Stefanica D. A Primer for the Mathematics of Financial Engineering
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124
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
ProoJ.
The
density
functions ft(x)
and
h(x)
of
Xl
and
X
2
,
respectively,
are
ft(x)
-I
0--1-1
~-2-1f
cxp (
(4.21 )
h(x)
1(J21~
exp
(
( 4.22)
cf.
Lemma
3.14.
Since
Xl
and
X
2
are
independent,
it
follows from
Theorem
4.1
that
the
probability
density
function
J(z)
of
Xl
+ X
2
is
the
convolution
of
the
prob-
ability
densities
of
Xl
and
X
2
,
i.e.,
as
J(z) =
By
completing
the
square, we
can
write
the
term
(x
-
(ttlo-~
+
(z
- tt2)0-f))
2
2°-I
o-~
1
(o-I
+
o-D
(z
- (ttl +
tt2)
)2
2(o-I
+
o-~)
dx.
Therefore,
the
density
function
J(z)
of
Xl
+ X
2
can
be
written
as
J(z) =
dx.
For
the
last
integral, we
use
the
substitution
Y =
x -
(ttlo-~
+ (z -
tt2)0-i)
10-10-211
J
o-I
+
o-~
The
function
under
the
integration
sign
becomes
exp
(
-V;).
Since
4.3.
INDEPENDENT
RANDOM
VARIABLES
125
and
using
the
fact
that
1:
exp ( -
~2)
dy
,rz;,
see (3.33), we
conclude
that
1
00
00
exp
(_
(x
- (ttl
o-~
+
(z
tt2)0-i)
)
2)
dx
2°-Io-~
1
(o-I
+
o-~)
Therefore,
J(
)
-
1 (
(z
- (ttl + tt
2
))2)
z -
exp
(2
2)
.
v/27r
(O-I
+
o-~)
2 0-
1
+ 0-
2
From
Lemma
3.14, we conclude
that
Xl
+ X
2
is a
normal
variable
with
mean
ttl +
tt2
and
variance
o-I
+
o-~.
0
If two
normal
variables
are
not
independent,
then
their
sum
need
not
be
a
normal
variable.
Given
our
definition (3.44),
this
requires
a
brief
clarification.
If
Xl
is a
normal
variable
of
mean
j..tl
and
variance
o-I,
then
( 4.23)
If
X
2
is
another
normal
variable of
mean
j..t2
and
variance
o-~,
then
X
2
can
also
be
written
as
the
sum
of
tt2
and
0-2
times
the
standard
normal
variable,
but
this
new
standard
normal
variable,
although
with
the
same
probability
distribution
as Z
from
(4.23), is a different
standard
normal.
To avoid con-
fusions, we
denote
this
new
standard
normal
by
Z2
and
the
standard
normal
variable Z
from
(4.23)
by
Zl.
Then,
Xl
=
j..tl
+ o-lZl;
X
2
=
j..t2
+
0-2
Z
2.
Therefore,
it
is
not
true
that
Xl
+X2
=
j..tl
+j..t2
+ (0-1 +0-2)Z, which would
have
meant
that
Xl
+ X
2
is
normal.
As a
matter
of
fact, from
Theorem
4.2
we know
that,
if
Xl
and
X
2
are
independent,
then
Xl
+ X 2 = ttl +
j..t2
+ J
o-I
+
o-~
Z,
126
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
since
Xl
+ X
2
is a
normal
variable
of
mean
""1
+
""2
and
variance
CTt
+
CT~.
The
sum
of two lognormal variables,
independent
or
not,
is
not
lognormal.
However,
the
product
of
two
independent
lognormal variables is lognormal.
Theorem
4.3.
Let
Yi
and
Y2
be
independent lognormal random variables
with parameters
""1
and
CTI
and
""2
and
CT2J
respectively. Then In(Yi
Y2)
'is
a
lognormal variable with parameters
""1
+
""2
and V
ut
+
CT~.
Proof.
Note
that
In(Yi) =
Xl
=
""1
+
UI
Z
I
and
In(Y2)
= X
2
=
""2
+
U2Z2,
where
ZI
and
Z2
are two
independent
standard
normal
variables. Since
Yi
and
Y2
are independent, we find, from
Lemma
4.8,
that
the
corresponding
normal
variables
Xl
and
X
2
are
also independent. From
Theorem
4.2,
it
follows
that
Xl
+ X
2
is
normal
with
mean
""1
+
""2
and
variance
ut
+
CT~,
i.e.,
Therefore,
which shows
that
In(Yi
Y2)
is a lognormal
random
variable
with
parameters
mean
""1
+
""2
and
vut
+
u~.
0
4.4
Approximating
sums
of
lognormal
variables
The
product
of two
independent
lognormal variables is lognormal variable,
while
their
sum
is not. However,
approximating
the
sum
of
two lognormal
variables
by
a lognormal variable is useful
in
practice, e.g., for modeling
derivative securities involving two
or
more
assets.
For example,
the
payoff
of
a
basket
option
depends
on
the
sum
of
the
prices
of
several assets
1
.
One
way
to
price a
basket
option
is
to
approximate
the
value of
the
underlying asset of
the
option, i.e., of
the
sum
of
the
prices
of
the
assets,
with
a lognormal variable,
and
use
the
Black-Scholes framework.
The
parameters
of
the
lognormal variable
are
chosen such
that
the
first two
moments
of
the
sum
of
the
asset prices will
match
the
expected value
and
the
variance
of
the
lognormal variable.
underlying asset of a basket
option
is, in general, a weighted average of
the
prices
of
several assets,
and
not
necessarily
their
sum.
4.4.
APPROXIMATING
SUMS
OF
LOGNORMAL
VARIABLES
127
Assume
that
the
prices
Yi
and
Y2
of
the
two assets
are
lognormal
random
variables
with
parameters
""1
and
CTt
and
""2
and
u~,
respectively. We are
looking for a lognormal variable Y such
that
Yi+Y2
~
y
To
make
this
precise,
let
In(Y) =
""
+ u
Z.
We look for
""
and
u such
that
the
expected value
and
the
variance
of
Y
and
of
Yi
+
Y2
are
equal, i.e.,
E[Y]
var(Y)
E[Yi +
Y2];
var(Yi +
Y2).
From
Lemma
4.3,
it
is easy
to
see
that
E[Y]
E[Yi]
From
Lemma
3.3
and
Lemma
3.7, we find
that
0"2 0"2
E[Yi] +
E[Y2]
=
e/l
1
+-+
+
e/l
2
+-f;
i =
1:
2.
E[Yi +
Y2]
var(Yi +
Y2)
var(Yi) +
2P1,2
vvar(Yi)
var(Y2) + var(Y2),
where
P1,2
=
corr(X,
Y)
is
the
correlation between
Yi
and
Y2.
We
can
now
write
(4.24)
and
(4.25) as
O"I
O"~
e/l1+T
+
e/l
2
+
T
;
e2/l1
+ar
(ear -
1)
+
e2/l2+a~
(ea~
- 1 )
+
2P1,2!
e
2
/l
1
+2/l2+ar+a~
(ear
-
1)
(ea~
-
1).
From (4.26), we find
that
(4.24)
( 4.25)
( 4.26)
(4.27)
and, by
substituting
in
(4.27), we
obtain
an
equation
depending
only on
CT:
128
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
This
equation
can
be
written
in
a simpler form as
(e""4
+ e",+4) 2
eU'
=
e2",+"'1
+
e2p,+2ul
+2e",+",,'1;',
(1
+ Pl,2V
(cull)
(cull)
)
and
solving for a we
obtain
(4.28)
where
e
2
f.Ll
+
2a
r +
e2f.L2+2a~
+2e"""'+
,1;,1
(1
+ Pl,2V
(eul
-
1)
(
e
u
l -
1)
) .
The
value
of
IL
can
be
obtained
from (4.26)
and
(4.28) as follows:
4.5
Power
series
In
this
section, we discuss
the
convergence
and
smoothness
properties
of
power series. We
note
that
Taylor series expansions
are
special cases
of
power series; see section 5.3.
Definition
4.4.
The power series
00
S(x)
= L ak(x - a)k
( 4.29)
k=O
is defined
as
n
S(x)
= lim L ak(x - a)k
n-too
(4.30)
k=O
for all the values x E
1R
such that the limit limn-too
L::~=o
ak( x a)k exists
and is finite.
4.5.
POWER
SERIES
129
Example: A simple
example
of
power series is
the
geometric series
00
n
~xk
= lim
Lxk,
~
n-too
(4.31)
k=O
k=O
which is convergent
if
and
only
if
Ixl
<
1.
Answer:
Note
that
the
geometric series
can
be
written
in
the
general form
(4.30)
with
a = 0
and
ak
= 1, V k 2
o.
Recall from (21)
that,
for
any
xi-I,
L
n
k
1-
xn+1
X -
I-x
k=O
Therefore,
if
Ixl
< 1,
the
geometric series (4.31) is convergent
and
00
n 1
n+1
~
xk = lim
~
xk = lim - x
~
n-too
~
n-too
1 - X
k=O
k=O
1
I-x'
( 4.32)
The
geometric series is
not
convergent
if
I x I 2
1.
If
I x I > 1,
then
1 x
n
+
11
---7
00
when n
---7
00,
and
the
partial
sums
L::~=o
xk
do
not
converge;
d.
(4.32).
If
x = 1,
the
partial
sum
L::~=o
xk is
equal
to
n,
and
therefore does
not
converge
as
n
---7
00
as required
in
(4.31).
If
x =
-1,
the
partial
sum
L::~=o(
-1)k
is
equal
to
1, if n is even,
and
to
0, if n is
odd.
Therefore,
the
limit limn-too
L::~=o
xk does
not
exist. 0
To
present
convergence results for
arbitrary
power series, we first define
the
radius
of
convergence
of
a series. Recall
that,
if
(Xk)k=l:oo
is a sequence of
real numbers,
then
lim
sUPk-too
Xk
is
the
highest value l for which
there
exists
a subsequence
of
(Xk)k=l:oo
that
converges
to
l.
Definition
4.5.
The radius
of
convergence R
of
the power series
S(x)
=
L:::o
ak(x - a)k is defined
as
If
limk-too
lakl
1
/
k
exists,
then
limsuPk-too
lakl
1
/
k
following
result
follows directly from definition (4.33):
( 4.33)
Lemma
4.9.
Let
S(x)
=
L:::oak(x
- a)k
be
a power series and let R
be
the
radius
of
convergence
of
S(x).
Iflimk-too
lakl
1
/
k
exists, then
(4.34)
130
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
Theorem
4.4.
Let
S (x) =
L:~o
ak
(x
a)k
be
a power series and let R
be
the radius
of
convergence
of
S
(x).
Then,
•
if
R = 0, the series
S(x)
is only convergent
at
the
point
x = a;
•
if
R =
00,
the series
S(x)
is (absolutely) convergent
for
all x E
lR;
•
if
0 < R <
00,
then
the series
S(x)
is (absolutely) convergent
if
Ix
- al < R
and
is
not
convergent
if
Ix
-al
>
R;
if
Ix
-al
=
R,
i.e.,
if
either
x-a
=
-R,
or
x - a =
R,
the series
S(x)
could
be
convergent
or
divergent
at
the
point
x.
Example:
Find
the
radius
of convergence
of
00
k
L~2'
k=1
A
.
It
. t
th
~oo
xk
-
~oo
k'
h - 1 k 1
nswer.
IS
easy 0 see
at
6k=1
k2
-
6k=1
ak
x
,WIt
ak
-
k2,
~
.
Note
that
(
)
1/k
. l/k
_.
~
_.
_1__
hm
lakl
-
hm
k
2
-
hm
(
l/k)2
- 1,
k--->oo k--->oo k--->oo
k
since
limk--->oo
k
l
/k
=
1;
cf.
(1.29).
From
(4.34),
it
follows
that
the
radius
of
convergence of
the
series is R =
1.
We
conclude from
Theorem
4.4
that
the
series is convergent if
Ixl
< 1,
and
not
convergent if
Ixl
>
1.
If
x =
-1,
the
series becomes
L:~1
(~;)k.
Since
the
terms
(~;)k
have
alternating
signs
and
decrease
in
absolute
value
to
0,
the
series is convergent.
If
x = 1,
the
series becomes
L:~1
-b,
which is convergent
2
.
D
The
smoothness
properties
of
power series
are
presented
in
the
following
theorem:
Theorem
4.5.
Let
S(x)
=
L:~o
ak(x
- a)k
be
a power series and let R > 0
be
the radius
of
convergence
of
S
(x).
(i) The
function
S(x)
is differentiable on the interval
(a-R,
a+R),
i.e,
for
Ix
- al <
R,
and
S'(x)
is given by
00
S'(x)
= L kak(X -
a)k-l,
V x E (a -
R,
a +
R).
k=1
(ii) The
function
S(x)
=
L:~o
ak(x
- a)k is infinitely
many
times
differen-
tiable
on
the interval (a -
R,
a +
R),
and
the coefficients
ak
are given by
S(k)(a)
ak
=
k!
' V k
~
O.
2It
can
be
shown
that
I:r=l
-b
=
~.
4.5.
POWER
SERIES
131
4.5.1
Stirling's
formula
Let
k!
= 1
·2·
....
(k
- 1) . k
be
the
factorial
of
the
positive integer k.
Note
that
k!
is a
large
number
even for k small; for example,
10!
=
3,628,800.
Stirling's
formula
gives
the
order
of
magnitude
for
k!
when
k is large, i.e.,
k!
'"
G r
V21rk.
(4.35)
More rigorously,
Stirling's
formula says
that
1
.
k!
Im----
k--->oo
(~)k
V27rk
1.
(4.36)
In
section
5.3, we will use
the
following weaker version of (4.36)
to
study
the
convergence
of
Taylor series expansions:
. (k!)I/k
1
hm--
=-
k--->oo
k
e'
(4.37)
To prove (4.37),
note
that,
if (4.36) holds
true,
then
r (
k!
)
l/k
= 1
k~~
(~)k
V27rk
and
therefore
. (k!)l/k . (k!)I/k
1 =
hm
= e
hm
--,
k--->oo~
(V27rk)1/k
k--->oo
k
( 4.38)
since we
can
use (1.29)
and
(1.30)
to
show
that
lim
(V27rk )1/k =
(lim
(yI2;)l/k)
.
(lim
Vkl/k)
=
l.
k--->oo k--->oo
k--->oo
The
limit (4.38) is equivalent
to
(4.37), which is
what
we
wanted
to
show.
FINANCIAL
APPLICATIONS
A lognormal
model
for
the
evolution
of
asset prices.
The
risk-neutral
derivation
of
the
Black-Scholes formula.
Financial
interpretation
of
N(d
l
)
and
N(d
2
)
from
the
Black-Scholes formula.
The
probability
that
a
plain
vanilla
European
option
expires in
the
money.
132
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
4.6
A
lognormal
model
for
the
evolution
of
asset
prices
To price derivative securities, we need
to
make assumptions
on
the
evolution
of
the
price of
the
underlying assets.
The
most
elementary
assumption
that
does
not
contradict
market
data
is
that
the
price of
the
asset
has
lognormal
distribution; see (3.52).
The
fact
that,
under
the
lognormal assumption,
closed formulas for pricing
certain
options
can
be
derived, e.g.,
the
Black-
Scholes formulas from section 3.5, is
most
welcome
and
explains
the
wide use
of
this
model.
To
justify
the
lognormal assumption,
note
that
the
rate
of
return
of
an
asset between times t
and
t + Jt, i.e., over
an
infinitesimal
time
period
Jt, is
S(t
+ Jt) -
S(t)
S(t)
Jt
(4.39)
If
the
rate
of
return
(4.39) were
constant,
i.e., if
there
existed a
parameter
fJ,
such
that
S(t
+ Jt) -
S(t)
Jt
S(t)
=
fJ"
( 4.40)
then
the
price S ( t) of
the
underlying asset would
be
an
exponential function:
When
Jt
-t
0, formula (4.40) becomes
the
ODE
dS
dt
=
fJ,S,
V 0 < t < T,
with
initial condition
at
t = 0 given
by
S(O),
the
price of
the
asset
at
time
O.
The
solution of
the
ODE
is
S(t)
=
S(O)
e
fLt
.
Since
market
data
shows
that
asset prices do
not
grow exponentially,
the
constant
rate
of
return
assumption
cannot
be
correct.
A more realistic model is
to
assume
that
the
rate
of
return
(4.39)
has
a
random
oscillation
around
its
mean.
This
oscillation is assumed
to
be
normally distributed,
with
mean
0
and
standard
deviation
on
the
order
of
Ju.
(Any order of
magnitude
other
than
~
would render
the
asset price
either
infinitely large or infinitely small.)
In
other
words, we consider
the
following model for S ( t) :
S(t
+ Jt) -
S(t)
S(t)
Jt
which is equivalent
to
S(t
+ Jt) -
S(t)
1
fJ,
+
(J.
~Z,
yJt
( 4.41)
4.7.
RISK-NEUTRAL
DERIVATION
OF
BLACK-SCHOLES
133
The
parameters
fJ,
and
(J
are called
the
drift
and
the
volatility of
the
under-
lying asset, respectively,
and
represent
the
expected value
and
the
standard
deviation
of
the
returns
of
the
asset. Based
on
(4.41), we require S(t)
to
satisfy
the
following stochastic differential
equation
(SDE):
dS =
fJ,Sdt
+ (JSdX,
(4.42)
with
initial condition
at
t = 0 given by
S(O);
here,
X(t),
t
~
0, is a Weiner
process
X(t).
If
the
asset pays dividends continuously
at
rate
q,
the
SDE
(4.42) becomes
dS =
(fJ,
q)Sdt + (JSdX.
( 4.43)
We will
not
go
into
any
further details
on
SDEs. For
our
purposes we
only need
the
fact
that
S(t)
is
the
solution of
the
SDE
(4.43) for t
~
0 if
and
only if S ( t) satisfies
the
following lognormal model:
In
(~i~:D
(I-'
-q -
~2)
(t2-tJ) + lTVt2 -
tJZ,
If
0
<;
t1
< t2,
(4.44)
where Z is
the
standard
normal variable.
4.
7
Risk-neutral
derivation
of
the
Black-Scholes
formula
Risk-neutral
pricing refers
to
pricing derivative securities as discounted ex-
pected
values
of
their
payoffs
at
maturity,
under
the
assumption
that
the
underlying asset
has
lognormal distribution
with
drift
fJ,
equal
to
the
con-
stant
risk-free
rate
r.
In
other
words,
(4.45)
where
the
expected
value
in
(4.45) is
computed
with
respect
to
S(T)
given
by (4.44) for
fJ,
= r, tl = 0
and
t2
=
T,
i.e.,
In
Gi~j)
=
(r
-q
~2)
T +
lTVTZ,
( 4.46)
which is equivalent
to
S(T)
=
S(O)
e(r-q-~)T
+ uv'Tz.
( 4.47)
While
risk-neutral
pricing does
not
hold for
path-dependent
options or
American options,
it
can
be
used
to
price plain vanilla
European
Call
and
Put
options,
and
is one way
to
derive
the
Black-Scholes formulas.
134
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
Applying (4.45)
to
plain vanilla
European
options, we find
that
C(O) =
e-
rT
ERN[max(8(T)
- K,O)];
P(O)
=
e-
rT
ERN[max(K
-
8(T),
0)],
( 4.48)
( 4.49)
where
the
expected
value is
computed
with
respect
to
8(T)
given
by
(4.47).
We derive
the
Black-Scholes formula for call options using (4.48).
The
Black-Scholes formula for
put
options
can
be
obtained
similarly from (4.49).
By
definition,
max(8(T)
_
K,O)
_
{8(T)
-
K,
if
8(T)
2
K;
- 0, otherwise.
From
(4.47),
it
follows
that
8(T)
2 K
~
where we used
the
notation
In(~)+(r-q-~)
T
d
2
=
~vfr
(4.50)
Note
that
the
term
d
2
from (4.50) is
the
same
as
the
term
d
2
from
the
Black-
Scholes formula given by (3.56), if t =
o.
Thus,
(8(T)
- K
0)
= 8(0 e 2 - 1 _ - 2;
{
)
(
r-q-s2)T+uVTZ
K·f
Z > d
max,
0 if Z < - d
2
.
(4.51)
Recall
that
the
probability density
function
of
the
standard
normal
vari-
able Z is
-l-e-f.
The
pricing formula (4.48)
can
be
regarded as
an
expec-
J2ir
tation
over
Z;
cf. (4.51).
Then,
from
Lemma
3.4,
it
follows
that
0(0)
=
e-
rT
_1_1
00
(8(0)
e(r-q-4-)T
+
uVTx
-
K)
e-
f
dx
..j2;-
-d
2
------'--_e=___
e
r-q-T
T+uVTX-T
dx
_ _
e_
e-Tdx
8(
0)
-rT
1
00
(2)
2 K
-rT
1
00
2
..j2;-
-d2
..j2;-
-d
2
8(0)e-
qT
1
00
e-4-T+uVTx-fdx
_
Ke-
rT
1
00
e-fdx
..j2;-
-d
2
..j2;-
-d
2
8(0)e-
qT
1
00
e-Hx-uVT)2
dx
_
Ke-
rT
1
00
e-fdx.
(4.52)
..j2;-
-d2
..j2;-
-d
2
4.8.
PROBABILITY
THAT
OPTIONS
EXPIRE
IN-THE-MONEY
135
We
make
the
substitution
y = x -
~vfr
in
the
first integral from (4.52).
The
integration
limits change from x =
00
to
Y =
00
and
from x =
-d
2
to
Y =
-d
2
-
~vfr
=
-d
1
,
where we used
the
notation
In
(
S~)
) +
(r
_ q +
~2)
T
d
1
= d
2
+
~VT
=
vfr
. (4.53)
~
T
Note
that
the
term
d
1
from (4.53) is
the
same
as
the
term
d
1
from
the
Black-
Scholes formula given
by
(3.55)
if
t =
o.
Formula
(4.52) becomes
1 1
00
y2
1 1
00
x
2
0(0)
=
8(0)e-
qT
.
rn=
e-
T
dy
-
Ke-
rT
rn=
e-
T
dx.
(4.54)
v
27r
-d
1
V
27r
-d
2
Let
N(t)
=
P(Z
:s;
t)
be
the
cumulative
distribution
of
Z.
Recall from
Lemma
3.12
that
1 -
N(
-a)
=
N(a),
for
any
a.
Then,
1 1
00
x
2
-
e-
T
dx
=
P(Z
2
-a)
= 1 -
P(Z
:s;
-a)
V2ir
-a
= 1 -
N(
-a)
=
N(a),
'v'
a E
JR.
(4.55)
From
(4.54)
and
using (4.55) for
both
a = d
1
and
a = d
2
,
we conclude
that
0(0)
=
8(0)e-
qT
N(d
1
) -
Ke-
rT
N(d
2
).
This
is
exactly
the
Black-Scholes formula (3.53)
obtained
by
setting
t =
O.
4.8
Computing
the
probability
that
a
European
option
expires
in
the
money
Consider a
plain
vanilla
European
call
option
at
time
0
on
an
underlying asset
with
spot
price
having
a lognormal
distribution
with
drift
f.L
and
volatility
~,
and
paying dividends continuously
at
rate
q,
i.e.,
where Z is
the
standard
normal
variable.
Let
K
and
T
be
the
strike
and
maturity
of
the
option,
respectively,
and
assume
that
the
risk-free interest
rates
are
constant
and
equal
to
r until
the
maturity
of
the
option
.
The
probability
that
the
call
option
will expire
in
the
money
and
will
be
exercised is
equal
to
the
probability
that
the
spot
price
at
maturity
is higher
than
the
strike price, i.e.,
to
P(8(T)
>
K)
.
136
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
From
(4.56) for
tl
= 0
and
t2
=
T,
we find
that
In
(~i~?)
=
(I'
-q -
~2)
T +
(JVrz.
(4.57)
Using
the
lognormal assumption (4.57), we
obtain
that
(
8(T)
K)
((8(T))
( K
))
P(8(T)
>
K)
= P 8(0) > 8(0) = P
In
8(0) > In 8(0)
p((I'-q-
~2)T+(JVrZ
>
In(s~)))
(
In
(sfu)
-
(/L
- q -
~)
T)
P Z >
1m
avT
P(Z
>
a),
where
In
(
sfor
) -
(/L
- q -
~)
T
a =
aVT
.
Let
N(t)
=
P(Z
::;
t)
be
the
cumulative
distribution
(3.41) of
Z.
Note
that
P(Z
=
a)
= 0 for any a E
JR.
Then,
from
Lemma
3.12,
it
follows
that
P(Z
>
a)
=
P(Z
<
-a)
and
therefore
P(8(T)
>
K)
=
P(Z>
a)
=
P(Z
<
-a)
=
N(
-a).
( 4.58)
Let
d
2
,/L
=
-a,
i.e.,
In (
~)
+
(/L
-
~)
T
d
2
,/L
=
aVT
( 4.59)
From
(4.58)
and
(4.59), we conclude
that
the
probability
that
a call
option
expires
in
the
money
is
(4.60)
From
a practical
standpoint,
in
order
to
compute
the
probability
that
the
call
option
will expire in
the
money,
the
cumulative distribution N (t)
of
the
standard
normal variable
must
be
estimated
numerically.
This
can
be
done
by
using, e.g.,
the
pseudocode from
Table
3.1.
4.9.
FINANCIAL
INTERPRETATION
OF
N(Dl)
AND
N(D2)
137
The
term
d
2
,/L
is very similar
to
the
term
d
2
from
the
Black-Scholes for-
mula.
If
we let t = 0
in
(3.56), we
obtain
that
In(~)+(r-~)T
d
2
=
aVT
.
If
the
drift of
the
underlying asset is equal
to
the
constant
risk-free rate, i.e.,
if
/L
= r,
then
d
2
,/L
= d
2
and
the
probability
that
the
call
option
expires
in
the
money is
N(d
2
).
A
further
interpretation
of
this
fact will be given
in
section 4.9.
The
probability
that
a plain vanilla
European
put
option
expires in
the
money
can
be
obtained
numerically from
the
formula (4.60) corresponding
to
call options. A
put
option
with
strike K
and
maturity
T expires in
the
money if
8(T)
<
K.
From (4.60)
and
Lemma
3.12, we find
that
P(8(T)
<
K)
= 1 -
P(8(T)
>
K)
= 1 -
N(d
2
,/L)
=
N(
-d
2
,/L)
, (4.61)
where d
2
,/L
is given
by
(4.59).
4.9
Financial
Interpretation
of
N(d
1
)
and
N(d
2
)
from
the
Black-Scholes
formula
The
Black-Scholes formula for a call
option
is
C =
8(0)e-
q
(T-t)N(d
1
) -
Ke-
r
(T-t)N(d
2
) ,
and
the
Delta
of
the
call is equal
to
~(C)
=
e-q(T-t)N(d
1
);
cf.
(3.53)
and
(3.66). For q = 0, we
obtain
that
In
other
words,
the
term
N(d
1
)
from
the
Black-Scholes formula for a call
option
is
the
Delta
of
the
call if
the
underlying asset does
not
pay
dividends.
Note
that
this
interpretation
for
N(d
1
)
is incorrect if
the
underlying asset
pays dividends continuously (since in
that
case
~(C)
=
e-qTN(d
1
)).
The
term
N(d
2
)
from
the
Black-Scholes formula represents
the
risk-
neutral
probability
that
the
call
option
expires in
the
money. Recall from
section 4.8
that
the
probability
that
a call
option
on
an
asset following a log-
normal process expires
in
the
money is
N(d
2
,/L),
where d
2
,/L
is given by (4.59).
As
noted
before,
if
the
underlying asset has
the
same
drift as
the
risk-free
interest
rate
r, i.e., if
/L
= r,
then
d
2
,/L
= d
2
.
Therefore,
the
risk-neutral
138
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
probability
that
the
call
option
expires
in
the
money is
N(d
2
).
We
note
that
this
financial
interpretation
for N (d
2
)
also holds if
the
underlying asset pays
dividends continuously.
The
Black-Scholes formula (3.54) for a
put
option
states
that
(4.62)
Recall from (3.67)
that
./}.(P)
=
_e-q(T-t)
N(
-d
l
).
If
the
underlying asset
pays no dividends, i.e., if
q = 0,
the
term
-
N(
-d
l
)
is
the
Delta
of
the
put
option.
The
negative sign represents
the
fact
that
the
Delta
of a long
put
position is negative; see section 3.7 for more details .
.
The
term
N(
-d
2
)
from (4.62) is
the
risk-neutral
probability
that
the
put
option
expires in
the
money. A simple way
of
seeing this is
to
realize
that
a
put
expires
in
the
money if
and
only if
the
corresponding call
with
the
same
strike
and
maturity
expire
out
of
the
money. Since
the
risk-neutral
probability
that
the
call
option
expires
in
the
money is
N(d
2
),
we find, using
(3.43),
that
the
risk-neutral
probability
that
the
put
option
expires
in
the
money is 1 -
N(d
2
)
=
N(
-d
2
).
4.10
References
A brief
treatment
of
the
lognormal model complete
with
financial insights
can
be
found in
Wilmott
et
al.
[35].
The
same
book
contains
an
intuitive
explanation
of
risk-neutral
pricing, as well as
the
financial
interpretation
of
the
terms
from
the
Black-Scholes formula.
4.11.
EXERCISES
139
4.11
Exercises
1.
Let
Xl
= Z
and
X
2
=
-Z
be
two
independent
random
variables, where
Z is
the
standard
normal variable. Show
that
Xl
+ X
2
is a normal
variable
of
mean
°
and
variance 2, i.e.,
Xl
+ X
2
=
v'2Z.
2.
Assume
that
the
normal
random
variables
Xl,
X
2
,
.•.
,
Xn
of
mean
f.L
and
variance
(J2
are uncorrelated, i.e,
COV(Xi'
Xj)
= 0, for all 1
:::;
i
=F
j
:::;
n.
(This
happens, e.g., if
Xl,
X
2
,
...
,
Xn
are
independent.)
If
Sn
=
~
L.:~=l
Xi
is
the
average of
the
variables
Xi,
i = 1 : n, show
that
(J2
E[Sn]
=
f.L
and
var(Sn) = -
n
3. Assume we have a one period binomial model for
the
evolution of
the
price of
an
underlying asset between
time
t
and
time
t + Jt:
If
S ( t) is
the
price of
the
asset
at
time
t,
then
the
price S
(t
+ Jt) of
the
asset
at
time
t +
<St
will
be
either S(t)u,
with
(risk-neutral)
probability
p,
or
S(t)d,
with
probability 1 -
p.
Assume
that
u > 1
and
d <
1.
Show
that
ERN[S(t +
<st)]
ERN[S2(t
+
<st)]
(pu
+
(1
- p)d) S(t);
(pu
2
+
(1
P
)d
2
) S2(t).
(4.63)
(4.64)
4.
If
the
price S(t) of a non-dividend paying asset
has
lognormal distri-
bution
with
drift
f.L
= r
and
volatility
(J,
then,
Show
that
(
S(t
+
<St))
(
(J2)
In
S(t) = r - 2
<St
+ (Jv6tz.
ERN[S(t
+
<st)]
ERN[S2(t
+
Jt)]
e
rI5t
S(t);
e(2r+a
2
)15t
S2
(t).
( 4.65)
( 4.66)
5.
The
results
of
the
previous two exercises
can
be
used
to
calibrate a bino-
mial
tree
model
to
a lognormally
distributed
process.
This
means find-
ing
the
up
and
down factors u
and
d,
and
the
risk-neutral
probability p
(of going up) such
that
the
values of
ERN
[S(t +
<st)]
and
ERN[S2(t+Jt)]
I
140
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
given
by
(4.63)
and
(4.64) coincide
with
the
values (4.65)
and
(4.66) for
the
lognormal model.
In
other
words, we
are
looking for
u,
d,
and
p such
that
pu
+
(1
p)d
pu2
+
(1
_
p)d
2
(4.67)
(4.68)
Since
there
are
two
constraints
and
three
unknowns,
the
solution will
not
be
unique.
(i) Show
that
(4.67-4.68)
are
equivalent
to
e
rOt
- d
u-d
p
(e
rOt
-
d)
(u
- e
rOt
)
e
2rot
(e
a2Ot
-
1)
(4.69)
(4.70)
(ii) Derive
the
Cox-Ross-Rubinstein
parametrization
for a binomial
tree,
by
solving (4.69-4.70)
with
the
additional
condition
that
ud
=
1.
Show
that
the
solution
can
be
written
as
p
where
e
rOt
- d
u-d'
u =
A+VA2_1;
d
A-
VA2
-1,
6.
Show
that
the
series 2::1 b is convergent, while
the
series 2::1 t
and
2::2
k~(k)
are
divergent, i.e., equal
to
00.
Note:
It
is known
that
00
1
1f2
Lk
2
6
k=l
and
In(n) )
(
00
1
lim
L-
1,
n-too
k=l
k
where 1
~
0.57721 is called
Euler's
constant.
4.11.
EXERCISES
141
Hint:
Note
that
n-1
1
rn
1
n-1
r
k
+
1
1
n-1
1
L k + 1 < In(n) =
i1
-
dx
= L
ik
-
dx
< L
k'
k=l
1 X
k=l
k X
k=l
since
k~l
<
~
<
t,
for
any
k < x < k + 1. Conclude
that
1 n 1
In(n) +
;,
< L k < In(n) +
1,
V n
'2
1;
k=l
For
the
series 2::2
k~(k)'
use a similar
method
to
find
upper
and
lower
bounds
for
the
integral
of
x~(x)
over
the
interval
[2,
n).
7.
Find
the
radius
of
convergence R
of
the
power series
00
k
L k
~(k)'
k=2
and
investigate
what
happens
at
the
points
x where
Ix
I = R.
8.
Consider a
put
option
with
strike
55
and
maturity
4
months
on a
non-
dividend paying asset
with
spot
price 60 which follows a lognormal
model
with
drift
f-l
= 0.1
and
volatility
(J
= 0.3. Assume
that
the
risk-free
rate
is
constant
equal
to
0.05.
9.
(i)
Find
the
probability
that
the
put
will expire
in
the
money.
(ii)
Find
the
risk-neutral
probability
that
the
put
will expire
in
the
money.
(iii)
Compute
N(
-d
2
).
(i) Consider
an
at-the-money
call
on
a
non-dividend
paying asset; as-
sume
the
Black-Scholes framework. Show
that
the
Delta
of
the
option
is always
greater
than
0.5.
(ii)
If
the
underlying pays dividends
at
the
continuous
rate
q, when is
the
Delta
of
an
at-the-money
call less
than
0.57
Note: For
most
cases,
the
Delta
of
an
at-the-money
call
option
is close
to
0.5.
142
CHAPTER
4.
LOGNORMAL
VARIABLES.
RN
PRICING.
10. Use
risk-neutral
pricing
to
price a
supershare,
i.e.,
and
option
that
pays
(max(S(T) - K,0))2
at
the
maturity
of
the
option.
In
other
words,
compute
V(O)
=
e-
rT
ERN[(max(S(T) - K,0))2],
where
the
expected value is
computed
with
respect
to
the
risk-neutral
distribution
of
the
price
S(T)
of
the
underlying asset
at
maturity
T,
which is assumed
to
follow a
lognormal
process
with
drift r
and
volatil-
ity
rJ.
Assume
that
the
underlying
asset pays no dividends, i.e., q =
O.
11.
If
the
price of
an
asset follows a
normal
process, i.e.,
dS
=
p,dt
+ rJdX,
then
Assume
that
the
risk free
rate
is
constant
and
equal
to
r.
(i) Use risk
neutrality
to
find
the
value of a call
option
with
strike K
and
maturity
T,
i.e.,
compute
C(O)
=
e-
rT
ERN[max(S(T) - K,O)],
where
the
expected value is
computed
with
respect
to
S(T) given
by
S(T)
=
S(O)
+
rT
+ rJVT Z.
(ii) Use
the
Put-Call
parity
to
find
the
value
of
a
put
option
with
strike
K
and
maturity
T,
if
the
underlying asset follows a
normal
process as
above.
Chapter
5
Taylor's
formula
and
Taylor
series.
ATM
approximation
of
Black-Scholes
formulas.
Taylor's formula for functions of one variable. Derivative
and
integral forms
of
the
Taylor
approximation
errors. Convergence
of
Taylor's
formula.
Taylor's formula for multivariable functions.
Taylor series expansions. Convergence properties.
5.1
Taylor's
Formula
for
functions
of
one
variable
Our
goal is
to
approximate
a function j (x)
around
a
point
a
on
the
real axis,
i.e.,
on
an
interval
(a-r,
a+r),
by
the
following polynomial of
order
n, called
the
Taylor Polynomial
of
order
n:
(x
a)2
(x
)n
Pn(x) = j(a) +
(x
- a)j'(a) +
~
j"(a) + ... +
~!a
j(n)(a),
(5.1)
which
can
be
written
more
compactly as
Pn(x)
= t (x
~!a)k
flkl(a).
(5.2)
k=O
(Recall
that
k!
= 1
·2·3·
....
k;
by
convention,
O!
= 1,
and
j(O)(a)
= j(a).)
To decide
whether
P
n
(x) is a good
approximation
of
j (x),
the
following
issues need
to
be
investigated:
1. Convergence:
when
does P
n
(
x)
converge
to
j ( x) as n
-+
oo?
2.
Order
of approximation: how well does
the
Taylor polynomial Pn(x)
approximate
j ( x
)?
We present two results
that
are used
to
establish
the
convergence
of
the
Taylor approximations:
143