The
BlackScholes
Merton
Model
In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major
breakthrough in the pricing
of
stock options.
I
This involved the development
of
what
has become known as the BlackScholes (or BlackScholesMerton) model. The model
has had a huge influence on the
way
that traders price and hedge options.
It
has also
been pivotal to the growth and success
of
financial engineering in the last
20
years. In
1997,
the importance
of
the model
was
recognized when Robert Merton and Myron
Scholes
were
awarded the Nobel prize for economics. Sadly, Fischer Black died in
1995,
otherwise he too would undoubtedly have been one
of
the recipients
of
this prize.
This chapter shows how the BlackScholes model for valuing European call and
put
options on a nondividendpaying stock
is
derived.
It
explains how volatility can be
either estimated from historical data or implied from option prices using the model.
It
shows how the riskneutral valuation argument introduced in Chapter
11
can be used.
It
also shows how the BlackScholes model can be extended to deal with European call
and
put
options on dividendpaying stocks and presents some results on the pricing
of
American call options on dividendpaying stocks.
13.1
LOGNORMAL
PROPERTY OF STOCK PRICES
The model
of
stock price behavior used
by
Black, Scholes, and Merton
is
the model
we
developed in Chapter
12.
It
assumes that percentage changes in the stock price in a
short period
of
time are normally distributed.
We
define
fJ..:
Expected return on stock per year
0:
Volatility
of
the stock price per year
The mean
of
the percentage change in the stock price in time
.6.t
is
J.L.6.t
and the
1
See
F. Black and M. Scholes,
"The
Pricing
of
Options and Corporate Liabilities,"
JOllmal
oj
Political
Economy,
81
(May/June
1973):
63759; R.C. Merton, "Theory
of
Rational Option Pricing,"
Bell Journal
oj
Economics and Managemelll Science,
4 (Spring
1973):
14183.
281
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282
CHAPTER
13
standard deviation
of
this percentage change
is
O'.;;s:i,
so that
(13.1)
where
/::;.S
is
the change in the stock price S in time
/::;.t,
and
¢(111,
s)
denotes a normal
distribution with mean
111
and standard deviation
s.
As shown in Section 12.6, the model implies that
InST
lnSo
~
¢[
(tL

~2)T,
o'vT
]
From this, it follows that
and
In
~:
~
¢[
(tL

~2)
T,
o'vT
]
In
ST
~
¢[In
So
+
(tL

~2)
T,
o'vT
]
(13.2)
(13.3)
where
ST
is
the stock price at a future time
T
and
So
is
the stock price at time
O.
Equation (13.3) shows that
In
ST
is
normally distributed,
so
that
ST
has a lognormal
distribution. The mean
of
In
ST
is
In
So
+
(tL

O'
2
/2)T
and the standard deviation
is
o'vT.
Example 13.1
Consider a stock with an initial price
of
$40, an expected return
of
16% per
annum, and a volatility
of
20% per annum. Fr()m equation (13.3), the probability
distribution
of
the stock price
ST
in 6 months' time
is
given by
In
ST
~
¢[ln40
+
(0.16  0.2
2
/2)
x
0.5,
0.2.JQ.5]
InST
~
¢(3.759, 0.141)
There
is
a 95% probability that a normally distributed variable has a value within
1.96
standard deviations
of
its mean. Hence, with 95% confidence,
3.759 
1.96 x
0.141
<
In
ST
<
3.759
+
1.96
x
0.141
This can be written
e3.7591.96xO.141
<
ST
<
e3.759+1.96xO.141
or
32.55
<
ST
<
56.56
Thus, there
is
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 Spring '11
 DonBlasius
 Math, Normal Distribution, Options

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