11.1 introduction to futures and options markets, 3rd edition © 1997 by john c. hull the pricing of...
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11.1
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Pricing of Stock Options Using Black-
Scholes
Chapter 11
11.2
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
Black-Scholes Model
• Black-Scholes option pricing model was developed in 1970 by:
• Fischer Black
• Myron Scholes
• Robert Merton
• Their work has had huge influence on the way in which market participants price and hedge options.
11.3
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
Assumptions Underlying Black-Scholes
• Black-Scholes assume that stock prices follow a random walk.
- This means that proportional changes in the stock price in a short period of time are normally distributed.
• Proportional change is the change in the stock price in time t is S. The return in time t is S/S
• This return is assumed to be normally distributed with mean t and standard deviation
t
11.4
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Lognormal Property• These assumptions imply ln ST is normally
distributed with mean:
and standard deviation:
• Since the logarithm of ST is normal, ST is lognormally distributed
ln ( / )S T 2 2
T
11.5
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Lognormal Propertycontinued
where m,s] is a normal distribution with mean m and standard deviation s
ln ( / ) ,S
ST TT 2 2
11.6
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
Problem
Calculate the mean and standard deviation of the continuously compounded return in one one year for a stock with an expected retrun of 17 percent and volatility of 20 percent per annum.
11.7
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Expected Return
Two possible definitions:
• is the arithmetic average of the returns realized in may short intervals of time
• – 2/2 is the expected continuously compounded return realized over a longer period of time
is an arithmetic average
– 2/2 is a geometric average
11.8
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Volatility
• The volatility of a stock, , is a measure of uncertainty about the return provided by the stock.- It is measured as the standard deviation of the
return provided by the stock in one year when the return is expressed using continuous compounding.
• As an approximation it is the standard deviation of the proportional change in 1 year
11.9
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Volatility (cont.)
• As a rough approximation, is the standard deviation of the proportional change in the stock price in time T.- Consider the situation, where = 0.30 per annum
• standard deviation of the proportional change in:–six month
–three month
- Uncertainty about the future stock price increases with the square root of how far ahead you are looking.
T
11.10
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
Estimating Volatility from Historical Data
1. Take observations S 0, S 1, . . . , Sn at intervals of years
2. Define the continuously compounded return as:
3. Calculate the standard deviation of the ui ´s (=s)
4. The volatility estimate is
uS
Sii
i
ln1
ss
*
11.11
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
The Concepts Underlying Black-Scholes
• The option price & the stock price depend on the same underlying source of uncertainty
• We can form a portfolio consisting of the stock & the option which eliminates this source of uncertainty
• The portfolio is instantaneously riskless & must instantaneously earn the risk-free rate
11.12
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
Computation of Volatility Using Historical data
Day Closing Stock Price Price Relative Daily return0 201 20 1/8 1.0063 0.006232 19 7/8 0.9876 -0.012503 20 1.0063 0.006274 20 1/2 1.0250 0.024695 20 1/4 0.9878 -0.012276 20 7/8 1.0309 0.030407 20 7/8 1.0000 0.000008 20 7/8 1.0000 0.000009 20 3/4 0.9940 -0.00601
10 20 3/4 1.0000 0.0000011 21 1.0120 0.0119812 21 1/8 1.0060 0.0059313 20 7/8 0.9882 -0.0119014 20 7/8 1.0000 0.0000015 21 1/4 1.0180 0.0178016 21 3/8 1.0059 0.0058717 21 3/8 1.0000 0.0000018 21 1/4 0.9942 -0.0058719 21 3/4 1.0235 0.0232620 22 1.0115 0.01143