Present Value Analysis
Theorem 8.4.2 Assuming a general distribution for the size of a jump, the
8.5 Estimating the Volatility Parameter
Whereas four of the five parameters needed to evaluate the Black–
Scholes formula – namely, s, t, K, and r – are known quantities, the value ofσ has to be estimated. One approach is to use historical data.
Section 8.5.1 gives the standard approach for estimating a population
Estimating the Volatility Parameter 149 variance; Section 8.5.2 applies the standard approach to obtain an esti-mator ofσ based on closing prices of the security over successive days;
Section 8.5.3 gives an improved estimator based on both daily closing and opening prices; and Section 8.5.4 gives a more sophisticated esti-mator that uses daily high and low prices as well as daily opening and closing prices.
8.5.1 Estimating a Population Mean and Variance
Suppose that X1, ..., Xn are independent random variables having a common probability distribution with mean μ0 and varianceσ02. The average of these data values,
¯X = n i=1Xi
n ,
is the usual estimator of the mean. Because
σ02= Var(Xi) = E[(Xi − μ0)2], it would appear thatσ02could be estimated by
n
i=1(Xi − μ0)2
n .
However, this estimator cannot be directly utilized when the meanμ0
is unknown. To use it, we must first replace the unknownμ0by its esti-mator ¯X. If we then replace n by n − 1, we obtain the sample variance S2, defined by
S2= n
i=1(Xi − ¯X )2
n− 1 .
The sample variance is the standard estimator of the varianceσ02. It is an unbiased estimator ofσ02, meaning that
E[S2]= σ02.
(It is because we wanted the estimator to be unbiased that we changed its denominator from n to n− 1.) The effectiveness of S2as an estima-tor of the variance can be measured by its mean square error (MSE), defined as
MSE= E[(S2− σ02)2]
= Var(S2).
150 Additional Results on Options
When the Xi come from a normal distribution, it can be shown that Var(S2) = 2σ04
n− 1. (8.8)
8.5.2 The Standard Estimator of Volatility
Suppose that we want to estimateσ using t time units of historical data, which we will suppose run from time 0 to time t. That is, suppose that the present time is t and that we have the historical price data S(y), 0≤ y ≤ t. Fix a positive integer n, let = t/n, and define the random
Under the assumption that the price evolution follows a geometric Brownian motion with parametersμ and σ, it follows that X1, ..., Xnare independent normal random variables with meanμ and variance σ2. From Section 8.5.1, it follows that we can use n
i=1(Xi− ¯X )2/(n − 1) to estimateσ2. Therefore, we can estimate σ2by
σ 2= 1
n
i=1(Xi− ¯X )2
n− 1 .
Moreover, it follows from Equation (8.8) that Var( σ2) = 1
2
2(σ2)2
n− 1 = 2σ4
n− 1. (8.9)
It follows from Equation (8.9) that we can use price data history over any time interval to obtain an arbitrarily precise estimator ofσ2. That is, breaking up the time interval into a large number of subintervals results
Estimating the Volatility Parameter 151 in an unbiased estimator ofσ2having an arbitrarily small variance. The difficulty with this approach, however, is that it strongly depends on the assumption that the logarithms of price ratios S(i)/S((i − 1)) are independent with a common distribution, even when the time lag is arbitrarily small. Indeed, even assuming that a security’s price history resembles a geometric Brownian motion process, it is unlikely to look like one under a microscope. That is, while successive daily closing prices might appear to be consistent with a geometric Brownian mo-tion, it is unlikely that this would be true for hourly (or more frequent) prices. For this reason we recommend that the preceding procedure be used with equal to one day. Because the unit of time is one year and there are approximately 252 trading days in a year, = 1/252.
To use this method to estimateσ, consider n successive daily closing prices C1, ..., Cn, where Ciis the closing price on trading day i. Let C0
be the closing price of the security immediately before these n days, and set
The sample variance of these data values, S2=
n
i=1(Xi− ¯X )2
n− 1 ,
can be taken as the estimator of σ2/252; S√
252 can be used to esti-mateσ.
Remark. Ifμ and σ are the drift and volatility parameters of the geo-metric Brownian motion, then
Because μ will typically have a value close to 0 whereas σ is typi-cally greater than .2, it follows that the mean of Xi = log(Ci/Ci−1) is negligible with respect to its standard deviation. Therefore, we could approximateμ by 0 and, with very small loss of efficiency, use
n i=1Xi2
n
152 Additional Results on Options
as the estimator of σ2/252. It is important to note that this estimator can be used even when the geometric Brownian motion has a time-varying drift parameter. (Recall that the Black–Scholes formula yields the unique no-arbitrage cost even in the case of a time-varying drift parameter.)
8.5.3 Using Opening and Closing Data
Let Ci denote the (closing) price of a security at the end of trading day i. Under the assumption that the security’s price follows a geomet-ric Brownian motion, log(Ci/Ci−1) is a normal random variable whose mean is approximately 0 and whose variance isσ2/252. Letting Oi be the opening price of the security at the beginning of trading day i, we can write that the ratio price change during a trading day is independent of the ratio price change that occurred while the market was closed – it follows that
This yields the estimator ˆσ of the volatility parameter σ:
ˆσ =
Estimating the Volatility Parameter 153 Equation (8.11) should be a better estimator of σ than is the standard estimator described in Section 8.5.2.
8.5.4 Using Opening, Closing, and High–Low Data
Following the notation introduced in Section 8.5.3, let X∗= log(X ) for any value X.
Let H(t) be the highest price and L(t) the lowest price of a security over an interval of length t. That is,
H(t) = max
0≤y≤tS(y), L(t) = min
0≤y≤tS(y).
Assuming that the security’s price follows geometric Brownian motion with drift 0 and volatilityσ, it can be shown that
E[(H∗(t) − L∗(t))2]= 2.773 Var
Now let Oi and Ci be the opening and closing prices on trading day i, and let Hi and Li be the high and the low prices during that day. Be-cause E[log(Ci/Oi)] ≈ 0, we can approximate the price history during a trading day as a geometric Brownian motion process with drift param-eter 0. Therefore, using the preceding identity, we see that
E[(Hi∗− L∗i)2] ≈ 2.773 Var(log(Ci/Oi)).
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Any linear combination of these estimators of the form αE1+ (1 − α)E2
can also be used to estimate Var(log(Ci/Oi)). The best estimator of this type (i.e., the one whose variance is smallest) can be shown to result when α = .5/.361 = 1.39. That is, the best estimator of Var(log(Ci/Oi)) is
Consequently, we can estimate the volatility parameterσ by
ˆσ = Remark. The estimator of σ given in Equation (8.13) has not previ-ously appeared in the literature. The approach presented here built on the work of Garman and Klass (see reference [2]), who derived the es-timator of Var(log(Ci/Oi)) given by Equation (8.12). In their further analysis, however, Garman and Klass assume not only that the security’s price follows a geometric Brownian motion when the market is open but
Some Comments 155 also that it follows the same (although now unobservable) geometric Brownian motion while the market is closed. Based on this assumption, they supposed that
Var(Ci∗− Oi∗) =1− f 252 σ2, Var(Oi∗− Ci∗−1) = f
252σ2,
where f is the fraction of the day that the market is closed. However, this assumption – that the security’s price when the market is closed changes according to the same probability law as when it is open – seems quite doubtful. Therefore, we have chosen to make the much weaker assump-tion that the ratio price changes Oi/Ci−1are independent of all prices up to market closure on day i − 1.