Valuing Stock Options:The
Black-Scholes Model
ECO760
The Black-Scholes Random Walk
Assumption


Consider a stock whose price is S
In a short period of time of length Dt the
change in the stock price is assumed to be
normal with mean mSDt and standard
deviation
sS Dt
 m is expected return and s is volatility
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The Lognormal Property

These assumptions imply ln ST is normally
distributed with mean:
ln S0  (m  s2 / 2)T
and standard deviation:

s T
Because the logarithm of ST is normal, ST is
lognormally distributed
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The Lognormal Property
(continued)

ln S T   ln S 0  (m  s 2 2)T , s T

or

ST
ln
  (m  s 2 2)T , s T
S0

where m,s] is a normal distribution with
mean m and standard deviation s
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The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
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(e
s2T
 1)
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The Expected Return




The expected value of the stock price is
S0emT
The expected return on the stock with
continuous compounding is m– s2/2
The arithmetic mean of the returns over
short periods of length Dt is m
The geometric mean of these returns is
m – s2/2
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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
Dt is s Dt
If a stock price is $50 and its volatility is 30%
per year what is the standard deviation of
the price change in one week?
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Estimating Volatility from
Historical Data
1.
2.
Take observations S0, S1, . . . , Sn at intervals
of t years
Define the continuously compounded return
as:
 Si 

ui  ln
 S i 1 
3.
4.
Calculate the standard deviation, s , of the
ui ´s
s
The historical volatility estimate is: sˆ 
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t
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Nature of Volatility


Volatility is usually much greater when the
market is open (i.e. the asset is trading)
than when it is closed
For this reason time is usually measured
in “trading days” not calendar days when
options are valued
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The Concepts Underlying BlackScholes



The option price and the stock price depend
on the same underlying source of uncertainty
We can form a portfolio consisting of the
stock and the option which eliminates this
source of uncertainty
The portfolio is instantaneously riskless and
must instantaneously earn the risk-free rate
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The Black-Scholes Formulas
c  S 0 N (d1 )  K e
 rT
N (d 2 )
p  K e  rT N (d 2 )  S 0 N (d1 )
2
ln(S 0 / K )  (r  s / 2)T
w here d1 
s T
ln(S 0 / K )  (r  s 2 / 2)T
d2 
 d1  s T
s T
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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
Tables for N can be used with interpolation
For example, N(0.6278) = N(0.62) +
0.78[N(0.63) - N(0.62)] =
0.7324+0.78*(0.7357-0.7324) = 0.7350
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Risk-Neutral Valuation



The variable m does not appear in the BlackScholes equation
The equation is independent of all variables
affected by risk preference
This is consistent with the risk-neutral
valuation principle
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Applying Risk-Neutral Valuation
1.
2.
3.
Assume that the expected return
from an asset is the risk-free rate
Calculate the expected payoff from
the derivative
Discount at the risk-free rate
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Valuing a Forward Contract with
Risk-Neutral Valuation



Payoff is ST – K
Expected payoff in a risk-neutral world is
SerT – K
Present value of expected payoff is
e-rT[SerT – K]=S – Ke-rT
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Dividends



European options on dividend-paying
stocks are valued by substituting the
stock price less the present value of
dividends into the Black-Scholes formula
Only dividends with ex-dividend dates
during life of option should be included
The “dividend” should be the expected
reduction in the stock price expected
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Dividends –Example
• Consider a European call on a stock with exdividend dates in two months and five months.
The div. on each ex-div date is expected to be
$0.50. The current share price is $40, the
exercise price is $40, the stock price volatility
is 30% pa, the risk-free interest rate is 9% pa,
and maturity is six months.
• Calculate the call price
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American Calls


An American call on a non-dividend-paying
stock should never be exercised early
An American call on a dividend-paying stock
should only ever be exercised immediately
prior to an ex-dividend date
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Quiz
1.
2.
Calculate the price of a three-month
European put option on a nondividend-paying stock with a strike
price of $50 when the current stock
price is $50, the risk-free interest rate
is 10% pa, and the volatility is 30% pa
What if a dividend of $1.50 is
expected in two months?
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