1
The Pricing of Stock
Options Using BlackScholes
Chapter 12
2
Assumptions Underlying
Black-Scholes
• We assume that stock prices follow a
•
•
random walk
Over a small time period dt,the change
in the stock price is dS. The return over
time dt is dS/S
This return is assumed to be normally
distributed with mean dt and standard
deviation  dt
3
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
lognormally distributed
 T
is normal, ST is
4
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
If T=1 then ln(ST/S) is the continuously
compounded annual stock return.
5
The Lognormal Distribution
E ( ST )  S0 e T
2 2 T
var ( ST )  S0 e
(e
2T
 1)
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
 Notice the geometric (compound) return
is less than the average with the
difference positively related to .
6
7
Expected Return
• Suppose  10%
•
•
and  =0. Then annual
compound return = 10%
Suppose  10% and  =5. Then annual
compound return
 – 2/2 = .1 – (.05)(.05)/2
= 9.875%
Suppose  10% and  =20%. Then annual
compound return
 – 2/2 = .1 – (.2)(.2)/2
= 8.0%
8
The Volatility
•
The volatility is the standard deviation of
the continuously compounded rate of
return in 1 year
• The standard deviation of the return
•
in time T is 
T
If a stock price is $50 and its
volatility is 25% per year what is the
standard deviation of the price
change in one day?
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 * 

9
10
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
Assumptions
• Stock price follows lognormal model
•
•
•
•
with constant parameters
No transactions costs
No dividends
Trading is continuous
Investors can borrow or lend at a
constant risk-free rate
12
The Black-Scholes Formulas
c  S N ( d1 )  X e
p Xe
 rT
 rT
N (d 2 )
N (  d 2 )  S N (  d1 )
ln( S / X )  ( r   / 2) T
d1 
 T
2
where
ln( S / X )  ( r   / 2) T
d2 
 d  T
1
 T
2
Properties of Black-Scholes
Formula
• As S becomes very large c tends to SXe-rT and p tends to zero
• As S becomes very small c tends to
zero and p tends to Xe-rT-S
13
14
The N(x) Function
• N(x) is the probability that a normally
•
distributed variable with a mean of zero
and a standard deviation of 1 is less
than x
See tables at the end of the book
15
Applying Risk-Neutral
Valuation
1. Assume that the expected return from
the stock price is the risk-free rate
2. Calculate the expected payoff from
the option
3. Discount at the risk-free rate
16
Implied Volatility
• The volatility implied by a European
•
option price is the volatility which, when
substituted in the Black-Scholes, gives
the option price
In practice it must be found by a “trial
and error” iterative procedure
17
Implied Volatility
• The implied volatility of an option is the
•
•
volatility for which the Black-Scholes
price equals the market price
The is a one-to-one correspondence
between prices and implied volatilities
Traders and brokers often quote implied
volatilities rather than dollar prices
18
Causes of Volatility
• To a large extent, volatility appears to
•
be caused by trading rather than by the
arrival of new information to the market
place
For this reason days when the exchnge
are closed are usually ignored when
volatility is estimated and when it is
used to calculate option prices