Estimating Stock Volatility
Submitted by
Chris Kinder
May 7, 2002
1
Estimating the Volatility
1. Introduction
Volatility is a great concern for investors or for anyone dealing with money.
Investors like to know how much volatility, or risk, that they are exposed to. This is why
we are interested in estimating the volatility for stocks. Knowing the volatility helps give
a feel for what the possible range of values that a stock will be in. Thus, when an
investor knows how much volatility he is exposed to, he can make informed decisions on
his investments.
This paper is organized as follows. Section 2 gives a description of what the
volatility is and what it is used for. Section 3 describes how to model stock prices and
shows the two methods that were used to estimate the volatility on simulated data.
Section 4 explains the results of the simulation and the test on real data. Section 5 is the
conclusion and deals with the good and bad points of the two methods.
2. Volatility
Volatility is a measure of the range of an asset price about its mean level over a
fixed amount of time (Abken and Nandi 21). It follows that volatility is linked to the
variance of an asset price. If a stock is labeled as volatile then the price will vary greatly
over time. Conversely, a less volatile stock will have a price that will deviate relatively
little over time. EBay is an example of a relatively volatile stock since its price can vary
just a few pennies in one day or it can move several dollars. The amount that it moves on
any given day varies greatly. Over the last year, the stock price for eBay had a range
from $40 to $73. On the other hand, a utility stock is an example of a stock that has small
volatility. For example, Ohio Power Co. had a stock price that ranged from $23 to $26
over this last year. As we see, the more volatile that a stock is, the harder it is to isolate
where it could be on a future date. Since volatility is associated with risk, the more
volatile that a stock is, the more risky it is. Consequently, the more risky a stock is, the
harder it is to say with any certainty what the future price of the stock will be (Criss 9596). When people invest, they would like to have no risk. The least amount of risk that
is involved, the better the investment is. Since almost every investment has some risk,
investors have looked for ways to minimize risk. One way to reduce exposure to risk is
to make use of financial options.
A financial option is basically the right to buy or sell something at a certain price
within a certain time. For example, farmer that buys feed for his cattle is exposed to a
risk of rising feed prices. The farmer can buy an option on feed commodities. If the
price of feed goes down, then the option does not produce any profit, but with the lower
feed prices, the farmer can more easily afford feed for his cattle. On the other hand, if the
price of feed moves higher, then the farmer makes money on his option and can now use
the profits from that option to help pay for the feed for his cattle. So by buying the
option, the farmer has reduced the amount of risk he is susceptible to by movements in
the price of feed.
What is Volatility used for?
In basic financial engineering there are five main variables in option pricing that
are important: the current asset price, the strike price, time to maturity, the interest rate,
and volatility. The first four variables are relatively easy to estimate. The strike price and
time to maturity are known constants, and the interest rate, for small time periods, can be
held constant. Current asset price is the asset price at the start of the option. The
2
Estimating the Volatility
volatility of a stock is the only variable that is not readily apparent (Abken and Nandi
22). In this paper we will present two methods for estimating the volatility of a stock
price.
These two methods will incorporate a mathematical model that is used to model
stock prices. The model of stock prices that we will use is a probabilistic model of stock
prices. The uncertainty in the world markets, be it due to economic recessions, war, or a
myriad of other factors, necessitates the use of a probabilistic model. In this context,
probabilistic means that precise statements cannot be made about what the future price of
the stock will be, only that an assumed distribution of the future prices can be derived
from historical data, current market conditions and other pertinent data (Criss 94).
Therefore, we are not trying to predict the value of the stock prices, because that is
virtually impossible. Instead we are attempting to provide a good estimate as to the price
range over short periods of time.
3. Computing the Volatility
The estimation of volatility comes from a mathematical model of stock prices.
The mathematical model we will use is based on four assumptions about stock prices and
their movements. The first assumption that we will be using is the volatility is constant.
The next assumption is that stock prices cannot be negative; once a stock price reaches $0
it cannot go any lower. The third assumption is that the price of a stock is a normal
random variable. The fourth assumption is the present price has a Markov property, i.e.,
the present price is independent from all past prices.
As stated before, we will assume that stock prices are random. The graph in
Figure 1
Fr
Figure 1 is a graph of the stock prices for Texas Instruments over the previous year as of April 12, 2002.
Figure 1 is of the stock prices for Texas Instruments over 1 year (data is downloaded
from Yahoo! Finance). Note that the jumps in the stock prices in Figure 1 appear to be
random in nature. The random behavior in the jumps of a stock price movement brings
to mind the concept of Brownian motion.
3
Estimating the Volatility
Brownian Motion
Brownian motion is named after the English botanist Robert Brown. In 1827,
Brown described the unusual motion exhibited by a small particle immersed in a liquid or
gas. In 1905, Albert Einstein showed mathematically that Brownian motion could be
explained by assuming the immersed particle was continually being subjected to
bombardment by the smaller molecules that surround it. In a series of papers originating
in 1918, the American applied mathematician Norbert Wiener gave a mathematically
concise definition of Brownian motion. Wiener also derived some of the mathematical
properties that were being used in Brownian motion (Ross 35). The Wiener process is
named after Norbert Wiener for his work in this area. More will be mentioned on the
Wiener process later.
Robert Brown’s observations were derived from analyzing the erratic motion of a
large particle suspended in a fluid. Initially the large particle has no movement, while the
movement of the small particles compromising the fluid is random. On the random paths
that the small particles are traveling, one of the small particles is bound to crash into the
large particle. Because of this collision, the large particle is moved. The direction and
size of this movement is independent and random from every other collision. The
resulting erratic movement has become known as Brownian motion. To relate stock price
movements to Brownian motion, imagine the large particle as the stock price and the
smaller particles as the trades that move the stock price. As we see each of the small
particles move the large particle, i.e., the trades move the stock price. Each trade moves
the price up or down and each trade is independent from the other trades. Thus stock
prices do display behavior similar to Brownian motion (Criss 96-97).
Brownian motion is an effective model for describing the motion of particles in a
fluid. However, Brownian motion has a few fundamental problems when it is applied to
stock prices. The first problem is that if a stock price were to be governed by Brownian
motion, then it can theoretically become negative (Ross 36). As discussed earlier, stock
prices cannot become negative. Another problem is that stock prices often change in
proportion to their size. Brownian particles do not have this property (Criss 97).
Both these fundamental problems can be addressed by using a geometric
Brownian motion model for stock prices instead of a plain Brownian motion model. In
the geometric Brownian motion model, we are now looking at the Brownian motion of
the return values, not the stock prices. A return is defined to be the change in the price
divided by the original value. The return values are a proportion and thus can be
negative. For example, if you purchased a stock for $100 yesterday and today the stock
5
 .05 .
closes for $95, then the return is computed to be
100
Now we are ready to simulate stock returns using geometric Brownian motion.
We will present two methods: the discrete method, and the continuous method. Each
method presents a slightly different way for computing the return value of the stock. To
simulate return values for testing the methods, we will use the stochastic differential
equation that corresponds to geometric Brownian motion,
dS
   dt    dX ,
(1)
S
4
Estimating the Volatility
where S (t ) is the stock price at time t,  is a measure of the average rate of growth of
the asset price, dt is the change in time,  is the volatility, and dX is known as a
Wiener process because it is a random normal variable with a mean of zero and a
standard deviation of dt (Wilmott, Howison and Dewynne 20-21). The   dt term in
equation (1) is called the drift term. The drift of an asset price is akin to the interest
earned in a savings account at a bank. The return on a fixed savings account is
predictable and can be computed with a high level of certainty. Thus, the drift term
represents the stock price would in the absence of volatility. The   dX term in equation
(1) represents the random change, or jump, in the asset price. The parameter  is the
mathematical representation of volatility, which we will estimate using two different
techniques, the first of which is the discrete method.
Discrete Method
The reason that this process is the discrete method is because of the way that the
return value is computed. The equation for the return value is
Ri  ( S i  S i 1 ) / S i 1 ,
(2)
where S i is today’s closing price and S i 1 is yesterday’s closing price. This is the way to
compute the return value over discrete time periods. Compounding annually is
compounding the interest 365 times a year, a set number of times, which is not
continuous.
For the numerical simulation the initial asset price was set equal to 100. In terms
of the model S 0 = 100, and S i is the closing price for day i. We will assume that we can
obtain a closing stock price for 365 consecutive days. To generate each of the next 365
asset prices, we use the equation
S i 1  S i  S i    dt  S i    dX ,
(3)
where, S i 1 is the new asset price that is being generated from the previous closing stock
price, S i .  ,  , and dX are the same as in equation (1). Equation (3) also has dt as
the discrete change in time. In this simulation, there are 365 time increments
corresponding to the 365 days that the closing price is obtained from. The discrete
change in time is computed by taking 1 year and dividing by the number of time
increments. The constant values in the simulation are:  = 0.10, dt = 1/365, and
 = 0.20. It follows that  can be expressed as a percentage, i.e. 20%. The model
represented in equation (3) will be used to generate the subsequent stock prices. Recall
that dX is a random variable with mean equal to zero and variance equal to dt . The
Random Number Generator feature in Excel was used to generate the random dX values
for each closing price. Notice that the volatility constant is a scaling factor on the
variable jumps in the return values. Each closing price has an independent random
number generated for it. This process generates 366 asset prices, S 0 to S 365 . Figure 2 is
a graph of all of the closing prices from our simulation.
5
Estimating the Volatility
Figure 2
120
closing price
115
110
105
100
=
36
0
=
32
0
=
28
0
=
24
0
=
20
0
=
16
0
=
12
0
=
80
=
40
0
=
95
day
Figure 2 is a graph of the 366 generated stock prices that were generated to test the two methods.
Examination of the graph in Figure 2 shows a similar random pattern in comparable
graphs of stock prices from actual stock exchanges (Criss 93-94).
The discrete return values, R i , are computed using the formula in equation (2).
This calculation generates return values R1 to R365 . Notice that there are 366 stock prices
and 365 return values, which is a result of how the return values are calculated. Let
1 n
m   Ri , which implies that m is the average of the return values. Using this value
n i 1
for m and the variance estimate results in a formula for the volatility
n
2
1
(4)
 
( Ri  m) 2 .

(n  1)  dt i 1
2
It follows that the estimation of the volatility constant is:    (Wilmott, Howison
and Dewynne 25). The estimation of the volatility for the simulation in Figure 2 is
  0.1941641.
Continuous Method
The continuous method corresponds to a continuous rate of return, which can be
computed via
Ri  ln( S i / S i 1 ) .
(5)
In the study of finance, we see that continuously compounding money is a function of the
exponential. Thus, equation (5) corresponds to the continuously compounding rate of
return.
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Estimating the Volatility
The asset values, S 0 to S 365 are generated in the same manner that was used in the
discrete case using equation (3).
In addition, the continuous method applied to the simulation results in 365 return
values, i.e., R1 to R365 . The next step in estimating the volatility is to compute the
1 n
 Ri , where n is once again the number of returns,
n i 1
365. The standard deviation is used to estimate volatility, (Criss 104)
average value of the returns, R 
1 n
(6)
( Ri  R ) 2 .

dt n  1 i 1
The estimation of the volatility for the simulation in Figure 2 is   0.1940812.
In both methods, the volatility was computed via dividing by (n-1) instead of just
n. The reason for this comes from statistical analysis. If we divided by n instead of (n-1)
we would have a biased estimate. The estimate would be biased to the time of day that
the prices are taken from. If the estimate is biased to a certain part of the day, say the
closing price as in our simulation, then the estimate is not a good representative of the
true volatility. To make our estimate better, we remove the bias, which simply results in
dividing by (n-1) instead of n (Criss 105).
4. Numerical Results
To further compare the methods, they need to be tested on simulated stock prices
with a known volatility. Both of the methods also need to be tested on the exact same
data sets. To generate the data sets, equation (3) was used in the manner described earlier
in Microsoft Excel. The volatility was held constant for each trial and its value was 0.20
or 20%. S 0 initially was set to 100, and S 1 to S 365 were generated via Microsoft Excel.
Equations (2) and (5) were used to compute the respective return values for each of the
365 days. The return values were then used in the volatility estimates given by equations
(4) and (6). The results are summarized in Table 1. There were 10 independent trials.
Table 1
Continuous
Discrete
Trial 1
20.5682%
20.5386%
Trial 2
19.4223%
19.4143%
Trial 3
19.9802%
19.9829%
Trial 4
19.3858%
19.3697%
Trial 5
19.9448%
19.9030%
Trial 6
18.8463%
18.8398%
Trial 7
20.8533%
20.8196%
Trial 8
21.0336%
21.0098%
Trial 9
19.3493%
19.3322%
Trial 10
19.9602%
19.9158%

1
Table 1 contains the estimations of the volatility for 10 trial runs. The Continuous column displays the
estimates from the continuous method, while the Discrete column contains estimates from the discrete
method.
7
Estimating the Volatility
As we see from the results of the trials in Table 1, for each trial, both the discrete and the
continuous method both resulted in estimates that are relatively close. For both methods
and each trial, the calculation does not vary from the true volatility by more than 1.5%.
So both methods seem to do a good job at estimating the volatility of the simulated stock
data.
Testing the Methods on Real Data
The methods were also tested on some real market data. The real market data is
the data of the stock prices of Texas Instruments from the graph in Figure 1. First the
data was downloaded in a spreadsheet off of Yahoo! Finance. After the data was
imported into Microsoft Excel, the calculations were preformed on the data. The
continuous method computed a volatility estimate of 60.072104%, while the discrete
method resulted in a volatility estimate of 60.06792%. Again, both methods come up
with essentially the same result. So it is our estimate that the volatility for Texas
Instruments is just over 60% or   0.60.
5. Conclusion
Both of the methods make use of straightforward formulas for volatility that are
easy to use. Our limited experience with the formulas applied to our simulations suggests
that the estimates produce very similar results. To further test the validity of the similar
results, more simulations need to be generated and then more statistical analysis, like
confidence intervals, can be done to the results.
One key weakness to this approach of estimating volatility is the assumption of
constant volatility. Volatility might be very close to constant over very short periods, but
research in financial engineering suggests that volatility varies with time. For example,
the period of time before the Federal Reserve decides to change the interest rate may
cause the volatility of the stock price to fluctuate (Abken and Nandi 32). So although
assuming that the volatility remains constant eased our calculations, it is not very
practical in other applications.
Another problem with the methods is one of the assumptions from the geometric
Brownian motion model in equation (1). Recall that the random component to returns are
normally distributed. But studies of stock market returns show that returns are not
normally distributed as the model predicts. In fact, large movements in stock prices
occur frequently and they occur more often than the model predicts. Stock returns show
what statisticians refer to as leptokurtosis. Leptokurtosis is when the returns are peaked
near the mean and have what people refer to as “fat tails” (Criss 115).
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Estimating the Volatility
References
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Dec. 1996: 21-35.
Chriss, Neil A. Black-Scholes and Beyond, Option Pricing Models. New York:
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Haug, Espen Gaarder. The Complete Guide to Option Pricing Formulas. New York:
McGraw-Hill, 1998.
Person, Ron. Using Excel 4 for Windows Special Edition. Carmel: Que Corporation,
1992.
Ross, Sheldon M. An Introduction to Mathematical Finance: Options and Other Topics.
New York: Cambridge University Press, 1999.
Stampfli, Joseph, and Victor Goodman. The Mathematics of Finance: Modeling and
Hedging. Pacific Grove: BROOKS/COLE, 2001.
Wilmott, Paul, Sam Howison, and Jeff Dewynne. The Mathematics of Financial
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Wolfgang, Paul and Jorg Baschnagel. Stochastic Processed From Physics to Finance.
Berlin: Springer, 1999.
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