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Chapter 18
The Lognormal
Distribution
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Φ( ; , ) x e x
µ σσ π
µσ≡ −
− 1
2
1
2
2
The #ormal Distribution
• #ormal distribution $or density%
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),;(),;( σ µ µ σ µ µ x x −Φ=+Φ
x N ~ ( , )µ σ2
z N ~ ( , )0 1
N a e dx xa
( ) ≡ −
−∞∫ 1
2
1
2
2
π
The #ormal Distribution $'ont(d%
• #ormal density is symmetri')
• *+ a random ariable x is normally distributed ,ith
mean µ and standard deiation σ
• z is a random ariable distributed
standard normal)
• The alue o+ the 'umulatie normal distribution
+un'tion N $a% euals to the probability P o+ anumber z dra,n +rom the normal distribution to
be less than a. /P $z a%1
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The #ormal Distribution $'ont(d%
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The #ormal Distribution $'ont(d%
• The probability o+ a number dra,n +rom the standard
normal distribution ,ill be bet,een a and 4a)
Prob $z 4a% 5 N $4a%
Prob $z a% 5 N $a%
there+ore
Prob $4a z a% 5
N $a% 4 N $4a% 5 N $a% 4 /! 4 N $a%1 5 2N $a% 4 !
• 78ample) Prob $40.& z 0.&% 5 20.6!9: 4 ! 5 0.2&3"
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The #ormal Distribution $'ont(d%
• Conerting a normal random ariable to
standard normal)
*+ then i+
• And i'e ersa)
*+ then i+
• 78ample !".2) ;uppose andthen and
z N ~ ( , )0 1 x N ~ ( , )µ σ2 z x
=− µ
σ
x N ~ ( , )µ σ2 z N ~ ( , )0 1 x z = +µ σ
x N ~ ( , )3 5 z N
~ ( , )0 1 x N − 3
50 1~ ( , ) 3 5 3 25+ × z N ~ ( , )
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The #ormal Distribution $'ont(d%
• The sum o+ normal random ariables isalso
,here x i i 5 !
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The Lognormal Distribution
• A random ariable x is lognormally distributed i+ ln$ x % is
normally distributed
*+ x is normal and ln$y % 5 x $or y 5 e x % then y is lognormal
*+ 'ontinuously 'ompounded sto'= returns are normal thenthe sto'= price is lognormally distributed
• Produ't o+ lognormal ariables is lognormal
*+ x ! and x 2 are normal then y !5e x ! and y 25e x 2 are lognormal
The produ't o+ y ! and y 2) y ! 8 y 2 5 e x ! 8 e x 2 5 e x !+x 2
;in'e x !> x 2 is normal e x !+x 2 is lognormal
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The Lognormal Distribution $'ont(d%
• The lognormal density +un'tion
,here S0 is initial sto'= pri'e and ln$S?S0%@N $mv 2%
S is +uture sto'= pri'e m is mean and v is standard
deiation o+ 'ontinuously 'ompounded return
• *+ x @ N $mv 2% then
g S m v S
Sv
e
S S m v
v( ; , , )
ln( ) [ln( ) . ]
0
1
2
0 5
1
2
0
2 2
≡−
− + −
π
E e e x
m v
( ) =+
1
2
2
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The Lognormal Distribution $'ont(d%
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A Lognormal odel o+ ;to'= Pri'es
• *+ the sto'= pri'e St is lognormal St ? S0 5 e x ,here
x the 'ontinuously 'ompounded return +rom 0 to t
is normal
• *+ $t s% is the 'ontinuously 'ompounded return +rom t to s and t 0 t ! t 2 then $t 0 t 2% 5 $t 0 t !% > $t ! t 2%
• Brom 0 to ! E / $0! %1 5 nα" and Var / $0! %1 5 nσ"#
• *+ returns are iid the mean and arian'e o+ the'ontinuously 'ompounded returns are proportional
to time
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Prob( ) ( )S K N d t > = 2
A Lognormal odel o+
;to'= Pri'es $'ont(d%
• *+ ,e assume that
thenand there+ore
• *+ 'urrent sto'= pri'e is S0 the probability
that the option ,ill e8pire in the money i.e.
,here the e8pression 'ontains α the true e8pe'ted return onthe sto'= in pla'e o+ r the ris=-+ree rate
S t = S
0e(α −δ −0.5σ 2 )t +σ tz
In(S t / S 0 ) ~ N [(α − δ − 0.5σ
2)t ,σ 2t ]
In(S t / S 0 ) = (α − δ − 0.5σ 2
)t + σ tz
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S S et L
t t N p
=
− + −
0
1
222 1( ) ( / )α σ σ
S S et U
t t N p
=
− − −
0
1
222 1( ) ( / )α σ σ
Lognormal Probability Cal'ulations
• Pri'es St $ and St % su'h that Prob $St $ St % 5 p?2 and
Prob $St % St % 5 p?2
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E S S K Se N d N d
t t t ( | ) (
)( )
( )> = −α δ 12
P S K r t Ke N d e SN d rt t ( , , , , , ) ( ) ( )σ δ δ= − − −− −2 1
Lognormal Probability Cal'ulations
$'ont(d%
• ien the option e8pires in the money ,hat is
the e8pe'ted sto'= pri'eE The 'onditional
e8pe'ted pri'e
,here the e8pression 'ontains a the true e8pe'tedreturn on the sto'= in pla'e o+ r the ris=-+ree rate
• The Fla'=-;'holes +ormulaGthe pri'e o+ a 'all
option on a nondiidend-paying sto'=
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7stimating the Parameters o+ a
Lognormal Distribution
• The lognormality assumption has t,o impli'ations Her any time horiIon 'ontinuously 'ompounded
return is normal
The mean and arian'e o+ returns gro, proportionally
,ith time
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7stimating the Parameters o+ a
Lognormal Distribution $'ont(d%
• The mean o+ the se'ond 'olumn is 0.00693 and the
standard deiation is 0.0&"20"
• AnnualiIed standard deiation
• AnnualiIed e8pe'ted return
= 0.038208 × 52 = 0.2755
= 0.006745 × 52 + 0.5 × 0.2755 × 0.2755 = 0.3877
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Jo, Are Asset Pri'es DistributedE
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Jo, Are Asset Pri'es DistributedE $'ont(d%
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Jo, Are Asset Pri'es DistributedE $'ont(d%