8.7Estimating Volatility

The only parameter that requires estimation in the Black-Scholes Model is the volatil-

ity . This volatility estimate also may be of use in estimating uand din a binomial

model (see Chapter 7).

There are a number of ways to estimate , assuming it is constant. One method is

to use historical data, as shown in Exhibit 8.9. We now analyze this issue.

¹⁷A

numerical example illustrating the Black-Scholes valuation of a European option on a dividend-

paying stock appears later in this chapter. In terms of the above formula, one merely substitutes the

value of the stock stripped of the present value of the dividends to expiration for Sin equation (8.3) to

0

arrive at a correct answer. This is simply the current stock price less the risk-free discounted value of the

dividend payment.

Grinblatt579Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw579Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 8

Options

281

EXHIBIT8.9Computation of the Volatility Estimate forthe Black-Scholes

Model Using Historical Return Data on Dell Computer

	Return	Gross Return	Logged Gross
Year–Quarter	(%)	(%)	Return

1990–138.57	138.57	0.3262
1990–265.62	165.62	0.5045
1990–3 30.72	69.28	0.3670
1990–4111.53	211.53	0.7492
1991–154.07	154.07	0.4323
1991–2 14.05	85.95	0.1513
1991–336.25	136.25	0.3093
1991–4 23.24	76.76	0.2645
1992–141.96	141.96	0.3504
1992–2 25.26	74.74	0.2911
1992–357.94	157.94	0.4571
1992–467.67	167.67	0.5168
1993–1 26.83	73.17	0.3123
1993–2 46.61	53.39	0.6276
1993–3 11.33	88.67	0.1203
1993–436.07	136.07	0.3080
1994–111.60	111.60	0.1097
1994–24.46	104.46	0.0436
1994–341.96	141.96	0.3503
1994–49.50	109.50	0.0908
1995–16.71	106.71	0.0650
1995–237.43	137.43	0.3180
1995–341.36	141.36	0.3462
1995–4 18.53	81.47	0.2049
1996–1 3.24	96.76	0.0330
1996–251.86	151.86	0.4178
1996–352.83	152.83	0.4242
1996–436.65	136.65	0.3123
1997–127.30	127.30	0.2413
1997–273.66	173.66	0.5519
1997–364.98	164.98	0.5007
1997–4 13.29	86.71	0.1426
1998–161.31	161.36	0.4782
1998–236.99	136.99	0.3148
1998–341.68	141.68	0.3484
1998–411.31	111.31	0.1072
1999–111.70	111.70	0.1106
1999–2 9.48	90.52	0.0996
1999–313.01	113.01	0.1223
1999–421.97	121.97	0.1986
Logged gross return standard deviation	0.303808
Annualized standard deviation, the volatility	estimate0.607617

Grinblatt580Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw580Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

282Part IIValuing Financial Assets

Using Historical Data

The appropriate volatility computation for the in the Black-Scholes Model is based

on the volatility of instantaneous returns.

•	First, obtain historical returns for the stock the option is written on. Thesecond column of Exhibit 8.9 (“Return”) reports the 40 quarterly returns ofDell Computer from the first quarter of 1990 through the end of 1999.
•	Second, convert the returns to gross returns (100 percent plus the rate of returnin percentage form, 1 plus the return in decimal form), as shown in the gross return column of Exhibit 8.9.
•	Third, take the natural logarithm of the decimal version of the gross return; thus,before taking the log, divide by 100 if the gross return is in percentage form.
•	Fourth, compute the unbiased sample variance of the logged return series and annualize it (as in the last row) by multiplying it by the square root of theratio of 365 to the number of days in the return interval (for example, formonthly returns multiply by the square root of twelve and for quarterly returnsmultiply by the square root of four).

Using Spreadsheets to Compute the Volatility.Spreadsheet standard deviation

functions typically provide the unbiased estimate of the standard deviation.¹⁸Remem-

ber to annualize the standard deviation obtained from the spreadsheet because the

spreadsheet does not know whether the returns were taken weekly, monthly, daily,

and so on. In Exhibit 8.9, which reports quarterly returns, this adjustment amounts

to multiplying the output from the spreadsheet by 2, which is the square root of 4.

Frequency Choice.Exhibit 8.9 uses quarterly data to estimate the volatility of Dell

Computer for the Black-Scholes Model. Statistical theory suggests that one should use

returns that are sampled more frequently to obtain more precise volatility estimates; our

preference is weekly data. The use of daily data may be inferior because the bid-ask

spread tends to make volatility estimates overstate the true volatility of returns.

Improving the Volatility Estimate.Procedures, similar to those designed to improve

beta estimation for the Capital Asset Pricing Model, can improve the volatility estimate.

Consider the spectrum of historical estimates of for a large number of securities.

Those securities with the highest (lowest) estimated volatilities from historical data are

more likely to have overestimates (underestimates) of the true volatility because of sam-

pling error. This information can be used to improve volatility estimates. In particular,

an improved volatility estimate can be derived by taking a weighting of the average

estimated volatility over a large group of securities and the historical volatility estimate

for a single security.

The Implied Volatility Approach

An alternative approach for estimating volatility in a security is to look at other options

on the same security. If market values for the options exist, there is a unique implied

volatilitythat makes the Black-Scholes Model consistent with the market price for a

particular option.

¹⁸For

example, STDEVin Excel or @STD in Lotus 1-2-3.

Grinblatt582Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw582Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

	Chapter 8Options	283
EXHIBIT8.10	The Value of a Call Option as a Function of Its Volatility

	Market Price of
9	Call Option
8	Black-Scholes Value

7

6

		A
Call option value
	5

4	Implied volatility
3

2

1

0
0.05	0.070.090.110.130.150.170.190.210.230.250.270.290.310.330.350.370.390.410.430.450.470.49