τ Z + ν) − x )+ ; τ : = T σ; ν : = T µ + log s .
log x −ν
From this we may calculate, since e τ Z +ν − x ≥ 0 if and only if Z ≥ τ ,
2

e −z /2 log x − ν
Z ∞
E (ST − x ) =+
e τ z +ν − x √ d z ; w := .
w 2π τ
The latter integral expands to
Z ∞ −z 2 /2 Z ∞ −z 2 /2
τ z +ν e e
e √ dz − x √ d z.
w 2π w 2π
The second term reduces at once to −x (1 − 8(w)) = −x 8(−w), and the first terms permits a completion of the
square in the exponent to give
Z ∞ −z 2 /2 Z ∞ −(z −τ)2 /2
τ z +ν e ν+τ 2 /2 e
e √ dz = e √ d z.
w 2π w 2π
Changing the variable u := z − τ reduces this to
Z ∞ −u2 /2
ν+τ 2 /2 e 2 2
d u = e ν+τ /2 1 − 8(w − τ) = e ν+τ /2 8(τ − w).


e √
w−τ 2π
 LOGNORMAL MODEL FOR STOCK PRICES 9

Taking these components together yields finally

2 /2
(5.2) E (ST − x )+ = e ν+τ 8(τ − w) − x 8(−w),
log(s /x )+T µ
and after substituting back for τ , ν, and noting that w = − σ√T , this amounts precisely what was claimed
in (5.1).
We shall show later, based on a “no arbitrage” argument, that if the bank interest rate is r, then
σ2
(5.3) µ+ = r.
2
If we substitute µ = r − σ 2 /2 into (5.1), we find
log(s /x ) + T (r − σ 2 /2 ) p
log(s /x ) + T (r − σ 2 /2 )

(5.4) E (ST − x )+ = se Tr 8 √ + T σ − x8 √ .
Tσ Tσ
Even without the argument based on “no arbitrage”, a simple argument may be given for (5.3). First of all,
let’s switch to stock prices based on present value, which is to say that we take the interest rate r to be the rate of
inflation, fix a present time t = t0 , and let Ut := e −r (t −t0 ) St denote the price of stock “measured in dollars as of
time t0 .” If, as in the preceding discussion, St /St0 is a lognormal process with drift µ and volatility σ , say (with
B denoting a standard Brownian motion)
St
= e σBt −t0 +µ(t −t0 ) ,
S t0
then
Ut
= e σBt −t0 +(µ−r )(t −t0 ) ,
Ut 0
so that in fact, Ut /Ut0 is again a lognormal with the same volatility, but with drift µ − r. However, in the
financial world in which Ut represents a stock price, the interest rate is effectively 0. By our formula for the
lognormal mean, we have
Ut 2
E = e (t −t0 )(µ−r +σ /2 ) .
Ut 0
Fix t > t0 . If µ − r + σ 2 /2 > 0, we would have E UUtt > 0, so a stock investment would be guaranteed to lead
0
greater wealth over the long run (by the law of large numbers). This is not consistent with the reality of a market.
Similarly, if µ − r + σ 2 /2 < 0, a long term loss would be guaranteed, and no rational person would invest. For
these reasons, one should assume that (5.3) is valid. (A “no arbitrage” argument depends on knowing more
about trading possibilities, which we have not yet studied.)