Download The Black-Scholes Analysis

The Pricing of Stock
Options Using BlackScholes
Chapter 12
Assumptions Underlying BlackScholes
• 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
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
is normal, ST is
lognormally distributed
 T
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.
The Lognormal Distribution
ST
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 
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%
The Volatility
• The volatility is the standard deviation of the
continuously compounded rate of return in 1 year
• The standard deviation of the continuously
compounded return over T years:

T
• If the volatility is 25% per year, what is the standard
deviation of the continuously weekly compounded
return?
• It equals .25 times the square root of 1/52:
.0347 = 3.47%
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 riskfree rate
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
The Black-Scholes Formulas
c = S N ( d1 )  X e
p= Xe
where
 rT
 rT
N (d 2 )
N (  d 2 )  S N (  d1 )
ln( S / X )  ( r   2 / 2) T
d1 =
 T
ln( S / X )  ( r   2 / 2) T
d2 =
= d  T
1
 T
Properties of Black-Scholes
Formula
• As S becomes very large c tends to S-Xe-rT and
p tends to zero
• As S becomes very small c tends to zero and p
tends to Xe-rT-S
• Put price can be determined from put-call parity:
p = c + Xe-rT-S
• On a test I will give you the Black-Scholes
formula for call option. For a put you should use
put-call parity.
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
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
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
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
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 exchange
are closed are usually ignored when
volatility is estimated and when it is used
to calculate option prices