Stefanica D. A Primer for the Mathematics of Financial Engineering
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84
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
(iii)
if
Ak
E
JF,
k = 1 : 00,
then
U~l
Ak
E
JF.
The
set
S represents
the
sample
space,
any
element
of
S is a possible
outcome,
and
a
set
A E
JF
is
an
event
1
,
i.e., a collection of possible
outcomes.
A
probability
function
P :
JF
---t
[0,
1]
defined
on
JF
is a
nonnegative
function
such
that
P(S) = 1
and
P
(Q
Ak)
=
~
P(A
k
),
V
Ak
E
JF,
k = 1 :
00,
V Ai n Aj =
0,
i
of
j.
A
function
X : S
---t
JR
is called a
random
variable
if
{ s E S
such
that
X ( s)
::;
t}
E
JF,
\;j
t E
JR.
The
cumulative
distribution
function
F :
JR
---t
[0,
1]
of
the
random
variable
X is defined as
F(t)
=
P(X::;
t). (3.5)
For
any
random
variable X
considered
here, we
assume
2
that
X
has
a
probability
density
function
f (x), i.e., we
assume
that
there
exists a
function
f :
JR
---t
JR
such
that
P(a
<::
X
<::
b)
= l f(x)dx.
(3.6)
For
an
integrable
function
f
(x)
to
be
the
probability
density
function
of
a
random
variable
it
is necessary
and
sufficient
that
f(x)
~
0, for all x E
JR,
and
I:
f(x)
dx
=
1.
(3.7)
Lemma
3.2.
Let
f(x)
be
the probability
density
function
of
the
random
vari-
able
X.
Then
the cumulative distribution
function
of
X can
be
written
as
F(t) =
l~
f(x)
dx.
(3.8)
Definition
3.2.
Let
f(x)
be
the probability density
function
of
the
random
variable
X.
The expected value
E[X]
of
X (also called the
mean
of
X)
is
defined
as
E[X] =
I:
xf(x)
dx.
(3.9)
1 A family of subsets
IF
of S satisfying
the
properties
(i)~(iii)
is called a sigma-algebra
of
S.
We introduce
the
concepts of continuous probability
without
formally discussing measure
theory
topics such as sigma-algebras
and
measurable functions. For more details
on
measure
theory,
we
refer
the
reader
to
Royden
[22]
or
Rudin
[23].
2It
can
be
shown
that
the
random
variable X has a probability density function if
and
only if
the
cumulative density function F(t)
of
X is
an
absolutely continuous function.
3.2.
CONTINUOUS
PROBABILITY
CONCEPTS
85
The
expected
values
of
a
random
variables is a
linear
operator:
Lemma
3.3.
Let
X
and
Y
be
two
random
variables
over
the
same
probability
space.
Then
E[X+Y]
E[eX]
E[X]
+ E[Y];
c
E[X],
\;j
e E
JR.
(3.10)
(3.11)
The
following
result
will
be
used
throughout
this
chapter,
and
is
presented
without
proof:
Lemma
3.4.
Let
h :
JR
---t
JR
be
a piecewise
continuous
function,
and
let
X : S
---t
JR
be
a
random
variable with probability
density
function
f (x)
such
that
flR
Ih(x)f(x)
dx
< 00.
Then
h(X)
: S
---t
JR
is a
random
variable
and
the expected value
of
h(
X)
is
E[h(X)] =
I:
h(x)
f(x)
dx.
(3.12)
3.2.1
Variance,
covariance,
and
correlation
The
variance
and
standard
deviation
of
a
random
variable
offer
important
information
about
its
distribution.
Definition
3.3.
Let
f(x)
be
the probability
density
function
of
the
random
variable
X,
and
assume
that
flR
x
2
f(x)
dx
< 00.
Let
m =
E[X]
denote the
expected value
of
X.
The variance
var(X)
of
X is defined as
var(X) =
E[(X
- E[X])2] =
I:
(x
-
rn)2
f(x)
dx.
The standard deviation a-(X)
of
X is defined as
a-(X)
=
Vvar(X).
Therefore,
var(X)
=
(a-(X)?
=
a-
2
(X).
From
(3.13)
and
(3.14),
it
is
easy
to
see
that
var(eX)
a-(eX)
e
2
var(X),
\;j
e E
JR;
lei
a-(X),
\;j
e E
JR.
(3.13)
(3.14)
(3.15)
(3.16)
The
next
result
corresponds
to
that
of
Lemma
3.1,
and
its
proof
follows
along
the
same
lines:
86
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
Lemma
3.5.
If
X is a random variable) then
var(X)
= E[X2] - (E[X])2.
Definition
3.4.
Let X and Y
be
two random variables over the same prob-
ability space. The covariance cov(X,
Y)
of
X and Y is defined
as
cov(X,
Y)
=
E[(X
- E[X])
(Y
- E[Y])].
(3.17)
The correlation corr(X, Y) between X and Y is
equal
to the covariance
?f
X
and Y normalized with respect to the standard deviations
of
X and
Y)
2.
e.)
cov(X,
Y)
corr(X,
Y)
= o-(X) o-(Y) ,
(3.18)
where o-(X) and o-(Y)
are
the standard deviations
of
X and
Y)
respectively.
Lemma
3.6.
Let X and Y
be
two random variables over the same probability
space. Then
cov(X,
Y)
=
E[XY]
- E[X]E[Y].
(3.19)
Proof. Using (3.10)
and
(3.11) repeatedly, we find
that
cov(X, Y)
E[ (X - E[X])
(Y
- E[Y]) ]
E[
XY
-
XE[Y]
-
YE[X]
+
E[X]E[Y]]
E[XY]
- E[
XE[Y]
] - E[
YE[X]
] + E[X]E[Y].(3.20)
Since E[X]
and
E[Y] are constants, we conclude from (3.11)
that
E[
XE[Y]]
= E[Y] E[X]
and
E[
YE[X]]
= E[X] E[Y].
Therefore, formula (3.20) becomes
cov(X, Y)
=
E[XY]
- 2E[X]E[Y] + E[X]E[Y]
E[XY]
- E[X]E[Y].
D
Let
X Y
and
U
be
random
variables over
the
same probability space.
, , .
The
following properties are easy
to
establish
and
show
that
the
covanance
of two
random
variables is a
symmetric
bilinear operator:
cov(X,
Y)
cov(X
+
U,
Y)
cov(X, Y
+
U)
cov(cX, Y)
cov(cIX,
C2
Y
)
cov(Y, X);
cov(X,
Y)
+
cov(U,
Y);
cov(X,
Y)
+ cov(X,
U);
cov(X,
cY)
= c cov(X,
Y),
\j
c E
1R;
CIC2
cov(X, Y),
\j
Cl,
C2
E
JR..
(3.21)
(3.22)
3.2.
CONTINUOUS
PROBABILITY
CONCEPTS
87
Lemma
3.7.
Let X and Y
be
two random variables over the same probability
space.
Then)
var(X
+ Y) =
var(X)
+ 2cov(X,
Y)
+ var(Y) , (3.23)
or)
equivalently)
var(X
+
Y)
= o-2(X) + 2o-(X) o-(Y) corr(X,
Y)
+ o-2(y),
(3.24)
where o-(X) and o-(Y)
are
the standard deviation
of
X and
Y)
respectively.
Proof.
Formula (3.23) is derived from definitions (3.13)
and
(3.17), by using
the
additivity
of
the
expected value (3.10),
as
follows:
var(X
+
Y)
E [ ( (X +
Y)
E[X
+
Y]
)2
]
E [ (
(X
- E[X]) + (Y - E[Y])
)2
]
E [
(X
- E[X])2 ] + 2E[
(X
- E[X]) (Y - E[Y]) ]
+ E [
(Y
- E[y])2 ]
var(X)
+ 2cov(X,
Y)
+
var(Y).
Formula (3.24) is a direct consequence
of
(3.18)
and
(3.23).
D
The
relevant information contained
by
the
covariance
of
two
random
vari-
ables is
related
to
whether
they
are
positively
or
negatively correlated, i.e.,
whether cov(X,
Y)
> 0
or
cov(X, Y) <
O.
The
correlation of
the
two vari-
ables contains
the
same
information as
the
covariance,
in
terms
of
its
sign,
but
its size is also relevant, since
it
is scaled
to
adjust
for multiplication by
constants, i.e.,
corr(cIX,C2Y)
= sign(clc2)
corr(X,Y),
\j
CI,C2
E
JR..
(3.25)
Here,
the
sign function sgn(
c)
is equal
to
1, if C > 0,
and
to
-1
is C <
O.
To see this, recall from (3.16)
that
o-(CIX) = ICllo-(X)
and
o-(C2Y)
IC2Io-(Y)
Then,
from (3.18)
and
using (3.22),
it
follows
that
cov(cIX,
C2Y)
CIC2
cov(X, Y)
0-(
CIX)
0-(
C2Y)
ICII
IC21
o-(X) o-(Y)
CIC2
cov(X, Y)
I
C
IC21
o-(X) o-(Y) = sgn(clc2)
corr(X,
Y).
Lemma
3.8.
Let X and Y
be
two random variables over the same probability
space.
Then
-1
:s;
corr(X,
Y)
:s;
1.
(3.26)
88
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
Proof. Let a E
JR
be
an
arbitrary
constant.
From (3.24), (3.21), (3.15),
and
(3.18) we find
that
var(X
+
aY)
var(X)
+ 2cov(X,
aY)
+
var(aY)
var(X)
+
2a
cov(X,
Y)
+ a
2
var(Y)
0"2(X)
+
2a
O"(X)
O"(Y)
corr(X,
Y)
+ a
2
0"2(y).
Since
var(X
+
aY)
~
0, we
obtain
that
a
2
0"2(y)
+
2a
O"(X)O"(Y)corr(X,
Y)
+
0"2(X)
~
0,
\j
a E
JR.
(3.27)
Note
that
the
left
hand
side of (3.27) is a
quadratic
polynomial of
a.
The
in-
equality (3.27) holds
true
for
any
real
number
a if
and
only if
this
polynomial
has
at
most
one real double root, i.e., if
its
discriminant is nonpositive:
(20"(X)0"(Y)corr(X, y))2 -
40"2(X)0"2(y)
~
o.
This
is equivalent
to
40"2(X)0"2(y)
(corr(X, y))2
::;
40"2(X)0"2(Y),
which
hap-
pens
if
and
only if Icorr(X, Y)I
~
1, which is equivalent
to
(3.26). 0
The
elegant technique used in
the
proof
of
Lemma
3.8 can also
be
applied,
e.g., for showing
that
the
inner
product
of two vectors is
bounded
from above
by
the
product
of
the
norms of
the
vectors
3:
(
t
Xi
Yi)
2
::;
(t
XI)
(t
Yi),
\j
Xi,
Yi
E
JR,
i = 1 : n.
(3.28)
2=1 2=1 2=1
The
expected value
and
variance of a linear combination of n
random
vari-
ables
can
be
obtained by
induction
from
the
corresponding formulas
(3.10-
3.11)
and
(3.23-3.24) for two
random
variables.
Lemma
3.9.
Let
Xi,
i = 1 :
n,
be
random
variables over the
same
probability
space, and let
Ci E
JR
be
real numbers. Then,
E
[~~Xi]
n
(3.29)
i=1
vaT
(~~Xi)
=
~c;var(Xi)
+ 2
,<;t-1<;n
CiCjCOV(Xi,X
j
).
(3.30)
3A
proof for (3.28), also called
the
Cauchy-Schwarz inequality,
can
be
given using
the
fact
that
n
I)Xi + aYi)2
~
0,
'If
a E R
i=l
3.3.
THE
STANDARD
NORMAL
VARIABLE
89
If
Pi,j
=
corr(Xi'
X
j
), 1
::;
i < j
~
n,
is the correlation between
Xi
and
X
j
,
and
if
0";
=
var(X
i
),
i = 1 :
n,
then
(3.30) can also
be
written as
var
(t~Xi)
=
tc;O";
+ 2 L
CiCjO"iO"jCOrr(Xi,Xj).
2=1 2=1
1:5:i<j:5:n
(3.31)
3.3
The
standard
normal
variable
The
standard
normal
variable, denoted by
Z,
is
the
random
variable
with
probability density function
f(x)
(3.32)
It
is easy
to
see
that
f (
x)
> 0,
\j
X E
JR.
To conel
ude
that
f (
x)
is indeed
a density function, we would have
to
show
that
1
00
1 1
00
x
2
f(x)
dx
= .
f(C
e-
Z
dx
=
1.
-00
V
27f
-00
(3.33)
We
postpone
the
proof
of
this
fact until section 7.5
and
assume for now
that
(3.33) holds true.
The
following result will be needed in
the
proof
of
Lemma
3.10:
(3.34)
x
2
Note
that
x
2
e-
z
is
an
even function.
From
Lemma
0.1,
it
follows
that
(3.35)
(
2)'
2
Note
that
-e-.y
=
xe-.y.
By
integration
by
parts,
we find
that
(3.36)
90
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
From
(3.35)
and
(3.36)
it
follows
that
2
t~
(-te-'?
+ l'
e-
f
dX)
2 lim
rt
e-~
dx
= 2
roo
e-~
dx.
t-too
Jo Jo
(3.37)
Recall from (3.33),
that
J~oo
e-~dx
=
v'21r.
Since
e-~
is
an
even function,
we
obtain
from
Lemma
0.1
that
e-
T
dx = -
e-
T
dx
=
-.y'2;-
1
00
x2
1 1
00
x
2
1
-_
~7r2.
o 2
-00
2
(3.38)
From
(3.37)
and
(3.38),
it
follows
that
1 1
00
x
2
1 1
00
x
2
--
x
2
e-
T
dx
=
f2='1T".
2 0
e-
T
dx
v'21r
-00
V .£,71
1,
and
therefore (3.34) holds
true.
Lemma
3.10.
The standard
normal
variable has mean 0 and variance
I,
i.e.,
E[Z]
var(Z)
o·
,
1.
(3.39)
(3.40)
x
2
Proof.
By
definition (3.9),
and
using
the
fact
that
xe-
T
is
an
odd
function
and
Lemma
0.2, we
obtain
that
1
00
1 1
00
x
2
E[Z] =
xf(x)
dx
=
rc:L
xe-
T
dx
=
o.
-00
v27r
-00
To
compute
the
variance
of
Z,
note
that
where
the
last
equality follows from (3.34).
o
Lemma
3.11.
If
Z is the standard
normal
variable, then E[Z2] =
1.
Proof.
From
Lemma
3.5,
it
follows
that
var(Z)
= E[Z2] - (E[Z])2. Since
var(Z)
= 1
and
E[Z] = 0, see (3.39-3.40), we conclude
that
E[Z2] = 1. 0
Definition
3.5.
Denote
by
N(t)
the cumulative distribution
of
the standard
normal
variable
Z.
Then,
1
1t
x
2
N(t)
=
P(Z
S
t)
=
rc:L
e-
T
dx.
v
27r
-00
(3.41 )
3.4.
NORMAL
RANDOM
VARIABLES
91
Lemma
3.12.
Let
Z
be
the standard
normal
variable. Then,
P(Z
~
a) =
P(Z
S
-a),
V a E
JR.
(3.42)
In
other words,
if
N(t)
is the cumulative distribution
of
Z,
then
1-N(a)
=
N(-a),
VaEJR.
(3.43)
Proof.
By
definition (3.6),
and
using (3.32), we find
that
1 1
00
x
2
P(Z
~
a) =
rc:L
e-
T
dx.
v27r a
We use
the
substitution
x =
-y.
Then
dx
=
-dy
and
the
limits
of
integration
change from x = a
to
y =
-a
and
from x =
00
to
Y =
-00.
Thus,
P(Z~a)
=
~
1
00
e-~
dx
=
_1_1-
00
e-(-~)2
(-dy)
v
27r
a
v'21r
-a
-
_1_1-
00
e-
f
dy
-1-1-ooa
e-
f
dy
v'21r
-a
v'21r
P(Z
S
-a),
and
(3.42) is proven.
To
obtain
(3.43) from (3.42)
note
that,
by
definition (3.41),
P(Z
~
a) =
1-
P(Z
S a) = 1
N(a)
and
P(Z
S
-a)
=
N(-a).
o
3.4
Normal
random
variables
Definition
3.6.
The random variable X is a
normal
variable
if
and only
if
x =
/--l
+
CYZ,
where Z is the standard normal variable and
/--l,
CY
E
JR.
Lemma
3.13.
Let
X =
/--l
+
cyZ
be
a normal variable. Then
E[X]
var(X)
cy(X)
(3.44)
(3.45)
(3.46)
(3.47)
92
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
Proof. Using
Lemma
3.3
and
the
fact
that
E[Z] = 0,
it
follows
that
E[X] =
E[IL
+
o-Z]
=
IL
+
o-E[Z]
=
IL·
From
(3.13),
and
using
Lemma
3.3
and
the
fact
that
E[Z2]
= 1, cf.
Lemma
3.11, we find
that
var(X)
=
E[(X
- E[X])2] =
E[(IL
+
o-Z
-1L)2] =
E[0-2Z2]
=
0-2E[Z2]
=
0-
2
.
D
Note
that
the
constant
random
variable X =
IL
is a degenerate
normal
vari-
able
with
mean
IL
and
standard
deviation
O.
Lemma
3.14.
Let X = lL+o-Z
be
a normal variable, with 0-
=1=
0, and denote
by
h( x) the probability density function
of
X.
Then,
h(x)
=
~
exp
( (x
-IL?)
10-1
21r
20-
2
'
(3.48)
Proof.
From
definition (3.6), we know
that
P(a
:'S
X
:'S
b)
= l h(y)
dy,
V
Q,
<
b,
(3.49)
Assume
that
0-
>
O.
Since X =
IL
+
0-
Z,
we find
that
P(a~X~b)
=
(
a
IL
b -
IL)
P(a~lL+o-Z~b)
= P
-o--~Z~-o--
1 1 (b-J-L)/a
(x2)
--
exp
--
dx
V27r
(a-J-L)/a
2'
(3.50)
since
vk
e-%
=
vk
exp
(_~2)
is
the
density function of
Z.
We make
the
substitution
x =
Y~J-L
in (3.50).
The
integration limits change
from
x =
a-J-L
to
y = a
and
from x =
b-J-L
to
Y =
b.
Also, dx =
dy.
Therefore,
a a a
we find from (3.50)
that
P(a~X~b)
=
.~
lb
exp
(
(y_~)2)
dy,
Va<b.
o-y
21r
a
20-
(3.51 )
From
(3.49)
and
(3.51),
it
follows
that
the
density function h(x) of X is
1
((Y-IL?)
1 (
(x-
IL
?)
h(y) =
0-V27r
exp
20-2
=
10-1V27r
exp
20-2
'
3.4.
NORMAL
RANDOM
VARIABLES
93
since
0- >
O.
If
0- < 0,
then
(3.50) becomes
P(a~X~b)=P
--2Z2--
=--
exp
--
(
a -
IL
b -
IL)
1
j,(a-J-L)/a
(x2)
0- 0-
V27r
(b-J-L)/a 2
dx.
The
substitution
x =
Y~J-L
yields
P(a~X~b)
=
1 l
a
((Y-IL?)
--
exp
dy
0-V27r
b
20-
2
-l!~
lex
p
(
(y~;)2)
dy,
and
therefore
1 (
(x-
IL
?)
10-1
V27f
exp
20-
2
'
since
10-/
=
-0-
if
0-
<
O.
o
Example:
Let
X =
IL
+
0-
Z
be
a normal variable,
and
let
Y =
21L
-
X.
Show
that
X
and
Y have
the
same
probability densities.
Answer: Note
that
Y = 21L- X =
IL-O-
Z is a
normal
variable. Let
hx(x)
and
hy(x) be
the
probability
density functions of X
and
Y, respectively. From
(3.48),
it
follows
that
hy(x) =
hx(x).
D
We
note
that
the
sum
two independent
normal
variables (and, in general,
the
sum
of
any
finite
number
of independent
normal
variables) is also a normal
variable.
This
is
not
the
case if
the
normal variables
are
not
independent· see
section 4.3 for more details. '
FINANCIAL
APPLICATIONS
The
Black-Scholes formula.
94
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
The
Greeks of plain vanilla
European
call
and
put
options.
Implied volatility.
The
concept of hedging.
~-hedging
and
r-hedging
for options.
Implementation
of
the
Black-Scholes formula.
3.5
The
Black-Scholes
formula
The
Black-Scholes formulas give
the
price of plain vanilla
European
call
and
put
options,
under
the
assumption
that
the
price of
the
underlying asset
4
~as
lognormal distribution. A detailed discussion of
the
lognormal assumptlOn
can
be
found
in
section 4.6.
To introduce
the
Black-Scholes formula,
it
is enough
to
assume
that,
for
. h d . bl
S(t2)'
1 1
any
values of
tl
and
t2
WIth
tl
< t
2
, t e
ran
om
vana
e
S(tl)
IS
ognorma
with
parameters
(J-L
- q -
~2)
(t2
-
tl)
and
a
2
(t2
- tl), i.e.,
(3.52)
where
Z is
the
standard
normal variable.
The
constants
J-L
and
a are called
the
drift
and
the
volatility of
the
price
8(t)
of
the
underlying asset
and
represent
the
expected value
and
the
standard
deviation of
the
returns
of
the
asset; q
is
the
continuous
rate
at
which
the
asset pays dividends; see section 4.6 for
more details.
In
the
Black-Scholes formulas,
the
price of a plain vanilla
European
option
depends
on
the
following parameters:
•
8,
the
spot
price of
the
underlying asset
at
time
t;
•
T,
the
maturity
of
the
option;
note
that
the
time
to
maturity
is T -
t;
•
K,
the
strike price of
the
option;
• r,
the
risk-free interest
rate,
assumed
to
be
constant over
the
life of
the
option, i.e., between t
and
T;
•
a,
the
volatility of
the
underlying asset, i.e.,
the
standard
deviation of
the
returns
of
the
asset;
• q,
the
dividend
rate
of
the
underlying asset;
the
asset is assumed
to
pay
dividends
at
a continuous
rate.
4Examples of underlying assets are stocks, futures, indeces, etc.
3.5.
THE
BLACK-SCHOLES
FORMULA
95
The Black-Scholes Formulas for European Call and
Put
Options:
Assume
that
the
price of
the
underlying asset
has
lognormal distribution
and
volatility
a,
that
the
asset pays dividends continuously
at
the
rate
q,
and
that
the
risk-free interest
rate
is
constant
and
equal
to
r. Let C
(S,
t) be
the
value
at
time
t of a call option
with
strike K
and
maturity
T,
and
let
P(S,
t)
be
the
value
at
time
t of a
put
option
with
strike K
and
maturity
T. Then,
where
0(8,
t)
P(8,
t)
8e-
q
(T-t)
N(d
1
) -
Ke-r(T-t)
N(d
2
);
Ke-r(T-t)
N(
-d
2
) -
8e-
q
(T-t)
N(
-d
1
),
In
(~
) +
(r
-q +
~)
(T
- t)
avT-t
In
(~
) +
(r
-q -
~2)
(T
- t)
d1-aVT-t
=
~
.
a
T-t
(3.53)
(3.54)
(3.55)
(3.56)
Here,
N
(z)
is
the
cumulative distribution of
the
standard
normal variable:
N(z) =
.~
jZ
e-4- dx.
V
27r
-00
For a non-dividend-paying asset, i.e., for q = 0,
the
Black-Scholes formu-
las are
where
0(8,
t)
P(8,
t)
SN(d
1
) -
Ke-r(T-t)
N(d
2
);
Ke-r(T-t)
N(
-d
2
) -
8N(
-dd,
(3.57)
(3.58)
(3.59)
(3.60)
Note
that
the
general rule of
thumb
that,
for a continuous dividend paying
asset
with
dividend
rate
q,
the
Black-Scholes formulas (3.53)
and
(3.54) can
be
obtained
from
the
formulas (3.57)
and
(3.58) for a non-dividend paying
asset by
substituting
r q for r, is wrong.
The
reason for this is somewhat
subtle: while
r represents
the
risk-neutral
drift
of
the
underlying asset
in
(3.59)
and
(3.60),
and
therefore (3.55)
and
(3.56)
can
be
obtained
from (3.59)
96
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
and
(3.60)
by
substituting
r - q for r,
the
term
e-r(T-t) from (3.57)
and
(3.58) is
the
discount factor from
time
T
to
time
t
and
does
not
change
in
the
formulas (3.53)
and
(3.54) corresponding
to
the
dividend-paying
asset.
In
general, after a result is
obtained
for a
European
call option,
the
cor-
responding
result for a
European
put
option
can
be
derived
by
using
the
Put-Call
parity. For example,
the
Black-Scholes formula (3.54) for
the
Eu-
ropean
put,
can
be
obtained
from
the
Black-Scholes formula (3.53) for
the
European
call as follows:
From
(3.53), we know
that
C(8,
t)
=
8e-
q
(T-t)
N(d
1
) -
Ke-r(T-t)
N(d
2
).
Then,
from
the
Put-Call
parity
formula (1.47), we find
that
P(8,
t)
K e-r(T-t) -
8e-
q
(T-t) +
C(8,
t)
Ke-r(T-t)
_
8e-
q
(T-t) +
(8e-
q
(T-t)
N(d
1
) -
Ke-r(T-t)
N(d
2
))
Ke-r(T-t)
(1
-
N(d
2
))
-
8e-
q
(T-t)
(1
-
N(d
1
))
Ke-r(T-t)
N(
-d
2
) -
8e-
q
(T-t)
N(
-d
1
),
since 1 -
N(a)
=
N(
-a),
for
any
a E
IR;
cf.
Lemma
3.12.
If
V(8(t),
K)
denotes
the
value
at
time
t of a plain vanilla
options
with
strike K
and
spot
price
8(t)
of
the
underlying asset,
it
is easy
to
see, from
(3.53-3.56),
that
V(a8(t),
aK)
=
aV(8(t),
K)
for
any
a >
O.
Therefore,
Thus,
the
relevant factor when pricing
an
option
with
strike K is
8-K
K
8(t)
- K
K
which represents
the
percentage
by
which
an
option
is
in-the-money
or
out-
of-the-money
at
time
t.
Example: A call
option
with
strike K = 60 is 15%
out
of
the
money.
Find
the
spot
price of
the
underlying asset.
Answer:
For
an
out-of-money
call,
the
spot
price is smaller
than
the
strike
price, i.e.,
8(t)
<
K.
The
call is 15%
out
of
the
money if
K -
8(t)
=
015
K . ,
i.e.,
if
8(t)
=
0.85K.
For K = 60, we find
that
8(t)
= 51. D
3.6.
THE
GREEKS
OF
EUROPEAN
OPTIONS
97
3.6
The
Greeks
of
European
call
and
put
options
Let V
be
the
value
of
a portfolio
of
derivative securities
on
one underlying
asset
5
•
The
rates
of
change of
the
value
of
the
portfolio
with
respect
to
various
parameters,
e.g.,
the
price
of
the
underlying asset
or
the
volatility
of
the
underlying asset,
are
important
for hedging purposes.
They
change are
called
"the
Greeks"
of
the
portfolio,
and
are
denoted
by
symbols from
the
Greek
alphabet.
The
most
widely used Greeks
are
listed below.
•
Delta
(~)
is
the
rate
of
change
of
the
value
of
the
portfolio
with
respect
to
the
spot
price 8
of
the
underlying asset:
~(V)
=
8V.
88'
(3.61)
•
Gamma
(r)
is
the
rate
of
change
of
the
Delta
of
the
portfolio
with
respect
to
the
spot
price 8
of
the
underlying asset, i.e.,
the
second
partial
derivative
of
the
value
of
the
portfolio
with
respect
to
8:
r(V)
= 8
2
V.
88
2
'
(3.62)
•
Theta
(8)
is
the
rate
of
change
of
the
value
of
the
portfolio
with
respect
to
time
t
(not
with
respect
to
maturity
T):
8V
8(V)
=
7ft;
(3.63)
•
Rho
(p)
is
the
rate
of
change
of
the
value
of
the
portfolio
with
respect
to
the
risk-free
interest
rate
r:
8V
p(V)
=
8r;
(3.64)
• Vega is
the
rate
of
change
of
the
value of
the
portfolio
with
respect
to
the
volatility a
of
the
underlying asset:
vega(V)
8V
80'
(3.65)
From
the
Black-Scholes formulas (3.53)
and
(3.54) for options
on
assets
paying continuous dividends,
the
following closed formulas for
computing
the
Greeks
of
European
plain
vanilla call
and
put
options
can
be
derived:
5The portfolio could simply be, for example,
the
underlying asset,
or
a plain vanilla
option
on
the
underlying asset.
98
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
8(C)
8(P)
~(C)
e-q(T-t)
N(d
1
);
~(P)
_e-q(T-t)
N(
-d
1
);
e-q(T-t)
1
d
2
_.-1
2 •
r(C)
--e
So-VT
- t
V2ir
,
r(p)
r(C);
1 d
2
vega(C)
Se-q(T-tLJT
- t
--e---t·
V2ir
'
vega(P)
vega(C);
Sa-e-q(T-t)
d
2
_.-1
e
2
2J2Tr(T
- t)
+ qSe-q(T-t)
N(d
1
) -
r K e-r(T-t)
N(
d
2
);
Sa-e-q(T-t)
d
2
1
e
-2
2J2Tr(T
- t)
- qSe-q(T-t)
N(
-d
1
)
+
rKe-
r
(T-t)N(-d
2
);
p(C)
p(P)
K(T
- t)e-r(T-t)
N(d
2
);
-
K(T
- t)e-r(T-t)
N(
-d
2
).
(3.66)
(3.67)
(3.68)
(3.69)
(3.70)
(3.71)
(3.72)
(3.73)
(3.74)
(3.75)
The
computations
for deriving
these
formulas are based
on
differentiating
improper
integrals
with
respect
to
the
integration limits;
cf.
section 2.3.
Note
that,
as soon as
the
formulas for
the
Greeks corresponding
to
a call
option
are derived,
the
Put-Call
parity
can
be
used
to
obtain
the
correspond-
ing Greeks for
the
put
option. For example,
to
compute
the
Delta
of a
put
option, we write
the
Put-Call
parity
formula (1.47) as
P = C - Se-q(T-t) + Ke-r(T-t).
(3.76)
By
differentiating (3.76)
with
respect
to
S,
we find
that
~(P)
=
~(C)
- e-q(T-t).
Using formula (3.66) for
~(C)
and
Lemma
3.12, we
obtain
the
formula (3.67)
for
~(P),
i.e.,
~(P)
= e-q(T-t)
N(d
1
) -
e-q(T-t) =
_e-
q
(T-t)(1_
N(d
1
))
= _e-q(T-t)
N(
-d
1
).
3.6.
THE
GREEKS
OF
EUROPEAN
OPTIONS
99
While definitions (3.61-3.65) are valid for
the
Greeks of any option
on
one underlying asset, including, e.g., American
and
exotic options, closed
formulas for
the
Greeks
can
only
be
obtained
if closed formulas for pricing
the
options exist.
This
happens
very rarely for options
that
are
not
plain
vanilla
European,
a
notable
exception being
European
barrier options; see
section 7.8.
3.6.1
Explaining
the
magic
of
Greeks
computations
It
is interesting
to
note
that
the
formulas (3.66-3.75) for
the
Greeks are
simpler
than
expected. For example,
the
Delta
of
a call
option
is defined as
~(C)
= ac
as'
Differentiating
the
Black-Scholes formula (3.53)
with
respect
to
S, we
obtain
~(C)
= e-q(T-t)
N(d
1
)
+ Se-q(T-t)
:S
(N(d
1
))
-
Ke-r(T-t)
:S
(N(d
2
))
,
(3.77)
since
both
d
1
and
d
2
are
functions of
S;
cf. (3.55)
and
(3.56).
However, we know from (3.66)
that
~(C)
= e-q(T-t)
N(d
1
).
(3.78)
To
understand
how (3.77) reduces
to
(3.78), we
apply
chain rule
and
obtain
that
Then,
using (3.79)
and
(3.80), we can write (3.77) as
~(C)
= e-q(T-t)
N(dd
+
Se-q(T-t)
N'(d
)
ad
1
_
Ke-r(T-t)
N'(d
)
ad
2
1
as
2 as'
(3.79)
(3.80)
(3.81)
Note
that
formulas (3.55)
and
(3.56) for d
1
and
d
2
can
be
written
as
In
(~)
+ (r -
q)(T
- t)
a-VT
- t
d
1
a-vr=t
+ 2 ; (3.82)
d
2
= d
1
-
o-VT
_ t = In
(~)
+ (r -
q)(T
- t)
a-vr=t
(3.83)
a-VT
- t 2
The
following result explains why (3.81) reduces
to
(3.78):
100
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
Lemma
3.15.
Let
d
1
and
d
2
be
given by {3.82} and {3.83}.
Then
Se-q(T-t)
N'(d
1
)
=
Ke-r(T-t)
N'(d
2
).
(3.84)
Proof. Recall
that
N (z) is
the
cumulative distribution of
the
standard
normal
variable, i.e.,
1 z2
From
Lemma
2.3, we find
that
N'(z) = y'2ife-
T
.
Then,
1
-dI
--e~'
V21i
'
1
-d~
--e~
V21i
.
(3.85)
(3.86)
Therefore,
in
order
to
prove (3.84),
it
is enough
to
show
that
the
following
formula holds true:
(3.87)
Recall
the
notation
exp(x) =
eX.
Since exp(ln(x)) = x,
we
find
that
Se-q(T-t)
K
e-r(T-t)
!ie(r-q)(T-t)
K
Ke-
r
(7'-t)
exp
(Ill
(:) +
(r
- q)(T -
t))
.
From
(3.82),
it
is easy
to
see
that
In
(:)
+
(r
- q)(T - t)
and
therefore
Se-q(T-t)
= K
e-r(T-t)
exp d
1
(J..jT - t -
(
.
(J2(T
2
-
t))
.
Using
the
fact
that
d
2
= d
1
-
(J-VT
- t, we
obtain
(3.87) as follows:
d
2
( d
2
(J2(T
2
-
t))
Se-q(T-t)
e--t
= K
e-r(T-t)
exp -
21
+ d
1
(J-VT
- t -
K
-r(T-t)
((d
1
-
(J~?)
e exp 2
d
2
Ke-r(T-t)
e---f.
D
3.6.
THE
GREEKS
OF
EUROPEAN
OPTIONS
We
return
our
attention
to
proving formula (3.78) for .6(C).
From (3.82)
and
(3.83),
we
find
that
101
1
(3.88)
Using (3.88)
and
Lemma
3.15, we conclude
that
formula (3.81) becomes
.6(C)
e-q(T-t)
N(d
1
)
+
Se-q(T-t)
N'(d
1
)
8d
1
_
Ke-r(T-t)
N'(d
2
) 8d
2
8S
8S
e-
q
(7't)
N(d
1
)
+
Se-
Q
(7'-t)
N'(d
,
)
(~i
-
~~)
e-q(T-t)
N(d1)'
Formula (3.78) is therefore proven.
The
simplified formulas (3.70), (3.72),
and
(3.74) for
the
vega,
8,
and
p
of a
European
call
option
6
are
obtained
similarly using
Lemma
3.15.
The formula
for
vega(
C)
:
We differentiate
the
Black-Scholes formula (3.53)
with
respect
to
(J.
Follow-
ing
the
same
steps
as
in
the
computation
for
the
Delta
of
the
call option, i.e.,
using chain rule
and
Lemma
2.3, we
obtain
that
vega(C) =
8C
=
Se-q(T-t)
N'(d
l
)
8d
1
-
Ke-r(T-t)
N'(d
2
)
88~'
8(J
8(J
v
Using
the
result
of
Lemma
3.15, we conclude
that
vega(C)
=
Se-q(T-t)
N'(d
l
)
(8d
l
_
8d
2
).
8(J 8(J
Since d
2
= d
1
-
(J..jT - t, we find
that
d
l
-
d
2
= (J..jT - t
and
thus
8d
l
_
8d
2
= ..jT _ t.
8(J 8(J
.
-4
Then, uSIng
the
fact
that
N'(d
l
)
=
vk-
e~,
see (3.85), we conclude
that
1
dI
vega( C) =
V21i
Se-q(T-t)
e-
T
..jT - t,
-----------------------
6Note
that
the
formulas (3.71), (3.73), and (3.75) for
the
vega,
8,
and p of a European
put
option
can
be
obtained from (3.70), (3.72),
and
(3.74) by using
the
Put-Call
parity.
102
CHAPTER
3.
PROBABILITY.
BLACK-SCHOLES
FORMULA.
which is
the
same as formula (3.70).
The
formula
for
8(C):
We differentiate
the
Black-Scholes formula (3.53)
with
respect
to
t.
chain rule
and
Lemma 2.3 we
obtain
that
8(C)
=
Se-q(T-t)
N'(d
1
)
a~l
+
qSe-q(T-t)
N(d
1
)
_
Ke-r(T-t)
N'(d
2
)
a~2
-
rKe-
r
(T-t)N(d
2
).
Using
the
result of
Lemma
3.15, we conclude
that
8(C) =
Se-q(T-t)
N'(d
1
)
(ad
1
_
ad
2
)
at at
+
qSe-q(T-t)
N(d
1
)
rKe-r(T-t)
N(d
2
).
Since d
1
-
d
2
=
o--VT
- t, we find
that
ad
1
ad2
---
at
at
_d
2
Since
N'(d
1
)
=
vk
e-T-,
cf.
(3.85), we conclude
that
1
~
0-
8(C)
=
---
Se-q(T-t)
e-
T
-==
V2if
2-VT - t
+
qSe-q(T-t)N(d
1
) -
rKe-
r
(T-t)N(d
2
),
which is
the
same as formula (3.72).
The
formula
for
p(
C) :
Using
We differentiate
the
Black-Scholes formula (3.53)
with
respect
to
r. Using
Chain
Rule
and
Lemma
2.3 we
obtain
that
p(C) =
Se-q(T-t)N'(d1)ad
1
-
Ke-
r
(T-t)N'(d
2
)ad
2
+K(T
t)e-
r
(T-t)N(d
2
).
ar ar
Using
Lemma
3.15, we find
that
p(C) =
Se-q(T-t)
N'(d
1
)
(ad
1
_
ad
2
)
+
K(T
-
t)e-r(T-t)
N(d
2
).
ar
ar
Since d
1
-
d
2
=
o--VT
- t, we find
that
ad
1
ad
2
ar
ar'
and
therefore,
p(C) =
K(T
-
t)e-r(T-t)
N(d
2
),
which is
the
same as formula (3.74).
3.6.
THE
GREEKS
OF
EUROPEAN
OPTIONS
103
3.6.2
Implied
volatility
The
only
parameter
needed in
the
Black-Scholes formulas (3.53-3.56)
that
is
not
directly observable
in
the
markets is
the
volatility
0-
of
the
underlying
asset.
The
risk free
rate
r
and
the
continuous dividend yield q of
the
asset
can
be
estimated
from
market
data;
the
maturity
date
T
and
the
strike K
of
the
option, as well as
the
spot
price S
of
the
underlying asset are known
when a price for
the
option
is quoted.
The
implied volatility
o-imp
is
the
value of
the
volatility
parameter
0-
that
makes
the
Black-Scholes value of
the
option
equal
to
the
traded
price of
the
option. To formalize this, denote by CBS(S,
K,
T,
0-,
r,
q)
the
Black-Scholes
value of a call
option
with
strike K
and
maturity
T
on
an
underlying asset
with
spot
price S paying dividends continuously
at
the
rate
q, if interest
rates
are
constant
and
equal
to
r.
If
C is
the
quoted
values of a call
with
parameters
S,
K,
T,
0-,
r,
and
q,
the
implied volatility
o-imp
corresponding
to
the
price C is, by definition,
the
solution
to
CBS(S,
K,
T,
o-imp,
r,
q)
= C.
(3.89)
The
implied volatility
can
also be derived from
the
given price P of a
put
option,
by
solving
PBS(S,
K,
T,
o-imp,
r,
q)
=
P,
(3.90)
where PBS(S,
K,
T,
0-,
r,
q)
is
the
Black-Scholes value
of
a
put
option.
Note
that,
as functions
of
volatility,
the
Black-Scholes values of
both
call
and
put
options
are
strictly
increasing since
1
dt
Se-
qT
VT
--e-
T
>
O'
V2if
'
(3.91)
cf.
(3.70)
and
(3.71).
(Throughout
this section,
to
keep
notation
simple, we
assume
that
the
present
time
is t = 0.)
Therefore, if a solution
o-imp
for (3.89) exists,
it
will
be
unique,
and
the
implied volatility will
be
well defined. Similarly,
equation
(3.90) has
at
most
one solution.
For
the
implied volatility
to
exist
and
be
nonnegative,
the
given value C
of
the
call
option
must
be
arbitrage-free, i.e.,
(3.92)
The
bounds
for
the
call
option
price from (3.92)
can
be
obtained
by using
the
Law
of
One
Price; cf.
Theorem
1.10.