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UVA-F-1522
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 article
1
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).
UVA-F-1522 -2- 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 75.201595115015
==−−=
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
UVA-F-1522 -3-
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%, R
f
= 6%, then the value of the call can be determined as follows:
( )
25.7110575.
1
=⋅−⋅
⋅
f
R
eCall
, or
( )
1
25.7110575.
⋅−
⋅=−⋅
f
R
eCall
, and 65.1125.7175.7825.7110575.
106.1
=⋅−=⋅−⋅=
⋅−⋅−
eeCall
f
R
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, R
f
: 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,
UVA-F-1522 -4-this potential price movement is critical to options pricing and the Black-Scholes model
2
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 volatility
3
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:
( ) ( )
T R
f
ed N X d N UAV Call
⋅−
⋅⋅−⋅=
21
where
( )
T T R X UAV d
f
⋅⋅⎟ ⎠ ⎞⎜⎝ ⎛ ⋅++=
σ σ
21
21ln
T d d
⋅−=
σ
12
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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