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This note was prepared by Professors Robert Conroy and Robert Harris. It was written as a basis for class discussion rather than to illustrate effective or ineffective handling of an administrative situation. Copyright © 2007 by the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send an e-mail to sales@dardenpublishing.com. No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation.

THE BLACK-SCHOLES OPTION-PRICING MODEL

Fischer Black and Myron Scholes had one of the last century’s most revealing insights

about pricing in financial markets. In 1973, they published an article1 outlining the first practical theoretical model to price options. Their Black-Scholes model harnessed arbitrage forces that ensure that two ways to create the same ultimate payoff will be priced the same in well-functioning financial markets. The model’s novel assumption was that an investor who wrote a call option and simultaneously bought a certain number of shares in the underlying asset could create a riskless cash payoff. Because the investor could also create risk-free payoffs using risk-free bonds, arbitrage forces would ensure that the risk-free-rate bond return would also apply to the riskless payoff involving the shares and call option. Once this was established, Black and Scholes could derive a practical way to estimate the value of the option based on variables that could be observed or reasonably estimated.

This note discusses the economics underlying the Black-Scholes model and then applies the model to the pricing of call options. We pay particular attention to procedures for estimating the potential for stock-price changes, as these possible movements are the key driver of option value. The Underlying Economics

To illustrate the underlying economics of the Black-Scholes model, consider a simple example. Suppose the shares of XYZ are currently trading at $105 a share. You are offered a European call option with an exercise price of $100 and time to maturity of one year. How much should you be willing to pay for this option? First, we need some assumption about how the stock price will move. Let’s make the simplifying assumption that the stock price will be either $115 or $95 at the end of one year.

1 Black and Scholes ultimately received the Nobel Prize in Economics for their work, which first appeared in

“The Pricing of Options and Corporate Liabilities,” Journal of Political Economy (1973).

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The value of the call option one year from now depends on the stock price then, as the chart below shows: Assume you buy H shares of the stock and write one call option. Your combined payoff in one year will be H times the share price plus your payoff on the option. Note that when you write the call option, your payoff is negative when the stock price increases and zero when the stock drops—exactly the mirror image of payoffs to an option buyer. Your outlay today is the cost of buying the shares minus the proceeds of selling (writing) the call. If we choose H carefully, we can make the payoff in one year the same regardless of what happens to the stock price. In financial jargon, we create a riskless hedge position. To do this, we choose H so that the payoff is the same regardless of the share price:

H × 115 − 15 = H × 95 − 0

or

75.2015

95115015

==−−

=H

The payoff from doing this is determined below:

Price Today Price in One Year 115

105 or 95

Price Today Call Value in One Year Max(115 − 100, 0) = 15

Call price = ? or Max(95 − 100, 0) = 0

Price Today Value in One Year Buy H shares of stock Write 1 call Combined payoff

H × 115 − 15

H × 105 − Call or H × 95 − 0

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Price Today Value in One Year

Buy H shares of stock Write 1 call Combined payoff .75 × 115 − 15 = 71.25

.75 × 105 − Call or .75 × 95 − 0 = 71.25

If we buy .75 shares and write one call, the payoff in one year is $71.25 regardless of the share price. We have created a riskless payoff! Because the combination of buying H = .75 shares of stock and writing one call option is a riskless investment, arbitrage in markets will ensure that this combination is priced to earn the risk-free rate of return. If the risk-free rate is 6%, Rf = 6%, then the value of the call can be determined as follows:

( ) 25.7110575. 1 =⋅−⋅ ⋅fReCall , or

( ) 125.7110575. ⋅−⋅=−⋅ fReCall , and

65.1125.7175.7825.7110575. 106.1 =⋅−=⋅−⋅= ⋅−⋅− eeCall fR In essence, we have simply backed out what the value of the call option has to be for the riskless hedge position to earn a risk-free rate of return. Any other value for the call option would allow investors to profit from riskless arbitrage in the market.

While Black and Scholes incorporate a more complex treatment of stock-price movement, our simple example captures the essence of their model. Valuing options is done by setting up a riskless hedge and then discounting the payoff at the risk-free rate to determine the option’s value.

Looking at our example, intuition suggests that five things matter in pricing call options:

1. The underlying asset value, UAV: the stock price, in this case ($105).

2. The exercise price of the option, X: we determined the payoff at maturity with this.

3. The time to maturity, T: we use this as the time period to discount the riskless payoff.

4. The risk-free rate, Rf: we use this as the discount rate to discount the riskless payoff.

5. The potential for stock-price movement over time: this is the 115 and 95. Note that if we made this wider (say, 120 and 90), the call option would be worth more. As it turns out,

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this potential price movement is critical to options pricing and the Black-Scholes model2 as that movement will drive the ultimate payoffs of the call option.

In many practical applications, we often measure this potential for price fluctuation using the volatility of the stock price. Volatility is just a statistical measure of how much the stock price can change over a period of time. Typically, volatility is expressed on an annual basis. A stock with a high volatility has the expectation that the stock price will change a great deal in a year. Conversely, a stock with a low volatility has an expectation of a small change in stock price. The stock price can go up or down, but what is important is the size of the potential price changes. For example, a stock with a price of $50 and a volatility3 of .20 has a 67% probability that the stock price one year from today will be between $60 and $40. Correspondingly, if the volatility were .50, there would be a 67% probability that the stock price would be between $75 and $25. As we lengthen an option’s maturity, the annual volatility has even more time to work. As a result, a two-year option on a stock would be exposed to higher chances of wide stock-price movements prior to maturity than would a 90-day option on the same stock. The Specifics of the Black-Scholes Model

The Black-Scholes model requires exactly the five inputs discussed above, together with some statistics. In terms of notation, let call be the value of a European call. In markets, this value would be the call premium or price that the buyer pays the seller to acquire the option. The Black-Scholes model states the following:

( ) ( ) TR fedNXdNUAVCall ⋅−⋅⋅−⋅= 21 where

( )T

TRXUAV

df

⋅

⋅⎟⎠⎞

⎜⎝⎛ ⋅++

=σ

σ 2

121ln

Tdd ⋅−= σ12

2 See the Appendix. In the Black-Scholes model, we would use volatility as the estimate of price movements.

For the example used above, the price movements over the year would be equivalent to a volatility of .105. 3 The convention is to express volatility in terms of annualized percentage rates of return. Thus, a volatility of

.20 (i.e., 20%) means that the standard deviation around the mean expected return for the stock is 20%. For simplicity, the example has assumed away the effects of the stock’s expected return over the year and just focused on the deviation from that. The point is that the higher the volatility, the more likely it is that the future stock price may take on high and low values. One must also be careful in assessing how volatility works over time because random movements up and down can cancel each other out. As it turns out, a typical assumption is that one day’s stock-price movement tells us nothing about the movement on the next day. Such statistical independence would mean that the volatility experienced over four years is less than four times the annualized volatility. Specifically, if we assume independence, volatility goes up with the square root of time, so four years of volatility will be only twice the annualized volatility number.

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UAV = Underlying asset value, or stock price, in this case X = Exercise price T = Time to maturity, in years Rf = Risk-free rate (continuous compounding) σ = volatility N(.) is the cumulative standard normal density function. In essence, the density function allows the model to deal with the nature of how a stock price might change over time. The exponential, e, is used to implement the assumption of a continuously compounded risk-free interest rate.

In using the Black-Scholes model, the first four inputs are relatively straightforward; they

are either stated in the option contract (X and T) or are outcomes of traded prices in financial markets (stock price and risk-free rate). The remaining issue is how to measure potential movements in the stock price over time. In their model, Black and Scholes assumed that stock prices follow a lognormal distribution over time. In this case, what determines how much the stock price can change over time is the volatility, σ, which is calculated as the standard deviation of the log of stock-price relatives.4 Essentially, this is the standard deviation of returns for the stock. The convention is to express the volatility on an annualized basis.

Like almost any other statistical parameter, we can estimate the standard deviation or

volatility using observed data. Exhibit 1 shows five years of price data5 for Cisco Systems. What we are interested in is the standard deviation of the log of the price relatives. Exhibit 1 shows the calculation of the standard deviation of the log of the price relatives using monthly data. The monthly standard deviation of the log relatives is .1219. As we noted before, the convention is to express volatility on an annualized basis. To do this, we multiply the monthly standard deviation by the square root6 of 12. For Cisco, this results in a volatility of .4223. Applying the Model to Value a Call Option

Now suppose we were offered a European call option on Cisco where the exercise price is $15 and the time to maturity is nine months. Currently, Cisco is trading at $17.48 a share, and the risk-free rate for a maturity of nine months is 5% (continuously compounded). We can use the Black-Scholes model (in Excel) to price this call option. The only input we are missing for the Black-Scholes model is volatility. If we assume the last five years are representative of the

4 The log of price relatives ⎟⎟⎠

⎞⎜⎜⎝

⎛=

−1

lnt

t

PP

. In continuous time, Pt is (Pt−1 )(er), where r is the stock’s continuous

rate of return. The law of exponents shows the log of price relatives is then just equal to r. 5 Note that we are using prices labeled “adjusted prices.” These are prices adjusted for stock splits and stock

dividends. When using price data, it is important to adjust for stock splits and stock dividends. Stock-price changes caused by these events are not part of the volatility of the underlying asset.

6 Note that if we had a weekly standard deviation, σW, we would convert it to an annualized value by multiplying it by the square root of the number of weeks in a year (52), assuming weekly returns are independent.

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volatility for the next eight months, we can use the annualized volatility estimate of .4223 calculated in Exhibit 1.

Using the five inputs in the Excel-based model results in a call value of $4.135, as shown below: Underlying asset value (UAV, or stock price) = $17.48 Exercise price (X) = $15.00 Time to maturity (T, in years) = .75 years (9 months) Risk-free rate (Rf continuously compounded) = 5% Volatility (σ annualized) = .4223 Black-Scholes call value = $4.135 Estimating Volatility in Practice

In applying options pricing, volatility is almost always the most difficult input to estimate. There are two basic approaches to estimating volatility for a particular stock. One way is to use historical data to estimate the volatility for a future time period. In doing this, we are assuming that the past is a good estimate of the future. This is the approach we used in the above example. There, we used the historical volatility in Exhibit 1 to estimate the volatility.7

The second alternative is to use market prices for call options and the options-pricing

model to infer a volatility. Such a volatility estimate is referred to as an implied volatility. For example, suppose we knew that a call option on Cisco with an exercise price of $15.00 and a maturity of six months was trading at a price of $3.574. We can use the Black-Scholes model to find the implied volatility for Cisco. The inputs would be as follows: Underlying asset value (stock price) = $17.48 Exercise price (strike price) = $15.00 Time to maturity (in years) = 6 months (.5 years) Risk-free rate (continuously compounded) = 5% Volatility (annualized) = ? Black-Scholes call value = $3.574

7 Note that in Exhibit 1, volatility is also estimated using simple percentage changes or monthly returns. The

result is quite similar to that obtained using the log relatives, which is technically appropriate given continuous compounding. Because the results are so close, a common practice is to use percentage changes to calculate volatility.

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The volatility that results8 in a call price of $3.574 is .400. This volatility estimate could then be used to price other options on Cisco or on any underlying asset that was assumed to have the same volatility as Cisco stock.

Implied volatilities have the advantage of capturing forward-looking expectations. The

call’s market price used to estimate implied volatility is, after all, based on the market expectation of the stock’s future. The disadvantage of implied volatilities is that any error in the pricing model used will affect the volatility estimate. While the Black-Scholes model works well for many options, there are situations where even more advanced pricing models are needed.

Whatever method we use, our goal is to estimate the volatility for a future time period.

Here, we have a historical estimate of .4223 and an implied volatility of .400. These numbers are quite close and thus the choice does not make a big difference. In other circumstances, they could be quite different and the choice would matter. The choice then must be made based on what you believe is the best estimate for the future time period under consideration.

Summary

The Black-Scholes model is the most widely used technique to price European call options. Using only five inputs, the model offers a practical way to price options: UAV = Underlying asset value (or stock price, in this case) X = Exercise price T = Time to maturity, in years Rf = Risk-free rate (continuous compounding) σ = Volatility While the first four inputs are relatively straightforward to estimate, volatility requires more attention. Volatility captures the potential price movement of a stock and is a key driver of option value. In practice, future volatility can be estimated one of two ways. If we think future volatility will be similar to past volatility, we can use the volatility of past stock returns. Alternatively, we can look at the volatilities implied in the prices of options that are currently being traded. A number of financial services provide volatility estimates for a range of stocks.

8 We can find the volatility either through a trial-and-error process or through the use of the goal-seek function

in Excel.

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Exhibit 1

THE BLACK-SCHOLES OPTION-PRICING MODEL

Cisco Stock Prices Date Adjusted Price Log Relative1 % Change2

8/1/2001 20.30 9/1/2001 16.33 −0.2176 −0.1956

10/1/2001 11.90 −0.3165 −0.2713 11/1/2001 17.66 0.3948 0.4840 12/1/2001 19.86 0.1174 0.1246 1/1/2002 18.11 −0.0922 −0.0881 2/1/2002 19.21 0.0590 0.0607 3/1/2002 15.00 −0.2474 −0.2192 4/1/2002 17.52 0.1553 0.1680 5/1/2002 13.70 −0.2459 −0.2180 6/1/2002 15.33 0.1124 0.1190 7/1/2002 13.09 −0.1580 −0.1461 8/1/2002 12.10 −0.0786 −0.0756 9/1/2002 13.82 0.1329 0.1421

10/1/2002 10.94 −0.2337 −0.2084 11/1/2002 11.61 0.0594 0.0612 12/1/2002 15.06 0.2602 0.2972 1/1/2003 13.10 −0.1395 −0.1302 2/1/2003 13.48 0.0286 0.0290 3/1/2003 13.72 0.0176 0.0178 4/1/2003 12.98 −0.0554 −0.0539 5/1/2003 15.12 0.1526 0.1649 6/1/2003 16.80 0.1054 0.1111 7/1/2003 17.24 0.0259 0.0262 8/1/2003 19.15 0.1051 0.1108 9/1/2003 19.14 −0.0005 −0.0005

10/1/2003 20.20 0.0539 0.0554 11/1/2003 21.71 0.0721 0.0748 12/1/2003 23.18 0.0655 0.0677 1/1/2004 24.23 0.0443 0.0453 2/1/2004 26.20 0.0782 0.0813 3/1/2004 23.53 −0.1075 −0.1019 4/1/2004 23.74 0.0089 0.0089 5/1/2004 21.12 −0.1169 −0.1104 6/1/2004 22.24 0.0517 0.0530 7/1/2004 23.11 0.0384 0.0391 8/1/2004 21.24 −0.0844 −0.0809 9/1/2004 19.09 −0.1067 −0.1012

10/1/2004 18.93 −0.0084 −0.0084 11/1/2004 19.26 0.0173 0.0174

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Exhibit 1 (continued)

Date Adjusted Price Log

Relative1 %

Change2 1/1/2005 19.32 0.0088 0.0089 2/1/2005 18.13 −0.0636 −0.0616 3/1/2005 17.50 −0.0354 −0.0347 4/1/2005 17.70 0.0114 0.0114 5/1/2005 17.26 −0.0252 −0.0249 6/1/2005 19.64 0.1292 0.1379 7/1/2005 18.99 −0.0337 −0.0331 8/1/2005 19.26 0.0141 0.0142 9/1/2005 17.65 −0.0873 −0.0836

10/1/2005 17.74 0.0051 0.0051 11/1/2005 17.48 −0.0148 −0.0147 12/1/2005 17.69 0.0119 0.0120 1/1/2006 17.12 −0.0328 −0.0322 2/1/2006 18.58 0.0818 0.0853 3/1/2006 21.06 0.1253 0.1335 4/1/2006 21.59 0.0249 0.0252 5/1/2006 20.96 −0.0296 −0.0292 6/1/2006 20.50 −0.0222 −0.0219 7/1/2006 20.00 −0.0247 −0.0244 8/1/2006 17.48 −0.1347 −0.1260

Monthly standard deviation Mσ 0.1219 0.1242 Annualized volatility

12⋅= MA σσ 0.4223 0.4302

1 The log relative is ⎟⎟⎠

⎞⎜⎜⎝

⎛

−1

lnt

t

PP

2 % change is 11

−−t

t

PP

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Appendix

THE BLACK-SCHOLES OPTION-PRICING MODEL The value of the call option one year from now depends on the stock price. Assume you buy H shares of the stock and write one call option.

H × 120 − 20 = H × 90 − 0 or

H = (20 − 0)/(120 − 90) = .67

If we buy .67 shares and write one call, the payoff in one year is $60.00 regardless of the share price.

( ) 00.6010567. 1 =⋅−⋅ ⋅fReCall , or

( ) 100.6010567. ⋅−⋅=−⋅ fReCall , and

49.1300.6010567. 106. =⋅−⋅= ⋅−eCall

Price Today Price in One Year Max(120 − 100, 0) = 20

Call price = ? or Max(90 − 100, 0) = 0

Price Today Price in One Year Buy H shares of stock Write 1 call Combined payoff

H × 120 − 20

H × 105 − Call or H × 90 − 0

Price Today Price in One Year Buy H shares of stock Write 1 call Combined payoff

.67 × 120 − 20 = 60.00

.67 × 105 − Call or .67 × 90 − 0 = 60.00