Transcript The Black-Scholes Analysis

11.1
The Pricing of Stock
Options Using BlackScholes
Chapter 11
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.2
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.
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
Assumptions Underlying BlackScholes
•
•
•
11.3
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
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.4
The Lognormal Property
• These assumptions imply ln ST
is normally
distributed with mean:
ln S  (    2 / 2) T
and standard deviation:
• Since the logarithm of ST
 T
is normal, ST is
lognormally distributed
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.5
The Lognormal Property
continued

ST
2
ln   (    / 2)T ,  T
S

where m,s] is a normal distribution with
mean m and standard deviation s
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.6
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.
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
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.7
11.8
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
Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull
11.9
The Volatility (cont.)
• As a rough approximation, 
T 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.
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:
 Si 

ui  ln
 Si  1 
3. Calculate the standard deviation of the
ui ´s (=s)
s
4. The volatility estimate is s * 

Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.10
11.11
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
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
Computation of Volatility
Using Historical data
Day
0
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19
20
Closing Stock Price Price Relative Daily return
20
20 1/8
1.0063
0.00623
19 7/8
0.9876
-0.01250
20
1.0063
0.00627
20 1/2
1.0250
0.02469
20 1/4
0.9878
-0.01227
20 7/8
1.0309
0.03040
20 7/8
1.0000
0.00000
20 7/8
1.0000
0.00000
20 3/4
0.9940
-0.00601
20 3/4
1.0000
0.00000
21
1.0120
0.01198
21 1/8
1.0060
0.00593
20 7/8
0.9882
-0.01190
20 7/8
1.0000
0.00000
21 1/4
1.0180
0.01780
21 3/8
1.0059
0.00587
21 3/8
1.0000
0.00000
21 1/4
0.9942
-0.00587
21 3/4
1.0235
0.02326
22
1.0115
0.01143
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
11.12