Stock
Price Volatility: a primer
By Dr. A. A. Kotzé from Financial Chaos Theory
http://www.quantonline
co.za
Abstract
price series or an economic indicator that changes a lot and
swings wildly is said to be “volatile”. This simple
and intuitive concept is the cause of many difficulties in
finance. Unlike many other market parameters which can be
directly observed, volatility has to be estimated. This is
difficult, if not impossible, because we cannot say that volatility
is necessarily stochastic or that it conforms to any mathematical
model; all we know is that the evolution of volatility is
uncertain. An accurate estimate of volatility is, however,
crucial in many applications, including risk measurement and
management as well as option pricing and hedging.
1. Introduction
The publication of the preferencefree option pricing formula
by Fischer Black and Myron Scholes in 1973 was a giant step
forward in financial economics. Since then option pricing
theory has developed into a standard tool for designing, pricing
and hedging derivative securities of all types.
The Black & Scholes formula for pricing
vanilla European options in an ideal market needs six inputs:
the current stock price, the strike price, the time to expiry,
the riskfree interest rate, the dividends and the volatility.
Of these, the first three are known from the outset and the
last three must be estimated. Black & Scholes assumed
a perfect world in their analysis and took the last three
as constants. In the real world, however, the correct values
for these parameters are only known when the option expires.
This means that the future values of these quantities need
to be determined if an option is to be priced correctly.
The most important of the three uncertain
parameters, is the volatility. Changes in interest rates (especially
in a low interest rate environment) do not influence the price
of an option as much as changes in volatility. Most options
(and especially warrants) are short dated (expiries less than
12 months). The impact of small dividends on the value of
an option that is short dated is minimal. Dividend risk can
be minimised through good research whereby the future dividends
that will be paid during the next 12 to 18 months can be estimated
quite accurately.
The importance of the volatility parameter
was highlighted by Black & Scholes through their model.
Practitioners now needed to estimate only one parameter, the
volatility, and input it into a relative simple formula to
find the price of an option. Of the three uncertain parameters,
changing volatility has the biggest impact on the price of
an option.
Volatility measures variability, or dispersion
about a central tendency — it is simply a measure of
the degree of price movement in a stock, futures contract
or any other market. Volatility also has many subtleties that
make it challenging to analyze and implement. The following
question immediately comes to mind: can we estimate this seemingly
complex quantity called volatility?
In this short note we’ll explore 2 different ways to
estimate volatility. Both of these are quite simple to implement
in Microsoft Excel.
2. The Statistical
Nature of Volatility
Black & Scholes assumed that financial asset prices are
random variables that are lognormally distributed. Therefore,
returns to financial assets, the relative price changes are
usually measured by taking the differences between the logarithmic
prices. These differences (the socalled logrelatives) are
normally distributed. A normal distribution is indicated by
a bell shaped curve. This is shown in Fig. 1.
What does this all means in practise? Stock
prices are usually observed at fixed intervals of time (daily,
weekly or monthly) and we then have a time series of data.
The logrelative returns are mathematically defined by the
equation
(1) In SA most of the top 60 companies pay
relatively small dividends.
where S1
is the stock price at the end of the th interval and
is the natural logarithmic function. We also assume that there
are stock prices in our sample. This equation can easily be
implemented in Microsoft Excel. This is illustrated in Fig.
2 where we used a few MTN share prices.
Volatility is defined as the variation or
dispersion or deviation of an asset’s returns from their
mean. In Fig. 1 we show two normal curves. Both have the same
mean but the dotted line shows a greater dispersion than the
continuous line. These two curves also illustrate that volatility
indicates the range of a return’s movement. Large values
of volatility mean that returns fluctuate in a wide range
– large risk. The most common measure of dispersion
is the standard deviation of a random variable.
But, what does this all means? If we assume
the mean of the logrelative returns is zero, then, a 10% volatility
represents the following: in one year, returns will be within
[10%; +10%] with a probability of 68.3% (1 standard deviation
from the mean); within [20%; +20%] with a probability of
95.4% (2 standard deviations), and within [30%; +30%] with
a probability of 99.7% (3 standard deviations) — according
to a normal distribution.
3. The Variance
Rate of Return
In their paper in 1973, Black & Scholes mentioned the
parameter which they said was the “variance rate of
the return" on the stock prices. Black & Scholes
took this as a known parameter that is constant through the
life of the option. Did they really know what this parameter
was?
In a paper prior to their seminal one, Black
& Scholes gave more insight into the variance rate of
return. There they stated that they estimated the instantaneous
variance from the historical series of daily stock prices.
They thus defined volatility as the amount of variability
in the returns of the underlying asset. Black & Scholes
determined what is today known as the historical volatility
and used that as a proxy for the expected volatility in the
future. In that paper they tested several implications of
their model empirically by using a sample of 2 039 calls and
3 052 straddles traded on the New York stock exchange between
1966 and 1969.
In analyzing their results they noted that
the variance actually employed by the market is too narrow
and that the historical estimates of the variance include
an attenuation bias, i.e., the spread of the estimates is
greater than the spread of the true variance. This implies
that for securities with a relatively high variance, the market
prices underestimate the variance, while using historical
price series would overestimate the variance and the resulting
Black & Scholes model price would thus be too high; the
converse is true for relative low variance securities. Was
this the first observation of a volatility skew or smile?
In further tests Black & Scholes found that their model
performed very well when the true variance rate of the stock
was known.
4. Estimation
of Volatility
4.1.Trading or Nontrading Days
To estimate the volatility of a stock price empirically, the
stock price is usually observed at fixed intervals of time.
These intervals can be days, weeks or months . Before any
calculation can be done, however, a question one needs to
answer is whether the volatility of an exchangetraded instrument
is the same when the exchange is open as when it is closed.
Some people argue that information arrives
even when an exchange is closed and this should influence
the price. A lot of empirical studies have been done and researchers
found that volatility is far larger when the exchange is open
than when it is closed. The consequence of this is that if
daily data are used to measure volatility, the results suggest
that days when the exchange is closed should be ignored.
4.2 Historical
Volatility
The historical volatility is the volatility of a series of
stock prices where we look back over the historical price
path of the particular stock. We previously mentioned that
the most common measure of dispersion is the standard deviation.
The historical volatility estimate is thus given by
(2)One has to be consistent; if the frequency
of observation is every Thursday at midnight, the returns
all need to correspond to such a period
u1 was defined in
Equation (1). in Equation (2) gives the estimated volatility
per interval. To enable us to compare volatilities for different
interval lengths we usually express volatility in annual terms.
To do this we scale this estimate with an annualization factor
(normalising constant) which is the number of intervals per
annum such that
If daily data is used the interval is one
trading day and we use h = 252, if the interval is a week,
h = 52 and h = 12 and for monthly data³ .
Equation (2) is just the standard deviation
of the sampled series . Fig. 3 shows how this can be implemented
in Microsoft Excel where we show the daily closing values
for MTN from 1 November 2004 till 25 January 2005. In Fig.
4 we plot the 3 month historical volatility for MTN.
(3) There is approximately 252 trading days
per annum
4.3 Implied
Volatility
A simple option pricing model (like the Black & Scholes
model) will give a theoretical price for an option as a function
of the implicit parameters — constant volatility being
one. However, if the option is traded, the market price might
not be the same as the model price. In that case one might
ask, which volatility estimate does one have to use in the
model so that the model price and the market price are the
same? This is the implied volatility. In a constant volatility
framework, implied volatility is the volatility of the underlying
asset price that is implicit in the market price of an option
according to a particular model.
We illustrate the basic idea by analysing
the MTNABA warrant. This warrant has a strike price of R40,
it expires on 17 March 2005 and the cover ratio is 10. Fig.
5 shows the MTN and MTNABA prices from February 2004. The
warrant price follows the MTN price but due to the gearing
of the warrant the swings can be wilder.
To calculate the implied volatility we ask
ourselves: on 1 December 2004, the warrant price was R0.44
and MTN’s price was R40 (the same as the strike price),
we now want to know, if we substitute this price (44 cents),
into the Black & Scholes equation, what volatility will
pop out!
Before we can do anything, we need to know
the parameters mentioned in the Introduction. Current interest
rates are at 8.5% and MTN’s dividend yield is 1%.
If you have an option pricing spreadsheet,
you can substitute all the parameters into that and use Excel’s
Goalseek to search for the volatility . If you do not have
such a spreadsheet you can use the formula due to Corrado
and Miller. They refer to it as the improved quadratic formula
where
(4)Remember to multiply the warrant price by
the cover ratio.
Here X = Krt which
is the discounted strike price, S is the stock price,
k the strike price, V is the warrant price multiplied
by the cover ratio, π 3.14159265
. . . Archimedes’ constant) and T is the time
to expiry. It is accurate over a wide range of strike prices.
Fig. 6 shows an implementation in Excel (ensure
that the sheet is set up as shown with all the parameters
in the cells as shown). We show the implied volatilities calculated
for a few MTNABA warrant prices. In Fig. 7 we plot the MTNABA
warrant price and the implied volatility time series.
If we had many warrants, which vary in strike
price and time to expiration, that were written on the same
underlying like MTN, we would observe a term structure of
volatilities and a volatility “smile" or “skew".
This is due to systematic deviations from the predictions
of the Black & Scholes model and warrants another broader
discussion.
5. Difference
between Implied and Statistical Volatilities
Implied volatilities should be viewed differently from statistical
volatilities even though they both forecast the volatility
of the underlying asset over the life of the option. The two
forecasts differ because they use different data and different
models. Implied methods use current data on market prices
of options, so the implied volatility contains all the forward
expectations of investors about the likely future price path
of the underlying. Also, due to the Black & Scholes assumptions
this method assumes that the underlying’s price path
is continuous.
Contrast this with statistical methods which
use historic data on the underlying asset returns in a discrete
time model for the variance of a time series.
6. Realized/Actual
Volatility
This is the historical volatility calculated looking “backward"
when an option has expired. As an example, let’s say
a trader wants to write an option today that expires in 3
months time. To estimate the volatility he/she might calculate
the historical volatility of the past 3 months. If similar
options are trading in the market he/she might calculate the
implied volatility. The actual volatility will, however, only
be known at expiry. Once the 3 months have passed, one can
calculate the realized volatility (actual variance) between
the original trade date and expiry because the actual price
path is then known.
This article is published for general
information and is not intended as advice of any nature.
The viewpoints expressed are not necessarily that of
Financial Chaos Theory Pty. Ltd. As every situation
depends on its own facts and circumstances, only specific
advice should be relied upon. 
