mcdonald 2eppt ch18

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    Chapter 18

    The Lognormal

    Distribution

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    Copyright © 2006 Pearson Addison-Wesley. All rights resered. !"-2

    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%