(' r' )' 1* .-,..
..... '. LE-..q'-'.'.q.'''r.'''.''.'' (.r..'.).r.@.+'..'r'ri.r'r'. ). .....:..:..............L..L.. '''.'''
''
.. y ''''(
L''q'''''.''-'''''r'''('''-)''''''(''
''''''''
''''
''
'
'
l
' l
....-.
2..
,.....
-
'
.
''''''''-''''''
'
-............:
g'
. .
. .
ri
.
.
(q'................ ()............... -...
. . ..
'''''''.'-' ' '''''' -''''
''
' '''''
'
'''
. .......
...
-...
E... .-
..-'.'-'-''-''-'-'-
-.
........
.
....:...... .) :r:' .....)( ....
..-..........................
'''
''
'''
'''''''
'
'
7r........ . ... .
..
.
-. '
: -
..);,-...
.L.:'')
.
'
'
q.'. ..Eq..q:''y(ry)'()'),'y) . ))7.Erf./. .. ..LLL;LL
()) yyyy.(( ry
'
(
j j
j'
' .
'' ' . . ..
( .
( .( . .
k E
.j) ; E
.'.. ..g .:g '. ': jj . ..g j jj . gE.E .
lj
(...( . .
.) gy.(jr.jk y
. ..
EEi ' q E I.
:
l. .
o
j
Q yoy y
*
u tyy y
g g
:4 a r k e t s
A
'
>
....'-'..---''-----.'.-----.
''
;....;..;.....,.L
:....t.., .... ......
-...................,,..-....-
..--...
-....-..-..--..--..-......-......-.........--..-........--.. --..-,-.-.-
...,..
.......'...'...'.
. ...'..''-.'.'.'.
-'....
. . . ..
y .--.zt=LL yyysyLLLIL.LL...L. .. ...........,,-...
.'..-'.'-...'-.'..'-...'..'.'.
......,....
.,...-
.
The Addison-Wesley Series in Finance
D e r i v a t i v e s IV a r k e t s '
Aladura Perstal
Copeland/Weston F/k'tpp')? and Colvorate Policy
Filancial
Dufey/Giddy Cses l International Ealdns
Marthinsen
Risk Takers: Uses J?7#Alpflle,
.#7??J??cc
'
Finallce: fnvd-nncn/-, Institlltions, and .#.f('l?7Jygd??7d?7,/
Eiteman/stonehillNoffett ultinational flf-undxx Finance
Gitman
Princles
Second Edition
F??c??cd
ofManagerial Ffnl//cd
Gitman
Principles t?./-lbncprrfcff'f?pt'lllcc -BriefEdition
Gj'tman/loehnk Fuldalnentals t////?,vdstf/prg GitmanNadura hztrodttctiol to Ffncnce HughesNacDonald International Banking: t-f?7# Cases Tcxt
oflnancial
Derivatives
McDonald
Derivatives Malkets
Megginson Colporate Fncncc Theol.y Melvin International ft//7dl?and Fztw3c: MshkinfEakins Financial A4't'I?-//J
and f?p-/fzfrfon.
Moffett
Cases n Iltentational
Ffnt-fnc:
Moffett/stonehilliteman Fundalnentals of Finance
Rejda
ofRisk Mc?pwd??7c?7f Princles and fn-llmncc
RO B E RT L. Kellogg School of Management
Solnik/Mctzeavey International fnvd&p?/d?7rl
:
Boston London Toronto MeMco City Munich
e
: !
.
.
Sart Francisco New York Sydney Toltyo Singapore Madrid PnriA Cape Town Hong Kong Montreal
.'
' ..
.
ve
%*<
Editor-in-cbief: Denise Clinton Editor: Donna Battista Selkior Senior Project Mt7/vgw-l Mary Clare /lcEaving Developntel't Editon Marjorie Singer Anderson Senior Prodtttion Sltpervior: Nancy Fenton #lT./ Executtve A/ffp/zclla.gjjtlallagel ; Stephen Frail Deslkn Managen. Regina Hagen Kolenda Text Dcspicr? Rgip: Hpgen Kolenda : Rebecca Light Covel- Des i 4777 f/7rt7gc:Private Collectiort/Art for After Hours/superstock # blterior Covel Seniol'Manttfacturing fffyt'?'l Carol Melville Sltpplelllots Editon Marianne Groth Project Mf7/it7gt//?cn/'zElm Street Publishing Services, lnc. upo/isorl/g
' Yz Jph te
,g77c?
ARy
2 2% OYa *
e
o%
'
'
'
Copyright *2006 Pearson Education, lnc. reproduced, stored in a retrieval system, All rights reserved. No part of this publication may be mechanical, photocopying, recording, electronic, transmitted, in any form or by any means, or the of publisher. permission written otherwise, without the prior
or
of material from this work, please submit For information on obtaining permission for the use and Contracts Department, 75 Arlington Education, Inc., Rights a Svritten request to Pearson request to (617)848-7047. Printed in the 211d f>x your Street, Suite 300, Boston, MA or United States of America.
Library of Congress Cataloging-inublication McDonald, Robert L. (Robert Lynch), 19.54Derivatives markers, 2e / Robert L. McDonald. P. Cm. lncludes index. E ISBN 0-321-28030-X 1. Derivative securities. 1. Titlez 14G6024 A3 113946 2006 E 332 64'.5-dc21 E 0-321-28030-X ISBN 08 07 06 t)5 E
E
'
.
'
.
'
12345678910-14-1--09
Data
?
For leaca Clail-e, David, and flapw
%
')' j'1* ,';' (','qL' (-' td jL' k;' q' (' t''t' y' :';'k' .t' .r' yyd jjd jjjd yjd $yj' ryd j'y.jrtd qk'Etd r' )' dl':td t';'jyjd k' .j' E)d ?' .;' t'k-:.' ;rkd j'td k' kr'r' :;;jtjpr...' $''('EX .-' $1* q' f' r$;;'-' q' :rd j.'Fi.' ,*5* ,'pjd kjjijp'j!jd jkjd y;' j'j' yttd ;;(' jq.'j'(' )' !j' .;' ;,' !)' r' lyjyljptl...rprrd jrd (' ri'-tt.-td -'))jq4))(4L.' jd t'?i' i'''ipd j.yyt'.' jld 't' k' :';'s' (' y' .y' yd )' qkd p' qjd ilrir.-,rttirr.llllikply:::iiil.--;-d l:rd fd #' j'p' !' jjjjj)jd ''r' j:'..' j.' ,',:' .'id y' k' 1* jyd t.d yy'.yd jqjd y$' q' j)d j'yjryd .y' t't' .;' ;(y' ;yy' (yd yrd .yj' .j' ljd y..jd jpd )' i'yrj'd E' j:' yyd yygyd jjd jyjyd yj.trjyd $,* $' y'jd prd ygkrd p' j.tjd yl!ld kjd y'yd lqd yjd j.yd j'' htl'jd ')' tr)d k;qd p' ;d :)* !' t@t(:r.-.,'t;;)yjjjp' :'t).fld ))' jttd 1* #' jrd @' ('(' ((' j'i.td jsjjl-ljjj'yd )y' ppyd ),' .j' fkd yitd 1* y' (' :tjjpq'.lpjslr'-kjd t'(' (jt' jjjrtd Ckd yyz' ;'lllllyyrll/-s-j.d jd y','y' ?' j:gd y-' j'j'y('(' k' yrd 'F' tq'pj).;rl?d q' jy;rd .y' yjd :j*yy* 4* j(tttdjlrrdyjqilq:.-;d .yt;t;';' lp)' )'r()r.' (.;'.' )' !-' t'-.' kjt;tld y' (' yrd yyd jrjjjydtrd k' yjjy'.' ygjd gj'j' yrjyd yjyyj:,' q' t)' y.grt';' yjgjj'yjd .y' )' q.!j)' td rd rid y' ktjr;'.' yyyd yjd r' 4.4* !' (;' E' g' tfd 1* ;@jr:;))'-,:' i'y''jljtijyi.d !?' tj-ytd q;' p' @' y)d yj'-jjd :'y.' 2* 'd!' g' yld j'Id lp.rlrdyjdtr'!.rd /'r'@'y' yy'y' jjd .(' .t' l-jd )'.y' )(' jg)j))r.' (' 1* liid j;d y-' )'(' (.(' y' tt'iiild ).ry' T ,J;,)2)j,*:7C11t1111%,,* (yj' r' ;pj(j,..;yygj(yj'.' jy-y' prd )'tj' j)')' $' p)' ,d jgd i'ttk'k' td .y' ;'.(y' ktd 4* tkk'y!d ttprj-.tkjkt.kljl..d jj.'.' )')' ;j'yt' t.ih'1(' yqt'il'yjd #' qid )' k' ..L-,L1''' (Ijjkr' r' gjjjl::d .!' jy,d t't.(' rjr'.' yyd y-ygyd (r)' jyr';ggy)' tjd ty;ygj'..' t-:' y;jjj)')-' 'g'(rd )j.-' tyd );'t.;;' r;')j' rrd )' ttd t'p(' jjj;)',' t(yjrj'.' .t' j;,j-y.(il)'-' ;jj;':' ifff-fltk:t:'LjijLLLqLkLi'.' ;L)-.);j3jjjj';);L:;.,,-;L:.(;;);;qL)...' kd jtjtly' r' llfl' rrd tjj..' ryd ryt?d r-kj' rpd ttyjjd r' yjyd .g' yjyry'yt-:;d )j;(''!' :IE'IS'))X IIIIIIII:::)I'X
..' ,j,,,' .,' jj,' .,,' ..,,,j' ',jyyd .-' ,'.'.;' '.tt:7' -'-',,,g.' ,jt.)' .y' .;' --'-yg('(' ',jy)' .r)' .k)' .d.l!-ld ,'.'-' '-.k' -k' '.'-'..' -' ,'.'-y' ,'-j' -' -q'gld ''..' ,'-t' ,,'.11)1$19f!$(*1p21$1111* .y' .'-'.'.' '','-y' .'-' 'q' ,'j,-,' -;,r.' .j' -'p' t';'r' ''j' j'-'-' t-h.d ;;' :'j' !''yd -F-' ,':((' y' :j' yyd r' j')' t'Ejd Et)' jrd ,'gjd 2* -' ijy' y' 1* k' jryy.yyytd 'j@j;kq.' ;''-,L' -(' ,t' (-.;jjrjpj;..' -(' -' -'y-;' g' -,' h' ?' ''#' ;'-'t.jjjyd :'i-kjd -' -(',' jpyjd :(' :-' kgj'.td ''h' t'-t' i'y' E'E' f':'gjgrryyyjtrtr.:,d .E' :'-' ,L'-.jb' k-' j,' jd ;'j-' (..' 4,-* (' ;'j-' E-:.' y-:--;-d ,'--;' y' t'(yd .y' ,y' :'Ej' s' (j' ktyd itd 'E(' y' jtd yjy'.' j'r' L'rd ,':'k' yjd fjd iyd i'q'yd ;'tgly'd i,d :k$tt' q' frd '
[email protected];d Ekd F' r.j;))jj(pr));j))':' E' ?pi)(' j''t'7)* g' pq)':(;(jtjjjy.(jj'yjyE.' j.j'.:' t'-''::kl)!-''' E' !' i)d ijl'd ry'!iE!' 'i;d .y' (' tyd 4* :Ejd ' ,'L' 'jFd ;''?' y' ;','!' irytjlrdjrppjjytljrdktgl.Ljjysttjjjtl'ljly'gjjj;d Ed k' 5* 'L'L' ,1* '':'y' f' ,'C'
.'.'r' .'.' ' .'.' ''''.'.'''
'. -' . pi-.dltrrl:tL.)sppry -.: -. -'. . ... . - -E -. - :.' . '- - !-E - ' -'.E. Ei'E .E - ..' .E' -. i.l ;'-;tE ' :.-''. . Ei i. E;.i -.. EE S.iljj'ply'j)kE ' -' E; i ! ' : -.. E .:. E E:EE.E E--' E .. E. E.. EE .-'... E.. .. - : 'jji j(E. kr'. i E . . ' :. -E;! E :- : . - -. E.-.- E.!.E . i.'-.. :. E.E - E.E- :..E:.. -.-.. ! . . 7:-..- E .. E :.E. . E'-.i' ;: E - -! . :.'E . .. -..!. - E i ;- : E .!.E : .E. E..q . . . E. E 6:*-4::y*. . EE-':E - E . E E E E' E! E . . FE E E. EEE::..- E:E.-!. - . E.. i . : E.;EE. -:E ..E- .E- iE! : -q.. .. -E.E .E . .EE E.E. : E E. .. i :E .- -..:.. . EE. !. E ..:' -'....... EE: E .g . ! i i . . E-q. .-! tE. - (Ett ! :E-E. ;Ei - . : - E: E -. : . -;!. -: . -. i .E. :i! E. : i . - -! -. :. E . :- . . -E ! -. : E. - r'i . E E .E.i E . . .. .. . E . !. E. E. : . E . .-. .i E .. E. ; E EEEE E !E. ).E .. 'i i -. -E ' EE .E..E. . E E.!';E.EiE E . ..E-. . . - E:.-. i- :. E E. .. E. qE ..E E . ! . ' - -E ' EE E q -.E -. .- kE .' : :. . . ... E z.pssrdkyjyj.jjl-i EE ''. : E .iE-.E: (yqljl'.g EE. i i : . . . -. .E.E E ! : E E ... . . . E -. E . E.E. . : E EE : : E!..E E:.. i : E - . :.. -E .i . EEE q.. qi E: i - -:. E I ! ,;:. E :E - (E E: ,';);yyE:,,j.-:.Eg: .. i E gy ! . - qE E.- -..-E. EEE - -E-. ;. . ..: E. .. - -. -.E i -i : .E:. E E E. i .y .:;iE: . g..i: : . - E. .y. . !.: : jjyqlr'jgjyjj-- - .- E- EE g ,E E :. .:. Eq.. E.... . ; E. E i ! -. -'. - -. E -! E EEE (E. .y . . : . ... . q . -. . . E : . . . : E ; . q -;'. . . . . -jt:- E . q E . E. . (E . . !. - E E :. E Ey -- ;'. .Ei . i .. ! EE .E. Eq y .y. - E. . E... . @E- gy. y :. g-'(E y . E --C.E r y yy .. . .- .E E. i !.. E : - . . EE.. ..Ej i. E - -. E . .-tE..EE . qEE. :E- E-.. E- @ .. iEE gE . - E y E E E!?! .- . 'j.g . E g. .. E -. !i .g-.. E . : E .g . g5 qg. yE. . E.. Eg. (y.g. -.y. r.E.....:...- E...y!. y. . y -!- . .- -.-y.- qEg Ey- .EEE i- Ejj E E yE E- - E q g! ..y E- - yE .. .. .-- . .j g tyz'i : -' . - -. E.i . E . . jk'y ;. -'jj!j(q.. : : i, . . . y -. . . . . : -:y y .. y yy y -. . - g ;jE 'E E. ! ji.jqiql?/q. yy yg ,.yyyr y . . -y :j.E . . . y t.l q g . .! - -:. g.-. . . E- . E.. y g i . j:yy- -: - IE -)j,'j4.,;)j'. E yt'E . EEE . E E . . :: :.-E : .y . .. y :E . g; E., i..... y . .. E..-. . q .. ..E. y. E.: . . - .. EE. E.. g y . .E q -E :.q. . .. .Ei--.... . . E..-E i . yit-'ykplq'; . . E E E -. ! -. i g - E EE . E: -- -'. : --.- . . . .. y. -Ey. -. E..E .-;p))y:. .. y y. . E :- - . gy . E EE - E y. g - :( y- :. y. - yq y ..- y..-...: -y. . . E y -. E yy. . E y -. ;y.y'. Ey.i - . (. -. 'y y. y.. .. y. ! q . g !- . E'ti'; (';. i.i .y y ;'y. -. r :.E y.r i E . E. i : -. -. .. .. i - . gy: ... y . EE. . r ... EE :.. : . .,;'E ?... -r-'.- .E . .E . . EE .-. : E :..E -y '; - E E . . y y ;.. y . .. ! . :. : .E; jL'fL. !jj i i -i E- E . ;', Ei .- E. E y .: . y y. . . E !-... E. E . :.. E!l))-'.; .. .y : .- E. EE E E- : E. : Ey'. .- E.-E.y.. . . . y .. y '.! y E . -. E.. . E! . . -y E : .. .: . - . . E ' :. . E .. . E . - , . E: E.. ,E E. . ; . E .. -. E qE . E.- E : . q k .. , -'. g . - . .y. y:.y.q - ! :....EE E.. g r g. . .. E i -. -,'.( .q.y.- . y. gg - ; E -.k . . . gy lLL.iiqjL'LLL.iLqtt.Lqktq3q3', y- y ,-1$3*. FE - ! --'t.;. gy yygg.y !- y yy.. -. ... . . . . . y .; E E - . - ygg -y. -..- . :. - r:-y . .Ey ...... E. -. Ey . iyj', '. i ij))'.- -E! E.-.EE E . :y-. - E. E.-E .q. .. yy y : i : y :.- .-. y . i- . jrty'. . . .:. . EE;Ejy -l.j . . . E y : -. E . : - . -.. E E E E. :-.. E . :. -y- y E. - Ey E . : ,;-iq :t'-. . -'. E-i i : . : . . : . (. . t;E . E E E yjytjjjgr. E E E i E y i ,. E -. . :. : ..k-. E E. E i! . : : E E : : . ! .y . E ... E . . q. - . . . - : . .. - -i'E E ; q. q .. g .. q - . E q q.( g- .. g .. r : E q- f'.E . . . : :. . . y- -,.-:.,-. .ijj jjyy:jj : . E . i EE ;. . !q..-.y. ,i.. g. .y. ,. y', g . ::. : ;iE E E ;'.E : :: - . . y. . E. . ;-(E . E y g ! g , .. .r'r ! ! - ... . : .(-r. ,. .... . . . q. . ... (E :.E E . :..E. . : E.: ..y'.jg. . : .. :- - : y.y. ; ( . g .gqq gy- . g.. j .-. : y ;.y - yL'. . E . .. . g q;: . .. . : . y '. . ;ti : . y-yjyj';. E E: ' g .: g: : : ,y: :y'j. . r y'. :: . g gj g : r. . E . .E r : y y .. g y: :. : E.. E. . .j ,,'-jjjrq;tjy yr . ; Ey : . : y yy y: g ; :' ! E :.. E. E : : yy j .. .. j... .y..y. y. : :yy . j(4'-(,'. .j;j. E yy y: !j'. ; :E Ei :. . :. . EE ,*1)!2. E . E . ::. y. y. E: . y!E y.. y . y g;g.: .. r. y. .yyy( : y y .. ,'.y . . )E(-IE . . .. . :. ( :. ::: , g i,-i E E ;.kyy'E . : jfL;. .... E . E E 2k;j'y ggjy.g y ;. j; : :y ;j'.. y'E EE : g .: .kE. E ::(1: E. LS6'L .E..E .E . q: E. . . .E.... i : ... ....... E E .'E . , :.. g : E . q E j E. . .. r jkt';g tq';yr : :. : E ! y yy y.y.: yyy....r . y... .. . . .. y. . : y . g y yy;y., g. : .E :. ' ;E. y g ! . . . .y g 5. g g:. qyi . . r . : i.E. E ! ;()'p('. :. : :. g :)- . E : I y : ; .: : : y. : ... g: E . : :. . . : E'. E :.EE.E Ek.: .E E . E t'- :.. : : . y y : ! k: gg iiE E: E . : : . : ,'. .. g. ..y yjy y . yy y ; y. E . . yy; : : EE, . . i- g: : .y. : y.. . y y jy y:, y y;k: y..y y . .. . E ; yy :.. : ;E q.i.. ;g q. Ey y: g ! :. :.q , E ' iyy'yi E.:.. : : . : g , . q?': . . E ;.jE ,'. E:.q: ... ptEy i qE :'. : . yE : .. . : y : .. gy g . . q y:,. q y t'. . g. y (y ;.g . y. :. y .: y : . g . .y jyr :. . y. . !y..j , y.. g j .. y yy. ,. j y yyy y.: y ; y y ... . E . : . jy. .y g y y .j y , g. ! Ei g :: .: .: . .. .E . y :.E ; i 5. (y y y jIE gEj. E :.: .!E E E. yr'y ;. . y Ej : : ! : y 5'gj'. i : !g''. yE . !. i Et'. ii i ;E :. ;)r.j. @r y y g.. ! y y :: y.y E . . .. : , ; y g j yry.g. y ..y y yyyy. ..y. y...: : yr g y . q . y y :.: E ... : , ::: q q !E . . . y y'y . : .q .: EE E . : g. .g q y ; y yj : . . . . . . : .. ; '. r q . y . .. g. . g:y y r y r g . .. y ryy-yj . . : ! ' : y ; . .g .. g.y. y.. rg y E :. . E .( r, ' :. . : : . jg gg . . g.: EE yyj . g. . g : q . .., y y . q: . .. yy. yy.. y .yg y g g yg : y . y: :: :k.:E .. E yj'E . E . . .yy :. y: y. y ... .y .. y . . yg . .E: E y . E .: : : q :. EE ! y .. y E.q :. y : i E! . .y.:yky g. : y E , :g yjyt!. .;gE p';y. i. i :. :E ! g E y. . y . :E , g g :.j-E rtj ! ig , !-,q:i .k g. q.y;i . :y. . :, jy; y.. g.. y .: . y.y: .t'r .: g g. yy ,y.,yy'y:y 2 . : y -.t)@-. .j.y : ,. .y.g .kyj'.yy. sg.g.y... y, y., . yy . : -.-.E E-i g E -.- q--. ! yq. :. . g! . E gE ! : .:. ::. : r : : . . gE-y :. -.y. y. -:yygq .. y. .g y. i y.y:y.. y'. t')l'.. .g:.:.. y y. :. . y.. ..y ... yy. yy,jy'.', , y ,j'. -t'r,. g j..y : . y. j ! y E i-i i t,i , g y y yy : . :y . ,. y yy y yg ,j . yr y y y y y. y: yy!'.gyy.yyy : j . y. k'.. .. .'.E:E@i. -. E ,'. ;jy . y -y,;'.. . .. . gy-'jjy;tj ,y.jj'yg . : : . ...E ... y. ; tsikiq E .:;:.).......z)E?;i!;2.....;: yg .. y y . . y.y.r yyyy y .. -Ej y . g y !y'. ..E. E-- .- E E.. E. yyEE -- - -- !E .. -y E E ;. - E - E E !! E ..E.. -. . - -E-.. '.-. -.-.. . .--. yy : -. .. yjy.. - :y'. ..E-.(t'. - --- . y gy y .E .E y. ,..y (. r'y .-EE-.gE --- -t:. -:jg- . .-. ?--E yy -. j g y y .- . .r. y ,--..-..... .! - - ,EE j'll'lt'. . ,.- -. --. .-- -E - .- .- E .- -.. i-... ...E-.... - - .... .- -.-. -..-.--5-. E - ., .-- E-- -E .- . .. , .E . . zlElhkbjtktir .. -. . ... -... , -.. ; . g.g y,gy. jy . . . -. -((t-;:j;::j,,k'. ,'. . . . .E . .. . . . :;.:.;@ . . . .. .. .. Lk,b.......L....,,.,..'f:. .!.....E.t.:tLi.i.;E:jjj..:.kfJ - .. . ... . .. . . . . . . ....- -2'E.ii!'.....'..:f.'h'...-'';.J.z...'.z>2.k:;i2E2.. E.......:;t....... F
'...E'.. .. ((.... ....! ..'. ; E.' ... ( i..' '. E. :rpli.. #' '.;(:(('.; '; !;kt'.i i;p!..q;: E;; . ;;!! ! rr':rl ;)'jLLLLL'-jLL)jL;).. '..j)'. ;333336. j. ; ( ;. i'..j;))r:. - '.E.. i.(.LkL,. kitlr1' @p?';; .... '.... i;! '. E.!...E.:i 'E ; r-rs Ik)'. g i;;,; kk .i.(! i i ; ;;. ....i 7.' '-.; :'..'.q )1r. ;((.-:.(r$$-. .:((yI(t.-prt,.. jijtdp.jlly. -L:;L., .r ; 'E1.....i'i;.' g ,.'j-'.'... ;'.';.t;r. ;...k..;.' @'' yI:5.g;l)'j(j14p,jj)))yy.. .(r4;*1; i():;)'rtj).. g. . .-..., $!. i(lpj,.);rp;'.. !rjjrj'(jj))';y))):t;. l'-iy'. ///4*. jk'. jLtk'. ..(;;;bL,, . k E'.EI (.;.. ;r ;. ;...i :. . ' '.:-'q, ;I.;. ;(-.(.CC:E .'..;. i; :);'r);()'. . ;Lq;Ljj;LL)L.. . i!t'-4'' ; E. .E ;);'. .. )jfq1'j'3;qj4))L. . i ('trLtq jjj))j((j)'. rti;)i t(';: 33;*. ;;'.j;h')()'. ();;;'. . .(4::): .. /331.1336621333313633. . )))'.qik. .. kt.'. ; pj)t.:l)'. .;.; ;; jE...t('. .. .I. llhl t. );(jj:.tl !yr i... ; . .. . (4;jj4,. (... .i ! ; .! . yjj)j j; ;( .. tl'fi '-I pq: itll'yj. j..i.. (. ;.( !.-; r.y.r y!;q'. (. ; i ! ; ..q.E. . ;..EEi. . li.;.! .(..i.kj.. j! (y'yr . @'. .?)q.. !j(4. .pjt4j(.; .ji. .! E jjl lr.jy.g.y'y.. y)y(y;. C@-j#1?.. y; k.Ii I.E.l.j.-. qt.. y.(.(.. g': . iki. .i .(i. j..y i qj)'-'jjjj;yy.k.rly j,---;lpl )))i'' :j(q. .(. jr!y(. q.;..(..i;.;.. q.ji.kj.!.-i ....i'.!;..i..-r..;.1.; . g.(.r.;Ei;.!;!;(; ;'!irl t(:. r.-. (...r .... ..y.;..jy': tjj)q.. (.4;qj?',jtLt-,.;jjLj-q))L,,':;L'. .))r)).. ( (rj.)'-...(...jqr (j)'yy,,';(jq((rjj(j'. . . ()..rr; ygg;jtj,.. y,'.:j):. I ;. I . ....!. t'.i. I.q E j;;)j'. ). .. ((!)-)E. rjj'. i'((;. p; ; ! ....y: rtyy-.))'. jj!yy')yjjrj.'-;);:j();'. . pi .-'.EE'.iEj p@:f: .pclit.illyq-.;lr.j.. jyg. rqy-'ss (jyjjy. ..(!;. ......r(..g@(.........- (. q((.;.;. E..E.......q...;!.. ;.y..y.j.j..yjjpjy'jrtyy.jl 7$119;j*-. -jj;L):',. jjj'. . ! ..j..;Ei ;;k tL'$''j)L,.-33L333',.. . '- ;:.;,. q.... ( .... .. ... .. !)jlrr(.. . .ry ...y( .))'L'. . y;jj. ...yI.i l.)'. L.. lr.r .y..yg;r.kj.; ki....;.;rii.yi. q).q..y.yjr.'.;j'. . . k5'. k'. j yj. (t: ((t);hq'ijj.-:t!'. y )':..; .- . . ;t'. yg.;t('.;. igy @.. .i.. ..E.('. r.;Iy..( ;;yj ;.....q. y;yjj', ..-..!.. . r:y'. ?.);;;.. . . ( . !. .)'. :..E ;'. . . 1: ; jj;;jy. .l. ;..j....I .j.; j. jjj-t'ti j .y. (.. y t'. ' ; ! j. ...r j . .ty . . q ; . q t h! I (qyyyyyy'. j .)'. . ! y p'. . . y jq . tfli p , ( ; !. ;Lb'. yg #i'; iiqi ( . t'q );'. y'. j . . . . . . . . . j ; j . plljjjjr':jjjll, ' . . L;,. . j . ( ' . . . r . . k ; j'j(j'. j . rl.y y. ; . 1*; y j'. y j( . j. y.y . ..i;l(.l. );(r'. li. j'..yj .. i t'. ksq'iq'k.qk'fb'.). g. ;i @)'. ;. ( j'-r . yy El'y E l . ! r j.. ij,!y . (. ;(-..y )jtgj'..'... !.. E. ; k j . y ).( tjyr;. (. . yjy.... .. ; ... j. . ... . .j . . ;p'. . .'.i.: p1?'. ; .!..!. !.i.i!i. ;..yj'. r. ()'. jr!y .y y tj'. . g )! !'/; i.);r;); r('. . ' ---L--'-.' !!)@ y..j... .. .!-;;(. r ty. r)ir)-'. kpi'. i y))y'. . E ;;L.j;'-L;'.. . tftdgjlldttjr,jrl.tl-.-. . y4(:yptjj'jjjyj!. !.. ; ,! E . , ;. k. ; j).;; ;r.j( . q(??L,.LL(L3Lk,331L31'. r t:p((t.. . (1t1/24!j(.:.. .ttjkt''rrtp. ! ' ,. . k ; ; )',!t'j$rt$rpyjr.'--,y(k,'.. ;. . . . . ; ( q . li r (gk)j-... .. ;'.. . . . . j. . .. .. j . ; . k'' y'. . y ( . ). . . q . .y y ! y j ; ft!E.:g'. . q ; .j:y ( ; . ( ( .y.: k .t.. ; j; yy'; (y. yy . ;.;.j..j i.. . rry ; . j;'j. .y ...q .....;.j( ;.''!y ; ; gy .ti i'rikj;y.' );,. .-jLLL,6)L, . . 4(4;j4;*. j)'.yE j. i..ry ..... jg.;...... j..yjj; .; ( . k!;.;.. yt'. j Lt', ;1 il'i ( L3$'. E. .( j... ;..j.y,.tjjyjyy'. (j.FI k'. . y'! jp'. . j ri(syy ...j'. ( )',;-'. jri'itpi. ... y,. . r (. (. .... ;. ..... .... . r(... j.... (.q (...... ... . y.,.y(.... ..j;.j. k.. E. .......y...........j! .. yg..j..j...yj...,.@-. . ..;.yj;.-.rj-. j'j.;...( r..;)'..q. . i'.j!. ;. ..; ; (y. .r')-j'. . .g t)'(r)!-yjyjy. yjy. rjdr.gj:;lji .ryy .ir.!-.;i' t. ; ; .(.q (-'E .. ;j)jj(jjj.L1;';kLL,., q'yy ( r!i:jhq'! . .y.. ; .. ;. ! .g'.(. .... (tjy'!)..t'. fhli. . t@l);.-jj'. i.;jjt yq rp.;. r.i It)k;.-.titjk t;ii .rjt;y.jt. k. (!... . qy(y(..y. ).. !@!. .lj;'rk..!k; it'. .g . . ;(t'.yj j j.@j'(j.. t .jj. . y ;'. #j;'. jyy'pr;;jj(jy'. . jgyyrp...;(..(. y'.;... . .y';(... y . (.. y..igyy. LL. . ..( . ( j . y'. .... . j . jj. j'....j...; ! ((y.l !. . g ..)'. ; j ;t';.j-y y .j lyy...(j.:gy.t'g jjs'ittl-lii .I;. .E....-!i.;E g.rt'.ij..yy .i' i'. y.E..)'. L.L k).. ): ! )).. .y;y...- j'.. jj'tyty;;qlrp't ( (. .E (. ;.; (.j... gi.,..y'. ki rj;..r(.kjI ..y;y ').3-,311*. . y. ;....(..r..ry.;.j. ;. r...y.... ;..i...;..;i.t'.. .) ;rjj:.E... ...y.;..((jj..-rr(.j y'. tq . pi.()t$'. ..(k'. y.. $,rtjj ; . ....y..... ( .j ,..';yyyy j .j'.yy jj.y(j.j'yy .;.y( gy. ('. LLk. . )jq.Eyy . ri. 'i i.; ';.))*. ; ilt'y. . (, . j . j (.( . .. q!; ; ;. y'i . .tsi. yjj,)( ; ... .;.( !. @E li. i j )r. pts .; tki ((:. .f'. y..,.-;.y.. .y......r !. .y'; )'. . y', ry(k.'(j:..'?k-)tjt.. q):q))jl @;y- q(rg((ty(p'. . .. . i .. . !.; . . r ;(yr. 1E'. j-yi ; I yj.. j, . ;1/*. . j..j,L3,.-jj)j;-. .j;;y. . . j 4 y)j)y-. 52 r .. j . '. .. (k'. .. . j y tjl .. .. . . ). ): pjjg'. C'. .. . . . . g . . .r .. j. . . . . . r j . j y j,-',y;r.)yj;t; y .. y r j y; j-t'. j . . g y . . j .. y . jj . . j . , y .. -f'jLjt';)kL.,L3);LLL;;;);'k)'. t;'. . y-'! . . yyy q ; 7j:*)9: 3t';LL3'. . yy jjyg$r'. . krljjj t'. ., t(!. (:).t).; )... . E. . )1! :?;. rjr j t:tp('. .t'- j . ; ; . . yjy .. j..j yj-r jE. yyr ryy.;jy ;j'. . . L;;. jj.y. jjjs.ti . ;L,L'. .-j.34L'; . j. .j.j . .. j ... .j ... . jgg . yjj,, j')k(i'.ik.('. ;(( .jry j'. yyyr'. y..))jt; ' il'l'is7.;:' . yyj Ejr'. y'. g . ; j yy;;'.jy yttyyyjjj-r.r, . . jpl'k. . ... .y;'yy.s;'; . yrs .y-,j. ; .......j....!j y ;.k.. j.j r.- yr jyj.j jjj..y'. 1. ;,..(!.yyy. .j'it.jL';'. .;j'. iit'(! $,. )y'. ..yjjy'. .kj.p,r . ... ty ''lE1; ( . q;.j . ;i ; r.( ij.. r'lt.! pi(k('. r(j'ji.ry.j:;'. !li;q', y yyj.. j .... p'y ! k ;; y.y: . .....j.. i;jjjj', pyr.. j;jyy(;t;. j r;kj'. kg jq..t,j.9*. .j.ijts .y. )'. j!jr E . ;yy .t'..j. it'.r ;. .E.; tji ..:'(. .qy ggk.( (jy);;jyj. y;(jj.(...yrjj ;y.. .t';'. ;.rr.;; y. .yj.y;. .(;'...j ry y... j .. . . !.. . .).. g; ;. gk.( . yg.yj.y y.y (..:i. y g.. ; j.;.;(.t: jy t. j).). q. j k .. ... .r.,j)('i. ;.y.. i. . r..... . j.. ...r'y .yjjy-. ..;.)'.. j('y.kj;y'. .y ;);.;,4*. qj.. . jty . j j.yy( jjgyjr. j jj.. jy'.jrtl .j)));jL'. ..(..j: .yky'jy ry: r! r.jjrtjyjy jj;..;(t'.; ;qt. .' j'.i'!.'i:jjt: ...; tjy. (g. ))k;j'. ;...j. (..y..y':..r.(.;j:.r........j'.j .yjj:...jj.. .y;yj. yyj. g.;. .jyg.jjy yryg ;y j.j.t'. jLj4,, )('. t!.ygyyj y('. ;; y: y.kj'r j.j..jj. y', .j y ..j.. .j. y . .. ... jj('. gj ;(j.qj;tj.. ))gj(. ;jjgt-. .y j. ;.y', j.(,jj'. i.y: . jjyjgr... ...jjjjjy.yj y...-jj;yjg ;jjy'. j'rj-'. jyt'g j...yy;! y! .k j )'k.@; .,yg .y jy g- ;.y. .y.. jyy yj'g ; y'jy .. .y. yj;. (g... . )!)t'. . t,y'. j ; k (.(j . ; ... jk y,;j;yy ; ;E'. i; .. ( . y .y j.y yj ..s..j....y.jjt ; y.jg j. .j . kj;... .jyyy,. y..y (..;. jq;'.LL'. yy.tj y.yj I. k jj.. (jr. jg. y': . . jy 1. y y': . ( j j j jj;rE . . . . . j.. j.. j r;((;'. y j jjytj.j. k j (;yj'. jg,'. . .yr'. jg . . . j g y . y y )( j )t', ; pI j j-y. . ;. . ; g ., y. . yy-pq: y . y;g.yy,'yjjj-y;jt'. ., . ;'. . (. y . j i j .. j y . . . j j y; . (j;.jyjy'. . ( j.( j y j'. . y j'. .. k . . gyy'. . ty'k.. r . ; . j ' .(.. (1 li. j k y . j kjj'. . gy q k,jj'. .t; y'. j y .. . . . ; q . . . . . . jg', ; . y yy g y. . j.tl ; y ( j'. ..; j y. .. j (.k.iy( .. r;. i.( gg. (. (:.. k; r jg. j . ; j.. ..k(....... . ... k;.; .yt.-yj.. jjy. qjllc'lil g. r. . . . y ..( .j t': y'.g.. (j, j. tg.g. y .y'r . y;y yyjy j.y(j'. . yr'gj'y . jj. )'. yy . .yyy,. y'. Fj,'.j)'. j', yy j'.k (. .kk'. g. ; ;y'. j(t,, jyyyj jy'tyy ( .y ; ... y .. yy'yj /i.p;l );)'. .jj.yyy J): y. yy y.; ;j y . .tj)q,. .;.t.y.. . j g.y-'g ly j ;.- .y;j yy.qyj yyj. ; ..'g jg ..(j ..;jy ;gtg );;,'j y; .rj. y, j:y yyj-jl'-jj. g;y', r. j.yty.(..j.jgq. ygyy,ys ryi jjdrystyyjiyy j'. j-i; rj'. j ;y'. .jjgy g..i j!..t.:' jy.. . r.y...(..j. E.;...q..y..;.::. .j y; y.;.jy.(. ( .j'.k (. . . @,'. .g(;.'.y,, ('. .j.;j.yjy; tjy jy,;jL)j. )jj)jL)', . yy. .y.yyjj: . ...y.. . y.g'; jr;'ib-. yk jjj'. j.(j,'. tp(j..(.. . . j.,j !((y gyyt'ry jy ..yjy! jg'y; j. ..r. ygy'j(yjjyy. y. j yj g .... . . j .... .g jjyjytyjl . (j jg yjy: ..iyj ... .y. yjj ly'iC! . qk..;'.,(.y;. jjy.. .jy......yy. j . . ..yy . ..y.yjyy yj y,tl . yj.. t;)j'. y., rj;...y;..jqj;gj.y',jy'. yjjy j;j')q')jj.'jj', . .. . ;.j-tj. y(jy..r ib;. j'. y . .jy,jyj y',yj'.yyyj jjLj'. .yyj,.yji k-.,'j(y.'. jy'. . yyjjyq';1p'y,jg;jjy., k.krj: ; .yjyj j . .jj.. yjg.jj.., C'pl y, jgyj'y .;.j;t:r.k i;.(. j j.;.. (. yg .j jy-'. t,'r, g,.jg E y.;g .t. jj'-jjjjj'-)(L,,,'. . j.,. j'j)j', . i'i'tti. sj'g t.t!.yy.. k). . ; ..... . ,-'-.. y.. j y.. . ...@! j ;r.gyy j',j.'. . ;jj'.jyr. yjjj;j);'; .jj-.. p,r.q.. ! (. .i.yr. ,.; :. .. .'-L ,r t'. . yy'. i .t(( yy'. :-;-. (-.. ..., . y...j..j'-y. j j y..-y..y. ;'. ry-;. ;gg ry.ykyyyr'. t-. y'. ;-;. ...(; .. ........ j. (.-..;.....,.j. j.y. y.ty r. ( . ! ..; r'. . jy',i r.g.y . tg y . . . . . jy', . . .. . . . . . . . r jjy', . . . . . . . . . . . .. k))'. . . . j'-'.'. t.2! . . '. ,:.'j!!:): . f-'. k(;:..r':. 3t;ALq..z....'t..f...t1. j)qi::'yk: ).(jk!.). )kj;;ty)l).i)jkltg)jl). 'tlzl'. k)l'.. Ljjj' 1'r(.kt.k.:t9;(r(;)kgL;' .'j.:.LLLtL.,tLiLLj ):.'h#()2:.)t;' ''.(?j3L.' . ''. . . . . ;q'.. . . . . . tt'. . . zt.yihfi?g'.(;j i'. . . E. ky(!.:)!;;j).i/)';ij:;j(t)?l:.')p; . :'.. r'. r.'. )r . jII. ! -'jL;L.jkLLL.
' :jy:-:E .-(()j-.
:
.
-
.
ii ! EEj-
:. .
'
:
qkj-;j,j;).. . jjrisllllliik'q'lilitlllllllt-'' '
-f$'.
-'
.
- .
.
-
.
.
. -.
-.
-
.
.
.
- ..
.
. .
.
-
.
..
. .
-
.
.
.
..
. .-
.
qt'
-
.
..
'r:
.
.
..
.
q
...
.
.
-
.
.
.
.
-
.
-
. . - - .. .
-,
.
.
,
.?.t'i,.I .
.: i : '
.
-
.
.
. --'ty
-
-:i
--
-
.
.
.
.
-
.
.
.
-
-()q'.
.
.
-
-
-
-j
..
-
,
-
.
,r t:.' ' j:EI '...i.i
-
. -
.
.
-
.
. . . .
..
.
-
. ,
.
:' (E .Elri . :
. . .
.
.
. . - ,'.. . -. . .
. .. . .
-
-)y
.
-
.
-.
. ..
. -
.
. .
,
-
-
.. .
-
-
.. . . -.
-'y
,
.. .
.
.
.
-
,itji :..(. EE,)1
..
.. .
.
-
.-
.
.. ..
..
...
. ..
.
.-
-
.
-
.y
. -. .
.'j . -
,. .
- ...
.
-
-. .
.
-
.
-.
.
.
.
-
. ..
.
.
-
,
--
.
-
.
..: :
.
. -
-
- -.
:
.E.,.Erj ..E,...!jj
.. ,)jl,j ' . .'.;' 'lli :jj
-
..
.
..
--
. -
.
. . -
..
-
. -
.
-
-
-
.
. -
.
. - - - .. . . . . .
-
''. ..
-
.
-.
-- . . .
. -
-
.
.
-.,
-.
.
-.
..
.
.
. .. .
.. --j;y';;y'. .
.
. .. .
.
-
.. . . .. -
.
.
.
',-
-
.
. .
-'.
. -11;1.
-
.
' . -. . . . . ..
-
. .
.
. .
.
.
-
. .
.
--. .
.
.
. .
.-
.
,,-jy;g'.
-
-
.
. -
. .
.
. . -
-
-
.. .
. .
.
. . -
.
.
-
.
. .
.
.
-
.
.
-
-
.
. . .
.
.
..
.(t'.
.
--
.. -. .. .. ... . . .: :...... :. . . .. . . . .. .... . . j;E.(fE..'.');'.75..E;;;:.'.'.'....:..k:)r.;:..'E:.::.:'..T:'E'26'.:rt'::'.:2.i:#.;.;i;ii.t'.iE;'a;':j'.';)u(: .
.
--
-
.
.
.
.
.. . .
-
.
-
.
..
..
.
.
-
-
. .
rjt, .
.
y
''!?!'.''
iEijrqj
1.4 .
-
. -
-
rijjt
-t-.
. . ..
...
.-
.',
-
.
.
.-'(
.
.
-
. , .
--
. ,'.,,.
,
-
.
-
.
. .
.
,
.- .. -. . . -
,
. . - .
..
..
-
-
.
-..
-
.
. ..
.
, . .
.:
.
.
.
.
,
,
.
.
,
..
iill) liij rjjrg
Eilll.i!,t' E(:Ej
l':E E
;))) 'vy,
-
-. .. .
.. -
--
. .
-
.
. .
. . .
..
1
.
.
o
-
.
,
.
-
-
- .,
..
.,
-
. ,
-
. , .. - , . . . -
. -,
,,;
-
-. . . -
-
)i!.j,k ;' :
.
E(!)';'t),
.i ( ...i;4j) .
. - ,
-
l/ljitylj ityjt
.
-
,
.
.. ,,. .
-.
.
,
.-.-.r.I.r.-.t)---y1.
Practlce
ln
. , . ,. - . . . . ... . .. . .. . . . . . .
-
-
.
-
.
. .
.
. .
,
-y','ty.
.
,
.. .
..
..
,
. -. . .
.
-
.
. . -. . . .
- -
,
,
. .. .
.
. . ..
,
-
.
.
-
.. . . .
..
..
-
. .
...
.
.,,
.
iitj?
opuon
optlon
..tj.
j .y.1.i(( ic1) E:':jF Eitj,
EttE,7''E
.
; ( /1j8t(':j'' EEllijji :
E()tly: Eilf'lilcsli't
vet,
.
.
y
:.
i!
5 0
(i)E F*A
t(!
,''q$E
.
:
:
.
E
.
. .
.
y
:
:
.E
Eg .jytj '.1 ; . : .. r'.ltt.r)ir.. i..fI ... ... . E
. j/gtj Ej
(,t. jj.
'i( 'k).y).),
,
E
q( yjE E j E( jt ' ';.jjy.j...y'-...... ...E;E..-EE.E E ! '.'. . Ljjj
,
:
,
:
: . E:.: '
:: -.:.:E..-.; ' -. .
'.
-.
.
.
.'.
stj
x.:
c
.
2.4
.
li (1
Fwmsz7jLlr7l) . ,
.
.
.
.
gyj#.gs...'
.
g
.
)
'
38
33
.
43 44 44
Lon B posi ti ons Short Posltlons
..;'
l
.
POSlt1OnS
.
.
.
y
r
.y
.
optj'ons
e Insurance 45 Insurance Is a Put Option 4.5 uomeowner's But I Thought Insurance Is Prudent and Put Optlons Are Rlsky 47 Also Insurance 47 callOptlons Are
L,.-
'.
.;F''X
.
.
.
l I
.50
1.
.5
.
Retldinl Problems -$W Appendlx 2.X; More o11BJflq/i. o j-tton 6 6 D vi de n ds .5-3
an d o p tio ns 21 1 Fomard Contracts
21
The payoff on a Forward Contract Graphlng the Payo jj On a y'omvard Contract 2.5
.
.
a Stock
.,.,.,
,,.
1
j
!
j
.)
'
.. .
:?'.
E
.
F
F, l SU NCE, 2.6 Example: Equityin'ked. CDs . 48 Graphingthe Payoff on the CD 49 IIEEEEEEE.. GING AND sl PLEST TEGIES lli''''''''lll of the CD Economlcs 19 Equity-Linked CDs7. 1 why Cyapyy ywp hapter 2 An Introduction to Forwards FJO-l/7Cr
.
ri E :. (.E.E/;(t@.. .;i.; ..(..(.jj ))15*;/;y;, .'; i ...E . . kj..3bI11L..; j': .. . . . '. ..lljsjl. ...
,..
.:7
.
.
.....
Qrii '.E(E:
.
aa
KK
.!!;.
,) EE.'EE%
.Eii''lE/)
..
j). i
jjjjyyjpjuu?
liiytj liiyj. iiEii't
,
Payoff and Prolit for a Purchased Put (; P ( j*o j.j g# Payoff and Profit for a Written Put UPtlOn 40 T jj e pjoneyness of an Opti On 43 Summary of Fonvard and Option .....
.
.
. ,'.
.
payoff and Profit for a Written Call
.
.
.. ( .)'tltrjj
t:;l;y..,l t
. . .
'
.... .
''''''
,,
-.
.,
.,,,yj,
.
.
.''!tl.'. 7litk tE)l'))y
-
-
.
,
.
.
Put Optlons
q
---C!'-.ryjjjj''
.
1
.
11
.,
.... Ijljtj
.
.
.
,
,
,
.
-
6
.k
.
.
,lt.Ey.
i q :i ri!gE. i.k
,
..
-.. ,
-
,
..
() P tion Terminology an1 Payoff and Prot for a Purchased Call
uk/appsw-),
::
:.
.'k.'ii,. j
.
..
.
,
.
-
.
,
-
-
. ,
,
,
.
-
-
.
..
..
.
-
.
.
..
'a
.
(. '.E'.jy'i ...!('. '...jyjr;.i
-
.
.
-
.
.
,
. - ..
.
.
,,
-
.
.
,
.
.
.
-
..
..
.
-,
-.
.
.
. . .
.
, .
.
. , , . , -
.
.g .
-
.
,
-
-
-
.
-
-.
, -.
. .
. .
..
-
.
-
..
-
-
-. -
- .
..
-j'.
.
.
...
.
.
. -
,
,
. . .
.
.
.
.
.y
. . . -. . -.
-.
-
.
-
.
-; .. .j'-
. .
.
.
. -
.
-
,
..j'-. .
.
..
..
.
.
.
.
-
,
-gj.
, ..
-.
-. . - . . . . . . - - -
.
.
-
-
.
.
.
.
,
.
.
.
-
.
.
.
..
-
.
. .
.
-jjjjy'.
Comparlng a Forward and Outrlght P urc 11 a se 26 Zero-coupon Bonds ln Payoff and Profit ojagrams 28 cashSettlement Versus Delivery 30 creditRisk 30
.
,9$44;.
E!E: :... iEli.tll).)ty :... ..
.. ... . ..
.
.-
! j!
Buylng an Asset 11 Short-selling 12 I'he Lease Rate of an Asset 14 Risk and Scarclt'y m Short-selllng . j t; cbapter Furtber Readvlg 16 dlriih (;.(( '.:t;lr ;,.,r::;) r''!r .j!r:' (iiip-
'.'l)t
..r.
.:. . :
.
-
.
.
..
-
.
. .
jj i
y...j
.
Assets
:'i'):#). l'.'k Ei'j...'ttl.j
. .
-.
-
j
.
,jjjj
E:t!E:E)
.
- . .-
.
.
.
.
Growth in Derivatives Trading 7 How Are Derlvatlves Used.) j. g Buying and Short-sellinao Financial
,F,E
'.r.E:.j.....t
. -
-
.
. .
.
-
.
. .
. . -
-
.
-
.
-
,
' . . . .
.
- --
'
. .
- . .
.
. -
-
-
-
.
-.
-
.
. -
-
.
. . . .
-
-
.
-.
-.
-
.
-
.
.-. .-. .. .
.-
,
,
-
-
-
. -
. .yt'.,jt'-
,
.
.
.
..
.
.
. -. . . -
.
.
,;y
-
. -
.
.
-
-
.. -
.
.
-.
,
-
-.
-
-
.
..
.
. -
.y
..
,
- .. . . -. .
,
.
.
. . - .
. ..
-.
.
.. .
,
,
. ...
. .... . ,. .
.
-
. .
.
--
..
.
-
.. .
.
.
-
.
..
.
.
.
.
--
-.
..
-
,.
.
. .
,
-
. . - ., ,.. . .
,
.
Denvatwes
('EE.jI .( : (
.
-.
-
xxl
.
q
'
-
- -
. .
-
.
-
.
'
.
-
-
.
. .
.
'
,'),t$
.....))yj.
..
-
-
-
-
-
-
,,$,22444.,
1.2
.
i,.
. -. . . -
. - -
-
--
.
.
.
.
.
't'i1
.
.
. -
-
.
-.
-
.
.
-
,
-: -.
.
yknancial Engineerlng and Securlty Deslgn The Role of Fmanclal ets ar tjjjgjj;;yys y'lnancial Markets and the Averages 4 Rlsk-shatlng
,E:21;
'
.
.
.
-
. -
-
.
. .
. . . . . .
-
.
.
-.
-.
-.
.
..
.
. .. . . ,
- . . .. . . . . .. .
.
.
-
.
. . .
. .
-
.
..
--
-
. .
-
. . .
. . .
. . . . -
.
.
. .
-;
.
.
..
-
-
.
. .
. ,
.
.
. ....; jjr'. :))'.i,j .. :.. . .,)'gr1
.'
.
'
. -
-.
.
s'i .. ....'. .. . . . .'. -
o
E.E.:ii
.
.
- ..
-.
.y
-
. -
-
.
-
. ..
. .. . .-
..
-.
-
-
.-
-y
-
..
. '.
.
-
.
-
.
.
.
-
.
-
. .
. .
.
.
. .-
.
. .jjj'.
,
-
-
-yjj-
.
.
.
.
-
.
-
.
,-
,
-
. . .
.
.;r.
.
-.
. .. . .. .
-
-
.
-
,
-.
...r5jt.*..-:4t*2.),,;,;*.
''
.
. -('. ..
. ..
.
-
..
-
-
.
...E)E)i;E
---.
-
'
.
-
.
.
,
.. .
.
.
-.
.
.
,-:
..
.-.
.
.
-
.
.
- .
-
-.
. .
.
-:
.
-
.
.. . . .
-
--
.
. . .
.
at Is a Derlvatlve? Uses of Derlvatlves
1.1
ii;
-
-'
. - .
-.-
-
. . -
.
.
:
. -
'- ' -
.
. - ..
.
jaapter 1. Introduction to Der'lvat'lves 1
'ii
'.Ej.
-;jpk!
-
.
'. .
..
-
-
.
i
:
'
-
.. . . ..
-
.
.
-
.
-.
.
.
.
,
-
..
.'.
-
-
.. , - ...
-
.
-
.
.
-)'.
.
Preface
'.jEjil'i
:
'
.
.
-'.
..
..
-
-
'
.
.
-
- ' '. .
.
.
-
-
..
. .
.
.
. -. .
-
-
.
... .
.
-(;p.'.
'
-'.
.
-
..'. . -
-
..
,
,
. . -
. --
... .
.
.
.
-
-
.
,
.
.
.
.
.
-
.
.
-
-.
-,
. -
.
.
-
.-.
.
.
. .
-
-
'
.
.
.
.
-
-
. .
j
E :j
:
t''.! ;:
-
. . -
'
'
-
'.
.
.
-
,
'(.E)' i
g.: !j!. :..).. iqE.:l '
-. --
.
j
y:.
.
.
.'jy,t: . .'yg;'( . .'. . .. --
..
. .
.
.ii@ . )!'. (.
-jy'ji
.
-
- .
.
.
.
.
. .
..
.
. .
,
.
'.
:
E E
.E
..
' ' ' ' .
'
. . . . -.
''. -
.
- -y
.
.y . .
-
,,
.
-
...,..,
'
.
. .
.
- . . . .
.
..
.
.
-
.
.
. .
-
-
.
.
-tj'.
.
-
.
.
.
.
.
.
.
. .
-
-
-y
.
.
,
..
-. .
-j
.
-
.
.
-
,
-
.
-
-. . .
.
.
--. . , . -. -
-.
-
-
. -.
.
-
-
. . .
'-
. .
-
-
'. ..
.
.
.
.
.
.. .
.
-.
-
-
-
-
- .
-
.
.
.
.
.
.
.
.
.
.
-(g:
. .:
-
-
-
.
. -r
''. ..
' '.
'
. . .
. -
.- .
-
. - .. .
-
.
-.
-
.
.
,
.
.j!-
.
-.
-.
.
-
'.
.
. .
.
.
.
.
-
. . .
-
.
.
'
. -
.
.
'
.
..
-
.
-
-
-
j).1t'C' i: E . :..jj)(g,,;*-61112;*kj,jj1;7):1.
.-
-
.
.
'
.
'
- . -.
.
. . .. .
.. .
-!))t'.
'' -.p#):(t';jrp4))q)(-.-..i;!kpts;;l:).'--..:-';gjjjjj()':)jjt,@)l''''' .
: .
-
. -
IP/jCE
'
..
-
.
-
@ * .,11;;,..
rjjjyyjl.
% Co N
'r
CONTENTS
ElqT s
Exercise 57 Margins for Written Options Taxes
57
Quasi-trbitrage
4.3
Strategies 59 3.1 Basic lnsurance Strategies
59
Insuring a Long Position: Floors 59 Insuring a Short Position: Caps Selling Insurance 63
4.4 Golddiggers Revisited
Synthetic Forwards
3.2 3.3
3.4
-
Put-call Parity 68 Spreads apd Collars Bull and Bear Spreads Box Spreads 72 Ratio Spreads 73 Collrs 73
62
66
4.5
.
70 71
Speculating on Volatility
78
Straddles 78 ButyerfhrSpreads 81 Asymmetric Butterfly Spreads
..ztjjj;j;jjyj)j;jjjj);.
.E. '. '
Readlg .&f?-#7c?P/-(7/J/?//i.
82
Introduction to Risk 91 Management 4 1 Basic ltisk Marfagement: The Producer's Perspective 91 92 a Minimum Price
a Forward Contract
Insurance: Guaranteeig with a Put Option 9) lnsuring by Selling a Ctl Adjusting the Amount of
'
9.5 96 tnsurance
Basic R-iskManagement: The Buyer's Perspective 98 Hedging with
a
Forward Contract
(.1....... .
.''
(....
'
'
. t Jwyjjjijrgjjjgklgkq.'re..'
O
.
.
.
98
1.50
Currency Contracts 154
Currency Prepaid Forward Currency Forward 156 Covered Interest Arbitrage
' ' y 'jj.yjtjty)j.y'..7j. 'jy . r)1j4;jj;4:1j:j/-*:15j;54J1j1E114154;51;14514551j!q7j:..... .''
.(.'.
.
x;.r
.
..
S.7
Eurodoiar Fumres Cbapter F/fr/'/pc?-Reading uifpp/tzo,
.5
127
Analogy 128 Pricing the Prepaid Forward by Discounted Present Value 129 Pricing the Prepaid Forward by Arbitrage 129 Pricing Prepaid Forwards with Dividends 131
.$.3 Forward Contracts on Stock
Uses of Index Futures
Asset Allocation 1.50 Cross-hedging with Index Futures
133
Creating a Synthetic Forward Contract 13' Synthetic Forwards in Market-Nlaking and Arbitrage 136 '
1.55
6.6 Carry Markets
Gold Futtzres
'
184
187
Seasonality: The Corn Forward Market 188 6.9 Natural Gas 191 6.10 011 194 6.11 Commodity Spreads 195 6.12 Hedging Seategies 196 6.8
Basis ltisk 197 Hedging Jet Fuel with Crude Oil Weather Derivatives 199 Cbapter 5';f77;772Iz?-.y 200 Readiug 201 f/rt/pcr Problelns 201
160 162
Chapter 7
199
lnterest Rate Forwards and
Futures 205 7.1 Bond Basics
,
205
Zero-coupon Bonds 206 Implied Forward Rates 208 Coupon Bonds 210 Zeros from Coupons 211 Interpreting the Coupon Rate 212 Continuously Compounded Yields 213
Chapter 6
'
181
Gold Investments 187 Evaluation of Gold Production
156
Commodity Forwards and Futures 169 6.1 Introduction to Commodity Fomvards 169 6.2 Equilibrium Pricing of Cornmodity Forwards 171 Nonstorability: Electricity 172 Pricing Commodity Forwards by Arbitrage: An Example 174
179
Storage Costs and Forward Prices 181 Storage Costs and the Lease Rate 182 The Convenince Yield 182
160
162 13?-olp/t?/pz.s Appotdix .%A:Taxes f//ot.f tbe Fortvard Price 166 and Appenix 5.B: Equating .Fo?-r&tz7Ws Flfp/rcs 166
176
Forward Prices and the Lease Rate
'00
5.5
ix
The Cpmmodit'y Lease Rate 178 The Lease Market for a Commodity 178
.500
Financial Forwards and 127 Futures 5.1 Alternative Ways to Buy a Stok 5.2 Prepaid Forward Contracts on Stock 128 Pricing the Prepaid Fonvard by
(7
with
.
Chapter
Chapter 4
Hedging
1
.'
142
Fumres Contracts
The S&P Futures Contract 143 and Marking 144 Margins to Market and Fotavard Comparing Futures Prices 146 Index Arbitrage in Practice: S&P Arbitrage 147 Quantolndex Contracts 149
108
FORWARDS, FUTURES,AND SWAPS 125
85 86
,s'/f//p//t-fr.),
.. w
PART
Example: Another Equity-Linked Note 83 Cbapter
,..j
5.4
Selling the Gain: Collars 108 Other Collar Strategies 112 Paylater Strategies 113 Selecting the Hedge Ratio 113 Cross-Hedging 114 QuanrityUncertainty 11 Cbapter Slfnc///t'Il3! 119 Reading 120 FJ/?W7cr f'?-t//p/t?a/.120
-
139
Does the Fotavard Price Predict the Future Price? 140 An Interpretation of the Forward Pricing Formula 141
Reasons to Hedge 103 Reasons Not to Hedge 106 Empirical Evidence on Hedging
lnsurance, Collars, and Other
Chapter 3
An Apparent Arbitrage and Resolution 175 Pencils Have a Positive Lease Rate
No-Arbitrage Bounds with Transaction Costs 138
..58
%.
.
7.2
Forward Rate Agreements, Eurodollars, and Hedgng 214 Fonvard Rate Agreements 214 Synthetic FRAS 216 Eurodollar Futures 218 Interest Rate Strips and Stacks 223
CoxTElq'rs
k: Cox'rEx'rs Duratiop and Convexity
7.4
Duration 224 Duration Matching Convexity 228 Treasury-Bond and Trasury-Note Fumres 230
Repurchase Agreements Cbapter s'lf//i/lzf/r), 2.3.5Readilq 237 Furtber Pp-o/p/E7p7s 237 Price Appendix 7.A: Ilterest Rate and .8077:-/ Co/il/Ewlon. 241 Bonds 242 Bills 244
'
Chapter 8 swaps 247 8.1 An Example of a Commodity Swap
247
Physical Versus Financial Settlement 248 Why Is the Swap Price Not $20.50? 250 250 The Swap Counterparty The Market Value of a Swap 253
lnterest Rate Swaps
2.54
A Simple Interest Rate Swap 254 Pricing and the Swap Counterparty 255 Computing the Swap Rate in General 2.57 Th e Swap Curve 2.58 The Swap's Implicit Loan Balance 260 Deferred Swajs 261 Why Swap Interest Rates? 262 Amortizing and Accreting Swaps 263
Currency Swaps
264
Currency Swap Vormulas 267 Other Currency Swaps 267
268
Commodity Swaps
The Commodity Swap Price 268 and Swaps with Variable Quantity Price 269
8.5 8.6
Swaptions 271 Total Return Swaps Cbapter
,J//?27nJ7'y
274
'
E
.
'.
. .
?.
.
.
.
.. .. . .
. . .,
-
.. .
;..
.. .
.y:;!t:y,t!:!;;k;;.:
. .
PART THREE
..
.p5L
(Ej).y)j#.?-.zs'.g')
%..gu' ... .. '.
... .......
. .
. .
$,.--r2E!j;!)?... . kjl .....
OPTIONS 279
Chapter 9
Parity and otheroption Relationships 281 9.1 Put-call Parit'y 281 Options on Stocts 283 Options on Currencies 286 Options on ljpnds 286 Generalized Parity and Exchange
Options
10.2
Two or More Binomial Periods
10.3 10.4 10.5
Put Options 328 American Options 329 Options on Other Assets
,slfp//atzry
Chapter 10
Binomial Option
Pricing: I 31 3 10.1 A Oneeriod
Binomial Tree
313
ing the Option Price 314 c The Binomial Solution 315 Arbitraging a Mispriced Option 318
A Two-periotl European Call 323 Many Binomial Periods 326
Chapter 11
Binomial
Pricing'. 11 343
1..1.4 11.5
330
Chapter 12
11.1 Understanding Early Exercise 343 11.2 Understandlg Risk-Neutral Pricing
11.3
3.46
The Risk-Neutral Probabtliv Pricing an Option tiing Real Probabiliries 347 The Binomial Tree and
Lognormality
Estimating Volatility 360 Stock; Paying Discrete Dividends 361 Nlodeling Discrete Dividends 361 Problems with the Discrete Dividend Tree 362 A Binomial Tree Using the Prepaid Forward 363 Cbapter &f77;D7tz?-), 36..$ Readlg 366 htrtber Problenls 366 Appendix 11.A: Priciltg Optiolls 1t,,9/7 Trtte Pl-obabilities J 69 Appeltdix 1 1.B: V/Jy Does Risk-Nelt 1n7/ Pric'g Fb?-/,:? 369 Utility-Based Valuation 369 Standard Discounted Cash Flow Risk-Neutral Pricing 371 Example 372 Why Risk-Neutral Pricing Works
323
option
The Black-scholes Formula 375 12.1 Introduction to the Black-scholes Formula 375 Call Options 375
346
12.2
Put Options 378 When Is the Black-scholes Formula Valid? 379 Applying the Formpla to Other
Assets
351
The Random Walk Model 3.51 Modeling Stock Prices as a Random Walk 3.52 Continuously Compouded Returns 353 The Standard Deviation of Returns 354 The Binomial Model 3.5.5 Lognormality and the Binomial Model 3.$5
M
Alternative Binomial Trees 3.$8 Is the Binomial Nlodel Realistic? 3.59
Option on a Stock Index 330 Options on Currencies 332 Options on Futures Contracts 332 Options on Commodities 334 Options on Bonds 335 Summary 336 Cbapter 51/771772t?ry 337 Fltrtber REutii/g 337 Pro/p/cais 338 Appendix 1 0.A: Taxes and Option Prccs 3#1
287
Options to ExchAng Stock 288 What Are Calls and Puts? 289 Currency Options 290 Comparing Options with Respect to Style, Maturity, and Strike 292 European Versus American Options 293 Maximum and Mipimtzm Option Prices 293 Early Exercise for American Options 294 Time to Expiration 297 Differenr strikePrices 299 Exercise and Moneyness 304 305 Cbaptel. Fffrl/Jc?Readiug 306 Prolllca/s 306 Appendix .9.42 Parity Soi/atfs JorAmerica'' Opfos.s J 10 Appendix 9.B: Algebraic Proojs of Strike-price Relations 31 1
Omptlt
272
A Graphical Interpretation of the Binomial Formula 319 Risk-Neutral Pricing 320 Constructing a Binomial Tree 321 Another One-period Example 322 Summary 322
lztt?-1/7c ;- 1 c/ tlillji ;!77-$ ? Iel31 177-() s 1!7:-$
223
k
379
Options on Srocks with Discrete Dividends 380 Options on Currencies 381 Options on Futures 381
12.3
Option Greeks
382
Deflnition of the Greeks 382 Greek Measures for Portfolios optionElasticity 389
388
xii
XhwC
'r
Chapter 16
Problems Formldas for Appendix 14.A: zarcn.g Eotic Options 466 Asian Options Based on the Geometric Average 466 Compound Options 467 Infnitely Lived Exchange Option .,/62
12.4
12.5
Ptlrchased Call Option 396 Calendar Spreads 397 lmplied Volatility 400 Computing Implied Volatiliv 400 Using Implied Volatility 402
Perpetual American Options Brrier Present Values Perpetual Calls 404 Perpetu'l Puts 404
cbapter
13.5
403
403
z05
Fttrtber Reading 405 Problems 406 Append 12.AJ T/7cStak'dard No7'/af/l Distribtttiol' 409 Option Appendix 12.B: Formttlas Jt??Grqcis 410 Delta 410 Gamma 410 Theta 410 Vega 411 Rho 411 Psi 411
Market-Making and Delta-Hedging 41 3 13.1 What Do Market-Makers Do? 13.2 Market-Makef Risk 414
417
422
429
..
' '
rr
...-
14.4
ajaopiju/
Using Gamma to Better Approximate the Change in the Option Price 423
,tf
R.
'
.' .'
.
.
'
.
.Uw'Sj.'l
jl.tulj)J..')') jrqlrjpkkj.. ' ''''. :'
''
' '.'
' S
'.
ts J1!. 14)/
'o.jll
Other Bond Features 522 PutWarrants
Compensation Options
16.3
Whose Valuation? 5l5 Valuation lnputs 5l7 An Alternative Approach to Expensing Option Grants 528 Repricing of Compensation Options Reload Options 532 Level 3 Communications The Use of Collars in
Acquisitions
459
455
15.5
Cbapter F/frl/pcr Readq P?-o/p/c//is 543 ,slf//p/cfp--p
482
488
Strategies Votivatedby Tnx nnd Regulatory Considerations 490
Cbapter Fl/rf'/7crReading fl?-o/l/Ev/i.s 498
,9//77/72/?-)/
498 498
490 49.5
542 542
Chapter 17 17.1
486
Capital Gains Deferral' Tax-Deductitnle Equity
538
538
Options in Coupon Bonds 482 Options in Equity-lwinlcedNotes 483 Valuing and Structuring an Equity-Linked CD 483 Alternative Structures 485
Gold-Linked kotes 486 Notes with Embedded Options
523
The Northrop Grumman--f'lkv Merger
Bonds with Embedded Options
Golddiggers
!20
16.2
15.4 Engineered Solutions fr
Gap Options 457 Exchange Options 459 ,9//?277;t?-y
''$.';'.*
Zero-coupon Bonds 474 Coypon Bonds 475 Equity-t-inked Bonds 476 Commoditplnked Bonds 478 Currency-Linked Bonds 481
Compound Options 453 Compound Option Parity 454
European Exchange Options 461 Cbapter Reading' 462 .lhf)-#7c?.
;..
Finacial Ehgineering and Security Design 473 15.1 The Modigliani-Miller Theorem 473 15.2 Pricing and Designing Structured Notes 474
4.50
Stocks Options on Dividendaying with Compound Currency Hedging Options 456
14.5 14.6
.'
Chapter 15
Hedging Problem 44.5 Oprions on the Average 446 Comparing Asian Options 447 An Asian Solution for XYZ 448 Types of Barrier Options Currency Hedging 451
.'..'.N;>. J . (
FINANCIAL ENGINEERINGAND APPLICATIONS 4 71
Exotic Options: I 443 443 lntroduction Asian Optims 444
Barrier Optiops 449
a*
503
Debt qnd Equity as Options 503 Multiple Debt Issues 511 warrants512 co'nvertible Bonds 513 Callable Bonds 516 Bond Valuation Based on the Stock Price
PART FO
Chapter 14
14.3
.
corporate
Applications 503 16.1 Equity Debt, and Warrants
'
Market-Maha-ng as Instumnce 436 437 Sttmman' 438 C/pt7pf'c?. FJ/rl/JD438 Readq p/-o/p/tu/ps 438 Appotdix 13.A: 'Thy/op.Series 44j Approximations Appendix j3.B: Greeks j;; tbe Model 441
413
An Exampl of Delta-l-ledging for 2 D alrs 41# Interpreting the Prolit Calculation 418 Delta-l-ledging for Several Dgys 420 A Self-Financing Portfolio: The Stock Moves One (r 422 The Mathematics of
Delta-l-ledging
Analysis
The Blackcholes Argument 429 Delta-Hedging of American Options What Is the Advantage to Frequent Re-l-ledging? 431 Delta-Hedging in Practice 432 Gamma-Neutraliv 433 lnsurance 436 Mafket-Makers
'
13.4
The Blackcholes
XYZ'S
Option Risk in the Absence of Hedging 414 oi Delta and Gamma as Measures Exposure 416
Delta-Hedgzg
13.6
14.1 14.2
Chapter 13
13.3
424 Delta-Gamma Approximations Theta: Accounting for Time 425 Undersranding the Market-/laker's Prolit 427
395
Profit Diagrams Before Maturity
-$,,77,'77,,3,),
k
colq'rErq'rs
o h1 E rqT s
17.2
Real Options 547 Investment and the NPV Rule
548
Static NPV 548 The Correct Use of NPV 549 The Project as an Option 5.50 lnvestment under Uncertaint'y 551 A Simple DCF Problem 551 Valuing Derivatives on rhe Cash Flow 55l
Evaluating a Project with a z-eear Investment Horizon 54 Evaluating the Project with an lrnite lnvestment Horizon 558
Real Options in Practice
Lognormal Confdence Intervals 600 The Conditional Expected Price 602 The Black-scholes Formula 604 the Parameters of a Estmating 18..$
558
Peak-Load Electricity Generation Research and Development 563
559
Commodity Extraction as an ' Option 565
17.4
Lognormal Distribution How Are Asset Prices Distributed? 608
Single-Barrel Extraction under Certainty 565 Single-Barrel Extraction upder Uncertainty 569 Valuing an Infinite Oil Reserve Commodit'y Extraction with Shut-Down and Restart Options Permanent Shutting Down 574 Investment Whe Shutdown Is Possible 576 578 Restarting roduction Additional Options 578
. . ..
.
,... .
.
..
.
..
lrjjjjkjr'ajjjK.zkjjj/kukr
PART F
...
.
jk.j.;tryqt!(,q()..;;.) !rqqq4j,j;.z;.
. . . ..
.z
587
The Lognormal Distribution 593 A Lognormal Model of Stock Prices 595 Lognormal Probabilit'y Calculations 598 599
Computing Random Numbers
621
Simulating Correlated Stock 643 Prits
Monte Carlo Valuation
P?-o/J/cp/.
19.6
Ecient
Control Variate Method 630 Other Monte Carlo Methods 632 Valuation of American Options
21.1
646
21.2
Geometric Brownian Modpn
The Shqrpe Rqtio 659 The Risk-Neutral Procss It's Lemma 663
about
msk-Neutral Pricing
655
660
Functions of an It Process 663 Multivariate It's Lemma 66.5
Valuing a Claim on Stl 666 667 The Proceqs Followed by
The Black-scholes Equation Verifying the Formula for a
681
Derivative 683 The Black-scholes Equation and Equilibrium Returns 686 What If the Underlying Asset Is Not an Investment Asset? 688
Brownian Motion and It's
Lognormality 65.5 Relative Importance of the Drif't and Noise Terms 6.56 Correlated lt Processes 657 Multiplicaton Rules 658
20.4 20.5 20.6
The Black-scholes 679 Differential Equations and Valuation uder Certainty 679 The Valuation Equation 680 Bonds 680 Dividendaying stocks 681 The General Structure 681
Defnition of Brownian Motion 650 Properties of Bmwnian Motion 651 Arithmetic roWian Motion 6.5'3 The Ornstein-hlenbeck Process 6.54
20.3
675
Equation
Lemma 649 20.1 The Black-scholes Assunptin Stock Prices 649 Brownian Motion 650
624
Monte Carlo Valuation
Pl-ol7/tD//,s
Chapter 21
A pE'a dix 19.A: Fora7i//uz.JorGeometric Average Options 648
Chapter 20
XV
Fitrtbel. Reading
,9f?/7?/;f7?-),
Monte Carlo Vatuation of a European Call 62,5 Accuracy of Monte Carlo 626 Arithmetic Asian Option 627
19.5
Jumps 643
Generating 11 Correlated Lognormal 644 Random Vriables C/7f7p 645 ter Fltrtber Readiltg 645
Using Sums of Uniformly Distributed Random Varia es 622 Using the Inverse Cumulative Normal Distribution 622
Simulating a Sequence of Stock Prices 623
Converting p.NormalRandpm Variable to Standard Normal 590 Sums of Normal Random Variables 591
Probabilities
619
Simulating Lognormal Stock Prices 623
Chapter 18
18.4
Monte Carlp Valuation Computing the Optio Price as a Discounted Expected Value 617 Valuation with Risk-Neutral Probabilities 618 Valuation with True Probabilities
THEORY 585
18.2 18.3
605
Chapter 19 19.1
ADVANCEDPRICING
The Lognormal Distribution 587 18.1 The Normal Distribution
19.9
.
19.2 .
The Poisson Distribution 636 Simulatirlg Jumpswith the Poisson Distribution 639
,sfap/pfp-.)#
Cbapter .Flf?-#pcrReadinz 580 P/-ol7/cpo.$'580 Appendix 17.A: Calcttlation of Opfi??C$'# 583 Tinte to Drill a?: Oil 'T# Appendix 17.Bt T/pcSoltttion tf/ff/; Sbtttting Dorfal and Rcsfflrff/i,g 583 .,
19.7 19.8
Multiple
Histograms 608 Normal Probability Plots 609 6.13 Cbapter lhtrtber Reading 613 Prola/eaos 614 Appendix 18.A: T/lc Expectation ofa fwogaongt'l/Variable 615 a Ntllwffv Appca-x 18 rB: Casfnfco-n,g . .. ( Probabilit Plot 616
jiib
sifp.n//7f/rl,
..
k.
colql-Elq'rs
%.CONTENTS
xiv
21.4 21.5
690
Interpreting the Black-scholes Equation 690 The Backward Equation 691 Derivative Prices as Discounted Expected Cash Flows 692 Changing the Numeraire 693 Option Pricing When the Stok Price
Can
Jump
696
Merton's Solution for Diversifiable 697
Jumps I
Cbapter 5'J/777777f7r, 698 Fl/r#pcr Reading 698 .'I'h-olalEv/z,s 699 Appoldix 21 .A: Mltltivariate Black-scboles Analysis 700 Appendix 21.B: Proof ofpropositiol'
701
.'1
668 Proving the Propositior @ Specifc Examples 669 Q?7 670 Valuing a Claim on
630
.%l
633
20.8
Jumpsin the Stock Price Cbaptel. 67# .9/77777:f7?7,
672
Chapter 22 Exotic Options: 11 703 22.1 M-or-Nothing Options 703 Terminology 703 Cash-or-Nothing Options Asset-or-Nothing Options
704 706
21.1
k
xvi
Cox'rElq'rs CONTENTS
Ordinary Options and Gap Options Delta-l-ledging All-or-Nothing Options 707
706
All-or-Nothing Barrie Options
710
Cash-or-Nothing Barrier Options Asser-or-Noching Barrier Oprions Rebate Options 716
Barrier Options Quanto, 718
22.3 22.4
22.5
22.6
Variance and Volatility Swaps Pricing Volatility 759 xtending the Blackcholes
763
710 715
Currency-t-inlced Options
724
Foreign Euity Call Struck in Foreign Currency 728 Foreign Equity Call Struck in Domestic Currency 729 Fixed Exchange Rate Foreign Equity Call 730 Equity-Linked Foreign Exchange Call 731
Pricing
Measmement and Behvior ' Vol at ility 744
Value at lkisk for One Stock 815 817 VaR for Two or More Stcks Poitfolios for Nonlinear 819 VaR VaR for Bonds 826 Estimating Volatility 830 Bootstrapping Return Distributions 831 Issues with VaR 832 Alternative ltisk Measures 832 VaR and the ltisk-Neutral Distribution
Model 790 A Binomial Interest Rate Model
24.5
The Black-Derman--foy Model Verifying Yields 802 Verihring Volatilities 803 Constructing a Black-Derman--fby Tree 804 Pricing Examples 80.5
PART SlX
APPENDIXES 871
Appendix A
The Greek Alphabet
Continuous Compounding 875 B.1 The Language of lnterest Rates 875 B.2 The Logarithmic and Exponential Functions 876 Changing Interest Rates 877 Symmetry for lncreases and Decreases
878
837
Problellls
Appendix C C.1
Jensen'sInequality
881 Example: The Exponential Function 881 Example: The Price of a Call 882 Proof of Jensen's Inequality 884
C.2 C.3
26.2
Credit Risk 841 Default Concepts and Terminology 841 The Merton Default Model
26.3
Default at Maturity 843 Related Moels 845 and Default Experience Bond Ratings
Appendix D An Introduction to Visual Basic for Applications 885 D.1 Calculations without VBA 885 D.2 How to Learn VBA 886 D.3 Calculations with 'VBA 886
26.1
847
793
Using Ratings to Assess Bankruptcy Probability 847 Recovery Rates 8.50 Reduced Form Bankruptcy Models
796
26.4
Credit Instruments
873
Appendix B
83.5 Subadditive Iisk Measures Cbapter ,Slf?/pplzp-)?838 Rfr//pcr Reading 839 l''?-ol//tDns 839
Chapter 26
The Rendelman-Bartter Model 785 The Vasicek Model 786 The Cox-lngersoll-ltossModel 787 Comparing Vasicek and C1R 788 Bond Options, Caps, and the Black
Zero-coupon Bond Prices 794 Yields and Expected lnterest Rates Option Pricing 797
of
Time-varying Volatility: ARCH 747 The GARCH Model 751 Variation 755 Realized Quadratic
779
Equl'librium Short-Rate Bond Price Models 785
733
Historical Volatility 744 Exponentially Weiglzted Moving Average
746
Chapter 25 Value at Risk 25.1 Value at ltiik 813
The Belavior of Bonds and Inieresr' .. . . . '. . .. . . . .. Rates 780 A.n Impossibl l Prtcing Vdl 780 Etation for Bonds 781 An Equilibrium for Delta-Gamma Approxirtions Bonds 784
24.3
23.2
and
Chapter 24 Interest Rate Models 24.1 Market-Maling and Bond
Other Multivariate Options
Chapter 23 Volatility 741 23.1 lmplied Volatility 741
Model
Evidence 771 77.3 Cbapter &fn;?77'I).)? 773 htrtbel. Rct-ltfia,g Problel'ls 774 777 Appendix 23.4
727
Exchange Options 732 Options on the Best of Two Assets Basket Options 735 Cbaptel. &f7?;/?7t7?''y736 ltrtber Ref.ltfb/g 736 Problellls 737
lhfrl/Jcr
758
xvii
C/gtzplcrSltnul'an' 866 Flf/-#pcl-Readtg 867 f'lrol-l/c/as 867
sfp7/ntp'),808 Reading 808 f-b-o/pkcp/s809 Appendix 24.A: T/JC HeatbJtefrrow-Mb?-fo/zModel 811 Clmpter
757
Implied Volarility 764 Constant Elasticity of Variance 766 The Heston Model 768
JumpRisk
717
The Yen Perspective 720 The Dollar Perspective 721 A Binomial Model for the Dollar-Denominated Investor
-
Hedging and Pricing Volatility
%
852
853
Collateralized Debt Obligations 8.53 Credit Default Swaps and Related Structures 858 Pricing a Default Swap 862 CDS Indices 864
Problons
'
884
Creating a Simple Funcyion 886 A Simple Example of a Subroutine 888 Creating a Button to Invoke a Subroutine 888 Functions Can Call Functions 889 lllegal Function Names 889 Differences between Functions and Subroutines 890
-,.' ..' .,' ,'--' .'-' .gjjy'yd ..' ',..'' -' .'.i.trd -t' --' .y' -' .j.-' -'..' -.' -(jy,' ,'.'jjky)7' .''.(' .t.jd 'y' .,-' ''..t)'t((' .'-,' -t' '-j' -'.t' -r;r' ;'7* ;)t'.' ,.' .-' ',' (' )?' ..--' i'f' y' :'2* .:' ;-' :(' ),'l(-!' !;' )' :4g,jjy'j' ;j' --' jtd ,jy' ,y(' ykd j'jFr-'!,'),d -ti' ',-,' -.;' -k))y''.' -;' -.'-kjrj)yj'.tjt'yt)..')jjjjpjj(jr' ?' -y' jjd --' -'-' --;''klt)l!pt@/llrk:;l;:i'' -(' -?' -;y'i' --' -;' -'t' :'-' -' --jyti'r'ypjd -,-,))y;$' y' -t' -y' y' 1* !' k' Crd (' q' zjkljiijyyi,lltFrd :'i;7l.-':l'k)j)qjj'@f' ''kd ikk)d j';'k.d )' y.jj(y' jjd '(,'r' '' t'''-'-)' ..' .ty' (:' y-y.,,d '.s,'d Ejd 'X ?5* ')'t(Ei1l' Lk',.Lk'jkq' (!' ?p' j'-'--')' (,' Ekd ,'Eyd fild y' --E' y-;-' y' (' '':j''-l'k.d F' !' :'',''E;' ?' ykl'yyd ttlr:E-rr'tkd i'-j'
t'djid p' y' r' jyd $' jtd j';)' )' k' f;-.;r' '')' tjd @' ;'j' E,jd tjr'td ,jy'y' !' y' yyd gtd ''.' y-d f' !t' ir.jg-yd g-' j'rd ,'')r' ''''r' E#' s'yd t'jd 'X ,!tt')' ;'!;' jjd t('j:rE(' (' jjj;(' )1* y' rjd j')d jtd j'jyyd g' q' (r' yd t'g' ,t' ?' ));' jsd ,)' $,* k' jjd j)'jr;)j);;'.' ,';' kr;i''t.)tl'r'k(#' yyy.jd ;)-' ktt'p!ktd yjd ..' i!d :tq' yyyd jtk'jyjyd :'(;.' q' jy' ;'jd 'jt' ?4:..* ijd k,grd fr!d C' 1* k;d p' (y;;t'.' $' jjgjj..' j(' F((' .)j'?,' (tr'.' r' (qjgyyj).y)pd !;' p.yd '(';'q':','t' jj,yjd jjd .j' j.(' tEd )y' y-,' yyj-d j'!-p?)'))' !' p@' p;d i')'.#' j.ry'jjjj'tyj'rj(.;yy)(jg)yj-' i,' j'td prd j'jd 'j' j,d k' (kI.' jy';d yjt' jgd jjj'yy'jjd jyy(tt,y,' j'yd ),' jjtd y' y,' y'qd jqjjd ;(,' jk(' '' 7!8'r2tE'-1...11f9r111rr97:* lp.''hd (jt-' jyd (-'-' :'i'jqd (' ?' T' 1* :);kpp';' jrd jpqtjk)ih'.' rjd ))lt'tt')'' Tq.)' yy:)yd yjd ltd jy'ljl'kd yyyrljyk.-jjjd (yy'tjd );' ()' ;#' )' yri'irpl/qi'jrjd 71')* tj?'.' j'ttj(jgjg;;)y'-' jy')' p'.r' jtd ;'(' j'j'y' )' $' $j)'.' (t' ((' )';;')$' 4* q(' ptqd yj)))))(yt'?' !q.y,'!t' @1' yjy'jjjd l';' j'yjd jj),j;k;-;',' tqrd )(:' g,'-y' tqd q' )f' /1* t;.' C)' @' j.p,d gtd )1* ltjrgd ('(' 'j' q(' t't.)td yyd )')' (-.' td !y' ytttd 4r,* j-)' y-' ;'y' y'yydyryjyd j-yydjdy'-yyd rj;d ;'t' ;-' ryd k' jtydqy'--j')d ytd qrd y.:g-' p-'-' C'qd 65* !j' (.'' (' 71* dt-fd )lt' pt'lkqtlpp':kd lijpl'l)d ;LL;')3)Lqt)L'.' (t(k' ))' g-t;,' jtk(..' y'))t-' p' !' qql)d $k)'..j)r;)((tr'.' )'.11* yd .)' yir' f)d p.j)d kjd Ljjjt:f' )t' ))t' i'y'td yjd );(..' ;,' ptytlt';d yky':;d jtd .y' yrd l'td t',j)'-' )-'-' y4* (t' y')' )'!r' /'7* jyly.djlgyrg.ryy'jtd )#' $:'!r,'ir' ,''td )))' ()' j'yd .j' yr'yy.;yd gy;d y)d ikd tjd )j' ..t((t)'..' p' tid jjzi.ttd ((j' rd r' y' y'yd ytf.'tylj-,tjyjj'!,,d (y' ;';' t.'.' $' )' tl'rid 4* prd tqspltk'd tptd y' tyd tyty'y(-j-,,jy'.' rjd k' .t)' y'y' ppt)d ?' )j)'k.jIr' yd ')' tkd q' t'-yd 1* CFX ##' trd (:r@/44),)* tpd yyd (';,),')i' (;' #' (' (* t'td 'yy';j';' t'' pd @' .)' .(' ;j' jrz:jjjjjjd jj-iiijylllkz,jijy'.sd t'(f'.' sjjd yjjj.' .;' tjd tt$l'.-' r('ryt' 11'119* td tld t't' tyd jjty'tjd t,d rp'r' .)' rr.d r' rtd ri'td ';' .r' )jj)('.' rflrpr'p'tytd r)'
'''' '''''' .'.'::d .'.' ..'. . '' .'..' ..' .'.'
r'r,yd
lLi'jLqlLlkkkqjjk'Liil..jjk. ..(.! .j(!1 '. ! : 'i.j )'':!;ilp'$ ! '?. ;('i ;tj-1! .$)j)(pj))))t!. ;(qy' 2))*j44 ;' .. r. ! . r;()';.(j.... !.i ; t;. j yj))y. jI(j)))'. jjjr.lhl':i LqLLqLILL. y'. ;. y ;ii.. j.;.i);)1*. p)'li. jj'.p')p)r'. k'i .y.lpj rt'. . jq;tt':.;.lr .k; jg)-.y'. .i 2.i@ iy ;. ..); kj': 'jj.. ;kj.j.y;..(.qi (p(;;)r--. (@pj-'$;ppjjj(())pk!t:.-(!jr(pj;. @. .E; r.@ g... ).j .j4$))).qj't;. i((';y.. . !)!t-.i j;(((.:(. ..iyg.; (; .;pj.(i.. ;q! (rr.j)#'-.)! . qy)). .-j. g.(.iq.! i;; i.(..y;'.yy.y': j!.(; . p.;;;';(!;jt'. qj.ky j!.jttt;k('. y.k t;j!.Ei ;Epi;i!(. jjjjjj)'. qqyryy)yyy jk.j-j. yg ; jy.:rj't'p ;LLLL'. jrr'. . 'gj'i .' q;qy)'' L;)', )j'. rjkdljtlldrlry.dijjjltljlrttjlyjy::r)l (rI ; i i(.E .j..!gg(..j(.;.y'. yrrjtyrgjyil'-;. r;..'r@).'..ij.r; :kj(t$'. .rt(j'jj))tr. ))jkL;btL', . gyyj))'yrjt.y)ykjjyjjjjj('..'rr)kjj-jy .y'i g ; .;j . E' tjjj'. -.-,6L..qj;'j3k3j)LLL:;);))LLLLj);LL;L'. rr)kjp)jj);. ; ;j.r1. 't)('. )LL;'jj3LL;fL1LLj.3(i6LLLf3L;LL'. t)'. t'. ).j.jj . y;j..gk;. !kjy. lr(ty!,k?'. 1',p)'. r;..)j;! ;. ; )...rg( k.jj ;.yyj q.j'. y.. jr!;y(j)'. .. ;r...jIj. ;r'. . jrj.k.r)i :t('Tti. ..r.jr j j.j'! q . p ryg j;j'. j!. @ r.. .yj-.r. tp'. .jyjy..;j)y; jj.y ; (jj',r')()yj. jI. y jp .j.. :!;jyy((yyy(..yy. gyj t;j,)))'yq)j. j#;j;rtjpljjyyjp)kjpp().. ;jL;;jL)-. ..jjj;'. 1: p' ll'k '('(tj'yy pyty. ) 1......(.j.E'.y jj....j;jj y.y.yj I ; .y; .;;y. )). lg j..j': j...y gy': ''. .. j.y; y! .tjjjji tk.. yy ;q'. .jr...rj.;.j,.j t.. . .,;j.yyyjy.yy.. jyj;;j'-. gyjj yl'j. r.y..jr jjt;'. yt(. )..tyyr... (k$'. j,)jjj(jr(((. jjy;;', t )jLL)jjjj4,,,. 41/*;)j)jj),,*r-j ;.tr). jjy . i!( y.y 1: .y. Ej.p .,l'r jtrjgjy'. j..jy((j.rir i yjkjj(y ;r. ,;;j)j?jL'jj)q)j)jj3L,.)3$ff:L)L(LLL'.. .y;.@ ly'y :. . . ., q;. ( ;. y.... jg..(y!....'. y!p. !./'.p. . i jyj.q.j(j. .i (.; j'..;j'y .;;yj;jj.'qyyi.y.. tj ;. (', t'. 77. .:I IE k r ( . tr'yy'; i .jLL;L.3(j;;;'. t'. ). '-'.'' '1!17(jj(y)jjj(;j-. . ((4))):t4(4kkjh,k*.... ! ; );;t'. .. l'y j'g ( ' . ; j . : ); tyls'j . : yrtjt'. l i . i t'; p .. i ' i '. ; p :(-'. . . '. i i . tlj'. r;. l ) .ppt'. . I r i kl . ; ?.(jj);:. . y ( ))tqilllqdrrrtl''dpk . . ! p q i i . , )'. y ,k'' ((. ; :)'! . ; ?:r i'. y.;. . ! i ... yj-; y)r (iq);-;Ii .;jq j! !;:p 2/*))(p. .p..r..k-; y... ! .-. !... .;... tr ;. q: . y .! . .(.; :;., j'. t!. l j:y.. ( )'. .: p. . ; .. .; ..j.gy;;. .i.(':; !. . ( i i. !..kj!:. k .. ;i.;. .,!(ty.: .;p)-.. i!. @ ); ; )tj(r-. )'i#tti'r)!t:t: l ; 2)*. --bjb'1LLk-Lfb1'. ! pi 7.17/*. ,': g;g . : . : . p. ; i '. . k . ( ).t'. . k tk. ;; i i 111-*p t(j.,(y).. . I i -LLL--L--.. j,k. . ;'! ;. LL.LL3Lt'. r, i , . fl;l r ! . . : ( -111*. . r ( ; . ; ! , ; ! ;.t... . j h'. t-:'. j-j ; $.. j.. ; qlqtLbllll'L'llllllL'.--'. k . kji.; . q jr : . jy-. ! )j. . , j'. . .)., . ry . (j..yr. y r ;jr.. q'?zi:i.j.!)'.y2; ;p'. jr-. .j;t'(y'r'. !q. k: jr'- !(;. kt ttt))'t.)jrt.i(k)q$t'. y; y..j-.-. (y j... , ;...q i.y.j.(tj'r Ijj-. j.i r .. jt-..)!:..jr.:qk., .j;...i..y;...rjk.jir;.jj.jj.,yr,yj.k.rjgy.j;r.y,y(y..;'!.y.qy.yj.jg. (.,-j ?'. !(' ; .t'...yjj.j. y . . yr;jtt,.-)y'. jjjyjj,t))j. !.y'g (y ;)qq-,jjtty'. . ilj-,-ikLs.Lk (;. y.(.. ,'-y g j...y.yj.y ( )'. '' '. 11.,pp;q).-i .. .. jlj. ;;i.r.j.r)?'. ; j ..jt)(jy.. .y-. : !p:$1LL($L$'. l?/. q. . ...! ....r.. j.r..j.t'. kj;, ,2 ')ti. I:ij'. . !;7. g(r)y( . ;y;'. )!. i!. tj'. 1k; ..jj. ..;( : t719*): r.y'.yr . jgr. . rp;'. .r .j..yt; . ?'. j)'. 41.811:1*'6h17*. !k-.k k.tl l'i.. )).),?LLLLLj?(L;-. ty'.y;yy'y'. : ..ry;1--*. . .p'(y'. .r.)---t'l:t'..jyr; ..yy; ;qy '.r,.r. .:rq jyt'.; ..,-.rt'(jgj ,k .j:,q ;. tg')tl):r'y. .; t'. t(-.-;;.y r..y.yjrg y.(:.j.. yt (yyy'. j). .)1*. tyy jj jky .t. j.yjjyty'i .yt. . ;... ry'. (iy. :j'jy . ;y'. .. . r.j.g.. i ry y-j r'( jjrj'. tikr:(pp'-l yy..!y (tp-ij'l-p!rtl rytyy. ygy..j....;y jyy..j. .j.. jg j.'. i. j.;'. .,,;j-,. y j,yy jj.y.y ; yr.j jy (.r;. ;!..-.y. y.(t-..yjy.;,..j.g.... j.yj,jj.j-.!ly'y (;. ji-jti 't'7(..j. jyy yy'jj.jjjj;,tgg':yyj:;y,'jj)(t!y('yyjkt$j'. , y.jt'.r:(...tyg. qt(t(@'. ilii'. . !k: lLI.. '.p: t'. : sk;ti i. ll')lk ;jj', : i; . )'. gryy j.jyj. k..jgjj jyyt. 'qi yp( . r$'; rl-gj.jyry'y . q'yk . . i . t' yr .. i r . ! !.y; r@. yy y i yyyjy.. j . titi .' ;!rtp i r jj(! )t. . . j : )41;*jjrrg; . . t' r . k'. . ;. j-. y ); . ;! j (j'. y ! lk . -;L. . ,! r)t);'. y q ; t). :. )-. . . . j i t''.l. y . . t'; t. t!y . ). ; )'. q ; j .jj. ( . .t, ; ; , . r ;'. r j . jy y ( .. ( r)k)'.. ; ,.. ; . j ry;-)yy. ; ; ; . . ; k /'. y 4t:y'.. I t ! kj)tj))'' ( q ) ; k; . ) ; ;')y,. ; . q y t! (;L,--L;L;jLq;;. ( . ; i ; ! @,r'. (yy'j.j jk .. y'. !... y.jj gjyy; .....jk.'..y.. ;y .$-(.t, '. jp!p!-jjj! t-;)p. l'i , j jy j j;jjyiy()'.j! jyj'. . j !. . . p j. . j,. y. y jg .yjy,..y ;,..., y j: .jl'y,..y.(.yy l.dqt'j. j. jy.. ;. .. y . yy..; y'r y; rj( t! tlyydyj-'ttrtjllj lk'r?qllp.g y y;ty)- . . ) )'... rp. ). t'. )'-,;-)j .. y. r t.j- ,'((, ilritrrr .; j(..jy. .j(.. k..ypikj-. b.y'. . l'.g j.tt.i jyj ;;j jyjy. jt))4. ,ky y.j.,)! lr yri ptii r#)'. q;.(y .y;. .q.rytk jyjyj.y.. .j.j...'y; yy'yy';y;y (rpyjj g(;y jy'y jy .. r. j('. . ,jy. yy'g py'tl ;y'. r:q,)q'. y.(,y..j.yg jy-y'trj3LL;';L(-;3-jLL;j. .t)'. . ) j y...y. . yy. j ljj'r j jtt'. r gg.. )r. )'. j;') .y..( .jyL,;))Lq, t'-. .ry; '1'Er''?i@.(pt.#8!ih,.));k.t:)k. q(. )'.)1). .. (rl))'. jy jyj.yjrj'tiyl y'. j.;j.jjk'. qi ti.gdp@drtd:t'l).;rtp j'.y.. :tp'. j. .yy'-'y . yy. 44:*. /pk'.?i l''-f'll' .. .(y. yt.. y.!@'. ).yy..Eijryjj. jjt(:'t(.--yj'..-j-; y..jyjjjy y.. yt'. .-j,l yjj.j.t'. !))'. j (-j.j j j....ij#(!jytj 'gy'. ... p'..(-. .....-y'....).y..yyj .j'yry. r.)( y't'. ...yjjy., .y'. yyjjj,, ));.. . )'. yi. g).....)'..jy .j.., y(; j gyg .q'g jj--lpi-jtg'. j..yy k;:.)1. . tjyj j. g.)'. yj;.i pr j; jyj. '.. .-jy'k j.yt'. . yt-, j , . ...y.y...k'...y @; (t.jy'. yj.jjjy.jj( y( tk. j.-j-t'y j.t'. (j;y.. . yy'y ; jlp', .jyy;y j.j!'.t'-.i...; (j..j'.;.y.r...ykty'. (. t!.:. j'. .$! y'1;LL'. ....y.,jj.yk-)'. -y.. ....jj..l'y ty r'ytk . .jy'.j,j. yy....jg ,tyj j)y'. yyy-. .-jyj.-'. .'' i;ijjitr('. .r r. ygi k . . . ,j);.qL'. t.. . /; )': t,r:r-. jj'i .. . jj ...... y.. . jyj'. t...1..!.)l'.pl!'. r y.. .,jy.g.,..jt..j.)..,.j.y.y t-t'j -L'. . !t;.t.j .j.jgy...g.. j''yy . (y', ri jyy')jryjyy-..,'. y'. '.. . y)'. j (j-; .. yj. jyjg rit':l . . !,y'. y'y'. . ;.. (.(j. ;.(y'g .j.y'(.j'y r'y'jpy.-i . jry.yj,.t...,,.yy . )tp'. . -If'.jj'. y', :'),j(j),')C ij(yylj... . .q'y y'. ;(;;j,-. E t(k)'. . y...jj j.k . i 33. @. i;! (jjjki-ir. t'. (jy'( (1*. )!. l(i, ; q k'. ! kjjy y'. ip)l j k g j j jy-. j g'. q . . . ;l jq(.. jq'-j()(: jyj)jj'rj-.jjj(g. tyjjtjjtr.... j'. .. . lt.r)j . . ygsj'. )i @L)(. (ltpi. qi 145. jLILLLIL?ILLL' (ryj)y. yj ( j-'! jy;)yy: r y 2 l:. ) ):: j!. (t. j!... @l; p:. LLL. )q!:.j,;;!. ll. ))1:j ( . : r . . jjk)','jjjj-;(;j.: 4 yj (;. j y . y lk y . y !;'r).jj))()).. t4j y y rys j j j yyyjyjy'q ( )(1 y . j jj)))';;'j))j)j),,..:..-,jjj);', . yjyk('; )yy-. j . . . . . jtrjrj)'. j ; . j . ty ( j . j g y . jyj',)j(tggy-y,(kyy;,(tt;-yjjkyj-,.,'. . jjtt-lkjlliki yjjjt; jyzylr:t) jyjy. g j'. ., jl'jjljyj'. r'-j yyy;'. ilds.,.tjkikgijjlkttllkjyl'.s.. jj'.. . t gyjjyyg ; ,)))-j;)Lj,j,jj,L)jj,. )ph).'. llykjyykjitjjklgsylll-,. t12*. ptak-z..dltlllltdjk-yqttjl: ' ( (!;'. y . jj'. . ttj'jj'jjy;. pl'.i ' yjllkylylilitls-y. j'y jk;jjj-'. t, y'. . )jyj'. ))'. . yjjdtjyjitp..'. t(:-,)k))-;j;j))j);)).. . j'. . . . jjj('. . jy-yj'. . tjtgy . gt'. . )jjjjyj..-j;jy;-k)'. . yjy'. t.jjjjj;yjtt..j'; )));j)k'. )!;,j;g;;j . )(L'. . . 1*. /t*. . .pt'. , kjkj'. (j',.. !j*. jjjlljjlkLL'. . .. y;y'. (;j,'))'.. . . )'.r'.)'.r;'. .1*, . y)y'. . . tyyy'. . jj iq'. $'. plljjdjd.r.llpll',r '
:
%.
xviii
CONTENTS
.
.- : . . '
-
'
'
Using a Named Range to Read and Write Numbers from a Spreadsheet 891 Reading and Writing to CellsThat Are Not Named 892 Using the Cetls Functions to Read and Write to Cells 892 Reading from within a Function 893
D.5
Creating a Binomial Tree 900 Other Kinds of Loops 901 D.9 Reading and Writing Arrays 901 rrays as Outputs 901 Arrays as lnputs 903 D.10 Miscellany 904 Getting Excel to Generate Macros for You
904 Using Multiple Modules 90.5 Recalculaon Speed 90.5 Debugging 906 Creating an Add-ln 906
Using Excel Functions 9om within VBA 893
Using 'VBA to Compute the Blackcholes Formula 894 The Object Browser 895
for Conditiorts
-checlclng
D.6 D.7
Arrays
D.8
ltemtion
Defining Arrays
lndex
899
i - !. ;:p!.. : !-yri t. .
2
935
.
.
.--'11*;-
--
'-.. -
p-
.
! y; i :! !
. ..
.
.
-.
;
-
.
. .
i
... kkllyljkig'.
.
(E
-
'
.. .
'
.
- . . - . .
-
.
.
-..
. :
t',)
E'EEE
t)
',E
itt ',))() ',E))
i')) ''')t' .. q. qltl. @ .
.q
.'E''l; 1
. ''iti '
'f't r i.' j
(''.(il1' ,
' g. EEi',,t
'.jj
I
..y..((.:(.j .)')!E!j' 'r
!j .'...
))) ; '
ijligl itttt! j.; iiili'li ) l 'j. .'!,i,.i!.t j .
')
E EE .E :
Eqi.()!
E i; .
:'.( .)Ejt.:
.$
E,!t;(
Ellr'''i
' jjilI :.' :
' lt)( ,.t)l).t (!j .g '''''' ',)(lrrj ,E;,,..),rj
:
(.' j j r
'''i'1 ,tl.lrj
(( ... .;j(
,yt't)i,j
i'''i/ll ..it, r..
j. i . .'. .gEilllt :
;.),t.j
E'E't7tl,'j:
E
. E . .EEllEljtp
E
.
yljtj j
: y: . E'E!!')' .
.
'yjjqjl
;EEE.. j ljttj q !), :l!jj E''itpjE -.
. .
, ..
.
.. .
.
-
E
: ' ' .. E .. :(E .
..
-
'.
-
:
:
.
:
.-
.
.
..
. .
-
.
-
.
-
-
. . -
-
.
.
-
.
-
.. .
.
.
.
.
.
, , ,
-
.
.
.
.
.
.,-
.
. .
-
.
. ..
.
.. . -.
-.
. . -
-
. . ..
.
. -
-
.
' :
:
: ,
,;.r.
. -
.
- -
-
-
.
-
. ,.
-.
-
.,
.
-
..
.
.
....:.((:jj..:.'.(.
.
-
-
.
.
.
.
. . .
-.
..
-.
.
.-
..
.
.y
-
.
-
-
-.
.
.
-.
-.
.
. . .
. .
-
-
.
..
.
. ..
. -
.
.
-
.
.
-. .
-
.
..
.,
--
,
-
,
. .
-
..
-
-
.
.
-
-
-
.
. . -
.
.
.. . -.
-
-
..
-
yjljijl-! y.y. j
. : j.
..i E.
.. E.i : i E...Ejy .
.
- .. ... .-
. .
.
-
-
.
,
,
'.j,, -.
-j'!
.
,
. .
. .
.
.
-'yj.'.
.
. .
,tj.j.
.
. . .
. .
.. .
. .
. . . .
. . .
. . . --y(. . -y . .
.
.
-
,
.
-.jyi
. .
. . .
. .
.
-.k.
.
-
.
,
,,
,,
..
. .. .
.
.-. . . , .
.
.
-
. . . .
, -
, , -.
-.
,. ,
.
....
,
, , ..
..
-
. . . .
. .
.
. . . .
.. .
',
--
.
. .
-
. -.
. -
.
.
.. .
-'.
.
,
jlsi
E
..g(E.y.j;!Iij.E jjjjjjj'.g .. jr...j.y(..jy !jjj(-. ..E.E..E.,.jj)#Ej'. iE q .11))11*:
.
. .
''
:
.
-
.
.
-
-
-
-
-
.
.
-
,
.
-)(-.
,.
, ,. ..
.
. .
,
.
-
.
.
. ,
. . .
.
. .
,
.
,
,
.,.-
,
.
.
.
...
. -.
.
. ..
.
.
-
. .
.
-
.'rj'. -
.,
.
jj
j
j
j
j
l r
)
.,.
'*'
-
-
.
k.
'
s.
sclqwAR'rz
.
...
. Eljltj q. . . .. .iq.tt-
.
.
:
.
: y -. .. ..
L)L),,,$; r.tj)'. ..
.
.
-
. . .,. . .
envatives have moved to the center of modern corporate hnance, investments. and the mnnagement of financ ial institutions. They have also had a profound impact on other management functions such as business strategy, operations management, and marketina. A malor drawback. however. to makinz the power of derivatives accessible to students and practitioners alike has been the relatively high degree of mathematical sophistication required for understanding the underlying concepts and tools. With Robert McDonald s Derivatives Markets, we fnally have a derivatives text that is a wonderful blend of the economics and mathematics of delivatives plicing and easily accessible to A students and advanced undergraduates. It is a special pleasure for me to introduce this new edition, since I have long had the highest regard for the author's professional achievements and personal qualities. The book's orientation is neitheroverly sophisticated norwatered down, butrather ligor that creates an inherent iexibility for the structuring of a a mix o f intuition and derivatives course. The author begins with an introduction to forwards and futures and motivates the presentation with a discussion of their use in insurance and risk management. He looks in detail at forwards and futures on stocks, stock indices, currencies, interest rates, and swaps. His eeatment of options then follows logically from concepts developed in the eadier chapters. The heart of the text an extensive treatment showcases the author's of the binomial.option model and the Black-scholes equatio crystal-clear wliting and logicz development of concepts. Excellent chapters on financ ial engineering, security design, corporate applications, and real options follow and shed light on how the concepts can be applied to actual problems. The last third of the text orovides an advanced treatment of the most important concepts of delivatives discussed eadier. This part can be used by itself in an advanced derivatives course, or as a usef'ul reference in introductory courses. A rigorous development of the Black-scholes equation, exotic options, and interest rte models are . presente d us in g Brownian Motion and It's Lemma. Monte Carlo simulation methods are a l so discussed in detail. New chapters on volatility and credit risk provide a clea.r discussion of these fast-developing areas. D e rivatives concepts are now required for every advanced linarlce topic. Therefore, it is essential to introduce these concepts at an eady stage of MBA and undergraduate business o'r economics programs, and in a fashion that most students can understand. This text achieves tu s goa j jrj such an appealing, inviting way that students will actually enjoy theirjourney toward an understanding of derivatives. EotlAqoo
.
,y,i.,,lj
:: . . .
.
. ..
.
'-
-
-,
-
Eirjljtlty
E
-
. .
-.
.
-
-
..
,
.
.
.-
.
-
.
..
.
-
.
.
.
-
.- -
.
.
.
.
.
-
.
.
.
-
. .
-
-
-
- -.
-
.
--
.
. ..
.
-
.
-
.
-.
. ,
.
-
.':jj;j .
.
..
-.
. -.
-
. .
.( .. .
.
-
.;.yj
.
,
-. ..
-. . .
. ..
.
. .
.
-
-
.
.
-
--.-.
.
.
- .
-y
-
-
..
.
-
-
-
'-
.
.-
.
-
.
.
-
.
.
.
.
-
.
-
.
.
.
.
. .
..
-.
. -
.
.
..
'.
.
-
.
-.
.,
.q
' .
-
.
.
.y
-
-
.
-
..
-
--
-
- -.
.
.
.
.
.
.
.
.
,
.
-
-
...
. .. -
-.
- . . .
..
.
. . . . . . -
-t.j-.
.
-
. -
. .
.
.
.
-
.
-
-
.
-
. ,
. .
.
-
-
,
.. . . .
-
.
-.
.
. -
lE E.E. (
j.. E. j g.
-
-.
. - . . . . .
. ,,
..*1)*.
.
-
-.i
.
.
.
-
.
.
.
.
,
.
-.
-
-
-.
.
. -
-
.
-
.
-
.
-.
-.
.
..y.
.
.
..
- - -;
.. ..
.
-
.
-.
.
,
.y
-
.
.
.
-
-.
.
. .
i;(,jiq.; (l(..E..(-) (.
E :
i. E.E 'E E:.'y @. g... j.. . kjE j g. g y. , :
-.
.
.
-
'
.
.
'*'
.')Eri
. '. '' '('(i(rii. ..
: ' .
..
.
.
'
. -. - . . . . -
-.
.
.
''''''
'.j('E: . . .. . .E,)',';'ii
:
-
.
. .-
.
-
.
-
.
.
'
. --
-
-
-
-
.. - . .
-
. . . . -
-
-
.
.
-
,
.
.
-
.
'
.
..
.
.
. . .
. . -
. . .'..(jj'. ..'.. .'.. . ..
(.j'. g .. j.....L j('j'..(jj(.j'j' j'!y,j(jj.j. (. gj(y..(jj.)..j.jy(.,..j:..j.g.j:.:;..:jr . . y..gj..j.j.j . j.jj. , .'j: j(.'jj j..
!
.
5'
-
-
.
..
.
. . .
'
.
-
-.
.!
-
.
.
-
.
.-
.
-
.
.
:'
. .'(..' ' '. ).:j'i
'
-
.
. .
.
.
.
-
.
.,'i)l!.
.
qi
.
-
.
- . -
. .
- -
(.
(
.
-
' -. -
..
.
.
-((!tj)')kpp'. .
-
. .
.
. - -
.-
'
-
. -
--(;))tt))g;j)'. ..
.
-
-
-
..
..
-
.
.
-.
-.
.
.
--
. -.
-
'.
.
-
.
--r.'t,kt
,
-
.
.
-
-
-.
.
'
.
...
..
.
-
-.!
..
.
.
-.
-
'
-
.
. ...
-
-
..
.
'.
-
-
.
-
.
.-'.
.
'
.
.
....
.
.
-
'
'
-;
'
l
'E . '(
-
..
.
.-
..
. . .
: ...:..1 . .
: :
-
-
.
'-
..
.
. -.
' ' .('i rj
:
899
jytlr'?E ))E
' .( j. .. ...j j '' .
(E!: . ;j
?21
.
.
-
.'
..
;
.-.
;
.
@,'.t.
Glossary
-
'
.
.
.
-
.-;:
-
-
'
-
E
-
-
.
.
.
.
'--''-'.$tl;)'-'@!phr.-))i)@iii1pi'--''' ' ''
-
.
. -
:
-
.
.
:E j' E E
.
.
.
-j(i't;....
'
. .
-
..;pj)j;...
'.
..
EE'E ;1)*. E E E ..l!L. E ' .':-.: '' : ' .. ! ' ' g.. ..;L.....). .'
jtjpjljppltrjjt.'
(','')
907
Bibliography
87
A Simple for Loop
896
...
-. -
-
-
.
.
-j'.
..
.
E.-.. E
'
.
.
E-
'
-
Storing and Retrieving Variables in a Worksheet 890
- TE ' - i' E-E .E i iE E- E i - . - i - :i EiE:. E i E: E i- : - E ' : i! E-- ' EE-- i E. iE -E'E i E tE. . E- Ei-'- E i (E!- -!! i- !! i!!EiEiEii - E- E;-Ei.E'-.-.. i!. :EE. Et.E .:; E-E Et:;E:; i. E.EE:.E:EtEE... E-i'E: E ;E:- !- ; ;;. Ektr'; . E E iE -E;E E! #(.! E . ; i ;JE I: Ei IEE i i E. E - E!i;.;i E.iE- E..E-- . Eii.E : E i ; . E ! E ! -yEEj-jE .iy. Eq. E . : . . .- . - E i.i.. ! ).jjjt'. - . ! : ij IE : ijy-';. :q . i - - -.: '.y:E -- - EE--E.i . ii : EEEi :, qEi E. . -'. E-, qE. :. qy'i E.. E EEE.E i qEE!Eq E: .-Ej.y-. ; ,; g( E. iydr'yj -. tiqy E i - -j'. E - y :. i - -; . E;. E !E-.;q..E .y-... -. ;.EE. E.q-. q. EE- i ;i.E ! jtE ..'!)'((p@))(--'E!iiE - k'lE ; . i- . . .E- i. E - . y.E E. !y('- !- E. -!, ;E E; yy. Ei i i I'E E E ,.E-. q ..E E y'. ( E- . EqEEqi qE E yy'..y.! E - ( , E;q-. ; --'yyjyjyjj'.pjljj-,tjss'jjjjjlyy. , -'E - ., y , - q , y. E -' .E' y-y..i' . (g . y E- .- - -. ,y ;; gy qq'jr-. j!E !!!qEE-'E -..y;.Ey y !PE q . . .- . ' . - y E: (- E y;E. -- - 7E!1 --i y y y. - i !- ,y - i . yj y. y y.-.q ,jj'q();t'. E . ( -- . . i - q y i E ( E E t'. E -E @ ( i E E i . .. . E . E y'E ! i E i i E E . -E . -E E . . EE. .- EESE E- - ;. -! - y . .- . .. E. - - E. E;q. i i'.r-- i !- ; - - -. - 'E E-. 'E- - . - - lE. . - i E.E -. . .. EE . -..- - -E!! ;. i i - E --; Et.tE ... - .. E E. .E .qE . i !E! E ; .- - E.- -;-'-E... .E.E i -. --;;,.E. . - -E. (qE EE'- E i.. . . .. E . EE E E'' -.. .E . - E E .-j. g ig E ! E .i- -. ! . . . -'- i. . !g . ! -.. IiE - .. -@. . . E - i EE E ); . E i . E E . . . y ! E. . . . i : E - . E-E - E i -. yEEj'.yy;.'E.. E. . i @ ii i E .. -y. . E . --j,;. --.. - .q -- . . ;. E EE E.!.E . . E . - E -. E. E; .y --'!jjj;, -y :E jt)Ejyjt;'. E . .-. . . -.. EE; -y'. F'E -E . -tE. Eyq -E - -- E - -. (.E , E - E . gi .j E.; -;! - -. ;rE . iE-. .! ''- EE i;jky)yj .. :.. .-E E ; E(. .-;! E. E - Eq --. -E!EE E ' jE - i i-. - g -.- PE- E-E2g-,. ( .E. Ei ;g q . (. E..E . (- ..E.- E.-;. yq- (y - yEE. k',- . kE.Ey yq qy -. E'. -i- - -. -:g q g; -qy'; ( qq --'j ' . . ;. . ,;.- - ,. - -.! :j... k,tjj-. g.. . --.. . ., --.qE i. '. i i E.E. . (-!E;q; , -, E -, y. -- - y y. q y yqE. yy( ; i. E yIEj-E i .y-. E- y ( E y - .- .y -. yE i -E '' EEy . . ! !-E -.-g.-. ; --j.iq.qyy E .--.yy . . E - -. -.E : -jq. g.E-q- -.-E. - -. . (. -. ,(,!q; j. g; gqE-E ! ! . q . y( -E. . - .- y'. E-q E ?E '- .Ei-.. , q . q.y . ; . g. . - -E q y yE i gy.5 . . .- E g- !E; i yy, q ,- ,y - !, . ; . E (. . . , - .. yq (-,. . yE i -y (. .E- -- .y., . E Ey E. j.. . -.y,y- y - ; yy y g . ,- , .. i--j E - - -.. . -y -. yy ,,I. -. - y;. -y- y-.. i .- -. -. (E ; . :t'..y.. E;:-E.; (S.E-; E . -rt'. . . -E - ; EE -. (r E E-; 5--E EyE - !E t.Ery,. ; ,y't(y- Eq . E; - y - E.. - - E- E Ey E. -. . - E ! ; E EE :E.y ! , .; . tE lj -. . ( . .y E -g E g E E i ; . . , , . y ! . q ; g y q : y . E -. I E y . 5 j y. y . .. ! -. -. - q . q y - y(.y . -- y-. . E yj.y.;-. y ! ;- E. !- -yyyy.g -. . q . y q y . ( -,q y(. . !.E( - - j k y.. j; yj. ; E . yjyy ;i;:. -sjkj,.py ;- j. -. :; ; 4, . E- g; - --F't-, --tj y'. -. .-.-..y-.y- -E. j - --- --y --. .y- ;.yq'. q ( '.t'i g, y y y. --g-. !( gy.-. ;j g.yq.-- -y.y.-. .y -y. y.-y - .g. y. . y j';y5-'q y. yy- (. Et'. -yyy-;'-. y..--'yj(y;'. --'q .. y. ,). !;y y.. .yg'-. j ... y .yy-.- y.-y y(- -y ('. j!- - j.y yq.y. Ftl- (,. . . -. (.y y -j'-( -.y -.y yF'gk( yy g ; -. - jlqlj-'j .. y ; - -.)j.(r.. iE;; ;- - Ek . y- .- . y y E. . ,'; .. j pt-'-rkjjlt-. - - i. - ..-,!. '.:f--.--.E .. y; - . y y-, ;.yg'g...- i-.. E- .-- .- --. - :.:-.!y. '- -.-..-E.,!st--. Ei-...-.. E;-! k; . -. y - -y. yy(yy.y- y..y. .y y,'.!.r - ; !';'. E t-. .- - Ey;y.y y.y- -y(. y',-..yyy. yyy.y..yy(- . y .y.. -. y. - .-.. -'-. . .y- yy : . yy.. . y-y(.. .y E. . .E... ;''. : -. . . - - y -. (. E .!.iE I; . ., y . y .yg -. . . ... .. y;y -.- ,'-.. yy. y;yyjyyyy. .j.E .. :.,;'- --. . . - . .--'. ....k...-E .: . . . : . -', ! . :,'jjj.; . :.E. .tjyy,E . E .: y.jIItg;j;E. E. . . q jyjjjjjE E . (. ... ( E (; E . jj'yjj,tj;E. ! Itj.E. E. y y ..;y. . .. ..: . jyyj..k . rj.. tys'jllliy'.E. .E E ..E. (E t'j()'jtt(. (. E: ;q q:.E. i: qE!.'..(: r E(i.E.. ( i .. EE ..!.. . qjyj'i islE..'l . , k. jjy . E!q i !5. i :. (. y(ryy)j(jyy': . ;, : g . q ; .; E tjjt))y'E. :(: ! '. E : 1:. i(.).(E y'.','. gq y ( ;j: . :;;j.. gyr; gy g ( :: E ; 5 E !ri :. . .:. ( : . ; ; y g.y. E i yjI;'E !. ,, ;. ,;: ;y ; jg ; ,.:y ;j,kj'. . . yt(. y kj y . (. . E k2(r. : ; : . y; ; Iq..q;y g . . j . ; j . g j . ; q y ,, : : . ; ;;. . j. . g ;'$)', . . , . .y y, . . . . .. .. . , ,. . .,. . .. . . . . .. .. , , y .. . . . . , 'tli
--.
'
'
-
'
'EE i E . .EE E- i' i . - ' EEE .E Ei . .
;...
-
.
. .
. .1111
.r::!iiiI::..
..:rjiIj::r.
I I
1* ;1 :
'.IE.
rrjd )'T-' j'lyd y' ,'L-' 'r'))' r' (' 'nLt.jjq.if,;' pfd j;' 1* .:' ;'.d yyy'jyd 'q' ((t:@' #' jyd yjyd yyd j-';k-d j'j(' jp'jjEyd .j' ((' y' yt'yd k' .t' t'ty'tyd )j..jj,' jjd (y' ;yjj))'.' .y'y' j':'.;' .'.$qq3i6Lki''''' 1* :j' k' r' ;'tllyd t'yrjj)')' (' ?' 'f' )' ,-' Etd ;t' y' k' f' q' Egd r' ;qq.-' tpfd jptd t'' .-' jl,r,;d r'.ildtr'irrki'qprqd jqd ))' 5* ;,' 'k::d jyd yyd ji'jyd kd jjd j'jd t':'yy'gs)d jjyypi't;d yjjd qt'tjjd sytjd ygd (t(' (pjlrjtjjjjyjljjytjgjd jtd k' jjjlljjjjtdyjjyyjsygjyyjjjyydirtillliijjjli-,dljkjjljji-.ylqgjjljkjjri-yzkyjjpjjjjjyjjjjyjjjjjjyj...-d yt',jyjd yd jjd ytd )' Ej$?'j(l);-' )1!.$* Iliji'jtd ;'))' f/pq)qI-.@h.$r('((!rlll@lll)rp)E''' kl'y'qd .y'.' q))jj;'y)):-' jyjtjl',kjjd jgd p' p';' jjtd (;(t;.' y' jj,yd (' yjd !' @' .(' .y' t--.rd 1p-:' y' (' ;'j'y'y' (('.' i'@' qjd lll'rd j.'.' 'g' ))' ;-;t..' ;.'.-' jjj)')' )t' (?' ))t' j')-p''' j!.'' :'k.jljtd y))')' ((' ij'.' ;-' g' yyd r' yy.yd )y' yy..' )k,' Sqptykjjyyyyy'rtjj'y;d j-,tpd j'kd yy'yyd y'-y(j' p(' ..' 4* jrd .yj' y..(-.r)t.jf' ili' y'yd p)jjj(jjjy;'.' j,yygytd y(ty'y' ,@' ))jr' pjjd lljjl'yd )'t)y' gj'jjd yy'!rd gjrd jjd yrjjyd yjd yyj;d jyd .j' j';d y;d )'j' yy'jd y'jd -,i'ktLL;.' jj;y;d @tt' jd ;?' q' jy'jd 1* j-' jjj;.' yt!j,d (r' ir'ir' k.L3Lj'.' j'j' jjjjyy.:jpj)!py)j!j);-.' .@' jy' 4* g'tyjd jgtd j)' q)'ry;jy:jj)yjjjj);j'-' jktj't.'trd LL;'.' rl' j,jr.yy' '' LLLL3$);LLLLL'L((kL'LLLj3(33L);kLL.' -.,'t(()I$r'.()j(;;k))'.' lrd jj-'j-' jtd k,td j:k)d t'rtd y,td r' T' ;d (fd p' )y' ip'i''')r' i,jd j'E.' pd /'yq;d (-.' ?t' )d )'-' 7)* )' q' p' 4* !' k' ',fd !)' tjsld-qqr'kid LLq6)L.k?3L'Lj(..;q;;Lk;1.' ptd jiLL3L'1LLL3;?LLL'.' iid ,'jLL'-.'.j' )t' (rd $)' tjd y' g' yy' ytyjd ((.p' yyd rtd r'!' t)sd y(' (' ;j)' y,gyd yjd prd y-'yd ytd ljytk'j)d ;)' yysjjg'yyjjtyd jjyd jy..yd y.'(@' pkd f'jyd j.jjd r-' !' ;'tyd jy.;jd .y' yytd y' bq--'ksqli,.'.' )'j' y'yd i'$:* q' pd ));-' )qp' (.)yj))yt;.' ,'..;'y' tqjd ggytd yd jj))'-' g' $' r.yd y'qd @' r(' t'rd )' y'(p' )(' k' j)L,jLL'j))jjjttj))q.-,k' gtd t')'t'1* jjjd r' )(y(',' tyyj;td ty'y' t'r!d sl-ktd trkd tyjttytggyd trd rjd lgjll'),d t;d tyj'jjd ':f;!,* .-3LLLLjLk'.?'.'--' t'jd r4* r(' .t' r' ygt.td (Ej;jgX .(' rjti':kkd ;)3.)L;LL,..' rjd y)(r,' rd r)' .g' (11)1* 4j))y'..)yj.' r' rrtd rjj'j' t@jjj))jjty))'.' rtd r?' ..' .j.' ...'' ,'';)' ..k4t.,.'ykk' -' d-.;(r.t.'r.tl.l.t@)t)))))t)?('-' -'-' --' -:j-.* .'..' -.' ,''-g' ,j' -)' ,;' .kj.tldkyryt'yd -(t' -' .'-j' -' -y' -'-' -j'.' '-'-y' -'.'-(' ,',.' .j',j;',j;' ..jy--.)(';j;;(-(())j;'y(t' ,.,' -(;*.)12t*,)* -' .y' ..;' -'-t' .j' '()' .t' -j' -'.'.jj-.' .t.q'
j:;d -.td :'-:'-' -' ;'?j' L.' -ls)td -Fh;' J' ,'jyd --' prd E!' yjd gyd -,' -j,-' y' yyd t'-r' ;y' :'-' Srri..' ,;' J' j'(' 'yjjr'syyjryjd jyyj,d q' ;j' -p' Eyd y' ;'-' ,'(' ,(' -r' ;';' -sf' ''q' -yd -y' jr':td ;j' rr' -'j' ?' :y' :':' yy'.' (' '''itd ?' .j'd ;'E' i''(.E' ''.-)jt--' ,'j'!' :'k' t'(' ?;:(,' yy.y' j;d t''2:' Ed r' y' :tjt)* jtd yjd jd )' pjd sll'jd tyd :':' .,' yyd (IE--d ;'yjd g;' E'.' yjd trd ytd r' p' j'y'y' Etd Ejd ,y'yyjy'.' j'?' '''5* ,'td ;';' 1* .:,)j((;r)*(5* 'y' i'Jrd F' Eyd /'5* krtyy' '-'-gj.j))'
.'.'$33.' . .. ''.' .' . .' .'''.' .
'
pkjgklg-jklrtr ; ;.;r. j;'jjy. yj'. . ; ;j:)jpi jjjjjjtzyjtjjyjltjjptjjgdljjjr.lkjjjjty.: .t:)k.! .;.. (t; ..!....j;-.j ; y.;;(y....:.!. ;.:.;(r.!yy.!..;(.j(.j.:.. ;y p.i. :. :!. rp. ..;.;.y)); !. . lt! '. . k,'jj;.!..ji.!.;.-( j;y (-;:t'. ;i'(jj';l .;'.i. ;.!t.(.y. (j.. . .;y'j: .r)'.-;... k. ;.. . (... ;...: ; .. ;;.j'..yjtyi jk. (p. jy(r. p. y..ly .(...... . ...; r)r'kj;r'2y'. . (-t; . ;tk'. j4)';t)tj('j.'t'. ti tt.. t!pf'. .;.(i.;j.g)'lkj (j:.rj: .; jk. LL., rt(.'.-'jjk)kj):.. r:g.j.4 rk y$r)-. r . );j ti.'.;.. i(;.rj.:rj'ir..ti.-.;.-. .li..j ...(...(.!...jr.r( .! ;.ii!.;.;j. ;. !jj.)'. )r'. rgj'! ;Lkq;?)L()jLL;:j3L'qLj);'.)jLjL'L);))-'. .lj'LLL;L6L3LqL);3)3))L(LL3LLLj.. 1j'. . $. . . t-(:lily rtyr))k ; k r; j-I)'. .t.t-jy ;Lq3;j'. ))'. . k;; j!):. Ejy y;tj$jt'. ).1bbbL1L)(3L;;L;'. tr'' ;jy ; .y': . ( . ; j.jjyyrj.;;.. (. : . ; (j)t'. qtijgtitj. (k),jrq.j.rq j'j(';(j:(-.;. .j;'.(ij'. ?ti ; r ;q r( (((:)LL;q;., ( ...;.y yj j )'. l; .j'! .!;( j i(.@i( .!;.: '-.; !. ;. . l.i;.i.y.p;r.r.@ yr.pjjry( . jtty. gj;j;'. yij. ir.. gj'.j(j.gy; ; . . k. ;(..;j ;! I...( . yj ...'..'.... . . . . . . E@l.)t$)jr. . '.... ... . ;kt'. ;j!'.. :t('tq t'' t;. ). kir! y, . ; .)r!)'. j p:g).)g.@ r-'. t:t'..(y;$y'. t)':t. t.j'. t-. j)'. ;rt'y )'.(-yr. k!.gi.:(; )'. ; r; j ; .-l'q )-'. yjtr;. . j!y,y. ttr .;t)'jk! .y.;y'.';Iyi !; tt. . yqygyy'y'(q. Egi. l-. y)(.. #'. . (;t'. @ .r; ).yi(( ; . (;.; .;. j.; ((.j. !..k( .. -:LLlLLLlL'rtjrq';-;., .I.!!.( ! )tjr 2.(1(. . !. j:(t)'. ' .j...j;. . ...j.:l y jq ( y k'. .j).: (r(. )'. g?'. j.. . .;.;..;.).j ;:;. j;.(.yitjjj$-! .jj'. ;;);)jjj'.. . (i jsp',p !y-'. j)jt',t;t'... .i. )';y( yp:y'yy'jg j.j;jtj-p!y @;';'rt!;;y'tjjyj'tjjkl. .j i;j . j ;.j'.. ;.iy.k.y .y. j!.. jjLjjjqkjjqjLiljjqlqjil'. k.- yyr (.yy ; )j. kjg,:kji jtq.-j . . ; yi r ((r, ttjt. . yrpyy ;.t.'yy (, y .... . (.(..;.;.j (. jg.. . r . r y )'gr . j.yjjl'i . ( '... . '.... '''''. .' ...' .''.. ..'..''..'' ...' .'. .'.' .'....'.'-'' j.i'illjlry j i y y ;'. .Lktfjit'''LqlbLt'-33;33*j3j33*,, . . . . L'LtL$L;LL3'j-;-'333?3tL., . yy. 1. j j rs-yj' . :!p(jj!t)jjj)jl))(:.':l(lt'-lpl#f;'''' . . )jp'. rg ;. ;.-jt'. ( Lf;; : yy)!g l-g y;1*. ; . . yyi ; j jjyz',y )'. . . y;jy'p LIILIIL', . . . i . . g ('. . . t'y )'j (LIL. jj (.rrrr. . r '-jt)j. ; ; . ; . (.j j j(j jjjyy .@'. jyj (.- .j jy.r jy)!j!r;;'.. .jyy)y .y'j q.jyt';r.:-jj'. t'--yy'i. ..jjktjkyj y'! ygjjj'! j p jyj r)y))'. !j j.yj yj jj.j .yjjj.j'.j.jy..jj .....j.j jjj j. .y'. . ; ..j.,jy y .jj'. j y'q jjj ..jy j'.j. (.;)y, jj.kj.-j;; j(';j . y;y. k y j.yj jj ...y'. .ry.. k .y'. . j.jr. jjty.)tq'. .. tjy't.: jj.yrj. r '. . yj'j ty. j'.y j y r'. j t: .. t)r. yy'-ly'. k y'. jy(j . . t)4j!.:.jjjj24)-*3. ( ;'t; y y kgyj.. kyyj,jy. yk. !. . j j j j t j .. tj;. j (;'. ; . . . jyyytyjj k j . yj. y-'y .j .. jy qll . y t('. . j j y yjt'.gj,. j ;'. l j ) Lb. yy j t'.y'y. . j (t.. . Cl'y j j . j .. .. j y y-yj. j . )'j y j y. j y'. j j pi r j k'y j j ; r j ' . . . y j ; (j .. .)';.yjj'qj;... ('. yj'..y .jj. . )'. .ti; ;.l'g j j j y..j j ..jj.,jyj.jg;.; ('yjjyy. )'-yjr;(q. ..jgjjjjg.. .yy . .(; gjyyty...jjjy;'y y. . ..j. j .t; . j . .. )'. . j .. j ..k'y j yg. j I. j .t;ity-. tty. . . j . j . . y .jyyy; y-... (. ttjk q'. y . j( . jtjydjyjyjtjjy'--jy . yy'. p'. yj jjjyj'y q. j,jjl.. j;.( . . ...,..yyjj., j j . .. . . ' . . . . .. ; q ; yyyyi'j . k. j', j j jj yj-. y', j'-jy('. . . yy'jj. . ,jL,j.jj... . y! . . t.'.'. . . . y jjy. t'. tyqyjyy'tjjjkj'. . . j . ; j . . . j'. r j ) j.. y jlll-g., r t: $ .. t)', ' . . kl))-).. j j'. j-. ;'. ( g j.. p)l'.'-gtk.rt.lj)@@?lt'. k ky'k . . y . j't'j yy. jytjy'-. . . . .-'. yyjyjy-y-,-yjj'. y'y ; ; y jyj j'j r;yk k t'. rydjy'jy'yy (()1,. jljqk'nL;. t'. )'. . ygyy; k ( r . y', . (t(p .441*j)j:.j.. rj'..y r ; . Lf)L'. .j'yy'j;'. j;.p. . .('. .y.yyr .r j),-'. j-(3. jj.s'yjj. yyy'ty jyjyjgg'-s j..y; tr', yy yyyyy jy-,--..j)yj-yyy.yy'-, yjjj.gyy .j.t.y.g ( jjg.jyjy'. y.yj. . y y yyy-y ?,j.g(ppp. p!(.;! t'.. t-yy's'. tr. llllsr'ry'. )()yI .,yy y jkyi..yyjyjyjy:y . . y-. ryyj ry. yj .yjt'j$:. yji j'yjj .., ;. . j.j;jt't(r())'. y. j( ..yyy t;'.y j. jyy.yj.ty-.. .j.j'.j. j yy'. yj('. .j . y', . j',. j..; j.yy'k .jyy .y..('. ('; y.jj...yjjy yiy'y ..j(;.t;(y)'.. jj(;j.. . .y-;', jjgy jikr'yjj (;. .gg t))j).gpy y.j... j)'.y .j yj....y ty'.. kj;jj. . ....Etlt ri jjjj .t'y . jyj.y . qL)')LLLLLj;('.b)'. .y. jy'y;yjy. yyyy. . yjj .;j! j.jtj'. y )'yyj'. y. ty y yj-y .jjgy.. . ... ..g . y j y'. .jjjj j'i-ty'. . j-), . y, j'..j j..jyy .; g jy-'. 14@. j y.. y . yryyy yyy'. j.. ty'ygjj rt'.... .(.. jt.j; . .yj ).t('j.i! .ttjjy'; y. y yjy yj y..jy. j,jg .t('. y ...yy LLjj;;)). yt.. t't-;'. y-jy'j yy j y'! ;..yjyy'g .y. yt'. yyjy'yyyjy-'jg,.', jr.jj(,j'j):);'. y'. .jj',))jj'j),jj)jj. gjy;jj.'. jj.,(yy..t-y .;EyyyjI ):.j tk jL',,-t. .yyy', gj y y'. .y.y'. tj jj'. j.jyy. jy'y . j)'ry'. jy y'. yj,j .r .rrt:j. yjy'. yjyg. rjj!yj. jy;(.(. y'g .y.y'. .jyrg. yyyggjj.:'(y y tyjyj'. j'yy. kty'-rty tyj. yyy'. .. . . .. . jjj, yy; yj;.jy'yjk.r .j,. .;j jyj(y.,!. j'yg jyjyjyyyy. yyk-yy-yjyygyyyyy . p'(tjj')y. .yj 'y j.y. . jy'. yy (;.. yjy yjyyp'yy . y... j.(....jj. ..g.jyy.j.gy.. j.ttjy)$)'yjj,. j;'. .g. .j,.y jr'y yjg ;'..yy-. . yjyyyyyy-jy j . y..y ... yj''is'ykij;tg'.j! gjyyj..'yydydyydydjjty .yy.j.y ryy ygjj. :('; . j ;'g.. rjyjjy'ti tt'. .y'yj'y .y.. jj..y..jj.j.. y'y .jy . yyE!)y'. yj .jyyy j', ;.. yt.. y'ji. .Ii; t'i . t.-jjjty )).jjj'. .p:.gjy'y y(!. j. ty-. .y'. )fjL. .tyy'..j )-r-. ; . !.. yk'jj)r yy'. .j)..y.yyr'r i.k jyjyyj.jj.jyy jq:$'. r;:. y;;)'.;j.... yy . ; j t!. !i..pptriyy.!:yrf Lj.)LL'LL'LL'j'. .yy. ...gj.y.yyy'y jygyjj.klr-jyljy.--yg .r( j.yy;...y. ;(k.j....j ; jyy j j y.yyg yy-'. j j g.ry.. lrtj;. y.jjryyyy.j.jj...y.rtyy y(y.t'. j;'kjTy. pi yyjg ! . @j'jjygty'y. jjiky'j .yyjg.j... ;..jj... j.yy..,jgy.. .y. ;.r', j.(j'y'.j y,yj. jg. j. jjy. g'yg t-jttjt.;--! .j J:.; ry': yytk ;k;; k jy yy .r'. yy'ky y . . yjg . jj'.. .. . ty yq jjyj;:j)l y. ;jy'q t,. j yy j j t);y., )jj($r'. '.'.' 1t1 . rsyy'jjyy'yy ;j'. . tgyj j.. , . y . y g;y'. jy jpI j'. . jjjjjtjy'.rj. y; j jj):. j j tyyjjj ty'. y'. y t'. . ... . j! (. y j y ' yt'. y tj. . . . j tjy r'. j . y'y', jy:'. .. -)L))();, j j j t(').. ; j y yj. j'. p'. j j y j j!. j .. .. .. j yy j j.. .yy', . . .. k-. j'. .'. lq j;'.. t'.,'.ykj'. j..r'y yj-jLL-. jy. g . . y-yy. r- trgjyy', ... . .. . y ;yI-tlt-'. j yj. y-y . .j . ;. :. yjy. rr'. )'-..-'.y. . . j.. .. .. t)t!. . ('( .-.. .. . y'. r- ;j).. .;.: ti . j. ... . tjji-jyyy'jj .@il;'. j jjs ..')).....'.-j'y . y 'r-i j'. ... .. 'y'. .(I!;.);p)q. . Lq6q3qLk3. .q;j:k$f3. y'. .. .q.;.yi:;.( .;.i'.., .t'y'. y'., .j.yyy. y-'. 1lp: ' .pj!. . yI tt..i .. i j. ..j('.. ...p;qqr)q i ; .. :;gi.. @j'2. yyg. (y'. ji!.. ;.;.i'i(( t'......! )r'' ...'y ' '(q'l. .t! ..('. ...y'. . @ ;jk.. '. ('! . ;q'qLt(bL.. (t.. .j.,'j . . jr. )!)j)j'.. j!)j.)(. ; q q)():. !pit. )4631. j tj;r'' LL.LLLL.. Liqjkkkflk, jjr. )j)j. .''(t;. ki !pr.. (4t(,*t172 1r.2. pqiq ; . l),.. ;.t'' '... ;.!. 1. ): .: (. '. Lif'jk'i';k )4*: .-i; !((. ' ;.i1 .. 9),*. LL'. )i. ' ' '. . :)). ).. i ;. . )i. ;!. ' :t;..'. .... (i. 'ri l'i LLlLL'k. ;L;?)3L-. ::; '1);. ' )'.i' I j /5 i '. ( ')'. p ; . ! ... ; . '. 't;! (( '. ..'.'.'i'.k))r. y t:. . . y tlti. t: j!@'rJ48)i):. j p ;; . y . ;p. 'ri. sp ti. ! (tjjj'tl . jr.))@r'. iqp.,ttIq!)jj:('. t. '. tj!. . tr).ttit(..ii'pp#(r'. . '-'. (3*, .g !('. j;'. r! j t)r):.'. t')qq'. . .. ;j;';i)'. .)'. . .. r. . .... . (rj'. q'.. .)'. (('. .rr'. r-. . . . . . . . . . ;rj'. . kl.fl'. . . . . ;r)'. . j(('. ;;!'. ;'. j/'. . (r*. . )j*. r).;j'. . . . )y'. t'. . .. q)j)'. . t'. '. . . . ll'. . rq'. tpj'. rp'. tt;'. . t)'. . . .
.
E ..
..*.--.-;9.*;2t:j);(j:;((jtt4)j@k1:,
'
.
'
.
. ..'.
...-....,.
.
.
..
.
.
.
.
.
-' .
.
-
.
. - . . .
'
-.@tyj(.. .
'. '
.
.
.
-
-'f).
. -
..
..'::tt!r!.
'
-
..
.
...
. .
..
...
:
. .
:
.
.
.,
.
-
. .. . .
..(j.j.yjj ' ' .'i'.'
E.
. .
. .
-
.
-
.
.
-
, -.
.. . -*1
.
. -
.
.
.
-
.
.
. ....'.-rp:.
..
-;))))))'. . . ..
.
.
.
. ..
.-'y. , .
-
. ..
-.
-.
-.;t:'.
- .
-
.. ..
.. ..
.
.
:
. E
' :
. '!: ) t,;
-
.
.'..(E: r. ...
i( ....:y(. iEiiE
r(lj
''''''i :':'g
E
.
(',')' y.E(.jj .Eit'i
:7l
i!E@ : E:E.(;.E . EEj,,i'
,,t,.i '') ,'') '
.
E'.:.! i
.',tE@
r Et)iq
).!i
( ..
.. ..
iil i i
' '((Ei ..)EEl
Erj
ltiiriy '.
' :... . j : . FE)!'i!'
g
EE!:
Ei ; . !E'?Et''i :
E
.
'lig .
..j. . .g. ,
E'l
' 'i'li') 'j'y'jj)tyj' '
( E .E'EjE .q : ..jEj '. . .
:E
'E.E:.I.IE
E .
:(':E
.' rEi')i,l ;
)
)'..E.'j..E.l
.)
,,''y'i)i. y
y. t'!'f', :......(j. t y -
...j
EEtjti
yttjtjyjtjj i ' .',tyy . .( ...tilj :
.
Ei:
-
'.j
. . . .
.
:
-
!-
..
.
.
.
-
-
.'k' --
-
-
-
-.
.
- . .
-
- .
.
-'- .. . . . ' '. . . -
-
.
-,
..
.-
-
-
-
.
.
-
'
-
,
. .
-
-
.
-
-. .
..
,
-
-
-
-
.
.
.. -.
-
'..
-
-
,
-,
-
,
-
-.
-.-
. -.
-
.
.
.
.
.
-tjyyr . .
,t?'.
-
.-
-
-.
-
,
-
.
-
, .
-
..
.
- ..
.
.
'
.
.
.
,
.
,
. ,-
. ,y
, ,
-
,
.
, -
-
, -
.
. . -
,,.
ij
-
.
. . - -.
,
.. .
-. .
. .. ..'p)'1..p)l!.
.
-, , ..
.
.s
-
..
.
.
-
,
-
.
.
-
-
-
,. ,
,
.
.
-.
.
,.
-
-
-
-
.
-
,,.
.
,
-
-
.
..
. . .
.-.
..
.
..
.
,
-
,
,
.
-
z .'; ,
,
. ,
, ,
,
,
-
.
'.
-..
..
.
.
.
-
,
,y
--'( ..
-
,-
. ,yyy.. -t'y-
,
,
-
,
,
-y
,
,
-
-.
.
.
.
-
.
.!'. --
-
-
..!-
. .
..
-
-
' -.
. -
.
'
'
.
''
-
'-
,
..
. .-'. .. ., -
,
'i.'gi. :i .
,'It . j
ll . .
11
I.1
I lj
i
.
j
a,
.
,
.
E
.
.
'
.
y
::.. ::. ! gE y;j. . .: ; ';)
jg;s
I
,
j 1
.
'
tl
y
.
.
'
.
.
:
',i.)) .
)iE E,;!(
'trtj
E ' i'ryyq .. . j
j'
This book grew from my teaching notes for two el'n Universlty s Kellogg School of Management. .
,
A derivatives courses at NorthwestThe two cotlrses roughly correspond
:y
i':i' i) ' ',..',g.Ejlit . . g. '''if.
:
l '.
tkj )
ljjjji . .y)s! .. (EElyjj zi j ,,,( EL',j'kkvjL '. !( .
:
q
g
.
t
LLLL,(L(
rjjjj
:5
:
E, ,qE y
kjyjj, kjy.yyk(;......;.,.
. . . g!:: y .. . .
E. jgytyr'.-E . .. .
j
i iE .
j(
.
.
1
i j
.
.
l
E
.
.
i
:
..
.
'
1 1
::j.
'
E E' E: iE E
1 I
PLAN OF THE BOOK
'
i jg,. y
.
.
1
j 16 jj.
. :
.
-
,
.
:jy
:
-. .
.
. .
--
.1
.
.
-
..')'. .. .'(
';''
- .
-.
.
llirty years ago the Black-scholes formula was nem and delivatives was an esoteric and specialized subject. Today, a baslc knowledge of derivatives is necessary to understand modern inance. For example, corporatlons routinely hedge and insure uslng denvatives finance activities with strucmred products and use den'vatives models in capital budgeting. This book will help you to understand the den'vative instruments that exist, how they are used, who sells them, how they are prlced, and how the tools and concepts are useful more broadly in snance. Den'vatives is necessarily an analytical subject, but 1 have tried throughout to emphasize intuition and to provlde a common sense way to think about the formulas. l do assume that a reader of thls book ah-eady understands basic fnancial concepts such as . Present value and elementary statistical concepts such as mean and standard deviation. Most of thebookshould thus be accessible to anyone who has smdied elementary fnance. For those who want to understand the sublect at a deeper level, the last part of the book and presents some Of the develops the. Black-scholes apploac jy to pjjcug (jesivatives .. . . standard mathematlcal tools used in option pncing, such as It s Lemma, There are also chapters dealing with applications: corporate applications, financial engiheering, and real options. In order to make the book accesslble to readers with widely varying backooounds . arld expenences, 1 use a tiered approach to the mathymatics. Chapters 1-9 emphasize present value calculations, and there is almost no calculus until Chapter 18. . . . using Excel spreadsheets on this book can be repllcated Most of the calculations in with the book. These allow CD-ROM that the comes you to expen'ment with the pricing models and bull'd your own spreadsheets. The spreadsheets on the CD-ROM contain option pricing functlons written in Visual Basic for Applications, the macro language ln Excel. You can easily incorporate these functions into your own spreadsheets. You can also exnmine and modif'y the Vlsual Baslc code for the functions. Appendlx D explains how to wn'te such f'unctions in Excel and documentation on the CD-ROM lists the option pncmg f.un dtions that come wit.h the book. Relevant built-in Excel t'unctionsare also mentloned tlurllghout the j)00 k .
.j
,
.
-.
..
.
.
-
-
.
,
. -rt-. ,yy'.
.
.
'
.
,
,
,
. . ,
.
-
. .. . .
. .
-.
-
-
.
-y
-
-
-
.
.
-.
.
.y'.
.
,
-
-
-
-
-
.
-
.
-
-
-
.
,
,-
- .
.
-
. .
.
-
.
.
.
.
-
,
-
-
--
-y
-y
-
.
-
-. -. -
.
-
.
-
.
..
- .
-
.
. -.
-
,
.
.
.
..
-
.
-
.
.
.
'
'
.
..
,yy'.-..
.
-'yyjj')':
-,
.
,
'
-;
.
-
.
.
. - -
-
,
,
-
-
-
-..
.
.
-tt'..
.
,
, . . , .
-.
.
---th)'.jy. .
-
. .
. . .. - .. . .
.
,
--
-
.
.
.
.
.
. , . .
. -.
-
-
-
.
-
-j..
,jj
.
-
.
.
.
.
.
-
-
. -
-
,
-
.
-'y
.
.
.
,
,
-
. -
. ..
-
. .'.'y - ,
.
. ,
-
-. .
.
-
.
-
-..
. ..
-
-
.
,
.
.'.
-
- -
-
.
. -.
-
-
..
' iii . .(..EjE.:::E E
(
'
-
-
.. -
.
. .
-
.
-.
.
.
-
-.
.
.
-
.
.(((j!q. :j .E't''E) . . : . i .( : E . q(E y
:
E:.
E .'.: .
.
. .'..'.. jE( . i,
-
.
-
.
-
.
-
.
.
.
.
.
. .
.
iI
:
i.(
.
:
'j..'
.-'
.
.,';
.
-
.
.
-
.
.r
-
-
.
.
.
.
.
-.
'
-
.
-
-
'y .; .
. -.y
.;
.
'-
.
. ..
.
-tr'' -
-. -
.
. -t''
.
'
i .' '
.
: : E: E .
-
.
..
-
. -
.
-
-
-
.
-j
.
:': .
'
:-
:
. - .. .... .
-
-
-
.
-
-
-
-
.'
'
-
.
,
-
'
.
.
-
.
.
,
;
:
-
. ' ! :
,
-
.
.
.
-
-
.
-
.
'. (jg.g: . .. E..r
,
. -
-
..
.
:;..y
.
.-
. .
: j::Eq
' .j
.
-.
-
.. :
-
,:,
,
-
-
.
-
-
-
.
.
yyy'y
-
.
. .
.
.
'
::
j... j
.:..
:
.
,
-
. . --
.. .
..
.
-
:
-
-
.
.
.,
-
-
-
.
.,yrg
-..
-
-
-
-
-,''
.
:'
.
.
,
,
-
.
-. .
.
:
:
:
,
: , ,. . qq gq.j :. j y.(j. ( g. ....g: . q:.q!.;;..g.g..jy... j;t!. .. y.. kyj g q..y.yyq..q....j.y( gg:.j'. g.. ... y. ;y.y'.. . .;'.y
k .EE
.' :
.
.
-
-
-
-'p
.;
. .
.
. . .. -
.
,
-
.
-
-
-
-.
. . .
.
y j jj jj '' .( (E E ( gEI(E (.'.( ii.E.( .jjj.p.j.g.y)': jgj . g( (gg..j:1). j. )'. . . E i EE E E Ei lE .. .. .; !. q(j-.. i ..; .y. (@
' :.(j .. .
.
,.
.
,j. -
,-
.
.
'
.
-
--
-
. ,
.
..
. .. .
. . .
.
,
.
.
.
.
.
-
-
,.
.!(;.yj
-
-
'.
..
.
-
..
.
...
..
-.ri.
.
-
. .
..
.jj)q.
'
-
.
.
.
.
-
-
-
'-
.
.
..
,-
-
.
-
-
.
. .
-y'-
.
-
. .
-.')jj));.
.
-
-
...
-
.
-
.
- ,.
.
.
'
.
-
.
.
.
-
- .
.
.
.
.
'';)j;...
'
'
.
-(t,'.
.
.
'
'
-
-
.
.
.
. .-,.yp4yjj)jj(gj))j,,kj..-.'jy((jpjt)));)-j'.
' ' --.
.
.
.
..
' E' . .
---
..
.
'
' -.
: - PE -. i: : : -. . - -.' ' . :. ! E: i . : !. E E . .y. EE; ! i i E- ; :E:. :. !. EE k; - -. ryE !E i !i i - E: - Ei E- E-. E . . -. .- E E. E EE E E ! E E q: -.E: EE; E.EE . . : @ -. !yi . ..j'. i - E ;!E i !si (?E-j;'E i: ,. I-'E E; !i. - - i E EE; i !- . E EEE tshtt..'-pjjjklpkjjjjjlkjllill'.. i:iPE !q: :i- (E!':: ;?':! ,;E E: E-i i E!i iE. EE.. EE E. E ! ; EE q.E-. : EE!E: (E.. i:EE: q- E..-E!. ii E i.EE.Ei.E. : . ik i Ej: :! . E. . -. -: ;E. ; .!E EEq q;!E; : E.EiEE E.q. ;L;L; q q. q; I.i.E- !E.-y : .!E .. E - ''.E' . . . E . q!-'.-.i.- IE EE: E. E(! .:. q q. . EE: ! -g ! E E';j'. ! (E q ' . r'iE :k q -'. ! : E . E -E -.- . E E .:E . :.- . E E IE : :E -- E-.- E - - :E ! iE.!.E::. --. ; .E-. EE @ ;.E.iiE.EEEE.- : . . E;-EEE. .E E. E .. .E: E. .i.y :i pil')tiqtjtjjyyylijltlyyjji EIi i. g ; EE . - . i . ii-. E i Etk ;E :. : !'..iE E E E ('E i. i .;: i:E ;:)tty))ry@i. . qii:..E! : E. - . . -q-. . .. EE.yq.- E-.,.E. Eq: ;E: 5 :( EE:i Ei i. E q Ei . : EE:-. E E..i E ! ;gj;i;E;:-Ei(E- E.!q E.- y'E Ei E. E . i ! -:. i E .E Eiqg ,4,;. !.. i. ;.-jj'. ! . EE .. . yi-E. . E . ! ; , !;jii . q! , . ,jjyr'. .i-.i.-:E:-' E. .-..-EE..;E. EqE. .i.-!EiE!.E : ( (tt)jjjqj'. q. . .. :.. : y '. - y. !. -. E ,E:EE . .Eqi.. :y!.iEE E E )i'. @ ( j. . q@ ::'.. ; . . E . q; yq.: q-E: Eq i ;.!iq- ! .!E.i - : E ; -. y. q : E . . ;. 5jE:(( : : : 'i -. qE r'ykjjlj. ;r;j;j'yjjj(j'..'. E (tij.yyjyy-yjjjgs. EI -. E i ;E E :.E tjypjjk-iliidrytkjjprl'lllillllllrlllpltly'.f-i iE.E-. ! . .! . ! i. y EE. . - gE. : ..- . !.: -kE--.! E. ;. .. -.. : E.: .. :-i..:. E E . - . ...-.. .- . iE.. -:. .f.-E ! E .E-E jj. :j ! E: !!.(:- !E . i ; EE- .- E-. E E: , i yy!:jj)j:y $. EEi . . q;. . gq( qE. .-.(i. . ! E . . Eq..y:.y E.. (; . IICgIjXE-ICIiCEg y y . i E : . : - -. E : ; q . .,.L,,, : . E . ; ; E.j(yy q -. . -j-t.. '. . . . . q E. ( : E E . -'yj-'; ' ' iE g : . ,q q q . E., . g E g, : -yyy. . q . Eyy -. i E;. : y-:; :. ; . i.(!Ey'l-yj q ( ( q.! . . EiE. j';.E. ... y . . Ey.. . : E: : . .:jytt,.- yy y. ky ('. Eiy E. . E E i y.E. rttf#cdplhllllllllipl-'t.... g;.;jg; :yyty'. - . Eyiy -j'. ( i.iy .Eyy . ; g . ( g ..EyE... !y-y E-( ! . . y.y Ej. E. ig .- qj'EE.j'. ;q r. -y q ! E (;, -. .k'. :).''j@(@2rj()';. .y jEyrE ;i E-- ! . : (jj.E y':k'E iyyj. .Eg .y. . ;'. E . j! i ,, Ey pj'. E . :. ; . .y qygj- q. .. . y . y;. E -. q . .y'. q q yE qE. . E. E . . , q. . -jL. q q E yj.. : q ( y y E E. E . i'y -, . . , g g . .y q . y ( ( y y . (. (j.!y ;,?'j , yy qy -. . yt'; qr'li. y-( - .j . .-y ;. .yyqy. -jq q! y :, ygj )'-( t!y y g. ( yy - y .:-.*4j4,7)p8..pl. I//I3LS'LLL. :tt;ttg'(y'jjjjyyryjryyy'tj;r('kyyy;y'. : .- . . j . y . : -. yy'! (: h'y .- . ,h,; . (jdyj y yy q..- yy !-..gyy. i'g;. g y y gj . q'y ;jygyy y;.. q. yy.. y. - g; y y .( y,;tE(-gtt.. yyk 2 ( ;'kj . @ . ,,))))'. . E: y yy g, y . --- ...y-y,. y. g:.yy , y j..yy . y, . . . y , . . jjI;E . y.. j , . --y ;'y E. qjjt'y y E i . y . y'y . tjq)ty g k j . y E E ' i. E , y y y i. E . !'. . - .! q j. !t;'. . y y.. y y yy - j-:.r';.g'E yyjj :.g.: E E j E. -j ; - . y E ! !g - g.. qE . (.E . . -.k .. .q .j -. . -, : . E : !q. -jj.,j). q.gy;q. yE y ( i y. ... ,'), rj; : . . .g yr- j y : k . . ... jq, y :. g y - y y(. k ;,j'.. . .. .. . . i'.. r ygyyg- y k .. yy yj. . y . y ; y!yy.y yy..-yj. ... . y.yyg-y-gy.. ,; j y r (, y yi g ,jj'- - (E i;,j. j jyE;.y yy.. y. y.y.yy.y( .yy-. y.. .;.yyy.-y. y . . .--j.E- qg q-kqj'-y...yy . y.-j y y y. y. -. ,j ! .. . yyyj.yy j .j E!iq: . ,y.. ,;jy. i; -. ;'. y.y . .ttyyy'y , j-.j')j-. yyj; . yyy. y;yy yg -y y .y..ygyy.yqj.y- j.jyg..y yyyy . y. y.y.. y g .. . .yyy .y,-j .jjy ( y , g . .y j g g y y . y yy y ! j . y . y . y .. :'.y', y . gjrE. -'-. y : ,- . y y j t'. y . E , I j ky. j . ( y -t-.'y Ft'?. . g. y y i y jyjkr ,. j g ; j . y , y ! . . y y ,tjjj;', . ( : !yy:jkp-. . yy . y. y yy y g . y. ; j y . !. . Ey.q. y. . .E. jti j. yy- y y y y y j ,. y gkg . - . . y'yy y.yj yy.'. k'yj'E y?!E !y , k.ytyry'yy ; .-y y yy.. y.y yyy-.yyyyy- yyyyy y,. y. -y ..y jy .y. y..yy.y.. g- y.y. - yj- ;(yy y.yyyyyy. ( - y gy yy'. .y yy. E y g (gy .-'yy. j . yg yE--q: qE.- - kE. -j y-.y. k.y,'.ygg y j(.yg-.jyygj'; , E y - .ttjyy q ; yy-yy y y.j .y((j;,,'. , . y . E q :', . . . ; .g gyq : y E. y g y y y ; y . . ' -.ky. j . i . . E ; ! y : . . j. y , y y 'y ,'. . . y y j g j j yy y y . ' . y . -j. g y . ( y y j ; . y y g y y . . y y . y y -k'. ty'. j .. y'. y y '-'(t) r y y. . yyjj .. y . jj.. yy y g,g.j - y E . Ey Ey j yy yy-y. ('. yks y Ey (,E ,$y; E i: E. ikjLLLL..L .y yyjy. y,,yy yy'. , -yj: t'E sy'j g y. . -jjy. ;y. yy. @):E !' yt(EyE . .i g-!.... j Ey.j .!.! ..i ..! y;;.. y.yy j yjy yyy..yyy . y yyy-j.yjyy y yyy, gyy.. y. -.. .- y q. ! ky y y jyy . ygy.. . y. . r'E ; EiE ;-y'. E ; ! E - g:E :LLLL E ; yy . k''('. y yjg; :- : yj y - .Ej.. Ej'-j y ;gy. - .yg y -. -.y j-j yyyy.y.y y E .; E i-i y q ,,.','. ;yyyy.y yj g.y -. .: y . qyg. . y - -y -.y . . g g. y . - .g.. y- - . y q- . -. -.. . -.yqg y;y (jj'. g - , (E. . y ' E .g : .y . - qy y :'g yy ('- y.-, y ; ;: y y'. yyg . .y. yy.yj'. .y.y y. y y. -y . y : E. E. - : -EE i yj (. . Etd.jitd-ly.st).)j yyy-y- j yyyyy.yyyyg-ygy;gjyj- y jj'y yy .- !. yyj', j;kyi --E. . i ;E 'r :.! ('E yy..;yy ; y .-. j. y.yy.y- ;, y;y.j y. y. yr , E. :@. . jr!- :.'iE ?E :.yy'- - :(@- r E-.E i .p. ,y'. i. i,,;?.b,-'..j$?.6t,,. l'tpt-.k)j. ; E ). yy-. qysE . E((.. E,. g-y'.j-.. i !t....';:- ....E.E., yE...............E.-.. . E.!!E.-.... . E.i . . yy-.g. yy. y.j yg-. j' yyyy. yyyyy....y -. y. .. . .. - y ... (. . y... ' (E' .y-..- ;g.... y.... y.. - yy.. y... . -.y y(yy j,..y(-.E;y.- . ( j... y. y g.gyyy.y. (Ej -yIEjk: ; j- t'yy'. Et .:E L.Li-'-'( y- . - y -.y -..y . ...y y.. :; -. . .E . i E-- L'-, . j)':. . . . )))ilE.. -. i . .- . . ...:.'' . . . -..t..j(jr. - . . (j'. 6LkL;/L. -,2)1 .. - :. . ..PE . -.. :', . . .. g -'... .,i-ji.'k . E.i E !. @ -: . j:*. (/'E y. . . rq'qsj.-. - -.-f()'E (....! ' :IE' EE.' ....--k-; E '. E ... E. '..'' E@ !E i.E -ji....'.'. .)E. . PE ! ! --. . y . yy- .yy.. y'.y.. E'iE. E E E ;E!i!i-.i..-y!. E. . .;?'.;-.: i. E'.. i )'- j.E.i y'.-i'.i E !.! ....:. . E ('E E;C.E y..... . y. yyyy. E. kj.'pj! i('..(l!. )E. ;qE. . --'.i :.gI ,', ?,i. I i;. i:*(:: 't'' ! . '!E j).. E iE. E .. ..: . E i j y y y . ! . .. -. !i' E. ,8.E . t.-. yC)'.E p.. i . 'j,' i E . E ... EEI . k-.. E E . .i ! E-'EE.y.E .;. ! - ii . :).-. , ; .. .. . ! .; E-. E..E. .. E i . -.;;5 .;E.t:. .E .. . . .!..E . .: i EE '.. .. . . . . .;'. r y . . k.'. . . . ?'. . ('. . Lj :.j!j . ' E..(...'E! . ': . .Eq:' . .E.' '.' !! . '.:.....(; .. E': ..E: ! . -
'
'
.
.r:jjiIj::r..r::!iijj:yr.
.jjj;
WHAT X11
%.
Is N EW
I N TH E SECO N D ED ITI O N
PREFACE
general introduction j third of the book. The srstconrse is a u to the 'Iirst two-thirds and last options, swaps, and structured products), t futures, products (principally to derivative The second course is for those wanting a markets in which they trade, and applications. and the ability to perform their own analysis. deeper understandilg of the pricing models basic statistics and have seen calculus, The advanced course assumes that students know option-pricing framework as fully as and from that point develops the Black-scholes scienMBA-level course will produce rocket lo-week possible. N0 one expects that a students cheating of derivatives and it would be tijts, but mathematics is the language to pretend otherwise. order than it occurs in the book, Yu may Want to cover the material in a different severa.l possible paths allow Qexible use of the material. 1 indicate so I wrote chapters to possible to hop around. For example, 1 through the material below. In many cases it i.s siniulation chapters on lognormality and Monte Carlo wrote the book expecting that the might be used in a first derivatives course. 1 introduc ihe basic building The book has five parts plus appendixes. Part Chapters 2 and 3 and call and put optins. blockj of' derivatives: forward contracts hedging and invetfent strategies. examine these basic instruments and some common risk manageent tools ad discusses why Chapter 4 illutrates theuse of derivatives as chapters focus on undefstanding the firms might care about risk management. These pricing. contracts and strategies, but not on and swaps contracts. 1:! these Part 2 considers the pricing of forward, futures, pre-speified pzce, at a fture day. The contracts, you are obligated to buy an asset at a Chapter and how is it detrinined? Iain question is: What is the pre-specified price, commodities, assets, Chapter 6 discusses 5 examines forwards and futures on inancial forward contracts. Chapter 8 shows how rate and Chapter 7 looks at bond and interest from forward prices. swap prices can be deduced intuition about options prior to Part 3 studies option pricing. Chapter 9 develops 10 and 11 cover biomial option delving into the mechanics of option pricing. Chapters formula and option Greeks. Chapter 13 pricing and Chapter 12, the Black-scholes used by market-makefs when managing explains delta-hedging, which is the technique hedging relates to pricing. Clppter 14 looks the risk of an option position, and how compound including Asian options, barrier ojtins, at a few important exotic options, options, and exchange options. chapters are applied in Part 4. Chapter 15 The techniques and formulas in earlier from the which is the creation of new financial products covers fnancial engineering, compensation and equity pricing, derivatives building btcscks in earlier chapters. Debt 17 studies real optins-the 16. Chapter in covered Chapter options, and mergers are valuation and management of physical investapplication of derivatives models to the lnents.
hedging in depth. The material in this part Finally, Part 5 explores pricing and assumptions underlying the standard derivatives explains in more detail the structure and model and shows how the Black-scholes models. Clapter l 8 covers the Iognormal discusses Monte Carlo valuation, a powerful formula is an expected value. Chapter 19 20 explains what lt means to say that and commonly used pricing technique. Chapter
%
XZii
stock prices follow a diffusion process, and also covers lt's Lemma, which is a key result in the study of derivatives. (At this point you will discover that It's Lemma has already been developed intuitively in Chapter l3, using a simple numerical example.) Chapterzl delives the Black-scholes partialdifferential equation (PDE).Although the Black-scholes fonllltlais famous, the Black-scholes eqttatiol, discussed in this chapter, is the more profound result. Chapter 22 covers exotic options in more detail than Chapter 14, including digital barrier options and quantos. Chapter 23 discusses volatility estimation and stochastic volatility pricing models. Chapter 24 shows how and binomial analysis apply to bonds and interest rate delivatives. the Blaclscholes Chapter 25 covers value-at-risk, and Chapter 26 discusses the burgeoning market in credit products. WHAT
IS NEW IN THE SECOND
EDITION
rfhere are two new chapters in this edition, covering volatility and credit risk: * Chapter 23 covers empirical volatility models, such as GARCH and realized volatility',financial instruments that can be used to hedge volatility, such as variance swaps', and pricing models that incorporate jumps and stochastic volatility, such as the Hestop model. . Chapter 26 covers stnlctural models of bankruptcy risk (the Merton modell; tranched structtlres such as collateralized debt obligations', credit default swaps and credit indexes. throughout the book. Among the
There are numerous changes and new examples changes are the following: important more
. An expanded discussion of bond convexity * An expanded treatment of computing hedge ratios * An' expanded treatment of convertible and callable bonds * Discussion of the new option expensing expensing proposal
rules in
FAS 123R and the Bulow-shoven
. Discussion of a variable prepaid forward on Disney stock issue by Roy Disney . ln-depth discussion of a mandatorily convertible bond issued by Marshall & Ilsley, including pricing and structuring . The
use of sirulation
to price American options
. Additional discussion of implied volatility . Enhanced discussion of te link between discounted cash flow valuation and riskneutral valuation
.
%
suppt-EuEx'rs
k.
xxiv
PREFACE 777:* )' :y' -y' .j.''.;;L'tj'. :(' j;jjjyjjf -' ''--'E -'' E IE F' E' 1F' Ejf : ' jj:gyyrjjf jj::yygyf jj:yyygyf jjyjjjgjf jjyjjgyjf jjrgyjgrf rfI'.--yjIyEE 11::2::1,* ' E .(!:.. : ' ' Ei E!!'l'Et?
)' i' p' y' 1* E j: ' ('E ('i(' ( ( 7* )rf JT)' (?' ));'. iisf jjr,,.f j'q'' k;yjyjjjf jjy,,.f )''pt))t))'-. !!i!;;' 7'7* q: ' (''EEi'' ! (EIj;. ':' (' @)' jy' .j' jf jy,,,f '.jj't' trjjj;f trjj;;f q' '-.jL')'.')LL--,iL('1 ;'tjjl'.f j'!j;li ! j.q;q)lrt7 (;rr '!!''l!Sr17T5'l7( )j' j'ltf tjf j'yty'. t'):' y'. ()!; E( ''. !..' (.'.( ' ; .'IE .' ' i (r?. i':'1* )' '(j;t. ;')'!'F(' y'. jjy,,,yjjf jj:yyy;jjf jjyj,::yf lili!!!!tf (' $' )f;', y))y':.'. y' .' . ' . r' t'.jt' 11::::1,* t)'. tjyj'yj' '.- . . . ' l ;r-.);!:j!.iE;yly.qk. ..,.q (..y(E.. q( j(gjgjy yjij . .;.y;.q.....j.(j.'. qjj ptj:. j.yjyg ()jjy ig j .-q!yyyyyy g; . y IIIIEEE!IIik ( . y y y y . . tt!tj y yy r j . .. .r. . i j..q.j j!E. jI1jL---k.E @.77'7IIiIr'77'( ,.t)-r E ..L,qq33bbqqL;. ' ..:.(..... j.. y.... .rr (. tt))-. . . .. r..rr....! g..(. y)q . .. ... . . ... . '' -y' ,jjggy' ,jjyyy' .,,' ,jjygy'
.'..' '
:
discussion of value-at-risk * An expanded functions for pricing options with * New spreadsheet others pricing, the Mertonjump model, and
NAVIGATING
E
dividends, CEV option
sxed
-
.-
.
.
-
-
-
.
!
.
:
: ..
.
-.
''
' '
E '!'E' (E E . . ..
:
-
.
.
:
E
E 'E '
' E
'' ' ! iE.. .
:.:.. .
:
. E
' ''E' EE'E ' E:' ;.EE: E: E: ''E 'E E . . ..E . . . .yi (,E E: . :
:
:
'
.
.
-
.
.
.
E.
.
.
i E E i .i. E E
.-.1111( .
-
..-
.
'' ' '
'7q77'l'7!7r'!
-
.
.
.
..
..
-
irli ; --
-
. .
'' E' .
.
.
-
..
:
EE : . '
'
E. . ; :. E E. r. ; 'jy -.i. i . E-- E E. E - . . . .... . . ... .
-
.
;
.
-
.
.
j. .
.. - y - E
.
-
.(
:..
-
-
-
xxv
.
.
.yjj::. jjj .jjj::.jj::: kyjgjjji krjjj;. jjyrgy:jjs jjj jj,..,yjj jjr jj,,,.jr,.,,jjj jjyryyyjj-
!!;;E
. -
jjyyjjgry
jjyyjjgrr
jjrygyyjjj jj:yyygjjr
jjygjyyrr yj,,,.jjj trjjj;.
jj:gyyyyyyy.
jj:gyygjjj
.jjyyy. kkjgyjji jjr,,,jjrgjjjry jjy,,,jjy,,,yjy,,,yjj
,jjj,j,jjj,;j,,
jjjjj,.,,yji
''*''
''y are strongly recommended those marked with a are fit with the track, but are recommended. and those with a ''
,
''''
optional. The advanced course assumes student.s havq already 5.1, 5.2, 7.1, and Appendix B are recommended background.for aII introductory courses.
-
taken a basic course. Sections 1
THE MATERIAL
.4,
the material in this book. The matelial is There are potentially many ways to cover mathematical dificulty, which means that generally presented in order of increasing For example, tixedincome is related-material is sometimes split across distant chapters. exotic options in Chapters 14 and 22. Each of these covered in Chapters 7 and 24, and neighboring chapters. As an illustration of one way to use chapters is at the level of the chapters I skip in the courses 1 teach (withinthe the book, here is the material 1 cover constraints): some specisc topics due to time 15.42t5.5, 16, 1-6, 7.1, 8-10. 11.1-11.2, 12, 13.1-13.3, 14, * Ineoductory course: '
13, 18-22, 7, 8, 15.1-15.3, 23, 24, 25, 26. * Advanced course'. possible sets of chapters to use in courses that The table on page xxv outlines some sections of the book that provide background have different emphases. There are a few short-sales (Section 1.4), conreader should understand. These include on topics every (Sections 5.1 and 5.2), contracts prepyid fotward tinuous compounding (Appendix B), 7.1). (Section an'd zero-coupon bonds and implied forward rates
8. Swaps 9. Parity and Other Option Relationships l0. Binomial Option Pricing: l l 1. Binomial Option Pficing: 11 12. The Black-scholes Formula 13. Market-Making and Delta-l-ledgng 14. Exotic Options: l
A
NOTE
ON EXAMPLES
assist you this book display intermediate steps to Many of the numerical examples in it will also be possible for you to crein following the calculations. ln most cases basic assumptions. and compute the same answers starting from the ate a spreqdsheet generally rounded to three or four decimal However, numbers displayed in the text are signifcant digits. This creates a points, while spreadsheet calculations have many morewould obtain using spreadsheet, a those you dilemma: Should results in the book match equations? displayed obtain by computing tle or thoje you would will provide tlze results the numerical examples in the book rule, general As a spreadsheet. The displayed entering the equations directly in a will be you would obtain by logic of a calculatiqn, but a spreadsheet caleulations will help you follow the helpful in reproducing the linal rsult.
SUPPLEMENTS
accompanies materials for both instructors and smdents A robust package of ancillary the text.
15. Financial Engineering *
l6. Corporate Applications l7. Real Options 18. The Lognormal Distribution ..
l9. Monte Carlo Yaluation 20. Brownian Motion and It's Lemma 21 The Black-scholes Equation .
22. Exotic Options: FI 23. Volatility 24. Interest Rate Models 25. Value at Iisk 26. Credit lkisk
lnstructor's
Resomces
Forinstructors, an extensive set of online tools is available for download from the catalog P age for Derivatives Markets at www.aw-bc.com/finance.
i
AcxxowuEocuEx'rs
xxvi
% PREFACE Mark Cassano, University of CalAn online lnstructor's Solutions Manual by solutions to Ohio State University, contains complete gary, and Rdiger Fahlenbrach, problems. selected solutions spreadsheet to all end-of-chapter problems in the text and Indianapolis, features of University Will, The online Test Bank by Matthew W. and questions, short-answer five approximately ten to Vteenmultiple-choice qtlestions, for each chapter of the book. one longer essay question electronic formats, includingWindows The Test Bank is available in both print and Test Bank are and Microsoft Word files. The TestGen and r Macintosh TestGelt fles available online at hup://www.aw-bc-com/irc. Cao, Pennsylvania State UniOnline Powerpoint slides, developed by Charles Emm, Georgia State Uniand Ekaterina versity', Ufuk Ince, Unikersity of Washington; Copies of the slidej 4an the book. selected from art versity, provide lcture outlines and facilitate note taking duling class. be downsized and distributed to students to the computerized Test Bank filej (Testcontains The lnstructors Resource Disk the Test Bank liles (Word) and Powerpoint Gen), the Instructor Manual files (Word), 'liles.
Student Resources A printed Solutions ilanual by
Mark Cassano, University qf Calgary and Riidiger all the ven-numbered problems Fahlenbrach, Ohio State University, provides answers to in the textbook. Solutions, by Riidiger Fahlenbrach, New to this edition, Practice Problms and and worked-out solutions for each problems additional contains
Ohio State University, chapter of the textbook. functions in Excel are included on Spreadsheets with user-defined option pricing functions are written in VBA, with Excel the book. These a CD-ROM packaged with Visual Basic editor built into Excel. These the code accessible and modifable via the the book's Web site. spreadsheets and any updates are also posted on
ACKNOWLEDGMENTS book in 1994 by asking that the Kellogg Kellogg student Tejinder Singh catalyzed the derivatives course. Kathleen Hagerty and I Finance Department offer an advanced notes (developedwith Kathleen's initially co-taughtthat course and my part of the course third of this book. help and feedback) evolved into the last edition, I received invaluable assistance from kiidiger In preparing the second material with a cdtical of Fahlenbrach, Ohio State University, Fho read much the new other mistakes and offered valuable suggestions. Numerous eye, and who both caught the first edition. Colleagues in sttldents, colleagues. and readers provided comments on with their time include Torben Anthe Kellogg hnance department who were generous Deborah Lucas, Mitchell Petersen, Ernst dersen, Kathleen Hagerty, Ravi Jagannathan, Many Kellogg MBA and PIA.D.stuSchaumburg, Costis Skiadas, and David Stowell. Arne Staal, Caroline Sasseville, and Alex dents helped, but I want to especially thank
k.
xxvii
Wolf. Others who reviewed new material include David Bates, University of Iowa; Luca Benzoni, University of Minnesota; Milhail Chernov, Columbia University', and Darrell Dufie, Stanford University. Mark Schroder, Michigan State University, kindly provided code to calculate the non-central chi-squltred distribution, and Kellogg student Scott Freemon implemented this code in VBA. A special note of thanks goes to David Hait, president of OptionMetrics, for permission to include options data on the CD-ROM. ' I also received help and comments from George Allayanis, University of Virginia; Jeremy Bulow, Stanford University', Raul Guerrero, Dynamic Decisions', Darrell Karolyi, Compensation Strategies, lnc.; C. F. Lee, Rutgers University; David Nachman, University of Georgia', Anil Shivdasani, University of Nol'th Carolinai. and Nicholas Wonder, Western Washington University. I would like to particularly thank those who provided valuable feedback for the second edition, including Turan Bali, Baruch College, City University of New York; Philip Bond, Wharton School, University of Pennsylvania; Michael Brandt Duke University',Chades Cao, Pennsylvaniastate University',Bruce Grundy, Melbourne Business School, Australia; Shantaram Hegde, University of Connecticut', Frank Leiber, Bell Atlantic', Ehud Ronn, University of Texas, Austin; Nejat Seyhun, University of Michigan', John Stanseld, University of Missouri, Columbia', Christopher Stivers, University of Georgia', Joel Vandeni Dartmouth College; and Guofu Zhou, Washington University, St. Louis. I would be remiss not to aclcnowledge those who assisted with the srst edition, including Tom Arnold, Louisiana State University', David Bates, University of Iowa', Luca Benzoni, University of Minnesota', Mark Broadie, Columbia University', Mark A. Cassano, University of Calgay; George M. Constantinides, University of Chicago',Kent Daniel, Northwestern University', Jan Eberly, Northwestern University; Virginia France, University of Illinois', Steven Freund, Suffolk University', Rob Gertner, University of Chicago; Kathleen Hagerty, Northwestern University', David Haushalter, University of Oregon; James E. Hodder, University of Wisconsin-Madison; Ravi Jagannathan, Northwestern University', Avraham Kamara, University of Washington', Kenneth Kavajecz, Wharto:i School, University of Pennsylvania', Arvind Krishnamurthy, Northwestern University; Dennis Lasser, State University of New York at Binghamton', Cornelis A. Los, Kent State University', Deborah Lucas, Northwestern University', Alan Marcus, Boston College', Mitchell Petersen, Northwestern University', Todd Pulvino, Northwestern University', Ernst Schaumburg, Northwestern University', Eduardo Schwartz, University of California-l-os Angeles; David Shimko, Risk Capital Management Pqtners, lnc.; Anil Shivdasani, University of North Carolina-chapel Hill; Costis Sldadas, Northwestern University; Donald Smith, Boston University; David Stowell, Northwestern University', Alex Triantis, University of Maryland; and Zhenyu Wang, Yale University. The following served as software reviewers'. James Bennett, University of Massachusetts-Boston; Gordon H. Dash, University of Rhode Island', Adam Schwartz, University of Mississippi; and Robert E. Whaley, Duke University. Special thanks are due to George Constantinides, Jennie France, Kathleen Hagerty, Ken Kavajecz, Alan Marcus. Costis Skiadas, and Alex Triantis for their willingness to
'
AcltNowLEocu
xxviii %.P REFACE multiple times and for class-testing. Mark read and comment upon some of the material which I used both to compute the Broadie generously provided his pricing software, calculations. Heston model and to double-check my own Will, and Charles Cao for I thank Rdigey Fahlenbrach, Mark Cassano, Matt In addition, Riidiger book. for this their excellent work on the ancillary materials and KlishNovikov, Dnaitry Fahlenbrach, Paskalis Glabadanidis, Jeremy Graveline. Kaplin and Andy book for the checkers namurthy Subramanian served as accuracy jrovided programming assistance. Burghardt of Can-Futures, Andy Among practitioners who helped, 1 thank Galen Jacobson of the International Alex of Moor of E1 Paso Corporation, Brice Hill lntel, Tradelink, L.L.C. of Securities Exchange, and Blair Wellensiek intellecmal debts. From the many, Iwant With any bok, there are many long-term several classes from Robert Merton at tale good fortune to to single out two. I had the book is deeply in his debt, and M1T while 1 was a graduate student. Every dedvatives 1970s the are as essential today as they this one is no eycepton. His classic papers from with Dan Siegel, with worling amount learned an enormous were 30 years ago. I also 1991 death in at the age of 35 was Dan's options. whom I wrote several papers on real personally. profession, as well as to me a great loss to the made it clearfrom the outset The editolial and production team atAddison-Wesley lucky to have the project overI that their goal was to produce a high-quality book. was Donna Battista. Project Edity, Finance and tireless seen by Addison Wesley's talented myriad details and offered exof track expertly kept Manager Mary Clare McEwing Editor Mmjorie Singer Development board. sounding' cellent advice when l needed a significantly. Promanuscript improving the suggestions, Anderson offered innumerable manuscript into a physical marshalled forces to t'ul'n duction Supervisor, Nancy Fenton Services Publishing E1mStreet at excellent teams book. Among those forces were the edition, of the design first the compliments on and Techsetters. I received numerous second. Kudos are due to Gina Kolenda which has been carried tllrough ably into the and cover design. and Rebecca Light for their creativity in text tried hard to minimize errors, including have The Addison-Wesley team and 1 of course, 1 alone bear Nevertheless, above. noted the use of the accuracy checkers will be available at updates software and responsibility for remaining errors. Errata find do errors so we can if lnow you 1et Please us www.aw-bc.com/mcdonald. update the list. revision using Gnu Emacs and MITeX, 1 produced the original manuscript and authors. l am deeply patef'ul to the worldtools for extraordinarily powerful and robust this software. wide community that produces and supports Through both editions 1 thanks go to my family. My deepest and most heartfelt and tolerance. This book is have relied heavily on their understanding, love, support, children Claire, David, and Henly dedicated to my wife, Irene Freeman, and 111.M '
Ehl'rs
%.
xxix
Robert L. McDoltald is '?3W?llj Ndp?l?/lcr. Distillgltished Professor tTFf/lt-I?gcc atNorthJs/c,re?-/l University :5'Kellogg School ofManagemellt, lWpc?-dhe has rt-Illg/?/' since 1984. He is co-Editor of the Review of Financial Smdies altd llas !7c77Associate Editor of Analysis, Management #?c Journal of Finance, Journal of Financial and Quantitative Science, and tl/'/ze/--jt?l/?-??t'l/.. He tu a fh ill fct/nt//??fc..-t???? the U/?ve?-.W/yofNorth Carolina at Chapel Hill and a PIt.D. ?? Economicshnnl MlT
; : .J tE. f L. ;;;i.E;i:: ' z' :.
'
'
.
I'E;'j ' ..r.
;
.''.
frt?s 7)7' E: Ei L .''
' '.'
'
.'
i' EEF'ir ''.. '. ' '
.
'
'.
''' .
; r ' '! ''E''''. ''
? ''''''''
'
'''
'''
'''''
wHAT
2
k
ltq-rRoouc'rlolq
'ro
DERIVATIVES
and is pejorative. Suppose your family grows corn automatically think the term Yo provides insurance'. to mi11into cornmeal. The bet your friend's family buys corn your income. Your supplements this price; low sells for a earn $1 if your family's corn family the high cost of offsets buys is expensive; this friend earns $1 if the corn his unfavorable outcomes. The the bet hedges you both against corn. Viewed in this light, both of you. contcact has reduced risk for price of of contract simply to speculate on the Investors could also use this kind /74)/ the is It point: is key contract is not insurance. And that a llt7l it is corn. ln this case the ltsed, and JW/t?uses it, lt'Il detennines JW7cle?-olitself bltt Fitnj?it is ct?pl/?-t'7J everything. risk-redttcillg. Context is if you are new to the subject the impliAlthougl w've just defned a delivative, be' obvious dght away. You will come to a cations of the desnition will probably not through te book, smdying different deeper understanding of derivatives as we progress economics. products and their underlying lvbet''
Uses of Derivatives
derivatives? Here are some motives: What are reasons someone might use
and other users to reduce companies Derivatives are a tool for Risk manasement seller this in a simple wa#: The fnrmer-a risks. The corn example above illustrates low. contrac' t which makes a payment when'the plice of corn is of corn--enters into a h farmer who we therefore ty is hdging. This contract reduces the risk of loss for t. e forbiddingly complex, but many deriktives are It is common to think of derivatives'as Automobile is a derivative, for example. simple and familiar. Every form of insurance around a have an ccident. If you wrap fourcar insurance is a bet on whether you will valuable', if the car remains intact, it is not. tree, your insurance is in investment vehicles. As you will see later speculation Derivatives can serve as make bets that m'e highly leveraged (thatis, the book, derivatives can provide a way to of maling be large relative to the init'ifll cost the potential gain or loss on the bet can that the S&P bet example, if want to you specic view. For the bet) and tailored to a dedvatives today, can be from 1300 and 1400 one year ,
500 stock index will be between constructed to let you do that.
effect Sometimes derivatives provide a lower-cost way to transaction costs wish fund mut'ual of may a transaction. For example, the manager other a particular financial and paying brokers this entails paying fees to to sell stocks and buy bonds. Doing which we will discuss later. It is possible to spread, bid-ask the trading costs, such as economic effect as if stocks had acmally trade derivatives instead and achieve the same transaction the delikative might result in lower been sold and replaced by bonds. Using and buying bonds. costs than acttlally selling stocks regulatory restrictions, lt is sometimes possible to circumvent o' Regulatory arbitrage Derivatives are ften used, for examby trading derivatives. taxes, and accounting rules eliminate the risk of holding and of stock (receivecash ple, to achieve the economic sale
Reduced
Is A DERIVATIVE?
%
the stock) while still maintaining physical possession of the stock. This transaction may allow the owner to defer taxes on the sale of the stock, or retain voting lights, without the risk of holding the stock. Thse are common reasons for using derivatives. The general point is that derivatives provide an alternative to a simple sale or purchase, and thus increase the range of possibilities for an investor or manager seeldng to accomplish some goal.
Perspedtives on Derivatives How you think about derivatives depends on who you are. ln this book We will think about tluee distinct perspectives on derivatives: End-users are the coporations, investment managers, and perspective derivative enterinto who contracts forte reasons listed in the previous section'. investors speculate, reduce risk, costs, or avoid a nlle or regulation. End-users have a to manage riskreduction) and goal (forexample, care about how aderivative helps to meetthat goal.
The end-user
The market-maker
perspective
Market-makers are intermediaries, traders who will.
buy derivatives from customers who wish to sell, and sell derivatives to customers who wish to buy. ln order to make money, market-makers charge a spread: They ate like buy at a low price and sell at a high price. In tls rejpct markt-ralts Marketprice sell the retail price. wholesale and the low who higher at buy at grocers makers are also like grocers in th a t their inventory reqects customer demands rather than their own preferences: As lng aj shoppers buy paper towels, the grocer doesn't care whether they buy the decorative or super-absorbent style. Mter dealing with customers, market-makers are left with whatever position results from acommodating customer demands. Market-makers typically hedge this lisk and thus are deeply concerned about the mathematical details of pricing and hedging. The economic
observer
Finally, we can look at the use of derivatives, the activities of
the marlet-makers, the organization of the markets, the logic of the pricing models, and try to make sense of everything. This is the activity of the economic observer. Regulators must often don their economic obselwerhats when deciding whether and how to regulate a certain activity or market participant. These three perspectives are intertwined throughout the book, but as a general
point, in the erly chapters te book emphasizes the end-user perspective. In the late chapters, the book emphasizes the market-maker perspective. At all times, however, the economic observer is interested in mnking sense of everything.
Financial Engineering and Security Design the major idea-is that it is generally One of the major ideas in derivatives-perhaps possible to create a given payoffin multiple ways. The construction of a iven financial ngineerig. The fact that product from other products is sometimes called linancil
'
TH E ROLE oF
4
k. IXTRODUCTION
TO DERIVATIVES
need to hedge their implications. First, since market-makers this is possible has several 'I'he marketworks. market-mnking understanding how positions, this idea is central in that pays position ofsetting and ten creates an maker sells a contract to an end-user, posidon. hedged This creates a laimif it is necessary to pay the customer. how it can contract can be replicated often suggests given Second, the idea that a risk, initial the change dials effect, to tarft market-maker can, in be customized. The These changes pennit the cteation derivative. of paymentcharacteristics a premium, and given situation. of a product that is more appropziate for a about a given derivative by realizing Thiyd,itis often possible to improve inmition already understand. that it is equivalent to something we arbiage create a payoff, the regulatory Finally, because there are multiple ways to code, or in stdp. Dijtinctions existing in the tax discussed above can be dificult to is since a particular secttrity or derivative that regulations, may not be enforceable, the replaced by one that is tzeated differently but has regulated or taxed may be easily same economic prosle. that derivative products can generally be A theme running throughout the book is constnlcted from oter products.
MARKETS
sayin: that the Dow Jones lndustrialAverage has gone up We take for granted headlines arld interest rates have risen. But why 100 points, the dollar has fallen against the yen, index (such zise and fall of a pnrticular snancial do we care about these things? ls the and winners trak lndustrial Average) simply a way to keep score, to as the Dow Jones where watching sports; we watching the stock market like losers in the economy? ls of sound atld f'ury,but f'ull by told journalists, tale root for certain players and teams-a signifying nothing? often underappreciated, impact on Financial markets in fact have an enormous, consider tll role of fnancial markets we will everyday life. To help us understand the and both children 2.3 and Sarah Average have Average family, living in' Mytown. Joe income pays for employer in Mytown. work for the XYZ Co., the dominant lef4 is over goes cloting, and medical care. What their mortgage, trarlsportation, food, retirement. children's college tuition and their own toward savings earmarked for their of the markets and dezivatives play in the lives What role do global finarlcial 'rheir
Averages?
Financial Matkets and the Averages
affect their of the ways in which financiql markets The Averages are largely unaware lives. Here are a few: rieed for money to finance employer, XYZ Co., has an ongoing * The Average's dependnt on tll local bnnk forfunds because operations and investments. Itis not stocks and bondj in global rarkets. it can raise the money it needs by issuing '
MARKETS
$
lisks. In addition to having property and * XYZ Co. insures itself against certain for its buildings, it derivatives markets to protect global casualty insurance uses against adverse interest commodity price changes. By and itself rate, currency, risks, able into bnnknlptcy, XYZ is less and less being likely to go to manage these unemployment. throw the Averages into likely to
* The Averages invest in mut'ual f'unds. As a result they pay lower transaction costs than if they tried to achieve comparable diversifcationby buying individual stocks. * Since both Averages work at XYZ, they run the risk that if XYZ does fall on hard times they will lose theirjobs. The mutual funds in which they invest own stocks in a broad array of companies, ensuring that the failure of any ope company will not wipe out their savings. * 'I'he Averages live in an area susceptible to tornadoes and insure their home. If theirinstlrance company were completely locak itcould notoffertornado insurance because one disaster would leave it unable to pay claims. By selling tornado risk in global markets, the inslzrance company can in effect pool Anytown tornado risk with Japan earthquake lisk and Flolida hurticane risk. This pooling makes instlrance available at lower rates. *
1.2 Tt4E ROLE OF FJNANCIAL
F1NANCIAL
Averages borrowed money f'rom Anytown bnnk to buy their house. The bnnk sold the mortgage to other investors, freeing itself from interest rate and default risk associated with the mortgage, leaving that to others. Because the risk of their mortgage is borne by those willing to pay the highest price for it, the Averages get te loWestpossible mortgage rate.
'l'he
(
In all of these examples, pnrlcular financial functions and rijks have been split up and parceled out to others. A bank that sells a molgage does not have to bear the risk of the mortgage. An insurance company does not bear a1l the Iisk pf a disaster. ltisk-sharing is one of the most important f'unctions of fnancial markets.
Risk-sharing Risk is an inevitable part of our lives and all economic activity. As we've seen in the example of theAverages, tinarlcialmarkets enable the fipgncial losses from at least some of these risks to be shared. Risk arises from natural events, such as eartquakes, Qoods, and hurricanes, and from unnatural events such as wars and polhical conQicts. Drought and pestilence destroy agriculttlre every yearin some part of the world. Some economies boom as oters falter. On a more personal scale, people are born, die, retire, find jobs, losejobs, marly divorce, and become ill. In the face of this risk, it seems namral to have arrangements where the lucly share with the unlucky. Risk-sharing occurs informally in families and communities. The insurance market makes formal risk-sharing possible. Buyers pay a premium to obtain various linds of insurance, such as homeowner's insurance. Total collected premiums are then available to help those whose houses bul'n down. The lucky, meanwhile, did not need inslzrance and have lost their premium. The market makes it possible for the lucky to help the unlucky.
j'yyjd y..tyd ,y' y' yyd k' 'yyd y'y' )' t' ;'t)d y' yyd yjyd )' j'j' y'y' jjjjjjjjjd y' jjd t';,;,j' k' y' yyd jyd yjyjjyyyyjyyd yy'yy' yyjd j'yjd (' y' yjjyyyd jyyd yjd yjd ,',yyy'jy,d yyyd y' y;y' ' ;'(' y' yyd ,;' ;':'.;' t':'J' Ctd y' yjyd y'yjd j'yjy.d ;';)' y'yy'y' (' ,y'
.'.' .'),j' jjjjjd
6
%.
IN-rflooucl-lolq
To
DERIVATIVES
and interest ln the business world, changes in commodity prices, exchange rates, dollar becomes lf down. the burning house rates can be the financial equivalent of a It makes others and hurt. companies helped are are expensive relative to the yen, some risk, this exehange that the companies enabling so to sense for there to be a mechanism lucky can, in effect, help the unlucky lisk. Consider an insurance company that provides Even insurers need to share A large earthquake could generate clnims residents. earthquake insurance for California companies sufxcient to bankrupt a stand-alone insurance company. Thus, insurance claims. aainst larje ???c?'ker insuranc reinsurrs, buy, from to often use the reilsurance Iisks, risks enablinjinsurance bcome more t thereby Reinsurers pool diferentkinds of widely held. catastopliebbnds-bonds ln some cases, reinsurers fnmhersharerisks by issuing earthquake, that the issuer need not repay if there is a specifed event, such as a larg tisk earthquake willing accept Bondholders can to causing large insumnce claims. is if bond tlle there no interest payments on buy these bonds. in exchange for greater those exactly by risk erthquake be borne to allows earthjuake. An eazthquake bond investors who wish to bear it. Although there are mechanisms for sharing many kinds of risks, some have argued lisk-sharing is possible and desirable. Tlze ecolmrnist Robert that significantly more creation of entirely new markets for risk-sharing, the envisions 2003) Shiller tshiller, risks associated Fitfl house prices), incometrade including home equity insprance (to be need fully repaid ifwages decline in a particular that loans not linked loans (personal with occupation), and macro insurance (contracts payments linked tp ntional incomes). inciusive markets for While these markets do not yet exist. there is a trend toward more created an risk transfer. #or example, Goldman Sachs and Deutsche Bnnk httve recen with based claims buy possible payouts wlzich is to it market derivatives'' in teconomic economic statistics. The box on page 7 discusses this market. on You rnight be wondeling what this discussion has to do with the notions of diversiable and nondiversifable risk familiar from portfolio theory. ltisk is diversiliable risk if iy is unrelated to other risks. The risk that a lightning strilce will cause a factory idiosyncratic and hence diversisable. tf jny ikestors to burn down, for exnmple, lisk, this it has no signiticant effet on anyope. Rijk tat dpes not piece of share a small nondiversilia b le risk. % e ri sk of a stock spread vanish when across many investors is nondiversifable. example, is market crash, for Financial markets in theory setwe two purposes. Markets permit cliverslfable risk By deEnition, diversifiable risk vanishes when to be widely shared. This is ecient: markets permit nondiverstfiale risk, time, At shared. te same it is widely snancial which does not vanish when shared, to be held by those post willing to hold it. Thus, the ctmcdpf- and ??yl?-kcl' discttsed i'n this the l?7#twnc?7M/ ecoltomic idea ll?p#cl-fl/iWq p'??d/f. p-l/c-,/lt7?-ii/g mechallsms evelyone. book is 1/2:71the existelzce of .is
1.3 DERIVATIVES
IN PRACTICE
Derivatives use and the variety of derivatives have grown over the last 30 years.
jjjjrijjyj. gjjjy,4jjjj,. . : .:. . . .. . g. ..... yjjjrj(jjjj'. jkjjgg;jjj;'.
jjjjg'j j.yyg g
j)'; .. ..:
. .
..... ('..jjjjjjjj('. .
jkjjjj;gg,.
jjjjys . jjjjyyygjyj: t;jjjgjys .jjjjj;jj gjj.jjgj.j. j jjjjj. . jj ...j.g ...... . j jjjjjgjjjjj;.. jjjkykjjjjj,yykjj'. , tyjjjkjjjkr jjjkj(jjj'. .j . .j'..j.... ... ....
.yjjjyjjjjj. .... .j. EE : .ii;.. ..2 E . ( ( : . . L. E((i : ( . (. .:: ( . : . . . ... .. . .E.. ... (... .. (......r : .. .. .q.. ( . g(.;gg;.. jy... (gg.gy.j. . ;.yj;. y jy..y.. .. . : E . E E E(( .. ( ( ( .
. .. E . ..
.
.
.
.. ... .
. :
y
(.
. .
yyk;jyj;jjg'. .
...
.. .
:
:
.
. . . . .
,.
.
.
:
.
.
. ..
.
,
:
. .,
:
. .
:
.
..
,
.
g
.'.. . .
.!.. . ..(..... . .
r . ... ..
.. ..y .. . . .
. r
.
. . ..
! 2
D ERlvAn-lvEs
y.y. y. y jy.j gj . . . . g g g . y. ygj y. g y y g. . . . . . . . y .y . . . . g. . . .. . .j. :
jyjjjyjyjj,.
j
.
kernnlentagencies
..... ....y..j.j j . .y
g.
....
j
y'j
.
y
g . ..j..y.y j.
g
g
yg, . .
g
g
.
g
.
;.. y.
:
g
,
dealers,
to
in pate .2c():2 possible to trade claims based on employment (U..S. nonfarm payrolls), .
.. .
.
.
. .
. '
.
.
.
'
..
. .
...
'. .
'
.
.
, .
.
.. . y.
.
.
:
g
g . . .
g.
g
g g g
..
.g
.
g. g
y.
. g .g
:. . . .. .
.. . .. .
y
g .. y .
.
.
.
E. .
Manai
.
.
,
yy ...
., .
g
.. y, .
-
.
y
v':'
Jn 2003 wlae t Jr F k; l ek' pavvv.oi,nza *0
.C''''o 'z
'
'p
t
,
t.1-1ttr1
.
.
. .
)('
-xw
k'x'x
'i
Yr
:-
es-,,
pavroll
Aar
tlf -'
-.-.;..-
'e
-.,
.
y;
)
Q!
'
t,j
'
'
E
,
tl
is
q',
.
E5
'''''
''
y
Xs : .
:
j.
.
On
il'..
Yj
.
j... . '. . . . . ' ... ..
.
(.(.. .
t. t.i)
E
.
,''
# #
nlmn'lE.t
,su)
ukizc, '2
..
y
.
('
E '
6zi&b
IC'
i' ' '
'
E'.''
i
'
7'
'
,k,'.i.,,.',''..'''''''.'.'i'''',.i''.'''.''''i (
.,.t,j,t','
E ( ' : lE' i' : E ' ( j j(( : .. : ... . . jy ..t''. j.' . ... . .' :.
q.' j . 'C' uction a m.jiip...t-y,. outeh tenjjj; axs b txaclfng (Ijfssut ujujltlfj.tju tpkjjtnl'lii. wilt (jiscussa j' 6f7'''7'''l''' :
.t')
we
smards caljs
Itk
'
pts
'
jpteadj
-
it
:
hajtt
j y . yirli .y i
'stljtjdlj
'
?
L'Y t'E''''i .
'
,
'
r
(
''''
( .
'
'
:
. y
.l !.
'
E E '
' ' E E ( .....)..)....... E....-...r( .E. ,
.:::11. kpkjl;;:iq .-..)...-.-.(.-.. . lr. ;i:.,.. ,!r ...-.. .......' an k!ii!jp si!::,r:es j, an til ).:(.'....r i::t,, '.-...... ..r....-. ....-.,. ... .:!j;: a jl op ,lr)tig (.. i .
.
..
.
1::j51.
(. .
.
.
k..-.
.
.
.
.
.
g
.
..... .
.-...
-.'-.-
..E..
E
( . . ' . .. ' . j . .j.. ........ 'j. .g. gg.. j. '. .. . .... . . .. . . ...
' .
. :
.
'
Growth in Derivatives Trading The introduction of delivatives in amarketoften coincides with an increase in priceriskin that market. Currencies were oficially penztted to float in 1971 when the gold standard was omcially abandoned. OPEc'S 1973 reductjon in tjye suppjy o ofI was followed by high and variable oil prices. U.s. interest rates became more volatile following inPation and recessions in the 1970s. The market for natural gas has been deregulated gradtlally since 1978 resulting in a volatile market in recent years. The deregulation of electricity began during the I990s. Figures 1.1, 1.2, and 1 show the changes for oil prices, exchange rates, and interest rates. The link between price variability and the development of derivatives markets is natural there is no need to risk .3
manage
2In a Dutch auction (here is single a price that is paid by buyers who are willing to pay that price or more, and that is received by sellers who are willing to accept that price or less.
..
j E
'
. gj.j..... j.y.. jg...
when
.
,
. .
,.
J
!
1
)5
!, i
!
''
i
j
l
.
' '
. . .
... . .
;
''
ttlt
..
'
'E
.
!
id i
i'f
'
.
'
ii
y, y.,,y
'
t
.
.
'
'
fiibflft'l I/ui. .
(jfi
.-w-z
.u-
'
q
,
jjypjfqsjgj.yylj.EYE.../.
detrnd
Ci '
i#'
J lJj f )J ujjfjk
matchd
'
) ))
,
) ,1,0Y1t
'
'
E
pefi
-
Y'Yain
.)
jVsX ido:jjyfkusj., j'ilitatrililt
(.
Y,,
..
,
klitf
Wa
EE E
.t.'),. ,).'
i
''
.
;
y. y
)),'t)i.yi
,.
.--..
nutrllv. -
q
.
=-
' E..;
.,
winIj tlo 9 i dlei
.
;
;
.,,tJ.;
..
'
'
.
'
,
-
11V '
.. ..
Yjsj.j)
.
.
:
.
))h )
s'-
y.
yayyjyjyyyya
yy.yy.
,
E'u Doye, 0-.!!
ViC,-l'Tu
''
goj
.
yyyyyyyyy,jj
,
y
,yy
;.
)E(111.EED. 1i 7 yFICVt0r
rajyd
t
y y. , - ; ..y. . ; . .
y
. .
.'
jyltjallj.
-.-.,
,
i z-, v nr Uell kfjTjfg'D/ Syk.
devlst
day btof
y
g
.
. . .
''
wjjreas jjj
xx,ur-u
.
g.
. .
y'.
.
=;t
2'
statijtit numblj
.yj..
y
.
, .
;
.
y
Yl
most eonlsriuci briefly bgjbf jo
y
;
.
'I R 7 li'
UtpYd
j. j j j.j. .
y
.
Wnsox'''r
Euro
'h
;.
,yy .
g
l
T,t
,v
,j'.
,y.
:
l-X'o
'
(thepurclaasin-!iilI7.
.
.'... .
yyyy.ttyyj .
beginxng statistics. specincally, became
j j .
g
provide clues to future government them--can Olicy. Consequently, money managers, p and other market participants paj. closeattention to these statistics; their release Often results in a flurry of trading activity and changesin stock and bond pricesk Because of the importance of these for nnancial mm.kets, Goldman j,statistics Sachs and Deu tsche Bank created markei in a economicderivatived in which Jt Js jossibl trade claims witlapayoss based on these
.
y
lj
g g..y g. .y j gy'. j.. jy .y'g jj ggyy.j yjy.y yyy yjj.jjgjj.j'jyj'. jjjjygjjg.jjyag jyy j.....g...yy..jjyyjjg..jy'..gjyjjyjjj.gjgjjjyjjjj.,'jjjy j.jyj.. .... ..gyj y..y. y y.yy...yjjy...yjyyj.jyj..jjj..jjyyjy....y'.jjy..jgy. y..y....yy.. ......yj,.yy'....j.y.;jjyy yy.yy...y...jjj.yjy.y.yj.jyyyjy..yyy.yyy. y j y. y'y yy yy y yy yyj.yyy y.. yj y. jj yyy.yyj.yyysj. y yy'y yy yjjjjjyyyyjjj jyy jjjyjy y'y .y... ..j y y. y . j y'. gy'; y g. y jy .jy jyy'. jyyyy .y yjy yjj yyy. . j..y......... ...y jtj jjjy'jyyjyyjyy'.. jj(.j . . yj . . . yjj.y jjyyyy y. y. yy.jy.y,y.. jyj'. .yjyy. ...g j'. y';.y'yyjy.jjy.jyy..yyjyyyyy yyy.y,'. . j'. jyyyjj.yy'yy'yy', .jy ........ ......... gyj'..y'. .j jjy...y';..q..q..yjj'. .yyyy'. .jyjy'yj yy .yy y. .jj. .yjyy y..y.yy. .gg..y . yyyy'y yyy. y yy j'. ... E........jj'..y...y.......j.ygEy.Ej. ... .(.. gy.;E..ij.E.. ...jEyj.jy.. yygjj.j.. ...y,'. .yyj'. yyjy y..yq .. . .. . . .. .. . .y'. y kj.k . jy yyyjjyjjjyyyy. y y. jjyy j..y .yyyy . jyy. jy ...y .yj y . . yjyjysyyyyyyyyjyy; yyjyy'ya yyjyyyyy.. yyjjyy.yjj y y jy .... sj.yy j'y y'. yyy. . .. ,, .)'iE', yjy. .
in most countries Peliodically announce economic statistics, suchas the number of jobs in the economv, producti on in diferen t sec ors and the le l Of sales. These statistics provide information aboutthe performance of the economy
it Production industrial
t
%.
l x pRAca-j cE
;;)2..* r' lrd y' jjd r'. 1lI1qqq5-' j)' trd j'.j' .j,'-. );' j$t'. yjqjtd )' 1111r*. .y' #'. t'. y' )', r' ttljyjd t'.y' t'yj 7i.' t)'
''IIIEE:PII:: .''. '' I
I
% llq'rlkoouc-rlorq
'ro
DERIVATIVES
DERIVATIVES
(E ' E(E i.yjj 'E E'(E:1: i .!ki '''EE:.' E'E'E' . !'i('' E!'E''. jt)qi i'EE(' '''';E'E.'E'.E'i'E'k j(y. @ )(E: tjp)q)E' 'E':('E' :EE :' ('((( @(i ! ''
.'..'
''' .--21)*. --' -)' -' -i' -'. qjjjy', !' 7* F' 1*
.' rrf ' ' :: :.
i' :! E
.
:
'E :
.
(EE : E' ' ' ' . :. :E. . ''
:
.
.
E .
i
'.
..'
:
: .
'
.
:.
'' :.
E!'
:.
: ' ' ' '' ' . : . .
'
: :
:
:
.
. . : . g. : .
.
..
..
.
.y :
E' .
.
. ...... . .
-
- .
.. ...
.
-.
'
-.....
.
E :.
.
-
-- -
:
,
. -
-
-
.
.
E
y .
-.
.
.
-. .
Monthly percentage change in the producer price ndex for oil, 1947-2004.
'
:y;: ''iE. k
:
. t ti
'
i . E .
30
-.
.
20
;
f % 10 a 1 t
.
' .. .. .EE E. E. E .ip!.EE I.;j;yi .. y! q. E. . .E).E.jE. jE. :.
. : ... .
::
.
-
.
'
Ei.:E -E E
:.
- . . -
.
-
. .
'
'
'
.
.. :. .
y -'
.
:
..
.
.
......................
...........
. . . y
..
-;;
....s.
-
............
.
'
:E
'
.
.y. .t,:
.. .
.
.
!!' E:'E!
'
.
. - E:.- . .
..
-
.: . -. y . -. -
.
-
.
:
.
- -
-.
. . -
.
.
-
:
0
-1o -20
'
t. .
.. ... ... . ..
-
2
..g.
-. .
1
a
o
t,
iH
8.
1
t1(
-1
Jb
fi
1
! l
-2
!
i)
-a
=
4
E
1960
1970
1980
2000
1990
1960
1980
2000
Sottrce: DRI.
Source: DRI.
.'.' ;')' ;:' :.' :'t' :'y'. 't'jjd ;r' y,;'. 7)774)*
11,1:::1,1.* j'j' yrrjd '(' )' q)' @' ;'(qp' ;)' q' i,6b''''-'', -;!jpIr'. ;'...-kj::!Ip1(j'. (' 144* jl!d 1(* q' 'q' j;;d qqd )' '1j11:111r*. 'jj'jkg, jjy'j.jj jy'2. jyd j'(' y' ))' 1* qjd )d (J' q' r'
i'(.)' (' y' j'1*.)* ktj.' r' td (y.'yjg t'(.).;' ?i ;'(( qj! ('('' ;' ;' ' )' (FIT@JTqIIIIIE)EII )'(qqy (:;(irk
.q.-'j' .y...t;..t;.'(':' ;'-' ';.'
.'
''. ''-'f' gyd -'y .t'E C' :' jtd 5*
rgl.'. rry-'. 'q' y' F' f' i'''r' ?' C' :'t'trd :;' i)id '(' j''i;d f'X )' 7* i'r')k''' y' :';'t''q' 741!(1:j*''r1)1*;11: 1* EEF;- -' ' E' E E- ' ' ' ' '' - ' ' E : E ' : .EE :;i i;' 'E i ' ( : ; EIE E((;: .(;. . ;''E..(i'!.-: i.q:EErEi;.:g.iili-.':. ..ii.';E..'.; ... i. . IE E..... . E-. -. )'E. -::. . ):.:;1 I . iE;(i! jq' )). -: .. E-'E i - !Eii 'i. .. q:-.. ::. -;. !. . E!.i.EttE . E -. - E -.i E !' !q-.g i-E.!t..jr(...........-..;.).,..-.). E -t.i........t..-.....',.'.''.'..'..:.' . q !E. ! . :r---...-)t?:.k,-....(;jk))k--.. ;iE. q;rg.yE(.. . k-.jjjjj y ;)jy( E ,;y! ':-:' ! i ' ' . . !Ep. . E . y.jq.y y.y. .yy.(g..;(jj yy j . . - .;. )E..:..i)..) . ... . . . gy . ys. gy,..j....-...y ..
.j' r?d dr' jyd j'y' (' y'. q' t)j);'. jyrl'yjytyyjjjd j/' )' ;'..2j;jrjjjjr;'. ))d r'p:':rr';'r::rf r' t)yjjyyyyjy'j))tkj.. r)d t'(,,;'. )' r7*; t'q'g r; . r . 1. . '.j:' ' . r. tyi )) $:' (!((' tj j)y)g. ..yj'.y..j r; .. j!)(':.. (' '' ). .....tIt:;itL. t; . jy. r i:g)-t.qi.t'..1iIiqF....... j '1)) . (. y . ....jy ..L.. .)--);k;)L...
.'
.
-
.
E
.
'
.
.
-
-
.
.
.
Monthly percentage change in the dollar/pound ($/E) exchange rate, ' 1 947-2004.
.
. .
-.
.
,
.
.
'
'
. .
. ...
..
S
. .
.
.
-
.
.
-
E . . . . .
i .
. . . . . -y
.
.
.
. .
.
. .
.
.
.
.
E
o
l:tl
-J % a i
-
.
.
' ' E : 'EE' ( E 'E' i'' : q' ( /E (. (.'i E *1.* E ' E'E.E'!'E'E'.'i.:.. ' i''' i ! (E;' !'E.( E. EEE': ; . .,y.. E. . ' . . .. .. E i : !E' ' i: i :' ': ' E j EE.... ('i '.'.':'.': E.(. j j '('j ! :( g E:.!.().E E! :: E7;i E' E; !(kIjllr(ykj(: . . . : 11.,.---. . : i: :. :. jji..; rj 5, . .. ((,. y g p : i.' E''i'.E.. . E; ; E(E': ' . a E......'... . . . . '
.
E
-
.
' '
.. . .y . . . . . .
-
-
..
.
'
' '' . . .
'
-'
' . E .. ' E ''.
' :
: '
ij''E'E('' TE' g . . q( .'. ' ' ('. (;q'r!')'rlrl:'q'tj'q')rrqqqqrlrqq7q . . .. y E yjjr' EE gj g' j('r j gE (.' j' (.( jjgi.kj. gjg'. '. 'jr'. jjy.):. j' jj jjj(y ..j.'. jyy:g'..',.('.gj(.j'.j.k j. y j.(j 1I!I:::j! ' . ; tj . y'.r ' . . jj r : .... u . . yy jj.(..y. Ey-.:..jjkrj(E.r...gj(g.yyqqyr.y.qyy tu n wx. . ymj.. yus,. . (. . . .r E
''
'
.
.
..
-1o
'
: ' ' . : . ..
.
. ....
'
:
'
:
''
..
.
.
:
: .
.
y-.
-.
.. -
., .
.. .
:
: . '. . : y -, . ..
:..
' ..
..:.....
.
, .. .
. .. . . .
: : .. .
,:
;: .;(:
-
.:
.
.
:
..
'y
' ' . . :
.
.
:
;
: ...
'
:-
,
.
.
.
wu . , of contracts Millions traded annually at the Chicago Board of Trade (CB7) Chicago Mercantile Exchange (CME), and the New York Mercantile Exchange (NYMEX), 1970-2002. .
,
-s
...,.r
,
. y)ji q(. lll@jjjy'. (@.'
''''
'
. .
:.
.
:
.
..
.
-
:
:. . . : :
,
.
yg
,
4sc
cBT
% 4oo
cuE
.l
.tz
...
8 aso
l
(
xyxux
,1
Y '$Cxn
(
4
J
k62raso
3. 4
E
3.
7 2oo
q
.t
j
ls()
;.'
y
$2 <
19 60
19 80
2000 Source: DRJ.
there is no risk.3 When risk does exist, we would expect that markets will develop to permit ecient zisk-sharing. lnvestors who have the most tolrance for risk will bear investors. more of it, and risk-bearing will be widely spread among
3It is sometimes argued that the existence of derivatives markets can increase the price variability of derivaives tlle underlying asset or commodity. Without some price risk in the lirst place, however, the market is unlikely to exist.
p+<
.
50 0 1970
-20
'
z
q
#
''i loo
-15
l
o
=
jj
CJ
1950
j
l l 1
.
%e
E
-30
l 1
E
' -. -. .-
Monthly change in 3-monthTreasury bill rate,1947-2004.
40
%
''qf'lqs!l
q'!' (( ii r'lqrtl'qrqrtqqqlq ''. !. .. .t: . .('' ;'(' E q E. ;. i. ;. .jyty. . :. y...r. .y;.t..rty jy.j j.y:tE!E!!2l;.. jkj. j.;y.jy- y j q.;.y gj .jyy.. ?l--. : ,1.1k---111! E 1111E85E:. ' liE:.. l . E. . :.-..;i'...--E-.-! ..... ''7iiI'. .........'''-.. ! )i.. @ '- .:.-E qEt-)i. . . . r.. . -. i.. ...... . . . . . E
:
:
.
,
..
.
-.. . . .. -.
'
':t)..
-
'-.
:' .
'
:.
)' j''(td 1* t'qf)f qpf j)' (' ' jhi E ' ;'r!' ykjjjj('. .)(' t'5* (tf j'1* r' (6r:('*:)7:'577777* E! ' E '''' EE i' 1111k--:111f*. tjd t'j' )' lf y' (::'r7:,r7* (rq''r)rqf '(t)f ftf qjf ' E' ' ( !Eq! )kE !'(EEi!E (q( ;! 1.* ,r)' yjf jjj'. (' : y; 1IlIi:!tF'. :--'-. ..' !''Ii:::;:k''. j.jjyy'. (j(' t)' 11118*. jtlt-'l----k--'pf 1IiIt5E!)'. -;tiiII'. t')'(t'-p;,'. ttd (yjj'. j'. (j' ).tjyf t)jd ' q(.( i'(t'(.!' ('i('.(..i((. E'jEEi. ;''E!'(.'!):.i(''.' ('( E! ;E... '('. . (i j.. ' ' . qillll!lqij' t')' .r' t'.,'-?'. .y'j. 'yyq'jy.yj..y. jj ( yj i..gE ''j E i!i ( i; rr gygyr gy. '.. '. EE!E(E@)1g jss;:' ' ''(E .j E!E(g(qry jg.jg.jqg..g(gy y j ! j . 4.. j ( j . j..y jj j.. , y y j jjy. . . j jj j j . . jyj )lk g.jy yj E E. j . . (. j(--),)k.. gk . (jj. yj 'pi)(;.. ((j '!jjyjq! E' .' ) 1IIi:tiiIijy (j ' .g . i gyy tttE --. -i -.( . i : . -:. . . .@ -.; -:. - ..L . :'fp;t.. !:,.. j: !q.Eq E - '. ' '-''.i' ; . ... @ . ..l. ' -. i . q i. @) . .. ... ..... . . . ... ... . . .
ik
k.
PRAc-rlcn
Ix
..
..x...>==x.,v,-..p.. - - -
1975
-
-
1
-
..
1980
..
..
..
.
4.
rl * *
1985
x
< *
>
vz
*
*
4: *
1
4, o
Kr
va
*: >
*
.a;
>
.
*
1
1990
>#
1995
I
2000
I
2005
Sollrce: CRB Commodity Yearbook.
A futures exchange is an orgaized and regulated marketplace for ading futures contracts, a lind of derivative. Figure 1.4 depicts f'utures contract volume for the three largest U.S. f'utures exchanges over the last 30 years. Clearly, the usage of futures contracts has grown signifcantly. Exchanges in other countries have generally experienced comparable or greger growth, ln 2002, Eurex, an electzonic exchange headquartered in Frankturt, Germany, traed 528 million contractsj the largest trading voltlme of any futures exchange in the world.
% Irq-rRoouc-rlolq
10
yy'yj'' (;'j))'. )' li'yi ykk-y'r ))' y' qlqjt''tjilr?f t'. )j'. )'-)* ;'yjy' yj-')(. tf tt$'. t-'.l' .t' t'))'' t,.)))-'' rj'. (. ' jjgjjjgji .t); . y.-..)y. l tjj . .y!':!.... -' ''lqlrr:f ;y;jj' ';' -'jj .'E dl:::::.,f dl::::::f :'t'jrf ')' ''' ,(' t'-'-n q;:f ;')' :'-'. dl:::::l,f 71* il::::::f .1:::2:11* dl!ii,,f y' dl::::::f y'tjsjjs'. dl:::::.,f ll:!!!,,f ;j).
To
TE E CE ' i ;(E ;'jrf ii!is,lf :!j;;' ;:'jL)j)LL. 1* ' 'E ('jr' )' g' j'y' (' jjf yjy'.,.jyjyj(tj,' @' lf 6* ))' 1)2* t!f ''' :( ri(i:(j!)j 1* ql' jf tlf qr'!f y' jjjjjg,f ljl! : !E iE.'i.; 'E..'.!('.E'!'iE'..E.''E((q'.Ejt...!.i'E(LE(t@i i Eq(.E'i..'.'.(. ,i.(E'Ei''(.'ii ( 77727(7. y' j'.j' jt.y'). yy'. ))))'. ;yr-jjjjr.:jy'.' .1It::.t!i!;;' yyy'(jjy;y$tr('. . !'.(..(Ei'.(.E (;. lgjf yjf rf .'::::EE:::' iiif jj'y ytf '-j' )' 1;* .!Ig:2' .!Ig::' jyf @17 . ; , - ' . -y $.kt .; -E. -.-. ?---jj(jjy .)2b3tij..i ,- .; ..;; -. p 8 g .-.i ..jjjjkjjk-, ( ii; .ti ;t; yy q : q r. kyj ..j;yq .. .- j.y; ,y-.y ; .qry.y y , . . yy . r y jj;jjjjj,j;.. .y r yy j . j j . j. E)!)!'iE(......l...,.-.;..-...... j yjy., . y .i........-.,.-...(.) jj.yyy.y. yyyj .j.jjy.y. . y:.y . .y yyy yyj ;.i. . . . t:......... . . . . .. y . ..yyyyj .y..y..yjy . . . . '(' 'q!'!?' '',-tj;jj.-'
.'.' i
: -
E
'
..
. E-
:
:
' ' : : :.
' ' E
:
'
.. : E:
-
-
-
. -
.
. .
''
'
.
:.
'
'' ' ' .
.
-y
.
.
:' .
' :
-
.
-
,.
,
...
'''
.
-
- -
--
-
...,..-.t)
................
:
': :. .
.
. ...
.--
Buyl lq G AN D SH
DERlvATlvEs
-
j..yt-; )j. -
.. --.
.
.
. --
. - --
-
.
-1::::211! dl:::::.,.
.'ItE:. iii,l.112::. ll....l,l Il.'.' Il...':ll ll'..' 11....11. !!qII IIEEE!!L.1It::. a 'Ir.'. .:I::e a . and Exchange (CME), Mercantile the Trade (CBD,Chicajo New York Mercantlle Fxchange (NYMEX). ijj,,, j,y,,.,,r.,.,,j r:,,,,::y jjj::r:,,-
fl:s!,,.11t::. Il..,,i,l ll'... l....,ll s
-1:::::11!
'l:::::),-
Ii....,II
-l:::::),-
o RT-SELLI
Nc
Fl NAN
CIAL
Ass ETs
%
11
different credit lisks. Hence bnnks could be expected to use interest rate derivatives, lisks in tose areas. Manufaccurrency derivatives, and credit dedvatives to manage turing firms that buy raw materials and sell in global markets might use commodity and cttrrency derivatives, but their incentives to manag: risk are less clear-cut because they are not regulated in the same ways as financial finns.
1.4 BUYING AND SHORT-SELLING FINAKCIAL
NASDAQ100 index
Namral gas
Eurodollars
Heating oil
Gasoline Gold
Wheat
Nikkei 225 Pork bellies Heating and cooling degree-days
Oats
Japanese yen
Electricity
Corn Soybeans
Copper
ASSETS
Throughout this book we will tnlk about buying and selling-and short-selling-assets such as stocks. These basic transactions are so important that it is Worthdescribing the details. First, it is important to understand the costs associated with buying and selling. Secopd, a very important idea used tlzropghout the bpok is that of short-sales. The concept of short-selling should be inmitive-a short-sale is just the opposite of a purchase-but for almost everyone it is hard to grasp at first. Even if you are fnmiliar with short sales, you should spend a few minutes reading this section.
Buying an Asst
ln recent years the U.S. Securities andExchange Comnzission(SEC), FinancialAccounting Standards Board (FASB), and the lnernational Accounting Standard Board (IASB) have increased the requirements for corporations to report on their use of derivatives. Nevezlheless, surprisingly little is known about how companies acmally use delivatives lisk. will be The basic sategies companies use are wemunderstood-and to manage perceived risk what of example, frction for described in this book-but it is not lcnown, do We companies the all frequently aggregate. in is hedged by a given company, or by hedging. hedging for rationale specific not or not lnow a company's We would expect the use of derivaives to vary by type of ftrm. For exnmple, financial firms, such as banks, are lzighly regulated and have capital requirements. They may have assets and liabilities in different currencies, with different maturities, and with
Suppose you want to buy 100 shares of XYZ stock. This seems simple: ((fthe stock price is $50, 100 shares will cost $50 x 100 $5000. However, this calculation ignores ansaction costs. First, there is a commission, which is a eansaction fee you pay your broker. A commission for te above order could be $15, or 0:3% of the purchase price. price'' is, surplisingly, imprecise. There are in fact two Second, the term prites, a price at which you can buy, and a price at which you can sell. The price at which you can buy is called te offer price or ask price, and the price at which youcan sell is called the bid price. Where do these terms come from? To buy stock, you can pick up the phone and call a broker. If the stock is not too obscure and your order is not too large, your purchase will probably be completed in a matter of seconds. Have you ever wondered where the stock comes from that you have just bought? lt is possible that at the exact same moment, another customer called the broker and put in an order to sell. More likely, howekr, a market-maker sold you the stock. Market-makers do what their nnme implies: They make markets. If you want to buy, they sell, and if you wany to sell, they buy. ln order to enrn a livin'g, market-makers sell for a high price and buy for a 1ow price. Ifyou deal with a rparket-maler, therefore, you buy for a high price and sell for a 1ow price. This difference between the price at which you can buy and te price at which you can sell is called the bid-ask spread-s In practice the bidrask spread on the stock you are buying may be $49.75to $50. Tllis means that you can buy for $50/shareand sell for $49.75/+%e. If you were to buy
lx'he table lists only a fraction of the contracts traded at these exchanges. For example, in June 2004, the Chicago Mercantile Exchange Web page listed futures contracts on over 70 different underlying assets ranging from butler to the weather in Amsterdam.
51fyou think a bid-ask spread is unreasonable, mskwhat a world without dealers would be like. Every buyer would have to lind a seller, and vice versa. The search would be costly and take ime. Dealers, because they maintain inventoly offer an immediate transaction, a service called imnediacy.
Table 1.1 provides examples of futtlres conacts traded aythese exchangesx Much commercii delivatives trading occufs in the over-the-counter market, where buyefs bnnks and dealrs rather than on an exchange. It is dicult ' gnd sellers transact with volume. However, in some markets, such as obtain statistics over-the-counter for to market is signiscantly larger than the over-the-counter the clear that currencies, it is market. exchange-traded
How Are Derivatives Used?
=
.
'Tstock
k
$
INTRODUCTION
To
BUYING
DERIVATIVES
commission twice, and you would pay the immediately and then sell. you would pay the bid-ask spread. difference between the Apparent rapid price :uctuations can occur because of the the bounce.'' bid is $49.75and the ask is If called bid-ask This is bid and ask prices. stock was last aded $50, a series of buy arid sell orders will cause the ptice at which the price has not changed, however, because to move between $49.75and $50. The changed. the bid and ask have not t'tnle''
commission Suppose XYZ is bid at $49.75and offered at $50, and the Example 1 100) + $15 $5015. lf you is $15. lf you buy 100 shares of the stock you pay ($50x Your round immediately sell them again, you receive ($49.75x 100) $15 $4960k receive from and what what you you pay tlip transactin cost--the difference between $51. k the bid and ask ptices-is $5015 $4960 a sle, not counting changes in
AND SHORT-SELLING
FINANCIAL
ASSETS
$
There are at least tluee reasons to short-sell:
1. Speculation: A short-sale, considered by itself, makes money if the price of the stock goes down. The idea is to hrst sell higi and then buy low. (With a long position, the idea is to first buy low and then sell high.) 2. Financing: A short-sale is a way to borrow money, and it is frequently used as a form of hnancing. This is very common in th bond market, for example. 3. Hdging: You can undertake a short-sale to offset the risk of owning the stock or a delivative on the stock. This is frequently done by market-makers and traders.
.1
=
=
-
=
-
arld come f-rom. Incidentally, this discussion reveals where the terms like it sounds bid price backward. 'I'he terminology is that the You might at lirst think price at should be what you pay. lt is in fact what the market-nlaker pays; hence it is th what sell it is will forvhence market-maker what the offer is plice which you sell. The reflects the perspective of the market-maler. you have to pay. The terminology happens to your shares after you buy them? Generally they One last point: What carefully, ' are held by your broker. If you read the ffne print on your brokerage coneact shares to another investor. Why would your broker typically has the right to lend your topic, want to bon'ow your shares? The answer to that brings us to the next 'Ibid''
Efask''
anyone short-sales. associAlthough we have focused here on shares of stock, there are similar issues ated with buying any asset.
Shortlling exnmple, When we buy something, we are said to have a long position in that thing. For sell if we buy the stock of XYZ, we pay cash and receive the stock. Some time later, we that is lending, in the sense we pay money te stock and receive cash. This transaction receive may not be of The return we rate back the future. in receive and today money stock price lnown in advance (ifthe stock price goes up a lot, we get a high remrn; ifthe nonetheless. of loan but it is a ldnd goes down, we get a negative ret-urn), entails The opposite of a long position is a short position. A short-sale of X'YZ cash. time later, receiving Some we the borrowing shares of XYZ and then selling them, the lender. A short-sie can it it, and to ret'urn cash paying for stock, the back XYZ buy ft'om a be viewed, then, as just a way of borrowing money. When you borrow money advance. of in bank, you receive money today and repay it later, paying a rate interest set the This is also what happens with a short-sale, exeept that you don't necessarily know : ;
rate you pay to borrow.
These reasons are not mumally exclusive. For example, a market-maler might short-sale to simultaneously hedge and snance a position. use a Because short-sales cgn seem confusing, here is a detailed example that illusates how short-sales work. Example: short-sellingwine There are markets for many collectible items, such as :ne wines. Suppose there is a wine from a particular vintage and producer that you believe to be overpriced. Perhaps you expect a forthcoming review of the wine to be negative, or perhaps you believe that wines soon to be released will be of extraordinary quality, driving down prices of existing wines. Whatver the reason, you tlzink the price will fall. How could you speculate based on this beliep If you believe prices will rise, you would buy the wine on the market and plan to sell after the plice lises. However, if you believe prices will fall, you Would do the opposite: Sell today (at the high price) ad buy sometime later (at the 1owplice). How can you accomplish tlzis? In order to sell today, you must first obtain wine to sell. You can do this by borrowing a case from a collector. 'I'he collector, of course, will want a promise ihatyou will return the wine at some point', suppose you agree to returfl it one week later. Having reached agreemet, you borrow the wine and then sell it at the market plice. After one' week, you acquire a replacement case on the mtrket, then ret'ul'n it to the collector from whom you originally borrowed the wine. If te price has fallen, you will have bought the replacement wine for less than the price at which you sold the oliginal, so you make money. lf the price has risen, you have lost money. Either way, you havejust completed a short-sale of wine. The act of buying replacement wine and remnng it to the lender is said to be closing or covering the short position. Note th a t short-selling is a way to borrow money. Initially, you receive money selling the wine. A wek later you paid the money bak (youhad to buy a teplacefrom ment case to return to the lender). The rate ot interest you paid was 1c)wif the price of the replacement case was lom and high if the price of the replttcement case was high. This example is obviously simplifed. We have assumd several points:
* lt is easy to find a lender of wine. * lt is easy to buy, at a fair price, satisfactory wine to remrn to the lender: The wine you buy after one week is a pelect substimte for the wine you borrowed.
BUyl NG AN D SH O RT-SELLI NG
TO DERIVATIVES
k. INTRODUCTION
14
'r')!77(TE'7. !( i';(.I 1( t'i';'(' qrf @' )' jtf ))' itf j'll7'lpqllll l 7')'q7* tql'' ( i '.' r' j'. '!()' .T1!' ;)'t(q: ;i qq ;'j'' )' .j' j)' ((1rtl r .((r' jjl . k! rrf )jj)'. ))' t'j'y')'. j;f )' )t'. y' ' ! ;.; (.'.'(( ;i(.. t.: !ii'.;!i ' q.q...j.'(( ' (.' ' ' q: r)' t'. 'r),rf tjf k;;j;!grj': ) ...;. 4:::). ElFj;ji )g. ; g.'. Ir..l!l), (i!. .t!jkk t- .iIEEkI . .. I..:lk: lp yii.iII777.-.---!4;'.i-... ..(..-..-:.--.-tiii rj. j. lr------!l :Ig. jr ,1.!1,k . . j jty jy.. tiy.i(..tt'q!i(.-.( 18..)' y , ; jlid!l;i y iitltr ,,L-,L-jj-;..-;.-;L-)...;-..;,..-,.-.,-..--...,i.----..L.-.L-)----.---L.-k-.;-; a lr..e d:::z,, t!!;. se 1iil,r..l'!;I-a s Ii..:, tt-l.-? iEi.s Ih..), lh..,' .Ig1: iss'-:,5;.t!i;. iiiiikt e . r. .L-.-,LL..,;;;.;.L:L;q.. t' . jilk:-...-.. y)t-.. r.. j.r.. d::E:g : dividend, D, T
'.i---)' 't' j',' ,)-'. '''--)'.' 777* @' 7* ((':E qif ')' $' : : E E EE
)' q' 'E' (' i';'f' ')('
.
:
-t'#.-)j E
.
:
-
1trE..:'--...:
:' : ' EE ' 'E : ' ' : ''' i;E EEE ' E E:: ' . EE pl' iE:i E E!': E ('E: EE . Ei...' E .i'. . E - - .(i ' ..E.. - : :-! - - .-.E.-. EE y. -@ E - i -;j..;y.r---y,.-.(.r)t),'. - !-. E -.. . E. ; Fq--i E. . -.tr--k;..-.. ; ..-t-))---.. q . '
'
:
( iE
..
--
'
'
:
'
''
.
:
:
: .
-
.-
- ..
.
..,(.....-.
qr( ..r -
-... ....-...
....
...........,..........
.............
-..-.....,..
.
.....
..
. .
i.y.i)..l. i ,
,
El..':.E.
.........E.. .'... ... ... . . .
...
.
.. .
((
-.
.
.
.
),;).q@,,,,
'
E' ! .
'
' :
:
-j
..
. - ... - --. E- . . .
.
(t( FE
:
.
.
. . ;;
dt::;ll 'k---i---.
-
-t:::,--k---p-'-.:,!;:
.
d::::'. d::::
-t:::,-
dt:::,-
,,..
.
days. Note that the short-seller must pay the the share-lendec
.
, r,i,,,,b,k#,r,r r
i.t,,,
............(.:
......;...
......
.
..
..
.......
jsjjy
tyt.tll:,,r,,tjjukjukj ,
j
,r
t
,,
.
Action
Borrow shares
Security
Sell shares
,
,
,
,
.
t, t ,,t r,
(
,.t
js
* The collectorfrom theborrowed wine.
tL,jj ,
,
,,rt,,tt)
.
t
y y;
Purchase shares -&0
whom you borrowed is not concerned that you will fail to retut'n
a payment in order to lend a bottle. This would be
The preceding examples were simple illustrations of the mechanics and economics of short-selling, and they demonstrate the ideas you will need to understand our discussions of derik.atives. It turns out, however, that some of the complexities we sldpped over are easy to understand and are important in practice. In this section we use the wine example to illustrate some of these practical issues.
credit rislt As the short-seller, you have an obligation to the lender to ret'urn the wine. The lender fears that you will renege on this obligation. This concern can be addressed with collateral: After you sell the wine, the lender can hold the money you received from the lender keeps the money selling the wine. You have an obligation to return the wine; ' . don't. that the in event you Holding on to the money will help the lender feel inofe scqre, but after thinking the matter over, the lender will likely want more from you thn just the current value of the wine. Suppose you borrow $5000woft of Wine. Whai happens, the lender will think, if the price of that particular wine lises to $6000one Week lat/ This is a $1000 losson your short-sale. In order to returu the wl, you Fill hve to pay $6000for wine youjustsold for $5000. Perhaps you cannot afford the xtra $1000and you will fail to rettlrnthe borrowed wine. The lender, thinking ahead, will be worried at the outset abou thispossibility and will ask you to prov tde lnole than the $5000th wie is worth, say extra $1000. Ius exea amount is called a haircvt, and serkes p proiect tlae lender against your failure to rettlrn the wie when the price lises.o In practice, short-sellers must have funds--called capital-ao be able to pay haircuts. I'he amount f capiti Places a limit on their ability tp short-sell. ''''
''
.
Now consider a short-sale of stock. As with the previExample: short-selling stoclt selt iq receiving cash Fhp you short-sell stock you borrow the stock and ous eyopl, it the original today. At some f'uture tiate you buy the stoc k in the market and rettll'n to of te stock you shorttoday, equal to the market value owner. You have cash coming in market then-current its sell. ln the f'uture, you repay the bonowing by buying the asset at short-selling of a.loan. Thus, price and remrning the asset-this is like the repayment interest rate you pay is not that the except borrowing money, equivalent to a stock is stock plice. The rate of known in advance. Rather, it is detennined by the change in the security falls in price. In effect, if the interest is high ifthe security rises in price and 1ow which you borrow. With a shol-t-sale,you te rate of return on the security is the rate at rather than the buyer of a security. are like the issuer of a security 1.2 depicts the cash Suppose you want to Short-sell IBM stock for 90 days. Table shprt-setler must in t'urn dividends, the flows. Observe in particular that if the share pays wtne! This with alise did not make dividend payments to the share-lender. This issue and dividnd ordinary payment, dividend payment is taxed to the recipient, just like an it is tax-deductible to the short-seller. opposite of the cash flows Notice that the cash flows in Table 1.2 are exactly the l/2c opposite of bttying. literally from purchasing the stock. Thus, short-selling is
The Lease Rate of an Asset make payments to We have seen tat when you bonow an asset it may be necessary to stock are an exnmple of this. We will refer to the the lender. Dividends on of the asset. This concept will arise payment required by the lender as the lease rate discussions of frequently, and, as we will see, provides a unifying concept for oqr layer derivatives. under some circumstances The wine example did not have a lease payment. But Wine does not pay an explicit it might be necessary to make a payment to borrow wine. seeing bottles in the enjoys dividend but does pay an implicit dividend if the owner 'short-sold
15
ltisk and Scarcity in Shortelling
-D
Cash
require
$
CIAL ASS ETS
to
Return shares
-
cellar. The owner might thus a lease rate for wine.
Fl NAN
.
'
an
k'
scarcity As the short-seller, do you need to won'y about the short-sale proceeds? The lender is going to have $6000 of your money. Most of this, however, simply reqects your obligation, and we could ask a trustworthy third party, such as a bank, to hold the money so the lender cnnnot abscond with it. However, when you remrfl 4he wine, you fnrd?'dJl. This raises te question: What rate are going to want your money back, pltts you? the will the lender Over of interest pay course of the short-jale, the lender can en'rning. 6%. The lender could offer to pay yo 4% on the funds, invest your money, say, difference the 6% ealmed on the money and the 2% between keep thinking to qs a fee What happens the lender and paid if borrower negotiate? the 4% to you. point: The of is Here the interesting interest the lender pays on the cpllateral rate people how going depend want to borrow wine from the particular vintage is to on many
''
f'Note that the lender is not concerned about your failure to perform when the price goes down because the lender has the money!
16
%.llq-rRoouc-rlohl
To
PROBLEMS
DERIVATIVES
practical matter, it may not be and prbducer, and how many are willing to lend it! As a wine, the lender will offer borrowed high demand for easy to 5nd a lender. lf there is willing for being to lend something that essentiily enrning a fee a low rate of interest, the lender might onelude the wine, is scarce. However, if no one else wants to borrow close to the market interesi of offer ' th an n othing and you a rate that a small fee is better rate.
different markets, the repo The rate paid on collateral is called different things in market. Whatever it is called, stock the in rate in bond markets and the short rebate another cost to your is of interest market rate the difference between this rate and the short-sie.
k
Jorion (1995) exnmines in detail one famous tderivatives disaster'': Orange County in Califonzia. Benzstein (1992) presents a histoly of the development of 5nancial markets, and Bernstein (1996)discusses the concept of risk measurement and how it evolved over the last 800 years. Miller (1.986)discusses origins of past Enancial innovation, while Merton (1999)and Shiller (2003)provide a stimulating look at posmarkets. Froot and O'Connell (1999)and Froot jiblelrl/rd developments in snancial (2001) examine the market for catastrophe reinsutance. D'Avolio (2001)explains the economics and practices associated with short-sales. Finally, Lewis (1989)is a classic, funny, insider's account of investment bnnking, offering a different (to say the least) perspective on the mechanics of global risk-sharing.
PROBLEMS the pric of plpthing Derivati.vesare financial ins%ments with apayoffdeterminedby lisk nd i ieduce speculation, management and else. They can be used as a tool for transaction costs or avoid tues and regulation. risk-sharing. One important f'unction of financial markets is to facilitate optimal d ith cpincid w an increase in The growth of derivatives markets over the last 50 years has shck, the ablidpment 1973 oil the zisks evident in variou markets. Events such as the created have a ew role of fixed exchange rates, and the deregulation of energy markets for dedvatives. ther cash) Ashort-sale entails borrowing asecurity. selling it, maki dikid:d (or coneepttlally the short-sale is payments io the security lender, and then retunng it. A of financing, used for speculation as a form opposite of a purchase. Short-sales can be details f short-selling in practic arl be undrstood of the hedge. Many o or as a way to short-seller and scarcity of shares ihgtcan be bkrowed. as a response to credit risk o f the lnderss' liig with Short-sellers typically leave the short-sale proceeds on deposit wit.h ,
1.1. Heating degree-day and cooling degree-day futtlres contracts make payments based on whether the temperature is abnormally hot or cold. Explain why the following businesses might be interested in such a contract: a. Soft-drink manufacmrers. b. Ski-resort operators. c. Electric utilities. d. Amusement park operators. 1.2. Suppose the businesses in the previous problem use f'umres contracts to hedge their temperamre-related risk. Who do you think might accept the opposite lisk? 1.3. ABC stock has a bid price of $40.95and an ask plice of $41.05.Assume there is a $20 brokerage commission. a. What amount will you pay to buy 100 shares? b. What amount will you receive for selling 100 shares? c. Suppose you buy 100 shares, then immediately sell 100 shares with the bid and ask prices being the same in both cases. What is your round-trip transaction cost?
collateral is called the short additional capital called a haircut. The rate paid on this the interest rate. rebate, and is less t11:,11
FURTHER
READING
this chapter. HowThe rest of this book provides an elaboration of themes discussed in 3, and4induce 2, discussion. Chapters the related directly to ever, certain chapters are derivatives, and show how forward and option conacts, which are the basic contracts in derivatives marketdetailhow in discusses 13 Chapter they are usedin risknanajement. derivatives can be combined makers manage their risk, and Chapter 15 explains how products. with instruments such as bonds to create customized risk-management sites for TheWeb thatlistthirconeacts. sites haveWeb The derivatives exchanges (Chicago Board of Trade), wwmcme. the exchanges in Figure 1.4 are www.cbot.com (New York Mercantile and Exchange), www.nymex.com com (Chicago Mercantile www.gs.com/econderivs. discussed at Economic derivatives are Exchange).
1.4.
Repeat the previous problem supposing that the brokerage fee is quoted as 0.3% 0 f the bid Or ask price. '
1.5. Suppose a security has a bid price of $1t0 and an ask price of $100.12.At what price can the market-maker purchase a security? At what price can a marketmaker sell a security? What is the spread in dollar terms when 100 shares are traded? '
1.6. Suppose you short-sell 300 shares f XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the seculity to cover the short-sale, how much proht have you made if you close the short-sale at a plice of $29.87?
18
k. IN-rRoouc-rlolq
To
DERIVATIVES
1.7. Suppose you desire to short-sell 400 shares of JK.I stock, which has a bid price of $25.12and an ask price of $25.31.You cover the short position 180 days later when the bid price is $22.87and tle ask price is $23.06. a. Taldng into account only the bid and ask prices and interest), what proft did you earn?
(ignoringcommissions
b. Suppose that there is a 0.3% commission to engage in the short-sale (this is the commission to sell the stoek) and a 0.3% commission to close the short-sale (this is the commission to buy the stock bck). How do these commissions change the prct ilz the previous answer? paid nothing c. Suppose the 6-month interest rate is 3% and that you are much interest do you lose dpling the 6 proceeds. How short-sale the on months in which you have the short position? '
'
. ..
:
.
.
.
.
.
E
1.8. When you open a brokerage account, you typically sign an agreement giving the b.rokerthe light to lend your shares without notifying or compesating you. Why do brokers want you to sign this agreement? 1.9. Suppose a stock pays a quarterly dividend of $3. You plan to hold a short position in the stock across the dividend ex-date. What is your obligation on that date? If of the you Jtre a taxable investor, what would you guess is the tax consequence deductible'?) dividnd the be would tax to particulm (In expect payment? you Suppose the company announces instead that the dividend is $5. Should you care that the dividend is different fromavhat you expected? 1.10. Short interest is a measure of the aggregate short positions on a stock. Check an service for the short interest on several stocks online brokerage or other snancial of your choice. Can you guess which stocks have high short imerest and wlzich have low? Is it theoretically possible for short interest to exceed 100% of shares outstanding? 1.11. Suppose that you go to a bank and borrow $100. You promise to repay the loan in 90 days for $102. Explain tls transaction using the terminology of short-sales. 1.12. Suppose your bnnk's loan officer tells you that if you tale out a mortgage (i.e., than you borrow money to buy a house) you will be permitted to borrow no more terminology of using ansaction the this Desclibe 80% of the value of the house. short-sales.
An
t0
an Cytipns
chapter introduces the basic derivatives contracts: forward contrats, call options, and put options. T hese f'un damental contracts are widely used and serve as building blocks for more complicated dedvatives tht we discuss in later chapters. We explain here how the contiacts work and how to think about their risk. We also introduce an exemely important tool for analyzing derivatives positions-namely, payoff and proft diagrnms. The terminology and concepts introduced in this chapter are fundamental and W ill be used throughout this btjok. ,
2.1 FORWARD
CONTRACTS
Suppose you wish to buy a share of stock. Doing so entails at least three separate steps: (1) setting the price to be paid, (2)transfrring cash from the buyer to the seller, and (3) 'ansferring the share from the seller to the buyer. With an outright purchase of stock, :11three occur simultaneously. However, as a logical matter, a price could be set today and the ansfer of shares and cash would occur at a specifed date in the f'uture. This is in fact the definition of a forward contract: lt sets today the terms at which you buy or sell an asset or commodity at a specitic time in the future. A fomard contract does the following:
* Species the quantity and exact type of the gsjet or commodity the seller must deliver. * Speciss delivery logistics, such as time, date, and place. @ Specifies the price the buyer will pay at the time of delivery. * Obligates the seller to sell and the buyer to buy,subjet t the above spciscations. 'rhe
The time at which the contract settles is called the expiration date. asset or commodity on which the forward contract is based is called the underlying asset. Apart from commissions and bid-ask spread: (see Section 1.4), a forward contract requires no initial payment oc prerniumk The conact'ual forward price simply represents the price at which consenting adults agree today to transact in the f'uture, at which time the buyer pays the seller the forward price and the seller delivers the asset. Futures contracts are similar to fomard contracts in that they create an obligation to buy or sell at a predetermined price at a future date. We will discuss the instimtional and pricing differences between forwards and futures in Chapter 5k For the time being, think of them as interchangeable.
21
k. AN
22 gt'. yj),y#,'' yt'. (' ,'y'jf )' )r',' (r' ;'. i-'' lf ('. 11.1:---211.*. ''. t'tiy)f q' yjt,f rqf trf ;-p)('-)--r-'.'.-L(LLL'. .:2t-*-* t(' t'(-'. .))'r' 1111!*.* ltt('. t'. r'.
frq'rRoouc'rloN
To
--' --') -yy'-.' -'i .-'' -'. -)?'' ----' -'-' ''i' -:' -' -'. j'-'-*.*9!*. E' 'j' :'f' ;'k' ,'j'. E.')' ,'?' i'if '1* ''r' q' :'t'Etf j'''j' )y,)' ..' ?' ;,17r(* p)'' t'lqf ;),' /(144q44172*:74()j4'j* )' i'. (' (rqf ,'y' !' p' (7* q' @' ';)' ((' 'E :' E' E: ' E EE E' 74 : : (': : ((; '(.EyE.;'iji!ii.i.'. . 'iEEE'(.('.'('. ''.. ('( 'E'' .' ';EiE': ('i'..E'(E:':(-(r('.i(@ (: . .'.. EE;;E:;.!: ' rtp:l'r?q!;)qq:r'!rqrEsqr)trl 'E:i i(.r.pti(i1:( lr. k) .'.11 .' - '. !j.'(;rEjg . -.. .- -.;-... - -. - i ! -.-. -E. : -..;-..-. - : ,j--(y$ EErk. ; ;. . ;.-. E.;- -.-.E;. -. . . (q - -- .! - (-... i .. -. . r .. y - .... )zt .-'' (i.!'-- . ,:;,q . . ( . f-----..k' -- ---iy j .E.!. ,i. ! .;.i--:.-.Elgtll:::iiiiit.-. --- ).' #. .E E kkk)... -. . : '.. ..-.-.. . ;. .. --.E --; -;.-. : . .. i i. . - -... .. . .. . .
.-*4 .' ttr)r'.. 'rjt((t' 11! .
.'.''' . ....
FORWARDS
FORwARD
AND OPTIONS
''
. .' '.y;'y)t)q' ' .' '.''.. '.. '-.(.k. '-. j,j-.-. ( '; ..j... . .y t, yE j ;jt.' ..-.. '.l tljtj) r . . . . il!IEii!!k. iI1i!:q!q.. '. '.. ! . i ki''-''- ..'11I1::tli'!.. '.
i
'
.
:
.
'
'
.
.
:
.
.
'
'
tr ;
.
.
.
.
.
. .::;:1.!1!:..
-. ' ' .
'
-
.r
..
-
-
..
.
. - .. .
.
- . ..
.. ..
..
. .
..
. . . . ..
.
.
.
-.
'
.
.
...
'
.
. .. .
.
.
'tl)-'
. .. . .. lndex futures price Iistings. .
-
.
.
.-
. .
..
-
.
.
-
.
.
Index Fulures
w
c22q Illlj
52.1.11:.(im
-
LIFEIIRE (lz.ll Illil w m.r
DJ lndustrial Avefage ((2n-$11x Index 9235 4>2 9953 19085 9951 1162 113 15 57 s'lt 19072 319 116 19575 8442 159 125 nacvc1 149; 1:08 lnt 4,6 Est vcl !n n 911;Qpan i6. /5/4, +123J712 p?l: B'l19193-132 99533 cnse n ldz Mini Dl Industfial Average (T)-$5 x laex ,9
,8
,7
995 10p2! 9919 +k>.10 62 5:/t Vnllue 1 7,7339npvnIat 18,7894
+131
113 19629 931
42,695
...
.r
...
-.19.
-t17.
Mini S&P 500 (r.MD4%x index
16327.5lpzs lgz .ept lrg !!nIlke 729.9(5;cponInt 612,116,+3M.
q5 :11.48%19752: 595759
S&P Midcap 400(D-$52 x kdp.x 575.1 87.($ F73.10 +17.15 61.552 598.79 B,Mq sept 56 Et vcl495:vQl l/ln 7p; Qpen Int 13,M +.63. I(k prl; l1l54.852l.c 511Jk f-lH 571.02, Nasdaq 10 (r.ME)-$1(11 x Index 79,921 lzs BD5P 8995 r59 :1.55599Bz Jept l'Wp Qpen 1nt7431J9 Estvn1144R vnI
,(1
'
.9
.g-2
+23-::.
source: J.Ptz//Street Jounal, July 28, 2004, p. C-16.
Figure 2.1 shows f'uttlres price listings from the JPcl!/Street Jt?lf?p7tz/for futtzres lndustrial Average (DJ 30) contracts on several stock indices, including the Dow Jones indices are the upderlying assets for . and the Standard and Poor's 500 (S&P 500). these examples the contracts. (Astpck index is the average price of a group of stocks. In 'I'he stock.) hrst column the price of just one we work with tls group price rather tan the plice at show that follow of the listing gives the expiration month. The columns settlement and day, duling the the the beginning of the day (theopen), the high and 1ow
.
The Payoff on a Forward Contract Every forward conact has bot.h a buyer and a seller. The term long is used to describe the buyer and short ij used to describe the seller. Generally, a long position is one that makes money when the price goes up and a short is one that makes money when the price goes down. Because the long has agreed to buy at the fixed fonvard price, a long position profits i.fpdces rise. The payoff to a contract is the value of th position at expiration. The payoff to a long fonvard conact is Payoff to long fomard
'rhe
plice, which reQects the last transactions of the day. and 1ow The listing also gves the price change from the preyious day, the high of number the which measures during the life of the futures contract, and open interest, seller, and both buyer a a a contracts outstanding. (Since each trade of a contract has buyer-sellerpair counts as one contract.) Finally, thehead of the listing tells us where the Mercantile Exchange and contracts trade (theChicago Board of Trade (CBTJ Chicago is 500 the S&P $250 times the index (CME1), and the size of the contract, which for detail 5. There are Chapter in in more value. We will discuss such f'utures contracts 2.1, both Figure those in than eontracts many more exchange-traded stock index f'uttlres in the United States and amund the world. September and December The priee quotes in Figure 2.1 are from July. July in for purchase of the index plices for tlze two conacts are therefore prices set 500 f'utures pliee is $1092.50and in later months. For example, the September S&P 'rhe
23
'rhe
DI-AIG Commodlty Indextnmg x inlpx 125.9 419.1 2,886 455.9 r. Est k;t Q;vQIMQn (); ())Qr1 lflt 2:26, tm:1 (lnse14.1.311$. 1.0111.9242 ldx )rl: Hl1M.7.13: x index S&P 500 Indexto-sa 50pt 10231p 199692 1 2329 199250 9 116929 7819 575,917 9e( 129352 19559 1,932' 199299 97A2 ll 21 72199 12,717 st kol 39763;v:l McR .t37876)Q)Cn1nt5.z8912+0.7l. ldx p#) Hl19.F2 : 18l.7l Oere 1021.83,
AQj
%
the December price is $1092.90.1By contrast, the current S&P index price that day is $1094.44. This is the spot price for the index-the market price for immediate delivery of the index. There are many more exchange-traded f'umres contracts than just those listed in Figur 2.1. As we will see in Chapters 5, 6, and 7, there are also f'utures conacts on interest rates and commodities. Fumres are widely used in lisk management and as an alterative way to invest in the underlying asset. Agricultural futures (suchas corn and box on page 24 soybeqs) can be used by fnrmers and others to hedge crop prices. discusses an unsuccessftll proposal for a new f'utures contract that was in the news in 2003. We will discuss in Chapter 5 how forward and fumres prices are determined and details about how fumres contracts work. 1.11this chapter we take prices as given more and exnmine proft and loss on a fonvard contract. We will also see how a position in a forward conact is similar to azld different from altemative investments, such as a direct investment in the underlying index.
.'-.l'. .j' -..' --.1111'*.. -'' .'' ..'
.
CoxTRAcTs
=
Spot price at expiration
-
fonvard price
(2.1)
Because the short has agreed to sell at the fixed fomard price, the short prolits if prices payoff to a short forwrd conact is
fall.
'l'he
Payoff to short fonvard
=
Forward price
-
spot price at expiration
(2.2)
these calculations, consider a fonvard conact on a hypothetical stock index. Suppos: the non-dividend-paying S&R (pecial and Rich''l 500 index has a current plice of $1000 and the 6-month fonvard price is $1020.2The holder of a long position in the S&R fonvard conact is obligated to pay $1020in 6 zonths for one unit
To illusate
l'rhe use and nonuse of dollar signs for fumres prices can be confusing. Many futures prices, in particular those forindex ftltures, are in practice quoted without dollar signs, and multiplied by a dollar amount to determine the value of the contract. In this and the next several chapters, we will depart from this convenuon and use dollarsigns for index fumres prices. When we discuss the S&P 500 index fumres coneact in Chapter 5, however, we will follow practice and omit tlle dollar sign. 2We use a hypothetical stock index the S&R in order to avoid complications dends. We discuss dividends and real stock indices in Chapter 5.
associated with divi-
$ Ah1 INTRODUCTION
24
FORwARD
AN D OPTIONS
Fo RWARDS
TO
.'
'''''''''. ..jjjj;..ryj'. .j;!ijjk,;k,;jjjj.'. '''''''''''. ''''''''''. '''''''': .'.-' '' ;'''''k''.''''''''''. :''''''. '''''''''''td -' jjjy:tjjj'' $1!2::::,,*: -' j'E' -'. jj::rr:,,f ':;'E:' dl::::::f ;'
CoxTRAcTs
%
25
tq .,jjj.,jjj*. )' ,''''Tt!'' (' !;!' jlr,..f :!!i;;' r'..(!'j' ;''q(''((@ IIF::CI:.X -l!iijk:Is:q' j'(' 'j;f ()' '( ( ( ! t;g!! jg(rj j-yrf ;j' )t'!.:j(::-jjy::' ,lr...'l.f llr...f .1ltE:' lly.'.!l.f t')' (7( j;.'.(..(( (.( Ej (.(.(.'i ).'1. y'. y' @' ;-y'. ,'(jI:qrl;qrtjl,'. .y'E 111::h:;*.i::12::' 1::::' 11::2::,,*' ij2!r,-'. 4(2@jii,'. L,bi-,,i.'. y-,jyjgyj,'. k,;'. ,'. jj2jjyj,'. 1I...,I...,Ik..,I...yjjyjjjjjj,jjyj'. yj-'. d1::k::,.'. y' dli::::,id 41::t:j.,* 11((:* .!jj:!.;I!' 11::::::* 11:,..11* I1i...1Ir.'' I2:t)i;'' 11:::2!' il!!illt.d .jII::E..' i:!!)1p'' .11:..11L.*. .j66LL''. .JIk.' 11t..1Ik.'' jtrjgyj;d .j3q)4,..'' j'jj)j))j4'. yjj,.d .jj,.' jjyjjjy'. illr)d iljj!)ll.d ';yjjjj;'' (1:(qj)' (jjjj!!:d l:jt:::d jj:gjq:jjd (jkjjrjjd .,,jj),Lj'L :')' j'jjjjy'. (' !' )' ,';'(' ---,L:-.k-j'''b'b,'',bL' ,*1111r.1k1112611111,* .j'. iki':i:k'kt'-' yrf jjj..jj...jj:jjjj...'. y' 'jjijf )r' jk-'iiiilk::'. jktt'tiili--kll'E' yE,y)'. !'. t'j-)d y'!y'. k','F' ;'i q.' y' r,f r' --,LL:' ))'--' j'jyj,'' jj:::,,,f )jjy..yjjt.d ...a' .;,.kjjy.(' ,',jygjjj,'. '.kjy' t')' yg' .j,j.jj,;'l ::ljlilk.d 7jt22* .i:2E::I,'' r',iig:j)'. :Ii(#i)iL'(.;:jjjjj(jjjjjyrgjk.'. y'. i::!::Ik'. k(,k;g;,;;'. j'y.kjyjjjj,.jjjjyj.'. -k:,,,''))jjjjL.j)j)'. .itE:Ii'' iit::t)iyr:1i.'. Ii!1!l'( 1* plt,zll.' :idi:i'. LbtLL,'-..,,t..'L,:LjLjL,.. y' lt...llk'y IIt:i::I;'. i'i'. I2i!i;' y'y j'.,lLq.'ld,'.x,llk.'y y' 11(j).1122(:11.* kjj;,k;'. (j' yjg'' (.,j;...' kj;.;'. t'r'y IJIIi,T';''' ......y' )jjjj)j))jq,'j j'y' jjjg;;d jjjjyy.d jjjjgjgjj,.'.kyjj,jjjjg.d ggjjgjj,jjjjgyyjjj.d jjjjjjr'd .jjj.jjj(.' y' 11:j((:Ij' Iljjjy;d lzjjjr;d iljjj;;d iil)jli..d j1j(r()1('' :lygyjl.d j'jjjjjjrrd ?' ,'(' jjkjjjyjjd jjyjjjgjjd ''r,' k!!!4;pp!rt'f' i:,y' ;'$,' t'1iii,;iiir'y' ,,tf t-t)f ;'y' j'y-'E ygjjjkiy'. ,g'. ..jjjgjgjjy'y' ('j3L'. ('.It:2::l,' j'. l'i 'E))1 (jttt''-jtt-. iIiI(k:-:'krf y' iiiiiEE!l.;;'' .rjj::-jII:r)' .1Irr:' tf tjf y' lj'y 11j:44444j:1)4* .''y '.y'. . (' y' :)I(jiIj:j'. iltEqd jj)jj:i'. (11tj:)1t.*. :,iEi;'. rllEj..d l,i!;'. 'i1Ei:i,;'. tI1q.:j1'. :j1j4qrj1t:11j*. tiIEt:i.)'' :1kEEI1,g', I:Elll,.'.llki:. jjI(j:ilk.'. (jj)d k:jEil,.d :t('. kliE!ri;'( j'(1j.. .ikEEi:.;'. d1:tE:ilk'. (...'. t4I:;rr;ii;:r'!!!!i;?':r(.tjjggjjjj'y' ,IEir;'. ..kk;k;;,;.'. yjjyytgjjyj'iI(i.i1k.'' iI):i.'. ;l:@i.'. kltiE5i:;'. j'('jjjjyjk'. dlr:zEjll.'y .11k,2!I(.' (211.(*. :)i(::i..'. Eliq:iikd :ll:''d E,l::r.:ili.d ggj(g,gj(.'' kI!EtI1i.'j yyd y'. ClEtl..d :ri!i;'. r!I4.'. yjjj.'. E!!pyd :--7* :jjgyjj,'. ajj,.,jjj'i Eirzqd :1!::* y.' (qIrE#:Ii.'. i1t::.'1IiEi:i,-'' :.11:.7* :lEii!lk.d kl::E::.,d :1!j:* :,EE;;' (.1(2..* -,i!E;;' .IEL:::7'. (llli''',;pjlirsr. $l2iEir:'' (lil:.:l.d :111)* :!1(j7iik.' ,y' 11::2(j:!k;,,24*. .1Ii,:Ik.'. Ell::'.d :iiE!I'. !1t:E'. :111#-.* :)i::';'. 1I@i!!!:'. (!ItE.:.i)'. (lIE:tIk.'. IltEili..lll:r''ll!:;d (iii,d Ellijqt'd (' ilzi!r:d 41::E::11* (11(:7.* .jj.,y'. qiil.d ;1!El11;.' 5115'1111* 1* (1lE:'' !ii(jq11)i1i)(Ii:22'' k' ..js' yjjgsjgs' j'y gjjyyjjjjd jyj,jy'g ()1j.'. .rj;' (jlr''d (j1r;k'' jj(q'q11.'' jjtj'qjld 'jjjj(j;jj'; k;ygjgjk': :,(jj...jj..'. .jjgy)'.,,:,,.. iliEjj;d (Jj((y.' jjt,;rjj.d j'('. ,''r' t'tjjjjjjjj.'. t'Lk:.k''','2'jjL' tjf q' .yjjyyyyjj' ;'ij'. tjj(.(yjj('. $' jjjgjd t'' ((q' )q!' :j;' )',-'. (' :-'g )'! y' ,y-t',' 4y'rryyk'. ,(y' lj-,'. j'kyjjj;j'. !i(::I1.'' ,)'. kjjrjtgjk.'. 1kiiEii1)'' Llii:jd iiEiikk'' )' j'. lti)kd :lEi@I1.)' ilrzE'il..jll:.d )ii:'ilkld llE:''d li.):)i.d '1i(:1IL.'. kt'. iiElih'giiiEq''t!ll:st;ij:d j.ykj'. qt'g j'yyf y'j )'; lli..'!lkf ---,'1El .:!15E;;* . . y' tjjly..d tjj,jyjy'q $k;j:k;,.'(..k..j'( j-jd jjrjr:jijd ;L66'i'. krjjjyy'y gjd t'i'q'.kk;yyjjjjyy'. .ijy. .;. rq (tItEp'E't(.(', .y' ;'y' ilisll-.'. jjggryjjd :,,4j4)y,,.. j1l::7''. j'. ..' j'y' kj,;,,kkk'. '1ill)'. :!jE!j;'. ' jyjjy.'. i y' jjjj;fd jjjjjjj;'. jygjgjjg'i .jj(.' jjj(jjgrj': jjtjggijk'y :y'.j y@)' yrf k' y'tf ii!!i;i:', jjgjjjj.jjjjyy;,d rk'.i q'. (yf ;p@'.r'. yjf y' jyy'. jj;,;,,'. ..,jjry'. kii:it::d jrjry' q--'j' j)-'LLL33L;;;L:LLLLLLqjLjkLjLLLLLLL:j)'--' ;'----j-j(jjy'. y'. )' jqt'j'. j'j' @' y')' l'g y' (' 7'i j'g:jjqjjr.d j.' y' yjjyg,,'rjjjgjyy,'rjjkyd y' .,' r' r1Ii:q::!!,-#r.r'. r!ll..ik.d iitjgjrjg,)d i1Iiii'rIi!q2''. kl:::::ihd ;i::j:j'. ii:@ii'jjlqjqjji-'. j!qrq:ryd r,.,,:rr,d :1I:i' i.iijiik.d )' rlli?'d illirlp'.'l::!i,?i-qd zlrt'rll.'y :1It::' )''Eyd jjlll:::lid 'iI1:)' j'j llljji''llliljll,l j't'y'gyd j'. jkjjjjjjjjgjjyd jjjjj;jd jjjjkjj:jktjjt.d jjjyygyjj,d jjjj.d jjjy!y!kjtjjj''d .jjjjjjj'. rljrr:jjjd rjj:jrjjj.d 'jjj..(!j(.jjiik.' jjjjgyrj'. .j.ry' jjyjj,;jr,d )j' yjjj;y'. j;'. 4* ir-'l-'iillEiqtk'' 7* )))'--tjj.. jjjjjjj;jyd )' ..j,' jj@(')jy'. y'. yy))jt'-'. j-t'. yyjf .-' ?' lpjft-.rqyqy'.ypyyk (' jy' )qj'.j')'. y-#r'. jkf-kbbb,i'-.'-,:jLjtt.. ('. kl::l)d k(jjjj:j;'. .y'@.t' ;'. j't'. '71.1!:1kk*. t'y'p'(y'yy'. .yf j)'. t),'; jj'. j't'y')'L'y'k'k'r'i.;)f )j''y-' lj-y'; !'!'. @''. ..' (jjj.d j'. k;jjj,jjj'. )'. (EII::..tiiI.X ;ikii!i)1lL)'((11:((((::)1.* iI!q?''11(it::jii. yjltqri,i.d -ii::iiI(.'. i!iiil;d t1ItiE!!!!i)'. yyjj,'. (1I((g;i?s' tjgygjjjj.'' .i1i)'. t'yy'. (11(.*. (11(:4::,.* (11)* yjjj:,,-..,.-.,.'. t1:::(!IL)'' qII:,t--).'.''..' .yyjj,.' q:;.i ((1Ii(,,.).'. (jjI((!:.,' illi-liil'' kiiiiik'' (;1.4;tj,,* ((114r::)1* ()1t(4::,1* )' (112r711(g* 111(:.'* j('. (' jy..' t'jjryjjjjj.d (jjj,.gjjj.d jjjg,yyjjjggjjjd ')jyj::jp' jgjjjjjj;d jgjjjjr!jjjjjjjj,'' jjjjtjjj,d ))' )y)' y)' )(kqy' )': rf :iiii;;'y)-'. jjjjF:jky'-' ('yf (t.j4ty(tg.' )'.-y'q' jj', j)'p.tttlf )'. (t(' )jjjj'. y'. ;', (' jyjjkij'y ..' ti'. .I:iI2!q' ljll'tf yrjjij,;),,.'jj::yr;j,j itf jjyjkj;'. k'p ..jj;ytyjjy'(!iI)(::li'. lllkjjpjyjg-lk..f i1i)EjEI1i::!ii.'. kj2yjjjj;'. qElliiF::;r'd j:jjyjky..' t'(!!!1,;r-)' y' l'y y'tky'r (-.ykkjjjjj'. t'. .-''6;'kiiL-'. ; ;. ' rgy-)yjjy!.i..; r. gjjj4;g'. .tkjj2jr,;'(j:jjjjjjj'. k1tiiji,'. 111.1#111(52:11(*. tyjjy,jjkj'y rj!d l:ljjlljjqd y' jjjjjjjyjjd )' tyf ttf tkjgjjjjjg'y yjf .(' t'.'i'. )y'.' r(tjj;'ti;t-t)jj'-;yjy)jjjjjjkjjt'. tjjjk.d t$r'. y;' jy;;jq,:..:jj:-d jLfqq;,-'. jjjgjji;yd .jjryy' (k:yjg:j;'. kl:tzird jjjr.'-. .,: .j!.;.j.k t'y'( iii'iiiir'r';yf :- ' kt p k),kttl E q. E ,) tjjgllirgy' t')'tk:i::!:d '-' kiiti,:l'. l,i!k;k'. ..jjyyy,jrrr'. t:,d ..' tjjjjjgyjyd tyygyj;yjy,jk'. tjllriqll.d (jjjjj,'. .((,'. (4jjjj,4*. (jjjjtjjjyjjyky'. tjjjjjkg.'. (4444j,,4*. yy)-'$jjjyjjjj'. liltt,kd t-y'. ( . !). '';g. j),.yg..1:ii?q'. 11* .;.. .!q:.j. ,y. ; .. . ;; q ); :( .- E E.(y :(,. q g. (g.ry j lljt-yrll-.ljrj)i .y. .gjjgyg;j(j'. t,ijgjr,;d (11i#:ik..'. ltiiiid ))jjjjjf)'. t)p)f ttf r' .(' tjgyj;j'. t.'y'( l''rllltlq''-'.lziirrd t'pjty;.' tj'. j!;..t;qj.(,i !,q :jk-. ;q.jy.ryj '.,r i ))j; tt. .gjjjjjjgyj';..q;.:-;.; . .:jly y Iiii!i!il. '. tjgjgkjj(.'. '-' ' !' :r ',q!::!li...'-;;! .r.:i.k;-:... ..i . :.. :y. i!.!.!. .q-r.. E!... r.i'i)'!-'r': Lj?3b'if''. ! y i. ! .. .... .j.Iyy t!lr.d tjjyjyjjs'y tji(gyj!jjqj,,.', tiit.d .';'y4j! r.(E.. qyr(.. (yiEy tyjjj;j'. . E.p !- .q(.i.( ).;rj rrj. ;'jjj;j)ty; p. . ;y'. ;y ... .. ( y ...r .r313j1314-. r .1It::.' . )( j . . i. r.j.qy! 1. ( y: )y.. '; . Ekt!. .. y t. j i. . j t ( . .. .. j y y .i . i,;yi(jji ; . ti . y l j . . E. i. i. iy i. y r ) ty. ky. !.,-#!. ;. y :..! r . ; q . y . . y. #:, E yyj;r E .(. . y !. E. j ) . j . ( tt tj tt; . y t! r . j . ) . . . E. ry . r r ; ( .tiy. j,.,..,lr...,., E . ) . . y!. .t!. .Ig::'Il'.'..,r..'!ly y. yj izj,,,-e dlr''' 11:...-.1-,---..-,,-. E.i i. ,q. y(;.jry . .... !q. ..( ( . ; y :y .jjy;y.y(yk.y )-.;g:jy. t;. LLLL.. , r.yj j.. .@ :LLL;L;. ((;. ..j yj;.,.;.. t!l'r-l,g.k;-!..:;,y a . y : 2. a 11(!!(1,' .!I::' a I,.''' Il'.'.ll, akr!:,,--,. j r. ri .''-;- ;. ' .. ..!. .... j . yg.-j-ry: jjj.. y(yaya.. jkyj.. .... yr. .......;jq.(.;.ry. ...jy(... (. )-.'. ))!. jy.y ;i.i''r!E(!; r . . 'r.;.j.;ill'rrrr' tr-;.y.yq... ry....-j.g;......j..((.q(jg..j..q.,y,;;.j...-. . yy!ttly.y ...)yy..jy.)y gsjggjsyygysggggyjyjjjsgsgjyssj.ssjsgggssgggjt.. ...k.,:y. ?i. g.(. ,y .-.tt-.L)t)-. j.jy-y y yry.j y;j.j ... ... . , q.k g; q. .yry i ,g; qy . (,); r... ;ji ; ,jyjj yjjyyjyj. y..j. .. ,. ..k-. j....r.,j(...q.-.. .y.. j.k......jy..,,r;.... ;y,-L). jjy ).).,.) . ..yjyyjyy . .j)yj. jsasay . ... yjyj. jjjj.yjy j yj.jy..jaaaj yassjsjasjj .. .y.. . .y. .j .y.y .y.. yj.y.jyy .. y . . ..j..( . t;. y .. ...j ..... .y r..r. r..y . . . , ... . y..y. ..... .. . .. . . . . of and a short S&R forwird contract at a forward price . . .g.y ..j.., .. ...j.).j..)... ' '.(.yt.t...()..iyy(. . ......j;,j. . jsy..(..(. y:.sy.....,gjjjjj..j(gj;j. . gjjj;k.. j.(jjjjgjjj.. . gjjgjjt. .j . . ( .yj.jjj.. .j.ytggyjyjj.jjjj .jkjgjj jjjj),; gjgyjjgjjj.y.. Ljjjq,. ..jy,yj.y ... yj.ky .j.y jggjjj,.j(ggj;. jjjtj));, gjy,.gj .y...j.yggjjg.y.yy....g.,.gyygyyyy..jgg..g.yjgg.g .. jjjjjjj ..j4q)4f,. .jjggy( . . jjjyj. .j ...( jjtjj;jj .yjgy,. .yjyjj,. yj..j... .$jjjjj. j.y jgjgj:. jkjgj;. .jkjjjj.y. . jjjgg;.. . jjj..ggjjjj:j(jjjjj:j(jj;jggjjjjj b/h ... .jjjj;jy. .jj(y,. .j. j.... j j. j:j jt $1020. If the index price in 6 monthj ij yyy t ijjjjy gyj .qfj4),.. g j .gjgyjj,y jg yjyj j . j jg g .gjgy,.. y jyyjj y ytj j g ( j tjjj.s y y yj tgjjjs j j j. . pi, gkj t yyyyyjjy . j .. j j . (jji,E tkk.t'l;.(..;ryr(.yyy,gyjy,.(y,yyj,,jryy,y,,y,,y .. .. j . . .. ..j. E.j...k . ... tj and hav payoff. If index price is Iong short 0 jtfitd ftt E the the Mtjiy r. . dlt p. a r q udobtdty t t 11:2211). 'iii!f'i1.1!iL.' Ell!!ipil.:z'l:. .IEiii:.. '1111:.!19-. '' 1Eipii-' ' I1i!q''. E ! t1!!!Eij:k. iiii:iri' ' lli!:ll:t'' ' ' ''. i1t:t:!::. k1(::it).-. 'q '-.'; EE....E.(.y...i ; ( q; ' and Iong makes :11E51.):11():2. .1iiI2i: '11.,(iIL.:ii!lk. E.. dl:irr;. I1iE'iI..' th ' than the kl:E::il,. 1EIt??''k;-'r:jiq(' ' '. greater . ' '' . money ' :iI:r:1IL. . l42ii!:). ; ' tlli:i' i . ( .;.. :1EE1l,. (2111E::.,; .; .. ..; .. . . . .. q ( q..r , . i. : i ...qiE. r. g. ; (....). q... '(11EEE::..j'. y; y i (114::),. .:jjjjr; . .....qq.. -''' jjjrtr,,. y.. .r.. .. . .. ........ . . ..... . .jjjjj:j;,. .jj(.yjjj,. gjyjjjj,,. . ...ljg.(....g. .y..(...yj.( ;(;,.. .yiq..j..j;j!j2!i,k,)yy....tjjjy!jj(.. . . j,LjLj;.. r(..j...y......... .. .. k;kjjjr,;. shor't Ioses money. I.fthe index price is Iess than $1020, ((d1!!r;::E jjrjy y ..j..j .. .. ... .... . yjjjrjjjj y.(. . ..:r;;;(jrr,. q.. yjjj(y:jjy. j,,. yy:!jk?;,-.yjjj.s .y .. . y...yy.. yj . ...yyjr.jy.j.. .1:tEE:j,. .-,.. :IEE!I.yyj.. j r,,. .. . yy.j. gy y..y .tgjj;k,. ....j .. r,,. q. . .. . y..y :Ii:r'. .:1EEj1,. jy. .kjjjjjjy;y-y yy yy.(. j...y .,... y y... . q . . . y .. y r y .. y . . ...j..y.... j .(11::-.:11:#.. . y y y y . . y . . . . . g . . y . . . ... . . j . . .. . y. . . . . . .. jjygjjjj.. . ( ( ( .. .. . q.. .jj...jj...jjr... . . ... . . . j jyj . . . . ...yy.. . . . . . . . . j. j. . .. j . . jjjjjyjj((. .. ( . . .. y . y .,,LfLf3?jj.j. . y . j ( . . . y the Iong Ioses money and the short makes money. jjjlks . . g . jj jj.g.jj. . . gjjj . gjjyjjjj;j jjrggyj;y . . jjjjgs . tkjjjygjj-. .jjjg, .. j tjjljy,. yjgkjljyjs .. g'.j ;. . jjjgtt gjjgyygjjj .. j..y.j....(...(....j.. .. ...j ..g ..j(. j.gj..j.....y yjy .jjgrj .jjjjjjr;:. gjjj.j .q. j. .j jj...j . .. .jy.j... . '' ... y..jj..y... j....gjj.y. gjjjjjjji j... ;..g j;. jjjgyyyjj,. . .. j j( (.q.... jyj .g. y (. (.(..j .g j.. .j . j .j . ( (. (. ( ... . j. ..(j..jj.... !g ' ;.. .. . g j yy j . .. j tjgljj y . yj. .. j . .j.... j. . j. .y j.j j .y.. jjy y j . g . tjjjjy . . . .gjj,ggjjj, ' g . . y jjjjjjjjjtjk.. . . .))jj))j)j, . .. .. . . . .ggj. .. .j. . ... ... .. . ... .. . .. . . .. gkjjjjjy.. . . .y.. jj(yjkyjjj. . . . .. . . g . y . .y g.y..j .. ..j.j. .y ksuu y j . . y... . (. ..... y.;.y. ; yy(y...ygjjjjj;. y yylliy) x o yyuo z uy yyy . . y. ( y. ... q. ygjyy .. . . . . . .y (.. ij gjjjjjjk, w yjj.. ggjjjygj;jjjrgy. y. .(.; .j.jkjjgrj;. r:jjyji;, . q . jy. .. . . . ... . ( .. ( j aK w mor W arul . ..jj .. . ...j. .y... jjjj.gjjjj. 1% mlxyyryyy .j.gy. . .. .. . . j.. . jjgjjjskyjjjjgy,g j;.. (y. 'ijj.(..g.yjgj.gg.jj.(.j.jg. .jjyjjj. j(jj...ggjj.. . . tj:rjjjjt. :ji;j(ljkjjrjjl, g . . j .. jl ...1 . y.. j g . ( j i . . ' ( ( ( . y' .. y ( j(jy ! il i' (' ( .'' ;, ( '. E'. (k;. ' ; j. . '('' ij l ' ' '' . .. . (.i ( ( yjj.y.jjj j .j.jj j ' . i :jjjj:k ( . j j jj;j;j:' (. ( ( li:jllij;. g j ( y y . . jjj;,,jjj. . ' . . ' . :ljlqlj,.qjjgr''jjil. ' y ' . (jjj),. .qjjj,,j;j ( . j . . . . (1j4r.j1(, y. ',jjgjj. j .. .y. j jjg.jjjyjjj jj.jy.j(j...j.. iy. ljjj!) ... . (yjjjjyjj,. .. . .. . lj j. y.gy. y gjjjjgj. ' y..' 'jjjjj;,' ' '.'gj...gjygjgjjj.q . 3(3;jL. '.y.. . j .j'gyjjjjjy;,' j .. y.j . j y..'.':. .)j j...l j..y...y.j..y..;,jjjj.,. ' . jgj(gj,y( jjjjj,,' kjjj,.,jjj. j'... '' .jjjj,jjj. . yjjjjr,. ..... '. . y Eyy'.yl( y Iljlqllr, .j tyjjyjj;jg: jll!il:jlj 'lI..' 114:4:1). . ' .. . ... j,jj;;j;;;r . . ' ( . . j..(. . . ... y('.. y.. . ' jj j . ggjj(jj,,' . . j ' j j j.....j.j y . . . ( . . j ' ' j . . j j '. ; ' . . ' j. ' . . g '. . i.( j. . j ' ( . j j j. j .jjyj.ygjj.jj..y.j.j y.j.j (.( ....j j . . (.' (.'.'.. . g.. ('.l(.. '...!. gjjj.j jjyjjjjyy. jff,. ''''': ''''''. j. .jj j j. jjg. . .j g j . . j. j j ggjjjj gj j. j,jyj,tkyg. j j yjj.jyjjyjky.g . . . . . ..yjy,.,yjg. j.j y j . j . . . . .ijjjj;j j kk.,tjjjj . j. j . j j . . g . . . . . . jjjyyy,.jjyyj . . j,,,t,tg,. . . y j . y j g . . j j . ajjjyjjyjyg j y . . . j g yyj;jjjyj., . . j . y . ..y . . . j...y . . .. ... . .. .. . . .. .g.. j j. ; y.... j..... jjj .jj .j.. . .jjggjjjjjs . . . jjyyjyyjjj. . .. . y . . .. . ....qjjj,. . jj.j..jjj.. jgj.jj..yy.jyyyj.y. jjjjj;g ggjjgjj,j ggyygyy .. . .(. j. j .... .. jyy y..yy... y.. yy..yyj j.. tyjgjyjj,. . ( j ..j.yyjjyj jyyy.y.j. y . j jgjjjyj. j. y j yyy ( .. 900 $120 : .. yyj.jj.y . .. . . . . yyyyyyyy;jyy. y. jj yyy . . . 70 950
.'.'y .'..
.'li-''',iik.;''d E'('.(. y'' '''lqllllq''i'.-lik;;.si,?,-. 1iki2!:;' -',' -1jjgjr,j'-'. jjjjjj'. '.kj:rjj:j;'E y' ll'..,.,'. ,1::.81...11..*. t'Ejd ,,Lj,j'. jj,rrjj;'. :y' .:jr...k.y' j;jd j;;d .j:j,,' ,ygggyy' jy;j,gyj,d jjj,,;'. ,4j.,,,,,* .1jjj;)j;j;jr.' g.jjjjyjyjj'. :jjjjgjjjj'. ''iryljllq'l''lqjlgrk. :y' ;'jjyjgyjj;. -.' '-.)'. y' 'g' -$'. ,'-' -;' -jk' -;,j(jgjy;y(jjjyyjp;jryjjy.')' --y'!'-...' kjjgj);d jjg:g:jj' :rrr::d zjjgrjd jd '.-jjjjj)'yd j':jj...y': :'!11::@:,k*. :.....'. .!!t.;;Iik;,.' ,12:::::* di:::::i,'. ,k,;'. jjkyyjrry'q j'-';IIt.X 1iiiji!d(.'. -' Ik-p'l'--r'l-,,.-pp'gjjjgkjjj,kgjjy'y ....,'...',.'jjjj)j,('. --''. jygg(jjj'. 11:15:2:;* '.;,k,.': 11,,.*. lliis!i;d jk.,.jjkj'E y' 1iiii!!)l'. lli,..:lk.d 41::214,..,1..k1..*' g' 'jjjjjggr;'' 44j(jg4,;*. yjjjjyjjjyjjyd jjjgjgggd jj2jjyyj;'. jj,.,kjj,,,jjj...j;:jjjj,.d jjjygyjjd j'.yjgyjjjjjt'g (gjj.j'E E,jjjjg'. .jjjjjjgjjj'E ..jjjgjjjjjgj;'. jj:yrg:jjd jjrjjgjj;'. .jjjjjjjjj.' j'4jjjjjyj,*, -' .jjjj.'E Ey'' .,,'E jjjjjgj;;'. :'-rr'' jjrrgy:jd jj:jy:;d jjyjjjr;yd -y.--j'. --' '-#jk' -'r' jjggjjjj(.'. -)j)-y'. y' f' y't:yjgjjj,'. jjjjjjr,d ;'-'IIygjg:jj'E dl::::::d yjjjr,jjjjytjjI,'E -' ,--'.-' 11::1::1k*. y'' -'. ,'-t'. jjjyzjrjj'. -j'.
.'.' .'.'...
E ;E !( ! jjj;jjj;.d t'y' j)'. ,'E-'' f' 41::255;;* F' 11:::::,.* 41::::::* d.:::::llf dl::r:rlp'' .';';q' ((' -
.' r'
-' i,jkj'i .;;',,, 112:y:412*. ,k;'. .'.11:)):1127.11:)*. 4)j)364,,..', 4,.'' .-,' -' '''bi;'i'i'''. .'bb.v''''j ..;.jj,k.jjj...!;'. '(' :(' ,,.j.,:'. ,.j!y''.'!jj..'' rI,' jg::rr::j,''''''''' .111(12.** :j' ,yy..gyj.'. ,,gj,j'. ,yj,...j,' .gjj' ,,;' ...*'(11()* ,;,' ...,' ...* ...,;j' ,jjjjjgjjj'( ,(jgjjj;j;'. -' '' .'.11.;;* .k.' '1l));'1i::' ,'.yj;y'. -'.;j'. '1it::' ..jj(('
.'.'. '''..' .'.' ''.' .'.' .' .'.' ','..'
.*k11...1:,/*'.
--.
- ..-
-
-
.
.
..
-
. i--.
-. .
.
..
-
2 --.
.. .
-.
-. - -..
.
..
-. --.- .. . . .
.
.
.
(
,
;
-::'
..
.
. . .
..
.
.
.. .. ..
.
.
.
y. .
.
.
.. -. - -. .E
-
.
.
. .-;
.
. ...
.
y
.
. .
.; - --
. -
.
-
E.
-
-. -
. .
.
-
y g. .
. .
:
:
g
..j
..j .
.
y
j
ggyy
-
-.
.
..
.-
. .
,
..j.L...L.
...y....
r
,,t
-
. '. E. . .. . ..E . .
-
:
-
.,.
.
. y
. .
-.
.
.
.
.j
g
.
yy y
,
- --Ej -.. . . .g
..
-yy
-
.
.. --
.-
..q ..jjg....,jj.yj.
: yj
g
.
-
jjjjIj;jjy,E .. r
.....
j
g
y.
y.
.jy.yj,.yj... , ..
..
y
y
y
y,y,y
y
g
g.gjyyy.yg.y..yy....gj..jy..y....jgy.g.j.jg.g.
,yyjyyy
ygy
yg
.
-
. . .
. .
-
.
.
.
.
.
.
.
-
.
-
.
.
..
.
.
(. t .
-
.
-
.
.
.
.
. .
.
,.
.
.
.
:
.
:
.
.
.jjyyyjjj;,.
.
.
.
.
:
,
,
g .
.
.
'
:
:
.
.
..
...
$:4j44,,
: '
'
.
.
:
.gjjjj.j.
g
g
. .
.j,....g
g
. .
.
.
.
g
g
g
. g .
. .
.
:
.
:.
.
:
.
.
.
.
. .
. .
.
. . g
.yyyyyygy,yjjjjyjyy,..
.
.
.
y
y. y .
.
.,
E. g . E( . : g. j: . . . g j.. y. y..jy j.... .. . j j y j.g . g j .
djq:jj:qt'.
( . . g.j j ..
j
j
g g r j.j(.g.yjg.yyjj jg.jgj. ..jjjjjjg.gj.g.tjy.yjjjgj g.
g
.
,
g .
g
.
:.
. ,
.
.
.
.
. .
.
..
.
.
:g
j. t. j
.
g
.
.
:
..
.
.,..
:
:
. .
.
.
.
.
.
.
.
.....
. ,
.
. g .
'
' :
'
g
.
. .
, .
-
.
g
.
:
,
: .
.
.
...........y
:
-
:
-
.
g . g . .. . . . .. . . .
.
:
.
. .
,
g . .
.
:
.
..
.
.
.
. ,
:
g
.
g
.
,
.
:
-
-
jg;r;rr;jj
. .,....jjjgjjjjjyj;jyj.
.
.
.. . .
. -yjjg,.
. -jj::j,,jj.
-
-
.
. . .
...r
. . .. . . . . .. . . . . .
. .
-
.
:
.
.
. ... . '
.
,,,.,,,,,..
-
-
.
.
.
,
,
.
jj... .j g j. j..j. j.. jj. ..jj.j jg.j ( j jj.. j. ..( j. jj.. .. j.. j.... . j. jj.jj.. ..jj. jjj jj.jj. .jy..j.j. g jj..g ..... .j j ..jj(.jy.jjj ..j j j. y jj j j j .jjyjj...... . . ..jjj.gjjyyjjjj . . . ... .. jjyj jyj jg. .yjmtj tjjjj;i . kjj t..jj. j. j jjjj 4.,..: . yjgjy . ..j jyjt . . ... jj. jgjjjj . .j ( (g... gj . .j..jjy.j.j.j .yjy. gjjjgjs . yjjyg jjyjyjjjjj yj jjjyj y y jjtjj, .j yyjy gjy.;j,.jj j j jjjjg . j...... jjyajj (jj. ... .( jj ttjg;j (:. . . yy,,jyq ( liit 1i lit l.. ; fj y 'kjyly...y.,,yjyy,y,,.jyj,rrjyyy.,yyyyyyyy ; (k:t.kjj yIIE!Tii., ; (;j()L.L ;. ..;jt.y j ( pp,kl npntjilli!li.. ( (ki)' .jt it.t;t. (. jj. ( .;j y jj)2!:.' j k. . . 11jj71i(.' s,t... k . .. (ii:rllll. j ;1iEli.l: . (lIi511ii.. ( . ' :!i!;i.: ' ' . k' 'lliliiili''y 1 j ( . 'iIii( . . t. . . 'iji:q' . l ()iEri1Ii. .. .. . .. .. . 1i::EilIl.(1I(. . ... l'' ll(l''''''q''' ly.' 1 .. . . . . ' ' IIIEII)IP' ' ...j(..g( jj)'j'.y. . g ( .'.('.' . j.j'.j j'j j..j'j'.j y..j.(j.j y. .. jj'.jy. j j'j(jjjj( ;j' jj.' y.jjjjj.. jyjj(jj' j. j.j;jjjyjj''.jj..j.'.jjjjyj.jjjj'..q.j(j. j(.jjj..jj(......jj.(jjgj...... (EIIli1'' j !.... .. ..j.j .j ' '''(j11((::1i' ''' .j.t:jj: ijgyyjj.sjjj,y,. '. y .. j..j.jj. j. . .jjj.. ...j jgj'gj .'...jj.j... .(..(..(jjjjj...; jg j.. '. .... .j. .j'. j:jj . . . ... jgj.g j.'g j j . j . .. ..jj.y jj y . t. ... j... j.. . . , . . . ( ( j. . .j ( . ygjjjj.j. .yyjyyyyt j j.g .y.. j . yj.jjy .y.j j .. iii;:l!,. 1i2;;j11. :liiilkIlj!ll.g:ll::'' :!!i41; 41::E:21,. y.j.j..y.j.j.y rEllEE' .j. . . .j . . .y.j.yyyy..yjyy jyjgj y.j; .yg... y...j... ..yy.y.. k...j...(..(...k ...jj.... . g . gy . g.y .. .jy ..; .j.y.j... . .. 'i@5Ej).. . ... . ( j..' ....y.y. ' ( .. (. ' ''. .(jii... .y. ... .. ..tIj!r:ii... jjy..j.j....(..jyj.qy. . ((jI(!1(E::ii.'. . .... E ( ..j....jjjyyyyjjjggjjjjj ;'' (...'.. . .'y . (. 1i2E!r:; 'iIt::. . t!lElr:i?l .....;. yryy ;g..r ..y ( .. '. . !1ItE:2),-'I12EE. '1IiE::),-. :!p!l' tp!jij!j;r ... . . r g (.r. ..'... j rl!E: ' ; r j j .. ' g . '. j . CIEE!;) )( .. (11). j . y.g ..)' .. yj ..J. . . iiiiE!F''t1i:i.(( .r.. . )j.S.:Ej:i::. .. ..(. ' . .. :. ..( .;.ii... . t.' .( j.. -itiE@!:-. ' .y ... .)..t. .. . ttiiEijii.l. ' .'. . . .. .. .'Iti:kltii.' ' .iI:!E. .. '' . . 1..1)1..:1111:-:. . . . .(.)...j.. ..i.. .. .. p... .1:111!::11@@11).. k.(.'...ry.. :rIpr. .'.'. . .. 1!iii-! tiiiijikt).. ' jy. ' '. ' . ' '. '. .'. '. ( ' . ' q . .. ') ((.)t ' . . '' t '; '. ... i... ;. ' .g'-.. ..(.. .'(.) y . . g .. y . ... s.1I:i . .(.(.( i i . yyr.y, (.'.'.) ..) ... . : (. p .j.y.. .y .....g .;.y. ..gy..... ...j. ; .'. ;...i,jyi1I)2g1;:::.. 4I22jij:)( .. qj:iilt:y. .jjkjEil)gjiE:t.. t..) . gj:jj,#rr.. k1:jjj::,. ....(..(i..1.j2j:j1j:j.. . . .)'( . . r . .. .. .. . ) k(.L1Ijy:. (j,k-. . ... .( . ... y... .. . j.. ' ' '. ... . . iizjgi:, ... .. ... .,yyjyjj,,. . ryjjyjyrj,,. .. . . . :jggrjj.,. tiirrjli:r:)l.. .y .;y .. . ... !. . ).! .. ...y.. i ..... . .j. q.;y......yj.yqjq...yyqj... pj;;-. .... g....y... . j j.... t(ig:j,.. . rjjyyyjjj,. .. ...... . . .. ........... .II::EEq!E:!! . ;. (1i.. :jjj;;;. . q . ; ; .. . . ry...-. . . ;. . : jyggjj,,. ....y...y..,!gkjj!yyyjjjr,j.. ..iii.. r .. .1Ir...1i.'.!II .11r..!1E...!1. Jtiir:..y.. jjjg;,. y . y y y.y.. .. ....yj.. 1l2E:r:. .. . . . y (::!j.?h-.. . . . . . k,:yjjjjj.y. yj.y.. .. . . . . ... $,j2gj!;,r y .. . .. . . . . . . y . .. . . . y j .:Ir...1Ir..'1... y j .11:...111. . . . il22i2r:. .j .. y .. . . ;L:jjifi'... . . :,qqqFrr. .. . .. . ... (j1r.. j.yj . . .. ;yy . . y. .j yjjjjyjjrj. r ( y .yyj g..yy.y.j.g.jgy.y.j.y.....y...j..ygy.-..jy. .. gy. y. ...... ....y.. jy gg yyyy; ..y yyyj.yjjjyy. )...j.q ryr.j. .....y.y..y.q.yy...j..r....y......j.j....j..(( . (;ijy::Ii.(iiq.. . . . y. y. ,. y (ilrqrjl.y , . .. '. '. j(jiiql j. (j( .. . ..g.......r...yy......... y.. y y . ...y jiii.@11t.. E . . (Iiq1'(iiry:i(:.:ii:.. . yg .. yjjg,jjjr,. .. .l . ! ..j. .... . . .y......... yyjy;..yyy.. ..j..y kjyjyjyjjgjjkjj.. ... . y..i!!!k;. .. ..... j.ry(. y. g;jjj,jjyjj... . .k:Eli,. .........:iEijl,g. tk:::):i;:. ' . . (11:r.:11:r:21L. .i:jiII). 'Ik@22r:. . .;. . . . .' i. . k1i:)jjlI(' . . .:11Ej.. .. . . .Elllr'. $ir:h.. l12@!i: . 111::::,,. .y. yj!:r .11:2: . . . g j ( . . g j . i (: (y .,k;. (g ' . E)'E.g-;E:I.'' . (' . (' .. . y..'' '. (. .(.'. t;jj. .jj..j.j....jj..;y..yy..yjyjj.yy......jy...jj.-qI2);i#:ii.. (ji.jkg,i ..r . (. q .iiiE. ;IE.'.'ik)i;. LiLLi:. ..dii:!'. . iijii..y'. ..d(iE1i'. . .. .. . . '. ... .. .j.( ...'j.iilj. . 1EiL.. ji:7'kk)::j''j.y. y.(...... LiL'. '1itq. JIE.I.'.1tii!. ..( dk:!::' .)12t11::j1.. . iiI:'1(i2!:. ...j.. . . kii!:.. .j)3..,'.. ..tj..jyk,,j .lr1i.. ;Ip.. )''(I():k;I7k. i1il. iliE)ll 1:2j11. .' r.r:-. .jjt. . iki:i:..(iE. 4t121; ' IIEE (Irr-511.. ';(51l..ir.tIL. d:::.' .' .(ir7q1g7il...1(:@1p (lj;,'q ' 1(:.. t(t )'ij.t).(:i k irr;. lE:' .)!)) ((. ( ) . ((y(.. ' #* . .. p* 1 j (. . ( .. ( L'.. '. tjrpri,'. ; .:1irtif. .. (.(.. iiiik. iEj;. .1tjk... .i82j. . .12:1)11).. !It!k...kj;... k@j;. .,.) ..1lrqI',l. 11#1t.., 1.l'1,..14j), it. ittji. '.L...'L'qi. .)'. .... iEi;. ii)ii. ltzk:Eit 14;:. .).'.ii)1(:i). '. ltiL itglr'.....' ..,'..... irtj: .'. it,r. ... k (tj.. ' )((ii4k' .'jii. j:liti.. .ir) t.. '.t(1L ltiri jyg it:lL . (j 'Clilpiirk':E)i t....4tE(:).. j(( t. . L !.y jlyj ) (j!) EEEJ' ' ... ) (j qjjlj (g:r(. ..k::k. (.'.. (j . . i ! jj i(jkj '.CEpd.!1(E:::i,E . ! . ' .. . jgrjp i ' i.. iElltlr' p 1.. '.'.tyt 'lltiqtlli:q :' :.4E,:)It. .y , q; ' ' . ) ll:E:)l.l .lqql.. .. . 'iIi:::.,: , . . ' . . :1tE5!1..t.1:7:11.. , . . . , . ' t1ki!:ii,-' .:1@k'. k . . ;,14.( k iltil. ..., ( . . ;1tir1. t1::!::).-' . kl::E::,,((!IIEi1Ei,r.((.ryj.'.r ;. .. y jj!. yj jyjj..)..;gj. . yr , ; ' . 2.1:::1.. j 'I:EE:i:. .. ( . j . t!lr::t' ( j ' . r '(:!!;,-'-' (()1(2)::1::1,. ' ' ; . j j y j ( . ( . g k . . il!i.:llq. . . . . j. j.. .g y 'jj.. (.j . '( . . . . ( j . . . .. ..t j...j r j.. j mmitgiit'ikgkriiktk.'.qjiy'iyy' ' . . Ntjrth boiat jld i it (.'(r'. .ji j(: (t .gi .((.( . t(I1:;Iii::. y.j ( '!.g t.i; .11:1511:-. j.j'.'(g.'. (. k. .j..t(.i. (iE. k. ;k t t.(): . j (. (' (... k. t )..(.. (. iiEEk:-. r Ii.jEEII-. iy :i1i... ( . . . .iir::jI!:#''. iliiEi'... . 1:15E111)1111.11112). . (i1:.:!iL.. ' i ..1i2i@!,).. ' t.. '. i,i5iii2:!I.:7k'. .1!2EE. . '2!i!i::. . .i .:, ,. ;....i.. (r;-. ! . jI.;IEijIk.. .. .11tE. t(Ii... ; ;iE#E!i... -i;:EE:,.;. -1:::j!I).. . ' . :!E5!i-- .it:.(Ii.. 1I!E,i:-. . .(11(4:::11. . .. . . (:...(tj.g; y j g .. .. (14.. j r !pl.. r yyj jj.y.. y . .. .; ;...y. . gy.. . I!ili .. y ( '; ' ), r .. y '. . . .. EI) i i .. ... . . (, . t ) j . . . . .. (... ( r . . .. y . (: . k . (. . ; . . . . . .. . . . .... . . ( . ltakp, r. . tllll;::''.jo/, ... ttiitkiikktttrttyy, . iiijrjtkkt, l;t l y) . ... .ljerjadig it ..!.j (.7) 1( wi .)).. l.yyiyjitttkr ' .( .(.:)1rjil)..... i !. :ii)kj' tii!'..i''.. .E kp::);. j. .L ;!L:(it,. qlkjih!kl :Ijii.. (... !..iiEq:,k jjjL..j--j-j-j-..j..j. r!Ey'ktj:i;... g4j..jjt)l..-.. :Eik......-ijjk. 112j2:j;4;j. ..-.i.. (jgy:)r.?i()jk.y ' qj::ik,' .;jiL'. jlji. 44:(1(q:: $ij3l.ir.1i.-. -k....!ES2.it.. (1L. :!k;. :i)7.(!ijIj(' qlklt.' (jqr 4k2:g .'... q1Er)L.' :;.qIk,I(ijj;.. .-(:sjj,....... (ljlqrjrlh ' qlg.ilr:)k yjjy:,y .''.... Stlll.rlj.)ly ' it:r;p. ..i(j),...jtjjt. j::: jj y t..t..tj... (k:j. j j r (j j jjjgjj j yj j g j ( (:!;;,. iLs::y,t, y (Eltikdkttdk k (k j .... .' yt xif ) btc l) pni/ stp. aotht ....
.
jy..yj jyggyy jjyyjyj,.yj
.yjyj,
.
..
.jjj;. .
,k,,,jj'
. ,jjjjjrrjjjjj..r
,yty:
..j...
.
...j
.r..
:
g g
..:.
g
.....g.
..j
j
.
j : y: y .j . jg. ...g .
j .j :
.
.
g
......a..j
..
(E)
.
j.
g
j.... .
.
( (
.(j
.
.
, .
jfj.
:
j)jfq).
.. ...
. ' ' : ' ... '.yr . . . '. y y j.... ... g. . gj.. y
.
.
.
:
jqLjt.
jy,.jj
. .. .. .
..
:
.... . .gg.. ' .jjjjggjjj . jtjjjjii.jjkjjjj;g '.jjjj(jjjj(gjjj,. gjjjjgjjjy . g y y(yjjy. gk;gg. yj.yj. .yy.;yy. yj;.g,..jj.; jr;y,.. gyyyjyj.,j. yjy.jj. yjsyjj . gj;.jg j.. ... .jjgj;jj jf)j4j. . jjg .jjrjgjy t(j;yk. . . ky..g. ( . .. jjgjjjjjjy
.jjjgjgjjj .jjjygjjjj jjjjgg;jjt
.jjtjggy;s . .kjjjgj
gjjk;r jgjjgjj,y .jjjgggjjjksjjyjjygjs yjjg;j
jjjjjjrgj
y
:
. g.. . . (. rj. . . (jr.j. :. . g :yj. y .j .jj...j g y g. j g . j jy j gy j .g yg. gy...yg .j.y j. . gatj . g .yjj. j g yg... ggjjy: . jjg g . jjyg .. j. j .g :: jj joy
g.
.
,
.
:
. ,
:
, . . .
g .jjj.:.),.j jggg;y. jjs jjsyj.,yg
gy;yj.
. g
y
g
: .
y.
g
.
. y.y . .
y .
jjyy
y
,
g .yy,,,yjg,.
ykgj;;....
ayjyj.
,jyy,j(gj
ayjj ,j;.,,..
yy..jr yyjyy.,,yj,ya
,vj)).
.yj,
.,,,.
,y.
.,..y;..,u
.j.y . j..j j . j ,
.
.
.
,u.,,
.
g j j gj j ggg.y.yjg yggjg. g..yg. g. g g.gy.. jy.jjg.gg y. . .. j ... .. jjyyjgjyyjy . yyg. jj. . . ; y jaajja.yajayjyyjyjyyaajayajay j y..j;;; y.yyyyyyj j,. y. j
g.
y
g .
.
y
yyjyj
,y)q
q,.yyyy . ,yg ., , , ( : . :, yt E. y j ( (. @ (E ' E E i ' ' '' ' $1k,..,1p'' '' ' ' '' EEEEE E. ' .'y.E. ' E ' :. ' Ilkiz!ll ,jIk.. ' ' 112:::;;.' lli..,.pp'. !Ik:::;;'' ' '' ' 1i;::::;.. 11.,,,241. E yq . .. . : E. .y. gyy j... y. . ' i E '' E ' . .. ' E EjEj (: . : 21..11:.. ' : .E :. . ' :j :. 11111!1:. E. : g . ' yy: . .. ' ( . E y ( !. ( E ..y. .y E E ' : : . E. y .y .. : . ( (. i !. . ( E . . . g . . g. y g ' sjji,j.,jj,ij,j . E . : ;.jrjjk.(j,jjjy.:( ... : r y (. . y . : j . j: g . ( j.r (y. . E jyy.;yqj. jy. g. . . ,,;,,,,E . .y. . y.g.:. kkjr,kgj,.. : : j y .. . yj gg. y yyjgyg . y g jy .. . : y. :... . . . i . j j . . g y : j j j . . g j y . y y . g . y ; . . . . kj,...jjj..... y . . . .... y .. y. . . y . . j .. .. .. jj,skj,. ... . .. . (. .g:. g g: . :j. .. qj :. g gg.g g E j (. y. .. g . : g g q. . . . . g g (: .. . . .g. . '....,,'.. ,1;...11,.. ....IIk... .jI:?!...... ' : .. y yy.j.... g. yjy y . j. g . . . .. . : ;:2::.1. q..q. ..( E.... . : E.: . . . ;:!):i; 11,,. .1k...jpp. !1k.,..)I. 11;:;:::: . y. y r . g.g q yq: :. (. y .. . j.. y . : . . : yy yy..y y y. yyyyy . .. . y. . y .: . r y yy . y . . .: y: : ! y.g ... y... y :: y.y(. : . . y y.. . .. . . . y : . y (: . .. . . . gy j.y. . . y.Ey 11k:!j!i;. .. .. '' ' ' 11k:!ir6;. . ' '. ' ' . . . .'. .' '. . . ..' . i . - j. . . . . g -. . . 1i:if:;. ' E. y E . gE.. E . : . . . ( . . . ' ( . .. . . . . . . . . .. .. .. . . . EE . . ' . ' .. .'ilik;)kik,;;,. - . -. ' '' i . . . r .i:2:t2,;. . . . 1ik22!r:; -. . 11ji!r:;. . . E . . . . (. . q. -. y.qy.-......qy. . y E ilzi!!:i q ' . :))q . .LLL.L y -' . E. . E. y - yy jE'- .E: -. ..y i(: . (E.( - -. . () . . y :E' ' i i : E'Ei y . j ' . . ..) i i y . ': . E q j. . E (::.E y il -- . -. . .. y -. ' .Eg. .... ..:(!111.. : ( 'f.::)' ' '' .E ( i . : ... . .(.. -11:1.t,,.. E-,E ' . ' .F .. : ... '. . E :'. (j. .. . y .. E. ' .. -1:::::-. ... .: E . ... . . . ' - - -E .. ( -1:::1:)... E -. . 'q . ' .. -.. EE E-. --.. . . .. .. . . . . .. -!:::::q.. - - . ':'( .. . . . - . . -. - . ..)i().'()(((.('. . . -. -. . ):.'..:(:.(E -n ' :' . -.- - - - - . - (. - - --. . - . - ((1 . (k:jjj1().. - -. . . . ' - ..(E(.E..). '.' p. 'E' () . .- . . -. . q. 4kj2jr:k.':(. .. : ..E. jk E. ,k,;. . -- . .. . ( :E(.!E'E' . .- . -. -- . -.E .y q. -. . . -. .. . . -. . . . .-. . . . . .. q- ... ..- . q. -. . . q q . E . . 412t2:::, .( E . ) . . .. . ,j2:ryr:.. t . . .. . . . .. yq y . .. .. . . . .. . .jjyyyjjjj.. E. . ... .. .. . - . E .. . E q .jj:jyjjjj,. . . E.. -. .. E. . . . y. . . . . . . . E . -y -. . . -. !. .. - . .. - ;. y. E .j,jjjjr,g. . E . . -- - .. E... . . . . . .. .. .. .- - Ejyjjjj;,-.. i E. iy !.i. - .y ; jkzjyjjj. . ! . tE ;. jjjjjgyjk.. E :rE y E . . . . . . jj::j,;.,E E y . . . . i . . . 41211:::. y jijjgjjrj . .. y. .. y .- . . . . . .. . . . y.. E. E .. . .( . -. . -. .. . . . .. y . ... EEE .. . . . . . y; . ( .. . . -y .. -. E .. - y y . y y -y -. E. . . . E. . . - E. E . - -. -..y g E yjjry,jjjrjyjjjjy . E . -y -y , y y. yy.- yy y..y- . - . y. y. .. -; - q- y; . . . . .. . E .. E . . - y .. .. g- . ( y .. k- -..-yy.. ..y, -.. (y. . q.-.-y-... . y(. y y y y.jjrggjj,;y q . . - y. y - y . ... . .. .. -.. .. . . q . . ... . - . . - . . . jjy ..y- . - y. . y . .. . . .. - . y . .. . . . E. . . . y q -. . y.. y y E . Ey . . . . . . - . . . . . .. ( . . . y . ' . y . y .' . . E. . . y . . ' ( y . . . -y . . . ' .. E . . y .. ' ' . ' .. y dl:rrjrl,. di2!r;.. E. . qjy (. ..j .y.. . . ' . . . . . -- . ' -. y. .. .. . .. y. y. .. j g y. y. .. . - . y. , yy yy.. . y .. . y. . j yy. y y . E Ey.y : y .. . : . '. ' . j. ! ( .gE(. !... .:g.g. . .. y. yy . . . E( y yg i : . : !E . ! E y. . ! y!y . ii..,LLLLL.L :'.iriip. jjjr,gjjj . . . : .. .. . . .. . .. . ... y . . . . .. . . .. . . .. . . . . . j..' . r'' . . r(..'.).'. : . : . . . . .. .EE'.1..;1.**. tE'jrqqq?;'k.;'''' . ' . ' . ' . e.E....jy.jyE...E.y(y..y.y...gy . t.j;... . . . . . .. . .'.'j...'.-...E . k!i/h''.. . . . . . . . y;E..yj.gy.yjg--..-...yy.y-..---.y..yg.-.-gyy.. .. .i.E.. 1;::;:. . '1i,;i1ii,i'''
..
. .
.
....
.
.
.
'.
.
.
.
.
.
: . :
. . . . . . .. .
i
,k
''!ik!;'''
'11:5.
''1i.i7'ik,i'''
''
.
:
:
.
.
.
.
. .
.:..:-.-
'-ii.,;,''
'
.
.. .
'
.;.,,t.l .
k .
:y'
-
..
..
..':
.
.
..
.
-
,jjyyyyjjjyrjj(,. . .
.
.
.
-
-
. .
. :
.
.
.
:
:
:
. .
. .4jjrj. ..
,
.
.
.
.
'
:(
-
..
.
.
.
.
-
. - ..
.
:
. . - . .
-
.
.
.
. . .
.
-
.
.
-
,
..,,,.jjj.
.
. .
..151k.11#:1.-.
.1r..
.
..
'
.
.
..
.
.
..
.E. ... .
.
.
:
.
.
E ..
. .
.
.
..
' 1::j11..
I(i:::.
i . . ..
'.
.
. .
.
.
..'.
: .
.
...' i..
. ..
E
'.;.'.
r..
...
....
.
.,...
:
( 'lki:i:.
-
.
.
-
. - -
.
.
-
.
-.
.
-.
.
-.
.
.
..
- ..
.
-
... '
.
L
.
E.(.....
E .. . (.
E.
.
-.
.
-
.' . . .. . . . .. ... . ... ... . . . . . . g'sertd
E .. . .
.
.
.
'
. .
.
. .1:::::,1.
. il::,:i:. . :
.
.
.
.
. -1:t2,r:-
.. .
.
q
.i!i:(
.. . .
d;$;
.
: .
.
.. . .
.'
'..
.
.
.
.
..
..
'
-.
:
.
.
E
'
-
.
.
..
.
'
.
.
k
.. .
.(
.
... . . ..
:
,
')r?kjf;i:1r)qI:r,-.'...E. yE. ....... .
.
.
.
..
:
.
E.)
'.' ' .:
'
'( (
'.
j. ..E . g .E .. . . .. .4224,4). . . E. . . . .. q ..( q. q.... ,yy,' ..
..
. ..
,
..:
.'.
''
(
-
..
. .
.
.
....
.
.
..
..
.
.
..
..
.
..
.t . . .....
. .
..
.:
j
:
j
g
.
q y. i-y
:
.).
E..i
.
.
.
..
.
.
. ..
'. .
:
.
.
.
..
. ...
...
..
.
.
..
.
.
..
'f
..
:. . . ( y
.
i.
..... ...
.
.
.. . .. .' . '
j
t
. . . . ..
(
E . y. ( ... ( . . .
,E
.. .. .
..
-
.
'(
.'
:.E E. : . E..E . ..'
f'
:
-
-
,
.
E
,
-
.
EE
:
.
,
r.)'S.
!: ( (:iE
.
g
:
:.
' '' ' : .: : ..
'
: '.
.
'.
.
E(' E
j
.. .
i
.( .(.': .
..
..
:
.:
.
:
:' .
( E:( ..
'.
:(
.
::!
g. ,fk'E.
.
-.
-.
.
.
,
.
.
..
-
-
-
-
....1!$4#(....'.::1:1:.,
..
.
ELL))
.
..)
(:( E : (E ( . .
-
. ,L(.jE
,
:
.. :.
i;E.. .-.
-
.
.
-
-
.
-
:
-
,
E :'.(.. .
'
('
.
-
'j.:j.j
':..'.:(j'. ... ' .
.
-
i). .
.
((
(
:j
E ... ... . ..- q-E. - . ( g. .. .
-
.
- . :. E . . . .: . .
( .y.. .
. .
'.:.'.
.:
.(
j,...)jr,.,!j
,,,j,,,,,' . .
q
,j;....jj:'
,
'
'
'
'
...
..
: '
'.
:
.
...
::r
.
:
.
.
.
E
.,
:
.
. . y y
g
..
tjjjjjy,jj tjjyjp!jj, .
.
.
.
.
.
.
.
;
.
.
.
.
.
.
.. .
.
.
.
.
y.
... .
.
.
.
.
.
. r .. . .
...
j. ...
....
...
y
I.:::::,,.
.
.$1020.
.$1020,
:
uy
Ei : . ....
.jjjk)jjjr,,jjk.iIII
. ::
.
:. EE .
:
. .
E
'''''''''''''''':
E
''''''.
:.
::
. .
'''''''''i'.
.!. ... E.
:
.
.
: .
'' E .
' :jjj;.tjjj.. E. .. ' E. E. E.E.. .E:. E. :.E E... :. : .. .. . . ... . . .
.
.
y :
''''''''.
.
.
'
:
..
'''''''''.
. .. : .
.
.
'
:'
: . .. . : . .y
:
: .
-
' E...
$jjjj.,,,. . .'
E. . . .. : .
yy.
.
'''''''''''''.
.
' ' . . : . - E. E :. ... .. ..
.
.
. . . - . . .... . . .. .. .
: :. . . ..
. .
E
:E ....
E
.. .
.
.
:
.
.
.
.
.
.
. . . .. . . .
.
.
.
. . .
-$120
-70
1000
20
-20
0
1020 1050 1100
30
80
;lE
-
=
:' : :
'
.
-.
. .
.
-
-
EijEi(E)(IjE:j. E:.('(E j . .$...---E.y....yy..yE..-.yE-E-.....j..qj...yj...-.yy Edigj;jy..y..t.ik6j;i.. .--.......$(jjjj-..-
!'
..
-1:::::,,-
-l:::r:,,-
.
-
-
.
. .
.
.
(
yEj
(
gj :
y:
@E E rI(:::ii...E. iIi2i:r:;. E II....jjjgjjjjj,. lli::r:i;yb:::,t..,qjj)))). .E..y y. yy ;I::1I:.j1L.. IE(g..1I:2.. g yj k. .yy y.yj. y j y. yy rg.yy..y g. (. g y. . yy g y y j y y..y yy. . y y .y.yyy .yjj.gy.; y yygy. E( y.y yy. y jjryg.y y y.. y. .. yy. y. . .. y E.1..i( syn; ( .j.g i y ..yy. ..j.. yy... yy. . y.. ...j.y.y..y.y.. y. y yjjgg.;,. . . . y . . . ..qjjjy.i,,yyyyjjqgy,ryjilgyyq.yyyi,y,'qq.yyjqyg,lyy . . . . .. . . .. (nprE . . . ... ... .k. '( ..j.. . . y.g. . . .j.y g y.y y.y .. , ...(((:. , r E(' (.j.j(: y((j ((( ...;...EE(.! E:'.(.E:.. .q) ((( yj'i(E jj. y). . ji....l jj j.) ; tti%ldEr.i.Eql'ytjjylqjylly.yyygyEjEEgEyqyyyyiE ((:.(.(i.(''..(( g.j)k ).(..(( ..:.Ei ....:..( . .
- - y. y . ..
. -
:,.....:.
.
... .
-.. .
. tt wlt..ii '.'(
:. :
E
..r.y: ' ...
iiljljffri
y.
, y. y.
.,
:(g'..y......:
E ..:(i
:
.
ttkti
-1::,.1,. y
y
,
..
.
-
t . .(
gj...(.-.j j j j ... jjr. jjjjjj,,kjjjyjjj,-. jjjjjj. j. j(.. j....j. j . . .!'-,.!i?-.jjjjjggyjgy(gjj,.
..11:...11...11...
j. ..
<.j.
jj tyi' jjakj
)
y'
:
)-)-.'(
..
E((.jj.(.tj:
..:(
..j
4:
.q .s(
-
:( ,
( ...:..
.
-.
-
-
-
.
.zgjj :.:.:@
,
-
.
-
.
.
....i..j
-
E.
-- -
y-.
--.
y.
y
.
. -
k
.r..j:
k!yyt j.uy .. .. .g j (j..j '.. j..g . . . .. jyyyjjjj ..
! .j.jjt.;y yyy . Lj,), .gE y ((!j. ilyjq yljjyrjj, g( jyyl y,j rrttL;,, ! ,. y. . ( . jyy y. ryy ( .y (yjtgtjju: yjyyyy jj. @ EEg r . . E..y. ( yjjjgg;jj,E j. ( ).( y y ( (??3'lfL.. Ejjyggyjj,g l qyy ' .j . y jj. yy yyjy j y. j. y jj j;. .j y y .y.j . y .y j. j E .jj;..' .yjr,k,,' ( y g. j.yy y. '. ' j . y yjg . ' j jg. j yy.y y .. y . gy . j y j jj:,.;ryy. .jjgy,' g ..j.jjg j y ., . jjy;,.,jk,' y ' . ( g jy.(,jj;' j . . ' y j. . j . y . j . y $jyjyjj.(. j . y ( y ' . ; . j ;j;gjggjj,. . . y . . j j . y. . g j . y . '. ..jjr;.,jjs . . y...yy... j.y. j.... g...g.j.... .yj. ..j yg. ..yj. jyggj.. j..j.. yj.y.jj.j..j...j.j. jy..j.j .j.jg. . .jjgjj.ggjj.j.yj g.....gjg.j j j.gg..j.j... ..g..j j...jjj.yj.gyg. . gj.gjgy.y j gj gjg.jjyy )3:,,qj' j. j. j.j.j.jj. y.jyg..y . kjjiirijrj' .. (;j(jy,,. jj.g . yjjj.,yj:' . .. ..jjjjjj;(yrj.j.' .. jj.jyj..jgjy..jg g . j . . . j . . . ... .g . . .j. .. . . . . . ... . j... . ...g.. . j . . ......j..g..j.)jyyyyjjygyyj . .g y jjj ..... .j jsjsaa... ... ... . j. jy. . y. j . . .. .yyjy .y. ..y .. g .. ... y ...j yyj y . . .yj. . aaayayaayy y j .. .j. jj)j)).j......y g ..j jyj. .jj y .ayyaaaaj . j g y.. . .. j y.. .. y .yy aaayj j y j jjj..j jj:ryjjjt, yyyjyjjj j. ..y....j .j. jj:rgjjjj, j yj y.yy. gj.j .g yyy yyy yyyyyyjj j.. j jy. jyy .y jyyg yy jyj ..jjygj . .yyjyjyjyj j.y. yy jyyyj. jjkjjj!:. 11:2)):,), jjkjjtjr, jy. yyy. .g. . y y . . y y . .. . j yyjy, ijjjj;i y j g g .y .. j gg j j.y. (jjkr.rjj,st y. . g..... ........ .. . j. . .....gy.. j j .j j j. .. . g j. y . j. j gj. g . j .. .g .j y .j. . y.j.j g. .g ..... j. j. y. . . . g . . kgjjjljj;. j j j g j g y j rjjgrr!jjs . j y . j j . g . y y y .y.y . . .. . .. . j y. .. .. gg jjyy.. y. g jj .jy.. j y j .. .yj.. j .. . ...jy....y. y.j.y. .... y j.. ..y .yjyy.y jyjj y.y....y.yy..yyy.yj.y.yj yj.yy.yjy.yyy.yy.j..yyy . . y..yj . .yy..y..y j. . .. ...y. y.yjy.y.yy . jyy.yyyyyyy.yyy jj.yyjyjy.. y.jy.y..yy..j.y . . yy.. .y.. y.. .. ...yy jyy.y......y.... . .yyj. y. .y . . .. . . .. . .... . . .. .. . . . ,
.
.jj...,jjj.(
.
.
-.
y..- .. jj : : E( ;: tthnltktlkllld'q.qqEqriyjyEyyyj'.,j,i,i.,gj,.y,yrEj'iE,Eyyl.riyrgl.iE j jjE Er E :: j y E . jy jj r y j j
((
--
'
-
.
ys'l. )r' Suppose the index price is $1050in 6 months. Aholder who entered q Example 2.1 q long position at forward plice of $1020is obligated to pay $1020to acquire the index, @) .a iltE and hence enrns $1050 $1020 $30 per unit of the index. The short is likewise y .. k. k. obligated to sell for $1020,and thus loses $30. illstrates the mechanics of a forward contfact, jhowing why the money .When the price rises and the short makes moey when the price falls.
exinpleE
This
:k').)':.)'jj.. ::i EEli.gtE EjE .,.....-.y-ilj)jirijk)jj)...
1.:;
TE
l
.
...... (E .,. .,. .. g . . . . E
j
.
-
. . . . :
..
-
. .
E!EiEE(.j!)E
IIEqEE'qti
..
:
---
.
-
.'..
.: ..g...(jj E !.' '(.' :j.y:y...(..yj.yj.jEj.;E.j .......j....(j.j :.t.. 'jg.y j j': ( g. -yE .q- . . . . i. -.-. E . . -; -- i- . y -;. - -y . . .,j.yyyEy,,i;ry,j.yyiyyy,,',.rl..yj'yy . y
.i(:' j j'..: .
:.r
E
-
-
--
.
.E
:
.
:': ' ..
E k1i:!..
!iE E(EElpli , ..
(:E
.
:(E(.'.:J.!t..q'E.rEE
L:
:j
E)
.
. .
:(:.). ...t)t
...,.,k2i,.
-
. . .. .
EE!
:(
jg(E.. ....! .
.y
''
.
(
.
.
E.j E.':'E .
'.'
-
-
'
.
.':i
-y
-
:
:::( , jl E j j -jt ' rj jj j jjytjl'j:( !q-.. E-. E-E. i - E. -.' - .- E - E. -. E . i . - . . . E..'E: E ...E - . .-... - . . - . iE ( : . ( E y(... . y. ( (i .i -. ( .
..
((
,;. ' : : E.( E(.::.....: . yE. . : ..j . .
.' .'
. -
-
(
..
. . : . . :
--
...l2..l!.r'
:
(
:' : '.''.(
.i(.
..
.
.
Eij.E'
: '..
,. :(, .
. .
-
.
. , .
**'.1..;,.*.)
' .
.
.
-. . - . q
:...u.y.:
.
..
.
.
:..
.
..
.....
E
' . -. :( .
.
..
..
(
. .. .
-
.
long
.
Iales
.E''
.
E (i1:rtI1L.-
... :. :
-
-
.
( ) ) .. .. L
..
E
'
:
11:1::,,k..
r
-.
.
.
...
E.
f1:::(:)p-.
,(.
.. ..
:
''
''
-
' $I::22!j:. ' ' 'q E. q... ..g.. .... iljil!l;' . (111.. . ...E (jI(4r:jp(jj(r;::,j. ..1 .y.Ey. yy .. . .E. ... . .. . . asjsltcj artt, (.j j .. ( .
'
'.
'. . :
,
.
. .
..
and bet n ili(. k.. . . iirtli? jottl Arhrica ?? y y y .
..
'
:
21j441;. (j1(g;:)p.( ( .:
EE
,
..
yy.
: . ..
'
.
.
:
!II).L!iL.. 41:::::,,. :I1Er'; Ilti!!l.g:llr'. :!jE:-.1i22(.' 1lEj:i:.
..
.
'.
. .
.
.
.
.
'
askd? Eiii..
( . ..
EE E,
E
,,,
ilt::.
E
.
..(
.
. .
-
'
-
,
.
'
.
E
.1d::(
''
.
..i5i'.
:
:
:(
.
i'
',
:'
''
..... .,...E
:
. .. . . .E: ' : . . g
.
E(.''
..:...
... .. E( E .sE (: E liki!,) j 11i:i:i); ' dltz:r. E. . ' .. : .. .E. ( '. . . E:' ' .' : . .... i . ..' .: .g. . . j '.E' (. E . . j .y. (,gg (.g. (.. .. j (.. . g y. . y... g g....g. . .. . . .. . .g
:
:
-
-.-
:E
:
.
-
.
.
-
''lt!9;'.r
.
'.....
..;.....: ...
(
'
.
.
.
.
)(i Ej gg .i:((...(.. .).. .L.L. ,)t..... ( .. . . . ;::,'
.
..
-
:E 7.' E E'E''.' :' . . . . . . : E ' ' - . E Ey . - -. - -. - . -.. ... -. -..
-
.
-
-
.
-
.
.
...(.('.
...-
...
E
-:,.
'. .
jl
.
:
.
t
.
.
:
.
.
-
:
-
.
.
.
-. --
.
.
.-
.
:
.E'
-
-
.
-
.
' E'' ' ' : E' :E'' E ..
;. E...E'E'': . . .
E' ' '' :
:
.
. .
.
.
. .
-
-. . . . ...
'
:
.
.
-
(
.
... -
.
. . .
-.
.
:
-
-
. ..
--
-.
..
.
.
-
-
-..
! :. :E
: .
i
:
:
.
'.
.
'.
.
.. . .
,
,
'
-.
.
,
'
:
.. - . . .
.
-
-
.
..
:
.
:
..
.
-.
.
.
:
,
.
.
,
:
,;;.
.
- .
-
.
:
.
E ( E:.':' E:'Ei:. ''E'(E . ' -' . . '
E.
,
.
.
.
'
'
-.
-.
'
E
E
.. :
.. .
:
..
.
.
:.
.
. .
-
. :
.
..
.
.
:
:
.
-
.
.
..
.
.
.
.
. .
'..i'
-. .
.
-
.
.
:
:
sj't..,j't..j
-
:
-.
'
'
.'
:
.:
,
'
.
-
.
.
. . . .
.
.
. .
.
.
.
,'b....'..
g
:
.
.
'
,
.
.
,
.
i :.
,
.
.
..jjj..:..;,,,,'
.11::
-
'
.q.
..
.
.
E.
.E
''j
'
'
.
q ..j.j;
: . ..
E' . ..
: . . . '
..
:
.....!.
..
..
.
i,
,
::
:
.,j;,,,y.,:.
,,,,,.. .
.
,;,.k...
-, -
-
.
-
.
.
,jjt,.tj,.(j,g;;...jj.j,,...
.
--
j...j.))..)j):.j)
..j.j..j..
:)..
..
..j....
,-g ---.--,.
-
.
,..... ,
,.
g
'
. :
'1124,
.. .
..
...
..
.
.
g .. .
gy
g
.
jjyjjyjjrjjkjjjyr
..
..
.
y rj
y,
g.
.
-.
-
-
-
,
-.
-.
.
-
gy
-
.
-
,yytj-y
-y;.
-
.
.
......jjjgy.. ..
-
-
.
:
.
j:jgjjgjk;rgj;..
lljjj!:i
- - - - -. -- -- . y,E -.
-
-
-
. .
:
g
..yg
.
-.
.
- -
--
.
-j)yyr(! krj.ppltrEt:rtt;r- -. tryE-t.prttttr,j --
..
-
-y
-
.
.
.
:
.,;..
.
-
. .
.
.
.
.
.
.
y. . .. .. .jjlrjyyjji
j.
.
.
.
g
-
:
. . .
- -))-.
.
.
.
.
' .. . . .
. . .: .
.
:
-
'
. .
.
.
. .
.
j . . .y .
-
-
-(
-
-
.
.
.
.
'
.
.
-
:
-.
.
.
-
.
-. .
.
-
..
ii::2:k,.' E. '' ' -E E . . E E. ' . . - -' ' '- ' .. E.E.. . . E. -. . . . . . . . -. - - . E- . E. . EE. . .. E E-jjtjyy:j,,; -. -. -. . E.. - E y E.. . . - -. . . . .. ! y - -.. .. .EE.E . .. y. .. . . - E..E .. -. . - .. . . . .. .. jjjjjjjjy. E. - E y jjjryryr:. .... E ,jjjj:jjy. . . .. . . E . - E .. ... . E - . . .. . .-. . .. -. .. . . yE y - . . .. . -y . . y . . E . - . . E . . -. y . . -;jyyjjjj,y y. . - - .. . jjgjyjgjjyj y g.. ... - -.,k,y . y. y yy . . . E. -. -y y. y . . . . . y. .. . . .. . . y . . . : E : . .. r .: . . . .. .. E . : .. . y y y . g .. r ..g . . g . .. . . y g . . yg . . g. . . g . .. . . .. ggq g. g. .g . .gg. g.. g . . .j . g g g. .: . .g g . ' g.yj$(jgjjj E :. g . .. E. . E ' .: ' . g g . . .y j . .. . gyg y y: : . : . E jjtjjgjgjjj. yjjjg.E .. E. . : :. . . . .g q . . ; .gtjjjjgjjjk : .. (. . : . .. : E. .E E.. E.. . g . y g... y. y . . . . . . y. yjkrgjjj;,. . jyjgjggjg. . . :. g . . y. . g . . . .y g j. . . .g . y .y y . : g . . .. . . .y . y.... y . y.. . y.. .. . . . . .. . . y . y .. y. y g... . : y . . r . .j j ..yy k. . y . yy. . .: .p,,gkjjjjy,!rjjjjjyjjj.. yr . : : . . 1j::jrjj; . : .y. . g. .. . . . : : .y jjjgy,. :y .jjjggj;jj,. . (yjI,jgjj,E : . . : j .. . . :. . . : :: . q .. .. y . : g. . :. g y. . . . y... . . .y .. . : . . y . .. . . . . y . .. . y yg . . . : :. E (jjkjjyggkE . gE g .: ' . : : (gjjt.Igjjjjyy.E E . Ej . (jjjIj.jj.tjjjj.E:.' . . : E E . E qg.: E E. :' .' E': .. E' E : '. ' y ll:jjr!!q . . .g ' 'E E ' y :. E . j r .. . . y . y .. E. .. . :. . gy g g y . j . .. g . E . j. y . . . g. y . . rj . . ...... .. E g.jjjjjjjyy.. j . . .. .. .. . . j g g. g g gj g g g g. . g g y . jgtyyjty,. g g . g g g . . . . . .. . jj;,,,yjj,. g j g g. g . . y . . g . . g g g g g g. . yyyayjy.g g g g g . g . .g . . g . .g. g .g g. ... . . . g gg g . . ( .: g g .. .y . . . . j y y . ;y jjjtgj .... .. . .. y. yjyy. y. . . qg . ... . .. y. y j gg g . .;jj.. y. y.. g g . . g.. g y g. g jy;gggg;s . . yyj,..,jjy jyg;gjy,y yggyyyy yy y g g . .. .yj..yy,. y. . y .y . .. . y . y yy . . y yy.. .y.y.. . . ; . y y y . . y ; y ; ; y . .y ; y . .
-kyjjj.... . . . . .. . . . . .
jjjjgyj;,
...
:
.
.
gy
,y
g.
g
t.
--; -. E- . - . - - -.- . - -(E.E ..- . . . .. . . . . --y E -. E -- . -
-
.
.
y
:
,
-
-
.
-
-
-
-
.
.
...L...j..j.....jj...L..)...L..j.....
g.
.
.
.
.
..
-
g.
.
-..
-
-
.
-
-.
..
.
-
- -- -E - .. - -- . -.-y--. - E E-. -.. . -.. -. ;-. . -
. - . . - .-
i
.
-
.
''1j(yj:)j,
jj:yjgyjjs
-
. - .y.
- -.-- .
.
-
.. -
. .
..
.
. gjgjgj. . . .. . . . . . . . jjjjgj:
''
:
.
..
.
.
-
-- -.- . .
.
-.
..
. . . . : . . .
. .
--
.. -
-
.,,.
.!''jjjjkj'!l(jjjy,gjj..
11::::::.
:
. .
.
-
-
-
11i2r:;
y :
.
.
-
.
. yyjj
-.. ..
-- -. --.,.. .-
--
.. ---. --
-
- .
-
Etjjs
.. .
..;!it!!!!!.-(
:
-
.(y
-.
- -
-
- .-y.; . - -
.
.
-1111141;(
: . . .
'
- .
..
-
.
-
.
-.
-.
--.-
..
-
..
.
E
.11::.
-.-
E -- -- - - --- -. - .
.(jjjgjy.
..t
.
.
.
.
'.
-. -
.
.
-
.
.
. - - 2.-)y ;-. - . . --
.
:gjj;;. LL;ILjLjjf.L
....
-
.
-
..
.-.-. .
-E . .-. . .
-
y
. y
.
g
.
.
.
..
.
.
: :.
j j
.
jj ...
y
.
. .
.
.
.
.
.
:
r g.
.
.j
.
.
.
. y y
g
.
..ajy
.
.
.
y
..... ,.
..y
g .
:. .. y
. :
y. .
,
.
.
g
.
. . . .. g .. g .
.
y....
r
.
. .
.
jtgk!
k,g
.y
.. ........j. j.... ...jj....j ...j....,.j.. ,y
:
: g
:
:. :g .
,
gg . y: . y
y
..
,
,
:.:. y . . .
g
g
g
yyyyjy
:
g
y yg . g
. g. . . .
.
gg
.
g
g.
g .
g
g .
:y:.
y
.
.
. .
. . g g. . . y ..
y
.
.
.y
,.
g
y. y.. g .
:g
y
g .. . . .
. ;
:.,.
.
y
. . .
g .
..y
., :.
y
g
.y
: qg y y, g .g
g
g
..
.
.
. y
.
.. .
..y
..
, .
; .
g
;
.
.
.
. .
.
.
y
.
.
t.. t .
. : y g . . .
:.
.
yy. .
,
:
yg
.
.
.
;
:.
:
yy
.. g .
y
yy
y
:
:
..;..jy....yy.yyy.jyj. ;
.
g.
;
:
.g
.
.
.
. y. . .
sell one unit of the index of the index. The holder of the short position is obligated to possible futtlre values various for for $1020. Table 2.1 lists the payoff on the position of the index.
g
.
g
.
y
.
.
. .
. .
' . . ..
(' . . . . ..
.
.
Grphlnk th Payos n a Frward
Contract
We can graph the information in Table 2.1 to show the payoffin 6 moths on the forward contract as a function of the index Figure 2k2graphs the long and short positions, with the index price at the expiration of the forward contract on the ho/zontal axis and payoff on the vertical axis, As you would expect, the two positions have a zero payoff when the index pdce in 6 months equals the forward price of $1020. The graph fr the short forward is a mwr image (aboutthe x-axis) of the graph for the long fonvard. For a given value of the index, the payoff to the short is exactly the opposite of the payoff to the long. In other words, the gain to one party is the loss to the other. This kind of graph is widely used because it summarizes the risk of the position glance. at a
i
%
26
AN INTRODUCTION
(' :5* ;'k' ))* y' 1* E (.(:jj;jjjjjg'-., ''@' jyflflftrjijjjjjj:jr'. k' E)r:f ('' ,'?gf t'#' E E!'i( (E r' rt!fktE'E.i( ;tFt. 'r' :rjjj;;f qsrq!f rrsE''r!f )'r' t:)f jtf f' ;'!' )' !E('E.!(i';' E(E'EI(EE(!'(i..'E. Er.!!i.q(q' i i;' i!((i. ((.'(q.(' E!'iE'(' ':'' '(rlqrqrFrr:qErE .j' tj'i'lli ()1T: ;1!:!' 11( ' (;l (''.T q!. ;1 ; . !q't: 'i' ; E(@ j)y)yy.-'-.-' t-1* ;'y'j'y' y'. y' y);-yjjj)-' tryl')f yyf jppptf (' .. . . ' ((. E. .' . ,:.(:;r!.q r.. ; . ! g. p y( :,E,j:jjjjj;;;;;jjE .q.(y.!..gjjy, (jqyjj;!. y. ; ;j.-;r,. ygj.g.q;y .y jyyytjtyttti.. :')E'l.:)!'........;::.......t:Eu.;. y jjjgEggjky. j .. . . jjj. q ,.y yn,i yy.y . . .....,L;. j lkzLrjii . 1fy k;:;j: y q. q y y q j,,. ; rjyyj :2.:. .. .. .. .,. . . .
)' y' )'. ;')1* y.jt-y.)r-' yjf #)' yy'. t.'y$jjyy);'. y'. )y' t'tt)-f )1*t(4*. jljl:!f t.yyillyfykjytkyyjjjj', t'y ! .t' 'g' (. )) y . )lt yykijtkkkjjjj.t;q ltyyy . tryy. j yj. y.y .. ..jjjjqiijjrjj
-' --s-----yj'if jj::g::,,f :'''''(iEj)tj)'! j'-';'y'
.'
E. LE '
'' '
'
:
:
--
:'
:
.
..ji
'
:
:
'
:E :. :: .: : .
'
. :
' ' ' ' : . ..
'
' :
'
'
. .
'' .
E
'
:
E':
:
-. -
.
.
..
.................,.......,....
.
-
-
- . .
TO FORWARDS
AN D OPTlON5
FORwARD 1/11154!! : 74: ($';#' ;k' 1* itf ))fl: q' i'(2;157(42,6(J7/j* F7;TT7T:ITI:)Y (r' j''(q' 7771:77:7:7744:7:! !.(;! p.j'(q' t')'j;'. #' (6* (f' q)' /'t:' ((' (i'lrj (' @' i ( (l !! ( (;'t' (' q T'lqq:qi j. (!(j1 . .!!! ';( iEEi?E ')q');; j'yjf jjf k:kjkjjjj;'.. j'r jqf .yy' ),f r'. )'. j'yf y'. tyt'. y' r' . (yj )' (('p)yj'y,yyj;j));. . .' 1(;'(i.i'(:(:( . .q..' . ..ir.'...(.(! tj'. t'jyy'. 'j.y billl.' jjjr.i.yk ., q!!ii::::kr..... :t!:(;i!,-... .IltitiiL ; tk ' :ki - ;,p; ) ' ,.;E ;. q. jy; ; ) llli/t); y ( . . LcL,jL.-.--.-j-j.;;-.L-L-jkL;LLj.,.;j--.L-L..kj,..q,;.j-L;-.k;.; j j ltyyy-y--jjj-iyy ))r));j. tr . j j. ' . ( .. .l !.!rE!.Ji..qi......:):.?.-...,.s..........................-.... ...r.........-..y.-y-y--.u).),s. . .),:jygj,.... . . 1!1) iitELlt.-y,; d::z)p ............E Sllk.:j;;-. :@1;; ?:Iq!I,(E (;). . rjgy,--' ;'f' ',F''. :'C' )' ??' ;''-'-':'kf t'(' )' 1* #'
'rryt)jy)))))'-)jyy' rllt,jip. '!)titt.:.; '''
'
'
.
::
.
..
.
.:
...
,.
-
.
-
-
.
.-
.
.
.
. Long and short forward positos on the S&R 500 index.
.
-.
,
-
-
.
-
-.
- -
-
.
...:.r).).....l......;...;.,......
' E
:
i'
;
-.
'
. 'E .
-
.
:' E ' : :' E . . .. '
.. -.. --. -.
i E :'
'
'
JE ! IE i .
'
.
:
.-E .
.
.
-
:
' E :: :
E '
.
i
.
'. -.-jj Ey.. E - - .(yyr.(tE -E E .E.E .
,
gj;jq:kjjyy. jt:rj ;:,,,. y)jyy:,,.g:!;,;,,. y)j-
i'
:
-
-
-
. -
-
-- .
: : E:
.E- E
Comparson
of payff
2500
after 6 months of a Iong positon in the s&R index versus forward contract in the S&R index.
150 1O0
50 0
ztoo 1500
1000
forward shortforward
.+.Long
.+.
500
-50
.+. Long forWrd Long physical index
0
-100
.
-200
-1000
1020 800
900
850
The S&R forward contract is a way to acquire the index by paying $1020after 6 months. 'An alternative way to acquire the index is to purchase it outright at time 0, paying $1000. Is there any advantage to using the forward contract to buy the index, as opposed to purchasing it outright? lf we buy the S&R index today, it costs us $1000. The value of the position in 6 months is the value of the S&R index. The payoffto a long position in the physical S&R index is graphed in Figure 2.3. For comparison, the payoffto the long forward position, is graphed as well. Note that the axes have different scales in Figures 2.3 and 2.2. To ee how Figure 2.3 i.s constnlcted, suppose the S&R index price is $0 after 6 months. tTl1is isjusi a thought experiment for te purpose of constructing the graph, but fims in if you would like to be concrete, imagine that the S&R index contained ltret the year 2000 which would be bnnkrnpt in 2001.) If the ipdex price is $0, the physical For index will be wprth $0; hence we plot a 0 on tlze y-axis against 0 on te I-axis. :11other prices of the S&R index, the payoff equals the value of the S&R index. For example, ifwe own the index mzd te price in 6 months is $750,the value of the position If the index price in 6 months is
-1500
950 1000 1050 1100 1150 1200 S&R lndex Price ($)
Comparing a Forward and Outright Purchase
is $750.
$0, the payof'f to the forward conact,
using
equation (2.1),is =
0
-
$1020
-$1020
=
lf instead the index price is $1020.the long index position will be worth $1020and the fomard contract will be worth $0.
-1020
jiza 0
500
1000 1500 S&R Index Price
2000
2500
(s)
Wit.h both posidons, we own th index ftr 6 mpnths. What the sgpredoes not reqect, however, is the different initial investmepts required for the twp positions. With the cash index, we west $1000initially and then we opn the index. With the fol-ward The conact, we invest $0 initially and $1020 after 6 months; then we own te payoffgraph tells us how much mony we Enancing of the two positions is different. end up with after 6 months, butdoes npt acctwntfor the initial $1000 westpenywiih the outright purchase. Figtlre 2.3 is accurate, but it does not answer our quesiion-namely, whether there is an advantage to either a fonvard purchase or an outright purchase. Both posidns give us ownership of te S&R index after 6 months. We can compare them faidy if we equate the nmounts iniially invested and then account for interesi enrned over the 6 mothsk We can do this in eiter of two equivalent ways: 'lkeasury in zero-couppn bpnds (forexample, bills) along with the 1. Irkvest $1t0 forward 'coniacts '' in which case each position initlallv costs $1006at time 0.
tndex.
'rhe
''k.
. .
2.
. .
Borrow to buy tlle physici $0 at time 0.
.
:
''?'
'
(
..
. .
S&R index, in which case each position initially costs
Suppose the 6cmonth interest rate is 2%. With alternative 1, we pay $1000today. $1020. At that point, bopd ij Frt.h $1000 x 1.02 6 months ihe kro-cofo bonct proceeds to pgy the frward price of $101t. We then own the index. we use the The net effect is that we pay $1000inially an (i own the index after 6 months, just as if we bought tlle index outright. hwesting $1000and at the same time entering a long forwardconeact mimics the effect of buying the index outright.
Aer Payoffto long forward
..+.
-500
-150
-20
.
y -
..
200
%.
Cox-rRAc'rs
=
'
28
%.Ah1 IxTRooucTloN
'ro
FORWARDS
FORwARD
AND OPTIONS
cox'rRAc'rs
'k' )kplf :5* (:y;'ykjjyjyjyq)' 1* lf qrf 7* ilf (' trjf ' 'i'p' qlq)f 1* ifjf 1;* hiill..jqrkifibi.jp'it: )lq't!(()''t)j)qrl@))plr'7$;lt''@l1iET!!rr!i!5lq!71(!qiil!ii;' i El)): r' jj':)f yy'. ('(.'ty' ;'tjf t!l;f ()qf (jf j'yk p)' jjjjjj-z'jjqj);j;fj))4kjj..jj))))j:jLjjL.jjj)LLLj)',tLL,j);Ljj)j'Lbbb6-.' j'jyf sf (l:;stf E'ii . i . ; E ' E l'?!(E(5i y(' rif 'ilii ;r(E: t'y' ' . (: I!EE IPPTE r!lr ' ' .rl:E(!'j. liEi(t-i(j;i. 'i. EE:.E frl7l ; .i E Ft; i.j.t. ;; i Ejj /i. : ' lll$r'-' , . ( q('.. ; -1?.7F. . . jj)y -jy)y....(:;;!!::.. yyj.j (!yytIIi-,-j.41I::1;k'E;;---.'. t y;.j .. j.y.y.y lllk-)kl (jj. --q-k(L...L.jL.)L;L;;L-j;?,, IItE!L...) (((..r.tF )ir.:j,)t. )!(trpty!.-1Iiq5q.--., 11!:.1:: )..ttp(j.(.-)..y,jt.:-t.(r,t.-;.y--:t.). ,t.j,rpIIk-.-------k..tjtjp.. (@,,rjyqj7y. d:r:,j iq,kl:j;;,. 4:2 :!iq; (1Eq:1r (IEF), '':' EirEiiqllijftpqljjjtyf.qjljjrlyfih'jytjjji'-.gyE :''' s' !:''7h;(i 'j'()!' ;'t'.q' )'
k.
29
('
With alternative 2, we borrow $1000to buy the index, which costs $1000. Hence months we owe $1000plus interest. At make no net cash payment at time 0. After 6 we $1020for the borrowed money. Th ne# effecy ij that time we repay $1000 x 1.02 also osvn te index. that we invest nothing initially, and after six months pay $1020.We into a lone forward Borrowine to buv the stock terefore mimics the effect of enterine =
contract.
We conclude that when the index pays no dividends, the cmly difference betaveen of a payment that will the forward contract and the cash index investment is the timing using the interest be made for certain. Therefore, we can compare the t'wo positions by conclude that the forward example, we rate.to shift the timing of payments. In the above in the timing of contract and the cash index are equivalent investments, differing only other. the cash flows. Neither form of investing has an gdvantage over the This analysis suggests a way to systematically compare positions that require differentinitial investments. We can assume thatweborrow any requiredinitial payment. At expiration, we receive the payofffrom the conuct, and repay any borrowed amounts. differing Wewill call this the net payoff orprofit. Because this calculation accounts for than rather payoff profit primarily use initial investments in a simple fashion, we will diagrams and prost the book/ same are Note that the payoff diapams throughout the initial investment. requires it no because conact forward for a point in To summazize, a payoff diagram graphs te cash value of a position at a in of investment value the time. A profit diagram subtracts from the payoff the future
i
:E
:
.
.
.
:
(iE '
'
:.i. . .
.
.
.
:. : .
i
.
' . .:
' .
:i : .
!
..
:
.
. -
.
.
,#yt,.
........,-,.
...
()p
,!
....
,... ..,-.,......f,,.-,..,
.......-,,,-,
.-
.
.
Payoff diagram for a long S&R flward contract, together with a zero-coupon bond that pays $1020 at maturity. Summing the value of the long forward plus the bond at each S&R Index price givesthe Ine Iabeled
2500 2000 lsto
$1020
1000 500
+ $1000 bond
-*
-5 00 1000
Long forward
-1-
0
''Forward + Bond.''
Fonkafd
+
bond
1020
-1020
--
504
0
1000 1500 S&R Index Price
2t)0
500
($)
'
the position. Tls discussion raises a question: Given our assumptions, should we really expect value of the index? The answer in the forward plice to equal $1020,which is the f'uture until explanation Chapter 5. this case is yes, but we defer a detailed
Zero-coupon Bonds in Payoff and Proft Diagrams purchase The preceding discussion showed that the long forward conact and outright value time take o money of the physical S&R index are essentially the same once we and into account. Buying the phlsical index is like elgering into the forward coneact point thij same simultaneously investing $1000 in a zero-coupon bond. We can see graphically by using a payoff diagram where we include a zero-coupon bod. Suppose we enter into a long S&R index fomard position, and at the same time tThis was purchase a $1000 zero-coupon bond. which will pay $1020after 6 months.
by holding the index paid a dividend in tis example, then we would receive the dividend 5 how to will Chapter We in contract. the see forward entered into when we physical index, but not '; take dividends into account in this comparison. 4The term protit'' is defined vmiously by accountants and economists. Al1of our proht calculations with another, not computing profit in any absolute sense. are for the purpose of conlparing one position
31fthe
alternative 1 in the previous section.) Algebraically, the payoff to the fonvard ptus the bond is Forward + bond
=
Spot prie
at expiration
Forward payoff
-
$1020 + $1020 Bond payoff
= Spot price at expiration rl'hisis the same as te payof to westing in the physical index. 'l'he paypff diagram for this position is an easy modifcation of Figure 2.3. We simply add a line representing the value of the bond after 6 months ($1000x 1.02 $1020), and then add the bond payof to the fonvard payoff. This is graphed in Figure 2.4. fomard plus bond looks exactly like the physical index in Figure 2.3. What is the proft diagram coaesponding to this payoff diagram? For the forward contzct, prtt is the same as the payoffbecause there is no initial investmeni. Profit for thefomardjlus bond is obtained by subeacting thefuture value of the irtitial investment. The inidal investment was the cost of the bond, $1000. 1ts f'umre value is, by desnition, $1020, the value of tlae bond after 6 months. Thus, the prost diagram for a fonvard conact plus a bond is obtained by ignoring the bond! Put differently, adding a bond to a posi:on leaves a prost diagrgm unaffected. i Depending on the context, itcanbe helpf'ul to clraw eitherpayofforprotdiapams. Bonds can be used to shift payoff diapams vel-tically, but do not change the prost calculation. >
=
'l'he
i
$
30
AN INTRODUCTION
TO FORWARDS
CALL
AN D OPTIONS
'l'he foregoing discussion assumed that at expiration of the forward eontact, the contract called for the seller party short the forward contract) to deliver the cash S&R indek
tthe
buyer ttheparty long the forward contact). However, a physical transactio in a broad stock index will likely have signi:cant eansactioncosts. An alternative settlement lnstead of requiring delivefy of the procedure t.ha t is widely used is cash settlement. parties make a net cajh actual in dex the forward contract settles hnancially. The two occurred ad bpt.ilparties payment, which yields the same cash flow as if delivery had example. with this an )'. had then closed out their positions. We can illustrate
%.
contracts are transacted directly between two pnrties means that each counterparty bears the credit risk of the othens Credit risk is an important problym with a1l derivatives, but it is also quite complicated. Credit checks of counterpalties and credit protetions such as collateral and bnnlcletters of credit are commonly employed to guard against losses from counterparty default.
Cash Settlement Versus Delivery
to the
Op-rlolqs
,
,
2.2 CALL OPTIONS
''''''
We have seen that afomard contract obligates the buyer ttheholder of tle long position) to pay the forward price at expiration, even if the value of te underlying asset at expirathe forward price. Because losses are possible with a forward contract, tion is less t.11:.11 wonder: is namral it Could there be a contract where the buyer has the right to walk to away ffom the deal? 'I'he answer is ys; a call option is a contract where te buyer has the right to buy, but not the obligation to buy. Here is an exapple illustrating how a call option works at
Suppose that the S&R index at expiration is $1040. ecause ie q Example 2.2 !)! forward priee is $1020, the long position has a payoff of $20. Similady, the short yjtj positipn loses $20. With cash settlemnt, the shol't simply pays $20 to the long, with tl .4)'q) . . ansaction costs. lt is as if the long paid ljt no trnsfer of the physical asset, and hence no ansaction $1020,acnuired the in4ez worth $1040,and ten immediately soldit with no
exp/aon.
lE
'''''
j . costs. )y
.
t.j 1)1
lfthe S&Rindex price atexpiration hadinsteadbeen $960,thelongposition would and the short would have a payofof $60. Cash settlement in this 7 have a payoff of tlj k tij case entails the long paying $60 to the short.
Suppose tlwt the all buyer ages to pay $1020for the S&R index in r Example 2.3 11 months but obligated to do so. 6 is buyer has purchase j a ca jj option.) If in 6 not (jt t'1*he tl mont: the S&R ptice is $1100,the buyer will pay $1020and receive the index. This is ,).) 'l' a payoff f $80 per unit of the index. If the S&R price is $900,the byer walks away.
-$60
'
$
Cash setllement is feasible only when there is an accepted reference plice upon which the settlement can be based. Cash settlement is not limited to forward contractsdelively virtually any financial conact can be settled using cash rather t.1::.11
Now thirlk about this transaction from the seller's point of view. The buyer is in coneol of the ofdon, deiding when to buy the index by paying $1020.Thus, the lights of the oydon buyer are obligations for the option seller.
Credit Risk crdit risk, arly derivatives coneactvhas Any forward or futures contract-indeed, who fails to make counterparty owes money which means there is a possibility that the the price and lixed price spot a payment. J.f you agree to sell the index in one yea.r at a the obligated buy to the is countepmy t'urns out to be lower than the forward price, will for the that counterparty risk some face the index for more than it is worth. You the index. Similarly the counterparty fades the for forward price fail te pay to reason risk that you will not f'ulNllthe conact if the spot price in one year turns out to be higher than the forward price. minimize With exchange-traded contracts, the exchange goes to g'reat lengths to ultimate counterparty and being the participants a1l collateral of this risk by requiring in a11 ansactions. We will discuss credit risk and collateral in more detail when we discuss f'umres contracts in Chapter 5. With over-the-counter contracts, the fact that the
)'
T Example 2.4 If in 6 monts the S&R pri ij $1100,the seller will receive $1020 ((i !' . lf the S&R ).) and gike up all index worth mote, for a loss of $80 per unit of te index. ll rice the will less the seller obligation. is has buyer than Thus, $1020, buy, not so )ll P no ')) at expiration, the sellr will have @payoff whih is zero (if te S&R price is less than k1 $1020)pr nega tive (ifthe &R pric is jreaterflaan $t02O). k E @
'
5Of course, credit risk als exist.s in exchange-traded contracts. The specihc details of how exchanges are strucmred to minimize credit risk is a complicated and fascinating subject (see Edwards and Ma (1992), ch. 3, for details). In practice, exchanges are regarded by pal-ticiplmts msgood credit rislcs.
i
32
%AN
INTRODUCTION
To
FORWARDS
CALL
AND OPTIONS
Does it seem as ifsomething is wrong here? Because the buyercan decide whether make money at expiration. This situation suggests tat the seller to buy, the seller cannot tfblibed'' to enter into the contract in the flrst place. At the time the must, in effect, be buyer and seller agree to the contract, the buyer must pay the seller an initial priee, or prenlillm. This initial payment compensates the seller for being at a disadvantage at expiration. Contrast this with a forward coneact, for wlich the initial premium is zero.
.'''. .'-
,1::..* ':'tr--''l ,'.j' y' (' F'' ' )! ;:' S' ;'jllq')kf t'(' )' f-' (.' (' 1* i'' lqf #' ' 7* 1;:..* 11.,11* 1111k-:-111.*.* ytf j',;yjggjj;;j. (yt(y'. j'y'.j. Elqf tf'i);f t(' ')'! t'pyf 4* if E EE !5i!fpqsr li.llit-l-';i Iri.lrf p' q' )' '.' (.fj'tjj'y jyjyjfjjyjyyry' yyf r' lf ' (E E 'E'.' EE'11' (: E 71);. (EEEi'!E'!(i':Iii( '.i(.!.E.(('(E.( ('.'(.''(''E('(!' E7*(7,7:7:': ! q' q1: ;--k-)')-tt.'. )!-' Ilii!iii:f y.'. yyjyjkf ;'. 'j' .y'i yjlf rf j'1*. (E:(7('k5. qhr!r'!ts!':' ' EEE ' ()'.('q' --L'. 11111*11II:iFq'. )' y'yrj'y' yyf rlf t't.'. ' ..' '. ' . '..'(. ' r' '.- '. ; q E(E '' j tj.. '.'yj .(y 'y.. '.qg.(.(.yq. pq jg).(! E il (' !j:(;g. q qEE (, ; q ))j)L. qk r .-.. 'E.'...E )..... : tj. ..j.jy. y ,j. .jyyj . y . y . yg. . g . .. g'9iI::::;:k,-. .. . . j qE.... '' ;tl')l . y . . E' . -E'' yj;j...Ey-j.E.. j ;. @yE.-i-EiE. . '' 1111F:11.12. . '. ; E.... r r.@ '..... .... '). rl '. itri (-.i(;k. .' . .. .itrlrl . .. .. . . . ..j. . .. . -' .'' -.' .'..-' .:7;11!!!::*.. -'. ' -' -'. .-.' -.---(;' -;' -'y -'-' t'?' y' ;'. @@'-.Fp'p' 5* I'E j'. y.'E q'. 'y'.'E ?y' f' '' ELE E .
r'lil'i 1); p. '
'' .
:
-.
'
. .
. ...
. -.
.
.
!:
E
'
-. .
-.
.
.
-
'
' ' E
' '
'
:
:
'
:
... . - .
:
'
-
! -. -
.y
. . - .
.'
'
-.
.
. . .
-.
:
'
.
'
' :
:
...
-
..
.
.
.
-
.
:
:.:
-.
. . .. .
-- . .
Closing options prces
for 5&P 500 index Options f rOm the
.
.
-
,
-
-k--i)((.. ..
'
..
-
yyy.
i : '.
: : '' . . : . i .. . .
.
tjjy-.j)Ey---y. .
-
-
'
.
-
.
IIEEE':IE'' IIEEEE
-
.. -.
opvloxs k.
.
.
SKIKE t. t5: (%G lms & pstmlspx)
B 975( 417.1 0.? 6M Aug 975p 72 0.29 66,613 lv 1.15 lrzu nc z-o 5Qp 9:57 311 3.70 8711 5:p 995) 41 5 aMl 27,5. s:p105) 1.5 5.29 oct 115) l 19.29 1,195 94l u9 1(12()) 218 l'3p Aug125 ( 1,33372 1.1.2C4,M8 Augllsp t2(I ll5 /,556 5e) 1025( 1,155 n 9.5() 4,965 5Qp llsp 391 8.10 51,251 oct 19l5( 1 83 lt(N 13 ()tt 1625p 5 16 82.9 Il5 se) 1232( 15 79 ls 5Qp 13p 1,41919.% T -:.59
sQp
Chicago Board 'Options Exchange.
-120
-19()
.
-z12
-.1
Option Terminology
-119
-175
He4e are some key terms used to describe options: Strikeprice: The strike price, orexercise price, of a call option is what the buyer strike price was $1020. The strike pays for the asset. In the exnmple above, the price can be set at any value.
-l3
-1(1,
-1.29
Source: Wl? Jrrecr Journal, July 28, 2004, p. C-15-
Exercise: Th exercise of a call option is the act of paying the strike price to receive tlle asset. 1.rlExample 2.3, the byr decided after 6 monthj whether to is, whether t pay $1020tttzetcike plice) to ieive the exercise the option-tat S&R index.
practice. While most exchange-traded options are American, the options in Figure 2.5 d are European. Later in the book we will discuss American options in more etail.
Expiration: The expiration of the option is the date by which the option must either be exercised or it becomes worthless. The opdon in Exaple 2.3 had an expiration of 6 months.
Payoff and Prolh for a Pttrchased Call Option
.
We can graph call options as we did forward conacts. 'I'he buyer is not obligated to buy the index. and hence will only eyercise the option if the payoffis greater t.1)a11 zero. 'Fhe algebraic expression for the payoff to a purchased call is therefore
Exercise style: The exercise style of te option govet'ns the time atwllich exercise could occur only at expiration. Such an can occur. In the above example, exercise option. European-stle be tf the buyer has the tight to exercise option is said to a at arly time during te life of te option, it is an American-style option. lfte buyer butnotforthe entirelife of the option, the can only exercise dllring specifiedperiods, by option. (The Bermudan-sle terms 'European'' apd is option a Berfpudan and Americgn, geography. Etlropean, with nothing do the way, have to options are bought and sold worldwide.)
Purchased call payoff
Tc''
'
''
maxlo, spot price at expiration
-
strike pricel
(2.3)
The expression maxlc, l?qmeans take the greaterof the two values a and b. (Spreadsheets contain a max function, io it is egsy to compute option payoffs in a spreadsheet.)
Hytmerican,''
To summarize, a Etlropean call opion gives the owner of the call the right, but not the obligation, to buy the underlying asset on the expiration date by paying the sfrike price. The option described in Exnmples 2.3 and 2.4 is a 6month frlrcwctw-./-yfeS&R call with a strike price of$1020. The buyer of the call can also be descried as having a long position in the call. Figure 2.5 presents a small portion of the opdon price listings for the S&P 500 Index option traded at the Chicago Board Options Exchange. Each roW represents a or a P to denote call or different option, with tlle expiration mont. strike price, a the last trade of the day, premium day, that at the put, the number of coneacts traded wit futures, As and every opion trade open interest. the change from the previous day, buyer-seller pairs. of number the interest measurrs requires a buyer and a seller, so open option. buying of mechanics the of an 'Fhe box below discusses some For the time being, we will discuss European-style options exclusively. We do this because European options are the simplest to discuss and are also quite common in
=
'
!p Example 2.5 Consider a call option on the s&R index with 6 months to expiration 11mad a strike price of $l0 00 thb index in 6 months is $1100. Clearly it is suppose ylit lt worthwhile to pay the $1j00 stlike plice to acquire the inctex worth $1100. Using .
) equation (2.3),the call payoff is
'
(l'''
l..
m axp
).
IE
)(j'i
$1100 $1000j $100 :::z
-
,
@'ji!E)
lf the index is 900 at expiration, it is not worthwhile paying the 'tj buy the index wortll payoff is then $900. tj
$1000 stnke price to .
'l'he
()
maxlo,
$900 $1000) -
=
$0
%
As discussed before, thepayoffdoes nottake account of the initial cost of acquiring For a purchased option, the premium is paid at the time the opion is
the position.
k. Ax
34
IhlTRoouc-rloN
((j(' (' gjjj,?'f 'y' !kkii,likl.'. j'. C'. ijii'. y'E t!d j.' F' )' ;'j' j';' y' y:.yf jjfyyjty:jjtg (' y'' -i1iiiiiiI'' y' ,-' ,'' Ejf 'q jt'E )'. ,'k.. )' (:;i'I;:.' 112212:)* jyrf j)j44* t'('y' ;k!1kk.;kI.-jIi)'' 'j' ii'li'rIII'',kkiI:!iii,;')'-1il:t:Iiik'. .y' ;'. ,t'jj(yyyy' ('pq'..-yyjjy. ('(' r' rd ryd )' y' k')jjf (jf (r'yjyj (' j(':.jjkj', yf '.'' ' E;! EEE ;E'E.!(j( t'iiili'iIkk'' g'. jrf )g'. j'jy-.' jy' y;-'y),jy'.. :t';)-: ,'y'y' .y' i E . .. . ! j(i(j((( 'j( '.j jjyg.j jj.tibiL6'itLLL ' (E jiE.. .'' . (.y; j( . ' t'k;j j jj(yttk ' '' j--k-. ' . ...' ; EEE. EE .(q -.-. yE ;j. !(I). :jEiq EEi Et;@.(E i q i . t.rr. . .. . . . .;2., .... .r . i.- r. . ..
:(' '.' --(j)'-'. .'-j'. .t-t(-', -',-' ;'t';':';':' jj44yy'.' (';'';' -' ;j!' 5* -111::11k,k*--' 'iiit:!:iiii,'. j-'' y' :'(jf -)'. F'. -' ((.)t'. -'. klyld y' ;';)' y-'
jjlll;::l'd (.' yyjf jjjf yjf )j'. iiIi:1f'. y(' j'. k'. l-' )'. )yy' )-' (' y'( j)'. j'y' )' ;'t' yjjf ytjf ('. kjf jtyf jjj'. (j't(jjjy'.jy jyjj(j'ty(j. j'jy'tjy'jj 1* llilt:!tlili,f )' .iliiEEE!1ii2'(' r' 'tL'''. (;' k,' ;);jyjkry--' y'. t'l'y iy;'. 4r'(.((j)' yyf y' i-' t')j'. i.'. tjry)yyrjj('i;f ty'ty. tt'q t'. iiiii-t-iilltf tjf r' t'j t'y; i( j.. ' .
.-.ty', .,' .'
.'-. -
'
i Ei
:
-
-
..
-
:
;.
,
' .:
.
'
:'
..
.
.
..
-
. .- . . :. :: . . ...
-
.
-
.
.
.
..
.. .' .
.
.-. . .
.
.,
'
.
.
- -
- .
..
' .
.
.
.
.'.
.
' . ... ..
. .
g ..
.
.
.'.. :
.
:
EE j g.jj g.: ' . .gy .g -. - . q.. -
.
'. --
.
-
-
: ' k''''IE (.(...;::: ' . E.E''!k(jg(.((jj!jr''.Er.: . gyygikyE ..y;.Er; :
' '
'
q'
' '
'
.
.(
. .. . . .
:
.
.
....
.
E
.
-
.. . .
. . - .- . .-
.. . :
.
. . . .
(
E
.
.
.
.
:
. .
(:
:
,
-.
.
.
. .
: (.g;j.(.yy:g : j ! .' '
j. E; . ' .E .
::'.' ir. :.. . . ..E
'
'
EE:''
j (j.--Ey(r g.. ----yj tj .E, -y EE -. . - .. -E. E E :.
y.
'
.
.
,
'
(Eyy
gy q j
:
,
-.
--
.-
.E..y. .. .
.
:.
. .. .
E y.
qg....
-.
.
: .. E.: E...
.
. .
.
.
-
g..
.. . .
.
: r .' . . . g' E . ... (
..
...
:
.
g. .j ... ....
'jliit
. ....
.
-
-
y
.' E ; '..E- -.((g .. . ' j i . . E -.
;
.
-
.
..
-
..
.
..: ..:. q. ..
'
;q
.
E
,
'
.-
(E
'
'
.
..
:
.
.
,.
-
..
.
-. .
.
. .
.
-
-
.
.
.
. .
.
.
.
.
.
:.
.
...
:...
.
.
k/Ex.k'
..E
-
.
:
.
.
.
:
!
-
. . , .
.
Y
r
.
-.
.
- . .
.
' : ' . .
:
..
. ..
.
.
. -
.
:.
.
.
.
-
.
-
r .
.
:
::
y.
,t-(: ,
:
.
-
. -
:
:
:.
,
-.
.
.
:
k
:
.
.
:
.
-g
.
.
.
-
,
;(E
.
.
.
.. . . . ..... .... . . .. . .. . . E
Ej
,
.
- . ..y.
. .
E :' : : i : ! y.(y((yjj((k, ( .: ;j. 'j y.E. .j.r E.;.g.: gEy.j.:g g((q ;' (.y: j! j E.(gi jg...( ' g' ( Ey y.:g': :'.( j :-E ( :: (j.. . jgr': i j.(.E.''((. (:(EE'(E-'Eg !( y, j g. .--IE j y Eky jy,E . E. .-- .-. Ey yI;yE. .(yE . E . - !. - - . -. E . . ,;;. . -.-y . -. ; -. E. . i - ---- --. - - E.-. -...; -. -.. -. . g. . - - . ,. ,. q q q . , q . q q . q . . ; q.-. y- - . . . k( . .; ; . y q - y : .E. E:.; .. - g . -. -q: y . .. . : ... . . y . q. . . . y :.. . y . . . .. . y .y y . . y y.. y .y y yy . yy . . . y y r . . . . . ..y y y: y.. . y.. .. . yy ..... g..y.. .. y..y y' .' .y.. . .y.y.. . . y.y.... y.. . y. . y.. E .. E' : ' ( :. '. . ' . . . . '. E E:.j:.(i.. . E' . E . : .. ( . .: ... . (. k'. gj : r : . ( . . g. . . g g. . y y.. . j . .. q j.: ( . .y .. . y g. ...y. . g. : j y..y.j.g. ... .y.. j :... . .y.g. g(...y:y j. !.. .(..;y.E qg... .: y .. ... j. . :. ... y . y. .. g. tjhiitziliitklE (.. : . y g.. . r g . g. .g . . j. y. . . j . ( . ..... . . . . . . q ... ,
.-
.
.. .. q
.
.
.E
i
IL
E!
tqEyt;jy)t:,..
qyy
,
just
.1
,i
i.y..jt')
tryt',t).
''
'
'
(
(.
:
:
.
op-rloxs k
CALL
''(j. j. j j . (... (..' 44.j44)jj4j j. jjj j(y 'j' (:j' ktq. k jg j;.y jjyl r; j j yjj (. j (jj.;'. j.(.jttr j ;j' '..iikiEilik;. jtg ' ' j.'. j' . t.(y kj.. . . j. j j.'(!.jj,.' jjjyj .tt.;ktjjyrty (yk yg j(':. .. ...E.. i . .i .)(g (4. jjtji. (.' .y.r-j.. ..jy..irj.i ykq .j j'jgjjyjj j'4k .j:j.yy. 'jjyy)). .tjytj:j !.j.; . i(.q y( j y.!1Iikll'iiIq.(;.'jk '''r. g..... . i.ip yj--. ;;r..y,jj; ;;;..) .. .)j)y.;.;...y.,j,. . .j:(.iy ( yy . jy;g:.(,y..ty j.j.yyyjjq. 2'-. ''Ilpi' '' :i. yjk;;(r7t.E tg ly !jt-. ..., ,y p-. .....r.q....y..)jyk... ..g.,.j t.. q..y.y jr.,)j(yy-k,. !(.y) j jjj.. . j..y ygr .. yj. .j g.. y yr). ; ) j. ))k. jyjjjyq. yg k .yt.. r ( y j-. ..y . . . . . y j yj . j . . y y jy .. , . ). .. . .; . y y;t)(yj:)-. .. . .y j . . . . . tyy. . ) .j k r . . j . . y . rj. . . y.. t y. . r . . . .. . . .. .r..... . ;. y...iljli:pl!,' j . . . ''-jr... . j:33i','. j.. y.g.. y g.. ..( j...... .. . j(.((( y..' ' j..y... y.... ... . ................. . y ... r y.. .. ......... ...j.. .y.. .j....... . . ( . .. . .. j..jj......... .. . . . . ..... ( ( . g...j. ..j . .j .. . ( ( . .' 11 y ((. l g. j j. j... ..... j....... ... j j.. y.j .j.. ... ... ..((.jy ((...l .......!. ( j'..... .'...... )j. (..;, j...g(...(j .k.;(. '. ..y( ..)).E; (. ( .' .(.. .E... . . y. '. .' . '. ( . . .ky.(..;.. . . . . . . . ... j . qL.j.j.j . . . . ... ... . ..... . lE . )11 htj/' ljiif) wout ht R'''''X: 'CV y yotl aciuatt# bpyVitj' jlix...t y yiktt .q l /(!( uk.uyuk.q )()(i s jzy oo xu xtu vk./-j)...q t s u; ( tt o vvo lte suguwcu uv,tw atpu uputgubtyy wzua The quick answer is tht buflng art opuon ls ( jqy kttatk f kkttitutik,lk,itptttqikidtytj.,' like byiiv , stock- tjittjftpftto t, , twfifkt; y'!, E jkro ji pqt t t k ji jst like stock pnkyj. ihzf (: ( . quoted . . .. . . . lltltij tidfl#ljjsj iily b lihi ( jotj sj/ Provides an example. (For it 1 () i kkb; jttjwwwwyrji.zjii i.ij sr exalju, ttvti EE ik) ittil jilitk ftt tik: i Bofl' X'kfltk . shows bid and ask pricej t t Clitj ( ; .. . ). . j j.. .. j . kfpjttimii . titkr. elir ikltw jftk kjtttkt' tjpti . optionstxchae.)Usin f t(.jy' t rl ). j. (..'j(;;. jjjjj. jjjg. (j;jjj (..t jjj;:jygj jyjj. j(j(. jjjt .'jjykjyjjgjgj jjjt jjk. .:j.:(::..;....jjjg 4.i...jy.(.'..i.:i'.ij(j(....'jy...y(..'........g(. .jgjg jjts,o (jjjj... gjjjkty;. j;. tjj;. .jj;: jjtll. jy;j dkizri il::lL lli::,l jj.. kjj .!1rjp.. g. CrEll. jss.sy-yjj..yyjy lL. (I'q1. lE:qlL .y r....... t!i;i tjjyjj.' tyjt 'tjjt. .i;..y.....y... trj. y... y. jy.tj(jgj;y-.-. . y-.... (.....(..........-.t y.. ...-j.j..;j..)j.....j.-..)...-..;...-. y. .y. . (jyyg,. .:gjj;-y..t titjit : g tviihstkkt buy an ytt. .t(yjj(.. t -skf J ( ( ..j. jg ( ( g to (;.kg, (. .( .(.. q... (. '. .'. y '' .;.;y..(.jk.;jj.. .j. .g. tk......k.:;. y....... t. . yy,,gj (jt t0 th optiv Prrxutll, jjy.t #ayp,y ( ( i y) y zzuy ytp mgjyj oj jty..jqijyjyyi jtjy y tqytyy skgjj j f commisjij (j tjus t j j t Ygtt udul r yy yy y y u wxuuztqu,su auao totjjjyyjy yjtj ojjjjy yuu, tjjjjjtfal jjdt y bttlutu'i k.Jpllops On nullmttlus zto uctuy u ( y y yyyjjl yyyjjyyjjj y y jgjy jyy g jjjy y yy j tyyyyjy jyyjxy j 1.g7#).i ktti/ 4tlj exchangesi d fcir ay $k4kitlt tt gq y p ! (.y mon t t k (( ( jij xuuuu i jll jj j j ltyy uujj l jy y ( p tvtil oi J k u t u u ux ouu index, ttiik t xu;j zuyj iauooxuzxuu.ru J tl ut7 v=t y uuuuluu jtk. i tk fitt i tflklabl,g dtfffi 1. tkt tljsgt '' EE
'
AND OPTIONS
FORWARDS
To
: .
'
':
.
.
E
. .
. . '. .. . .
.
.
. ...
:
.
.
. . '.
.. .
.
',.y,j,t
,t.yy..,j
q');)ty')iE
l;)(E
E
E
E
AEmE
: E tkkEEiititE
:
:
E
E
E
:
:
E
EEIEE)
it
E
j' 'll prot ,tr'..
.'
is
)..t
'y !;j ' t. yjj jt If tlle index is 900
)'@FO p) )# j)
35
$1100 $1000j $95.68 $4.32
maxp,
-
=
-
. at expiration, the owner does not exercise the option. It is not rthwhile paying te $1000strike price to buy the index worth $900. Profit is then
.
E)i
$900 $10001 $95.68
maxlo,
-
-
-$95.68,
=
.
Llreflectijlg
%
the loss of the premium.
..
.
..
....
.
1::111.
.
.
.
. .
.
.
.
j.
,,.
.: . . .
. .. .
.. .
. .
(
..
E.g
(..j..
,(:2),.
....
. .
.
..
j.
.
.
.
.j.
.
jj .
.
..:
.j..E.%. ' '
-.
.
:
.
...
...
..
.
.
.
...
.
.
.
.
..
.
:
. .
;.,.;,.
.
(.j...jk..... .
. ;,k....
..
..,g,;.
.,..i
.;..y
....
Eg
...
?fgu
gi..:.... ..
'y
..
E
..
g .:
E
.u
..
.
-
.. !, E
,j
.w
E
:
y
..(
.
.
( q
,
y
iy
,
y
...u
avti.
g
kzlgtauuo
'
'''
. . qiyg E
rin: .
;$r(1.E' .'
.
.
.
-.
.
.
E '
: .. ..
.!,.
'
'
.. . .
.
'
.
.
..
' .
..
. .'
. . .
(E
utg ,
y
y
( .:..y.. (. ...,(.j .y ytEyyjiytyyqty
.y.ky
uuy
.
. .
'.
. .
'. ((. ' . . .. .
E .: . : . .
. .
l ( . .. . '
.
'' ' : . . . .
' .
. .. .
. .
(
.
' :
. .
. . .
.
,yEjq
u ,
wuw
.w.
aa.x
ijjl
.
qyyj y
(
.
'
(.y . '
yy,
y
,((, yy EE
wwvyxxsswxaww..a
uwxuua.u-E
jyy
E iy E
,j
y
.
E
,jy
y, y
.y
..y
r.jjEy
' yE'.EEy
,
E
g
E
'rhe
y,,
'. .
t&optionality''
'
i ' E.(' ' ' ' ' ' ' ' E.' ( E '(' j( ' j.('. ( 'j E!E. . .. . j.. j.g. r.. (. . j ' ' . jj'' . ' . jj (. '. ' '.i. .' . .'.. (. (.y( (j '.. (' ('r . ' (' ' j E ( (.j'.E.. '.' .E.....E.g '.( g'.:q. . . ...LL . .E. ...( . .. . ... i. ;.... . gj j g j .E. (.(..(..1 j g(j.. . ..E. .!.:. . . .Eq .(.. ..'.''.E.j(. ..E...... .(..-. E(. .y .. .'.. .E.'( .: )....j ..... . E. q.:.).( .) (.. .q . q( ... . E . i. .. . .E... ..... .gy.... ( . ( . . y. i . . ..:. . .. ; k. yj. j gjy.j gj jg...(. jj.yy.j.y.j yyj. j y.. .y y. yrj yj.. . . y j g.. .. .. y j j y( . . y.g .. . . .. . .. .. .(y g.. . . r (. .. . k. . .y. y. q . ..j y. y y . . . . . y . . . . . . . .. . . y y .g y .. . j. j. . . . g. .jyg . . . . . . . . . . . .. .. . ... . . . . . y. . y. . g... . .... . . . . . . jy j.j.. j. y ... ..... j. .g. g (.( i' . ' E . . ... (j ' . ...' ( (.j. ' i (j E.j.. ...j.(.E(.r ' (E (gj . .y. j j E.. g. . jg.... j . ' j g.(( ( l ' ( . ' .. . .. . . . .. .. . . ... . . ... . .. . . . . .. . . .. . ....... .. . .. . . E
. E
.
. .
... .. j..jj. y....y lrjrE
(
E
.
.
.
oy
We graph the call payoff by computing, for any index price at expiration, the payoff on the option position as a function of the price. We graph the call proht by subacting from this the future value of the option premium. Table 2.2 computes the payoff and profit at different index values, computed as in Examples 2.5 and 2.6. Note that because the strike plice is tixd, a ltigher market plice at expiration of the S&R index beneits the call buyer. Figure 2.6 graphs the call payoff that is computed in Table 2.2. graph cleady of the option: Below the stlike plice of $1000, the payoff is shows te' zero, wlle it is posiive and increasing above $1000.
.
E...y.y .:y..E.y..........
tjjyjly
.s
y
pr.,
:.
( .:...r.
.y
..
,
.
t......-
..
..ykkgjjj.. .
'
.y
.
.. .
..
.
,yjg.;)
..
,,
y
.jg.. . '. .. ( E'
..
.
'
(. .' '.E ' '. .
E
.
.
' . . . .
y y
:
E
''''''
'
.I.
(
:
-.yjjjgyj......
,jjpjtytljttttygvtj
y
w
)kjj;. Ejj;. .
.. .
(
:
..jjjgjg..
..jjjj;yy, ..yjjjgjjj.j;jjjjg;...-y ..j).. - .y
..jjygjj...
. .E). .E. .....:...rE .... (..i...... j..t.:'... (.r... (.t.....
(.
j . .. ..
.. . .
y..j.....,,jj,.,..
.y
EyEty
u,y yw:
: :
.:
.
..
..
:
g
g
'
4t((jj,' 1:y'(jg;;rt' $1jj1. dkjz:y4:yy(qI()gIj.1L.' qlk)jl, 4((1)): .'' '..' '. )t.jjk 4:4(2)(14. 4(:2;). jrEll. jlyl jlrr)t. jiljjlg.. k:yjjl, 1!4)L. ,g)I'' (1(, iil gyj,hr rrrjj ..'' y . . ...(. .. . .'
..
yj
g i;t
l.J
E
.
..
,..k..
E yy E
E
j j..
yy
(
vz.g
.. ..... . .
g,j.... (j.t..iy''. .,;k.;,;,...,,,k., ( j .k,r, . ;;.. . .:. ...g yttuyjst:tj
,
E.iij
,
.
..
uyut
y Ey :;y E
,
......-au
E,,..k.;.k,
:
.. .
k.....fjjj'jj.
....j.j
....
. .
ttitikiktsj .
g .
. .
.e)(
y . . ... y
...(
,jj;
E
.. . . . . . .r .g .
.
:..(.
y
.jjkt;.
E
..
)'
..;
.
.( . . . .
: .. . . . . .
.
'i..
'.
..
.
.
.
.
..
'
. ...t.'
(
: ' . .
:
.
.
.
:
..'
'
. .
''
'
. .
'.....
.
..
E
.
..
'
...
'
: i.
.
.
.
.
.
.
.,.
: . : . . . .. . .
g. . .. . . .. . . .
:
. . .. . . . . . . . . .
. . . . .
..
.
.
'
.
' . . .
:
'
:
:
.
:
:
:
:
.
.
. . .
.
.
.
j.j..
.
' : ' . .. . ...
.
'
'
.
.
..-.... .. :. .
:
:
:
.
:
. :
: . .. : . . . ..
,
:
j. . y.
.
.. : . : y . . . r . g.
r
:
.
. y.
:'
' '.
.
:
.
.
..
:
: .
'
. :
E. .
.r..... j... j (.
: :. .. . .
.
E' ... ' . .
:. .. .. .. ..
.
.
y
.
.
.
.
::
:
..
:
:
. : . . .
......
.( j.. .g.. .... .. ....... . :
.....
.
:
,
g
:
..
..
:
.
.
:
.. : y. : .
.
r
:y .
. y . . .. .
.
..
.. .. . .
.
.'''' .'..' .'.'
'lllr.'pll'q'll''. '''''''''. '''''''' -.' -' ;')' t':')' '''''''''''. ''i''''''''''''''.''''.'. '''''';;''''''''. '''''': F' 1....''.1'*
iiirjii1l2'. (' .'' f' .'Il'..il,'.:lli'. ;i!ii;)' .y' 'hrl'ljritl'' 1* ?llf ?' .j'i'-'. i)Ii::lk')'' (' )?' qf T' )' ()' 7* tiqf llllfesilplqf.ky.lltfi/ji',,tyjjjyj''l. r' tf (tlIit1Ei'''.'. i1IIq!iii)'. )'!1* ;( !E' ; !5 i:E j)#)1.1!q1 i( 1: t'i:lpqii pqq1( '(i' !:.(!(i.!' i I .'. E IIPIE !
t--....
Purchased call protit
=
strike priceq maxgo, spot price at expiration fumre value of option premium -
-
The following example illustrates the computation
(2.4)
of the proft.
: E :: : itlEkiifiiEi E': E.' ' E: :. iE':.E1: i TE E ' q:i )qj' !i . i ; E E . ... E..(.'E . .' . '. .'yl?)-))jE).jt.))y .ItE.i.. . E:!E((. . . -yy.,(rgj-.-yj.--.g.-...,(..rygt..-j-)..yyry-t.--y-t.;.;.,(E...).--y-yyj( . :,y.(.-?!--y)-.:------..i:-.jjy..yy-),,.;.:t-(:ty.y;y.,y,,-:I,.;,-y...
--.-d,4tiikl)''.
1!11(/;;.-;
.@rEr:!..
-
-
-
.
(pp,y
i,. ;,rk2:rJ,.,-.,,....--.....:..;.
-
.
-
-
)p.;)-(.-..)..2t.-;.-y-.r-.-E.!..#.E.--)-yE--....y)(@..-.-g
Ir..1r..,L''
.1y:.
jl:::,-
d:::,j
..-.-..;....
't::;,.
d::::.
E .. ' ( ( (. .':.E.q . (. . (.qII.j.r.tI2. l.tE.).'. j..( j.'.''g'( j . .(. j . . . . . .j..( .g..E-.(. g(.i.......... gjil;t:i .,1i:;!i;t...E (..E(:.. E' (. (,,,Lj1,1. j l ..'ri.E... (.. j... E' . . ((j. .tllr:ij!-(r(. j:( jy. (( ,;:kg. .. .-....-. k..;/:1It .(.-y........ tj;j.,. ...ks,b-. E...;....;...;L. ,!:,.,. .. tfiv.-'. . .y.... Li'66f,'L. .......... .-jlrqll jll'qjk .(l (E'lr. ,,r:i,.jIk.,krt;,'jj;.:,. k,k. .. El E. E l E'. (1 ' (E@ .. ( . l l E ' ' ' ( ' ' ( ( '. i E . ( ' ' ' ' ' E . . ' . ' '' g .(j (' ' .q . (' ..'( ' (-tl'.(j ( ):: (yy( .IiiEII.'. ( .j.jE. . . (y yq) (-1iti!i,. . ( g. j... . . j j' g j g(j E y(.. ; qy.. ; y. y. ' . . . j j. . . g . . y . . j . . j k . y . . t . . ' dkIti:i,k,. . . j E- . (g((IiII)t)i';.. i. .1ikEi)ii. (111t.:,11::111.)(. E (iII:ik.. :'Iii:. (1111E1E:11.1llliir!lllr-' .'...'.- . ;.try 1IiI:i. .111r .11... .111, '. . -.. i - i. E .. (::tji;;-'' ..'.. . q......' (1iIi!rk' -. ..... @,11:@--.. i. . .. .. i. .. . l(E... . .......si.i.'. i..... !'..' i i E' .E'. E E... ' ( . .... . Ei ..-'..y.. .. E'. . .. ! ..... ( . ... . .... . . ... . . .'.......'... ... .. . . . .... . . '....... . .. .... . .:.(
:. (
E(
Ej
E( E( ( :
(
E
..-...(t.. .' :. .
::
.?k1.
L.L E'.. (.
E
. .
..
(
...y......
EE
:
:(
.,14;k;//..,jj,;/j11.
:
i
-r . .
!: ' .
.-
@ (i.
future value of the call premium is $93.81 x 1.02 = $95.68. If the S&R index price at . expiration is $1100,the owner will exercise the option. Using equation (2.4),the call lii
6It is not important at this point how we compute this price, but if you wish to replicate the option premiums, they are computed using the Black-scholes formula, which we discuss in Chapter l2. Usng the BSCal1 spreadsheet function accompanying this book, the call price is computed as BSCal1(1000, 1000, 0.3, 2 x 1n(1.02), 0.5, 0) = 93.81
(' tlllr:til'lt:ili.. t:iiill:l. t1i::ijik-.
'''''.''''''''''. '
.
.
.
: .. . . ..
.
' .: .. . . .
.
: : . .
.
' . . .
4.1.1..',;11..'.1....
: :: ' ' ..
''''''i''''
''''''b'.,
:
..
' . . .
: . ...' ..'. : . .'' . . . ..
' ' ' .'
:
:' .
' ' . . . . .
'''.'''''''''.
'
' '
.
:
:
''''.
''''.
.
:
'
.
.
:.
:
.. -. .. . :
. . -'
. .
.
.
.
' .' . : ' :. . . . -. '
'
.
:
.
-
.- .
' ' .. : ' : . : . ' ' .: ' . . . .. .
: ' . . . ...' . . : :. . . :' ' ' :. ' .:. : ::'. ' ' . . ' .' . .. . . . . . . . . . . . ..
800 850 900
$0
950 1000
1050 1100 1150 1200
0 0
' .
.. . '
.
... . .
' E
'''''''''.
''''''i''''.
.
.
.
.E
.
. .
:.
.
..
.
:
'
.
.
.
.
. .
.. . ..' .
' ' '. ' . ' ' ' ' . . ' .' ' . . . . . . ..
-$95.68
.
:
.
.
'
.
.: . .
: .
.
: : .... .
. ' .
:
. . . . . .. .
:
. .. . .
.......
'.'...' ...
' ' . : .. . . .. . y. y
'''''''''.
''''''''k''''''..
.
.
: .'L'
.
....
.
Use the same option as in Example 2.5, and suppose that the lisk-free 1)pExample 2.6 tt) rate is 2% over 6 months. Assume that the premium for this call is $93.81.1Hepce, the
dir4:::j1.*
5&R call option with a future value of 1 premium of $95.68. The option premium s assumed to be $93.81 and the effective nterest rate is 2% ovr 6 months. The payof'f is computed using equation (2.3) and the profit using equation (2.4). .ooo-strike
-.
-
'L.' :''' .
. .
-
.-.).;-....-...-.#-.---..-...----..).)#ssk.
: .
j'1
'
rr !' '. . yjyytty' . . ,.q,tq(;1L1L1(,, )lrt').. )jjjt.yjtyj--t)).),)--.ttjt. yjl.tyry.yyyjjjy )l7l@l:77iiI77:-Ir,'' iii!,lki jg:;,ij,lIr..,l lr.. qlqr Ir..'t:::,- a Ig:::,. lr..ll l....l .Iy:jj:: ,(:y:I; 1h))--''''..1k..,. .IL:. ayj,,,, e Ir..dt!;kIr..,r..,p . s .Ig:' a it!i ti..ll ..jr:r:,-i; a.Ig:-1k:( .
acquired. In computing proft at expiration, suppose we defer the premium payment', then by the time of expiration we accrue 6 months' interest on the premium. Th option projit is computed as
:
:
..'.
''''''..'''''
:jj
.. . . -. . .. : : '. . : : . : . ': .. . ... . .. .. . .. . . . . . . .. . ... . .. . . .
: .
..
:
. . . .
-
.
' . . .. .
:.' . ' ' . ' . . ... . . . . ..
E
: .' .
:
.. -
. :. . . . ..
.
-$95.68
-95.68
-95.68
-95.68
0
-95.68
-95.68
0
-95.68
-95.68
50
-95.68
-45.68
100
-95.68
150
-95.68
-95.68
..'
.
''''''''k'r.'''....''... .: . . . .
-95.68
200
:
kjr-r..kr
4.32 54.32 104.32
. .
.
.
.
. -
%
36
AN INTRODUCTION
.'..' . ........
,'-' ..' .tkty'lgy -' ';'''''.-' ;'y.gjjyjf gyjj', (;)'t,;; rr'rrr:'rf '');q' i':'F'
: : y' )' k;ygjjj.f (' ')' :'q' j'$i' Fkf F'((E 1! ' E :' ( ' E: ' ' ' i'' E i ' ' ' ' E' : EE E : : t'y-,f p' y' f' F' ff ''p' (;jjjjj')' jjyy---y-jf ;;i' 1* i@ !;EJE ;!E (i'l!q!ri . !:!('rq (' yjf y' 77:4(7;7* (i''rf 1f' (.(E.(E.-@j(qEl'rrr'pq:' I E(1: ( :'(1E ; E E.i'#.;: ': : . '!(!.'(!. : E!'7EEEEi(iE!.i q''iii'E(. E(((.: ;TE.E.(.''.i..(: .. E.'..q .Eji'.(.!i.qqq .! .: ! lli . E ,j. ' .' . ( - ... 'qq. ()E, y.;( yy y .j. y.j y. j- ''i');EE 5. E q q;yq;.Eq-. y. qqg.yyj)),33jqqjj. y .,.-.jj,,j)jj . y . y yjyy- yg . yy . yy .. . .--j--. y . . (.. yyyy - .. y E:..y. ' E .yyy g-j.j y ; . . .. y yykyy.y-.-.. y. yj l;E.E ..i E y. . , . ; i.. . E . . . . . ..
)r' tjf jjf j't',,L.'j;. 'ty')! r;)f ild ygrj;,;,,f yyjjy::,-f tjf y' r' j'q iilti!E )' ;yy' jyyyf j(' j'. tf'jjjj::::;:j;.. (y' jj'--.. ' jy,yy';..,yj' y' yjf ))' 11.*. iff '.' qr '-t(.( jj;jyjtjjiy'. 't' t'j'j)' yyjyjyt'tf j1jj(kjjjjjj'. . ;( j((j r .. tyjj. yjjjjyjjjrj'. .jL;;, . (yjyy,. ! yj y. . yy ... jyyj j jjjjj!jij jy j j .yj j!. j!r! . );.jj jj;j:kjj. ytyjjj,. jjystyjyjj .yj..y. y.g.gjj.jyjjy.j.yt. . r . '. . l. r . . The payoff at expiration of a purchased S&R call with a 1000 strike
.'
'
'
'
:.
AND OPTIONS
TO FORWARDS
-
.
.
-
-
.
. . .
.:
..
. ..
..
s,j
,
.
''
-j .
'
. .. .. ....... .... .
. ..
.
.
.
-
-
. .
. .
,
- .
.
.
yjjygyrjjjg, jt:yj $:jjj4;,
jg)j-
j,yyyr,,-
200
Purchased call
-1-
1sO
price.
37
If the index rises, the fonvard conact is more proftable than the option because it does not entail paying a premium. lf the index falls suciently, however, the option is more proftable because the most the option buyer loses is the f'uture value of the premium. This difference suggests that we can think of the call option as an insured position in the index. Insurance protects against losses, and the call pption does te same. Canying the analogy a bit further, we can think of the option premium as, in part, re:ecting the cost of that insurance. The fonvard, Which is free, has no such insurance, those on te call. and potentially has losses larger t.11a11 This discussion highlights the important point that there are always trade-offs in folavard contract outperforms the call if the index rises and selecting a position. undemerfonns the call if the index falls sufliciently. When a11contacts re fairly priced, you will not find a contract that has higher profits for a11possible index market prices.
:
.
:
.
.
%.
CALL OpTloNs
100 50 0
'l'he
-50 -100 -150
Payoff and Proft for a Written Call Option
-200 -250
.'' .'..'
,'E' y'. !' :'r' 6* 7* ;')' C' '-'@' y' j','jjjjj-j-jjjj'p.. y' t'p'f (' :F' 1* )' @' ql'ql. ' Ei ' FE ' ' E E ' :' tliE i-t;k)igt)d q' ,';'. ).' p' i'(1* (' 1* (qrjf qlf (!' ;11* ()' )q' '()*14)* t1t''p)t'. pir' qd E E E1: E...'. iE E.:( (l ;')'''?' )1* y' '?tf .y( i ii EEE'j:E. . f'. '.; FE'! @ jy' j'ylyty't)). qLLlqti'n r:7qqt!l!!:l:'':f ... . Eq (kl. EE l (J: 1( iil('''l'E'' (Ei(' 'E(i EEE.EqEi !q' . . 'q' :'jl : .. . --: ,.-j-- giI.qp;.r.' -. . . '-.. .E.......... ' E'.-.'.'; . E' -:;-''- . . jyt(' . .)j; ;.q.r. qE--yqrq . r.. jjjy ;)r. . q y . . . ( . j j . yj j $j. . ..-jj.. E . y.-j.. ...y.. EEyy . E ;:).. i 11111.. : it;! ). )E; E!lq . lE a:--- s.;. p.....EE :.i.). q(li . . .. .. . . . ........ .
-'-i' -.' ''S' ;'s' C'
tgf r' y' y'kjkj'. 1* lif ;'. )19-)-*. 1r' t'tt$'j::;jjj!r;'. yyjj'. r' ( )())
850
800
900
Now ley's look at the opton from the point of view of the seller. The seller is said to be the opdon writer, or to have a shprt position in a call option. The option writer is the counterparty to te option buyer. The writer receives the premium for te option and then has an obligation to sell te underlying security in exchange for the strike price if the option buyer exercises the option. The payoff Mdprotit to a writlen call arejust the opposite of those for a purchased
950 1000 1050 1100 1150 1200 S&R Index Phce ($)
-y'. '.'..' .'
E
'
'
.
'
. tltltjt : jj -.
-
.
'
'
. ..-t-i
.. . .
.-.-. ..
.
'
-
. .
:
'
'
:
'
--
-
!
'
:
'
:
: : ' .
E
-
-y.-
- -.
-
' ..
:
'
.
-
)E . .
.
-
. -.. -.-.. -
.t)!.
.
......... .. .
:
' .. ..
-;
:..:
: : :.
:'..
yyj' - .-(. yy '') .y )t.yy. . yyy i.1. r .. .. -.
.
.
.
'- -. -
-.
.
. -
.
-
-
call:
-
wojjt ($)
-
.
-
-
-
Profit at expiraton for purchase of 6-month S&R ndex call with strike price of $1000 versus profit on long S&R index forward
200 15O
Written call payoff
=
Written call proft
=
-
-
max(0, spot price at expiration
-
maxgo, spot price at expiration
-
1O0 Index price
50
position.
=
--'u'
qn
Profit
strike pricej
+ future value of option premium
$1020
-*95.68
=
lndex Price
100
=
$1000
r'
p Example A.7 Consider a looo-strike call option on the S&R index with 6 months jl. expiration. At the 17me the option is written, the option seller receives the premium @to 1! . tl of $93.81. 1'1 SuPP ose the index in 6 months is $1100. It is worthwhile for the option buyer to )) ) pay the $1000strike price to acquire the index worth $1100.I'hus, te option writer will (t@ . . sell worth the stlike price of $1000. Using the index, equation (2.5), $1100,for ) have to l'1 written call payoff is li) the yty i lt maxp, $1100 $1000) )1.5.) . j.' jtt prernium has enrned 2% interest for 6 monts alzd is now worth $95.68. Prolit for tt. the written call is 6!. ''
Purchased call + Long fonvard
-150
+
-200 -250
800
(2.6)
This exnmple illustrates the option writer's payoffand prost. Just as a call buyer is long in the call, the call seller has a short position in te call.
0 .
strike price)
850
900
950 1000 1050 1100 1150 1200 S&R Index Price ($)
.
p
.
,
-$100.
The last column in Table 2.2 computes the call profit at different index values. Because a purchased call and a forward conlact are both ways to buy the index, it is interesting to contrast the two. 'Fhus, Figure 2.7 plots the profit on both a purchased call and a long forward contract. Note that prot and payoff diagrams for arl option differ by the f'uture value of the premium, whereas for a forward conlact they are the same.
-
=
-
'l'he
i
.
E ?. . (i 1
jj
y
! ..
E
-$
jtlg +
$:5.6g
-$4.gg.
=:
.
'
%.AN
38
INTRODUCTION
To
FORWARDS
)' :'4* j'Tl()' rjf ?L)qjj'. r' 5. !ii' ((' .'?)' ir'if 1* (' r' ;y' pl'k i)plf Irjliljs'i t'jy'. k' jf )' )rf ';'jj?f q' .'(' yjy'. j f:i'.( !i'' ( q ((i iyq( q:?: i;i ! k i (. .';:i ;.(. ... . .I . jij!( (('(E (* 1)* rilll'elq).lhj)Eqf r' 1/1@* ..')'k;;;;;jjrj;'q.'. yyg'sgjjjjjjyijj E 1!'. 3j4E ( E.((.' . lili jy'y' j')yj'. f(;' rf yt)tyt;j.' . .(.qq .y'. jyyj;jy'. . t'y' ()())))))'. i(; i i : . 1!i . . . q . ... .jyy r(( .i. yy,.. .q.q ri yy;jg.!.j.j..yy .:yg jyjjjtjyy. j?#a-)..-t..-iiyik-.-;p.i;.,i-.j.)-.;)
[email protected])t.: j.'/..:r-.r.C-rtht..:..-.-....'i.!. jjjjy!...,: )(yjyjjy. -yyyrjyjj, yj .IiisEl .J) ()(t(:yjj. y.g.y.y.;(y lilb?sLbbqkb?.s6'iL-l yjt;. j qllii-glllikkkr.. . .-..i-G-.-....-'.'..'.....':..(22. qrr i.).-.. .,,--L-. .' -' ';' -'. jgjy)f 'JJY F' j':(';q' y;'yyyy'.,yyjjjyy. ('
.'
E
''
..)lliE7' E
t#-
kqEi ' : ' ':' : ii E. i .. E : E ..:-
!
-...litiiii:
..
.: :.. q
.
.
-.
. .
-
.
E- E E
..-..r..-
..-:-.
'
:' E: E . .: i :
:. :
-.
.
.
E
EE : E(i - . .
E.
''
j( E .E .
.
'!'' ' :.5. . :E
.
E. . ..:. . E....
.
. .
.-yjy
-.
E y
:
. -
-
. E qg . q .
..
(,::d:::), .!r:4:::i;;;):I:ji. lE?;-
-..-.
.( Pr6tit for Writeroii .......
.
. .
.....
... ..,.,..,..:..,..
........
A put must have a premium for the snme reason a call has a premium. The buyer of te put conols exercise', hence the seller of the put will never have a positive payoff at expiration. A premium paid by the put buyer at the time te option is purchased compensates the put seller for this no-win position. It is hnportant to be crystal clear about the use of the tenns and in the above exnmple, because tere is potential for confusion. The buyer of the put owns of r/2cpttt is a a contract giving the right to sell the index at a set price.Thus, the @puyer seller of the indexl Similarly, the seller of the put is obligated to btty the index, should the put buyer decide to sell. Thus, the buyer of the put is potentially a seller of the index, and the seller of the put is potentially a buyer of the index. (1fthinking through these transactions isn't automadc for you now, don't wony It will become second nature as tI'IiIIIC about options.) you cmtinue to Other terminology for aput option is the snme as for a call pption, with the obvious lnparticular, ihe sttikepriceis te ajreed-upon selling becomes change tat price ($1020in Example 2.8), exercising the option means selling the underlying asset in exchange for the sfrike price, and the expiration date is tat on which you must exercise the option or it is valueless. As wit.h call options, th e are European, American, and
E: q:
i
.
..+.
250
d-month S&R call with strike o f 100Uversus profit for short S&R forward.
+
call written frward Short
'buyer''
E
200
150
Prof'it
595.68
=
Index price
100
=
$1000
so 0 -50
Index
#rice
=
10 )()
'
-100
tbuy''
-,150
-200 800
850
900
950 1000 1050 1100 1150 1200 S&R lndex Pdce (5)
1)'The
-
maxp,
$95.68.
%
Put option payoff
Figure 2.8 depicts a graph of the option writer's profit, graphed against a short forward contract. Note that it is the mirror image of te call buyer's prost ilzFigure 2.7.
2.3 PUT OPTIONS
=
maxlo, stlike price
'
'
''
'
Suppose that the seller agrees tovjell the S&R index for $1020 in 6 but is not obligated to do so. lrfheseller has purchased a put option.) tf in 6 ).ttymonths gnd walk away. If ')' months the S&R price is $1100,the seller will not sell for $1020 will kvill sell will and at for $1020 enrn $120 that tlme. l(' the S&R price is $900,the seller
spot plice at expirationl
-
(2.7)
t')' () Example 2.9 Cnsider a put option 1171 and a stnke price of $1000.
on the S&R indx with 6 monthsto expiration '
!1 ll Suppose the index in 6 months is $1100. lt is not worthwhile to sell te index i! i worth $1100for the $1000stlike price. Using equation (2.7),the put payoff is .j j.j
E)-y!!
.
.
'
E '
.(.. '(
llj
@!
maxlo,
(
.tiy ) lf the index were 900 at epiration, lrtj;t lj payoff is then
)) k'
maxp,
$1000 $1100) $0 =.z
-
it is worthwltile .
selling the index for
$1000. The
.
$1000 $900) -
=
$100
%.
..''
,
.
t
ti!
k.
'
The put buyer has a long position in the put. Here is an example.
-'
We introduced a call option by compnring it to a fonvard coneact in which the buyer need not buy the underlying asset if it is worth less than the agreed-to purchase price. walk Perhaps you wondered if there could also be a contract in which the seller ould option is a is put A sell. The answer ys. away if it is not in his or her interest to eample is obligation. Here arl contract where the sellr has the right to sell, but not the worlcs. illustrate how option a put to
r Example 2.8 ).lt
-'
'
'
rf'he
=
option writer keeps the premium, for a protit after 6 months of
'.'
We now see how to compute payoffandprostfor a plzrchased put option. The put option buyr gives the put buyer the right to sell the underlying asset for te stlike price. does this only if the asset is less valuable t-1:a1:the stlike plice. Thus, the payoff on the put op on is
$900 $1000) $0. -
'
Payoff and Proft for a Purchased Put Option
.
)i) ).')
'tsell.
tlseller''
Bermudan put options.
)'@I. tlze index is 900 at expiration, it is not worthwhile for the option buyr to pay the t,.). t)) $1000 strlke plice to buy the index wocth $900. The payof'f is then )'1
11
op'rloxs k.
PuT
AND OPTIONS
As with the call, the payoffdoes not take account of the initial cost of acquing the position. At the time the option is acquired, the put buyer pays the option premium to the put seller; we need to account for this in computing prost. Jfwe borrow the premium
k
40
AN Ilq-rRooucTloN
To
FORWARDS
.jji:)' .'-' .k11j;2j;j;..'jj;::i,j.: jj;:!,;.d qlllr..lll'.rll''.
-' '-Eky' ' ' .Ijj:,.,,kkt,k;'. ;':';.' E '.y' r'ii illj'lf lr:' :111:)* (Ijjjjjy'.I tt;f :'k' f' ,'.
=
maxlo, strike price
'.'
spot price at expiration)
The following example illustrates the computation of prost on the put.
.' r'
Use the same option as in Example 2.9, and suppose thatthe lisk-free ratq is 2% over 6 months. Assume that the premium for this put is $74.20. The f'uture -) ( value of the put premium is $74.20 x 1.02 = $75.68. t'j If the S&R ipdex price at expiration is $1100, the put buyr will not xetcise the )y Clt )q OPtion. Usig etation (2.8),profit is
1j Example 2.19
E E
'
EE fi- i
.'.
'E i' E:' ' FrE ;' ' ! : E ' ':)
E IE:
'
E.ti - q -.
.1r-'
value of option premim - futtlre
j'
EE
j)ipji --t4iik2-.rr'E
lql:E
Purchased put profit
ti!
i( .y jg(
.
g.. reoecng the loss of the frenliup. .. ' expiration, .. the put buyer exercises the put, tt lf the indx is $900at tt t) for $1000. Prot is then ,lrj y(jj 1) maxgo, $1000 $900) $75.68 $24.32 jty ll ' ltt . . 11 (E .(
.
(E
'i
E . :: IE E . .. . ;i. . . . -. E- -. EEE E.EiEEi i - E E ; . yy
i ; E! ' :;. .. .?..: ..
-
. .
.
.t
-.
.
.11
y
...
..E E:. (. .g:( .E (' (((.(. E( E)..iIi(r:jiII;' ; . j (E(.E(()(j! j.j.(((.E.E ik ( E'j( E: ' ' (... j r ((.. E E... ('( j rE ( i . (j..E j j j ( j (. l ( k.. .(. . . i E. ( ...E( EE' i .. ' .. .. j). q gjt'. ( E. ..,,,j.i,,.. (E'.1IIE'.:;,,,,,;k,. ( .jjjjr:iii) . ( ! '( .. ((j ( !( 'L.6LL3i'L,L.. y (.. 'y .j( . jj;,:j,,k. ' klj;;::i!l jikrjkjjj. .. j.. . E ..j...... ...(( ,,*x'E ,Ah'E . . . . (.q;(... g . 1k11:.2 ..qa.'.g . . a, * .u; qzy ...(.k? . q,..E,.y(..E,.E.E. q x. (j,j,:.kj,: .. (. u., .. g .q. .. i EEqi ( E( j (g xf g(EiiItEFiII';E . . ' y E A**.! u, ...E.E)E-i ()( x u, t' .(i1Iiiii,..' .E, y jq Y ( ) r .II!,)iIik(. i. ' E i1Ii:!ik,-tki:E:ikk.' !. ' . E !. :1:EEiijL.' ' i (1111:7k11Ii#:1IIi7r'. .E . i.'. ' '' .iIIi:. E ; -. ' E' ' (:II(!!jii:. E ' !. ' rlllk'.' 1jIit''. .1111. '..(. ; E ' ' 'r E. ' LL:3i;,b, (y.-q.E-.q;..:.k. 'q.. k;. E'... rq..!...; q..... ' (;.'-.. ... r : q .. ( q .. y. y.. .q . r .. . . . y.;. ...q... .. . . .. rq.. .. r.. (.y .r . . . . . . . ... ... . ... . . .. ::
:.
.
..
.
t'
Et
''''.
'' '
' -
. -
.
.
.
..
,
.
,
-
''-
.
r
,'
'
..
y..
..
. .: : . .. ..
:
,.'*.,
,
.
-
.
:
t )'..'..'...j . $200
-
.
'
-.
-
'
..
:: i . . . .. . . . . .. . : . '
: . . .. y ..
. -
.
850 900 950 1000
'
.
-
'
-
..
.
j -.
.
:
.'(.
: :
.
:
,j,::j,jj
''
:
..
:
..
.';
.u,.
.
,
..
. :
,u.
:
.'..jil(jj)ij). .'
.
.. . .. .. g .
'
.
.
..
.
:
'.. E. . . .. . .
.
.
-
-$75.68
'-' . . .. .
. .
-
.. -.. . .
-.
.
: .
t ()111EEF:.1;-. kil:j:llk. . . '''
-r
--
..
,
-
.
-
-.
$124.32
150
-75.68
74.32
100
-75.68
24.32
1150 1200
.
,,
,
,
:
. . y.
.
,
-
50
1050 1100
' index llipg tfke
:
: .. '
.. . ..j114::j)k.1
'
.
.
-.
..
,;,,.jk,,..
:
.
.
$800
=
-
-
-
,.. :...,.rzr= ..........,;..o....w.,s..-
-$75.68
$1000 $11001 $75.68
.
.
.
maxlo,
.
)t:,.
.
.
EE@EEE
'
'
- -. E
y)Ey)
.:
41
jjj;;illifd t'(IrIFIII'iITI:'IIICY:jIIEiSCIIPi7I',IjjjjIJqjjjjjjjqjIjjjj'iI .jjjr' j;;,,k,;'. @' j:' 1* (EY !!' ?' )'. :'(!j' yrf ;!ii1;;' Fi ( ( !'; ! i'))' (1* qk' )' (' f' ;q' ' (47!!* .IiiEjIi'. iIIl)!iik:i1IL)'t1lIi):iI1)'!Iik:iIlp)'. 'tii'j pr:s ?p8! q:!7..i,t:j,.L'. r' .iIrq1i;;'i(E!)! i;lil! ,1J:12 t'(i'll'ijiii' ljjjjjy'-i r' )' rlf i)'. (' j'jjf q' jjyj'.' jy.jjlj--lf (qiklj))i''i''. L;', ');q':.ryjjjj.; (jj,';:'jjjjjjjjj;jtp(. )'' qrf (!Ir!!iik,-ri:iII(()'. (jf )'('y' q.L', . ' .'' 7:. (' !. . :il. (( tfyjlflljj'jyyjy' .. t'q')j))'(11/9:211)y)k1.)*. kr'q #.jy(jtr r:.).) ty( ( (j( . q. j )y(g. jy . j))yj:. g yj)y)j. . j yyyjtt,tjky : ..ELy(( )tt.....(......-.....-..,).k......:-..):.....-j ..i:s. . :7lIti.. ..stkk. . jjjjjy,;-;; . .,,,,,.,.$,..;,.':','(bL.(..,,.,.,.,).L.L,.,.,),(..LL).,,,.,j j j. .y .- 1F:. 1). . d::)l I.., rl.::),l,::), 1,. ..l).. !ilse d::.l 4:2)1 d::), it ahl-te 1..dt!skI..,..d 1#., rrt .. k:.i,---kk1q.t. . n t s 1..d::,pm a I::)h u I..,::r e t!ik!!q;t: s,t:,7.i1-:: put option with a future value of premium of $75.68.
nmount, we must pay 6 months' interest. The option prost is computed as -
op-rloxs k
PuT
AND OPTIONS
-t5.68
-25.68,
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-
-
-
=
reqecting the payment of premium.
y' F' F!'' 'ts;' EE E ! EE i: E qf ft' !' $!92 .iEE.)!Ej.i.i f:. i('E qlq:pp' )@)75)* 1* 14q11)*,t11 )!14r*/* i'i)rq' lisjgg'ttl''tq.)i q' ltpiiq '(q . E(f( iE'EE71(ICIE F' !f p1' qf'lpfilftppiqtllfeliltqt'ylllljtq'..r;slf )' t'j';@' lqrlq' . iq i;q Ei ! )kl rs!r' (pr (r ' rv?r;pi,)#-i..EE'.i!..i!-E-E;TE)E).-.-.r,):.ityi.). '(i EEti . . . . .. .. (.. !:41)7r11 ..'' . . ' . syrtE-t;IjEjEt..).ijt.gEEt.i,.q'.j.y).;)(yt.y-;y-.riy.tjr;:y(r.i.)j.
qjlljyyrf )qt@t7... -'--!(;).,!!;;''!--.--....,-;#:.T'k).,..p,,kt'. 1l:2. ). . .4))( ;.yysyjjjt-j.sjy.j. .).rrrt,t)!l1Fi').-..-.. .11!141,:. -l..,...-E,.G.,-----.'..'..,.'.'.?b.E-y:')-.:-t--..-.'.2.'..'.,'-..-.-. i:s,t-.-#?.:.-:-,.-...:.;.r.t,p...-)
;'r' t'''7*
'
:
.
:71 '
:lE
E :-i!.)y)y;..
-
: !.: ' . .
.'
: .
:
.-
.
.
rqIgj.i()y()yt(FtEyti?.)))y)t(jt..--..
k.
Table 2.3 computes the payoff and prot on a purchased put for a range of index values at expiration. Whereas call profit increases as the value of the underlying asset creases, put proEt increases as the value of the underlying asst decreases. Bcause a put is a way to sell an asset, we can compare it to a short forward position, which is a mandatoly sale. Figure 2.9 graphs prost from the purchased put described in Table 2.3 against the profit on a short fonvard. We can see from the graph that if the S&R index goes down, the short forward, which has no premium, has higher prot than the purchased put. tf the index goes up suciently, the put outperforms the short forward. As with the call, the put is like an insured forward contract. With the put, losses are limited should the index go up. With the short forward, losses are potentially unlimited.
--
,
[email protected]'.Profit purchased
(.... .
-h-.
...s........
.-.............,.,,,;;....
q
.;,
.
yytyt..jy)ty.j-yyy; .
..,.
.
on a S&R index put with strike price of $1000 versus a short S&R index forward.
jtyg;j yjjytjgjtj2r4y; kjgy jjjy;j g;j .jyy
-.s
250 200
y(s
=
Index price
50
=
:1020 k.
0 -50 -100
Profit Index price
=
-$75.68
=
$1000
-150
Payoff and Proft for a Written Put Option
''This price is computed nsing the Black-scholes formula for the pfice of a put: BSPut(1O00, 1000. 0.3, 2 x 1n(1.02), 0.5, 0) 74.20. We will discuss this formula in Chapter 12.
purchased put short forward
150
-200
Now we examine the put from the perspective of the put writer. The put writer is the counterparty to the buyer. Thus, when the contract is written, the put writer receives the
+
.
800
850
900
950 1000 1050 1100 1150 1200 S&R lndex Price ($)
prenziup. At expiration, ifthe put buyer elects to sell the underlying asset, the put writer must buy it. 'I'he payoff and profit for a written put are the opposite of those for the purchased put: Written put payoff
=
=
maxlo, stlike price
-
spot price at expiration)
(2.9)
%
42
AN INTRODUCTION
Written put prost
TO FORWARDS =
-
AN D OPTIONS
maxgo, strike price
spot price at expiration)
-
The RMoneyness'' of
+ future value of option premium
Consider a looo-sttike put option on the S&R index with 6 months Example 2.11 expiration. At the time the option is written, the put writer rceives the premium of to $74.20. Suppose the index in 6 months is $1100.The put buyer will not exercise the put. Thus, the >utwliter keeps the premium, plus 6 monts' intertst, for a payoff of 0 and profit of $75.68. l.f the index is $900in 6 months, the put owner will exercise, selling the index for $1000. Thus, the option writer will have to pay $1000for an index worth $900. Using equation (2.9),the written put payoff is maxlo,
$1000 $900) -
2.4 SUMMARY OF FORWARD
$75.68. Profit for
AND OPTION
Figure 2.10 graphs the prost diagram for a written put. As you would expet, it is the mirror image of the purchased put. -' ,'-,----' -r' -' )' 'k.'. q'. ()l::'d Cliilii.d #' y' )' );' r' y' ('
j'. ;'.-' (iI(E::-,' jy'. !!riI:::i;ik-'-. y'. jjyjd y' ,jj:L)jjj,j'k. .'i1I,iiiq'..' r-' rl'p,l 1IpIl'..' j'y' (' )' ild y;d ;'lyyyy'rd Tqq!tqqd ld t--);;'. ..!iiik:.iir'.. .',''j' -t'jj --' -' --k(:-' -d --...-----:---.------..--rt'-----:t' 'l:::::,-d
''-.' .'.'..'.'t;ki'. .''
EiEEEEEi(((:E'('qjEl('((fCf,((?i$t,'E(: .(E):!E)7E iE EIEIEETIEEFE:EEI: Ey 'f;iijlltfttllptlllpl)i) qTUN:rq('iFC(EE'@lq4(7(.'T .y . E E E-(.!. q . j ... g..(y.yyyy .(.yyyjjy;yy;jjjjjj.. . . jyyjjjjtjjjjjjyj gj y tty -irlhkilrkili-.-. ytg j!.jyyy.y.- yjjyyyyy.y.yj.gy.gy.;yyg.,gk-.'y-.-jj.-.y.jy(g;-ygyjy. ; . k.-.. . . EE ;i jr, ..)... :: r --. ..( -. -. j- .-:....; q -. 'rl... . .. (E
-
.
. -
.
.
-.
...
.
.
.
..
-
E - --
-
....
.......,.w...c .
.
-
-, .
.,j
--
.
.
...
-.......
. - ---
.--
-.
.
.
.
'
-
.
ds:: ;:ii5;;, .11::: (:),-
-
-
Written S&R index put option with strike of .$1000 versus a Iong S&R index forward Contract.
-.
'.
. . (ytyj(g.j.;j .1i1ii:i1i,i.. iliijii!:l ' . yyittyyk,y .)- . . -
-.
POSITIONS
We have now exnmined six different positions: Short and long fomards, and purchased and written calls arld puts. We can categodze these positions in at least two ways. One way is their potential for gain and loss. Table 2.4 summarizes the maximum possible gain and loss at maprity for fonvards and European options. Another way to categorize the positipns is by whether the positiops reprsent buying or selling the underlying asset. Those that represent buying are fundarpntally long wit respect to the underlying asset, while those that represent selling are f'undamentally short with respect to the underlying asset.
%
-7$24,32
=
yrty . . . . ..... qq. y. jjtrtjjj jjjqjtjllilissEl'i SkqLqjjqL ;y y. 8).(jIIEI. y
Option
nn
-$100
=
The premium has earned 2% interest for 6 months, and is now worth the written put is therefore
-$100 + $75.68
43
Options are often desclibed by their degree of moleypess. This term desclibes whether the option payoffwould be positive if the option were exercised immediately. (The tenn is used to describe both American and European options even though European options cannotbe exercised until expiration.) An in-the-money option is one wlzich would have a positive payoff (butnot necessnrily positive profit) if exercised immediately. A call the asset p rice and a put with a strike price greater tan te with a strike price less t.11a11 prfce bpth in-the-money. asset are option is one tat would have a negative payoffif exercised A.n out-of-th-money immediately. A call with a strike price greater than the asset price and a put with a strike price less than the asset price are bot.h put-of-the-money. An at-the-money option is one for which ihe strike price is approximately equal the asset price. to
The put seller has a short position in the put.
-
$
5U M MARY OF FO RWARD AN D OPTI ON POSlTl ONS
-
200 '
.' ..' ':' )' '''-' S' j';';'1111:::4(4;:1;.*. F' CII-'. C' j';' lq' ;fd .111r..,1,..711j*. .411r* .::,,11::..:114*. q--)tf .pll!..IIl..:Ilp'. (' ;')';' /'.111!.* !' )J.'qI''rq?)j)l' (?' !-' lj' ')'5757:::7:4')12* (' (1* y' :j,;jj::Ey:ll.)d j'-'i'.r't',f1*
t''):)rj);'. lllr:rqlll.d 7* )k' y'. p:' )jyr'. r' (:'. (' ;(' )' y' )'. gyf p-.t-yytrr-lq.-,f j't'. (yrr'. ((' (llllldlkp-p-'.:pllrj'd rjllrr:)iid (q1I1((().,k,'. g;l-'. yyyf )' tr'g t'))LLL).)k);'. r' tijlqiljgd yj--y-'!
''ji
' -y' .'s.jkj---k---'y
150
' E j(( i E i yj EEi.EEE fEE @! Eii EEE E ' ((! i j.(j..(i. jljE(' E EE;..'E(EE' (E((E'(( !iq?k E(; ( .E i:!p;liiElsqp ' ' !E..!. EE( (.E(EjlE;;): 1)71;2 (( jg. ( ( (EE. TE (1 . qi ' (' q!Eg( . ).?; )E(:4-- E:E ! E .E (j(. .;! i. i E.;E. i E ; -: - !i !. -. : ;)E!.;:.;).;i. .q(. . i.E!E).. j.y -j;qL.. . ,rtrr.j(.. . - . y t,. ; . yy. j ( q -i..;..((.g..yq.,..,.;..qj)y.,.)..j ,t. y k . ; k.. . . . . . . '
E
' :: :
.
50
.-
' :
E.
--
'' .
' :' :
:
: ' E' : .:
''
. : . . .. . . ..
:
.
-
ztj.
=
51000
Profit
=
$75.68
j
-
Index price
-100
=
51020
.
-l- Wlitten ppt + Lng ftward
-200 800
850
900
950 1000 1050 1100 1150 1200 S&R Index Price ()
-
-
.y.(ry)..t..,t(q..
.-..
j llitr:jlj.
: .
..
:.
.
: .
( ( g 411(::),1- E. . . . . .. . ' : :E:: . . . -
:
:
'
' ' '
j
- . :i . . E .- . . .E . ' .' .' .. .. ' . . . : . : : . .. . . . .
Long call
Short call Long put Short put
.
-
E'
: E. . - . .. . ..
-
.
.
j
. .
(
.. .111!.
(((
g
((
::
:
(
(
-
i
(E
.
'
E EE: . .
Ej - j . E. E. E .' . .
:
.
-
. ..
-
. .
.
.
-
. . .
.
. ' . . .. ... . . .
:
j
E. ... .. .. .. . . E.. . .
E
E .
-
-
j
: E( g: . . . E.. i. . E E E .. . E' .E .. E. . .
.
.
.
.
. . . .
.
.
:
'
.
-
g
:
-
y(; j
::
.
.
-
i
:
: . -- . .-
' ' E: . .. . ..
.
:
j
:
j
' ' ' ' ' : . . . . . . EE . .
.
;. j j . 1'y .' l'('( .' ' . .'
: ( ( E E :E EE . EE i . .. . E; E.' '....' ...' . ' E. ' . .. ' ' '. E
-
.
.
'
' .......'. .. ...'.
Short forward
-150
-t... . -
--
;:!iiiij. i. i j(E.E. E. g ) ! .. ;: .. i :iiiii;:.k ; ! (Iji:jtiil... ::,::1111. .. . .. :iji(;;;. .,111. . .. . . . .. . ;.i'(..'.. ;pq(q ( qlllljy:l,i. .;... (..i.q...j........E.!. (111:. .E.... 1 (. ...;... 1' ( (iI:!iii.,;ir:IIIqq.. rjjlrkrjjj::j;jt, ()11:41j,11:j11). . ... i:jjlll.ljll, .. .;.i(' ' ( (.' (. i ('.q( (.!l (.....!.(. (.... ...lq.. ..(. .. (.' r . ''. ' . . .$11t22,,;' )(' ('. ('.' (' ' '.. . . . . . ...... . . ... ..' . ... . u-Forwrd price Long fonvard Unlimited
rj g . . E . ..
l'..
ii i ' .l ('. ( (
-50
:: E :
.. .. . : . . -j
.
-
......,j),t)4j.
'-
Index price
:
-
.-t.-..4..t-..).,,,...-r.s. .
...
. . .
-x..
.
O
-250
' i:g .
.
jtljj, d::jl 4::j1 ,::?p t.y . tl: 1,.1,.. rj . Ihb-//l ql:i iikIr.)L ... .(.y ;yt. !;k( jkk-ti. d::)l d::), . 1t: 15t!iiI;r:p !,lih(i1..1..1 1..1 Ir.lr.ljr:)p rl:o1-.I,::;)1..1!((j; islIr.)L 1.. ;E,t 1..1 Ir.itl:lj,?Ida 14:), I . tsts. iEsl. ii!!;.....;.!...-t:ji:. l'.tl,li.. )..,,. 11E!1;. t. short forwards and purchased and written calls and puts. Fl/bt#/wnflf?nl denotes the future value of the option premium. -
-
.t).m:,..
100
i'
:
.
. . .
.
. . : :
. . . .. .
: .. . .. .. . . .
E
. . .. . .. ... .. .
: ...: .: . . . . .. . .. . .
: . ...... . ... .. . . : .. .
E.
:
.
:
..
.
. .
.
.
.
.
.
E:
.
:
.
. .
.
. .
.
.
.
:
..
.
Unlimited
Forward price
-lR'V(p?wnIf??l)
Unlimited
Unlimited
17V(p?z?nII?n)
-l7V(.p?z?nfIf?n)
Strike price
Fvtp/wnflfl?l
-
Strike price
-FV(p?-dp/flI?n)
Fvtprdpnflf/nl
.
.
%
44
AN INTRODUCTION
TO FORWARDS
(' ,(-' ,:' ;')' ji)f .'j'lr''i :q( jrf t'LjL3::;k3,'. tjf q)'fq' /')' i;f qqlrq/pli''qjr'plf ))('): 14J*)!;tl '$' q' i. ( ' ('! (qqll: (( i ii''tts' ;'. y-,jjjj;t)' (r' (' Iq)ri. i.Ei'i i' ! ' ( ( CE q; y'r j'r j'. .i q jjf j'jyyjjtyj;'yy)jlf )y' t'yy-.j'. .y' ;', yy--rjijjjjjyj-f ;'k''?''s'r'i.r)' . Eik- i- .;. )'' .. jyyyy'; .ikr ..(; . y!jtj' (')..;. !'))l)!. .)r,;j '(; ,(y ..;.y .lryry,!yyijy..yr;;. '. ' y y . j y r ..,.IIii!i:.. y . ( . j ).....T..-1'I::ii...-........i1t:iIi). . j .. j ) . . .)......tt,t.:Ctt-qq.r.'.ii:!'. y . . .y .j yyy)q . j .L-.L1..-LL----..L--..--L.-..--jjL. .. )..-.t) --.I1i....--..-!q!:;:Ik-.... . .yj.y.j.)---.;:!l:.i .y' -'y -' ,'-'s';'C' ''q' j'
: ' iE'E : ' ilEI ' j.t)yjyy'j----yt-yj'..'jtyyq; y'E t':';'E;' i'';'pI tE i ' :. :' ( 7. (:.(': ;'E'EEE!'.E EEE.:''. E'E'E:''..Eq'EEi ::E'EEEl i ( !E E I( . . . .'. E. ...(. . j.-. . )--.. -- E : ,--- - - !j. ; .:
'
:
'
:
.
.
'-
..
'lhtlitr-.. .
:
.
-. . .
.
.-.......t..--:.rf..---:y------jt--kkr--,--.
.
...
-
.
....
. . . . '..
.. .. .
E .
..
- ......
.
-.
..
l
---t!
..-.
' :
.
' E (. (.iE.E
::
. .. : .. :
':
:
-- -y. - - -.
,' -' .'..' -#' : -,' -' L-.;kd l'jjjjjgjgjj ;';'yjyf :'E' '5*
)' ,'trf g;f i'')q' 5* 't' ytf t'gyq.' !j' g' ,;' Ff 2* ;'/)' !7q7. 7( )' 1* ltf (' p' 17(1* !' ))' (t' j'j' J' Jqllqqrqii'');sf ;' ! (7757'7)!4:777!7. ( (!(( ' ' I' E( ( E E ;i' f;;'' '( : '( (E t'j-y'jjji'yf r' @' ;'r-;jjjjyi q!pf (i;' ((' IEITY t.'t' (' ljj.f ;'jj''. 'tj'''!.(l1)1(lk.;(rlEEq (* yyf p'. yjjyrj-tf )tt'-tt;. (j$' .):; (. . i(q.' .'i' (i' (rr ' . q' : ; 7!7*:77*::;57:717: jy).','rtjqyyjy'. ljjrjyyt'. ')'y' .. . E, . '(' .'('lt'-rf jl)f t'jjijgyjjjjr-tf E..(. )1 ! ,.;y i ;.. . LlLlLl jjy;. jjj ' .y yty. . q jj . y jji-,jji y yyy). . .. j. r y . jy g j y.j.jl .yj( ( .j.y . (j.ky. jr j yyyi -.))(it.t,t( !,(::;j;y........ry r .. ijl. .. ;. .... pi !!!.!llllqiqili (.. . jg;.,-,-,-LjL)j,jjq. t.. ; ... .
.''..'
'j
.
.y! -.y
g .
r-,.-
Elqqji. tl:s41@1;; ::(), p r d:::)-
:..
.
150
100
wriiten put.
-
....
200 ,
:
!'' E E
'
E E(E E
-.
y y, g
Profit diajrams for the three baslc Iong positons: Iong forward purchased call, and
ora-rloxsARE
AN D OPTIONS .
.
. .
. .
.
. , Profit digrams fr .
iEE. -
.
'
.
.
..
. ..
.....
. . -. . . . ... . . ...
..
.. . :
:. . .. .
. . . . . .. . . . .
... . .
gjyy., ..
j
. ..
...y-jjj-jy-jt.
....
.
the
three basic shor't positions: short forward, wrttten call, and purchased put!
50
.
-
j
'
jt:g;jjjjgji. jty :jj;i (jrr jgy,p g;j .jkr
-s
250
+
2O0
'.A-
writtencall purchased put Short folward
15(j 100
'
0
-50
-100
.
-+..Purhased call + Wrttten put Long forward
-150
8O0
850
900
950 1000 1050 110 S&R lnde.x Price ($)
-150 -200
1150 1700
Long Positions The following positions are long in the.sense that there are circumstances in which tey ' represent either a right or an obligation to bkt' the underlying asset:
Long forward: A n o bligation to buy at a ftxed price.
-
-100
-*-
-200
Purchased
--
50
0
-250
-
EE- yE i .. .tyjEj-..yg.y.y...yt...jgj..y.j.jj.(-.,yj.jj;jjr.yjyy.j.qy)yyyy y g
.jjjj;jjrrg..
-
:
.
-
' :
'
.
k.
;E
E TE '' '' E ' E : E' f( E (E: ' ' :' ' EE.E ' EE.' EE:.. E E EE Ei i : ! i' E !. .E ... . E i -. -. i EE i E ;. ! .-. jjjjjjjjjIj,E EEi
'
IhlsuRANcE
800
850
900
950 1000 1050 1100 1150 1200 S&R Ipdex Price ($)
Figtlre 2.12 compares these three positions. Note that the written calli short when the asset price is greater than the strike price, and the purchased put is shol't When the the strike ptice. All frce ofthese Jm-f/tm. benehth'omfalling asset price is less t.11:11 ices. #?
call: The right to buy at a fxed price if it is advantageous to do so.
Written put: An obligaion of the put writer to buy the underlying asset at a fixed plice if it is advantageous to te option buyer to sell at that plice. tReca.ll tat the opion lfydr decides wheter gr not to exercise.) Figure 2.11 compares these three positions. Note tat the purchased call is long putis log when the asassetprice is greaterthan the strikeprice, andtewlitten the when risingprices. lrcd ofthesepositions svike price. l7c?7c/r.#'t???l price is less than AIl te set
2.5 OPTIONS
ARE INSURANCE i
J.n many investment strategies using options, we will see that options serve as insurance against a loss. ln what sense are options the same as insurance? In this section we answer this questipn by cpnsidering homeowner's insurance. You will see that options are literlly insurace, and instlrance is an bption. A homeowner's insurance policy promises tat in the event of damage to your house, te zsurance company will compensate you for at least part of the damage. greater the damage, the more tlle insurance company wi.ll pay. Your insurance policy thus drives its value from the value of your house: It is a derivative. 'l'he
Short Positions The following positions are short i.n the sense that there are circumstances in which they represent either a right or an obligation to sell the underlyixng asset:
Short forward: Arl obligation to sell at a fixed price. Written call: An obligation of the call writer to sell the underlying asst at a lixed price if it is advantageous to the option holder to buy at that price (recallthat the ' option bktyer decides Fhether to exercise). Purchased
.
put:
The right to sell at a fixed price'if it is advantageous to do so.
Homeowner's Insttranc ls a Put Option To demonsh->t how hoineowner's instlrance acts as a put option, suppose that you own a house that costs $200,000to build. To make ttils example as simple as possible, we assume tat physical damage is the only thing tat can affect the market value of the
house.
'
%.AN
46
llqTRoouc'rlolq
To
AND OPTIONS
FORwARDS
op-rlorqsARE
Let's say you buy a $15,000insurance policy to compensate you for damage to the house. Like most policies, this has a deductible, meaning that there is an amount of damage for which you are obligated to pay before the insurance company pays anythipg. Suppose the deductible is $25,000.lf the house suffers $4000damage from q stonp, you pay for a11repairs yourself. lf the house suffers $45,000in damage from a storw, you pay $25,000 and the insurance company pays the remaining $20,000.Once damage occurs beyond the amount of the deductible, the insurance company pays for all furtherdnmage, up to $175,000. (Why $175,0002Because the house can be rebuilt for $700,000,mzd you pay $25,000of that-the deductible-yourself.) Let's graph the prost to you for tllis insurance policy. Put on the vertical axis profit the on the insurance policy-the payoff less the insurance premium-and on te horizontal axis, the value of the house. .1f the house is undamaged (the house vtlue is $200,000) the payoff is zero, and prost is the loss from the unused insurance premium, $15,000. lf the house suffers $50,000damage, the insurance payoff is $50,000less te $25,000 deductibl, or $25,000. The profit is $25.000 $15,000 = $10,000. tf the house is completely destroyed, the policy pays $175,000,and your proft is $160,000. Figure 2.13 graphs the proft on the insurance policy. Remarkably, the insurance policy in Figure 2.13 has the same shape as the put option in Figure 2.9. An S&R put is insurance against a fall in the price of the S&R index, just as homeowner's insurance insures against a fall in the price of the house. h'surance ctwnptwzd. are f?)the bttsiness of I$?r9?l.g put optiolls! The $15,000insurance premium is like the premium of a put, and the $175,000level at which insurance begins to male payments is like the strike price on a put.
INSURANCE
%.
47
'l'he idea that a put option is insurance also helps us u'nderstand what makes a put option cheap or expensive. Two important factors are the riskiness of the underlying of te deductible. Just as with insurance, options will be m6re asset and te Jzmount expensive when the upderlying asset is rislier. Also, the option, lile insurance, will be less expensive as te (ie du ctible gets larger (for the put option, this means lowering the sike price). You have probably recognized that there are some practical differences between a financi put option and homeowner's insurance. One important difference is that the S&R put pays off no matter why the index price declines. Homeowner's insurance, on ln the other hand, pays off only if the house declines in value for specified resons. decline real is covered typicl in by homeowner's particular, a simple estate prices not instlrance policies. We avoided this omplication by assuming at te outset that only dnmage could affect te value of the houe.
-
qjtf :'t' yf '().:r.f (' )' )'. rqltFif yqirllppqqllll'tF yyf )' r' 1* :767,*,))( .t' y'j-kjjjjj' (' ;;-'. j' f' (f (5* ''' y' ip':qlf 1* !' (rf ;(f'. i1IIk-1-iiIE'. ),' lI,IFi'''. !.i1I:ii:;;k:-' j'jjf :!iiE;;' .-' y' iiIIE'. t't'. yyf ti.f ;'jjy'tyjjf ilii5!E:..' )' tk-p'. 7;! E 7* .t' y' ili ' ''i'.(.' L:iL'3tL'. ;' ((. ( (( qE(''' ''li .tyj' .')4t:71* .('(i''' l.' ' r'i .'(.(.( .q((' EI .. ; ; . . ;. . ( . . ..y ( '; i .g(yg.y.,.q.y..)q,. .yy y. y. '. ..j..j(yy y jy y. t-.'y. )jy , yyy '' .j)..' . . . . y y y j j .. . j . y y y . . E 11iIiipIlii.. (: j ...yy j.i.E.Eyj..j . .. j . yy 5CE ---L-.' .-. l .()... i'. ..... ;-.j(-. :.. .. kk (t....t-(-.. . tk.. .t-i. .. . .. .... . ... . . 2i. r. .',''
-.' -' ----' --' F' .-' ytyf j't''Etf 'l:::::,-f -' '(F)7'r7'77')' ('
'gjggjj;y;j'.'.'..' .. ..
..#' t---'. : l' . E
-
..
)E
(E : E:
.
. .
' '' ' ' ' '' E 'EE'' ' ' EE' ' :' ::: ' : ' pi:': ' ' ' . :' ::E ' i : E:( .. ... . .. . .! . . :.' . E..' E
- T'rqi!(1().iiCE tjEt(()). jk!lI!I'E (1h!:!!)rtE.it 'EE. -
.
.
-
:.
-
'
.
-
-- ,...
--
.
..
.
-' ' . -. .....
-
.
.
-
...-
..
-..
-
y.. gy . . .---();.
'.
.
E''rE
E::: :' .
.
.
'
)j ..
--
..
E'E''' E IE -. ! . -. .y, . . . y- -. . .
-
E yyjyyj. y-ysy.j yy
-
.
. .
.
--..
-
-g
-
.
.
-
.
.
-. - -. ..
.
.
. . . .
11:: :1iqt!ii. .11::: (EI(E::,(ql,-
Profit from insurance
policy on a house.
$200,000
200,000
$160,000
. 150,000
Gain on insurance due to damage
100,000
.
.
.
lf we accept that insurance and put options are the same thing, how do we reconcile this with the common idea that buying insurance is prudent and buying put options is lisly?
The risk of a derivadve or any other asset or scurity can only be evaluated in the context. Figure 2.13 depicts the tisk of an insurapce contract without ctpnfcrnr This would be like owning insurance on your neighbor's house.. risk ofthe nwretfa.zf. because you would buy the insurance policy, and you would lose It would be entire investment if there were no insurance claim.B we do not normally thirlk of your Iike this, but it itluseates the point that insurance policy is put option a on an insurance the insured asset. In the snme way, Figure 2.9 depicts te risk of a put option without considering the risk of any other positions an investor might be holding. In contrast to homeowner's instlrance, many investors do ownput opions without owning the undertying asset. 'I'his is why opdons have a reputaion for being risky while homeowner's insurance does not. With stock optiops it is possible to own the insurance without the asset. Of course, many investors who own putoptions also own the stock. Forthese investors, theriskis like that of insurance, which we normally think of as risk-reducing rather than lisk-increqsing. T'risly''
Call Options Al.e Also lnsurance Loss of $15,000 premium if hous is uhdamaged
50,000
y
$)7,5 (;ctj Deducdble
1s#O0O - --
But l Thought Insurance Is Prudent and Put Options Are ltislty
-
-
-
-
-
-
-
50,000
-
-
-
-
--
---
-
-
--
-
C:.II options can also be insurance. Whereas a put option is inslzrance for an asset we already own, a call option is insurance for an asset we plan to own in the f'uture. Put dilerently, a put op tion is insurance for a long position while a call option is insurance for a short position.
-
100,000 150,009 200,000 House Price (5)
250,000
8Of course, in real life no insurance company will sell you insurance on your neighbor's house. The reson is thatyou will thenbe temptedto causedamagein orderto makeyourpolicy valuable. Insurance hazard.'' companies call this ''moral
i
k. AN
48
EXAM PLE: EQU ITY-LI N KED CDS
AND OPTIONS
TO FORWARDS
INTRODUCTION
''''''
* Is the CD fairly priced? * How can we decompose the product in terms f options and bonds? @How does the issuing bnnlc hedge the zisk associated with issuing the product?
''''''
.'
CDS
EQUITY-LINKED
@How does the issuing bnnk make a profh?
Although options and forwards are important in and of themselves, they are also comFor used as buildia blocks in the construction of new fnancial instrriij. mo-nly otfr piodts tht alloWivestors cpmpanies westmt example,bnnks and. insuranc ' . . ' . . prvide if the rnarket and stock ris beest idex i a guarazired retr tat from a a to rkd iikj t tools notd cDs can rever e-en gineer'' such equitplinkd '''''''''
.
.
.
'
:
.
.
.
.
we '' declines. developed.ms .:;jjgjjj.jjjyjjjj(.'. (:jjjjkj,tjjj.yyjj(j'. tjgyjjjjj.'. ggjjgjj,k(jjjjgjj,'.' kggjjjjj('. kjjj(jyjjjjgjgjj('. tj)f t't'. r' t't'y t'.y'. $jjj;g,jjjj':,' gjkjrrykkkj'. trf jjijjjjj)'- we 'q' tkjjyjkj.gjjjktjjj(,'. t'.,,fq.'. t.j;yj,:jjk(j'.. ljltd tl:ij:j))'. ti:gr)g'. $yjjjjg,j'. tjjyyjjj;'. j:jfq;jjLjqjjj'. tgjjyjr:;d yjjjyyjkj.'. i1!11l;7)';'.ty.yt'ryf tf .;' tjq'jjf .(jjjj.'. tii@::i:;'. yjjjjy-' .g,jj;(': ';jjjjjj(j.' tjkjjjgj;'. r' .;yjjjy;'tlr:l,d tyyjgjy'y tgjjj.'. t;jjjjjj,j'. ty)','
.''.
' .
. ....
far.9
have E,'. t'.. ,';'. y'y j'r'. .11...,j...11:,,....*. ..1i...jIt.'. yjjj.y,j;,,k,,;-'. (' ;;;' ;iitiji,.'. ;;,' .yyjgjgjgggggg'. jjjj,kjjyj'. jjyyyjyd kjygjyjjjj,jjjjj'. t'(' yjgjgjjjj'j l4:iir:).'. -'::;'Ik...Ik..''.';jjjyjjj,'y-,.jd Ii:::::;'gjjjyyjj(jygjjj'. ''i.'i,'jjfj,)f;,'. 111151!!,*. ,'j,!;yyj;)'. .,,,:--' ;'. y' ;;,' jjjjjjgjj'. y'. ii'ii' pit' t'. ,'t'. ;'!' E,' q'q' F'. ,yyr'. ('t'@' ''@-' ijf 'q' :')(',jjjrj,jy. j'kjq';f;f E:rf k;,d #' kI:jj::j;'. ;1:(j1j.4j1:.j*. 111jjj(11j4.:j;(*. jkjjrrr;'. y'; j-'y jyd yjjjgjd :1!!1ij.' (k,' :jjjyjj,'. ijjjg;;d g' y' k;;d :1(1131k)241)1-2*gygjjyr'. j'y' 41::(yy(4r:,1,k;,,#:41((r*. .yjy.,,.;jy.y.yyy'. i1t2g:'. ;')' $:jgyjjjy;jjjj;;'. y' ykjjyjjyjjtjjjjj,'. 'lI!:)(IiEr:)IL.'..,,),.'. ijjgjjjyj'q 'I.--'kj;jgyjI;'. F'j $jgjjjjj,'. 'bkil3'jjq,j'. j'j' i,'-'. jj,;'( j-;d y' yyd jj;(jjj;;'. j'gjjjyjjjj'. yjjrjjyrjj'.' jjjj,.gjjjjjy;..' L?q;'; ,'t-?d :(k(' ;'1-* p' y'. g' t'))' (' s' ij'. .r':', ,q' )'. k,,' ,y'jj:::)'-;-)' C' ;)' ,q;' ?' ?' :')',jjj,-jj:'t' )'. y' t'g y'. y'y t'jtd .(!II(!)iI:E:iI)'. --.*.11:2:111::.7(11L.*. 'i!!Ei;;' '.)i:,jiI)' ;I!Ii.'. (' (Izizrr:'y .I:EZ:::'. r:I;EqEpIL.'jjjggkkjtjjyy'. L,,,.,,L'L yyd ))Ii)7'' gjjyjyyj(.'gyjj,'k;;d gjjjjygkjj'. yjjytyjj(t'. gjyjj(yjjjj.'. jyjjj;r'. E;i5iIii'. ):iIj:r'. jyjjg;;'. tjIi#t:IL)'. kik!ii!:)'. j'. jyd k:!iijj;:;'. rjIjkirjjIqr.i1iy.qil.)'. g;jjyyd (pIp.' :lEtijl)jiil::'.d gjjj.d .jjgjyjjjy,yykjy'. j,k,'. (gjjy'. ytyjjygjyy'. :-;' syjjjykyyjjy.'y .yyjggyyjkgj(y;j.'y yyd jLj))'. rjrqjryrr-j;yjgyj,jd t'y' I(:':11.'' (lil:t:li.d q?lLyd jj2jgjj;j'. yjjjjy-' lllEE:ll'' (' EiI1iE1iii'. 'iI(E' 'IEii:l''y .yjjyytjjjjyjjjy'. .j,j'k'. .jjy2gjr:jj;jryr'. jjj:lkjjjy.d (jj(grq,jrrjji(.'. )' ;'yjjjy'. y' kjrjjjj;r'. jjgjjjjjy'. gyjjjgjjjj(j:,;'k-;d ijgjykkjs'y gjjjty;d (' j'jjjyj,,'. 1* r' jjjjjjrjj'. @#!t?t)'-t.-------.i7!!!!'@l-'-;tj(trjqtqljj.(q'1111i'!1iI'' ;jjl(l)'g' p' ('. y' jy'r );'j)q)j' jy,yqpy'. r' '('q'y q)' ,'r'. y'. (' )'.. )r.-.kkk;;' y)f ;',j,jj:rjjjjy)' );*,111:::::,,,4,*';?,' ;j!!jj;kg'i ;;'..).. )' kjijjj!j;'. j'j'r'j'y'';'y'y't' y'y .:k,' jy,d (' jyjjj;;d rjjjyd ..jjjjyjgyjj,;rg;'. '(jI(.'. ..jkjyyjjj'. )'yjjjyjjjjj'. @' j't:ijI1gjjiI(j-)'. sjjjjyyjggjjjgyk'y ('j'yy'y y' .,jjjj).'. jj:rjjj;.'. .;jjyjj.jjj,jjjy'. .jj(,jgjjy'jygjjj$'. y.rqgqq:ryd ..'; jjygjj;d gjjyd tjjjsjtjjjjyjjts'jk;d q'. !' qq'. iglllq:llilk' lllt;rd )'' y' yt'j'.' j'jd ;;'. 1*. rtd ))' i'(rd p)' t'. q' 1).*2-. qyt'y p)'. )))?'. k)f .y' (;t'. ;:'yy;yj', t'y'r' k' p'. t'jjjjj::jjjj'. f' jptji-.jjyrf )d tj4t)',,jjjj2j,j.: jjj'i y'. y' j'. t';IE(EId,..1I:...II..'. jjjjgjjyj.d j'y'j'j'g jjyjjjj'y (gjjj,y'.-j,;,;,k-' 3-;6-*. gjg;gjjj;y'. :-g'. y' .jjjjjjjyjjjjyjj'. jjyjjgyjj'.' jjjyjyk'yd .J1j..(jrr.rjl.'. .;gyygyy'. j'jjjgjjjjj.'. '(,;' .k;jyjjj,,'y '1It:i'. riiii'd 1l2iE!S;' j'gj:j;t;yj;'. jjfjj)),,)'.,6bLL.2''.,jj,,,,j. j:!lyd jj(jj'. rj!jyd jrjjjj-'y rjlt!,d jjjjykjkj,'' jjjyyjj,,';jjgjd j''jygjgjk,.'. jj(yyjjjggy:y'. .jjjjy': jjjjjjjy:k?;d y' (,j'. p' ll'i rq'!td jq'yr' k)d )' jkb'.. jjf )'. 1*. 'r' t'll'.'! j('. ty'yy j)f y' ;')'-y. );'. .j' (' jj'. ('. j'yj-' qq'. .t' j'y'yd r'. t'. j'q jy'jrytf ),' (yr$' j'k)' (ryjj,;r,.d 'iliE!i1,.q!I(..' ..j,t,.t,;g.k;' jyjks.jyjyjtjjjj.jj,d r;jjy'yr:r:qgqt-g,-y. ((11:.*. (::!j,;#r.' jjjjjgyjyd jyjyj,gjyjj,yyj.'. (' .gjj,.'. jijjjjkj'y 'b::1,.)Ljjjj,L.'. )' .yjjj.',;,' yjjjljr:jyd jgjt,j;,gd (211..* jjjgjjjd (jjjtd jjj,jjj'j y'( j'. ell7dprdehqkl'-. 1* ll'i (j'jyf ' )p' jjj)ryyyjjt))'. .(rtj'. #' ))1;*. )pq'-k-')):.(;l-'. 1*. @'. q'. jt'. y'y:.! )' q)' rj)')'.(jj't-'r yyr'. yj'.. i'j'. )'. yy)))'. )#'y)yjr. 1*)* t'(y'.;)'. lj-yf 9*. ).rd y'. .y' ).-y'. lyyl-jf y'j y''biil'iv'.Ltf'fbbbi-vL'. jgjj.jgjjyjjyjjtd .jjjjj..'. likjji!;d jjjyjykj,'. (jjjyjjjjj'. jjjjjygjjy'y tjjjyyk'j t'jjjjjjg'. yjjjyjjjj.'. '.'' ..,jr....rjg'. kij:jjjjj.d tjjjr,:d kjyyj:kj'. jjjjtjgyyj'. (jj(gyj;'' tjrggjjj(('. (jjtgjyj'. jj(j,jjjjjjykj.'. jjkjgjrj'. t'y'-' '))jy'. ))' td qr#((-'. y'r)' I'C.. yr)'.)ir'. ytjyf t'. .t'(' )'tyf ll'r y' ty'. )('. ((yj)' .tj' '(' jgjjjyg;jj'jjjjjjyj;d (jjryjjjjyjyjjj,'. .gjjjyj;'. jjjtjd jgjjjjggyj'. yjjjggjgjggjjy'. jyjyjj;'. ..'y gkgggkjjjjjjyj-yjjjyj'. ...:''-':;.,'.'jLjjj).. (jjjyyjjkj'. j:gjyjkyjjjgyjjjjjjjj'.
.
.
.
.
.
.
.
...
,;,j.j(j,'. ,;;' ..r'. ,'',!k,,,,,,'. ,j(.jj'. ,jgjj'. ,jr'qr'q''q.,j,,kkjykjj-,;,jyj'. ,;;',jjjgjyjjjjygjj'. ..,j1k...':;j1i.kjjljr;.. .j'.;,' ,jjyjjyjjjjtyjj.'i ,;,' -;,'. ,yjjjj'y .-'....--'jj:gjjjjj.jy ..1!.*.. ,11!(@72!!*. ..3,,qq3k:,qb' .,'..'':..'':bbb-..',:,),:.'; ..' ,;,' .;' .,,;;,,'( .,i)': .-,ljl,lj' .k;' .',;;' ''.'.y' '.'-'-'y '-yj' -' -.jijjjjj,'' y'1jijggj:;' -y' -'y' -li::!l,.d ''-vv..''',-'. gyd ,,j),'. r'y'y ,1...,.,,* 11.,.,,,,*. 11...,,1:.*: -kjjyjjj;'. -' -;,,j:j,*. :'sjjjykjyi,ij-' j,;.j,,,;'. ''';(jjt()''i'. ,jjijjj;g' .,..j,,'. r't'jj;jjjyj;'. -....11..--.::,-*j,gjj44,kj4jj:,,4j4yj*. 1':k.i';.'gyjjjjyyk1.-........,4j2jgj,,*. .-'I....',k...',jjj,yyjj.'. 'IttEt:::'E .-2,.-2*. -yjjgjjjjj'. y'E jj;yjjjj;'. -:--' ,jjyyy'. r' ;jjjjyjjjj'' ,,.' -'''-' ''y -y' -.' -'( -.);-)'j). -tjj;jyj,,,'-kkjjij:j'. -' kkjjil'. y'. y'yd yyd ,,' ;'y 11:2::::*. .y:*: .y'. y'. .y' $,g* j,,jj:'. i'ltd;i''d 11.,..11..J,:.* ;'qtd E..jjj,,,'. ;,,j(jk;jjjjj,,;,,k.'. jjgyyjy'j t12)I;.'' .k'E: jjjgyjjjy'y .;k,,..y::'. y' ,':',jjyyyyjjjkjjkj'. .jjjjjjyjj,'y Erk:,ki::)d .I::.I,..;k,j:jj,.'. .''...''k'. '1-'....''yjjyyj;jjj'. jrj::k,E-k::,;'. -;jj;,I'E ,,,,t;,;*. y' ,'y-ks-'k,;k;,.ky'y',jIyjgrI;'E ,jjyry'E jjjyjjrj'. ;jjjkyjjjj.gyjj;.'. jjjjjjrjy'-. ,'y' ':kjjjjy' .yjjgjgjkj'. jjjrjjjjjkgyj'. E' Elllil'l'ltlq7l''y --8jt1!!!!1-5-*. j-' -tjjjr'.' '--' ,'-':'. -;' ,d ':'-'''','-'y E-' F'Eyf !' j'jjjjrjijjj4ji'(' -pjjjjtjjjjj'q' y' ;'igf -jr'-f' 5*
49
A simple 5 1/2 year CD with a retllrn llnked to the S&P 500 might have the following stnlcture: At maturity, the CD is guaranteed to repay the invested amount, plus 70% of the simple appreciation in the S&P 500 over that time.lo We can ask several quesions about the CD:
Return to the earlier example of the S&R index. Suppose that the current price of S&R index is $1000 and that we plan to buy the index in the future. If we buy arl the S&R call option wit a strike price of $1000, this gives us the zight to buy S&R for a By buying a call, we have bought insurance against an mnximum cost of $1000/share. prie. the in increase
2.6 EXAMPLE:
Q
'
.
...
..
.
.
. .
To understand this product, suppose the S&pirldex is 1300 initially nd arl investor invests $10,000.Ifthe index is below 1300 after 5.5 years, the CD remrns to the investor the oliginal $10,000investment. If the index is above 1300 after 5.5 years, the investor receives $10,000plus 70% of the percentage gain on the index. For example, if the index is 2200, the westor rceives
.'.'$kgjjjj ''.'. .'.'j' . . '''g .'.' .'jjjjyyk'. .' . .. . '.' .''..'.''-.'. .'.
:
.
.
.
.
.
.
.
.
y
.
-
Ej E .. .j - .y : E. ;;; ; E E-jy. .E -. Ej E -E E rE j;.jjjjj,,j,-(i !. .. ,)! i. -. : - ;! iy- Ey ii E E :t!qkr : - jE E. !. . i ;.. - j,jq. )( - - .. -.E.y (q , E qk !E -j..t-- - i E:!:y:. .j .Ejy. . .;.E.. j. . E-ly g t).j.;;. ;.q--.E.---y.y.ht)' .y t-. -!., ;.j-. E- E ? . ., . jjjjjjjjjjj (-rj .i);tjrE gg!.;.j--;..!-.y.g-j;.:j- E:.-Lj;-LL;(--LbL(L;LLLL;L3L;. .y-jj.j!y.jyj.. yy))jk j.jtqj. !j(. .. -(ytr. y j( ...y.jj(.-jk.yg). E;..)r. jyj. yyj y..k;kq .t(. .i; jji .ljtry(k(4r!gq. .. ji..rE.j E.yit..y..E.. g.yj(y...)..ryjy.jy pjyj .l I . j5.. !)-i;.-:):;:.ri. t.! '. q!j)'qqEy.;( .kyq y.pllytyrl-tj.. ...;-r..;jyyqyy yj.;.yk). jj,E ;E...i..E-..y.g.Ejy yj.r..yy. .-.Ekk)tj qy E i'. t)yj .'..l-i'!.i yE.:!.!.y.--jq..( .EE.q:i' ....kyj!(!.p. 'E gj.. . . i. . kjjjj:jjjjjj. . . : . .y)-- -. i.i )).)q)((. y. ii:.;. (.iq!qijk! Eyk E :!.q .(.. E !-jy ;-q.(; E. ii- .). l :. . ji : i...- Ei .E:i.;.r LLLLLL3t:,. !y q #-. f(jr.. . -.:; l i ;q . i -E t -. (;. E . ; ;. li- ii)p:! ))t!. tE.-i. i E. ! --- ; i!j.y -.-. .1);2111j1:::91-)). !!!!'-. : :.y .. .y 'jE l EE. i ' . t.. E i ;. @-)ll-t: y #(i. ,r E. E. : E . . . . '. j . -y ( . ! ( :'.t'.i !j.y.i) !-y... !.t.j- 22q. .y .i .. - j( .q !.E..y . .E E. E. y.. .... . ;.k .( . . ! .. E y .( ql' )-'' q. .E ... '. ( . .. ) .j, r . ,. . r q.. i t'l.))p y r'E j : .( - .. tqti !!t.y r y .. t !.(. E E . j( . . y E EE-E r i E.E. !: .jq . ' -.. q qr. ( y . . ! . .. . q . . .. y ( ',!. ! r. . . . .y . . ( . . . . . E . r .j j E i . . .. . y . ' E E . . .. . r . . ;. '' Eq . ' i .. . . . E . ; . . i yy. .. .r)gttfq (y;y. kEqq .EE !..q.E..... q.(.-....r-.y...q... ..E. ..q...... (E.(-. E. ..qE...r.;...E.. . q r ;-.(y. ..(y-.t--yi..i.E yj!.- -. y( y...y;.j..gy.. r....E ... ..i--EE .t:ry. .:. E. - E.E i. . -. ..EtE. -..E.j....;..(E(;...-..;.-.!....qi..!-.-.EE..(.E;-...(y.....Eq.jyE..@ : j- yj.j.q .-y....'.('. E .E..E..'.-'E.'..E!E E';.!q;.'.:q.E.'.Ei.'.-;...E.E. ';-....--y.E;...rq.E -. ;i,-(.y..@. Er E:l.-: -i..:' '@ ))r(E.tti-. !.:..qE.- ..E...(i)E'. qr :''.E;.-E. E''@- 'E.E.........:.E ..... ...E;.....yE.-;. j E. yy . .' r . . E' r y.;.... i r . . (i..r.E.. r. y. i. . ...; j.( ..E.;.(E,.r........).;.. ; E... .. ;. ;.E.E'E.(.. !EEkyy yj.jy. yyy.-...-. , .- g.. . ..y ..!. '-rEj........y....;. .'E(.y...'-.;E!qq....r@ .. .E.. '. .. ry jqr;...y..., ..y..y .E.gy.. y..y.....yy.y.E.... (g..(-. .qy+p'ikt'')!.'i.''E!lr.'El:qE.'!(kC;'''),.!,!''tEE' ' .( . g jy '. ); .. . ;i . .j E l . ;r yjE.. 'E E..'.rE' . ' '.. ' .. . . y.E.. . .. . yr .( y.q E. y . i. i .r y... . .. .. .y:E..j.y.. . ,y.(.y......y..y..j.. y.y..j-.. y .. ..... ; . y ..y E'. .). .. . EE' t.;- l. . .E. r-. q . ( lkilitsEitikj . . ; ( r E. '. . . y.(....'...(y.'. r' .('. g . . y . E . . ( r . . y . .. . .. . . . . . . ( ' ., ( ; ( k . j .. ; . ( ' . . . ( . . ! ! ; . . yiij, .. '.' . . r . . .r.. .. .. r. .. ' . . .g'-r..'. i8Et'';j'!;''iy.y,,,'q'yyy . .( . . i.. E.. . . . . qq.. .. . r . j . . . .... .. r. ..... j. . k )..; ' . (lthtgjjljjjttikkf . . . ; . y . ... (. ... .r. ... ... . .. . rqEsr EirEu. .tu gygryyyyjyyq . .t( g yr ;(E..x.....Eq..o ;t. ; :..k...0ir .k!E k Et...........k.. ijy t ytq j .y i,t ruk.j.j,,.t . o.,,-r , .,(o E (('(lirqg-r ; rt . t q wsru,u sq.kj..k. o sjututkyku. E zzuu u u; tu- 5 y,iyyt'y jgz....ul E' yt taayj tyjoy;u oy wy yyyjk wjujj yuu. wwzjyx ojy,y,,kykyy i' C z tnrn.0 qsnyw. t l'u nu ol uaj pyz) E ymjjcj( n,tyj kpa....y. ikld tpaz , zm t y ,j ( .4 s.L.rEt j, yyr .t,( me t y E jyy ty y . )!.(y , y r y y q j q q,.y g ()Et,lkjtfiii) ; , y yy.y t.u.o gt r st ) o lrthe o yvo kzk ..r y y u y wy zuu wjy*m...E@ y tltllyyyyEy .y yyy jy qE.;y ldex declneskE Ejjjjygjjj ettt,r'tty y,y.g q ( .( jj;jgjjjj, ayyta, ij.yjk.jyyk. EiaqiEot ; , y E j q. j jt . (jjjjjjj(k(ggj(.. j;jjjjjjjj.. E q. y jjjjyjjjjj yy ktjtjtt ;, gjjjyyy;. j y .jr. ; j g . . yj gjjjttjjjjygjjjj. j y y j j gjkgggjjj.jjj(kj(jgy,. y y . j y r y j . j ( . j y . j jgtjjjyjul j j j kjyjgjrjj . j y j.y g. . . . ...kjgjgjjjj. jjjjjj,. . .g. .j .. . yj.. y.j jj.yg. . (jjj(jjyjgjjjjj. . .. . .. jjjyyjj kjjjjyj;j .jjj.. ..y.. . . ... j.jjjgj...jjjgj(jjyyjjjg,. g .j,;.j.... .EEk.ji..t.g j . y.jg..... jjyy j . .. j E E ;jjjjjjktjjjjyj,. . .. . E yj . yj g yg.. E l.i..'(l'...E. ...gj(.lg.i.j....(...g E .. .. ... y(..(.. E... j jEjjjjyjjjj-. E @ . E' 'jjgjtyjI,E yjkjjgy;;.. E (jjkyyjj;;. . (' ' '''yE' ;'.(E..( q jjkyjjjj;s ..l(jg .E' E ' '' '' ('jkjgggjjj,. E.(' . j E j. jy(q.... y... . y gjjjgjyjjty 'pjjjjrj;. (g. g... :yjjjjj,j( .'lE'yj11itjr:i;. E;''. (E' ' E ' . gjjjj.. ($jjyjjjjj.,' . jjjjj;;. .. .. ..jy..;.jjjgjjjkjjiyjgjjjji.y . (. (' .) ' !(!E.'.' (. 'y.'. i (:jjjgjgjjj' . E ( (. ...y.. . ...... .yy. yyy...yy. yy..gyjy ( ( (E( . (. 'jkjyyjy;;,. y. . (4jjjggj;,. ... . (..kjyjjrjj. . . . .. . . .. . y.. . j. .jjjjggjjj,. jjkgyjjjjjjjjjjjk..j. :'i.E.i.E.(y.. 'kj:jjj;k,' . ... .('. .(.. .'..;jjjgjj(jjjjj,. y.. ...jjygg. ; j ' (jgyjggjj,. y.'. . . (E.q..rE..... Ej.:...riqjEEyj.E.j.j...kkgjjyjjk.. . j... . ''... . y. . . ' j..',.'.' . j . . . . ... . .. jjkjyjjjj,. ' . .. .. . g.y.y.. j.gyj.g.jy jjjjgjjj,. ...E' E.; 'tjjgjyrj;: . . . j:k, E . . E E E E . .. E .(qjjjjqr. . E. ; .. . ; E. . q. E. qE ... E. . ... .q . E.E......y. . qq.. . .. . jjjjjytjjk.yjjjtyjrijy,y . . .. . q. .. ; g.; y .. E. (q. . E g. . E !. .y ..jjjyyjjj;-. . . E.. . . (. q.qjjjjyjrj,. ...jjfj,,,L-. yq.... .y. . ..j.. .-E.EE......... .kj:(ry:jjy. y. y.. . .jr.q.!E..rq....yq(!q...q.....y... ;. i,;yjjjjyy-k-. . . .. !..yy)(jjjyjyjjyjyjj(j. r .kyj......(..y...(..yj.yjyy.. y.q....y. . ..jjjki,q r... yy..y.yq.. ..y., ..... y..r......(..,j!. .y yy.. g... jjjrrj:y.. r qE tjjjtjjj;j ..r y.jjjjyjj. . . . ( qq..y..( .. . . ..r.r. . .. q .. y ... .. jjjjjjj:..2... . y.. Eq.r. r .. . r . r g y ; jjjjyyyjj,. y.y . . . . . . .. r . . . .. .. jyjjjjj.jjjyyj;.. y . . tjkyjgrji qjtryty:!j;;;r;. kjjj!jry . jjyjjyjjj... . . . y .t(jyjjjj,. . . . . . . ( . . . . y .. :jjjy;j jjtyjgyjj-. . . .. . .. .g tjjkjrgjj,.. . .E. E..y..y.. E. . . .. . . EE.y . . . ... girjtkjjjjs .:jj,..,j,y . . E E Ey..g.y.....g yy. E. .E E..y ..jj;,kkkk,E .y..jyyj.yjy.y.j..j.jy.. yy.y.yyjy.;j.. j. .. E . E ..E.....E .. j:,(,.:k... y j y. y... y. .y -yi'..-.(..(.'.r.'..Eq. .i..E...j--... .. E y;j!y, y.. E . .(....-. .(qtkkjkjj-,. . . (. . ! . .. y.y..y...y..y.. ....jjIgjjjjjjjgjjjjyjjjyrjkIjrjjjj,. j,,rjy,,.. q ; .ak.y.yr yt,j:r,k,;,. y j.jyy. g q..q (..g. jjjjk.. . . .. .. .. y..y.. g .. .gj .E.E((.r;..y. ....,--,.--.,., . E.4,,.. yyq$jkjyjjjj,, .....j,,y.... y.jj.y.jj .y,j,i,i.y ky,.krjjyq y..Ey;,-j.... . . r::,r.11.r..(r. . (-;.:k-.. . .. E..k!:yjk;;. . yE ..L..L.j.. .. . . .j;;j:j,j,( .E-E ...'iii1i1Ii'. '. .y y .. ..;;,yj;.. . ..... (E... ...;;,..(-........y. q..;.t.yg2..... ..r k.k.jy. ;j. ;yyjjgjj. j .... . .. ..j;2jyyj,.. jy ... . q .. . .... r..jjkyjjjj,. .kjjyjj:j. jy ..jj.yy....gyy..y.yjj.(..y . . r . rr.- . yj r;-.;j,-..!jjy;;,k.jy.. j...... Ej.E..j.r!-.(.Erqyj.yy.gyj.Ery.i...yEy!.yyj...;.,,.j,;..kk,,,!kk. .. . y.,;--.,. ..!1k...:l''. .!.y y.y ..... ...$.kjyjjjjj,.. .('.E(t(.j: . :-..,jj4jq,:. ... y . . . E...211.. . 1It... . .EE. -11EE.;.. . E llL..q ..''...''..jj,jLj,. E' @ . EE. . . . ::11... . . . bt'i',Ljjjj,. . . . E ; .ryyg E .y . . y E . . .g. y E kgjjyjg;yjrk. ( .y y yy.. . . ..''-..''...-L?3'.f'.. . . . yjkjyjjjjjj.y. .. g.. y l' '' ' . .(....us....r j..U:.)(j..:. .-o:qd!;. . q . y . .. .. ..yy ... yqy j..y . q. . y j qg jjjyjjrjj. . y k;,.y y.. .q.y.jryy..j... .E yg y . .. ... q. E . ( ( y . . .y;.y. . y.! ..(. y.. .y . E .. y y E. q y y . r . . . . y y. . . . . ( j . .. .. . . y jjyy,;y. . . . y . q r .. . . y y r , . y . y . .. . . . .. . . ... ..Ej...y..E. y.. .. . .. . .. . . . .. . yy . jg) . ... . y ( . jj(jjjrj;. . (....' (. . .y .. y.. . . .... y.y. jt1I::(iiid....j . ..y .. . . .jyy.yyyj .. ... . .. 'ElillEiltlqtliiill'll:jjz!: ( . .. . . . jkkyjgr;;. j . . . . . . E . . $ jjjjjjjj,. . . . y $jjjjjr;j. ' E .. . . . . .. i Ei' :Iij7i2i.(. ( jjryjjj(,. . j. :Ily.. . ... . ... . .(.E. .. ' kl::i!!lt. y.(j..q j...y.y. ..Ej.y. iq ' '' qjit.. . ygjj,s .y(. .. i EE Ey EE ! .;. E .E :!jlk. E. jjgygyjjj-j .y.. .. . . .yy.rjjjjy-g . y. y(. ( y . y jq.... y. y, . q y . .j .. jkjyjyyjjj. E '. . (. . ; y y .. $jj(gjj,,! y E y.y. yg j..yyy . jj;jgjjjj.. (.. y y.j j.yy(..y.. y.. . y. .yy.jg yy.jjjggr;.. :i j.j );. E. ....i..y.'. j y.. g. yy.. y. .j.yy..yyy.yjyy. j.. yyyy..... .. qq E.. q . ( ... y.. ..y....y.. .jy.y.(jjjjyjjjjjjjgj;,.. .y . .jy.. . . y.. .. .Ey... j. . (E q( q... . y y .. . . ... y. ., ..y.... yyjjjgyg;jr.. . yy . (4,jgyj4,j:,,(jj,jy,j4j. . !...i1iEikl... . yjjjgy,,. y. . . . ... tjjjjy;y. . . ...jjyyj. .. (..r.kjjjjyjjj.(. q(...2..jjjyyj...j......jj.j..... .y . E E.q y .. . . . .. . y . y . ... . .y.j. .. yyy.. . yyy . .yy gg... r . j . ygjj(.j. j y. r . . . y . y. . y yr y . ... y . r . ..... y .t'bj,iiL. r .. . y . y y . r . r r y ...y.y.......j......y.. . . . j . . . . . . .. y . . y . . . . . y . . ;j2jjgj,;. g . . E E . tjjyj,,yjjt..y . . i1i:q:iit..y.. .,...jjjggjy,j . .. . . .. .y . y y . .E yy. ...q..yg.y.j.y.jyy...g.j. . jy jy.y. y . .yyj . j. . y . . .. tk,,. .ky;. ...j.. yy..yj! y y... yy.yy..jyjjjj,,.. ..g .yj. y.j . E)I(.(. .. . E E . y y... q..j.y....;g q. . y. y . . .. yy y g . . j yy y y.. j. ;y . .qr. .( .1( ,),qjjjjjjq)jj,j. .. g.y r yyy.y.- y . j. yy.yy y jy.y. yy. yy y.yy yy.... ..y... ..y.j ky( j.y. yy. . . ..y . y y y y y g . ( .. y . . ... y. y . . y .. y, g y . . . g . g .. yyjg,.yjjyy,jkj,. '.i j ( (..g( (. E.... (. . ..(.(jj(...E' (.. E (........ rj.j j.j.. (. y(. ..y..ry.(..j.(j.j y .j . j.. . (jj( !. .((.... y( (.; j jj.j,j;;;,Ij,E jj . . . jj,y,,y,y.jjj::jk;,y. . . . j y j y . . . . y . y . .y y. ,,jjj;j;,.. . . y ... jjjjyyy,-. . y. y.y.jj .yyyjyy;y.. yy.g .. j . yyy. yy(-.. .. .. .. jjjggjr;j .. .j ..y .y .... yyyjjyyjj. . .. y . .yyjy,. . . kjtjgjyjjk . . . . .. yy. . #: (.(..... .....1j2ijjrg. .y.. . 11222:j2 . . ... . . . .E. E .EE. . y y .. . . g . . . .g y . jjLjq?L.. . . (j:jk(:jj2. .. y ij2!2rr:.. . . . . j . y . y . . y y . . . .. y .. . .,1:::2jIj,. g E . y.y (...111:)j ..y.yg..y-y...yy..... .yyI1;jI;:Ej. yy.y.y.yy.y ..y..jy.... q.Eyj...i..yq..i!:.y.iq.j g ...y . yy..j .. . ;yg.y..j.y...... y.yjy.g..y..yy.yy. ..yy.. .yy.j..yj...y y yy.y...yy...j ..y y yj. yy..jy.. ..y ( y.y.j.yyy yyyyy. yj .. ...... . .. .. ,,qjjfjq,,..jy y . .y jjjjjyjjj,. ..i. .: . Ej.E.y)().()..j(....E...(. y jjykgjgj.j . .j.,.)...j.j...jL .. y . . ..jjgjgggj,.. . ..EE.' .(.'r.l(...(.i.Eq(.(....!(.,.:..(Ei:!j . . Ei)j.k.g. Ek((.;q(. jy . yg y. yjjjggggk;y i......'.,,j.... . . yy. . y y. y. ......(..yk.y., . . .jkgygjj.. . . yy . . . y . y . ..) .. i . . .(.t y . !.(..:.. jl . . . g . .(.gj:( y .. . . . . . . . . ..gjjjjjk,(..E!(.. .j. . . . .. . . . j jyjjjjyjygs li(:ik.'. . . .;. . .yj . . ... . . kii;' .i.). y..yj.j.y.y.g.. yg ..jyjjyjj.,. r.. (Iik.j . . . .j;yjjj;yjyygjgyj.,. .1jii.. y. ..jgjjyjyjj jjygjjkjy. y. . ...... . ..i-(-)....t .y :i:7'. ikE2r). '' y.. .'t!IE1)j-.. ..iktij;.. .-iE)!)k. .q.. .jjjgjj;jkj ..' iihqjlL.' .kj1E.' (jt8,k/;r11:.. ... .g ,rk....E-. rI7..,I...,I. '1k::;,. g. ..jjfjj,j4f,. ki5!:''. i!ik . . .j((.'tE ji)i:Iy :iEji..-iit)J-. (IEy.. .yj..( .....(). E(LLL iiik. tiy21)ti.... .ii!i.. 'jjji iiiti... @E( .. . E.:E. y... . ' yi '. '.' . . ... .. E,. .'.. .'1;. (E .'. 1?i;. Eyqj ij(k EE '. (.) E.( iEE E(E iq iq' . .( j.)i$(1211-. Ejr .'t j ) E E ( i .. :E !( EjE ( ( ! ) d(!i:' ( i E: T (( EE )j (j, ! t ( .tE. E 'lI:2' 1kiid:' ( ( ( ( ( ( k ((.., dtiii.--'.. di::ii'iiiIli).'lii'' i )))(i ii',(iiif.tiih.I)'iii:jk' r'....'1iir;' (tdithE' k. .. ,kt!,. ..-...--E-E--.-.E..i;.......iE.-..... i( j.'i! dijElf t.......L.'....-......-..'-L' ( y .--j (,yg .# ' itEikiqtil/rlkilk.' pIi)!1)7ik)' i@il;' '.ii@:1 .-.. iiqzki'.'' '127... k:,1. .i;i(:q..!tq1Itrit1i)i1k.'. . kdiiik-'.' . tiE:il. ii:;'' 1t(.. .t iif:ii)....-. . 122... j: iLil.. l.iililil
[email protected];. k11,:9. iik' .--...-.1:112 . ( qi(Ii.I.. ktizjtitiiii.. .iiiijk' ik::ll ' ' ' iiiik .kk219).. hii:'k,,..-t--..k,i:'.k::jk)' .I):tIt71i). 81t:. .. E(ik2r'' .1r:-id;;#. .r .:.. (E. L....j.:..'j.-.r ''' ' 'k2E:i. !12. ..t. tt.. E )..)E E). .:'(... .E )' p E.' rT7 ' i ''j'ii::. .r. E'. ..(..( E(( .E..q (g' E. ): .j j ( E.r(i.kMltiy'ilkiEEd/ktEl.ji'An @t1@rikEq;'.!yy,E,ri,y'(.yyk;!:., (' x.'..L. ):.Ei.(.. (.g.j '.. )' ' .g'j 'yyj.'(..gL... E''(E Ej. Ei'.'.'g'q'.g': . (E.'...L .:t.. .(. j.j:k. E . jE.y j. .. E r . j..'( . j )(.j (.( .g( ) ) . t..'.j (:( ('.'( )..''y .'( t. ...' . . ().) ((.. .'T( j' k( . .
'jE.
-
'r ''
tpE .-1. pr-
-. -
.
-111.
tti
'
'
-.
-.
--
.
-j...
-
.
liqTt
-
'..) .-
. --
.
-
..
. -..
.
.
.':
.
.
-
:
.
' ' . . :E
:
:
-.
-
-
.
'
-
'
-
. - .
:
- ..
.
-
4
'...
:
.
...(j
.
. .
-
:.
:':
.
... ..
. . .
-
.
.
- .
:
E
-
.
.
-
. . .
.
. .
-
.
.,jjyyyyjj,. . . ..
.
-
-
E
:.
-
.
E : ..
..
-
:
-'
. .
.
-
-
-
-
.
-.
.
. .
.
-. .
.
.. y .
-
-
- E.
:. -
.
.
- . .
.... . .
.
.
:
:
.
.
:
.
:. . . . .. ..
:
.
-. -.
-
. . .
.
: .
. .
:
-
.
: . .. . . . .
. . .. .: : E!. y ... . . -
.
j
E .
.. -
'
:
: . :. i . . .. .
.....'
.
.
,.).jjy...):..
-
-
..-:k:11(11711)...
.
..
y
.
.
-
.
:...(
. . . .
rgt..j.r:..j
.
-.
-
. .
..
..)..g..
-,
,')y....
......
..
:.',,)
..j,
y.. . ... y
-.
.:
.y
:
...
.
.
-
-
. - .. . : : . ..
..ktii.
..
- . -- -.. . . . -.
. : .. - :
.
t -
--
.
.
'.
.,.j
-
'.
--
.
-.
.
-
--
-
.
-
' -
-y
-
.
.
.
.
.
. -. ..
-
. -
.
-
-
.
.
.. -
-
.
. ..
-
.
i
- . '. . .
..
.,;,,;-;j. .
-
.
.
:
.
.
.
.
.
... .- , E
,
: .
. -
y
.
E. . ..
-y
-
.
-
.
.
.
,
-
, -. . -
-.
.
-
,
.
-
-.
-
-
.
.
..'
.
-
.
.
g gg.- . . . g . .
y.
: -,,,-;. .. . . .. . .
-
.. g . ..
.
.
-
,j,jj,,.. .
. .
. . .. . -.
-
-
-.
.
. . .
.
.
-
.. . .
-
-
.
.
.
-
.
- . ..
.
.
.
-
-
-. .
..
.
.
y y.
.
. y
-
-
.iki2!)'
,..
-jjjjygj;.y .. . -. . .y .y
.
,
.
. -
.t
-
-
.
.
..
:
.
-. . . .
.
- .-
.
g
. .
. - .. - . -
.
y.
. -.
..
. . ,;, . . .
-
..
-
.
.
.
j.
-
.
.
-
:
g . .
:
.
.
-
.
.
E' -
.
.
.
. .
. .
. .
.
.
-
El.
g
..
g
.
- .. . .. ... . ..
-
y .
.
-
-
.
-- -
-
.
.
yj.
-.
-
-
.
.
.
.
. '
- ..
y -
g .-
j
y
-
.
.
.
.
-
,
. y.
.
-
-
. . .
.
y
-
.
,
-y
.
y
-
.
-
.
-
- - .
--
-.
-
-
.
.
- .. .
y .
.
.
y
-
-.
.
y
.
'::E
:
.....
y
'
..E(... ' .
:
.
'
-
:
:
.
'*. : .
.'.
:-*''
: '
E
.
E
:
:
tE;
.
E.;!')q.'
.
',
EjE
'.
'
:
,.jui'
':E ' :(
.
:
:... .'
jj'
E
-
..
:
'y
-.
'
:
-.
:(.(.'.
E
'.
'
''
.
:
:
..
:
'
:
-
-
:;
',
'
.
E
-
:n
.
.
......
:..k.jy .
..'. :.
'.
:
.....-...-...p2k.
'.
.':.
. . :..
..'j '
:
. 2-.-
:..t
:. ''
.
:.
..
...j.)
:
:
'
: .
.. .
:
:
:.(.. :( .':..E
...
:...
:(.
Thus, te CD pays
:
$10,000 x
j.j.ryy.jr.ij.iyyEyqryyyyyy.j.qy,
)..yk
,j
iyt..tt.
'.
.
.: ..
1 + 0.7 x max 0,
Sfinal
-
1300
yui.syy,yyjj..k.yj,q.rjyy.yyyy,yyj.y.jyyiy y,p qjjyyjyyy, yy.yjyuj ,, , yg y(y ylyly E.jE;j(kj..:. , (( j (:( yy..yq y(,jqyg ,y j' . .. .j...(;.(.(j( , j 'jIjEi:j;. j ..(.t 'jl.(kIrqi:''. ) y.. '' ''E E.((. (!IiEiikiE:)i.. kIi:'. t.CIiEiI;-. kiiEiiik. ;i!!Ii... E j. . EiI::)if)' iIi:t:j);-. .ti:EiiIt).' ii@!i:jk;. t1tiEii.. . k.. yqp.y E .yE-( . '' . i.ijtt):Ik:iL.' ((IItiF::;#'-. ' kIii:jiIt.' '' '' ' (jIt.)(iit.. ' . '.iiI!i.. . . )..;...; klljlllEiiEqtiLly ' ; y y . ! . . (114(4:),y . . y y y. y y . . . .. . j,ygjgrj,.. yy. yy.. y y. y. y y. j . . .... y. . yjjjyjgjrj. .. . (y q jjryyggjj,E . y.!(.y . yy g. . y y. ,,)jjjj,j. ?'tjjjq,L. j...jy.y.yy...yyj..ygy....-jyyy.yy.yy..y.... q E q Err. yy Figure 2.14 graphs the payoff at expiradon to this q q. E . q j .. .yj.yy(. . .y(jjjyyjj(,. .yy yy. j y y yg.. yg.j.. .jt,,qq. .jkyjjjjjj... . ...y. y q ! (.q..jkjjjjr;y. yyy... yjtL.. .y. ijjjjy;. . . .. .y....y.y .yy.jjtjyjjry. yy.. yyy yyjjjjyjjjjgkjjj(.. y. q.(.. q y. qj.y q..jq....q . . . . j;ygjjjjjty,,;,.jjjjjjg.,,. .yq.,.y.. ...j;yjrjj;,. . . y ygjjy.. . j y..ykjjjjjjg;y. jjtjgjgjj. jj:yyjjjtj .. .. ..r.. g..y .. .. gkjjj.jk;, . y .. y... .k,tt;;j;j yk.kjjjyjjjj . .. jj,,. . . .. . :jjjgjj,,. . . . . y . ........jjjkygjj,. .. . . . g. j . . . . y.. . . . . . y.y j y .. . . y . . . j jjyjgjjj,y . jgjy!jj(k;jrjjjjjygjjjjjj,. .. . .. .. . .yy..yj.g jy.y... y .. y . . . y y.y.j.. ..y. .;.g..g.gj.. ...y..j.yg. .jy. . . . y ... g.g . y.. j. . jjgyjyyjj,; .(E(.E'..E'.i..'.. . . j E. y . E . j . . . . y . E. . y . ... j. y..yyy.y. yyy. jkjjggjj,. y... y . . q.;E Eq 1112:. (.Er.y..q. .... q.. . jj(jjrj;. jkjgy(jjj,,. yjjgyyjjj,( qq (.. E.. . . . .; E. E. .. . E E.E .. E.. E.. ,,,,.. . .j. . . E( . .(.( q. . E q.. .( (... Eyy.y..jj(tjjyjgyy::. . .. ..EE...... (E. . discussion i 2.1 of incorporating a . . j E q..yjyjyyyjg.j. .j (..:jj;..4,,-,,j,. y Recall . gq .i .. . . gjjtyyjyjj,. y ' . . q . . ' . y y y . . ( . .. . yyayj. y . . y q r . . . . ( y .. . . .. r . y . y . . . jjjjytjjjjq . . . y ; . yjkyyjgjr;. y . ; jjjjj.. . ... .. g r. . . . . .y 112111111. .yy ( . q .. . (jiil. .. . . j. ..... . kjjjjjjji . . r.yy.. r ..... yy. yyyyjg. yE.. . . j,,),,,. ;jjyyjjj(j.. .y. . .y y .y . .y jjjyyyjjk,q ..y.jj. jjjrj:jjjj,. ...llk,.ll,,.jlk. '. ' ;Ej. jjjjjyk.. jg.. jjrjjjjjj, ..lzi!i . . . .yg ... (. .. ... ....$,,r;..g-,;kkk-... . y. .....j . . yy.;11.,,,,.,.,.: .. ..y .yjE..(jy...( ... '..qlilEr'' ..y .11..,11...11,. y '((EE .'. E.. 4122((:14:* E. 1.,,.. .. .y.jg. .gy;gjjyj. ..jjg. yjgy..y. ...yj..y... . .. ((q y y kgg... . yj. ..;;..... j. y jy . . . gy y g.y ..y . y.. y ;. y j...yyyy.y.j.jy.gjjjgggjyj: E. j.y .yy.gjjj.j q. .. ggyjj .y.g .y... j .j y j jj.jj.g ..jygj.y..gg.y yj j . . . y...jj.g.y...jjggy..g unit of qy E. diapam. g. yy j g... . . .Ej. q qr..:.y .jy j. .jy. . . . y. j.yj j. y.,jj;,,. jq.g g . q. y.y.j..j.(jj;,y.jj;,kj... g . g . . . . j j g y.. ..y r. g.y.. ...:E . j g . ... .. jy.g j. jjyjgjgjjs . g . j . . ;. .. g.yy...y.y...gy. . gj .gj.. jjj.j.jj.. y. ... y . . jgyjtjjyj,s .j... j ... ...q.y..g.. r...j,,..j,i.. . j j j g j g j jy g ... j ..gjy. ...(. j.... g. j,.j..j...yj.,.,j,4.jj,. j j j . g g j . j j are 10,000/1300 .r...E.E...g... . y . .. j j .j j . g g ..gj.y..y.. j.gjjj.y . . yj.ygyj. ..j.y.. .. y . g. y.y.j .j.j... .j.g g.j..gy . . . . .. . .j. yggj.. jg. ...yj. ..g.y. ...gj.. .g...y..jj. yj... gj. gg.j.y. jj.j. . .yjj....jj.jy.y.y....y.y j....11.,..1....j1:..E . . '' i . . 11.,.. ... y. (iirjr.'. Il:i::ii.y . .y y g.g. y y ..g.E...E.g.g j.j. jjq. j ..q. . . .y.. y.y. ..'-,..y.y. .y.y.. ylg.ggyygyjjj itktr::. yyy.yy.y..y.y.y.y. .yjyyjy.yy...y . !I2iE!::;. .. yilqy.. yy.aj : . ki:::!iltl u...uiu.EE .y..yy.yyy... .... r.gy.. y (11444:,,y, .y y . y.y. .. .. .. yy .L?,b','... .j.j...jy....y......y..y..y.y.y...y. ....y.y.g.y..j...yy..y.yy.yyy.yj.y..yyyyy.y..yj.yj..jy y . yy... .y..ll-a.,l... . . ..... y...yy. . y .... .y . yy ..y yy.. .... ry- g .. yjy.. !r. ..y. y . y . . .yy . CD ,y y . buyer receives 0.7 of an yy.y . y. j. y- y yy.yjyy . . y y y.. y y . y.. . . j y . y . .. . . y g y . . .y y y. yy .j . j . y y yy . . . y y y . .y . . y. y. . . . . (q .yy. y . . . . y y j . . . y . y jy. . . . y. y j,jyjgrj,y y g. y . . . .. . y . ko oroy yy.j..yy. j yyyyj. a ...j y.yy.yj y yygyy...y.y.. y..... j jj .y.ky yry. . y. y..y jj y.-y ..y..y.yj ...jj..yyjyyjjjy.y.j yy.. klu . . . . .. ... .y.y. ... . . . . .jyjyjj .yyy...yj.y.y.j yyjy. ..jy.; . j..jyjjy.,y,yyyygjyyygyyyjyyyyyjyjyyyyygy,yyjy . .y.y..j.. .jy. yy.gy..yj..y, j.. yy.y.y..j..y ......y.jyj.y.jjyy..yyyy EyEi,tt j.y.y..y....y.y.y.yj.yg.j....y.y.y.jy...j.j.....ty..y.. g.y-yj... yy.jyyy y.y y.. .yy.y.y.y..y.g,y,jytyygyygyyyyyyyyyyyyyyyjjy j-g . j.yy.yyyjyyyjyyyg yy y.y. .y . y ,yy, ...gy.... y yyy.yyy yy. . y y. jyjy y. yy. j.. yy eykayy a.xyyyyjyyjyyy . ryj j yyjyjjj ... j...yyy..jy ... y.yj .jy .y.yyy y y . . y. .... .yyjjgyyyyjgy,yyyyy,yyjyyyjyyyyy j yyjyjy,yy paying expiration. y y,yyyyyjyyyyj,yyyyjgyyyyyyyy bond y .y . y.y y... yyyyjyyyyy . y g. g j j.yygyyyyyy y . . . . E . at y j y . . . y y yjygyyjyjy . y. . . . . gyyyjyyyyyyyjyyyyyygyjyyyyygyjyyyyyyjyyyjyyygyyyygyyjy,yy,jjjyjyyyyyyjyy,yyyyyyyyyyjyygyyyyyytyyyyyyyyyyyjyyyyyyyyyjyyyyyjyyyyryjyyyyyyyyyjytyyyyyyyyyyyyyyyyy,yyyyyyj,yyyyyyyy . yjygyy y y yj yyjyyyyyyyyyjjyyjyyyyyyyygyyygyy,yyyj,yyy,ygyyyyyyyyyyyyyyy,yyyyyyyyyyyyyyyyyyjjyyyjj,yyyyyyyy jyjyjg yy yy,jjy yyyyyjyjyy,yy yyyyyyyyyyyyyjyyyyyyyyyyyjgyyyjyyyyy, y , gj.jj, y. jjy payoff to equity-linked CD yg t jj j y,y , yyy,yyyyygyyyyjjyygyyyyyyyyyyjyyjyyyg,jyyyyyyjgyy Table 2.5 computes r 1jkjjjj;.. g..g.((.( . .j ( yj g jg. . ( ( jj j jy j .j j .. jj. E . .. g (y... jj.. j. jjjtq: .k,.L63qj,-.. .rjjyr:jjk,. ,ik2kgr:(. E. . .. .(.j t( Ejtj.( igyy . . . . E j .jy i. .iI2jjj1jrjrjjty. ... y . . (. yyyyyyy q .y ( . q . yyyyyyj . . q q . . gy ( . gy ;..j yyqy. . j rj ( j . . y y yy y (j. . qj . yyyyyyyyyy y.. ... ..y. yy.. .g...yj.. ...j..yyygj y...yy.y... .. j yyy... j y j y y.. y jyyyyyyyj,yyyy yyy, . y.. . ..( yyyyyyyyy,yyyyyyyyjyyyyyyyyyyyyjyyyjyyyyyyyyyyyyyyyyyyyyyyy,yyyyyyyyy,yyyyyyyyy . yj.y...jy...y.j(.jy..yjrjyyyy..qy.jg.... y..j..jygy .. E Ey. . i ' ' jy-. y.. y.jy... jj . . yyyyyyyjy .yj..,y . . y....E.y y y y.. ... yy . . j y . .( . . . . . ' . . y . . ' . ! . ( . i . . . . . . . .y . j . . ( i E ( E ' ' j . . . E l ( y g. . E E E E y . . y (. . E ( E . i ' ' .. ('j (q (''j' ' g. ' .jg g ' . . g'y..i. '.E'Ej ( .jl.... gg.. ( .. (j(. .. ... .y. r y ..; y..y.yj @ ( ';g('(1y( j.y.(..j ..Ey.'.'j.(''j.' '. .. '.y..E(.....yj.yy. '..y.....y..'..'. y.(.(..'..j.'yj...y.(..y.g.j.yy...jy.y.(....yg....y..y....y...jq.yj....y....(y.jy.y.yy .y.j..yj.....y...y.yyyyj.g y...... j. y..j..j.gyyy.yyyyjgy y.....y.y yy j y y.y. y . y.yj.j.j....jy y...j.g.. y.. y.y.y.y.. yy.g.. jj. y.g..y...y..y...... y. . '... '. y........yEy .E... y. . .(. j.. j...y... y.y ...j .y.jj .....y j.y y... j y'.j...j.. . ' .....j.. j ..j ...j. . .q.j''.... jy ..y(.'g. .y.y........j.j..j.j..jy j......jjg '.j.jj ..j... . .. .(. .' (' j. . . . .' . .. . . ;'.... . .. . '.l... ............ ..... .. .i y. .... .. . . . .. ... . .... .. . . .. . . ...... .. . . . ..... ... .. .. . . ... .. ........ . .. ... ..... .. .. . ..... .
t.
:
'...
.
..
-
'.' '
:.
:
.
-
:y(.
'
..
'
-
-
kIiiEi:iI.
.
jtj
-yy-
jyjjjy;y .-.j.y...-..yj . j4: . .j,)f;f:qtjj,; . .. . ..k;jjjjjj,gjjjjyjjj,. . j-jjjky,y yjjjyjjjj,s . k..jgykjgjjji .. ... j;gjjjj..yjjryjjjj.y. .. .. .(j. yj. .. . . . yjjjj,(j-jj.. . .;jj,kgjjy,. tjjyggyr;. .. .gjjj,. . . . .. . tjjjg)r:iq' tkjjjj,.r j .. j j. g.. .gj j...j. j...g ).' .' . . y ....j yj j .. .yy. q ukks j . j j j .. . . . . j j .. . . j j .. j . . jjyy .. jj j.. yuyjjjyy . j. .. ..j...y j.j. .y jjj.-j.j jj... ....y.....-...,jj j.j yyjy . jj yj.-jj...j ..jj....jj.j y ..j tyg . .. y,jy . jy . .j jjyjtj fgtjtojyj j j . kj . y . . ry yj jyy kk:)q:,;-. ..)2jjjj!'(. j ..j j. j j.... . j. jj. yy ..rpjj,.qjjly. . . . . . . '... j. '.. ( ty ... ( (. (.... . .. -
.
.
.
.
-
-
-
E
.
j.. .... ... .jjjjy;,. kjjjjjjj;-. .
-
.
-
-
-. .. .-
'
-
.
.
.
.
. .
-
.-
- .
- .
.
-
-
-
.
. .
.
-
.
-
.
..
: . .
.
.
:.:
......
.
.:
:.
..
:
. .
:.
.
11kk::;;..
:
:.:....
....
. -
:
.
. . . . .
. . -.
-.
.
.
.
-
! ..
.
: : g.. . .
.
..
:.
.
.
,,.,sska..y . ..
.
.
..
.
..
...g
# qyyrrtlsy
tyijtj ..:.. :
..
-. . y .- .
: : E: ' '' ' .. .
. .,
. .
.
-
'
.
:y
. : .: . :--.3j334:. .. .
. .. E. . - . :
.. .
'
:
. .
y
. .
.
:j . .
..
.
.
.
:
: y y . . .. . .. .
:g
.
. y .
:
.
.
.
:
:.
:.
. :
.
:
r
.
.
. .
.
. .
. .
. . .
.
.
-
-
. -
:
. .
. .
. .
.
y
:
:
'
. .
.
.
.
'
'
.
:
.
y
.
.
.
:
.
:
. .
.
.. .
. .
.
gg :. . :y -
Ik-,,,,-.y
.. . .
:
:
...
-
E:
-
y
..
.
. -. y . ...
:
. . .
:.
:
.
i. . .
-. - .. .
.
:
:
.
-. . . .
-. .
- . .
-.. . .
.
: :. .
-
-yjjyyy;j,: : . :. .
,;,
.
-,,,fjjt-.
. .-,)tjjjj,j:. . .
-
.
. ,33::3j,).
-
-. -
- . jjrjyjjjt,ryjy,,y . . . .. . . .
.
-
-jyr,
-
. . .
-
-
-.
.
- . . . .
.
.
.
-
y
-
.
.
:
E
:
,
: . .. . . .
. ..
. y . .
:
-..
-
.
- .
-
- ...
..
.
.
..
.
.
.
. -
.
. .- -
...
.
.
.
,-,;,.
-
-. . .. .
y
.
-
.
-
.
g
.
.
.
-
-
-. -
-
-
-
.
,
.
y
.
.
. .
:.
.
.
:
y
y
,
g
-.
-
-
y .
,
,:
:
,.
-
,
-.
-
-
..
.. .
.
-
-
-
-
-- . -. -. .
-
.
. y . rr .. .
y
:
g.
g. g. . . .. .
-.
-
-
g g .
.
-
- .
-
-.
-
-
-y
.
. : .
: .. :
y
.. .
. : : :. .. . . .
y - ... . y
y. .: .. .. .. .
-
.
y .
.
, .
.
.
.
. :
..
. .
.
g g g. . .g y .
g
.
--
-
.
:
.
g g . .
g
-
.
-
:
.
. , y. . y . g
g
.
g
--
-
:
.
.
y y
,
g. . .
-
-.
-
.
:
. . . . . .
-
-
y
g
.y
g.
:g
. . g .
.
- - . -. .:g . r yy r
.
g .
.
.
.
g
y
. . .y .. . g g .
j .. .-j . . ) .
r ..
,
,
.
-
-
. ..
-. , -
--
-
.
.
. . . .
. : , . .
--
y
.
-.
. -. .
. g. . g . .. .. .
,
y
g
.
g
g
.
.
g g
.
g
g
.
yg . ..
y
-. . . - .
:. , ,
. ..
-
. .. y. g . ..
.
- -. -
y y . g
y
y
.
g
g
g
.y g
. .
y.y
-.
.
,-
-
-
. . y.y .
.
:
: y .
y
. -
-
-.
-
.
. . .
. .
.
y .. y ; y y. . - -. -
y
.. -. .-
.
.
y
1
(2.11)
inves%ent irl the CD.
-
-
-
. ..
.
vv,6ji,L.j : r. . :. . .
(ceruficateof deposit) is a ldnd of interest-beming bank account. You can think of a CD as being the same as a note or a bond.
9A. CD
.
-
.
-.
.
,
-. . .. -
-
.
.
:
:
. .
:
:y .E . -.. .. . E- . . . . . - : . .. . . -. y : y. . y : . ... : y .:. . . . . . . g . g. . g . . . -. . .. . . -. -y , . . . - .- .- . -. . . . . .-
.gyj(.. . j y.j...-j .
-
, .
-
.
.
.
. . . -.
-
..
.
-
:
.
:
y
.
g.
-.
:
.
. .: . .
,
uju
,
.11:2j
: ' ' .. . . g. .. . . .
.
-. -
-
.
y
-
' ' : . : . . .. . . .
-
.
.
)
.....
-
'
.
.
.
E:
. .
-
. . : : ,y;jg.,,jj. ,,..k,... . . . . . . . .
.
.
- .
- . .. .
y
,
.
-
.
.
..
.
:-
.
y
-
.
.
,1,f,.,.:..
:
g .r,:
: : . -:!!!:t.;:.
.
:
-.
.
.
. .
.
.. -.
-
-
-
.
.
.
- ..
'-
,jjy,,. . -.
.. : : .
-
.r..q
y
.
: .
-
.
:
-
'' : .
.
: -
.y
.
.
-.
'.
)
.
jjIyjgjjj;E . .
-
.
..,
.
on our
1300
-
.
.
$10,000,
snal -1
-
-
-
,
.
. .
y.
,
$14,846
=
As a flrst step in analyzing te CD, we will draw apayoffdiagram. Ifwe west we receive at least $10,000. If the index rises to sinal> 1300, we also reive westment 70% of the rate of retllrn
.
-
1) x 70%1
Graphing the Payoff on the CD
.
-.
-
At flrst glance this product appears to permit gnins but no losses. However, by now tgnins but no losses''; the investor must pay you are probably skeptical of a phrase like like this. somefhing for an investment
,
. -.
.
(1+ (2200/1300
- .
-.
y.
.
.
-
'r:(
.
. g
-. .
-
.
y
-
y. - . .
.
, .
. .- .
-
-
.
,jjjgy. . . -
.. .. . .
.
--
.
y.
.
. -. .
.
.
y
.
.. j ., .-( j.j tjryyjjj(... j. y j j.
. y.
y
.
. .
,
-$jjjjjr;;. . . - ..
,jjjyyjyy,jjyjtjgjjjy;,gjjyyjjjj.yjjyjyjy:j;,;,,,. -
y.
- . -
.
-
.
. .
.
-
.
-.
.
y
-
,jjjyj
.
-
...
-
-
-
, . .. . .
.
. .
-
. . -
. E E .. .
.. . -
tytttgy
g .
g
. -
-
y. .
.
. .....
.. .
. y. . . . . -,,))jj,) -.
E (EEEE
-
.
j
.
y
-
.. kii:' . tj..ij.. ' tkiii:.. tll.. 1-(.-.. yg. ytyqit iktjptjEqiti,'EAm' bltttkyiri . jy otytik tityit jg iy..ltj'r.r'j jjyt,).)j( jt j ' jjjtjrjjg' ) j) tjtyjjygtjyitt tjt j tgtyg k.k;g j' j j jltuytty,rljyygjyjyy nyny tjgjiyyyy j,ijg t jjjyjkuig: y.gyy tt'gy.t tj)k.y.r'kt ti,Ykk Liji)jt yyyryyyjyu rity,o,,yityyyyjyp,yy,jyyyjyyuy, .-
y -..-......)
..
.
.
y
-
-
-
. -: . - . .
. - .
.
-
,
- .
..
.
-
.
-
-
$10,000 x
-
-.
.
-
-
-
.
... .,
-... -....
.
.
.
E
.
.
y . .
.
-- ... -..
-
. -
- .
. -
.
-.
. - -kjgygygjjj-. -. y .. .
.
- .
-..
.
y
..
.
-
.--
-
-
.
-
.. tjjgjjjjj.
.
-
--
-
.
-
-
-.
y -
-
-.
.
.
,yygggy:y,
-.
.
.g
g.
g. . . . . .
-
.
.;j;yjy:j;.y .
.
. .
--
-
.
-
. E.
.
-
-
yjqy t yytyy yljyttjt,jttt .. tjjjjgtk;rrtjj . tjjjggyjjr
,.,
.
-
.. . . -
.
.
. .
,
.
,
.
aill::..''..jjyyj
. -
. .
jjr,,k-jk,
..
.
''-...'-.y
-j
.
. .
-.
-
- . . ... -
'
-
.
.
.
. .. .
.
-
, ..... -ki!::?.
.
-. .
-.
-
-.
-
-
-
.
.. -.. - . . - . -.);,,:.. . .. -
-
-
. .
-
-.
.
..
.. . . :
g .. . ,j
.
..
.
-j,jjj;k. . .
rs.
. .
-
-
,.;j
.
. .
- :
..j...jj... jj.:y .
,
.
.
-
.
.
.
-.
-
. .
.
:
.
-.
-
.
.
.
.
-
.
.
.
. . E.
-
-
.
-
-
.
: . .- . .. .
.
-
. :,jjyjj. .
. -jjjjjj,. .- . . . .
-.
.
-
-
,
.
-
: .
-
.
-
.-yj.
-
-
-
--
-.
.
-
- . . .
. .
.
.
-.
E
-
-
y
y . . ..
:.
-
-
':
-
-
titk/
tkfiii,
..
.
.
-
-
-
-
-
-
.
-j
-
.
.
-
-
-
-.
-
u
.
.
.
.
.
. . y. -
.
.
-
..
.
-
.
-
-. . .
-
.
-.
y.j j.. t1l.. t.-.t. .tk!2j;t tik'.,1r. .-.'....g..('...-.-.'-'.--.y.y'..--.y-.j.(.. t z)i -
-
.
-
.
.
.
:
-.
-
-;!!511!1..
-
-
i
-
- -. . y. .
-
. . . --
: ..
.
:.
.:
-
. yy . g
.
, . .
:
.
-. .
-
.
:
'
. . . . . . ,yjjjg;jryjj;;,,yy. .
y
.
.
. ..
'
:
'
.
,
jk:jjy:;;.
. . .
..
-
..
.
-
.
-.
-y
- - -.
.
yyznjy ct?j ung
g
. :
:
. . .. ..
-
-
't - - -. . . .
- .
.
- -
.
. . -
.
.
'
-
-
,.
. . ,
.
,-jk..r .. .
. . .
' ' '
.6'...j'.)jqjt.,.
-
.
.,,y.jj::,r,;y...;y.
-. . . -
:
,
- . . -
E. .
jjjyjyjjjy.. . .
-
- Ik1iiir:;
-
-'
-.-
.( . . .
.
-
.
.
. .
-
-. .
.
-
- .
-.
.
. .
j j. (gjjgy;ggjk.(. tjggjyyjk,.. ... tk:jgj:j;tyjj(,jjjjj,. .. yjjtjyjpjj tyjj(gqqgr:jij-.
:: '
..
-
-
-
..
-
.
E
-
. -
-
-1!,:9..
-
. .-
. .
E.'
,
- .. . .
E
. . . ,j,jy,. . .
-
.. .
. .
:
..
-.
'y
: ..
. ..
. .
.
.
.
-g
-
.
-
.y
.
-
.
.
. .
-?'- ' .
.
-
-- -
.
. -
-
.
-..
-.-
-
,
'
.
-.
-
-.
.
-:.rt.,j-
.
.
. .. . .
-
:.
.. .
.
-.
.
.
:
..
.
-
. ..
-
.
.kt.
E'
:
..
:
-
. . -
-
..
EE. '
E
:
'
.
-
E
-
. r
.:!111;..
' -. :.
-
.
.
.
jjjyyj:jj;. .
.
' -
' -
.. -.
j.
..
.
.
..
-
.
.
:
.
.
.
.
.
y.
. ..
,.;.:
.
.
:E
.:
.
. -.
-.
.
:
'
:
:.
' . :
-.
-
:. . y .y
:
:
.
:
-
E'' ' '. . .
.
. . . .
:'
(jjjy(jjjj: .'.'. :
-!qq;qqr. . . . ..
.
-. . .
: .
.
.
:'
:
-. .
-g-- -.-. -. -. . . . -
.
-
(
.
.
.
..
.
.. . - .
-
-.
tkjgjjjjj.,. tjjtt.jjjt.t ..
::
:
y
.
-.
- -
-
.yq
::
-y
-
-
..
) j j j b tjiltky .
-
.
-
-
-. .
-
--
-
..
.. .-
-
.
.
Ej
-.
-.
-y -..
.
-
-
-.
-.
-
.
.
. . E .. .
..
-
-
-
-. '.
..
.- -
.
.
..
-
. -
-
'
--
-q
- .
-
:
:
'g..
-.
.
-.
-.
-
.
the
Secion payoff Per the index (there in $10,000investment), the $1300 zero-coupon the the index.
loThl'sis the stnlcmre of a CD issued in 1999 by Ftr'st Um'onNatonal Bank.
zero-coupon bond into a 7.69 uits of the index index call option, plus a =
for different values of the
%
50
EXAM PLE: EQUITY-LI
AN D OPTIONS
TO FORWARDS
AN INTRODUCTION
EE : iE : !:' : i jE : ''' :
.
-
-
i
.. .
.
.'-;j..)-
.
.
..
..
.
E y
.y E E.
.
. .
17(:
E E q:rrqr!qqq!7';E!:
E
.
E E.
-
:
E LE (E : E:'. : E E!.
.
. .
i
'
. -
..
.
; y-
,.
-j -.
y
,:-.---.2,
.
i
'
.
:
q
$10,000x (1.06)
........
........
Payoff at expiration to $10,000 investment in an equitplnked CD that repays the initial investment at expiration plus 70% of the rate of appreciation of the mxrket above 1300.
payoff of CD
12 11 10
l l
.
X- J1300
9
j
.
1
8 600
.
.
1200 1400 1600 s&p 0 at Lxpiration
800
1000
'
..
'
'
'.'.'
;'j'ql'y q)f :'1t-*. l(qj' r' :t--''. r?;f iki,f )'. y' jjt!)f jjgyty.yjf il' .-' #' l!q! )' )'t'.-rq'lli (j)'. ..' i i'. !( !ii' ( ( 7. ! l')l')'. !1(:'prtrq (' i !l i (q((. .' tlkiiijiitk:)d likiiiiitrr'' jtiyi '. jtj'..). yi .. q . . )j):!:.(()t( ljj)lyjqjjjjyjjjji (jjyyt)..yt)tjj(yjyjyjj. )l)1. q. . ..y: ..ri. .ljjjjikl... . .'. ' tl'tiiIIi-t:.. . i kyjj .'. jg:;,, (rlti---'' kq. ; )ji ' '.:L.i-.1jiibiibiiL'., t);.!t-lt .. pi.i .i-. ... . . ..
tjiiiij!)jili;td llff ''-
''-..' '. .'.'iiiiiiij:.. ..'.' ;
:
-
.
-..
-
'
---..
-
-
-'.
.- ...-
-.
''
EE
' E
E. .
: :EE : E(E
.
.
:
IE
'
'
..
..
EE :. i !:(E E!. i ('E fiE' .;:.:. ; EE. ! ;I E:E :!?ttt(.E)y-(jyt:-jyyE . .q)@trF .:.p.......yy.yyq:yjjk.t-!gjj. y-jyj.yjj.ygjjjjjj:.-rF ------)itt--. ------:t(t.. -. -- . - - . - --. .- .-. . E
'
..
-.. . . .
.
.
.. .
-
..
. .. ..
-.
.
. . . ..
. .
- .
.. .
E! .
r!E
.. i E i ;E ( .ljyyljqylEyyjytjjyjll..j.-ljj-jjj-yyjj..j.j.jyyqsy .
. :. .. . : .... .
.. : :
lpE
-
-).
.
.
-
. - .- . - . . -
. .
.
. '
1800
2000
jj
...,
.
jgyg;jjjjr,,syyyjj jy j;..;j k;jyty kyjj jkjjyr jgyyy;jj jyyy;j-
jjjk jjj:y
jtygjjjtryyyyy jtyyij,.jj:r jkjyjr j;ry;jk jr..gjjtgyyr jt.,jjyjyjgyjjj,.
.jj:gj::
jgry:j-
-r-
.
.
.
Ej.(jjj-yy..:..-.y
'-.
.
.j4yj
.44y,
-,
):.
-E i. tIiiijtqjjiiiE. i ! E. E:. . .. ! E -!. . . . EEqEE -:. E yE q q.. - .. . y-y. . y ' ( . E: .E y. yyy: .y ( y y: y. j (.. . ..E : 'E.E.. j j j.j j .j. . yg g g j.y j . .. ... j y (. :..: .. y. g y j . . g jj. g. j j .. .. .
' .(j1111)j)j1;::k;;,. .;j'.((.1jti!jppii:j '. .. ((yr'!(y.j....jjiyj(y......qjy(j.jl(.j.li....j(y.(.. yj(yj..;jyj.jj..yy..yjy...yj.jjj..jr...jjj.y.yjyjjyy(..y..j..(jk.qj. ..y. y.... j .y...j. y.. .. i EE .. . .qE.. j. .
kIk)jiii((. t1Ik5l(((' (::!(!j:rr. ;.. !. lqjjlEEiiiiq y. q.y ( ;..'y.. .. ' y. g.. g. y... g . . .. .. . . .. -
'
-
-.
. . E .E E E- y . . :. . ..
.
E .
.
-
.
' .
E- .
).
( -.
-
-
. ttkfy'...#d.#.,.,J:,,..t))t!.rr),.r,.g,p@,,,r,,.,k. >. *.),.,,r)g.,y....,,yq,,,.l tj)','. j... (.:
j
. .
.
'
:
.
;
E
:
.. .:
'
g.
E. :. E. ' i. j' E ..
: : ..
:
:
. ..
q.
. .
.
EE
: .
:
:
:
. . . .
'
.g y
'y
.
.
:
:
,
: . .
.
.
.
.j.j..j
.
:
,
,
.
.
:
':
:.
' .
i
'
.
E.
500 1000 1500 2000
$10k000.00
2500
16,461.54 19,118.85
3000
10,000.00 11,076.92 13,769.23
Economics of the CD Now we are in a posiion to understand te economics of this product. TIAiIIIC about what happens if te index is below 1300 at expiration. We pay $10,000and we receive $10,000 back, plus an option. we have forgone interest on $10,000in exchange for the possibility of receiving 70% of the gains on the S&P. Suppose that the effective rfhus,
$10 000 ,
-$2
=
,
742
y Equity-lainlced CDs?
.
-' .': .','(' '$' -','t''5* -q-' --'i .-' )' --.)'r' -iiiii@5lll;'.' -'
('('(' :'j'F' -#' E-.'' 6k4* )' -.'----'---kj-----)-)'-' p' Ir' '((1* t..' ?' tlf @' ' ,'.--.-----;jqk.-' )' i'qqf ?' (' q' ' ( Ei; i lljjqj
-
Essentially, the buyer forgoes interest in exchange for a call option on the index. With this descliption we have reverse-engineered the CD, decomposing it in tenns of an option and a bond. The question of whether the CD is fairly priced turns on whether the $2742is a fair price for the index option implicii in the CD. Given information about the interest rate, the volatility of the index, and the dividend yield on the index, it is possible to plice the option to determine whether the CD is fairly priced. We perfonu that analysis for this example in Chpter 15.
la
.
51
-5.5
.-;-.)-
-s.
$
annual interest rate is 6%.,after 5 1/2 years, the buyer has lost interest of
(' y' :'(' )' yjty'r !r7E'r77rr7J;FqpqE(' 5* r)f 'y qr!((' yjjjjrrjjjjy--ljyf 1;.T): ;','q' t' i' :;1;l!(:1' p'.)1rk)' C' j.Cii ( )((;!'(!q!: (i' ((q' (7q)' (' (1* 1* !' 2* )' 1)* fiitff'lh.;llq!lsisi!!lliqi)if ?' ((' ql' jtf )'jl j(@' i:' k.'((E'i.' i;(1. y'. .y' gyt'r' lJ( 2 j', y'yt'. y'. jj-.)jjy('('. y' rjf t'i'tqf ::ti j.kyf j'j;'. y'i!)')' y' tyf )'(' ! . ' ti'.'. (...j... ..' . . . y(y'. jj(jjyjjgr;..' 'i '? (q't( gj;y;qj( g. ((qj. . .y,;.ttjy , j;tj-l.gg.yy F'lj yjyjg yr-jy y.. yqyy qbLqfb j j j !-y.....--:'.'...'..E.,b... . j .jgq . . . . . -kky.:g-kttqir--.' j . . . y ..y r tj y j )L)))))j)). );'))' ltlt' ..'........i1E''1:j t!. yy yyijjjjy.jy .Ii@Ejt.. .yg tj;g ':::)/jiq.l?d:::)l 4(: 11::1::,.. ql.;ls iEi..:,rr,k :!i; .(jjjyyy.J'krj;;-' ).k-r?. lEF,:))!tttr..-1iiF'q;.-..., . ,.....-...L,t-,L.-.-----i....,.,.. s dt::),!1F s ,::f1L ).!)..-)-...t.t..-).r-..-.t.,..i--q-., !)):... . . ()p '' ;'-','L,' plytj.ty'kyjyyf F' j.'' i :
N KED CDS
With reverse-engineedng, we see that an investor could create the equivalent of an equity-linked CD by buying a zero-coupon bond >nd 0.7 call options. Wy, then, do products like this exist? Consider what must be done to replicate the payoff. If a retail investor were insure to an index westment using options, the investor would have to learn about options, decide what maturity, stlike price, and quantity to buy, and pay transaction costs. Exchange-trade d opt ions have at most 3 years to mamrity, so obtaining longerterm protection requires rolling over the pitipn at plp pnt. An equitplinked CD provides a prepackaged soluiton. It may provide a pattern of market exposure tht many investors could not otherwise obtain at such 1ow transaction
.
costs.
The idea that a prepackaged deal may be attractive should be falrtilir to you. bpilding blocks, as it wer-and they Supermarkets sell whole heads of letmce-salad also sell, at a premium price, leytuce already washed, torn into bite-sized piecs, and mixed as a salad. The transaction cost of salad preparation leads some consumers to prefer the prepackaged salads. What does the :nancial institmion get oui of this? Just as the supermarket earns prost on prepackaged salads, the issuing bank wants to earn a transaction fee on the CD. When it sells a CD, the issuing balzk bonows money tthezero-coupon bond portion of the CD) and receives the premium for wliting a call option. The cost of the CD to the bank is te cost of the zero-coupon bond plus the cost of the call option. Obviously the bank would not issue the equity-lirtked CD in the srstplace unless it was less expensive than Zternative ways to attract deposits, such as standard CDs. The equity-linked CD is risk'y because the bnnk has written a call, but th bank can manage this risk in several ways, one of which ij to purchase call options from a dealer to offset the risk of having written calls. Using data from the early 1990s, Baubonij et a1. (1993) estimated that issuers of equity-linked CDs earned about 3.5% of the value of the CD as a fee, with l about 1% as the transaction cost of hedging the wlitten call.1
l IA back-of-the-envelope calculation in Chapter 15 suggests the issuer fees for this product are in the neighborhood of 4% to 5%.
k
k
52
AN INTRODUCTION
FURTHER
AND OPTIONS
FORWARDS
To
F' y' ;';,-jjjjr'.. k.:-jjjj(j,jjjjy''' -L' :'@..' (;' )' 't' k' r' iqlf t:t(' :$*)(EE (ll:r'f (1111q:11.* LLLL'-'LLL'L ('.'!' E' )!k,pkr'p(' )' t'. (r77' qsr)';f jj)f ( j! .11:::* .11t::* yy'. r'r'kjjjyjgr('-.'. )'. r' jjf q:)q''lq:!!;qql'f )' 't)f j'('('@j' ;q(I(qpE(((' 14* ;r; iq(!'! E;Ek '.Eq(. ( ('(: jj'. (' j):ytf jjljjqjLLj,' y' (q' j'. j(' lyyyf ;j;::k(1t1*. .kt:'$:;r' yjf j'-;,g'y (;'. rqf jjf '' . :$!().j 1. yfyjjj t'tyyj.'. yj.' y'. rf )' ;'j'j'. (' 1IiIE:F!q'. 1*. .. i.' ij.;. j .'( (.))I t'yf 5lt'. ' . )
'. .'''..'. . The basic profit
'k'..' -yjy'; 'Etf il:::::,'f .1:::2:,1* ''-'-..-' C' r' 'i'7* .!' ;)' y' E' .; i !:) -' ---.'...----).'il('. --jjjj----,r'yf .'-y'
ln this discussion we have viewed the equity-linked CD from several perspectives. Theend-useris interestedin theproductand whetheritmeets afinancial need atafaircost. The market-maler (thebank in this case) is interested in making a proft without bearing lisk from having issued the CD. And the economic observer is interested in lnowing why equity-linked' CDs exist. The three perspectives overlap, and a f4111explanation of the product touches on :11 of them.
: .
.
'' '':E' E'!' E E ( .. . E E .. . '' . . :y; E . .-... E.E. E . . ''
'
E
'
:
'
. .
'
.
: 'E : : ' i E (''' E 'E(C' : EE .. ... E. . EE. . . -. E jii E ! : ..,kE, E.i E ;.EE q .qyy.E.
:
' j.q (. j.1(
.
'
.
.
.
5i
i7'E
;
'.
-
'
E . Fj.i.
.
-
i
. -
- . .jjyjgggj(;. -
-
-
. . .
.
.. . .
-.
..
-.
. -
. .
E: . ...
-
.
.
E
. .... . . .
.
... .
...
-. -
.
.
.j
.
y
-:. .. .
..... ... . . . . . . .
.
..
:
. -
.
.
...
-.
.
.
.
. -.
.
Ey
E-.E. E . . .
-
.
E
E -. CE .E. EE;! ..
-
.
.
.E1(:'. (11EF:1.Long forward
diagrms: Ibng and short forward, Iong and short call, and Iong and
i'1i!:lii''. 11111111:7*. q' #' y:' ljgjljjjjf jjf .jjjjjjjjj' ).)'rjjrj(jjryy;.. .)' t'r' j)f !jIiiEt:i'i;'. illt:!r:d jL3bb''. 'jjiild i1!ii)1I;)'. tElilEE1:'i';'. ..' (' ytf j)'. yj)yjf j,jjjjg,jj.'. jyt'li7j':jqjtf pl!!rd )' 1t1*. y..tyj'j' tty.ytyyjfjyyyjyljyf t'j'tt'kg ('. y' ijjjkj;'. 1* tjjitjd 4)78 .t'----(. ..' ktjjjjjjyyk'r jr:!j;;;;.rj,,r.y,jjrrjjj,k'i1il'. 1i!iii;' yjjjyjrjjp.'. t)yy'. 1.t,'yj; (. ..j y;. . .j.. jy. ). y )ytjyj yy (. ..rjqigy; jy $'y. tj;,. i ;g. : rr 't'..: r.t:t-'. .jjjjt-jjjj ii y :. ty i. . .
.' .'..' .'tj;:jjjgjj'. ' .
.''. . ..... . itiE
.
.
.-
:
' ' i :.
:
--
-
.
-
.
E
:.
.
. -
-
..
' ' .:
:
-
' :
''
:
.
. .
-
.
. . . .. . . ..
-. -
......
. ..
--.. ..
.
. . .. . . . . . . . . . . .
.
-.
. . ..
-
.
.
-
.. . .... . - - .
. . .
-
-
.
'
.
.
-
.-. -
,;,,,,jj. ;jjyyj;;jyy,j, jjyjj,,,,jjj jjyryy:,,-
jj. jjj
,!jj::.
jjjj!:!jjj-,
j;,,,.yjj
kyj!,ji
jj,,.,(:jg;; ,,,.
jj:ryy:,,,
.
.
-. . . -. ---
jj:ryy:jjy
jjj::::.
,
E E--. -.
.
.
E' :
-
.
.
-
-
.
.
...
-. . . . .....
:
.
.
-
.
-
-
..
:
-
.
-,,,
,jj,,,jjj jj,,,,jjy,,,,jy jjy.,,,jy,,,,j.rg;;;
. forward and opton positions.
,,::2y:),,
(: ( j Ej ( j ( ((j(j ( (: (j j (( j ( g ( gE1iI:::lIi).. jjj jg jj( j:y g( t 11i;ii,,:. E ( j ( kj j jj j: y( (11::-:-. j( ' E. t1Iiii!'-' k1iI(:'::.E)!iik..IiEE?Ii:. 1iiiiiii: j.g iiiii!,-. ;,!ipi;:. jj;. !.j Ey (jIIEpt1i:i.. . ':!iiiii. 'iIII:i. '..ttki;f-k'k--;.(iiiq.. ' t.i-!.. ... (. .. ... .E E.y(:E. jj.(j(.y. . tt .'. ' (1114:--. '' 11it:;(iii).. . ..' E E. . rlilltEFlii'q .. i E. illi:i. ' .iIIi:.. t1i:i:1,k.. ' '.' .kkii;;'' ;.. .r '.r.. '.t(IIit((!::il,-q. ( j ;.;.. ( y. r .y. . q. . ..q . y. y. y ;..q.. jy y... ;. .. . ( ... . '. .. q; q. ( '. ' E . . . . y j.y. ..y ( . r .. . y y . j . qE . g ! . y .. ; ... . . .... .. yjy. .. k--. irjjjjr-k-qi. y .g.yiiltE!. j . j .. . y.. .y . .; ,.. .y i(. : . . y y . r .. y ... g . . Lt;iib'1. . . -i.y.. . ii'i:i'''. 'i1I:5Eiii2. . . . lll::!ii'l. . . ( . . . ! . q q q .. k1iiEiil''. . . . q.. ..r rkj,jj-k,. . ..j:;(jk,;j,. y E E' ..jjjjyy,yj;jj,.. .. .. E.E. .. . r..... y.... ... ..... ..jy yy kjjijjj-. . (yy;. .E-y.E.E..;'..t..jjjtjg:jjjrjl . kkjjj,jj.. ..y rijjjjpjrijk:rjj-i-yr ... (.-;!,,jjk,,jj. . kj,2;gj,.. . . . . ;;.. ; . '.. . . ( . . .. . 'r. .'..).'.'.'-. ....(...E...E.... .. . ... . . .i..'... . .. .. . . r...q.. . ....,.. ... . . .... ....... . ... ... .... . .... . ... . ....... . Always Long forward Long (buy) Guaranteed price t
E
)
' E.- .
.-iii);'k'
.
.
.-
. .
-
' ... E .
.
. .
.
-
. - .. .. .
. . .. . . .-
.
:
. E -
- .- . .. - .. . . . .
.
(E E:
' ' . . ... . . . . . -
-.
'.
.
-
-
-- - . . . .. . . . . .. -. E -. .. . . .. ... . . .
.
.
-.
..
.
.
E
.. . . . .. .
.
..
j,k:yjjjj(,. -
.
.
.
' ' '
'
- . .
- . . .. .
-
-
.
. .. . ..
..
- .. E .. .
... . . .
-
- -
.
.
.. .
-. . ..
.
.
. - .. . . . ..
-. .. . . -
.
. .
. . . ..
-
.
.
.
-
.
- .
-E
'
. -. .- . E .. .
. -. . ..
- - -jkjyy,,;;. .. -
. . .
E
-
-
. -
.--i
.
.
. . . . . .. . ... . . . . .
. .. .
.- . . . . .. ..
.
.
. . . .
... ..
.
. . .
-.
. ..
..
'
-( -. - .. . . .. --jk;jjj,,j. jjkkryrr:. . . ... . - -.. q
-. ..... ..
-. . . ... - . . .
Short
(sell)
Always
Long call
Long
(buy)
>
Short
.
-
Shol't forward
Short call
-. .
(
'j
.
.
.
.
.
.
. .. .. .. . .
)'
-
(j(
i: y
.. - . .. .- - -
-.
.jt
jE
-
. -. . -. . . ..
. . .. -.
.
-
Ey g : -. . -. - -. - ---. - . . y -
,jjjjjy
- .
.
.
.
- . . -. . E. .. . , .y. . - -
tj!jijj;jr .ijjj:g
.
- . -.
.
-
.
. - . . ... .
. . . . . . . . .
.
.
Ej
:
. -. - - . . . ...
.
- i- . . ..
jE E
::
E.
(se1l)
>
. . . .
-
. . . .. .
...
.
.
. ..
-. .
.
. -. . . .. . . . . .
-.
. -. .
.-
.
.
lnsures against
Strike
high price Sells insurance
against high price
Long put
Short
(sell)
<
Stzike
lnsures against
Short put
Long
(buy)
<
Strike
Sells insurance
low pric against low plice
....) . '.
- --
.
. -. -. . -
. .
.
.. . . . - .- . - . -. . . . . ...
. . . . .... - .
Guaranteed price
Strike
:
:
E -. . . -. - . .- . - - . . . .
.. . .... -
.
l
E
.-k!i!di1.k-)..
:
-
'
-
E
:
(jIIt(;1i,I.
' '
-
.
.
E
:
:
(1I(iqtii. EIIIF:)'Short fonvard
'
short put.
stockPrice
Forward conacts and put and call options are th basic derivative instruments that can be used directly and that selwe as building blocks for other instruments. A long fomard contract represents an obligation t buy tlle uderlying asset at a lixed price, a call option gives its owner the right (but not the obligatin) to buy the undyrlying asset at a sxed'price, and a put option gives its owner the light (but not the obligation) to sell the underlying asset at a fxed pdce. Payoff and profit diagrams re commonly used tools for evtluatinj the risk of ihse coneacts. Payoff diagrams show the gross value of a position at expiration, and profit diagrams subtract from the payoff the future value of the cost of the position. Table 2.6 summarizes the characteristics of fonvards, calls, and puts, showing which are long or short with respect to the underlyig asset. The table describes the
,' -.' ''' -' -r-';-.l)'..' -' ----' --' -).jjjjj!jjjj;'. :
Stock Price Profit
Long call
Shol't call
Stock Price
Stock Plice
Profit
Profit Long put
Short put
stok price
stockPrice
strategy asspciated with each: Forward conacys guarantee a price, purchased options are insurance, apd written ptions are selling insurance. Figure 2.15 provides a grapllical summaiy o f these posidons. Optios can als be viewed as insurnce. A lut option gives the owner the right sell if the price declines, just as insurance gives the insured the right to sell (put) a to damaged asset to the insurance company.
. ..
FURTHER
53
:.
i. . .y !. y .
Profit
)' --'. -' ,'-' -))),' 'jpt'llll'. -':).'t(' -i.t'' jjrjjy,,f j,:ryy:yf jj::ygrjjf ,'':'j' jj::yrryf ('E :'y'E ;'f1* ! E'IE EE E E ' i E '' ! E E dklritplk'y l1i!!!''. .y' )' ;f ' IC PE (lE i E. ' E. i. !E :';'k1Ittir('y' -'E ;,,)-' iiyjjjf ;(' k;!gjj;f k;jjjyjjf ;';;'y,j;jr)ijj,j$,'tj iijygjjf jjy.,,,!j,;,,!kj,,,,f k' ,'p' (k' 574* T(' 1* E E' ' ' TE i i'E((E :111)*. '!i1IE:.iIi)'. :iii1;' (111t*. )' iI''@:Ei'..'.' jjy.kyyy)j,,;,,f .jjjyy.:!j;;' .jjjrr)' ::y' y' ;'E'i'@' (' yjj,(rjjj;.'. q' :jjjyrrjjjj.'. jjjj:yd 2* 1* ltl;f )f lf r(' j(' ii (E!'i' (IE.E'E?(
[email protected]! . .. (. . (E( $. ))' 1iIj,!i'. 'j)'. C' r!jj;;f r' )' j'(;(i(.'qli;@ (' q;' q';F''qrrp'q'7;'ql .. i!' ) i . ':.!E:'i' y;) .'..' . ((('q:.t.')F-. . . E. . .t'i ... 1:1 . E. ! i. . ;p.E.E.E E ., . . .:! : !; j ;E;. i .Ei. ( ; E. , ; EE j --,-);. y yjy y y. y ; yryy . :!. -(i . r.r-(.y g ; . . y;. .y . j . . E E .
%.
! !q q!'qq'ql. ''j(' ;'.(q. (''):;'(1
:
' .. j q. '. ty;'yj. (y. j j.j(yjt..y. q jy. tjjt.tq jyj. )r.. r. . . .. jjjjy:;jjjjj. jq t)j 4y. j y.y . ..yrj...yjjjjj...j.y.lgyy j ! . @. y :p .;t 2 iqi )lr ... ' . . .li t! .. .. t... . ... . . .
READING
READING
We use the concpts introduced in this chapterthroughoutthe restof tiis book. Chapter 3 f basic option stratygie! which are Wilely used in practice, including Presents a'nmbef nd Qoors. collars, Chapter 4 presents the use of options in risk managerent. caps, A more general question raised implicitly in this chapter is how the prices of fomards and options are detenmined. Chapter 5 cpvers financial forwards and f'utures in dtail, qnd Chpptr 10 inoduces the basic ideas underlying option pricing. Brokefj iutinely su/tlly optios customefs with an introductory pamphlet optin titled Chal tacteristics :7l7# Risks of Standardized yptjsn'. vus js availabout You can also obtain current option prices able online from http://www.cboe.com. CBOE'S sites such and various brokerage sites. Web from as the
k
54
%
AN INTRODUCTION
PROBLEMS
AND OPTIONS
TO FORWARDS
The notion that options are insurance has been applied in practice. Sharpe (1976), for example, analyzed optimal pension f'unding policy tnking into account pension insurance provided by the Pension Benest Guaranty Corporation. Merton (1977a)obseryd thatbnnk depositinsurance and in fact any loan guarantee can be modeled as aput option. Baubonis et al. (1993)discuss equitplinked CDs.
PROBLEMS In the following problems, i.f the yields 1 + ?' after one year.
4teffective
annual interest rate'' is r, a
$1 investment
2.1. Suppos XYZ stock has a price of $50 and pays no dividends. The effective annual interest rate is 10%. Draw ayoff and proft diagrnms for a long posidon in the stock. Verify that profit is 0 at a price in one year of $55. 2 2 Using ihe sa'me information as the previous question, draw payoff and prost diagTnms for a short position in the stock. Venf'y' that prot is 0 at a price in one ' of $55. year 2.3. What position is the opposite of a plzrchpsed call? 'Fhe opposite of a purchased P ut? a. Suppose you enter into a long 6-month fo-ward position at a forward Price of $50. What is the payoff in 6 monts for prices of $40,$45,$50. $55, and $60? -
any advantage to investing
Why?
c. Suppose XYZ paid a dividend of $2 pr year and everything else stayed the same. Now is there any advantage to investing in the stock or te forward contract? Why? 2.8. Suppose XYZ stockpays no dividends and has a currentprice of $50. The forward pfi.ce for delivery in one year is $53. Jf there is no advantage to buying either the stock or the forward contract, what is the l-year effective interest rate? 2.9. An offmarket forward contract is a forward where either you hake to pay a premium or you receive a premium for entering into the contract. (With a standard forward contract, the premium is ero.) Suppse the effective annal interest rate ' is 10% and the S&R index is 1000. Consider l-year forward contracts. a. Verlf'y'that if the fonvtrd price is $1100,the jroEt diagrams for the index and the l-yellr fonvard are the same. b. Suppose you are offered a long forward contract at a forward price of $1200. How much would you need to be paid to enter into this contract? c. Suppose you are offered a long forward contract at $1000. What would you be willing to pay to enter into this forward contract? 2.10. For Figure 2.7, verify the following'. a. The S&R index price atwhichthe call option diagram intersects thex-axis is $1095.68.
c. Comparig the payoffs of part () and (b), which onact more expensive (i.e.,the long call or long fonvard)? Why?
b. The S&R index price at which te the same proft is $924.32.
should be
b. Suppose you buy a 6-month put option with a strike price of $50. What is the payoff in 6 months at the sate prices for te underlying asset? which coneact c. Comparing the payoffs of palrts (a) and (b), more e kP e j ive (i.e., the long pt or short forwrdj'i Why?
should be
.
2.6. A default-free zero-coupon bond costs $91 and will pay $100 at maturity in 1 year. What is the.effective annual interest rate? What is the payoff diagram for
thebond?
'
The proht diagram?
Suppose XYZ stockpays no dividnds and Vs a currentjrie of $50. Thefomard annual interest Plice for delivery in 1 year is $55. Suppse the l-yeaf effctik rate is 10%.
a. Graph the payoffandprostdiagrmns with a fonvard price of $55.
for afolward contract on XYZ stock
'
55
in the stock or the forward contract?
b. Suppose you buy a 6-month catl option with a stlike plice of $50. What is the payoff in 6 monts at te snme prices for the underlying asset?
a. Suppose you enter into a short 6-montll forward position at a forward price of $50. What is the payoffin 6 months for prices of $40,$45,$50, $55, and $60?
2.7.
b. Is there
k
call option and
forward contrat have
2.11. For Figure 2.9, velify the following'. a. The S&R index price at which theput option diagram intersects thex-axis is $924.32. b. The S&R index price at which the put option and forward contract have the same prost is $1095.68. 2.12. For each entry in Table 2.4, explain the gain or loss occurs.
circumstances
in whih the maximum
2.13. Suppose the stock price is $40 and the effective anual interest rate is 8%. a. Draw on a single graph payoff and prost diagraps for the following options:
$9.12. (ii) 4o-strike call with a premium of $6.22. (iii) 45-strike call with a premium of $4.08. (i) 35-strike
call with a premium of
$
AN #NTRODUCTION
To
FORWARDS
AND OPTIONS
APPEN D IX
b. Consider your payoff diagram with all tlzree options graphed together. lnmitively, why shotzld the option premiumdecrease withthe strikeprice?
2.14. Suppose the stock price is $40and the effective annual interest rate is 8%. Draw payoff and profit diagrams for the following options: a. 35-stlike put with a premium of $1.53. b. zm-strike put with a premium of $3.26. c. 45-strike put with a premium of $5.75. Consider your payoffdiagram wit :11 three options graphed together. Inmitively, why should the option premium increase with the strike price? 2.15. The profit calculation in the chapter assppes thai yolz bpn'owat ktftxed interest rateto finance investments. An alterntive way to bon'w is to Short-sell Whatcomplications would arise in alculating prot if you tinanceda $1000S&R index investment by shorting IBM stock, rathr than by borrowing $1000? 2.16. Conseuct a spreadsheet that penrnts you to compute payoff and proht for a short . and long stock, a short and long fonvard, and purchased and wlitten puts and callst The spradsheet should 1et you specify the tock pdce, forward price, interest rate, option strikes, and option premiums. Use the spreadsheet's mlx f'unction to compute option payoffs. 'stock.
.
'
' ' '' ' '
'
2.A:
MO RE ON BUYI NG A STOCK
$
OPTI ON
In June 2003, lomega Corporation declared a $5 dividend, payable on October 1. At the time of the declaration, Iomega's share price was $11.40.Since the dividend was 44% of the share price, the OCC reduced a11Iomega ption strike prices by $5, effective October 2.12 When we discuss option pricing, we will see that it is necesjary to take dividends into account when pricing an option.
Exercise Some options, for example those that are cash-settled, are automatically exercised at maturity; the option owner need not take any action. Suppose you own a traded option that is not cash-settled and not automatically exercised. ln this case you must provide exercise instructins plior to the broket'j deadlin. If you fail to do so, the option Will ekpire worthled:. When you exercise the option, you genefally pay a commission. If you do not wish to oWn the jtock, exercising th option would require that you pay a commission to exercise and then a cdmmission to sell the shares. lt might be preferabl to sell the option instead of exercising it. If you do wish to own the underlying asset, you can exercis the option. The option writer who is obligated to f411:11the option exercise (deliveringthe shares for a call or buying the shares for a put) is said to have been assiglied. Assignmet can involve paying a commission American-style options can be exercised prior to expiration. lf you own an option
AppENolx A
2.A: MORE OPTION
s'rocx
oN BUVING
The box on page 34 discusses buying options. are at least four practical issues tllat an option buyer should be aware of: Divideds, exefise, margins, and taxes. In this section we will focus on retail westors and xchange-eaded stock options. Be aware that specisc nlles regarding margins and tus change frequently. This section is intendd to help you identify issues and is not intended as substimte for professional brokerage, accounting, or legal advice. rfhere
Dividends The owner of a standard call option has the zight to buy fixed number of shares of stock at a fixed price, btlt has no light to receive dividends a paid on the underlying stock over tle life of the option. When a stock pays a dividend, the stock price declines by approximately the amount of the dividend. This decline ttle price lowrs the ret'unz in the owner of call option. to a F0r exehange-traded options in the United States, there is typically no adjustment in the terms Of the option if te stock pays an dividend (onethat is typical for the stock). However, if the stock pays an unusual dividend, then oficials at the Options Clealing Corporation (OCC) decide whether or not to make mzadjustment. 'Eordinary''
and fail to exercise when you should, you will lose money relative to foll6wing th optimal exercise strategy. If you write the option, ad it is exercised (you are assigned), you will be required to sell the stock (if you sold a call) or buy the stock (if you sold a put). Therefore, if you buy or sell an Amelican option, you need to understand the circumstances under which exercise might be optimal. Dividends are one factor that can afect the exercise decision. We discuss early exercise in Chapters 9 and 11. E
M argmsfor Written Options '
Purchased options for which you f'ully pay require no margin, as there is no counterparty
fisk. Wit.h written option positions, however, you can incuralarge loss if the stock moves against you. When you write an option, therefore, you are required to post collateral to insure against the possibility that you will default. This collateral is called margin. Margin nlles are beyond the scope of this book and change over tile. Moreover, different option positions have different margin rules. Both brokers and exchanges can provide information about current margin requirements.
lzlkeducing the strike price by the amount of the dividend leaves call holders worse off, albeit betteroff than if no adjustment had ben made. If S is the cum-dividend stock prce and S D the ex-dividend stock prie, Merton (1973b,p. 152) shows that to leave the value of a call position unchanged, it is D) 1 necessary to reduce the strike price by the factor CS D)/S, and give the option holder S/S additional options. An option with a value protected against dividends is said to be payout-protected. -
-
-
-
j'E' ,'k' 'j' !t' ;'!-' Eyd )' E-' y' 1* 't.yjjy'.,'jqj'qj'ryr.-jjj' :,t' ;'('t.jjjyj' jkrd -f' j'E(' ,;' --' i'i' jp:j))d q-' -.Lq' E' :':.j;' -,---' k,' yyd ''F' Jid 5.--* k' )' y' ,d t-yd q' yy'.' ,'''.-' E;' tfEyyyd F' L;' -' j-' ,)' :':' ;r)jyE,(g' ,':' .k)' ''' :1* ''i' 5* 1yj' (' Etj' jytd !j' jky)'-jjtr)'(E' E-' jtd Eyd Ed ;'F' yjtd ,yt' t'y' ,',rpiti-drlijrdpkrpprrd-iil''lllrEttr;r))d ,d ;.yy' ;'C)'.' k,jd #' Eyy?ijd k.-jd (' )' jjy.d s' !yj)' 'j'j' r'r' ''k' iyy'jgrd qyd ;rj' q' ,,' (yyyttt'.:jyd '':'?'4* !j;t' ;')' :.-' ,''Ci' y' jjd jd s' k.y)d p' y' j:*),* ,t' ,'/' j;gyy'.' .'.)' ,-'yyjyjryyr'-' jyd k;' $' .j' qi-.gd jy'tytj'-' (4* .j';yg'k' .yy(' p)' ;'r' .y' y;' t'q' yyd y' ;'-,y.)' j'j' ;t' E(' j(' ll'td :(' 4* ))d l'jd jjd ;:;!;;';;' lr#'' .:' kjd j(' )?' jy'.y' (' .tt' jttk.y-jpjd -F-..' ')' jrd j;' :'''pyd r' kt.-ttk' j't' ;'!';'t'Er;' d.,' jj';' jjd j'y'j'(' jrd T' g' .j' .g.t' jjji'iLkj-'jkiq')' itd ;j' t'(r' y.#' 1* q)))L'L3'3',,jj:L-.LL-;.b$'b..L' jjl'yyyyd t!.' )')' )q' ;)tjt$)'--' ,'rd jtd ytd !' g)' 'jj-;tj)gd )t' 'id ltprd y:' $' !..' (td ).p*:r* #' kkd ';lld !lk' (yyr'grj.td /d yjyd (jd .t' .;' 'j' q' (j).,...,t)I)(t((j;'''' )' yrtd k'j)d ird i'1* j)td ('.rt.;jtjyy)y-j;.'.' f' jy'yd y.tyj' ).)'' p' 1t'..kj$' '.-' .j' )j' (l';jd ;'):jd p.rd jkjd trd (t' jyyd :C' yjjyrt,jd llitkd tt';d Ctd p:d y' ijpt'jd 'E!,d ppt:tqft,d ljks-..d ,p.!!j)'t5--t.t' ))' (yrt(yrg.' l'td jr'l',rtd jrd gyld ;'-@-t' yyy)yjr)r.' .)(' jjr'-;jr.t;--' g' y'(' ;'t(' (rd jjd yyjd j-'-' j),.' ''jtld rkjrd .....-q;3LL3L...-j)L;j):;;'.;.63L3j';tt;33i3L.' --pkIqq'pjy(t)jrjjr2(jyjg(-.' iid k)rrd t)d F' ;(' yrd j(#q'(tt' jjrd ))!..#i' @' )'j'(111..* y' l't-d ryj,'--.'(jj;y;yy4-(jy-'y' r-'-' )!p).jk' (gd ;'tid pjt'tq.:)d :'(tdjipd y)' y())'.-.!th('.k;r' q,' @Ep' y'tijd kjtkjjd f' ;,d )' y'td lt...' .-' (' )p' pr)d (j' ;((j)'j)' q' p' k' ,,'LLL' (;)ji..' ' ttp:j)d .y-.' kj).,' t,' ;?' y,' @' '!)' 1#881t.8r71*.72111!711)21#)@7)*-th49):12:/pjtq8)15*5* y' t'-jd r'.d t'yyd ;)' ?'.' tyd li' tjd ..----(rC:;' 't(' ytd t'yjd )l' ($((' (r?d t'l!d 'y' jjgjyy.--' ,'lgy?d tjyd .r;' 1* .4* lkd 4* trd ?,' $)(* tr;d gkd !' (td (' t-sd t..rd ttd ts.-yd t'g' td q'Iilb' t'-pd tfd tjd t)' ljyygylykd t2'l' tj)d t!td t't' t.pptld r@' j'yd tihd ..)' ''t(' trd tg'yd t(' (tllyf'd t.id tp'.;d ((k'k' 't' tr..' tq'-.yt' rtd tp;yj;)kd jjjyi.tltid $;.)j';L.j);LL;L)'.' r' tz'z' r,td f';3LLL31)'.' .r!t(;pjjj-.tjr)(jkj...':ph)jrjjjjjrj::tr)y')y:yy))yrgjrjy..y)r'q)jjjjr)---.' rtd .-.' .'jjj(.' .'''......''' .y' ..,' +'.,,' ..yyd .(' :)' .y' -(r('ry' ,'-'-r' .;' .-'-jjj' '''jjjj!q,,-,' .q)' .j' .f' -'-' jjjppllllll,d ''.'-'-yd -t' '.'.l#' --' --' -' -,' '-)'' -' -j@' ...*'4j(44jjjjj))*.* .''-' .-' '''f' ''' --.' ..' -/' .'-(' -j' -y' '-t' ,''''' .-tq)' ' -' .l).)ptd
*' 't't.'..'j..,' :';'s' s.td ,),,-' r,' -y' -' -j' kEX -..' --.' .jj9I'IIpII,' .-.' ---.--' -' -tk' '-.' -',';g'!j' g-'-' ,'-'.' ''-'-' -?' --' '';,yyjj;,y.--$k-jj,',-' ---,'.---'---'.' --' -t' --'j))-fr' -'-' '.-' -)' -.t.-'' -y)* -,(.'.' -jj;';' -r' z--l-y,srkrydyktyrdzjtyjylpttyk.dtkgkjpi-k.pd -'-j' !rd -y)' ' -..(;' -(jr)rr' -pqtt'ird
''.'. .' r' 'r)' ttry;'-,' rt'jjttptjj'y;yd r-$' tj---1* tyjjty','
%
AN INTRODUCTION
To
FORWARDS
AND OPTlON5
.'.'(tt$rp;.' .' . .'.'. . . . .'.. .'. . ..'. . .. .'. .'y . .''' .'.'
. -. - i! -- . - -E . EE . -- .E- E PE . - -PE E-- -.-p ! . . - - .. .E- i ;. E.ii.t. E-i i - . ! EE i i . E--:EE- E- - -! E E E -- - ! .. E-. . EE i E - )! . iE jE E E E- jE!! '@- -i --i-. -i ;..- :E .EE!yq .!@ !EE- ;- EtE. i-EE . -.. ;-. E- ;..;'E E .E..--. iE!.E. EE-- -. - E-E. :;... -.... ,)g;ytjr:. E : I;rC'.: i..j: -E jll'tti ;E:ltr EEi .i E ;-i ! .- q E - )'. .@ .(yE (.yIkkt!yr qq: i ir! -!! i. - I p -:.:i..)y(jkjt;(T('. E ;.- . F-;yjpjty)jy;p))y'! jiy.. y'. j-.-'?!;;.! - )kk:. . :. r . ;E!.1t!' i. i k.-;L. .:,1,)))j44))*. ;:E-.EiE.E.i p. ...iE.IE:.;;. (!.-i .;EE-..E.i:- Eij'-. ft-'. - ;. ; . ;- E. .. E ((.. !Er.ij'-;. :. !Ei...(.-..- . ;EiEEi..!E--;.-.!:E .--;EEIE!E@ ;;; -!y y;riE;..'.- i-.-i ;'y)'. -...........-jL3)q3L);.(kbLL. ;: ))i;-Ekkj.L:'. y(E?l'. E- EIi .. :- .E . i!j)! jyl j ;E).E.!-E.i--jj-.E@ Ei . ; :E g- i ; ;.':(E j;i.:E y.i .jj . E - t)'. i1. E''!' 3;3, ; i . q! ;)k;F'. l-- .;E'.i. j.;;.E-E(-!. --.--. - .E 'g.E);'E - ;E. -:.jj Ekr'. i i E Ei (Ei - --.! .. (;.j ..g..- ;yj(;y ; ! IEE . . g .E.--. .;;.! ..EEE--- q . E. ., ((.... q--r;.!.-q.iq.gk.!Eg..;-..)p'. - -,E. - . yjy. '-.gEE . - . '! (. ,33(4-,j33334*. (p EE!.. 5j.jrj);t.)y'qE.E-. .. !q (Lj:'--j,;.. (E.t'q.g . ..jy ; i-. yj'. . E;. .Ej- Ei-.'.riii'Ei ! (.-. i- -'.E -.-;iE:;r.EEE..;.EI;E.E E.;.4.y-'.;-. j. -- --.jyl---p-j; . - jtEp--...-i.Egjq - ; k'. . Eyy.- ;E(q- i.!(;y'..y. gi.-... .E-... (.-.!(Ep'. ;:C((: ' ! ::,-:7:;;)214-'-.-j4/11jj4)4))()q ; . jpi-l-.. ; ;y,'..k'. E.!.IE.!;CE' - . -?;9ri. j-.. rE! ..E.Ey- Ey.(jiy.qi!yqEE.. q5*. pr-E y iii.. ;.-E..-;;Ei ;iE.qE ;))'. .! !-i - l.p. . !.ii -.. E.i- -EEE;E'. ;q- -t(;- i j.'. ).j'. .-i E: .;..:i,;-.;y;i;E).'- -.-!EIE..?ii''i:C . jE.!;.;- i ; ,(k'. (.-; -E;.E.@ .E.- !.-. ; . .- .E-k.(- qEi- . !yj'.. -# ,q. -.. . .j.y:.-k!.t;j't4)t).y((jk;y'(;j(gjj(k);k;.. .. ; - !E ..('.E - E k!j . - .! . ... - . '.E;E;EE.E E - yE - . . .. Iq.- (;. ; .. -. ((qy . E E qyl iE(. -- -i-. E.E....E...-y; E;. )'.; i.j-'qr'. js.'yjlpjjyki'. jt5'.j @ . .. E.EE;- - ! ;. ;i- @'; (r*. . )-'. - -. .. EI .. - .. ! ig E ).!-! -. -61*. E'' .- . E (-E ! .. - .'.-;.tq' . ; : -q - :( ; ( E E i ! E tt'. -EE' --. q : ! (! q ( ktyj kyy(;;. E (.-jjy'; . --;- - ;)(; j q i! sjyg-yyjjjyyj;;y ; ! E E C;E i . j'. ;. ; . y . y ( ! ? '. 1. k ; , . y . p ;L;L'jqjjj4j)'));L. . j; i q . , . y y . 2 q . t'. -. ' .. i . ! q ( q . E . E y , ttrrl''l:ll);)yy-'yyr . lg ; j gj . ;1' . . k)j. ; . ( y y . i E ! . !(k.. yE E y q-E j 'dtq-' kjjjjjydytjjjyry'zjtjjt')-: i- i. i . - .. E-. ; iE. E iE ir E E E -! . . - E E' .y- . . iE-E. ( (- - l -j;ji ; --ql).t. !i;-.E.qE. q;ii!-'!(:E. i-. ''CE !-j)'. . ; ; - Et;p -E .- . ' Ei .. E. '..' i'- El-j'; (E-.. ('.i. y(;'. jy.. (4y)j-ttjt;.jjg'. 'i - E-t: i 71;. -L---31L-. . E. -( y; . y - -. E.j..-. E- y- .!Ei - -- j-?.ljj'jk---ji.' . E .; .EE.-'i .. q E i-. ..E..-. .j-.-jyi.. jjiE;)'.( jy'z'. ; yl.'! .ji;yy-r):pq-;.. :..);..i.;;i-. @ . ;(E.;-;.E;EEEI'E -. Eljdlitpdltd.iclgti --' jyjyjj'. . E E k.'.-.. ;-('- . E- -j. EpEE jE- yjy'. '.#;--.. r.E.Eq-!- - E-..-:.!y.j;..-. j.;!y;-;.!q-y-.;j.-. -,--y!..!j.y. i.! - .!E;E.p..p- ;.y, - ;. qi ! ! .; )!;i))'. . . - .. . E .. E.: g E ' E jyi .i;!! i -)!''j,.):. .. i.-;. i- .-y'.i.))!' )'. yy';.;EE- ,'.E-.-!-..@E '''E E . -'.j'' t@ - -E .j.. .E E: . -k(. i.@. )E. ; ltyy i. - ' E E.- !E.. EE. )j'j-)'. j.. tElE' kl7'-11*.gy! jj. jjy. . . -'--;.E. . @52-;j.. j! j; tE'. 'i! yjj. !y y (jjjyj . ..y,-,yr'(yjyj'-. .()! E. ..;. .!:j.;;g. .i i - j'-'E.EEi.'!;g. i!i;!i-.- E.;E.y'-y;'E. --;t),'$y..yj;.. ..t'.@-r. (.-'q .,*$7*. i.j.jjy-r i-j.. -Ei..,-E-E.i.;E.!!..! ilj';!.(. --..i.-.- E!. l'i. jl@. iyiki'. ;p.:. . . .; ..).yy-'. .j. ; .- jj. ,Ir g!;..'.r(3,. -j.y ;.y!.-- -q- -- y jy-')y - E:E' !@; . ipq E jtr'. ltl.d'-i y'rrt... j'.. .i-;'.rjjjjj'jtt'.:jjy'j;;,,-yj@(-);kpp!. -. . L-.- ;',E.-y'.r;.y- . ..rEE.E);;.kq;! i. tEtE. -i ;-i yr'; i .. g. .. !t'. . . sy' .- y'.i;.;..q;qk!,-j.E-i..iyr.E,.-E yj.. -E'' ' i' !.lfl jt'jlr (j. :yyy . ;.E i j'j j'i k'. ,yk.. y,i jr yj; y j j-; -g,.-yy. .yy- E.. -y;.- ;.jyy.i .jyy-'jylii.. @ , . )'.. q. ! - -; jk . ,q;-:'..y-. ,j;j't'. yr ; --.y-. - j..jyyyy j.E y..j'. i , '- -j'. -rE. ig ! -'. y. q.g ;y -. -'y-'-:):-'. . j . g .jy yy j';;)),!)'yjt..yy'!)'. j . k(j. j i'l''. ; ' ' yg (q.,L,3LLLLL. E ..E.E;.;.E E- -. y y.y'. .. ,'.y yj.-. jjjgjr'yjjjyjly'. 111i'll$ll7)Cl'' ' j -- ( y,)('. q !. . i Ej'. y'q i j - y ---y 'ltt. @ y . rE E i . (((!j!(ry'. -. ; qt-t'y i E E . yy. y . -; g . . ,'Er E ! y j iqi. i y . y ')-.' E E -E , ; y y, kpr'))i :g E ! E y E y i i )'. yityi. -y ;. PE y j-y,.kill'; k..y'j ! - - .E -;.''. jgE. Ejlyr jy, ... yy..((q ; ipy ! yg; y y j'y jy'gk. yy. )@j@.)r'. 'E . . . :p,.71*-; E. ..yy i. . i.-);r-qp'. j)i jt'pqqri L'.@. E E:.!'.. - E l ! --$'. E y'j.i. ; - . :-j'. -.)'k iE j'. - E g ! yj-. (!j .,ryjjtg.g. j ..j,y.;.yjq( -. qqyy . . i .-'. ...(E.;...E!yq .'-;Eiq y. .. -')4,-)j-.)),--j'. i -.-ji E y . ! ( -!.- ! )()L'; ;)t'Eii i - y(r. ;.i ';'y. ;j;:)'y-. j,y! E LjbL..llllbL;' .kyy-j y)j;.. . j; yyy'. j.p(q*j21*. - j .. ( -. gg . :k y(). ))'. ; yk. (-. . !--.;...-iitEgj rf'.;. - i. ; E(E'E - ;i g-i-( gi - Ei.j,. (. i --- q. i E. - ((. E!E'E- :b'-'' q- .- j'yy .!--.;--.Ejy .y.j'-.EEk.EE.-EE.--;yyj. .. .y E q . - ! ; --. ; ;ryI gy-y-. ,E ?.Fi.r .:; . . -yy lj.. jjlj'. klp): j; j.. .. E.Eq- . .yyr'. .:;.; E..-.E.!(-.i-.i!qq-.;j.y'! - jj ;,j'. j-- jt'..j(-yyy ))- . k. yg jq. y-.y, .)! gI ;E'! - iEE:y'-jr'. r:;E: E.yj.g (.(Ey .-j;- - .y--yyy-.y..j.j.gy'y'. I.-'-E; ;:- E);'. ;-.kj- jE. ;lli. !.... -'.! -- :.'.. r; ij!;'-'- I -r31:*. E )t-E!;j!.j!t( ;ir-, .(t)'. ! i j;,y'. -.. . --'.-yq)'. ;j ;E .j- tjq'zy'i (E. . . j'.j; g jj'. ..-.!t$E. . (-,-.-. yyj . ,. t.jg gj.. -t. ;-y.- -. t'.y,;.;qy.- !.;- .(.!kly q.(t*5 q- . . . - j-y... --..(y .y - '. y E. -- y .jk-'. , .-)y-kr - g, ,--.!y(;'-- jj(,)'rj(g)'g)'jyjjj,jj.'--(rj))'. . .....i-.. ;-(.-q-q...;-.E...yE..yq.. -q(.(...-y-'.(.i.E.:.EE-....E.i.... E.)..E-.-.!..iE.1.!--.-.--.-.-..... -Eqr--..-.;..-!- !Ey..!.:E ;..; (EE--;y'. ...--.. -E. : . yE-y. yi -(:. . j lri. - . -. Ej- -;q-. -)(;12)*.. (.(jy . .E . ; '.El-. :f- ..-Eyy -. g-jj)--'. -li -. ; .---.-. ,.@'.. .' ! - z.,. :.E r.::'
[email protected]''j-.FkE . 5- - E ---. ;'. - - y ...,yy,-. , ;. y i -.-.- .E E- q;-.y. -E; . .- ;y . (.y...g ! E; -. jyyE. i. -. ;E ..--g E'. -. . .. --. (.. j(: ;j':;. ?.';@k!. yy E J-*.4)-.y y ! - -. .- )L;2L'. .....-E..-. - .. E i.. E.- . .--. .E.. -y 2--*.--. ! - E . .. . -. ; -...--..Ft,y.q'k; . .. -''-- -,--- ,pr --. ... . . ..... .-.-.-...; -........ ;'. .j. . ---j .E- ---. E tyi ''.jp.-?,'-(.. .- 59. . k $i .t! -. -. k'i ..- i'.. .. .. . . -'. . .. . .. . . -;'. -. .. . (. .. . . -. j'yyj'. -... . . -. ' i -k; . ,. . . . . . . . y'-1)*--.'. . . ! . ;. jj .. k'. . )t'. . -y), : ' -zr'?i (ft. . j'-'. . . . .... . . . '. .E ; yE (Ei . ! E . . E. !. -. y (j'. q(', --'. -. t'. . . . ..EEE: E... 1E ;y;E ;'-.- . . (. y -.. . . . -.E t--'. E y-. y'---y'. .,'yjj).E.!.ii . E. .j'--y-,'. .i jstj-i ,: i :.. E ' i i -. . i)kE;t(jy! g'E E I'E i ;;.EE ./4j1:2@rr2. kt ri l)q'. !.E-j. -j;....E:;E)E. . i E. ELL. . ir';i . j3)33;)i;L. -- . iq ..jjy'. E . j': . . (yt'. y,tIj)j)y. - : E ;. -.y q)k. 1!. ;. Eijjr!i k).). kjj,j,,--.E-.Ey.-@ -- 4.,r,4yj(1 E .. .... ty. ('. ..ky-'.-yt-. i(pi',. )))tI. i.!.;q. tjjI. qj)i jy.',. (p:;1 )((!);y.. ... j(q:y.. . ;r;);t (prg yyg .yg. .f'.t:,y;.E.t;:)jay...).kt;p.k! tj.-qirqpqi;ti )-. !j.tlq ;:'y .(t.. jg. lt'lr jtt.g jkj.i j-(-. (. ;i !.-...?h.L..j'1.Lb?3,.,...L....2. $:1 y'y. .- :,.2$E.::..:)....,,)..,.r.).?,r;,.) ;..yj(yj(j)'. -..-'. rpj'tytj ,'.q;yjy'j qj)!. )t; ;.y(!q. q. : j!; E.(.Et'. )j. ;(j;..(:,-.jij)E...k. -L))L ...,-'..L.tL.LL..Lk,Lj:.:j..jj,j.jk ! g. jj.--)'.k,,.L(;;:L:,Lf...... I(!);.. ;-i t@tt'. j.)-y;-.;.;-)j-k-j--. ty!. . grIj)j.. . ;:.- ?;. ktj(.qiljj'-jr );;L'.; )Ljj(L;.;:.:..L '.: :. E.. k. y'j.. yy'j j i ;.!i Ej'. (j jjq . ...yy-y .j'.:,j;((.(. tIt.;?j j.i;));. ;jkj!jjkj,,../:u).,(r .;6E i....'; -j;(j.. jt'. (.. .jLt..j,)j...), .j.- j.;.j...j',. . ;gj:k.... . kg,. ,. : ( i.;)j., . q.fLkLLj.j.LLLLkjj , t).. .i.q jj; (ljkr;. L...q p ..' g f; .-..4: ; r.r........:.?.)t! .)!.q;:;(2ijtjkr.. . )l--'. . LL;j'b?L)2.. k;., .(.: .!; q; j((.. ..'...;;,,;?.k;....tE.'. (tj'. jj'--'j:, . ;j,'-'- .; j.;-, . i... ?b)3....,..,.........r,;..u....,...:?.,,..?L..Lb$b..?.,.,t.... yjj. y'. . ..i:..j.; . y.. )k'. . i kyytyjt,. ,.,. ..j.y:y.:yjs.rjjjjljj.. ,j)j).jjLjjLjL3).?jk):t....,..f,j . .t?jLLjL)ji;;L.Lji()(.L .:.. .. , .y ..; ., j;;j'..- jjjy,j))4.. y j?)kL.LLj)kq k;.jq'. (jp'. (r'-,.,--y..jyg;,::. . ,,yr-. . .. r . ..).ki.yyg). . yj'. . )r'. i ..E jt))'.r'....r5. . ..';t', t(('. . ).yg.. . . 2 . y ; :);j')jj','. .ggyy(.y,jjyyjsytgk .. ?))E.:,i)i y'. tj.5'....:.!... . f'. . .y--y-,', yt(t'. . j'-,jyjjyj-r:-'. ., y.
yi I r. p.trq . ( ; y ) tr,. )))':,;pp';)jj)j((j))pt. ): .tr: t-g.. yr. tttp:tlp))q',y(..tryt;-)k')grjjjjp!'.j,jjkj$ri-jtj-.;rjk!. r. ( r);'.ty. . . ;i ;- rty ttt(':):r;yr,k)):!;:;):.. ( rk. -.'.--.--.'-qjqqqqjkLL,k,,.';;ikLjLLqj))))j3;L(LLL;j)-'..----'.jk;LLLLj;,bLL;. r. t. q ).;)qy)!: trrtp)'. y. ))'. pj(tjj-. i . r rp. .r )..g;;tlk':l . .. tlp . ; . .. @.yt. .r '-..p . .; )'l#i.yyr ; t'. (.?'. .?t;. ' i kr t: .; @. l t(tt;LL33L. l/'pl . t.rtt')p gt:. ..r. . tyl p. yt;-j)', . ;'.. l. ... j r ; .rli' . y;.yiy .-i..y'i ;yy..-. .. qjjiiijli,. jytg;(y.--tj.kjjy'-.y;y(ttyr;,kr)?'.k'(jyj'yt,.ktjj.(j . ;'..-(. . r r. j)tk)@j)ytt'. L;Lj)t'-,-. .,.-i-,bLL;))3:;L3L$LL,6qL;;;;L'..-.'LjqjLLL)3LkbLk;'. )'. ti. (. . .i y!i ;. )'. tj..yryly r)y-. .. .y.i'-. it; t. yr 4. '1-*. ktp.r! .y. .. ((. r i ; . r r.. pr;; r. 6t1 r . l t)'!j;-'. -'.' '.''-19118. .tt;. ! iiiii' .(; ) ; tl. ty'i LLLVIII/''LC'' ( . . rtrl . l ( . . j(-.. . tli pi t r r; 1. . q ). @ y p . r-. j). @. . tt'. ; q l k ;--ili''''-'l'll'''' t.;y'. y... I)r'. . tjkrjk. tyr t . .yr.rj.y ,1L. . '- )'.r r t. .. r! t:'. r; ' . ( ).i kr )--. y.y'jyt;)-! )'. . y. . t-. rrr j)-r?j.)'!t'. . . r LL:L..;L)L3L)L-;.-($LLLLL-'. .. . . . p, ' . l !; i . . 7).$t-' ('k! (jq '' ' t-' . i1l))'77;lC:' ! ..-'i 1bb't'33''jq-jL-. r.q . ' I i j.. i . . ('.j'. . r t'. j p. ';)'p)(.. . . ; (i . . . t)'( yyt. j y;. (.;( ..yj (()'.t)'.r.)).. r-'))))t'. ))jjf'.-j)y;t'. yg t)q'-tt'. 111*.y t'tjj .jg. jj',j(j. ..i j. j, r ty, y ..j .;f;jqL)(jL;', .. .. yy t'-y-tk'yyyq. .p)q'- 1b3f3b-. r y,'.r)j-..jkjlyq'.y j .' .. y'. )'tq yy'j,yy'-. ygkjjjyr.yyjjy'(',, .; 1*. ; ;',. ;.y, y)t'(tj'. #rr y'y k l y, . . . y;'pi;:.i t rjyyy yy y t'y. l.'li'()). ;',k. r'(t'('! ! j /'. k' k )-'. . t:'. . y $'t-'i'j . i r r'. rkl L6.1'L3', ))t-r pyyjpt'. y q ti . ; j t.('. .ti (?'y pt '''..8$.::1'7)--' r S1. ti,p!-!''. yjy'. q'. t.r. ll--yt.-'y t. . )rt-t'p . ( rt ( (ly',j tyrtk).. r #-.)y'lkLLkij'. rrjjjr'ytqi 7. rt.. rr!pr .-.rpl'-l rr'y .r.j..-.r(.. t. r'..);.r.t(. . ,tjqq;. ((. ..; tty!. .ky,q )! i --LLkL(;)jj'LL)ft3;'LLkLq.q3LjLj y rjyy-,yjy-':'-;yjy';p. ;j; k.r .- ttgp!-jy tz'i . tpit.-t- yyyy,y . j.yt;.. .. .k)'. . '.)!.rr.. . .p tr j rr @r1. . tj k. y;q;);L. ;.yr jy.tr .t-. . r t't:.. )'-.rr t'.t-.t.-;y; )yyy;'.@t-. tyjyy'. t- t!' . 1*. tji lq',y (,i yj .-y'iy, .;'. yy'. #'. '.' t; yyjjyy't'. sy')j tp'... r'.t. t! t!.. jj .j .. ).ytttj)y. )'.. jtr-! ;.. . .-j:jL;; jjjjjjjqjtr )r. ;k. t.@2:*. ).q-tq., q:;r:. (j:. (- )r'-itt Tl;k q;yrt'q; lji.p)lr LLILLL/L-;ILLLLLLLLI;. LLqL)q(.qLj)Ljjjjjq4LL;;;LLL, L;))LL;.'. ..!i((. . tt.;y. y. L31-L'. r tti )p';) r t. yyjyyyi.jr jjjjjj)))y))y))'. (t'.'-k j yt.; j j.. t4. ... h$)''. . ;-.rr. -
Taxes
.
.
'
-
''
-
rules for derivatives in general can be complicated, and they change frequently the tax law changes. The taxation of simple option as transactions is straiglafbnvard. ,If you purchase a call option stock and then sell it gain or or loss on the position is treated like gain or loss on a stock, and accorded long-term or short-term capital gains treatment depending on the length of time for which the position has been eld. I.f you purchase a call option and then exercise it, the cost bass of the resulting stock position is the exercise plice plus the option premium plus commissions. The holding period for the resulting stock position begins the day after tlae option is exercised. The time the option is held does not contribute to the holding period. The rules become more inicate when forwards and options are held in tandem with the underlying asset. The for this complexity reasons TM laws in m'e not hard to undzstand. he United statesaccord dierent tax different kinds of inome. treatment to The t?zx code views interest income, dividend income, and capital gains income as ydistinctand subject to different tax rules. Futures also have special ruls. Hbwever using derivatives, one kind of income can be turned into another. We saw in tltis chapter, ' Or example, that buying zero-coupon bonds and a fomard cpntract mimics a stock investment.
One category of special rules governs a constructive sale. tf you own stock, entering into certain option or forwtrd positions can trigger a constructive sale, meaning that even if you continue to own the stock, for tax purposes you are deemed to have sold it at the time you enter into the fonvard option or positions. By shorting a forward against the stock, for example, the stock position is transformed into a bqpd position. When you have no risk stemming from stock ownership, tax law deems you to po longer be an owner. The so-called straddle rules are tax Iules intended to control the recognition of losses for ttx purposes when there are offsetting risks as with constl-uctive sales. positions often arise when investors are undertaking tax arbitrage, which is whysuch tlae positions are accorded special treatment. A stock owned together with a put fs a tax straddle.l3 Generally, the straddle rules prevent loss recognition on only a part of the entire position. A straddle for tax purposes is not the same tlng as an optiop straddle, discussed in Chapter 3.
It is probably obvious to you that if you are tuable and transact in options, and especially if you have both stock and offsetting option positions, you should be pmpared to seek professional tax advice.
'
'r).
. ':E(!E.Ejh:: .
'
.
. .
.r:
.:
:....... .
....
.
. .
(E
'lrEl.liij
q
.
.
-.
..
.
'
.. .
-
-
-
-
.
...'.'
-
q'.
' '
.
'
.
-
.
.
. . .
.
-.
.
---.23$$3.-?--3$;--.
't-'
- . .. ..
. .
.
.
. .....
.
..'
'.
.
..
.'' .
-
.
. -. ..
-.
-
.
.,
.
..
..
.
.
.
.
.
-
. .
.
.
-.
-- . -. -
. . -
-
.
..
. - - ... ... . .
-
.
. -
...r....t
.....
E
.
.
.
-
,
-
.. .
.-
-
.
-
...
.
.
...
. .... .
,.
. ..
-
.
. .
-
-
. . . . .. .
.
.
..
. .
-
.
.
.
.
.
-
.
-
.
..
.- -
. . - ..
.. .
,
.
.
. .
.
-j-
..
t'. . .
-
-.
-
.q -y
..
. .
.
. .
.
-
.
-
-
-
.;.r -
.. .. .
..-. .l'ty '-
. r'.t'. .
..
.
.
.
-
. -
-'q
.
,
.
, , , .. ,.
-t-jj'. .
,, . ,-j.k;'. .
pt'.
,
-
-
.-
-
.
,
-
-
.
.
.:.j--
.. .
..
-
--
.
-
.
. '.
.
..
.
,
)'.
jj.
j r
y)
)
--
.
,:'.
- .
.
N
-
-
-''
.
.
-
-
,
. .
-
-
. .
.
.
.
.
.
-
.
.....,.....a.;q..o.a,..x......?.i..=
-
-
.
.
-
-
.
.
. . . .
-
-
.
-
-
.
.
-
.. .. ...tj.
.
.
-. . .
. -
.
-
.
-
.y-tty'.
.
tjjy))... rjL)jjL'.. )(t(j)j'. .
-
--
.
..
-
.
.
.
-
-'ytj;'-2
..
. ... . .
- . -. .
.
-
.
-
- . .
. .- ..
.
.-j'.
.
-
. ..
.
.
-
.
-
.
--
.
-
-;';
.
.-
.
..
.-
-
..'tyj('r::);y):;j;'.
.
. . .
. .
. .
.
.
.. .
. .
-
.
.-.)jkj
.
..
.
. .
.
.
-
- -
.-
. .
.
-
-
-
.
.
. -.
.-
. .
.
-.
.
.
. .. .
. ..
..
.
-
. . . . . .
. .
-
.
.
-.-'--;)rk----j..
-,jr'..
.
.
. . -
. .
.
.
-
-
.
.
. .
-
...
.
..
.
-
.
.
.
-j!-yr'.
..
.. . .
-
,.
.
.
....
.
.-
-,
.
.. . -t,i .
-
-
.:
.
. . ..
.
...
. -
.. .
.. .
.
-.
.
-
. --
.
r.tt'q )'. t. g-k.)y.. ...-L,,-L:,jLL,j--.-j,'. .);',.t'. t'. .,j'. .
-.
.
.
.
-
-
..
.
-
-
-
-
.
. . - . . . ... . .
.
.
-
. ..
(j.
-
-.
-
-
.
. .
-
. . .. . .
.
-
.
.
-
.
.
.
-
. -
. .
.,yy-
..
.. .
-
. .
.
.
.
-r-.
-
-
--
..-jj'.
..
.
.
-
.
.
-
-
. .
.
,
. -. -
. -.
.
'...5........
..
-
- -. .
-
.
-
.
. . -
.
-
..
. (j'. j;--:j---kjjjy),;jr',
,
..
-
.
.
-y'--
.
-
-
.
-
-
'
-
. . . .
.
.
-
.
-(
.
-
..
. .
. . -- . .
.
tt';i
.
. .
.
-
-. . .
-t.
-
-
-j
.....,..(.
..,.:)r,.;
.
,
,
.
.
-
-
.
.rj
.
. -
.
'
.
.
.
.-
.
.
-
.
-
-.
-.
. . .
.
-. . . .. .
..
...
.. . t',))jL'.
.
. . -.
..kpr
-
.
-
-
.
. ..
.
-
.
..
-
. . - - .. .
.
-
. .
.
.
. .
-
.
.... . .
-
-
-
.
.
-.
-.
. -
.
-
..
.
. -. . .
.
..
.
. . .
-.
.
.
-
.
.
.
.
-
.
.
-
'
. . . . . - . - . ,'-
- .. . .
-
- ..-
.
--
.
. ..
.
..
-;
.
.
- ..
.
-
.
,
: i qi:E ' : : ' . . . . . . .' . .:':i( .E(:EiEi ' .. ( EE j'. . (I !.I.E1.r : .... . ' i E .'..: Ey j( .. j.. .. ..::y....j.ji.:.(j.I! .. : .( : ' ... ' ' i i. .' :. . .E( . . . . ..:jj(.jjj.g . .. .. .:. '..L : :g...: i(.i @: . . .. . ..' ..'..:.!i : iiEi .E.' . '.( ' (: . y. ...'.:..'E: E.( iE( ! ...'.' ..E. . . .E i : . i ( j;gqE.E; . .. . ( '... . . . (' . ,'Et)EEi'.ll' ' :
:
. . .
'
.
'
: ' 'j' j yj.. r .
,,j!!jjjjjj;;;(
'.
'.
.- .
.
:
'
.
' '
.
.. .
ljt '
..q.ji
:.
.
..
:
.
.
. . . . :
.
:
.
p l' $*. (j)
' : . . :
:y
:
.
.
.
:
.
.
jyjjjjyjygjy yrjyjyryygyjj.kyggyygtyqyrrrtj : jy. ( EE. yjsrj. E ; EE;..l;q.r y.j y. y. ..(qy.(.qjyEg.gj.jy. Ek . .( tiqq. ; g.y; j(.!.Ek.. ( .(.E: r jjjj . . ...j(E.j.E( EE(yi:. j.: E. r.y... . . . . . E.rE. r.. j. jy. . .. y;yj..y...;j.q.yyy. ..y . (q .yjq y j ;.. . '. Ey . ..y'..(..;;.j .jg.. g.jy..y k. gk.. j. (y...!.. .j. .;y . .jy . ygjg.yyqqy.jj g g... .; g ..2..g r r yggj g'. . .. y'. .y; ''. . . ...... .. :j jgyyjyj
qyg
,ryr(y
;
. r. jj r . : . . : : :. : : : : : :y:: : .. : :': . :j
;
.
:
. :
.
:: :
.
.:
: : : . .
..
.
. . ..
:.
. ..'j . :
.
:
.
.
,E
: :i ::E :' E ..( r E: .' .r.!:
[email protected] .( j . j (..:. . ,t't'jE)ir.y.) . :
E'illtlili
E
E
;
j j E.'t!)
E
:
E
.
. . tlitt' (t
E. E .2.....r,.Ejt(!1:.
(......:.,. '.)..... . . iLl.Lj..kjjlk.j:j .j. :
llllill)
(',iE)'.'I1li
j, EE irr.ii .,
,
:.
(
';L'.'qLLL))'jj1.' .)'g'..'..'(
' 'ttttt)ii ;lj ((
j.
...g.(.j
jy
. ty . t'y
the last chapter we introduced fomards, calls,
!
nnd
puts; showed that options are lnsurnnce; and looked at nn example of how options can be building blocks. In thls chapter we continue these themes showing how the use of options cnn be interpreted as block approach, examining buil!!h'ii'' in.Vor selling insurance. We also continue t: . pn'ces and option prices and looldng at some common option tje tinkbetween forward strateeses, includlng spreds, straddles, and collars. Among your goals in this chapter S h Ou jd be to understnnd the reasons 1rr uslng One Stl-ategy instead Of nnother and to become factle with drawing and interpreting prost and loss dlagrams.
r
.
;
.
'building
,
.
;;'
' 'r. i i iE. 2 . . : : ..E. E'. .(;.(jE.'; (E E. ;i . :. . E .: ..'j' (...:'..g.. (. .; ....E...)i!@!E : E (E.'..:..E!E:ll;!) ' '.(.'.'.' E ' . ' (.(.j(.jy ' ' ..y .j(j. :. ...:.. E(.(E. i :iE;!I@ EEl)itj
.
.:
.
,Eq
E
:j.gj:
..
:: : g : . :: . gy : 2. .;:. . . ! , : . . ;'. J'. :. : : . ...y;. . : :. .. y y . r. . . .:', .. . : .. .. . .. . : . . . : g ... . . . : ... . . y
.
E
j ... .E'...'' E.g.' iEEj E(' y .'. .' ' .. (::!E(j;j. ,. E.: .. . :
:
. y : ... .
..
.
.
gi.
',
:
.
.
:
'
'k'j E
,,.
'
.
yy
::lE(!(
:( ( : ( g r . . ....E.( i E5.. . ' :E EE: .' .( .r (.jEEj j' (g'..E'.(.. y . ' .::'jE '. r i ': LL. ..
.
'
:
.
.
'....
'
g
.'.
:.
't,i(.tt'.t.i)) '
.. ,,
q'tlt)i/)rE
E'
,E
qEEE(EIitl:j i:( i'fEjilt'
('E .'Etit,iiE '
: :' ..' ..'
..
:
lljijjgl tyEEr'fj.i: (. : (.' 1... .:l.. . '.EEE,!,iEE@E'r'.i'.l.!i!'(,E
..
' E '.E ' (.E'.j'i'(5gkk:lE ji'j(j'E : .EE'..'. ( j .' ' : 'E i.E,liljtji,
4
..
!
E, ,)E).!..E :, ! .. : :: Ci : .. : : .. ',.,(
t..t rji .'E,tt),'r,) jj .:
'.
...
E
ji j ' y,( . .:. i1.jg.ji(. g.( g' . EEE...Egj;(jj. j.j.j.. ... . jr; I' ' q ' .. . ' ..''.E(!j!.i.. q.. . . g . . .ggr '.. ;. . . .' (!y(..;. jg ( E..ly............ E ' ' i('.(' i' 1. E(' . i ( ' . ' . .. ( 'j(.'j '. (E
:. : . ' : .. E : :
. . : . . . .: ..
.
E y
: . .
E
?E. .i
:
.
E: . y : ' : E E' 5. . lj '. :(.:.!'.')j'j(.
E. ''j
.
j.
11!g:EE(j (q q( q . : ( .. .. .. ' E'. . !i i' E EE.'' . .. . '.' (.('j..:;i!(r. (. g'r. l'i. . . . . '.EEilqt (' . (.(T! j ..! ( ...q:: ......;E.i.j . ((. '
.
'
:
'
.
.
.
EEEEEi,E;Eq'k!,,l,i,i.,E.,,qy ijsjlgj..jy'j).jEy.
..'
' ..'j ) .g.. . EEi)ltiit.'t,; j 'E'i E E i iE EEi !(: i . : : E EE .'. ' . .E. ' ' )i:;)jj::I ( 'i...i.: ''t;,!I;rp,.(,q rs'.j .'y
:
'
:
il'. 'j
:
E' .. ' ( i (j j) .. j(. : ).). ''
:
:'. .tt .
l ,t'l)q. ), iltkpiptylr! . .
.
'E
Ek))i
..)'...).j 'ttitt.
.q.
..
)
.t
,E'
,
,
.,t .E... E'
' .rjjj!.j((E 'il). .j,,
Eq q
'iiitlrtE
..
E: E 'i7 i '
''
:
.
'
.'.E,:..j.':i%.. .:.... (.
''
E)))!))Y 'r.
;,,,,jjjg;.E).,.),r;yEyqy,y( y
j).!. tlij; (q
:: : EE ' .ElijjkEtyiqE . g.gl. . .. (' '
'
'
.
.
.
.
.
.
.
' ''
.
'
.
.
.
' '
*
.
.
.
.k
*
.
.
.
.
.
.
.
''
.
il::lq:,p.
. . .
'
.
.
.
'
.
lnsttrlnq a Long Posltlon: Flotjrs *
.
'''.....'''
.
*
' '
.
'
.
.
.
.
analysis in Section 2.5 demonsated tllat put options are insurance against a fall in the pnce of nn asset. Thus, tf we own the S&R index, we can msure the posltion by purchase of a put opon ls also called a Eoor, because buymg nn S&R put option. sale pnce for the value of the mdex. mlmmum guaranteelng we are a To exnmine this strategy, we want to look at the combined payoff of the index 1)osition and put. In the last chapter we graphed them separately' now we add them together to see the net effect of holding both posltlons at the same time. Table rjj summarizes the result of buying a looo-st.r1'ke put wit.h 6 months to explraon, m conlunction with holding an index posltion wlth a current value of $1000. table computes te payofffor each position and sums them to obtain the total payoff.
'pje
.
....
'l'he
.
.
.
.
.
.
,
.
.
,-.
-,.
--.
rfhe
'
.
. .
(
ltttijEtly: :E... q.... ; j yj. .i j.)k) E E. . ......E..).,y EE(. t j ))j.E.E.E.E ....... . .j. . ::E)!jl.ll) ':.':.Eplii' .:.t.: '.-'E E E ' . '
:
'
.
' ;........ 'rylrll.,k.y,,qi -
..
.
'
.
:
:.
.
.
:
'
.
.'.
,EEqEEilr(:jj)t
: .'
'
:
.
:
. .. :
.
'
.. :. . .
. .
:.
'
.
,
:
.
. .
...
..
'E'E(
.
.
:
,,( !i)yk(
'
E
There are insnlte wavs to comblne ootions to create different oavoffs. ln this section '' lmpoftant linds of strateooies ln whlch the option is combined with a we examine two underlying osition in the asset. First, options cnn be used to insure long Or short asset P options posltlons. Second, can be written agamst an asset posltlon, ln whlch case the Wliter iS Sullling instlfance. In this Section We Consider four Positl'ons: Being long o Ption the asset coupled with a purchased put or written call, nnd belng short the asset coupled wlth a purchased call or wntten put. .k . . ln this sectlon we contlnue to use tlzeS&R lndex examples presented in Sections 2.2 and 2.3. We assumed a 1% effectlve 6-month mterest rate, and premlums of $93.809 tjje lloystvike 6-month call and $74 201 for the looo-strike 6-month put. .
E . . . '..(..E.rE . Ejjj' .g .. (g : ;j.r .j.' . gj.k . Eti'y.tt)!,i :
. .
. '.
.
!',;,.
!tE)EE,(il
'.
STRATEGIES
.
'
J(.. '..tt'i)) ..'E'::!IE1!El.
'...
.
.
3 1 BASIC INSURANCE
.
'
. .
timarried
-
-
r(j)))'. rl'q . r!
-
-
.
.
-
.
-.
-
: E 'E j.j : '.: j .. . ' .. .
( '. .. '
.
-
.
-.
-
. .
.
-
-
-
-
'
.
.
-
.i'.'.
'
.
--
-
-
-
'.
. . . .
.
-41t4)):*5
.
''
'
.
.
.
-
.
'
.
.
'
'
.
'
.
:
l3For an illuseation of the complexity in this patcular case, an exception to the straddle rules if the stoek and put are a occurs put,'' meaning that the two are purchased together and the stock is delivered to settle the ptlt.
-t!.
-.
.
.
.
-
.
-.
.
. -
.
''
'
T:.X
.j;'.
. -
'
'
-
'
: :' : . . . . .'..... .. : E' .
). )j
..L.. ' E' : EE: ( i ' '.)')lj . ;.EE'.;E q. .) :):' jt.. .)j.L..;.j :....::.E...y '. ....yJtj. . .iE41t>.!1.).::....:: ...j.''(j.tlti f :.Ei : ( k.y .: ..:.
5g
Ci'jtti'ij,' kb 62 %)Ittikt j'. jjjjjgy'j.... )yj('' jjd (' j'y' j'y j))jj;'. jyy;'. g' ('' y'' jjd yyd )' (ij!!jj;;d '@p' r' y' (' ),' rjjjg;,d (2-*. !1jjjiqqq'. yyyy(', t.,,'. 1.( i i ; ( (''( q)r'!q j : j qttj (i(;( (q ( 74:* '.'((.q' .tj ' ( ((i'J; @ .r(t; ( ' . 'y (.. . . (.' i ..''.(.( .rqlqq'qr:'qrltq:crrq' t'l.ri-i(.t.. y.jjj ' j.jg j k jyyjjittl.i:llyil.. y.j''y yj :i . . y y .yjjy.y yyyyjl ll' gjyj y.gjjj. . jyjyj .,;jy: j. jj.. ....j.jj !jI1:t:j;:;;.. l. .(. jLLj;;i3jLL. 11I1j..,jjip': Iliis!!jg.tj . jljl::;jjlj. ''q l j(qr j y .jjjyj . ; y jjjjl .t; y)j . yjjjr r yyyj(jyyj r jy.. jjyy-. tj.ggk;ttyj-. )! . . . .......t....jy.. yj.. ' .ti ..
OTHER
STRATEGIES
IN S U RAN
BAS 1C
CE STRATEG l ES
$
63
;'k' ,''l''.j' --' -' 7* S' j't' ijlf' jjyryg:j,d ,jkyy' F' E' #' yjd y' )' j'(;' C' )'' E' 5* jy'. j'gtd .jjgyy' F' (4::7'717* q' jd '$' 7* 1* (' 57r147141q*(44(((5* r' t'j''ild j'' ))' q!' q' (q' );' ;''j'()' Jq'i ()' )' (' qq')
.'.' .'..'y ..' ,'.'-.'
!' ! IE E :' :E i'E!E'E' ' E! E E ' E7( ( :EEi !. I i i ( E(' E'E'...i@ . ' 'yEEEE. ! E E''' . E fi-EyyEjjrir-p-ipsrrlit.rtitrs - .-.-- g... .g ; y j . . y ky y yy y g ... :
.
:
'
.
.
:
.
-.
..--.
-
--..
.
'E' ' 'E : . E'
:
.
:
:
E(' i : qq .
:
.
.
:
' jgky.yg..gy:t.
i E .
-
..
.E
' E' !.FE E(E: E ( ..:.
:)
'
E
-
EIE(E)'E !'E
'
y
. ..
..
E-
.... . . . .. .
. . . . . ..
-.
-
. -. . ....
---k-
.-
.
..
-
jy g : : gg jy.: y.q qE y . ;.. .. y. . , y y
,
:g .
-
.
1i'''3if'. i'ttt'-:;f:i!!))). r''. ?! (EIIIEEpkpI::'f'. 11IEliiii'. :;i,i!i::i(!IiE;:,ii' ))--)'' y' EIIIEEF;'l::..-' )'. f' .'.( ).: r r.' ','t36b11iL,:'. (' '1* t'Lii3i',-'. t.l!!!iil:k,:::d ttII1(iEiI)::)'. :2:,,-* .11(; tllli::kii.d (111@121111111*. t)) (.'t-r.t''q y .j)(;; jgyyjj jrryjjj yjyys jygg,r j,jgj;,ry jtry;j-jj kgjj jsjy,r jkyy,r jr,.yjyy.,.,. jr..yj jkjy,j yjjj;r j ,rj;r jy.j..(!j jj..yj jjjj;jjr..yj .jg:gjy: jj..yj tty g(tr.t-ytjs.yjyjyjyyy jj jr..yjjr,.jtgg;jjr..jr..yj j;:;,y jy jjjr,jkygjjs kygjj jg..jj jryy;j jr,j igjjjjtry;j jgr:,.ygj .4,y,iyjj:y jjjjyj tjy r;jj;y jrgy;j qjjjjjj;. :! ( ;E:. ' ! . E. l l i r((kk;ry.yyj). (ltl-i'l tyjjj,, . 'ty . ' (-(i. . .!.. ;)'. . and . .. buying a 1000 strike call option at a premium of $93.809. The payoff is the sum of the first two columns. Cost plus interest for the position is (-$1000 + $93.809) x 1 -$924.32. Profit is payoff plus $924.32.
,
. -,
.
y!jjyytjjss
yjjyy, jg;jj rjjkg:,,.
'
-
- E. .
...
zto c10 150,000 100,000
lnsurance
.
. -
i
'
..
XN
. xf
.-%
xxN
()
sh
lnsured house
x%
'':r77.
: :'
'' ' :.
'
-
.....
.
.
'''
-
......... -
... . .
'
.
..
.. ..
E
.: . -. . . . . .- E. . . . :
.
-i1:i:)',k
-
. .-
.
.jy:
. .
=
:
: .
.
' :
k'
150,000 Hottse Price
250,000
($)
we exnmined the insurance policy in isolatiop. However, in practice, a buyer of homeowner's insurance also owns te insured asset tthehouse). Ownig a home is analogous to owning the stock index, and insuring the house is like owning a put. Thus, owning a home plus insurance is like owning the index and owning a put. Figum 3.2 depicts te insurance pohy frop Figure,2.13, togeter with the pninsured house and te combined positiop. lntrpreting the hpuse as the S&R index apd insurance as the put, Figure 3.2 looks exactly like Figur 3*.1.An 7w?-d# house /?t'I' a #?-///. diagran; #?cIr looks like a call (7p/t?/z.
' '
.
' -. . - -.' . . .- .. . .. . . ... -
.
. :
:
:
:.
'
:
'
E
.i
'. :
:
.
.
.
'..
..j.
..
.
.
.
-
-200,000 50,000
jtry;j-
.02
-1100 -1150 -1200
.-150,000
jgry;j-
.
..2950
Uninsured hotlse
.jg:
.4,),
' . tE ...i.i.. -
-
.. -$900 . . -
-
E ( E ( ( ' ' ' ' .' . ! ' ... ) .('.:.(..( .' ( y . . . (. g. (E(lkk)ik,.. ::IiE)ki/. :111t):. (.(. E. E y . 'Iit::.i1(.. qj.:E. .'... .'..E' ..j. yE.E j.(... .. (!Iirr:iI,;(. E 't.''.'.. E. kiI::,i... iiiE.. E. ()111EEE:111-. :l,E#!I,k.. ; . . . ti.EiEiiC.. . !. ' .p... ; . .. q ... . ; i ;. . j g ;...y! .. r....r..r'(....r.. (. ...q... .-.;l......... (.......... :::!1,;,:' .. ; .. , . . . . . y . .. .. . . . . r . . . .q... . j..g ,.....(... r. ...j..j.q.........r......r. r .......q........ .. ....... ;.. . r l. ( . . . ( . . ... . ( .... . (. . . . . . . . . .. q q . EIIIEEEF'Ii'. . E. . . . .j. . . . . . -'ii!,i . . . . . E.'Ii.'-'. (E1IIiE.1Ii7:i''... -.i:E:),,k (,'i::--'. . .. :i'iE!'iL.. I1!i:.. '11It:: .........'E.'q. .. . . ;t!iiiii;l.-.111::EE2:1:2..:IIr::. 'iI::i''. .;1!11181;1.. ( .E i. ' ...:.. . ti.IEE:E:EIII!. ... -iI:i:.''-. I:!iii!1Il(111:.. EllI. . -! -.. ''lIt:: '.. '-iI:iillIL.. ......... i....CEIII1F7''''. .. . . . ... . ... . . . ( . . . . . (... ..'.'.'. .' ........ .. .... .. . - '... .....- .. .. .... ..p.,....i........ '.I:EI'. .. .. . .q...
..(
-j'.'
.. . . . . .
-1000 -1050
se'
-50,000
.-100,000
:
tj;jjjjjk.q)...'
E.
. . ..
..g
sxN
.
:
-
..
...
.. .
S0,000
i: .
-
.. . . . . - . . .. - . . . . .. . .
xx
.
-..
.
.
.
'
.
'
'
E
...qty. .
- -
. y:
Payoff to owning a house and owning insurance. We assume a $25,000 deductible and a $200,000 house, with the policy costing .$15,000.
''.' .'
r .y yy j
..
.
-
. .
EE ti ' . ).lE -11:::111*. -1i:i,'i'. y' !(' '1* :)' E E E ' ' ''' E E E 'L.;-:';'k';:L'k F'. ;')')' :11i::-'. j';r' .' (E(EEi!iE1'!'pr! !'.(':.' 5.jyyytj. ( -E.--i .'.E.;i('(.!'i.''!'!'.(E I1kEE!''i.'. llliq'''Elli!!t'd .. I ; :. ''i'.' (.'Ey E' (E !-...jt(yq Cliii;ld EjIIiEtiii(iiI'('. s'. !;E(E ','.y' .'EEi'I'. (lIi:r'''. k' t'k' (' )))d $' .yyy..yy..j..y.,y.qyjyj...g-yyjy.-..-.-...-.jjj;.y-jyjE.j..y...j.yy!gy..yy-.yjq.y..gy.yj!,.yyyy.t 7::'7:7E (''(.('(EE(.:'EEE(E''. p' '.IEi,!,' i'.f'J'?'ilijk-'llld r' ql' ( ii kplj.lhlFrE!!;rrql E.qt . . . . . Etjyyjjyy .... . E. Ejjjj(,.i.j.j(--jjjjjj:... . ;)),ytytyy-..,)Efrjjjjrjj.EjE? r( )!jjjjjj;)h..!..y..jjjjj,j.-.!. ..
' '' ..::.::.-.:.2*. -'--' ;'..-.11q..!p'.'' ?' i'. ;'F' :'j' :';'L' 6tjk'' ?' :
.
g gy j y. .g j y. :.
:
... ........ . ..
. . ..
.. E .. ... E. . ...
.
. -
:
.
..
-
.
.
. . :
. . -.
-
.
:
-.
.
-.
. -
. . : . . ' . . . . . ..... . . . . . . ..
.
(
..
. . .-. .. . . . . . -. . -. . . - . -. . . . . ' -
..
).(.:
((
'..' .
.
-
.'..(''(.
...
.
.
E
. . . - - . - -. . . . . - . -. . . . . : . . . . .. . . .
.
-
.
.
- . - .. . . - ..y .
.
.
.
.
.
.
. E
:
:
-
-..
-
... .. E ' . E. . . . E. . ... . . .. -g . .y -. . . - . . . . . . . ... . . . . ... . :
.'
.. .
. .
-. -
.'. . -.
-. .
- . . .
. . .. . ..
--
'q.''.. .j'' .. ' . . . . .. . . . .. . E --. . . . .
. - -. - . - . . . -. .. . .. .. . . -
.
.
.'.''(
E
.
-
.
.
.
.
.''
.
.
-
-
.
..
.
.
. -
-. -.. -. .
E
-.
.. .
.- . -
. .' r
-
-
.
-
.
$0 0 0
50
-. .. ..... .
.
'.
-
-
-
.
' .
.
.
.
.
. .. .
-.
'
-
.
'
'
EE
E
- -. . . . . ..
-950
-1000
-1000
100
150
-1000
. .
--
.
.
-
'
:
.
.
''
.
. .
.
. .
.
'
: :
E
$924.32
-$900
-1000
200
..
-1000
E
.
. .
:
' . -. . .
.
.
-
- .
.
E i
.. .
-
E . . . .
$24.32
924.32
-25.68
924.32 924.32 924.32
-75.68
924.32 924.32
-75.68
-75.68
-75.68
-75.68
Table 3.2 presents the payoff and proft for a short position in the index coupled with a purchased call option. Because we short the index, we earn interest on the short proceeds less the cot of the call option, giving as the f'umre value pf the cost. of The payoff and profit diagrams 3.3 columns graphs the Figure Table 3.2. with insured index position in Figure 3.1, we of purchased As the put. resemble those a Fiyh ash flows. payoj. in caref'ul in dealing The panel (c) of Figgre 3.3 is have yo be this coupled with ln purchased borrowing. like that of a put case, the payoff diagram call equivalent that from buying a put and the and buying is index to for shorting a diagrams of valtle borrowing the present are unaffected $1000($980.39).Sinc ptotit is exactly profit diagram the in panel however, the by borowing, same as that for a (d) with this comparing panel Figure 2.9. Not S&R index by You put. purchased can see (d) position look like has the insured it loss the short only does smne as a purchased a ptlt, of the price is bove which is value fumre tlle $74.201put $1000: $75.68, put if the -$924.32
premium.
Insuring a Short Positih:
Caps
If we have a short position in the S&R index, we experience a loss when the index rises. We can insure a shott positiop by purchasing a call option to protect against higher a price of repurchasing tlie index.l Buying a call option is al.socalled a cap.
j
.'
'
Keep in rnind that if yo have an obligation to buy the index in the ftlttlre but the price is not fixed, thenyou have an inlplicit short position (if the price goes up, you will have to pay more). A call is insurance for both explicit and implicit short-sellers.
Selling Insttrance We can expect that some investors want to purchase insurance. However, for every insurance buyer there must be an insurance seller. ln this section we exnmine strategies in which investors sell insurance. It is possible, of course, for an investor to simply sell calls and puts. Often, however, investors alsp have a position in the asset when they sell insurance. Writing an opt ton w h n there is a corresponding long position in the underlying asset is called covered writing, option overwriting, or selling a covered call. A11 three tenns mean
%.llqsuRArqcE,
64
COLLARS,
AN D O-rH ER ARATEGI
BAslc
Es
(' t'y' jf 1* 71* (' i i E'('' @: E@ t'r'y' i'r' 'f@' :'('E.Ei'( (;liiI'j!.1* y;f q' t,' k' ;'jrf (' jr'q!:l' j'!' l'lt' )' i ' (' (' q'(' (E ' ( rrr' ( ! ( (' g' (r' ;1* 1* !T)!l':Er'77'7' ('/:'97* 7* (.jE (.'((E!(p( (. y'jjjj::::;j;;'. rt/;qt' j'yyt'.tyt' tytf gyjjjj;jjy'( jjf y' j'. (j' y'. y'yjf 'y'. )t'. (' 'y 'g'. tiqi!..(i . ;.;,tqk., i i. k(.' '..(....;.'.'!...( .(.'.j.(,j;.! .i. 'j;.'j.j.(;(..' q kyyyrr rj't;(-jjjjjkjkjjjj'. .. q . t)yyjj'. (yq. .yy y . r yj j j ..yyy.yy g y)r;jj. y.jj. . j . j j ; j, yyr-.jyj.,.g....y)j...--.--..;y. . -i;Iiqq j .g.. .....,...........:LL.(L((LikLLL;3L. y.I1i Ijt:tli: j t . ; j j y tyttj . j 1ktj(... 'j...:.'q' j . . y . drg:,p jlytjq4:y :!@; ....E.......... L.L.L?111 .y . iE,k.lys;;,. jgy,j .k;-;..-).-...--..-.;..,-j-,.... ;l . ':)' S' Eyf yEyIjy'E :('yjjjjjjjjk. (;' r' 7r11* E -kk;;''''ltf
.
:
'' !
i
Eii
:
.
-litti' -- -- ClEpti-jiilE' .
.
.
E' E' E. E '
;
:
.
:
-. .
''
-. .
.
''
'
. .
:
'
.
:' ' '
. . .
-
.
''
' ' !' ' : :.
.
-
--. E g E Ey . . -.. ( y ( .y
-
g.
..
......,.....
.
.
. (a) shows the payoff diagram for a shol't postion in the index (column1 in
(1(y,, Short S&R Index
'
Panel
shows the payoff diajram for a purchpsd index call with a strike price of $1000 (column2 in
Table 3.2). Panel (c) shows the combined payoff diagram for the short indx and long call (column3 in Table 3.2). Panel (d) shows the combined profit diagram for the short index and Iong call, obtaned by adding $924.32to the payoff diagramin panel () (coItlmn 5 tnTable 3.2).
Clii (;)
d:::), llq'jli4:y illklyj;;,.
........
2000 ltotl
ltoo
$1100
($1000x 1.02) + ($93.809x 1.02)
-
I
zlotl
-
O 500 1000 1500 2000 saR Index at Expiration
0
500 1000 1500 2000 S&R lndx at Faxpiraon
(a) payoff($)
xto 1000
(b) profit
combinedPayoff
($)
combinedProfit
xotl
-$7s.68
ltotj
-1c00
/
.
covered puts A covered put is achieved by writing a put against a short position on the index. The written put obligates you to buy tlze index-for a loss-if it goes down'
-2000
-2000
0
500 1000 1500 2000 S&R lndex at Expiration
0
500 1000 1500 2000 S&R Index at Expiration
(c)
$75.68
=
<:
1000
-$1c(0
$100
-
which is the future value of the prernium received from writing a looo-strike put. 'I'he proft from writing the looo-strike call is colputed in Table 3.3 and graphed in Figure 3.4. If the index falls, we lose money on the index but the option premium partially offsets te loss. If the index rises above the stlike price, the written option loses money, negating gains on the index. Comparing Table 3.3 with Table 2.3, we can see that writing the covered call generates exactly the same proht as selling a put. Why would anyone write a covered call? Suppose you have the view that the index is unlikely to move either up or doWn. l'l71isis sometimes calle j a neutr al', market view.) If in fact the index does not move and you have wlitten a call, then you keep the premium. lf you are wrong and the stock appreciates, you forgo gains you would have had if you did not write the call.
-100'
-2000
65
Profit on Written Call
Proht on S&R Index -1000
$
ES
Because the covered call looks like a written put, the muimum prot will be the prost is same as with a written put. Suppose the index is $1100at expiration. The
Long S&R Call 2000
ARATEGI
selling the call. A payoff with limited profit for price increases and potentially large losses for price decreases sounds like a written put.
.
-y
-
.
.
..:....Ei;.!.(.....
;rf........iq:.........
' EEEE . .
:
-
......
.. ,.
'
.
:: : '' ' : rj i ' : 'E E ': : E E.: E . E E iE .E . ' -y E . E.. --E . E.E -- -. E
INSURANCE
(d) .,1:,,*.t11j),.* -.-j' -' C' !f :;' f' L' piy-.lt-pf 11:::::11* dl:::::.,f 41:::::11* Il::r::,,f :''j-'tf Ftf )' Cllf 1:11* (' :i/d ?' iyf i;!,.tf g'. JJX p' /'F't'(' 7* it;ff )i E y' ;'t'')'L.llii::ikk!'d llr...!l.f -l!iEk;IE2:' llr...f l!...f k' ''q';'!qqf ''r!''':''7''q'T: )' tj;)..j'. t:k))t..'. .y'' (' 'E' ' E: E EE 'E E 7: E ' j'j'1* t..;:l;,.)t;k,jj-.-.d ii;f kr)f t'---yjrf ii''Iiiii#i#'. .-It::)' iiIiEiji;'' .y'; r' i E!E ' i 'E l iEE!(iE.' !'(r(E(' Ilt..':l.f (,!1EE;)* 114::.::.** lli...!'.f .q::,,--,,' lljyjf t'jjEjjj;jj;'. p' y' i@ . .(.r..'('(' (. ('.i ((((i'.! ;.q . '( I( .!..'!! ( (' E''!('.' ('.' (' (' . . -- .y-y)(y' -- ; 2 (' E.-;;; E. r IijI(kkk.. .j ; . . . y ; g .. q y j--.-- .jyyryy .-yr(,y y. ry.(y- .y jy. !, . j. .. .)y . . y ;,.lt-.... i. -. y ,; (; i .: ,) .( .; jr :i.t--.i . . r. .. . . . . . ... . . ty. ... ...... ..... . .....
.'''
j'(' qf tl')' )@'pkl'rf ..r''' )' y)f y-'-' y' qq!f 7' pfl-. tj)yl tt-f'. ftp'''q yyyt'ykjjr jjjy' L.-,iiLLiiLL'.. '-' .t' t'y)yy)'. rt'. jy( p-, )-g' Il....l.. (:),'.::. r t)t(. .. @yjjjjjjjjjk. .. .-j. jj:::,,y) jj't; ll:.,,lll j:,-':::.iii .,It::. lr.'..llr..'!l:...,,',r.'' a k!si;: .tIt::. iiq!!qIi ii i i '.iiiiiiiiiii(' jlq::::,,e a ii 11.r..,., Ilq::::),. a p,.,..:ljjlq::::,,. a .jI--.1;--) r. . y. trjy ' .j....j..-..y..-. and . . . selling a looo-strke call option. The payoff column is the sum of the first two columns. Cost plus interesy for the position $924.32. Profit is payoff Iess is ($1000 $93.809) x 1 $924.32. -j;k;,.k)' .:1(:-* ''67776)':7'!5* L' -' ,'.11:::* :'F'
.'
ltrf ''))'.
'
'-
-
-
essentially the same thing.z 1.rlcontrast, naked Avriting occurs when the writer of an option does noi hay a position in the asset. In addition to the covered writing strategies we will discuss here, there are other insurance-selling sategies, spch as delta-hedging, which are less tisky than paked writing and are used in practice by market-makers. We will discuss thse other strategies later in the book, pmicularly in Chapter 13. call writing f.fwe own the S&R index and simultaneously sell a call option, written have covered call. A covered call will have limitd prostability if the index a we because option writer is obligated to sell the index for the sfrike price. increases, an Should the index decrease, the loss on the index is offset by the prernium enrned f'rom Covered
z'Ikchnically,
ovenvrting'' refers to selling a call on stock you already own, while a wfite'' entals simultaneously buying the stock and selling a call. ne distinctioa is irrelevant for our
purposes.
'option
ttcovered
.
-
.
'
-'
'
.
' '
'
.
.
'
'
:
.
-
-
'
:
'
'
..
.
'
.
:' :
' ' ' :' .
'
E
: '
:
'
'
:
.
.
.
.
.
.
-
.
.
--
-
.
.
-.-
-
-
. .
...- - . . . ..
-. . ..
.. ...
j
.
.
.
.
.
.
.
'
E
'
-
-
.
': ' .
-
-.
-
-
..1;-:-
-
. . ..
-l:::::),-
-l:i!,,-
-j:y::r,,,
-
d':i!s,,-
'.::::::.
.02
=
-
.i.1k)f.(1;. ,kkpj,E L,.k,..j,. .;).;;;. ( q ((E ( !. )l l (l. iE E.i( ( ; .(.(y ( q E.. E .( (. i . (. ; ( i E. ( k(.. !( ;Ej((( E. ( ii q.;.( . @ (.'(2.E.(.(.. .y.(. (i. ( ii.(E.ralull .. .yq. y ( . .. . .(. yy... .. y .. y utnxl-uuuu tjktkk). ..( ) ; ...(.... l'q) j j. jy.yy (y... j )...j.y..j)y.(......y... ir..utg.. . y .j... y.j. .)..( j(....(y.y.(y .. (.. .y jyg..).y.....jjyyj (..(...j E. ) j...y j. q ). ). .( ( jjj j(. ... y .. y ... . yq.. jjjy jjjy. .. jjjjjs jjjy.., .j. jjjj.yj... . t . jtdkr' y,r, r; , r E .
:
.
E'. .. . .q :
..
t $900
950 1000 1050 1100 1150 1200
E
kklr;j)::iiE :,kk;.;.I;
,l;j,kl-
.
.
.
g.
.
.
..qq
:
.
EE .. .
: . .
E ' . .. .
E.
-
.
E
:.
. .
.
E . q. ..
.
.
-
.
. .
.
.
:E
.
-
-
. E. ..
. E .
:
.
.
E . EE . .
,.)
(....E(
#tikigkttttty
...(.
..
.
E
!
.: E y . -
-
.
.
.
.
E
i
...
E: ..
E. E. r
E
- ..
-
E
E .
.
.
E
-
.
EE . ..
E
.
.,
.
. . .y
.
,.
..
....
,
$900
0
950
0
q.
E E E. E . . . . . .
.
..
.
,
$0
:
t(gj;. .y. tjjjgjr t(jj;.. . uj ..k. jjj;.. j;j:. . .,jk..
..
,
-$24.32
-$924.32
,-924.32
25.68
75.68
1000
-924.32
1000
-924.32
75.68
-100
1000
-924.32
75.68
-150
1000
-924.32
75.68
1000
-924.32
75.68
-50
.
-200
%)llqsuRkrqcE;
.'..'..'
oLLARs,
1pi4r'. j''t?' t'tt'jf 11IIk,::iiir'. ,ll::,,f )'j(y. jyj'. jy': (1717:':7!7*)471'7: t'. 1Ili1i.' r' .-' jf y' jytfytjjtyyyyy'ry jjf )j' rqpf .:::4'7/E '1iiiF:iljIi)'1,:*7/. #l'r... yyf (j' yyytyfyjjyjttj yyjf tjf l-:jf (' )' yy'. 'i.!ii ; !t;; r. y' r ;j y.yj.yy.jq.!. jyrjyjjj.( yy.j .y (y; j jjyr . y y,j . j.j.y Ii.IItttiitk-. . jjjj.j.Ey.,-p. -t7; ty yjj 1i. y:.j) -r!qjj-',,:. . . . -.;!p!.. E '-. ' ..;.. !... . .. . iI..IIiiC-. . . z .. r..r. r.
--' -'?' :)' -' --' q' E' -l:::::,-f -l-'-l --' -' t''fpif f' (' )' j';r' ))))')))')(j))L. fpf :' IIETE p' y' lrf :
!.'-' ')' LLk.k----i'. t)f )' j'F (llq-f r' g;f q' yllq-:llq-f 1* i'q' (' ()' litqE (tEi i EE (SEyjjj))Y Tl:qj'l1i' (i;;if 11* J' t-' 1* t'44)* ! E(:E ,'rpqjlltyi.'dtrtr-;k (p' @7* 'i.L3' r' (' :,ii;;' q'' 'yrj--1.! i!ijl. $' iii ''i(: '' ..'E((' . .'i((.';E'(EEi q .E !; ij .! 'EE7'(..(.E!E'.E.Ei iiE.i(' .t:. .q'iir . !k(.r E ( .. : ( rj ! ;lkj.yqi!jq .. E. . g..;.. E !gjjyjjjjj,; . y
Alqo
OTHER
syx-rlqEn-lc
STRATEGIES
k
FoRwARos
67
-'.''
ttjyf
'
'
' E
:
-
.
-..
.
.
'
:
.
. .
-
... .
.
. . ...
.
--
. ..
. . . .. . . . .
: .
'
::
.
..
. .. - . . .
-
.
.
:
' :
:'
.
.
.
:
'
:
:
57:):14)'44484:
' ''
. :
.
. .
..
.
.
- .. -
-
. ..
. :E y
. y . .. --- - . .
.
-
--
..
----
.
-
=
(' -' :'''
-
- . .
-
-. --.-. . -
.
.
11:: rlEa(:!j;---.. (:;,Long S&R lndex
i!!,.:. EllEF:),11:E2 ::,:,j;;--..
2000
lcto
lclo
-1000
xlo
-.-...-..-...-
-1000
-p,to()
0
500 1000 1500 2000 saR Index at Faxpiration
0
500 1000 1500 2000 S&R lndex at Expiraton
.(a) payoff
(b)
($)
prof'it
combinedPayoff
xto
s1000
Profit
$75.68
t
.
.
..
.
!
.
E . -.
..
0
500 1000 1500 2000 S&R lndex at Expkaon
'
E'':'' -, :
: ' .
E ' ' : : '' . . . .. . . ,,.. .
.
.
- . .
.
-(yt((-.
,,,j .
.-
. -
.. .
.
..
-y
.
. .
.
..
..
.
:
-
,y.
-. . . . .
-.
. -
.-..-- ,
.,
-y
-. .
,
:j.y:.
-
-
..
-
.
.
ry
,
.
-
-.,
.. .
,
ff diagrams for writng a covered .S&Rput. Panel (a) is the payoff to a sh o rt SszR p ositio n. Pan eI (b) is th e p jyof to a short S&R put with a strike price of $100.
Short S&R Put
2000
2000
lct)tj
lolo
0
O
' ''''.''''-'''-'-'''-'-'-''''-'''''''
--xotl
-xx
0
combined jayofffor the short S&R Indx and writlen put. Pnel (d) s the combined profit, obtained by adding ($1000 + x $1095.68 to the 1 payoff in panel (c).
0
500 1000 1500 2000 s&R lndex at Expiration
payoff
,$74.201)
(b)
poc
profit
($)
combmedPayoff
1000
($) Combined
Profit
2(00
-2oco
$95.68
1000
-5 ltotj
i
-1000
'
-2000
00 o
1500 200 S&R Ipdey at Expiration
0
500 000 150 2000 $&R Index at Expiradon
(d)
(c)
(d)
It is possible to mirnic a long forward position on an asset by buying a call and selling a Put with each option having the same strike price and tilr to expiration. For exnmple, we could buy the looo-strike S&R call and sell the looo-strile S&R put, each with 6 months to expiration. ln 6 months we will be obliged to pay $1000 to buy the index, just as if we had entered into a forward contract. For example, suppose the index in 6 months is at 900. We will not exercise the call, but we have written a put. The put buyer will exercise the right to sell the index
.
=
500 1000 1500 2000 S&R ladex at Expiradon
FORWARDS
500 1000 1500 2000 S&R Index at Expiration
(a)
-1000 0
'.--'-''''---
-1000
-1000
(c) is the
Panel
gjjygjjy jtgg;j jtg yyyj.yyyy;j jjjyjj tjjyy (;j
-.. - - .. ... .
-1000
in price. Thus, for index plices below the trike pric, the loss on the written put offsets the short stock. Fr index prices above the strike price, you lose on the short stock. A'position where you have a constant payoffelow the strike and increasing losses above the stnke sounds like a written call. ln fact, shorting the index and writing a put Produces a prost diag'ramthat is exactly the same as for a written cplt. Figure 3.5 shows this graphically, and Problem 3.2 asks you to verify this by constructing a payoff table.
,
' E' ' ' ' ::: . .. ... . : .
:
:
- :. . : Ejryijjjtjjjjqq
-2000
(c)
3.2 SYNTHETIC
..
.
.02
1000
-1000 -2000
Combinrd
zlotl
10)0
(s)
:
j t. .. ....jjjyssyjyysj jgggjjjjjgjjyy . jjyr ... jj.j)yy.jyyy..j i.i jjyjqt.jjt:y:jjij): jtjyjjjy gyyjy ; gjggjj r.( r. ... , . t). (;j tyjyyy ..... r. . r .. . Shol't S&R lndex Payo and profit ..
Shol't S&R Call
()
-...............<.+......-.-.-......
E
-
-.
-:!ij;-
2000
()
'
!
-
-
-
Payoff and profii diagrams for writing a covered 5&R call. Panel (a) is the payoff to a jl rjj; j; k k(j) (lsiticjrj. Panel (b) is the payoff to a short S&R call with strike price of 1000. Panel (Q'isthe combined payotf for the 5&R indx and written call. Panel (d) is the combined profit, obtained b# subtacting - ($1O00 593.809) x 1 $924.32 from the payof in ppnel (c). .02
r' (j:' j')' j'. Jl' t'.1r' -y',: .y;' .',--'
:
;E E. .y y
.'
y' (-y' y.' ,'ytt'.' 'lrf F' C' ;'lqlrf :r'' !'' '''7* F' ' y' y)f Lj' t'jjyjjjtytjjjjjjjjsi' !' 1'* 1* 7j:141* '' ' iEEE E g' j-.' (' J' Tl'' )' k;f pf .))'''Eq. ;'E' q' t''1* r' i' ;tllk ( EEt.li1* ! !Ei'.';'i(('.(''.(! ;'' EEE(E.( E..E''E.'..(i..'. 'E'..Eq.''.(.'i:''.'E. ('. (E' (' ' ::'ST;';':F!I'/' . @(!..!. .' i . ' . ijjjjli '..)'... E.jjjjj((E j. .. . r yyy. y. yyjjjjjjjjqjj.jfjj ..:jjjjjjjj: (''. ;.yj .y yg..y..y.yy ( q .iy.I(jj(j;jj..i)iE yy. yy..y.yy..yy...... ygyy.yy . j:)j:;.((jjjjjf... (gII.rIkIIy.j;k.C-jI--j--E IPIIF:!F,E ysjyj y ... .y. .jty. . gyj .. jE i . . . E jp,r 2 .. LklijjjL,. : j. ;E! j . 2 jjjy . !. q. . rr . . .
'
-y
:
y-kkjjyjjjjjj. .
-
:
E : i!
: i
.
,
for $1000;therefore w af obligated to buy the index at $1000. If instead te index is at $110t, the pt is not kercised, but we exrcise the call, byig the index for $1000. Thus, whether the index lises or falls, when the options expire we buy the index for the strike price of the options, $1000. The purchased call, written put, and combined positions are shown in Figure 3.6. The purchase of a call and sale of a put creates a synthetic long fo>ard contract, wlzich has two mipor differences from the act'ual fonvard: 1. The forward contract has a zero premium, while the synthetic forward requires that
wepay ihe net .
.
.
optifl premim. ' '
'
.
.
(
2. With the forlard coneact we pay theforward jnce, while with the synthetic forward we pay the strike price. If you think about it, these two cosideratiops must be related. If we set the strike price low, we are obligated to buy the ipdex at dijcount relative to the fomard price. plie than the foioard pric ij a bne:t. In order to obtain this beneft Buying at a tower we have to pay the jositiv nt tjti Ijfmium, which stemj from th call being more
.
$
6%
jxsu
r;d qqd 'k,l::'f )' -' t'E Ejjljjgjtkrf j',' i,k';f' C' q' @' F' ;7:6rq* 'L---;L;;LLk,LL'.;,LL;:' ,.'. p(' tf lf p' ,---,L'. q' ;')'i';'y' ,'@' ;',j,j:::j;r;j:. )' J' (' 1' qqtl)f @)' (777* ' E : ' :' : E'' E : !': ' r:E i@E '! .; F((i: EEi(Eii'1.E(J!);)IIS' ;':!:EEEE((:!: ' Ei'@ . (' E.i: E.E(;!
RAN cEy CO ('.NARS, AN D OTH ER STRATEG l ES
SYXTHETIC
FoRwARos
$
69
--' ,'' ':';'-l:::::,-f -.' -' S' -' Eyf ;'');t' ,':yr' C' rrf
.'.''
t')1* )' (' ,ir.;kk):,)f C'. t)'j.,( r' 1)* (jI(:;-,' iliiE!!f -jjjjjgEj;;kr'p' )j'. y' j'';' j--ty!Ejjj;'. yyyyy)).-' )';jr' j(t)'. j'. r' f' )-'' y)yf )' ly'.f yjryf yy'; y;tyf ('
5r17:777:qE7'77r: p' !' .' 777,7'5: tt' t.yt.jjjj,k,jjjjr'-' rr '!:.l tif:j.( '! r ,-)-! .!:(t-(.. ( q(r.;. (ri (j(.!!. ( ('(.''y.)yq .; .'yy)y';;..g .(-.ti: )yt;. y j ) .jjjjji;jjjr,. ). j ryj f'. Ip;p. y . j 11lyly y . t' .tryj tjytty-y .!,...t.yqyr,.j.;.q;,y.).y..:...r.y.jy.q. iy.. ;.rpi..)- j,,(,.p,.!y (()-.t. !;;r( . . f.;E ' : :'!.:'E''.;'' ' ' : E .
:
.
-
i
.
': :
. :
-
j-
; -E
-..
.
.'
- . . . . .-
-.
.
-
; E'. : ': . E
'
.
--.
'
. .
-.. .
-
-
q -
-
:
EE. E
'7:
:'l
IEE : ;1 .
..-
-
(. .
-
Equating the costs of the alternative ways to buy the index at time t gives us
E
.
'
.-
..
.
. - .. . - . . .. ... .
' '
:
-. -
'
.y --E Eiy- . -- ,.- ( . - - -. - -- -- - --,. -. . . . .
-
-
-
-
.
(.iitjii.14::. ;,ii5;;. .11:::
(rl,-
. ... .
.. . .. . . Purchase of a looo-strike 5&R call, sale of a 1Ooo-strike .S&Rjut, and the comblned position. The combined position resembles the profit on a long folward contrat.
E;... ..: .
;.
......
.
- . . . .. .
.
... . . . . ... ..
. . . . - . . . . . .. . ... . . . . . .
..
. -..
--
-
.
.
.
.-
.
.
.
PV(F(),w)
2O0
-
Puttff, F)1 + PV(ff)
We can rewrite this as
15O
Callt#, F)
100
so -50 -100
Ptlrchased call + Written put Combined positon
H-
150 -200
-
Puttf,
F)
=
PV(F(),w
-
K)
In words, the presentvalue of the bargain elementfrombuying theindex atthe strikeprice (the right-hand side of equation (3.1)) must be offset by the initial net option premium (the left-hand side of equation (3.1)1.Equation (3.1) is lnown as put-call parity, and one of the most important relations in options.
slozs N
O
-
gCal1(#. F)
=
.*.
(' )' q
Example 3.1 As an example of equation (3.1),consider buying the 6-month 10001)l@p ' S&R call for a premium of $93.809and selling the 6-month looo-strike put for ).1stnke )))) premium of $74.201. These transactions create a synthetic forward permitting us to C a '!'1 buy the index in 6 months for $1000. Because the actual forward price is $1020,this synthetic forward permits us to buy the index at a bargain of $20, the present value of !q.g . )ll which is $20/ 1.02 $19.61. The difference in option premiums must therefore be ). .'tq In fact, $93.809 $74.201 $19.61.This result is exactly what we would get )t.,$19.61. .'
(E
-250
800
850
950 1000 1050 1100 150 S&R Index Pdce ()
900
1200
i Et
(.'E
=
=
-
expensive t.11a:1 the put. ln fact, in Figure 3.6, tlzeimplicit cost of te synthetic forwardthe price at which the profit on the ombined call-ppt position is zero-is $1020,which is the S&R forward price. Sirnilarly. it we set the sike price high, we are obligated to buy the index at a high price relative to the forward price. To offset the exta cost of acquiring the index using the high stdke options, it makes sense that we would receive payment initially. This would occur if the put that we sell is more expensive than the call we buy. Finally, if we set the strike price equal to the forward price, tlen to mimic the forward th: irlitial yremim ntust tualzero. I this ase, put nd call prmiums must
jipwith
t
equation
(3.1):
$93.809 $74.201 PV($1020
i
-
=
-
$1000)
-
be equalr
'
.
.
.
.
.
' . .
.
.
' .
A forward contract for which the premium is not zero is sometimes called an offmarket forward. This trminology arises since a true forward by desniitpn has a zero premium. Therefore, a forward contract with a nonzero premium must have a forward market (forward)price.'' Unless the stlikeplice equals te folavard price which is price, buying a call and selling a put creates an off-market forward. Iloffthe
We have seen earlier that buying the index and positions prost buying a put generates the same as buying a call. Similarly, selling a covered call calll selling generates the same profit as selling a put. Equation (buying the index and a (3.1) explains why tis happens. Consider buying the index and buying a put, as in Section 3.1.. Recall tat, in this and the index price equal to $1000. example, we have the forward price equal to $1020 equals of forward price the index price. Rewriting equation the Thus, the present value (3.1) gives Equivlente
Put-call Prity . .
.'
.
.
'''
.
..
.
We can summarize this argumet by saying that the ndr cost t/-lllyf??' the f?7#c-'t: ttsing options ??2II'f eqttal (he n'f cost ofbttying #?dinde,x ldWlg afonvalzl contract. If at time 0 we en terinto a lonjforward positton expiring at iime F, Fe obligate ourselves to buying the index at the forward price, F(),w.'I'he present value of buying the index in the ftlture is just the present value f the forward price, PV(F0,r). q lfinstead we buy a call and sell a put today to guarantee tlle purchase price for the index in the f'utwe, the present valpe of the cost is the net option premium for buying the call and setng the put, CalltA-. FJ Puttff. F. plus the present value of the strike price, PV(#). trfhenotations Calltf, 7') and Puttff, r) denote the premiums of options wit.h stdke price K and wit t periods until expiration.) ,
-
'
of diferent
PV(Fo,w) + Puttf,
F)
=
$1000 + $74.201
=
Calltff, F) + PV(#)
$93.809+ $980.39
That is, buying the index and buying the put cost the same, and generate the same payoff, as buying the call arld buying a zero-coupon bond costing PV(#). (Recall from Section 2.1 that a bond does not affect protit.)
% Ilqsu RANCE,
70
AlqD OTH ER ARATEGI
COLLARS,
Similarly, in the case of PV(Fo,w)
writing
F)
=
PV(#)
That is, writing a covered call has the same proft as lending PV(#) Equation (3.1)provides a tool for constructing equivalent positions.
and selling a put.
In deriving equation (3.1), and in some earlier discussions, we relied the that if idea two differept investments generate the same payoff, they mgst have on the same cost. This commonsensical idea is one of 4he most important in the book. If eqpation (3.1)did not hold, there would be both low-cost and high-cost ways to acquire the index at time F. We could simultaneously buy te index at 1ow cost and sell the index athigh cost. This transaction has no risk (sincewe both buy and sell the index) and generates a positive cash flow (becauseof the difference in costs). Taking adkantage of such an opporlnit.y is called arbiage, and the idea tat prices should not pelmnit pricing.'' We implicitly illustrated this idea earlier in arbiage is called'tno-arbiage showing how owning the index and buying a put has the same profit as a call, etc. No-arbitrage pricing will be a major theme in Chapter 5 and beyond. a No arbitrage
.'
t'y' q' t'yy;'yjyy') Ti.-tli. ;'. try ry ty.. d:::: jj... . . ...t-. d::::. . 1EE1:, d:E!,r Ir.. t!!;k lk...l 1r...g..,. :!!; t!!;: Ir.. .1k: Ip4:: isd::::. ii':::;,lr..,.14::,, 11 l..1k 1( .1k:: iiIr..lk ii e 11r;4(:E:,).-(t)yyy,.:,j#---.)j-.-.(jy. llgli . .) iik-..-$kd:i1i.... lit.'t,' s a t!!i e t!!;i 14:::,Ir:::,a d::::Ip4::....' t:.-.#.jyy. . volatility 309% effective annual risk-free rate 8.33% ' (89%continuously compounded), dividend yield $0, and 91 days to expiration. '1r' ;'.
:: . .
:
..
'
:
!
:
::
.
C.
'
''' E' '. E. . :
' ' .
E'
.
.......,.......
.
...,....-...
...........
....
:
: : .
-
.
.....
' E.
'
.:
.
:
.
.
Ec)E '' .
- . ..-
-
.
,..
.
.
-:
.
.
-
.
.
.
.
:. : . -
'
.
-
-
-
. .. : ..
:...: .
.
.
-. y--;.j-,..;...yr...-.;-...-.q...-...-y.,....-.-q.-y..(q....)...(...j......-..-r.y .
--,:14.
dr:::,.
,t:::,-
-
dt:::,-
..................,...........
I::::.
---------.
...
=
=
=
r .'
:
:
E E
E jjjzigrl:i j ( .j . .yi
!(...j:!jjjjj;:-.. .
-
!
.
35 40 45
y
.
-y .
-
-. .
.
.
.
.
.
(.(
.'
it
yy. . q...y. .
...
EE
E( .:
E.
.(. E j
: '
:. S . E. (. E( :jji,kjjj(,. . E.. E.
. (jjjt, jjjjyjj;ry tiji:gl)lygrjr ......j.y. jyjjj(jj,!,-. tjjjj. .y...y. :
6.13 2.78 .97
:.
..
:.
.y. .
. .y . . .. y
. .
-
. .
.
.
. - .
...
71
You might ask: Is there a lower-cost way to speculate tat the stock price will rise, still has the insurance implicit in the call? The answer is yes: You can lower the that cost of your strategy if you are willing to reduce your prot should the stock appreciate. You can do tlzis by selling a call at a higher strike price. 'T'he owner of this second call buys appreeiation above the higher strike price and pays you a premium. You achieve a lower cost by giving up some portion of prct. A position in which you buy a call :nd sell an othemise identical call with a higher stn'ke price is an example of a bull spread. Bull spreads can also be constructed using puts. Perhaps surprisingly, you can achieve the same result either by buying a low-strike call and selling a high-strike call, or by buying a low-stlike put and selling a high-strike put. Spreads constructedwith eithercalls orputs are sometimes calledverticalspreads. The terminology stems from the way option prices are typically presented, with strikes arrayed vertically (as in Table 3.4).
There are many well-lnown, commonly used strategies tat combine two or more options. ln this section we discuss some pf these strategies and explain the motivation for using them. The underlying theme in this section is that there are always trade-offs in designing a position: lt is always possible to lower the cost of a position by reducing its payoff. Thus there are many variations on each particular strategy. All the examples in this section will use the set of option plices in Table 3.4. We will assume the continuously compounded interest rate is 8%. ';'7* j'f)' ?' T' 'SX kjjj(r)'. j'rjf )' f' @' ;'(' )' j'g' 1* ;r' qltqqqr;);f y' yjjltatjjljijiiz'k'k. tjjzyt;jjrgr'-d r' y'. j': t;f q'lrf j'k)f (tf ytf tt'':'p' ' E' i ! : (q55:'7* j'y' '')'()' ('. jjj'yy (': (jf )' E i jEE ' E E 'E '' E'' E EE'E'' ' : '. (('E..(. E..E(.iE;.q.(.E(.q. ('!:'.lj.(! (! ; (q'(... .ti(i.ij'ir( ; ;1:.j(.'(' (..('(.(j.((?'iE'EE.(.( @ . (.!.((!'( .q 1* .(( Eyrj,j . ! .y i. E! E i .EE . E E.E. i. E E E -L,)q.;-;L-t3qq'.;.. i. !t.EIE.-.t-.I;ftii-.... E. .E .rt-' E .E E . . g .E .. .( ( y y . , r r . .. . q r . j y . . j g j ! ( j . j j y . g . j ;y. . ....y.y.y..jryr. yj gj. yj.;j j ;j..jy,;r)r-))j. y.. .j.kj.yj. .;Igtii;l j. yyy .i:k(..... . .))t!lkk IjEjjL...?.:)t-.j.y.-j . .j ..
k:
price.
3.3 SPREADS AND COLLARS
'
COLLARS
An option spread is a position consisting of only calls or only puts, in which some options are purchased and some written. Spreads ar a common strategy. ln this section we defne some typical spread strategies and explain why you might use a spread. Suppose you believe a stock will pppreciate. Let's compare two Fays to speculate belief: entering into along forward contract orbuying a call option with the strike this on equal price to the forward price. The forward contract has a zero premium and the call has a positivepremium. A difference in payoffs explains the difference in premiums. J.f the stock price at expiration is greater than the forward plice, the forward contract and call have the same payof'f. If the stock price is less tan the forward price, however, the forward conact has a loss and the call is worth zero. Put-call parity tes us that the call is equivalent to the fonvard contract plus a put option. Thus, the call premium equals the forward the cost of the put, which is insurance against the stock price being less t11:,11
Puttff, F)
-
Alqo
Bull and Bear Spreads
a covered call, we have
Calltf,
-
SPREADS
Es
(. ..
0.44
1.99
5.08
SAnother way to express the plinciple of no arbitrage is using profit diagrams. Given two profit diagrams, there is an arbtrage opportunity if one diagram is everywhere above the other.
-'.
(' y' t' L'
r Exam' pIe 3.2 To see how a bull spread arises, suppose we want to speculate on the 1i1 # stoc k p rice increasing. Consider buying a 4o-strike call with 3 months to expiration. lil
jt) From Table 3.4, the premium for this eal1 is $2.78. We can reduce the cost of the )' Position-and also the potential profit-by selling the zz-strike call. ).1 l'1 An easy way to construct the graph for this position is to emlate a spreadsheet: :;( l)( For each plice, compute the prot of each option position and add up the profits for the individual positions. lt is worth worling through one example in detail to see how this ',! ;( . . is done. . t') The initial net cost of the two options is $2.78 $.97 $1.81. With 3 months (!r'l ) . t) interest, the total cost at expiration is $1.81 x (1.0833)0.25 $1.85.Table 3.5 computes 'i) the cash flow a t expiration for both options and computes profit on the position by .,! .# )ij subtracting the future value of the net premium. lt Figure 3.7 graphs the position in Table 3.5. You should vel'ify that if you buy the )) %. f 4o-strike put and sell the 45-strike put, you obtain exactly the same graph. 'i: :
.
.
?
IE
E'
=
-
E
=
.
%
INSURANCE,
'F' S' Etjytf ''-'$' y'i ''':')' q' y' !q).;')' j''p'r':E iqf :-;jj!jj;;;j'. ,))-'t'E IE fp' : '!
COLLARS,
AN D OTH ER ARATEGI
y' ?' i!if if t';'r' '(' p' 1* t'jr'l':l h' q' )' i'!' F' 5* qjf ?;' '1* ;y' .jjj;Ejjj;;')' r' ryf j'. j(' )y'. ''!l')1!* tf l;jr' j)' (' )' jk-,tf jj'r yj'k E'E;! ' E(E'iEE(' '' ;( TEs '7)7. !f;qiliE;'?( ; E(.EE' E('E(:;;i('!E. E.i(EiE!' f: E...'i' (E.. EE ;i: E ' i q))! ('I ((i . . .. '. . '. (.' . .... . 'E iE .E. E!tj E E t)E i E .. E i E !.. EE.:Ei: . ! E .y E jjytyyt,. ;j.. !.E.jy(r(!r:jg..E;.-yj(E.jy (j ;'k t))#,.E,,y',()igyir)jj)(,...(.),',(,'.lE),j y '.! ! .lFi. ( jE;:j.j ,1y !.((. . j y.. yEq.;. jjj. jj)j i. .(-..ErE)l((:()-....-;)....p.;:.............1)lE . .....E;f................... . . lt..ujs; .. . .
.'',
jjjjjjjk'. jjjjjj;;'. ).-jyyyyyy..' )')' (' )'. t'y' )' t.jt' jjqjjqjqjqjjjjf t'yyt.tf yyjj)y'trl ) 11 . trtt.i-jijjjyjjkkk...)y tt r . . ... y t. E
'
.
:
.
E : ..'. '.
E
.
.
. - . -.
.
.
-
E'
Ci :' '
''
- -.
'
'
:
-
.
' :.'
'' :.'
: '' ' ' i :. .
--
'
:
:
:
-
. -
.
E
:
.. .
.....,....
....
...-..-.
.
. ..
......c.,
-
:.
.....
.. ......
. ....
..'.
.:...; ... . .;.... ... ...... ... . .
..:
..'......
....;
.
. .....
.
....
.
...
....
Profit at expiraton from purchase of 4o-strike call and sale of 45-strike call.
.
tktjijtt tttt ). 4,
k''.!'E .iiy
-).
.
;y.t',; rg
,
.
.....
))'t.(.';( iki'ikik...ygjyyltyl)
y.y&#hiiitl:
.y'
)ll':i ;t.' (...
itl)l
u... Ilt' lt)t(, ( (!z,,cIJ',.t').' . -.;.,,r;
,t.;,i..
...
.
..
,);
g
. .
.
. . . .....
...
. . ..
E.,
.
.
-.
..
'y',4pkcy,
,t
.
......
....
...
$35.0
$0.0
$0.0
37.5 40.0 42.5 45.0 47.5
0.0 0.0
0.0 0.0 0.0 0.0
2.5 5.0 7.5
50.0
-2.5
10.0
. .
...:.
.
.:..
-t.y.i..q
),t . ttliiElntft ; .. . .
. ...
.......
. .....
.
.
t
-.ytrt
..L...
..
....:.
..
EE..E. E. :...;.......T:
.
-y
. . ;.
,E,)',..(!'j:.Eq)...i;')ij;y.?;.:;r).r.,tj.ly:E)' q.j ,,; .yu. ...r .!) ttE',t q ,t2d,y,qj2I.y.y(yy;''rEy
E:....E.i'
-$1.85
.
'.L oT, y .
-$t.85
k.
r;
:...:
.
.
(
-1.85 -1.85
-1.85
-
-1.85
1.85
0.65 3.15 3.15 3.15
-1.85
.
:
-jq: .
.
'
!
.
i .
.
:
E
.
. y :
.
:
.t,-
:
.
:
:
: . :.
:
. ..,..
...
....................
......
...
..
..
:
:.
.
..
.-.. -
.....
-
.
..
:
! !
. . :. .: .. : .;. . . . .
. : .
( .-ty.j. . .
-.. .
for a
-
.
-
-. .
bull spread: 40.-..41 bu/ng a 4o-strike call andselling a 45-strike
call..
..
.'
.
call and sell a zo-strike put, 1 B Py a m-strike tll ). ,)) 2. Sell a 45-st.rike call and buy a 45-stnke put. t) q jj l)j) The srsttransaction is a synthetic forward purchase of a stock for $40, while j'4 1, transaction is the synthetic fomard sale of the stock for $45. Cleady the the second '(r((. payoff at exp iration will be $5; hence, the transaction has no stock price risk. Using the '(.'; . t assumptions in Table 3.4, the cost of the sategy should be '
.
E. (.
g
E
(
.('
.
.
-0.25
El
:
,
.y
.
enter into the following two transactions:
Suppose we simultanously
=
. .. -..-
. . .. . . .
. profk iagram .
..
' .
:
-. .
-.
5.5
E
F; @E ' E 71 ijq ErTTSqITETI E!Eli!E!. i ! )! E'E(i'(.':q;'::7qtrr':':T(' !'qlil !i' ((i' )'E qq EE' i' 'E j)y !i j;'(qiqlr F' ((E (.q (.' . !' ( q: .:('7..yj' .EI.! q:: r;i i-.E .. . ' )r!):E -! (;'r l (jEjEt;.)'.('.(.yE.. ...Ei.(...E.(q.q@. . qE E.. j- EE. . . i ;..j i.q.y.ji.;g;.y . . ... .. jj g(i jjrE yyjg,! j. (.y . y j iliii:iiiti. jy. j y . jg j yjty j . . . )t. iq j . -2 ... j ....yj-qy-,.-.-jyy-. q.j; g.yij.tj llliipFk.( )!71(. . .(r. (t...y:.....,....;........ -----Lj--. t.(. . .. .r.. .,.. rr :
1* q') xample
lt 5 x (1.0833) $4.90 ;) ..) i.1i l'' the initial cash flow is t...jj In fact, using the premiums in Table 3.4
r('..' .i'.
''''''
E
-1.85
--jyjtyyykt)'. t----ylt-yl')-. -' :'y '' )' -' -l:r:::,-d -;'t#'rt----,.' y'E 7* F' 5* f'
.' .'.'
73
*
k' tll-E'd r' ;'r'r' t'(' j'j',jjy 6* C' i;' @' jq';l q)td )(' f' :'::' (,l::.d )' )gg-'. y' jjjjf4jj'. qj,,g'(' p(' (' jjjyjy'. yd qd )' 11* )tq''lr)' r' q)rd y' q' i)d ky))'.' 11iIkk:kiiIE';'. t/ftqtl'.)ltd 1* (td prd r!!iI::::;;ki'. t'r'.'y'. y' yjjjjygjyytyjy.lydjyjdtjlgyj-ijyyy.ty jjy.' kjjjjjjjj,qd t(';l' rt::)d r?' 1111(* j'(' y'. ('(' 'j' r' @,'1* ltd tt.yjjlrd .jjjjgjjj(y:(' t'-'@t@t''-; t.j.'(j' ''
)'' )' j)'. jd
'.
-1.85
-5.0
(
-''
: 'E : E . : . . :' : .
%
cash flow in the future. Hence, it is an option spread that is purely a means of bormwing lisk. The reasons for using a box or lending money: It is costly but has no stock price spread are discussed in the box on page 74.
-.
.
-
.
,'
i
-
't
AN D COLLARS
'
.
-
-
.
...........-
l..jitjk (tsl ; t i.. E#itiijtyi
.'
.
..
.....()j)..
z,
: .. .
..
.
..
.
.
.
SPREADS
Es
-
.
-
-
.
!
'tit
14:4. ;,jis;;. ql,qqr:,(,Iqjii. .11::: (zl,. .+.
4
($0.97 $5.08) -
-$4.90
=
ilAnother way to view this eansactionis that we have bought a 40-4.5 bull spread using tt) 4.0 call, se 1145 call), and bought a 4(:.4.5 bear spread using puts (sell4.0 put, calls t).! )'1buy 45(buy put). k
2
Ratio Spreads
1 0 -1
20
-
ij'
Iun spread
3
-2
($1.99$2.78) +
25
30
35 40 45 XYZ Stock Price
50
55
60
($)
The opposite of a bull spread is a bear spread. Using the options from the gbove example, we could create a bear spread by selling tle 4o-strike call and buying the 45-strike call. The prost diagram would be exactly the opposit of Figure j.7.
Box Spreads A box spread is accomplished by using options to create a synthetic long fonvard at one price and a synthetic short fonvard at a different price. This sategy guarantees a
A ratio spread is constnlcted by buying ?n calls at one strile and selling 11 calls at a different stdke, with a11options having the same time to mat-urity and same underlying asset. Ratit? spreads can also be constructed with puts. You are asked to constnlct ratio spreads in pfoblm 3.15. Also, a ratio spread constructed by buying a low-stn'ke call and selling'two higher-strike calls is one of the positions depicted in the chapter summal'y in Figure 3.17. Since ratio spreads involve buying and selling unequal numbers of options, it is signicanc of this may not possible to construct ratio spreads with zero premium. be obvious to you now, but we will see in Chapter 4 tat by using ratio spreads we can construct paylater strategies: insurance that costs nothing if te insurance is not needed. The trade-off to this, as you might guess, is that the insurance is more costly if it is 'he
needed.
Collars A collar is the purchase of a put option and the sale of a call option with a higher strike plice, with both options having the same underlying asset andhaving the same expiratio date. lf the position is reversed (saleof a put and purchase of a calll, the collaris written. The collar width is the difference between the call and put strikes.
Q INSURANCE,
AND OTH ER WRATEGI
COLLARS.
SPREADS
ES
41;:::::;* lligzg!:d lllrqjjjl.d y' -'. -;--'--'' -' '-' j'y' 412::::* ljjjr::d t'j'11j@111)* .gry' 11:(!:11* '-.:)'-r-,,451,,,$*-(' -p'. ('E r' )' f' y'. Illiil.)d .jIt.'. y' j,t'' y'-jf yj',':'t' ;''-'. y' ;t'..' ,'. (' )' :',' !.-'-kii:i!k'k-.t,' (t'' j'j.' Lj6LLj-'. (:!4E11j,)112)..* 111E4;)* (11q'7)1:* lllr''d t'. l't'-' '-t'. y'; )' 'iiiik'I'k'. y'r t',-' -iiiEEiik;'j'))'. i')f E E EE' E E E' . E '! E .' ' (. E (' j E'' .(E ' E il!lld t'. y' r' j)t'. t'y'. lt'y yq'jjjj tjyf j'j( r-'iiiit1'i'i.'-' ?' i;)f .'k' t'#',' j(' jtj'ttf jjf ! ( i jtjj i jE(y y' E i ! ' . ;; i ! (r . jjl jjjlllr:,id illrjd .1I(r)(ii.' kl:l:ri,id ,'('E-'' )'. p' j')')' ('. r-.'. j;))','. LILLLL'. ))'. )::-'.' r..,rf y' )' (' jtjtjf )')' ;..L'. h' jyf t'f:'''q'@' j'. (.jj (jj (E ' iE jy'. ,y' )'.(' jq)()'. y,'. i-' tj.tf )k'. jtf )('. 7*. (4jr' y(' @'yjttjf yyjf jjf (j'; y)'yk(j. -.iIiii!iiii;'-)))t-)k:--:--tjt'. )jt-.' ('. )i-q't' IliIqi:i1i-'. )'. #' j(' . (i jtk ' E ' E'' i Ei ( i'. E..'..E'..' ' ((.' j ?: tkki)i;k'L7. ;.y y. jy -.('j'j(( . (:jj. 'gy'jE('y(j'E'.' 'j) jgyjj.j.2.ry(j j k ;ygE. (.gqj.j(...qjy.E- y (.j((gy y(i. jy (Ek: jjgq.j. t( j;('j(j.: ' .!i('j--)(;( .. (I!'. (.(' .(EE..---'ji ';'('''.-.j-'!.g'':j;.k .j- q(.j--yg g(.q ( ( (j (( j. ;j'jgyjg '. j(t jj(!.(y.j.yyy kq. . - (; j . y! yg'.k j'.yjj j. jy 'yy .... ... ' ' . . j.' (j!..(;: jy . jjj(-j (j(kjj. ('.E .. ' @ yyyjj-(..;.qg...-..E-...-g..y.y.-.-.E.g-.......(..(....j jj-:.-(.yj . . .'l. -.-. -! .. 7.j r (. j. ..j-.jj.y.j.y .yjg.y(.lij.'--.i(j.-(.--j..yjj.. (jt): . . .E i:-'j . (; ::1k-k1i2-.j';----.(.....-q. '' .. . ?r.-. -....tj .j .E --(';. ;-.-. '-;-..''. -. . -.p .... - - - -.--...;,...---.'y j..-.j......q.j.---E..,jj--.-....,.-'t( - -.;. ... ......-...--... .-.-k.)-.. - -.; ).:---.;.'.;.;,; - :r;.jyt.. - . . ; t).-),)-p . - --g . i !..-. ..j..(-..,.- --..--.r:. r-.(..t-.:- -:y. - j.t,(-. . .-: ;..--). :.. :- i.-... . r. --- .. .)i. - . - ;. -. - . --,. . ..y . )q)j)q;. --i ;- . - $: ...q ; .-..-. -.- . - -.-;: y /,):. ; :. -( ; , . ; . . ( .. . q gy..-)..-------- ----. i -.------..- . ).-Eii.. y, . -. r g . . . ( .. . . . . g ( . .) y . . . . . . . .. y,- - ., -. ... ... . . . q. . . . qy - . . jtgj . . . . -.....r.. . . .. -, .j .j . g - y . .. . .. .. . ...(r.. - y ,j...r.y-..k. .k-y. yy. . (j. . .j.(..:. .. . .. kj g - :( . q . (. jr. . y . .. . . . . . . . . . r r y . . . . .r. . E ) ...' .''..' . )y... ...... 'E ..' . . .'(. g. .... .... .E...t ( .. . ..y..j .. . . ..........,..y(.... .. ...L.. .. ( .........jj. ..... . ) . . ... .,. ...; uy....E ., .y.... .j.... yy. .: .. ........t y(.j( .(...y .y....j... . j..g .)..,.....j..... . ..y . . ( . . ( . ....,;E..j (.. .(. .:j.(.. j.. E.. y ...j .( . .' j . . ... gj.. ' ..'. . .gj..j jj .E.... E:(.q. jjj,jj:Ej..:...(,i . ' .... ;ygjjj(;;jj;.(q . ; k....' g.. . .;.tkgjjj. .E. .) g......,.r..jjj,yjq .. y.ygy..g..g.y..yj..gr..jy.g..y.j. s.y(j. .!.j .j(gjj.y .yyk,.: ... 'j ... ....y . . y.y.y. .L.,.,..... . (.(:..:;..jj.kjky ..(..)y:.....g:r j.....(..;. t..:: .y.y...y.yjj. j '....:..r(( . .. .. . t .y.jjmt--k . .j....... ' . .. ... j . 'g (. . .... ... .g. .'. ...(. '.)(.k. (..'.y.y 'y ..y.... j . zxtEjjjk . . y.jjjgy;. . lk ..... .. Jk '..... ,y,,,yj'j .'.y.' x..,,,,'. .1,...,..*. di...,.d .1.,,.,,* -'-;' -.-)'-l;'. ,'.'' -' ;'.1122* .:Ik.' 1;,..,$,..11k.* ;'j'E -', --' -'rpf ---.' --;'. -'-' --' -r' -'.'''iiii1f'-' ?' S'
ti::qjlll.d .y' ('')' ))' ,'! )'. (' ?':' 4* ikiikkp''y' ((' yy': yj'. tll't;f .-'yjkyy';r t''iiiii'i'. tkkiijidif'-.'tjyyf iiidiik'. t'. .-'. t':-'t'y-'! tjjf 'i'iE:Eikk)'--' l'iikkk2'. r' .(' 't''-' l(r-t': 'ri'k1i'i7tiii' ' '. . ..
' LE )' 1* j';F47q'11* ttf k' ':'p' jtf 1* @' (' !' jf'yjf 1(* y)'y;)'. ;'gjjjgqy4rr'( :j-,'' ),' j'gqf j'l!f j?' i (: ;! '(.. ''EE.E.'( EE.E...(77:7E (rsiqp;!ll;E E' !i! 'E E! E E(iE'j6*iE(!j(!'((()' (.qjE!((.!(''lj'('('(E('i'E'.(.' !i'q! ;'!'Eq' 4))* ';'q' qqf j'(7' ti'jjr .)''.' )i'' 1* t')'(jjjjjjy;j'1: jr'. )' !.' (' jjf ' i.!; (' liE qllqq rjjjjjj(jjj;'. ry'y'. jjy-'r 'l'y .);))j);,,;)t'jL;)q'-j))'q;i;L)'.,jj. jjf yy'. .jy'. (', y' j'y j'rp'srjjjjjjjrj.pf .li.! . (.i(. E . i . .' . yjyf ltjr-yjt'-,t' )' yj'yr-,k;'. . ! ' .. t( . . Ey Ey! E t'yyf'y-i lq r!. . .. . ;. EE1. E !.: '!.qtEk! i ;. y.( . y). . .. ; . ( q jjj r,i . . ; ;y)!k q ; ly .. , . . -t.!q!)-. j yy.j j j j t(. yjy j . tt;!jyjj:::;;j;y. k jjy yj . yy r ly t, j ...iiE!itl....,...t-::.. y -.-'......-..k...-..........-.............. . t.. j . y r-(.--;... lt:t )-.qrt-r.r-....---'. ... t.(.y(.).)p,.t)-... 2rp,qJq. ......-.-,--...LL..-.-.---,L. ....-;-3LL.L.. . ilh... -'' '' -j'. :j' -lqfj,ktly'-?,ljip'jj : : (' ?' j)qE,yjy'. :','j'';F'' . : : :
t'y jyjyj', ;j.)l' tt)'. t, t) 'it-'-q.)
.'
.'.'g .''. '. ':':'
'
.
'.
: E :'
'
'
. .
-
.
'
. . .. .
.
.
.. -,.
-.
-
.
.
.
. .
;.
-. ....:;
- -
'.':.:gj.(/..:.'yEj'@(y(.j(!.!..(Cr,.rE..l((.(t
' . . ..j.j;j.L.'L';..;LL,,.L.,.jLj... .
'.
'
.
-
-
-
.
.
--)-
..
..
.
- .
.
.
:
-
- -.
.
.
-. .
. .
.
:
-
'.'-
.
.
- -.
.
:
-.
'
'
-
'
-
.
.
y
:
E
. .
.
,
-
..
:..g.
. - .
. .
.
.
..
-
-
.
- - -- . . r . .
-
.
...
:
' .
'
E
:
-
.
. -
.
.
'
-
-
.
: .
Ey
.
:
:
-.
-
-. -(
.
--
.
.
.
.
)
- --.
.
.
.
-
..
:
:
,
.
-- -
.
.
..
:
-
y
-
..
.E:.y).
'.
--
'.
-
:
.
.
:
.
.
.
. . ...
:
:
:
-
.
.
:
-y
-
-
.
:y.
.... y.
.....
:'
.
.
.
.
-
-
.
-
g. .
..
).
-
--
-..
.
.
y.yj ) )pkt,-. !.t-y-. . '
:
::
.-
..
.
,
..
.... ..
..
'
-.
..
.
.
-
.
-
.........
....
.
:
-
-
: -.
.
: ..
.
.-
. .
:
.
:.y ..
y.
'
E
-
.
. -
-
....
'
: . :
E
:
.
- -
-
..
...
.
:
-.
-
. .
.
..
-
-
-
: .
. . :
.
.
-
..
.
.
-.
.
-.
-.
. . - .. .
.
(jg i- tj'ti.l y ). . '.... .-.;.-r.. ......-.-. . .-.... .t. rt.. ...- . . .
'
E
'
: '. : .
t:. .' ... .-...; .. '- t:iikkki.. ' . ., . . y;....... . .. . '
t.
'. .
-.-. -.
'
IE .
:. : ' ' E ... '
,,.
--
.,-
.---
'
E
'
.
:
.
.-
. .
g.
..
g
.:
..
..
..
g
. . .
.
..
.j;
....
..
tyjjyg.(.. tj,gjj.jq . .(yjyjy,. .Jtttttk .jjyjyj,. E kal Jjt tk . ) ij y tq r y ju ( js u. ..t,.. g y. yyo. xg j z.t...jy y js:y x mt yy; y r jj u. t Eyi y s.y u lpsstt u jyy irirki-rqakt o/uony loss f jam i an q capllu. on.( pnrtllt bond. Ojtift . (.t or )t (..( ( y: yt . ). tlsaiiikl' ( . .. . ( (). (. . !it'. .'d. . . itt 42,9.. tt. .,1.... '.' .ltlk it:j;. '...''4t:j? .. .1Ek 3:1:t1:. itE. .1!:)k it .',:i:. 1E)1:11 lii, ,r drili i::j;.'.' ' 'ii)t)i' :,4:2 IECIL ' '1Ei;, (ir it)l2.. j())k. t..'. lL.L 1I( iii;.' .z (1. k. jlj:k 'J.' 1E@ it:)i 1r. lElkv i::j' j..'.%'. ''4t:i.1E!2:1:!... 1. t!l!i d:E)h 1l:ir:' iix t .'.'.) ).-..,.....'...i..t.. ))....)....,,.'...1:. !r VE . t ( l ( .( ( '' ( E ck ( ' %;(; ; ( j. vi ( V:t . ... ! ( Q q i l. V V. '. t. . ( o j j( ; . ( ..VL.L ( ) .0 .0. lyi 5 u C/lzj t . )Glnj w i y o Dfrllsvliv ...i DOX tllvaliit otild jliietls,y. Wnln IS taxeu ai. a to LLgL T! xituui g yy( ( (y jki u l q n kx wuu kkk yiujvugg xaayktyu.j zti iu; o j t d:t:!r 11:2:14k2::: .-!k;;::... Clslk./t:jil. .j6..jj.,. .I4.$jjjj..r Ik.jjggjj.t. yE Ik....-.j.ydt;1l..1k.jjgjjy...... y4k..;p.. 112:11::1:. ,6..6.43::,4,.. .yjjgjj,jq:yyy.dr:1l..It.gjj.y. kIjjjjjy. g-.-y.yjy...-..--yy...j...yy...-yyy..j..y.........y.y.j-y... 1(:2)). !l.r1L...y ;::,igjjj;j-. ll.llllll:r'/t:::l dkiz:i ii;1l.(jj.jj.gjjy-..6L..?.i.. 1L... j6L.j)j)jj),; .1!,;.. . !(2t:(Itr-ljjrkllit'dtEzlp.llg:'dii!r: - Iill.1!;........!.Ik.I..tjjjrj.... Ittj'jyyjjj...-g.. ::..jI:i'...jyjj;jj.....y... :IE:. dl:zll j....y :!I: 1E1:g)1:711. d:Ez:k .jqk,!..yyjj,. j1::4!-ykjgj;.. 1t::), (;$i;E) :5!1; (lr'tkz:l !k.t;: y( .L.. . (111111k (Igp:,' ... y .g.:....jyj;;. .j.... ..:. ... ... .'L.,6.jjjjLj. j y. j . j(gjg;y..y..-. .kg..;( lj yrkldd In the jst, bok jpia iktjiikrktE'.''EE''! ittEi 'C' ftmdsj iErii EkE#' (),'.t (1Fi;. i.kjtijlitiiigh IIEE CEEII. jr )j.:j.....yt....y.yy for jy............ (( ((j; y y..... .y.(j... .y.;;,;.. k...yy .:.....y. beneut .dkii.:. . . . . . . . . . .. . di::)i..lilkq . . . . gy ... '. ((j .(. . .di):?. . ,k2,if.'ikiii.. lEiE.(1:(114t1::. .. 1r1t.. .ilq q.;...;..-r....y-...,..Ej.,1hy.. iIi(jE.riL. ( .. . .('. q:rEjI.j. . . :15. iEji.y ik!r. :!!il2.. .i4!IC. iti!k. ..I#t,k. (.-,;.yt. .ii::::1tr1L..:!,L( .:ir..:I:-)... ,j!I-. ,,-.k:jit).ItE'..4k:E, i!!ki. ik,i: :::!), . tIr.il#:)L :liiisir;. i!Ei..::ii;E). :Itr!i.1it . lrr)i. .k(::1i.. lr. iI. :EEij;. 11::. kk@:9. .-t i!:;;.-. (t!$;#r 4!4;;,. lt: .......t... .11! ... ili:ip ..t. ily:lt. )... (1r. tqj''. ; ?4. . ..y..).-j (IEC:?' aplrlt e ...qy(jjjy.. ..y.j....y......y..y::jtj . (. .... iqF:q'j(gyjj. gjj.gjgyj(..). .). .( y.;...jjjy:. y:y..j....yy..t. d:iili(. jkjjgg. . . .. . .:jyt...( .( ..(: .i(s,L....r. jjj(j(,:. .:.. j.j y.. .j(jj4j.. jjygyy. . .. y.....:... dt:::: Eiiy'.. .(....kt:sri-y gj,jjj.. 1it,t). .jjyjjjkj.: . . . Ssii.rli: .jjjjtjjygj..... :It:)l..-.y.... (Ii:-ik!,:. gjj;t . . .yjj(... yjj,..$jj;;... . lLjikl:lE:ll::lkii. q..(....j-.).. dt::,. lli:;k jtgg;jj 4k1!:: rlrzlltil.tlL. . :EEi; ,t.2IL (:94::.. jjjj;j: (gjg,,. !!iI;.jjjyjjggj;.. j d:E:lp EIE:)L. iii:::. d:jIL.. ..-.)y.,.yj (Igljlt . lii!: q. . . . ..q,....y y. .j yr. yt.(.y....,-.,.-..y. ... ..., .y.!.j q.y( ...;. jyt. . .-... ..y ,. q y y .,y .y y.. )y... y .. ( ... . .t .y y . . ..... . . .. .y y yjjjjusu ry . ri!;. d:z:i jjjuj rr..... oy j jy . . . y y muxy; i.gj;yyjy yyu jyjlty j y y yggj( jjyyyjtj u,u w J jyyjjjjujy jy (j v jj j( j y g ( i (jy yjyyjjj u-uuwjxttku jj yjyj (y kokvtiii g) ki.,k.;! y (j( kxw Eytw jyjyt ; j Ayrjyjyy jyyjjyjyjtjj jj uuu ( ( ljg;j x. IJI/UL/ILCI..LIZi ltu Ll.lc ti;ta uutlwutmoj ( j Crcatr y yj y yyyj yjjy j y j y jy jy yjjq jyj j ygj jyjtjyjjq ygjjj y... jyy y( y tyq ajjjyy(.(q (yyjyy Considf tikabt ilkjt =. Wb j juj ...jyjt yyuyyjjj... ..jyyljkq y j k. . .L. (o. .;..t;.tii-.j)-.42:,. g. .-. .. .. q.. q. .. . r. qq. .. .. ..i6b6'.. E.i,.E....,i ... .j..'.E., li:)'. 1::i;. ki. 1l:l#)(')k. 1:t),. )()...) 17I.. It:i! ' :.,.,.t.'....,E,r......' . jiL 1i:ik. ' ' ' l.. ,.L......j....LL 1t7)L. Ek;. 11:); 1kj4:)ti.. jt:'. 1k:. ).1:.. 1;;. :..i(ri 1r2..!..),'..L...(' 1tr)(7)C. 1t: .j(1: ij ii.. 'i,.'.. i:t)i .r.ji..iti;.1t). 4::: d:jlL 1071:. iz1('ik1;;JEt .. 't..'.'t..) itzitlk. ti..1lj(.. qljt: 11.. t;k ii.lrrlL tbtl1t21171t12 (1.1jiE;s')Iq-I#!li(.. ..'''. .:51t;i:E)p .t!;;. w. tkt'kttkEitj.t,t Er) .. . .t.. .. ... . . . ''hkk'd .btiitk'' 1j. Jtkoi i ljjp ibi.tk ytvpczk .
'.
..
:
jjf.
.. .
.
:
..
...
..
...
ttir
..
.
k
..
.... .
g.
y
g
.
..
..
. .
.
..
.
.
. . .
.
...
.
.
.
.
..
..
...
.
k E.
=
.s
(
.
k
y'
,
....
.,
.
.
.
:....
.j
..
.
: .. .
..
q.. .
E
.
,
E, E
' E:
E
.(
:
:.
:..
..j
.y
.
j
q
.
.,
..
:
.
:
...t.
.:..
.
.
..
..
sE.%
: :u)
:
..
..
.
,w
.....
g
EE,'E
EE,
..;
u
,
.. .
.
.
EE
X.E
.(
..
.
:yyyj(....
y
. .(.
t.' :
..
..
g
,
y
0.( : Esitiads
ni:.l
y
E :..( .
...
.''.'.ik;.
gl/
E
.
y.q..
y
..'.-bi''.
k
:qg
...
wv.j
.....
..
:.
:
.:(
g ......'.. .
. .
:
'.Ew
E
.
:
..
.1
:
E
E
:
:
'.....'
.....
E
:
E '
:
yEkqj
-w-
,
.
.
..
-..r....;.. ..E.yq.y...
:.
!;2:;;jjg-.......:1;:;;;.(j(,. .
..,k::::gjj;jj....
-.y.....
..t-...
..'jlitki
...-
....
..
, .
y
.
..j. ygjg.;; .
;
.
..
y '..j,.j'.jj)j:..L;....;...j.-.jj !k..-.t(jgjjj. 'i..,'.jjj)j)j).... ,k.k;-jjgj.. .,L.j'..,,.j)jj.
,jjg;;j....y
.:
-
-y ..
:
,ik;;ik;''-
.
..
:
.
.
.
.
.
..
.
. .
. . . . .
.
:
<'
-kw
.
.
.
...
tiwilk
-.
..
.. . . .E. ( . .E . . .. .
.
:
:. . ...;....-..-.--.
.
:
.
.... .
.
.
.
.
y.
..
-
-
-
.
.... - ..
.
kjuE
.-(
...
...
y. : .. .... $; .
-.
..
.. ..
..
.
-.. ....... .. . . .....,ii!1k,ik!,,r.. .. . .
:.jj(j(gggr.
- ...)jt ...! (jyjjj,E; k(jgj;.E
..
:
.EE
:.
...
.....
.-.iir;ky
..
E.
E.-..
-
..y.
..y..y.. ..
-..
.....;
E
..
.
.
:.
.
..
.
..
.
..
.
.
,
yj
..
..
-
g y
yyyyj
. ..
.
..
:.
'?.'.
.
. ..
.
..
-
'
=
'
. .
g. . . . .
. .
ikd, '
'
.
.
.
.. .
.
:
'
':
PPIg..L
.
..
..
-
.
q.q (
.
......
.. .
.
...
...
...
..
..
t
: .
..
E
...
'
..
E... .E.. .
...
.
.
j. .. . .E....E.'..-) :
.,-
.
.
.
.
. . .. r.. . . . . . .. m....t.. . . .
.
E
:.yr.sjqqyysjsyjsj
.
L.E.... . ,
..
..
..
.
.-,.
.--..
,.q
.
.
..
.
.,......
. .
...)
..
.
.
...
-
.
E . .. ..
' ..-
. :. . .. . ..
.
yL.-.(
: :
:
y i yEk
yq
..-... , .
..
....g E-:i
..-E y..
:
i
.. .
',bb'.'
..;.-.
.
. .
.
. . . .
...
..
..
.
. . .
..
.
.
...
.
.
.
. ..
: :.y
.
..
''
.
.
.L('' ' . . . . .
:
' .w'' ...:...' .
:
(((jj.. . .' /flft
EE ( .
. . il/e-at
.
E
:
.
.
.
..
:
.''
.
.
.
.
.
.
'.
.
' . .
.. L:
.
'
.
.
zifi
.. .
iii.
'd1i.?/'''
.
. . '.
.
..
.
: .
..
......
.... . .
.
.. .......
..
....
. - .
...u..:
''
:
:
:.
..
E. ': .
.'' .
'.
'
X.E
...... . . .
.. .
..
...
.. .
E
. .
.
.
.
.
.
.
E:
.. .
..
E. .')7a
:
E
..'.
'
. . 111:45111
lliii!).
11111.
'11:.
(. . . . . . . lljiiiqli
.
.. . 11k2222:2 11::2:::2
.. . .. .
:.'
n
.
.. .
..
:
'.
... -.
..
..
..
.. . .
.
,
j
,
,
t.t
,
.
.
-...
..
11i.,,,,.
'
z
y ,
,
,y.k.
jyyuy yjjku,j
,
.J
g
y
E
..
...
.
:
'
) .. ..:
:
.
.
: :
:.
..
E
..
..
..
'i-it'..jifqj;t-q
y
.
(.
'.. . .. . .
. .. . ... . .
-.-'..i$';.'
'
'.
:
.
yjtxvE
..
.
. . ... .t;;gj.,..yj;jj.jj jLjj).
11,,.11,.
jgj:::::j.
11,..11,..11...
.
.
yj
.
:
.
..
.
.
..(
.
.
'''
'
:
.
.(: .
:..
'.
j.
'...'
:.
:
..
.
..g
: .. . .. ..
.
5
.. .
.
.
:
.
.. .:E
.
:.
:
(
..
:..(
... ... .
'
gE'i
:
.'
.! .i
E!
::
.L'
q .
nujj'
:>)
E
:
:.j7
''
.
:
..
E(.: E
'L-.
. .
!
.g .: . ...: . . . .
.
.:
(.(.. q
i:. .
:
..
.
.
.
i''.).E 5..''
;
j yj, EE(
:
:
:
(
..
E.E. ' E'.... .
:
. ..
E
:..
.
E sg q SrlivliitltEs/biktki yg E ;
:
g
.E '. ....
.
'.
..
'..
''.
..
:
j.'.: .
t
'12:211..
'.'..
.
.
j' (
E
:
j
..
.
g.
g
:
.
.
E:
r
g:
gy
'('.l
.' '
y
.' ..'.' ''. ...
:
t gjj
r
.
.
. .11kj2
.
..
.
,1....11.
1..,,1.
g
.
yy..y.. E.jgg:jj. k,.:...........
jj,jjy:. : yj.jjjy. j;(y..( .
.:..
'.
''lans
''
:. '..
..
(.:
.j. '
y.
:j
j..L.''.
'...
.
'.
:
,
.. . . .
.
'...
EE
.. :
:
( p.-.'
''
.,,..,,,.
j
j
: j . j.j . . .. .j .. .
j. .y
.rj,
..
...jjjg;yygjj...
(.. .. .
'
..
-10 -15 -20
.
.
g
.
.
.
y. . . . . .
. yy .
g
.g
.... . . . .
. . . . ....
Suppose we sell a
20
30
25
35 40 45 XYZ Stock Pdce
.. . .
..
'
.jjj.,.
:
. g.
:
y
y
y
.. .. .'' ... . ''..'- ..t-.i.. .
-
..
.
.
-
.
,
((.'' ' .
.
,
-
.
y.
.
.(
: .
'
'
.
'
:
'....
.....:
:
'
.
E.gE..E
y
t
...y.
. .
'
g
.
:
..
'
y yy ECrEE
j .g..
t)
j(.'r ( . . .: . :. (( . . .EE
''
....
'. ' .: ',
:j
j '( . . . .
.E. ... .. ..
'.
i
r
.j
.
:
.
:
.
,
g
g
. .
;
. ..g ....
. . .
.
,
.
y
.
.
60
(*)
.
-
-
55
'rhe
y......jyy.... yj.....yj...yg...:....j.y.....:yyj...j.y ..j..y.j.g..jj....g....y.........j......j g:....g..y..g.qy..jy.yj...yj.jyy.......(.. .j..
.
.
. .
.
.
Ek
:
5t' i5) 'E
.j' '(.j :y. E.. .
:.
E...... :(
y
'.. ' : : .. . .
( g . . j . r E .g y(y. y. .. g . .. .. . y gj.. . .. . j.g. .g g . . gg y gg. j y . . g j . . g .. . gjg .. . g . g g. . g.g.. . . g . . . . . .g yy. . . .. . . . . .yy g .. y.g y g .. .. . . .. . . y ; .; y ; . .. yy . .. . . y . . .. .y y. y y.y g y y. : .y y y y y . . y y: ; y. y(. y.. . . . r . y . y . g y.. . . j .y. .. . . . . . . .yy . . y . g. yy. . .y. . g . . ..g.. g gy.. .. . g gg. . g. y.y.y.. . . .. . y . . yg . . . . . . .. .. . . . ..
j.
.
.
.
.
r
,
. .
,
-.
(
''
.
. . . y . . . .. . g . y. . y. . y .g gg. j . g . . . . . . .g . . j . . . . . . j .. . . . yg.g g. . . .. g.. g g g . . . g. g . g y... ..g. g. . . . . .: . . .. .. . ( q. q ( q . ... y.. .. E. .. y. :. : !..: y..yy y . y . .. .. . y y . j . y r j . . j .y g j .y y. y . y. y ... . . y . y
....
..i
: .
.!
rl :
:jj
.. ..
: .:
50
lf you hold this book at a distance and squint at Figure 3.8, the collar resembles Eonomically, it is like, a short fonvard contract in that it. short fomard conact. a is f'undamentally a short position: The ppsition behets from price decreases in the underlying asset and spffers losses from jrice incremses. A collar differs from a short forward conact in having a range between the strikes in which the expirilon payoff is unaffected by changes in i e Value of the underlying asset. In practice collars are frequently used to implement insurance strategiesv ( or example, by buying a collar when we own the stock: Tliis position, which wetwill call a collared stock, entails buying the stock, buying a puti and selling a call. lt is an insured position because we own te asset and buy a putk The sale of a call helps to pay for the puychs of the put. The collared stock looks like a bull spread', however, it arises bull spread is created by buying one option and from a dtfferent set of ansactions. selling another. Tl'he collared stock begins with a position in the underlying asset that is coupled with a collar.
t
y
Er'
LL;' 'ji. ' SE)
EE
( : ( ' Ej.
..!g
ti'gttiE
j.E j. .
g..;
jE.:y..;....yjj:..gg:..:..g...y. .j..q.y ,.....;...L..j.......,..j .)L)..)..j,)j......j.
.
-
.
.
45-stIike
'
of $1.02.
5
-5
.,g
call with a $0.97 premium and buy a lo-strike put with a $1.99 premium. This collar is shown in Figure 3.8. Beause the purchasedputhas ahigherpremiumthan the written call, thepositionrequires investment
3.4
10
0
j,yy,,,,yj
yt
.'.
:
:j
.:'),
.
L.L
.
11,,..1,.
j . .j. .. 11::2::,).j.
:
':(
):.'.'ictE-.
''
j E.jj(..jkjjjj:: (.gjj(;jgE ..... ..jy....j......g..r.g.j..........(.j .j ...;jjggjjy Ljjjjg. ....gjjjgjygj(,. .
purchased collar . '. .
.
Elgyt
j y yy j
'
.
.
(
:
..r..
.
E
.
E'.
E
q ....
'...
.'' ....
:
!'.t.!...,:.j''.E'.'
'''
y jj y
...
.:
...
;: .' .
E:
.
Example
lj
.
..
.
q
.y..y
y
'/tik'..
.
.
E
''''..
'.
..'. :. .' (.
.
.. . .
.
...
'
'.:
.:gryj;yjy. ....jj.y;...jj;jgr
:yj,,jj,. .
.
..
'
)EEy(
,y,sj, ,
,
*#.
E .. :..
( .. '
:.
..
..t
.
:.'
11,.. .jj::y,p,. . .. y . .... j,jjjg!g;r
-s
20
(EjE Ejl
yy
:
.: jj.jty
!Iiqji. .1t:d:: tisi ())
(IEFI'dt::)p :.:7
.
E'
E.
ZEIE kjkE ,.atj,wcg
qj
-.'
:
:.
:.
.(E
:
..
'
il:il.
j!!j!j;q
.
.
:
.
:
'
kEigEuit
rnuuw
aaukE
E
.
. .'
'k:@,:'
j
E(
=
:
. .
:
E . .. . .. ' . .
.
. .
: ..
.
'.
E
( .. .
..
'1L:!1I
.4df1!!ik.
,L..
...
.. .
'E
'.
:
.
:(
. . .
(E E i E. .
''.
:
:
.
:
.
:
E
.
.
.
,c<
(
.
.
.
--j--,jjj),
!E . .
) ty
yy
...
. ..
...
:
.: .
..
'',..,...
i:
k
.. .
(
:.
-.yyjj(t.)r,
Profit diagram of a purchased collar j constructed by e Iing a 45-strike call ahd buyng a 4o-strike put.
':)'EE
.
t(' :' :
:
:
.
..
q Skltth k l tttti b ( d( i tt .. . .(... ittE!q;;E ;k-k..'::--i.tfiiip''b-.ii'' .j.('( -E'...t.'t').'.-..; ,Eli;i . .'..' .. .:r#,t). . (kr#ii.' . :,1,711!t1.). E.'.)(t...t..i .lli:'.t:,,' iEt,k' i'b' '..q . . . t.' ::Eil' ' di:E:,-' jIEt,i::i.'. ..tlEt,i. L;.''6,L((3t-.. :diii ik5i')(ItIL.' Jti,t. '. i(:t,i.' t,i'' .iiq.. il::l.. (t!;;i.---'..-.'! .t'.. ).--' L:jil.' '':itlk,':!;k i!!ik ' iL. 1I: ' k' ii!; ;pEi'.tiE. .-6iLL--,.;.''1--.L.. iik. Iriil.lL. igrq..' . '.. 1 (1L. :Ir: .:ik!;g s ...'':iIEq-,il..iiL. a 11:::,e lE. a lEs,t.llgl:)e .:EIL. i/ikkbt'k kkdtt Lii tloj) risksounds; A duzzu' krttt tnbhiii it. (( ( .j( .q q)i ? ( (. j. .t . ) J ( k .. . . itk j) ii . ... .. . ) k jkti Wtkk tr yty'.'y erii. i af/eii kr i i'avtsf l tii t' (. )( ' )' 4( (j. ( l ' ' ( whi j. . . .( gt l)1y. ' ) . ' ( ( ( ( ' ' . y: .;.. ' . ; .u7 t .. . . . 1 511 ltlnti L ' tii'i )'((1 rl) j (j j:jtg'jy): j ii/iil- U ) ( ; j:( IIQZJSSJJIIIIf llv'yj j igiriej Wtln ) t jj (j) jt jjyj j y j y biilt iiiljt tvt...j...((j. ik .1.:.(... tj.. jd jo (jftke ti Vl i .) y.j(:j..'.j( .j( ) )..'. YJ ).j(..'..t j j:... )) r. ' .'gy...
: .
....
.
.,,...
.... .
.. .
' . p' ' . . .
<
.u(' '
.
'
',
E
(E
,
.
.
. .
.E
. .
. .
Euxxax
g
..
' : :.
75
gyjjgj
)(u. ' j' ku '
g
..
.
ytkdjy
i'L .:
.
W
.
: .
.
y...j
j..
-.
'rl,tyt?
..
xjjjt
..
i. .
LLLi?LL
y
y
.....
.. .. , ..
.
. .
.
.
.
g
'.d:::::
:
-
..
.
.
..s
t jrt mechanisn ,;;'
.
...
.
g. .
, . ..
.
.
.
.
........g
g
.
-ii'i
:
:
y
y
;
.
.
%
COLLARS
E.-..... ..-.
,
-.
.y
-
tjjgjj
..jjjk;;
..
.....E
..-r
-
-
. iiE
..E(:: .
... ... . . ... :
.
E jktjjtjkksgjjkttijjtjj. jjjjj;;..E..j.y
.-
::y. ..-2k.:j11. . :
.
..
E E
.
uyjjjyjjjy ujjyyj uy sjyy uyjyj ujujjjjjjyyyyyjgjyyg u
E )cL
y
.
.
:
.
-. . ..
.:
....q
...
...
-
..
..
--
--
...y
j.....)....j
...
. .
... .
-..-
,,.
u ujyyj
IAuz
Y Auwo
..: .. :E .
.....
..
. ' . . . .. .
.
.
..)..;
..
..
.
. . .
...
.
'. .
.
.
:
.
.
.i
:
--
-
:-
.
..
.
:
-. -
Axo
%
q Example
Suppose that you own shares of XYZ for which the current price is >t optiri.' A way to pu wijhtp buy ipsuranc. voud this bf out-of-the-mopy Theprofitcalculations thisuraneis sell all. reduce the . f cost )') to an .j . . 4o-stnke of ansactionsvbuy stok, bpy sell this 'tl')fo te put, set a 45-st1ike call-are a 1) shown in Table 3.6. Comparing this table t9 Table j.5 demonsates that prost on the )(! . spread. Note that it is essential )Jj collared stock position is identical to profit on the bull 1) to account for interest as a cost of holding the stock. %
1))$40 ai ).)
3.5
'prhkig
'.
.
.yj
,.
. .
...
..
.
.
.
...
.
.
.
..
.
..
. .
. .
.
.
..
.
.
..
.
.
.
.
.
.
#,
.
. .
'
$
IlkskkcE,
.' .
;';' -' -' -t' y' 'j' '-' ''i?f '''' f j)y)E F ;,)' lyk'y:. ,,#,' E
COLIARS;
ND OTH ER STRATEGI ES
j'p' @' '(' ltf ':' $rrr!!p4rrrq)'.' i'5* f' t')'(' 171E* q' jff $1* @' /'1* y' !' pts !i C' E E ' r' kkktyfjjir'.trj.kk)gj,)f l;;f ;j' EE i'E''(''!'!'. .'E ' ( iE('!';. ('''E.i(i'E. E.'q' ! i'E! ;! ':p' i'E ' :Err.j.()j).;.:j-?:-...7.:-E):;yjt. ;!;Ei.iPEiE' j! !E.q(.y.r.:jjjjjg,.-'.;E q' i.h().... ETE . . . '.tptj)y)j)t)yyyl . . 'q' ....).;j(y.r..yE.qj';g:j.E..y-yyyjE....(..y.y.;jj.-q((-y...y.j..t);t..:yrjy(-y(j.-.j.yy.yy.j;.yt .ir.jjjyjjj;!):.. .'.! .y(j)y.j)rjy).jjjjj;).):E .;!liE@ . '.' rj)jt).' . . ,.iLjjj)Ljj;)-L,:)Lb3L E (rrl-pi,i .g (---,:,)jrEi'ltl' gjy: jtrr;,, g-j-.. . . jr,j . yjyr iyjj jryj,r kjjj . (;,:y jr,. . - . - ; , .i - E,..L lllt:. ) ( ).. .. .. .
spREAos
AxD
k
COLLARS
77
.'..' .'''-'
t'iIIil:-:'. yf i'r'jjjqrfjljqjqqjjqqjjjjjq'ils')f /;' C' t-*1)* ILLLLLILL. r' r itjy)l i(r Etl' f7.' yjtytyjy. k. . .y. ;.r .. . t! . tj. .
.'.'lif -
'
.
.
'
'
E .
-
'
'
'
'
'
:
.
.
:: .
-'
'
''
!. .
.
:
-
-
:
-. .
.
: .. ... : . '
E :
-
tlkkkiiiiiktlr'--t-.
jj::,j jy
..
-
....;..:....;....,....................
.....
.
-
......
. .
- - . . . . -. . . ......
-
-..
.. .
.
E xtscFsyu-okss
.q
!;
-
-
. - - . --. - ..
.jg:
.jjy
).; zz svswwE-vujt
(rlyttwsE
t.k..,.-ntE
.
E. E.
..l'.. E
: .
E
..
q(..y.E..(.y
...
E ...
. .
-
- . .... .. ..
E
. . .. . ... . . . . .. . ... . ...
jjjjj;,
:
. . .. .. .. .. .. .
. ... . . .. .
E
..
- .. . . . . .. . ..
E.'. . .......
37.50 40.00 42.50
.
E .. .
.
.. . .
. . . . .
.
.
-
. .-
.
:jjjjj
EE.
...
' '' : E :. . . .. . . ..
.
-
'
'
E.
-
.
.
.
..(.
-
-
.
.
.-..
E
.jg::
.
.yE.wsvos..Et
. . .. .......
0.00 0.00 0:00
... y y E: . .E. E. :E -.. E. .. - . : . : .. . E ...... . ............ . .. . ...... . .. .. . .... .. . . ..
0.00 0.00
0.00
0.00
. . . ...
0.00
0.00 -2.50
-5.00
j.),;tE)((t....'(:'iy.i.!!
q.y'y')
i;(jjjjykjjtjjj.jjjy.
jj.j,... ....jL. y(Ej i E. .. ( E ' .......... . ... . . . :.y
..jy.j
.)ygj.,.j....:, ( jjjy jjjjjj, . jyj .y...j i. (. i ( 1l .. '
' 'E E .E E. .
.. '. . .. . . . ..
: .. .. . . .
-$1.04
.
.
E. ...
-
.
. .. .
yg
.. ... ... .. . .
. .y , y E - :. . - .. - E .......... .. . .. ... . . .. ... .. . .
-1.04
. . . .. . .
.
-
. . . ..
E. .... . . .. .
. . . .. .
.
. .
.
. . .. . . .
: :- . ..
. . .. . ..
-. . :.... .. ... ..
-$5.81
.
.
-
.
. .... ..
. :.. . .
-.
-3.31
1.04
-0.81
.
i:. . .E.;.. ....... .
:.
1.85
-1.85
-1.04
1.69
-1.04
4.19
-1.04
-
.
-. . ..
. .
-$1.85
-
-
6.69 9.19
1.04
.
''jt )tj,j,y,qjg;y(,.j,)ytyy:;)(i)E((y(..'.y,(y;yj(kr:yqy),;,jt'y.ytt.i,..E.(
y(......j..
( ........
.
.
.-.E7:!l7;E::iq)k#7k/t.;:t'EE:i'J)'T7i''r7:='ppiX l).rii:';)(()k2t'!Ijqq''.i)q!iip!C7:!qIyl.' 5k:yiy)k.-.'jfyjj-k.t.:)yi;.j);t. yjk-tijj il..z6,...)L(tnppyn1.t,.L..,...LL-1;L.kLL;LL3L.;;3j;f)1;j(';:f;;. 1r: jjjL)-jj(;,,jj1L.L?j'1jjj'..jL-;k. t!.( r;.)#(yyjg4jjjljjt)jjjjjjjj,-.. )j).;;)j(j:);.tjjj.. I11)-;r,I'#j1::1;k()..'.' kp.y/t....;.y,)j,.,).t;.iE.iti.(-Ijtyi.jt.) 'l.llt. 1IEC1Il) tl-y;@t:ii!!1l;. ..qjj(k..jj;)))jt.. 1115!.-?7. rjyg jtj:jj;j gj:, rjjrji. ilqti (.: ')FyrjjjE.k!IlFq(. (jgy;l j;j . :';;k.kkr.;y..j;!!!;1p$h...,,....jj:.
'klllkkjllr.
:-
n?gyyyjktti. ..j.jjjj;.i)r((.yyj;.jy-
..,,......:.;t.;.:.))j;
..::
t,:,,).:L.
JE;;;...........ic.....,... .
jt:r;j
1E;
:..,,..
.
'
. .
.
20
created by buying XYZ at $40, buying a 4o-strike put With a premium of $1.99, and selling a 41 call with a premium of $1
15
10
.7z-strike
5
0
.99.
0.65 3.15 3.15
-5 -10 -15
3.15
The collar depicted in Table 3.6 entails paying a net prezium of zero-tost $1.02: $1.99 for the purchased put, gainst $0.97 for te written call. It is possible to find strile prices for the put and call such that the two premiums exactly offset one another. This position is called a zero-cost collar. To illustrate a zero-cost collar, suppose you buy te stock and buy the 4o-stn'ke put thathas a premium of $1.99.Trial and error reveals that a call with a strike of $41.72 ao has a premium of $1.99kX'hus, you can buy a 4o-strike put and sell a 41.72-st.1.111e call without paying any premium. 'T'he result is depicted in Figure 3.9. At expiration, the collar exposes you to stock prie movements between $40 and $41.72,coupled with k downmde protection below. $40. You pay for this protection by giving up gains should the stock move above $41.72; For any given stock tllere is an intinite number of zero-cost collars. One way to is to lirst pick the desired put sfrike below te forward price. It is then possible this see find to a stlike above the fonvard price such that a call has the same premium. collars
-l- Lonz XYZ stock Stock + Zero-cost collar
-20 -25
If you have a short position in the stock, you can collar te position by buying a call for insurance and selling an out-of-the-money put to pnrtially ftlnd the call purehase. The result looks like a bear spread.
'
.)tE''
Zero-cost collat on XYZ,
tl' u'vt'kk.i,i'.' vj joyntyyjy ytjt
'ty'.,E ..t) t gy,,jj,yEEyy;y..qj)j..,y;j
Iuy;
..
,,.,
tyr tiyytrjtkjgrj r ty.,(,!.'
: . ' . ..... . .E.
2.50
45.00 47.50 50.00
jt:gjjr :jyy. ,(jy,j jryg;j yjyr gjj;i jj.,jr jr..gj tjj;y jjjs,jyg;;j jk jr,, (jjjjs jjjkgj!rjt::;jijyjj
ryyeyrmzjt yy ljm, j jjjjrr jjjyjj, j.... y.: ....LL... t..q. y... jj;jjjy jj(kgyygj. jjjj).ytlj:.jy..j jjkjjjj. jjkyjjj. jjyjjjk ..jj.....y t. jjjjj EE . 1 .. ' E ( '' ( y:yjj yy tjjjjyjjI .,;,-(.E
.jjjg.. l 'l tjjjkryjjjjyjj,.. l('.'..!..... .. (.. . . .... . ..$35.00 $5.00 . $0.00
-.-
i..(
(
yyytyjyr'y!
.
. ,,,:j4.
45-strike call.
rujlttejy rurchsed..y''.'y'yiyy,'j nj g t Ntocl gjjygjjyy,,kp.t.j.( ..j... jt.).i gjjrl ).)j.! y... )jjjjjjjyi..sjjgjkj. .(..::jjjjjjjjjsjjjyjj;..kj;jjj, ):...(j.... rrtceE
.jg:
jryy;j jtry;j jkyq,r jtry;j. jj.,jy jr..jt::ry jj.,jj jr..yjjr,. j; jk kyjyj jr..jr,.jj tjyi jgyy;jjrgr;j
.jjyj
+
20
25
One aspect of the zero-cost collar that may seeril puzzling is that you can snance te purchase of an at-the-money put by setling m4out-of-tlie-money call. ln the above example, wit.h the stock at $40, you were able to costlesdly buy a zlo-strike put by also selling a 41.72-s11-1e call. This makes it seem as if you have free insurance with some possibility of gain. Even if you are puzzled by this, you probably realize that 'Tree'' insurance is not possible, and something must be wrong with this way of thinking about the position. collars
50
35 40 45 XYZ Stock Price
55
60
($)
..j'
'I'his puzzle is resolved by taking into account financing cost. Recall that if you ?351 brokel d1?cn. You have lost. pay $40for stock and sell it for $40 in 91 days, l?plf have beause you av forgone $40 x ((1.0833)0.25 r 1) = $0.808in interest. Thus, money, l4ce in xnmple i; this $40.808,aout halvay betwen $40 thetrue brealoeven stoc k p
d $41 72 of collars, considerqn executive who ownj a large To illustrate he us adpcing using executives stock. Such frequently hedgetheir stockpositio, positionin compmw tvt mattllitv yeis exnmple, several for ctgarj witll Sup/os? Vicrosoft cost to zerohak a price of $3t/shar and >r executive wishes to edge 1 rnillion shares. If the ekecutive buys a 3o-strik put with 3 years to paturity, what 3-ye@reall will have the eftectike annual sk-free rate of 6%, a zero dtvidend snme premium? Assuming an 3o-strike Blpck-scholes price and 40% voladlity, fr is the $5.298 a put with yield, a with a solver), option numerical call maturity. and trial Using 3 years to a en'or (or a highly agnin, ollar the premium. of Once zerp-cost has tll snme strike $47.39 seems Itwevr, thif cpmparispn doe not take into account financing cost. asymmetri. . J . . stock in three years for $30/sharewill in fact have lost three years executive selllng of interest: worth $30 x ((1.06)3 1) $5.73. an
.
.
q
,
'l''he
(
.
' ..
custo
. .
'
oe
'
' :
.
cullar
vli cpstcoltar in which #o
.
=
-
Understanding
30
' .
.
tI,e vorwaraprice skyose#ou try to construct a zerofet th jtrike of th >ut option at the stock price pls financing .
an
'
4For arl account of this, see 'Yxecutive Reh-et'' The fct/at/lllf-l, April 3, 1999, p.'64.
$
INSURANCE,
COLLARS/
AND OTHER
STRATEGI E5
cost i.e., the ftlttlre value of the stock price. In the glrday example above, this would require that you set te put stdke equal $40.808, to which gives a premium of $2.39k) The call premium at this strike is also $2.39!Ifyolt p'y to ??.If?'d against aII Iosses tp?) the stock, fncfllfll.,g interest, J'/?e??. a zero-cost collar 7itz-zero w/#/'. ' This is an implication of put-call patity, equation (3.1).lt trns out tfiat is also the theoyetical folward price. $40.808 lf we set the strike equal to the forward price, the call premium equals the put premium. ,
3.4 SPECULATING ON VOLATILITY The positions we have just considered are a11directional: A bull spread or a collar is a bet that te price of the underlying asset will increase. Options can also be used create pojitions rhatare nondirectional with respect to the underlying asset. With toa nondirectional position, the holder does not care whether the stock goes up or down, but on1# how much it moves. We now examine stradles, strangles, and buttertly spreads, whih are examples of nondirectional speculations.
S PECU )' y' illijj'--' )'' 'j))d k.jjtd yyd f)' 1* ;'''f;' itf !' :1*1)* '111* (,1::.* yllqtji.f t'q' )' 1* js'ct j'j' yj('.j(k.. qf (' )-' i'j'('j'.y' J' r' q' ')'1)' )' ((' lf E ' 'E j)l '.' ;)' 1* l i( llrpllq;qjqlll)i qi .j( (! l'fJT q(! 'k'.E.'E(.'.( E((..E'k.('. ('E((i''(1(i(i' (ti'.'(l'l(E'.E.E. E'.(E.j!(.'.E('.('(('('; '.('q' ; g'E(.'.. E....... (..( E;(... '(k t:i'i (i i1.ljq ' jy' .) (.) j, . k!iEi E k E(qE; (EEjE'E( . j. g (E ( ( E ,'( i i (jjE .. j . .... jjjj gjj( g. . yjjjj,;gjyjjqj gygjyygj jjgj!.g .g (.j ;j El).t..t g E . ,. E jE.j j . . E. ! E;. E . i lili : .... ; ; 7)0k ;! . .!-:.i. !!El.!q :. .r..... r. . ). . . ... . . . . . .
tijf qrf (:1EE::.,* ykd '.,'' jgyj'.' IIill'. 1IllF!!i''. (' (j' li1ii!i:'. j'r'!!(t' y' kyjd ykjj'. yjd yyj'ty .'tyqf t't,'.?j;))...
j . (. ( ) k.4j.. y.(...(( . (7:11411-(---: ''.'l(jqjtg j.. .y i k .tijj!!!!471711@!.. ' . . . . Combined profit
.' :
:'
.
.
'
E
E
.
. jE
:
.
..
. ..
:
:'
''
'E :
:
.
''' ' : .
:' . .g)t.( :
:
:
'
:
Consider te sategy of buying call and With thC a straddllft a Same Stlike price and time expiration: to strategy is called a The genral fdea of a straddle is simple: . ,1fth stoek price riss, there will be a profit on the purchas'ed call and ij' tlae stock price declines there will be a prot on the purlaajed put. the advantge of straddle is that it can proqt f'rom jtock ptice .1 both directions. rovs 'The djsadvantge to straddle is that t has a ltigh premium because a it requires purchasing two options. If tlae price ystock at ezpifation is near tlze srrike the two prejjliutns are lost. The .diagramfor a 4tutlike straddle is graphed pce prot in Figufe 3 10. jnjtjal cost ojthe straddle .ata stock price of $4Ojj $4.yy: $,.,4 sr tjje eaj.l aud yj gg sr oe put. Rgure 3.10 delonsates that a is a bet that volatility will be high; The buyer of azl at-the-money straddle is hoping that te stock price will bt does not move abqut the direction of te care move. Because option plices repect the market's estimate of volatility, the cost of a straddle will be greater when the mrket's perception is that volatility is greater. H at a given set of option prices a11jnvstors found it desirable to buy straddles, then option prices would increase. Thus, purchasing straddle a is really a bet that volatility is gzeater than te market's assessment of vlatility as reflted Option prices. in
$
:
..
E
.
.
::
E :.
:E
'
r.
:..' . ..
y
..
.,
.-
. . ... .
-......... .. .. . . . .
.. .
-
.
.
.
.
.
...
.
.
: :
:
: . ..
E .
'
:.
.
j
..
y
-
.
'
'
( jiy'jlj .t (jj jj jjjjjjjjjjj, jjjjgjgjjjj.: j. j. (. . t). i.. . . ,
' '
'
-..
.. .
.-
:
:
-
-.
.
.
:j.:.r
Eg :
y
.
-.
-
dagram for a purchased 4o-strike straddle-i-., purchase of one 4o-strike call option and one 4o-strike put option.
.
-:,1j1:(r:)I
11:(( .11:::
- -.
.
.+.
20
.
.. .
-
straddl
ls
10
5
0
-5
20
Straddles
LATI N G O N VO LATI LITY
-' ..(tjy'. .','..' ..'' .-' :(jjjjjjjyj'(' y.ttd '' -.' y'y :'--...----1-;13331.*. EgjjjIyjgjjjj'E 11:::2:.,* !):q' j'r'@' y'....j: 'J1' 6* 11* 1*
25
30
40 45 35 XYZ Stock Price
50
55
60
()
'l'his
'rhus;
rae
'straddl
,
strangle The disadvantage of straddle is the high premium a cost. To reduce tlz premium, you can buy out-of-the-money optiond ratherthan at-the-inopey pptio. Such a position is called a strangle. For examjle, consider buying a 35-stlike put and 45stlike call, for a total premium of a $1.41.with a f'uttlrevalue of $1.44. Wese reduce your muimum loss if the options transactions expire with the stock near $40, but they also increase the stock-price move required for a pro:t.
Figure 3.11 shows the 4o-strike seaddle graphed against the 35-45 strangle. Tlzis comparijon illuseates a key pointp ln comparing Any two faidy priced option positions, t.here w ill always be aregion where each outperfonus the other. lndeed, this is necesiary to have a fairly priced position. J.nFigure 3.11, the sangle outpedbrms the saddle roughly when t stpck prie at expiration is between $36.57and $43.43. Obviously, there is a much broder range in which the staddle outperforms the sangle. How can vou decide which is j.jy e jiet. ter investment? The answer is that unless yo tt have particular view on the stock s performance, you cnnnot say that one position is preferable to the other. An option pricing model implicitly evaluates the likelihood that one strategy will outperform the other, and it computes option prices so that the two strtegis aie equivalently fair deals. An investor might have a preference for one sategy over the other due to subjective probabilities that differ f'rom the market's.
.
-
,
Written straddle What if an westorbelieves that volatility is lokver than te market's assessment? Because a purchased saddle is a bet that volatility is high (relativeto te mlket's assessment), ?. m-ittn straddl-selling a call and put wit.h the same sttike price and time to expiration-is a bet tat volatility is 1ow (relativeto the market's L' assessment). Figure 3.12 depicts a written straddle, which is exactly the opposite of Figure 3.10, the purchased saddle. The written straddle is most prostable if the stock plice is $40 at expiration, and in this sense it is a bet on low volatility. What is strilting about Figure 3.12, however, is the potential for loss. A large change in the stock plice in either direction leads to a large, potentially unlimited, loss.
'
.
% INSURANCE,
80
COLLARS,
(jI(:::.-' )'. )' yf )p)r' ()'(IilIk,,:1iIr'. #'.'?'. f' j'1* t'y' y'' :ti!E;kf 'r' lllll'-jf IIIIEiq!q'. (iEEj!lll;'. t'. rtr/..jf yjy(', ;),'.yjjyyjy'. r: ri.' tk-''. '.. yy l yyjy i#t.) jyyEyy. ,9111:::;7;1,21l1IF::iII:. fs):'. .:. ;.)(i;yEtj @;j.1. jpii.i'.y: 7-.1111.--. IIIIEEEE:)-L ;.q !0. ...'. . . r(;-.... r . . (..!... .. .rrqqlllll.---'zm-strikestraddle and strangle composed of 35-strike put and 45-strike call.
.' .'
!
E'. : . ! . : . '
:
.
riE : .:
'
'
'
.
.
: -.-..
...:..,--......,-
:
:
.
.
'
E . . E --i - !:. -.- yy
.
.
.jyyy.
.
.
E .
:
:
:
:.
:
'
' : :: . . .
:
. .
.,.....-..,...
,,..
..-...
.-.-
...,--
.....-
--. .. ......
.
!
-
.
y
,.. .
y
:
.
-
:. ,
(::,1
-.-
E -- E
i -
- .
.
$
81
,
y .
...
LATI N G ON VO LATI LITY
Butterdy Spreads
dl:::::,-f j'r'' --.-' !' .5* jj''.q/ifjiE (C' F' 11::)* ''. yjf y' y'y ;(j'. )' illjtsrfitl'llif (y' 4* j't(' (rjtrjjytytjjjyy'' (' ?' E j?!,rf (,l::.f :1iiqtiii.' )'' j'ty $' lf y' (p' j)f )' i'.ljf EE'.(E'(@ E ilEjts )'--).. .11:::* ''':rk,'kE,q',)t.tq;q((-,,'.. 7* 11* q' (:jrf IIEEjqIt''!III:j)I)!TISI J1' (' f' )' y' (jyyjjjj'jjjy.;tjjjjjjjj.-' i ::!)lp' .))' t'j'j)' ;)'.. 7(71* j'. jjf qtpt't. 'E!'(. i' !. i. E.(.(... iE. i. .i.E.t:(i'i i'742( E(t)p( i(!l.''i'E''(!.'!. !.(.''i'. '(!!fi;:i (j1iIE! @ 'EEE(.E(E(jj .r . ((.qlql 'l'(.))l ' ,jyj,. y. 4!,., . .g; yE yE j g y ;.).E ''h-qirqhdtllEt rt9'?;@FC!/$E(EE-tE'.E L'ikllbs ;g.y yy . . . jyj.j yj j y ; ..y.. qyjj. j(..4.yy jy .yg j jty '1..j yyy E. ''-jj'. -' --)y' ;'
S PECU
STRATEGIES
AND OTHER
.
.
20
The straddle writer can insure against large losses on the sladdle by buying options io protect against losses on both the upside and downside. Buying an out-of-the-money put provides insurance on the downside, protecting against losses on the at-the-mony written put. Buying an out-of-te-money call provides insurance on the upside, protecting against losses on the written at-te-money call. . Figure 3.13 displays the saddle written at a strike price of $40, iong Fith the of options to safeguard the position: A 35-strike put and a 45-strike cil. The net combining tese tlu'ee seategies is an insured written saddle, which is called abuiierfly spread, graphed in Figure 3.14. lt can be thought of as a wlitten stradtle for the timid (or for the pnldentl). Comparing the butterlly spread to the written straddle (Figure 3.14), we see tat the butterfly spread has a lower maximum pro:t (dueto the cost of insurance) if the stock at expiration is close to $40, and a higher proft if there is a lrge move in the stock price, in which case the instlrance becomes valuable. spread we gain imWe will see in Chapter 9 that by understanding the buttey portant insights into option prices. Also, the butterlly spread can be created in a variety of ways: solely with calls, solely with puts, or by using the stock and a combination spread in Figure 3.14 of calls andputs.s You are asked to verif'y this in Problem 3.18.
saddle Sangle
...s.
+
ls
'esult
10 s O
-5
20
25
30
35 40 45 XYZ Stock Price
50
60
55
($)
rt'he
:')' C' '':' E..': -.-.'. (l;f qFp;;f 5* y' (' (tf ,';' '1* 1* pl'l'l)!f Ctf jjll'lE )tyjf li'tr ff yjjrf (rjjrf ((' (il' pt'' )' jj)jf )$'pi;'iE)' (!' EEi 1( i(E E E : 'E '''' i E (* tyy'' 5)5* !j' !' q' S''' ;((,'(i fif @i';r;'jj'' ))' f(' i;jf j'r,f t'yl!::.f '-'Iilii:111t:.. ' ;' E '1iE 5 E .'77* tf t'j(') r' y't'yj' )' yf i))'rk)j())--'... .,.,...jf'3'LbbLL$.' i E(' 11;rt. ' 1* ' .E.(!';:E.. :i.i . !' i.' EE!E((: ;(. qp E' Eql 1(2;IECE yjf rif .-.*111E!5!!-.. ))' i)' .-..'-?!rlI::::;;k:. (' p' .;::''1IIk---lIIp'. ...: ( 'i.it.'E( );' y'; y' )': tjj'tr:jjjjj!r:-', jy)j'. g-pq('. . 'E. ; (' jyjtyyjjjjyr;' ')rj:!.i E..(i (.jg; E !. ILLL; j. i E. i .yyy.)gjqrjy.' ! ;.(E.EE.. EEEyEE.;Eg'.p)i(-yj.j:.:jE lfJ: .@t): jj yq . . jyyq jy-! r;(.;.E y ..( jjj.. jg E .. yj-jjjjgt''- E '-..-. l-F--- y .y g Ckj : . .11!. . p kg: )ypy.: E .(rk..;!. .-.. ( . ' .ygygjjjy;;'; .')' .,,.' -' .'.-' ...-)::;:2:::;));;-)*.* t'':'llr'E -y' .j(.;'' -))' -' --)(y'. --' -.')t'.
il'lli' rrtyjgtt'.l .t' tjjtl'kkj'irf t't'' r' ))--' .'(yy'. .t;yp!!!qqist.. . . . ..
''. .'.'..' .*1111:*-* . IE
'
; :
'
'
E
:
.
!
.
.g .
E: !
:
.
. .-
-
,-.p -j)!
;.
:
'
-
: ..
:
:
. .
i...
.
.
.
:
' ..
:. . .
-
:
-'.;r,rp,
.
. ..
-.
t; . . . .tig .
-;(r. --. ..... .
.
.
. . .
-. .... .
..
.-
.. .
-
.
.
-..
-. .
..-...
-.i
.
:
.
-y .
.
..
jjEE' !
(1Iiiqjii.. (11(E::1(rl,dl::::).-
.
.
11:22 ktqiji;, .11:::
'
:
-1-
15
writtenskaddle
10
:
'
'
:
:
:
E
'
.
i
'
i
.
-.
..
.
-..
.
'
E' . .
:
' '.
'
.
'
E'
.
.. .
-
'
. -... .
ti ..
.33;.
..
.
-. -....-.
.
. -
-
straddle, purchased 45-strike call, and purchased 35-strike
P
'
.
,
..... ..
:
.:.:Ei)(qyE:q(y!1( . .y . . g jj.Ei:
:
Ut
s
E
,
-
.
.
. (;y' iI1ii:;iIi(... 1IIIEiE5t-. i. '. -.F'kiIi!-. . . ):)r. kj;. ; )-. t.. '). i . llllisisr. . .. . . Writlen 4o-strike
jy'E()t.
-y
-
.'. .'.'' ..' .
..' ,':iE!j!l!kk'. r!f r' rif :@jEE!i.-'-. )'. ..iIIIl'. i'('..k'. tl'-!f '''i rtz'.
.
.y
E
... Profit at expiration from a written straddle-i.e., selling a zlo-strke call and a 4o-strike put. -.
--.
E' 't)' (' )' t'-f :'-'p (llgqii.f (,ll:.f ..1.::* 7* 1* ;':'1)* f' 71* i(' (' '.'. !l'1::t:;7;;'-)' :.'. :')'' qqh!lq'l'rrf qtf t' (E:E! 1iE i'(1*,11.*))(* (1* @' (tt'lqf lplllliittif )(' j';';''1* !1' ' : ....*1:1-. ..' .r' ''r' (''' E' '' ' ' EE' E EiEE'(E EI(i E;( i'' ( E (; l (.tE'!(Ti E q! jjyy .. . .' '' ..j(i(.' p. ' .!T!TTTL7'7T!!CT ((j: (' (i( j'Jil . (41)1 . q.('! EE.(.''.'i' iik1@ . (-t-- .' (').i.7.: y)jy)j.yjyjjit(.jy.y(jE.j . .jrt.sjjrjjrj . j..(j. . ..)(.. . ..E((.E(EEir.(: I E)))jkyk.. (kyllj jygssj.jsjrrgsjjsj.jjjygjysjgy,jjjygjjsjsyj.j . .. j)j?)/(1?LLLL.LLj:jjj)L'3(LLL'L'' gj.jjy E'IIIk---i1II.. E---------t'. -;. -.- y '. . ..- .E.f-i . Lkk.bkkkq. i . . -.- - (:..-q -l::::),- 41:2 ;:!ij;! ii -. -.. .; r... -;. (. .-' ',' -' -.'.j-)'ilt' -'-' -' ;'SFX -.' -' !' E' f' )'
'
E
..
.,
..
'..''
. -.--.
'' E' E : r' : : .
(E
:
-:
.
-
.
-
(11:q::,-
-.e.
...-
10
These positions
Written saddle purchased call purchased put
s
k
c
.
-5
-10
-10
-15
-15
-20
-1-
15
combihed generate the butterfly spread graphed in Figure 3.14.
-5
(::,-
20
25
30
35 40 45 XYZ Stock Price
50
55
60
($)
lt might occur to you that an investor wishing to bet that volatility will be low could write a saddle and acquire insurance against extreme negative outcomes. That inttlition is correct and leads to our next sategy.
-20
20
25
30
35 40 45 XYZ Stock Price
50
55
60
()
snchnically, a true butterlly spread is created solely with calls or solely with puts. A butterlly spread butterfly.'' created by selling a straddle and buying a strangle is called an t'iron
:2E
k' IrqsuRAxcE,
t'. t'tjf jjjjjjjjj;'. tt'yi z'yyyr 1:* ti
doLLARs,
'' E -' --'E j,' t';'').hf i';E' EEE' E FE ;1)E E EE (jiq)jii.f hjf k' E' j'y' ;'q(' :);* q' lf f' ,')' .11:2:* -l::E::l,f ;,ii;;' (.*.-*!!'1:::::21--. 1*. );..(kj;jjjjjjj.'. )' (' .yjy'q j'jrf ((' $' (p$EtE' (rl' q' i'5i' ((f' i;f E ;1!ClEi:(Eli (';! iIlp.E;)EE! !! !((.r!'E. E !'(..q( q' q. 'EE(E'E( (.. i.. .r(.!(''E!'' E);'(.E(7! : ii:qjjqj J. ,f .,tj------'. ;')'yyf jyfttgtj-tty.tlf q'qrf t''itf .,' ytf . ' '! .' i...( E!'. Ei'EEi.i ;'(.'(( (.(.(.@ .j' j'('(' ''''. t'tilltlpy'y' 1pI1(' iiilE:'i-f iIitEi!i'. jjjj'tjyy'yf yrf r' lti'; ''. (),-gjjjj', . . E ).E!;(E . . ('. (j,;'. ' 111 y' ! !.:!;!!E E. (j(.. !.; E : E . ; ..!j .j. y,)q r tpq. !.(.. (;... !yjy. ,(y; yj y yyjjj -t)-j ..j-jyjq(..jyrjy.(qky.E-.jyy jy.)yy jryjgj.jyyj y . IiiE:iiIlii.. 111t1-. .jjij -. i i-. . . -. '.gj.)rr. :,-... y-- k(i-!-.yy i . .)j)..t!.....-.-.;.:(.... .: ,-j-,i.)-. !: q...q r....y y .rry....i..y .. . .. . . . . .. . -.' .-' .'. -jjsjf ...);j' '-' -' ,.---....-,--' --'-' ;'':t' ----' -.()';rk. -.' ('E j''yj.Ejjjjt-kjjjjj';g lEtlf jEf
.'..'
-
'
E
.
'
::
. . .
:;
E : : ?E : E
'
. .
.
--
.. .
..
i
.
'' :' ' : :'. ':' ' E: .. : .
'' : :
'
:
'
' :
:
.
-
'' . .
'
.
' :.
' :...
.
. ..
:
.
.
-. .
-.
-
.-.----.
. . .. .
j
'
.
.
.
.
. .
..
.
!
:
'
.
j.
-
. .
ARATEGIES
EXAMPLE: ,''';' !j' r' kttplllryjEjrrf ,''l'jiih)(i1:(hiis''''':iii):!E ''?' .
E
-
Compar son of the 35-40-45 butterfly btained by sprra addlng the profit diagrams in Figure 3.13-z-with the written 4o-strike straddle.
E -
-s
15
..#-
-0
writtehstraddle nutterfly
10 5
.
E: E' . :. !
. -
.
.--..
.
'
'i:. . !. : i i' :E E E ..E .
:
--l336.
:lE:r. ,l:F:--14:E. (::,-
-.. -
ANOTHER
EQUITY>LI NKED HOTE
yy-'. j'))t'ryjj'. Cr' :(' (' jj'. yyry)f (jyyy.f gjjjyjyjj,f t'r' (E' i:' ?' Thrll' 11q5r* iE' i'(1t:*1qEr1* ?)!;kry'.r( ,f j'. yy'r;yj. j'r j'j).ff y'.y'. sliitlr'k) jyjy' )!ifi' )))r:l)'tkC:r!!p(!!qrii7l )' (' 'tj)f 1* (:8)* yq' q' (lh'rggr''jyjk/))jf ikrjy'.jqltll i)' i':.(ii i'li: E r::;:q:7rlrq!:rp:!/rr7rqrF::7rrrq7rrErqqr!7srli'1. !' 1))8.9118p)(1: y'y' y.pyf $' 7.77(:1:* 1).*: ' . q't.! :q'1 iti ic: t't' y' r;ll'. yjl.ll'jy:rji )''q' )q' r'. rrjjy;;f r' r;)' tjt)..'r!iitl1q!. .. 'Ejt:.(. ! .. EE.i'.r.y.E.)1p.tti. (;@p! !E gky.)l ytjj: .y. r'yji..(g. y.(y)i (jjg (jq.. ;)t,:jLikjLjjflj.siqjlqttfjjlkLjj). . q. yjyg tT):-i..j.-!ii!11k'. .tyj.j. IIE:PI:L 411::;;k....-jj:yj . y;jy.j!..y.ygri 7-iIi-.......)y)...tt!iiI;!,,. ()j.:1f-:1f-;l)L;);-..kk.. -;L!11k;kII'. y.. . ..jjjjyjy,( (...Lq. tj )iIIE!SL. .1IIq!q;1iIl.-:$2: -'Lq..-..t.. !h.'#, ())(...jjyyy jjjgyyy,j jgjjr . . . . . -y' ,''''
:
:.
-
: E y
-
'
.
-yt....-..
-
.
E
OTHER
Alqo
.
.
:' iiqE: E -
E!
83
'
.
: 4)rE !..i (E ..-,-tEtj!(-. E . E .
i
.
.
EE
.
.
.;.
'
.: . : :. :.. E . -j. E !; ,jy :. -. ;
-.
.
$
Mymmetric butterfly, created by buying two 35-strike calls and eight 45-strike calls and selling ten 43-strike
calls.
,
.
.
;ry,
.. .
jyyyyg:jj
,
.
etric butterfly
.+.As
10 8 6
4 2
.
O
O -5
-2
-10
-4
'
-6
-15 -20
-8
20
25
cqn also be created by simnlylmenusly
spread.
30
35
40 45 XYZ Stock Price
50
55
-10
60
20
25
30
35 40 45 XYZ Stock Price
($)
huying a 35-40 bull spread and a 40-45 bear
-
50
55
60
($)
In general, consider the strike prices ffl, Kz, and Ks, where #1 Define so that #3 Kz -
= Ks
Examine Figure 3.15. It looks like a butterfly spread except that it is asymmezc: The ' . peak is closer to the high strlke than to the 1ow strike. This picttlre wgs created by buying two 35-stlike calls, selling ten 43-strike calls (witha premium of $1.525,using and buying eight 45-stHke calls. The position. is like a the jiuljtin iri Tble butterfly in that it earns a proft if the stock stays within a small range, and the loss is the same for high and low stoclt prices. However, the prolit diagrnm is now tilted to the right, rather tan being symmetric. Suppose you lnew that you wanted a position that looks like Figure 3.15. How would you lnow how malzy options to buy and sell to construct this position? In order to obtain this position, the s'trikes cleady have to be at 35)43, and 45. The total distance of the way from 35 to 45. In between 31 and 45 ls 10. Te num ber 43is 8t% (= 43-3/ jo ) fact, we can wlite 43 as '''
'
'
'
'
3,.#,E
.
=
#1
or
Asymmetric Butterdy Spreads
43
-
(0.2 x 35) + (0.8 x 45)
This way of writing 43 tells us our call position: For every written 43-strike cil, we want to buy 0.2 35 calls and 0.8 45 calls. Thus if we sell ten 43-strike calls, we buy two 35 calls and eight 45-strike calls.
Kz
=
ffl +
(1 k4Ks -
0.2, as in the above example. Forexample, if #1 = 35, Kz = 43, and Ks = 45, then for evel'y Kz call we write, we buy #1 In order to constrtlct an asymmetric butter, K5 calls. calls arfd 1 You should venf'y' tat if you buy two 35-st1ike puts, sell ten 43-srrike puts, and buy eight 45-strike puts, you duplicate the profit diagram in Figure 3.15. =
-
3.5 EXAMPLE:
ANOTHER
EQUITY-LINKED
NOTE
lion by issuing In Jgly 2004, Marshall & Ilsley Corp. (ickersymbol h41) raised $400kml* 2007.6 maturity Instead of maling matllring in August effecdvely bonds payment a irl cash, the bonds pay the holder in shares of Marshall & llsley's own stock. A bond that, under some circumstances, pays the holder in stock instead of cash is called a
6ln'orsimplicity we will refer here to the Marlhall & llsley security as a Chapter 15, it was actually a bond plus a fonvard contract.
''bond-''
As we will dscuss in
'
% INsuRANcE,
84
COLLARS,
CHAPTER
STRATEGIES
AND OTHER
%.
85
stock. Both graphs ignore dividends and other distributions. Based solely on compadng the mattlrity payoffs, investing in the bond is inferior to investing in the common stock. on the stock (an However, distributions on tlae bond (6.5% armually) are greater t.11:.11 analysis it is impossible valuation withoutperforming a annui dividend of about 2917);so Accounting indifferent. other, the or be to say Whetherinvestors should prefer one or bond to lead the upward and payoff lines shif't would the for the distributions would bond the outyerforming with stock prices, stock the at 1ow outperfonn 0.6699 shares of
Marshall & Ilsley note always settles in stock; hence it is called copvertible bond. bond. We will discuss this particular bond more in Chapter mandatorily convertible a interestinM but bond is 16the at this point because the payoff strucmre resemblej a rl'he
CO
SUMMARY
.
The bond pays an annual 6.5% coupon and at maturity makes paymets in shares, the numberof shares dependentupon the Erm's stockprice. 'I'he specifc terms pf the with mamrity payment are in Table 3.7. To interpret this payoff, note that when the Marshall & Ilsley stock price at maturity is between $37.32and $46.28,the payoff is a varying number of shares, selected so the bond is worth $25(e.g.,0.5402 x $46.28= $25). Figure 3.16 graphs the mamrity payoff of the bond as a function of the N.!Istock in price three years, against the payoff of owning 0.6699 shares of Marshall & llsley
at high pfices. The Marshall & llsley share price was $37.32on the day the bond was issued. The bond was designed so tat, at issue, it would sell for $25, which is the same price as 0.6699 shares. The bond was also designed so that bondholders would forgo 24% of the appreciation on the stock above $37.32. We have $37.32 x 1.24 $4,6.28,and 25/46.28 0.5402. Tlzis accounts forthe flatrangein thepayoffand the numberof shares exchanged above a share plice of $46.28. Finally, in order for the bond to underperform the stock above $37.32,yet to sell at the share price, it is necessary to compensate bondholders with additional payments. The 6.5% coupon accomplishes this. How would we pzice te bond? The p'aph in Figure 3.16 should remind you of graph is $y (1) owning 0.6699 shares a written collar. One way to construct the same with a strike price of $37.32,and calls selling 0.6699 dividends), (2) of stock (ignoring algebraic expression for the price strike of $46.28. (3) buying 0.5402 calls with a payoff is .
=
=
,k,,jj,,'. --' -' yi.f :('')tr'r' Cf .111:.*. 1Ii.,,pI,'. .112:)::* y' 7)* .1IIk..lII,'. (' j.;..t.)'. ir'.:f i''i' j''C' y' ltf )' : lqf)E j.-kr:j,,;'. ()11.* ;'f')'' f' (r' .iI!j);,ji.' j.;..-' j1i(r--2'. (' :ljikj;d :!gii;' t')lf':jEIIE 17* q' )' !I E .111.*.'-y' -''''ttf
E '' 1kiEi!,:'. j'';')' ...j..'. @' rijitd i;f I E. 2(!!:.qp'lE (1lIiI1E:'ii''. .,k;,k,,.'. )' j'j(yr' jyjj;jf jjf y' qqjlltqq:ii,;d 'jIi(' ((jjIIqq(i1i(' )('. y'yyj:i lllytt))tl jjjqtf tl;l7:''.'( Ei'.(''.;' !'.(.iE'(ittii.;.q.Ei.EE..(t.(. @(.''.(('E;'.i(;.'E. ;'7.( ('. )' j!!ijijjjgd r .. q((t))).:y!iyj(;,g-)).. E . . .i.. ; q jq qqi ; .i E . . i ;(r .'.j 'tr'ly r ..(. E . .()i) . )t)),)('(..''(jrij(rtq..'') ))12 y yy q.jqy yj q.y . y j . t).@(t( ...... ll....jiiijg. . . jjIjk(. ..rtkE)..... jyrtyyz . .Ej.. .. .y.j.k.yj jyy..jj E E ! )-)(.. ..E,. ).E:i ( ,. ( ; ! . yqj. ........ . . .rr ... . . ..
)' t!ljr:;!-d tpyrjtlyttrqll:l''yq!'q)'7:qrrqr'srp'l:yf t:ggj!ll.d t,j)illk.d ttj. tl) jj..)j
.' ' .
'
-
'
''
'
E' : . :
h.#E-;;jiijkgyl. : -tt'ijijjlj). ..t .
:
-
. .
.
'
.
'
'
:
'
E
' :
: .:
: : E.
i :
''
:' ' : . . :
-q)((;':(!i()#t)y()q...
ijs,, jkEE!r tjl; ;51. 1r,.tj;ijg:y;j. jr.. .1L:,ij,l.tjyr.Ir.. ijtiij. jI 425E,: .1kr,jg..,j !k:::)! ii.1kgg!j,,,. i; iqyl 1:..kk,j LLI.LLLL-L-LL.'IL.jLji;(q..... . t. :
:.
-.
--
:
...
.-...
. ...
. .. . ... .......
. . . . . .... . . .. . . .
--
-
.. .....
: : :
- ..-..yy.j).j..y-.g.y..yy.y)y-.y..y.yy.yyyj..jy.,-j.;-j-.,;.-jy.iyj..yyyq)j.y,y
...
.
-
: !: :
i
.. . . . .. . .
.
-. -
.
.. .
.
.
.
-
. ...
j::gjly
.jg:
jg..jr..lj jr.,jl:::)jk.,jl
rt.yi
.
-
...
4;51,,
jr..lr..lj jk..jl
j::r:)-
j::y:)-4::)j1
.jg:
:!5;,(:::)-
..
.
.
Marshall & Ilsley bond. 5'ul is the Marshall & Ilsley share price at maturity of the bond.
:;kj;ly. E 1kIji!it:qlliqik :gijllig,:!jii;j:' ' E ((Ir!!1p1;?,::1i1(q' L'LLLLkiLLiLLLL. '1iI()jlik. (jI1(. qlllrr-k. (jIIy. j;jikl;;, ' '.. llljj:llij (rrjj;jqrr. j .-y.j(...... y tj,l:j:il.j.i,lt. yy .. ; q;!.y. (.; '..) tljii:: . . . . tj1j.ill,. . . .....y .j-;y.y yy..jkj.. ... .(.....y...yj;... y.. y...j.y.. y. ......... .-'.'.'j.'....'.;y...r(....(' ' ;.. yy.y r. (iIji::iik.. y.. ...j. yy ..j. ' y y j-,jL,.. . . . . . . .. iIE)i!t:. y..... . . . . y . . . . . . . . y . . . . . .. . . . . t-;,::jjj.. . .. . . .... . .. '1112::. . .. y . . . . .y... ( j.. . y. .. . y.... .. :p11E:''' . .IIL..II.i. ( . 111k,.1IIL.. E y(' . 'j' . (111::--. . ( . . . '. . j r. . i.. . y ). .. .r..yj. .. .....'.j.. ... y .. '.... ...'.')... y.....y.)j'..... y.k1iiiiiiij.. y'y.. )y ... .... j. .y.y.......y... ....y... . . .. . .(y.'......y.. ............. ... . .... .. . .. ...... . . . . ....... ... .. 5'MI .. . . .... .... ... ... 0.6699 . < 37.32 -
. . .
-.
-
. .. E' . . E . .. . q.. .1*. ' E.... - .
-
.
.
....
E .. .. ..
-
' ' ' . . . . -. . . E. . . E
'
E. .
.
...
-y' ' ' . . .. E . . . . .
E E -. E. . E.E: - .- .. ... .. . .. .
-,j.
. . .
..
E '
y ..
. .... .. . . .. .
:
. .
.
..
..
.. . . . .
'.... ...
-
. ..
. .. . . . .. . . .
. ..
:..'.
..
.
.q.... . .
.
..
.
.:.
37.32 46.28
..'......... .
.
-
: .. .. .
:.
.
....
. .. ..
.' ..
..
...'..' .
. . .
-
-.
- . . . . E.
. .E ' . . .. :. . ... ..
111(1j,,: E. .. . y.
.
'y . . . .E
'. . . .. E .
..
.
. . .. .
E .. ..
E. .
.
.
-
.. q...yq
tqlt!'.kk';jjj'E
. . -
E . y. .. . . y. .
fll:lrjip-
. . . E . ,,,j,j(-j.j.,;,,.rj,.. . 111,,,.11... 111:...1.-. iirrll,-. I'Il:::i.. . . . . y... ;. y y. . y- .. .. . y . . . .. . .. . ..... ....... ... ... . .... ..... .. . .. . . . . -
. -
..,k,;:-,,kk-. . ..
.
.,111::::j11-.
.
. . .jjj..
.
.
.
.
. . .
.
.
. .
,lli.lllt.y
.
.
.
,
,
. .
. .
. . . . , . .
. . .
41k:jj111t.. E. ' ' ' E. ' ' ' E.E ' ' 'y ' '.' ' E . .. .E . . . . E . .. :. E q. . . y . . . . ... - .. . . -. - . E .. -....... ... .. . y y . . y. . . .y. . E. y . - .. . y . . . . . .. . .
. . . ... . . . y y. . .. .. .. y . .. j. g . . y . y . .- . . .. .. ... .. . ..
-
.
.
.
'rhe
$25/51 0.5402
-
5'ul
.
q. . .. . . . ..y . .
.
. . ;. yy
:. . .
5'xu :; 46.28 <
.y
-.
-
-
.
Payoff ty'-grjyd jjjjjd t)j'jjjj,kj-jjjj-' y' j', j'' .tjEjjjj:,'(' '-' j'rg',jjj:j::j:;k;. (t)r'. )' yj,yf jly.rf y-'. t't' E E ' y' '(' jjjjj:qqrd td tr'' EE
=
s'Ml
0.6699 x
-
max(0,
5'MI
-
1
37.32) +
.' .
1*(():q *' ' (i'i.' i 1.7::(45 (.'' (!. ( ((.' (i'(. j.'q'yii.((i('i i1* y(.. !' ( .itqlj):!;llr '(iI ( . !ii '.'.' ; ....r :.,;jjjj). q ! . pky : r- 'jjj;j:;jj:j.. !ryg.y , ' , ; .jjy r j r g . . .. q)(. . y j j y j . )-1.. t-t-tyy jyr ... kkjkrjjj;;kj. ' )1i . ... ......1t..))' .yj.j.jy .jjg. j.y..jj.jy(.j.y jy . (.; ty .yjtyr .;ty . .. Maturity payoff to the owners of the Marshall & Ilsley bond, compared to owning 0.6699 shares of .-
(IE
' '' 'E 'i E E' : E' ' ! : : ' !E ( E ( E ' ' j ( .. ! : I'. ' !.-. -. . . q. -y . :)E7itl'..:-;;.;l'k.-.... .ltE: '
'g; il ( -.;-.
'
'
.
' '
-
-
E k.-r
-
.
'
.
-
.......-..:.....k.)
-....--...--......!.......;.....
............
....,...
..L
'
(.''.'l.''' ' .. .
:
.
y ki:.
2111$115! '' '' 'EE''iE:'' :' ': ( IE :'lr7J77''1!qT/1r E' . E . ': E ..
q '-
E.-
. .
..;::-,,.,,j
'4'q
'(
x maxto, 5'xp
1.24
i'p' j'(' (' y'' jyjgy'g;y t'r' '(ld 1* j',,' jt/d )' y'q' y' q' l'Ff 1* )'' 1)* ltd ljqlltl.ir;d (j;' ygy;jd ytf 'jj;jg!jj(g'
;';' ,'-r(-' ,jk;'. (' j':':' '(-' -' ''d
:
'
j
-.
-tjy
-.
'
. - -.. ---.. --,-- - -q . ;-q- -. -. - --.. .- ; y .. - y -y gy-yjyy,yyy y,y yy.. .yyy. ; -j y g ;gg . -.. y ...yy . . ..y.- . -
.-
.
.
.
.
,,
. ..
.. .
Marshall & Ilsley common stock. The payoffs exclude the 6.5% distributions on the bond and dividends on Marshall & Ilsley common stock.
-
.+.
-+-
0.6699 MI Shares Bondholder payoff
% y =
stl 40 30 20
1?.. >. .. .
.....
...
. .
.
.
..
0
20
40 60 80 Marshall & Ilsley Stock Price
100
i
LLLLLLLLLZLLL' .(:L.('j(IfI( .'..J''.;t.Lk.:J(()!!'5.., ..:pJ(E L.zl. (5. q'L'L'L'.Lk.LLj)L'j')f'./'LLkILLLL.1.TL7ZL',L.?LIkLLLLL (r': .,'L..1...j'?'I1. .7.klTl.lrlllhlt:jlqtl.lirqillllliqFqsl L,'qL..LL'.LLL.'L.L'';,LLLj,'L'(..L.2 .4jq(li((l)1i((('gl.ii1l(():'(Eyg1l?r5E!.'J'l'. .. L1LLIjLLLLLLLikLj'LLLLjjIL1' ... .. .....lJ572:I. (yg(2IIl2.iE7.q!h117..
.
CHAPTER
10
0
(3.2)
'rhese
60
g;
46.28)
1/1.2i high Note that the bond holder is implicitly selling a low strike call and buying divialso forgoes premium. bondholder option owed 'Fhe strike calls, and is therefore effectively wllich distribution, is the explain 6.5% dends on the stock. twf)factors the dividend plus the amortized option premium. By plicing these options and valuing the 6.5% distribution, we can arrive at a fair price for the bond. 'Fhis discussion leaves unanswered the question of why Marshall & Ilsley would issue such a bond. We have also not discussed a1l the details of the bond's strutttlre. We return to these issues in Chapter 15.
-
7o
-
..
..
.:?.:.?.....g'..Ji(r. t.
.(l5jEJ.;Ir!ii2((5q'.(!!qk!ij7;q;r7TqL!r!5!r7!:757!'17r71:r!:!7E7!.. .,..
. ...
.
,.
,.
2(27((p(q!jr'@E.l2' ' .(j('E('ljqjE'(E.t..;kt)( LlkkLL;Lk;l
SUMMARY
Puts are insurance against aplice decline and calls are insurance agaibst a price increase. Combining a long or short position in the asset with an offsetting position in options (for example, a long position in tlae asset is coupled either with a purchased put or written call) leads to the various possible positions and their equivalents in Table 3.8.
k
86
INSURANCE,
AND OTH ER ARATEGI
COLLARS,
:21!,,* ji;f jl:::::,,f y'rj )t);'. jjjjjEj!jj;,-:kf' (t(j' .:!1E4;2* .!Ig::' rrjjg;;f y' jj'. ()'' j'. ;') ))' j'r' 1* ' jjjqtt'tyqjjjjkjjqjtjyj r' )'. tt'yf y'-' t;.))yyj-'t)tyyj)'.,;jjy;t),. ..' yyyfllllllllililllqqlfjyyfjjyfjyjjrij'.jjrj (i q' (.(q '!T'rq;qqqi' ')i t)t)pf rtf )))((44.11(r' ..;. .'r ' ';,.'' !jy( )#))i:':I(Ig-r:rr. : r t;fr ' jjj. j ( lj lii j .. )(;y. j ;. ( jjjjg,-jkj. j;.):1!.)2r1..---.. 't;.i:.. jj.. j. yyjyyj!yykyjjy ,,,.,,,,y,,,,,. ,,,,,,,,y.,,,,; t))lr .(!r--....y t! ....L.-.L..,..... . l.. t. yy, . .(...... .. .
k
PROBLEMS
Es
87
.,,' .-kk,;j,,f .'('
fl:::::,,f E-'. jj::rr:,,f ky-jjigyyy---yyytkyk-yjyyfz'jjl',f y' 't' :'C' F(' ;'k-j----g-gCjj-----I;jI.'. 41::2::,,* .kjjy)'-.) j'kgjjjjjiik'rf ''q'. F' (' ;;!i5,,,' jg,.,,,jf yyjgy'lti'. ;'tf q' 1* i''l;l' ?' ,lr,..,,r...:lrf ::E;;' Ipr...-l.f Ilr,,'!ljf 11:E!,,* :!ij;;' p' j,,,,yr!:j,,;,,f -;Eyj,j' jj:E,,f ))' q' '-f'-z'L' )' (' ITqT!!FL!;I;X i'jif iizf (;' jjf 1* .trf ;!.' y'(y'y yyy'. ''lfjl';i :!(E E' ' 'E' E : 'i 5 'E; iCi'i l'E' ;)f1EE! iE E E ' l '(E!''(E' ( l .E lljjj !li.IE'i(E.'ri.'.'1* E)f(' '! E)')**11 . .. E . E .' !jjjjjjjjkk. . . Ei-.tj E;i.i. E.-jj,jijjj.:.. :-;. : #. :- 'i.. (. ,yy . k.............)..........., . '
.
:
'
E
:
:
5
''
:
-
.-,6,i1kkii(;.
--
. .q
-..
'
'' :
:
'
.-
..
-
...
-. ...,...
kj
i
.....r;.
Ei
.
..
; E! j
..- ,
(( r.l.,.ii'k. Index Put :
((
.
-
p-Ej -y.. - -iE. , CEE) ;-) .,yj- . . ..(.
. .
...
. .
:jj
,
,
Call -lndex + Call Put -lndex -
E
,,k,
(
j,r...
1,,.,,:,, .rItry, .
EE: E(E
gj
,..,.,,,.,ii,,,.k,k, E(E
i
...11j
,1,,,2,1::::::.
,,,.'q,.,,.,,,(k'...,,.,k,..,(..,.;..,q,.,'..),..i,.t('.,'.(.,.,,,..,k,..,.,..,, ((jE FE E E qE(j ; ( j i j j yEjjljgyj yj jrr j (E(
jjAtu.. k,.it,t 4 j q( (i
(((jtkjj
+
Index
jii 111.!ItE:y iii .jIk:2y iiq
,j::gg:jjj q,...,,,j
.
41:::::,,-
:(
y...,..i
yyljl i(j
,
Zero-coupon
Bond + C:.II
lnsured Ass:t
Zero-coupon
Bond
Covered Written call
-zero-coupon
-zero-coupon
-
-
Put
(floor)
Bond + Put
Instlred Short
Bond
Covered Written Put
-
CaII
:i -
: '
.
; :-.... ...
.
.
:
:
.
..
EE :
'
. :
( ;'
E: 'E .
.
.
. . .
.jE. Eyl .
''
'
!..(.('t.E. '.':E'i5.(.' , ..... : :..
..
-
-
'
'
-
- -j . .
.
.
.yjE
. y. E. ! E ,; .g .(j.-.,r.t...--.j.-y..).-.t(..yy).-..yt..)..y..!..-.y,()
..
.
i
,
jy:,j
...-.-
.....;.;.).
........................-..
(cap)
dr:z)j
Bull Spread
.
Calltr, F)
Puttf,
F)
=
PV(Fo,z
-
K)
Collar
Stock Price
.
Stock Price
Profit
Profit
Buying a call and selling a put wit the same strike price and time to eypiration creates an obligation to buy the asset at expiration by paying the strike price. This is a synthe zic forwarct. A synthetic forward must have the same cost in present val terms as a trtle forward. This observation leads to equatin (3.1): -
,1:: (I:y ,Iqji. rllq;'
-1::
.
Profit diagrams for positions discussd in the chaptvr: bull spread, collar, straddle, strangle, butterfly, and 2 : 1 ratio spread
!ij? ( rllyljjjj .'(.;r.,,qykti..,,,..ir,.,;'..,.,y,,(, (E((jr()
-
:
..
:
;'
.E
41::22:,,-
:' E :' ' '' Ei: ' EE EE i 'i i' :: E (. ' : E( (' @. E : i i ; (E . .E! i E : . - .. rfji !:-- . ....- i .EEiE.. - . - - ;E.E.E. .. .Ei !E;.i; . . E
.
.yjj::E
'
'
iE (Ei
--
) (y tyjt ;..':...;.@,;.'..,t,,'kt.:,i:,..,.,.,,,..',..
s
fE;j' :'''-i' Ejj,jjjjj'., E 'E i))E CE' :'J'@' ''fCj' k'
E E:
-
-
-
....................,
j;f l'i;f 11* rtf 'i' j)'j)' ;'q' f' yyf )' ('i' tprf ('(!' )' y' @' jrly'sy'y' (' i );' 5: (.! ); r'tili. '('ijr( rjf llqjqltlplr';f tr' t'r'lj tf k' j't' j'j,i :j'.j' q' !! r!(il(!.' ('((.' ' 1.' ' j. r' (-(yjj;j;;;'j' jy.tt' yy't) y' )' y)' y'jyj;f t.)i . )!r ''yttt' ytyrjt:'))'. :; (rr l'.;t. .( . ;.iq :.L't-i ;; i . qi..gl-pg k'-t): : .y .jy. .;.j.y (jj. )(yjyy. ( . y . j ; y ( ( y ' . .. y jr . j .tj . r t; y . y .(.jyy t.t. l1I:-j1t. Iltilir 411::;;ky.. . tkkjjy(-:);ift.... .(!-!y);t).., yjjy. IIIFF! i.-;;-j-.r--.........-..r..t,-, g,gy:1yji. .)yy)y:);kk;j).....-.. .) ,----...-..----);qbfL.-;-. lgy,y !'....;. !p!-
IFIE'
''
:
'
.
-
.
. ..
'''
(3.1)
Skaddle
Strarfgle
stockPrice
jtock Price juoyky
Profit
explains the difference in call and put premiThis relationsllip, called put-call rf7p#@, othemise options. lt is for identical one of the most important relationships in ums
Butterfly
Rat'io Spread
Stock Phce
stockPrice
derivatives.
There are numerous sategies that permit speculating on the direction of the stock the size of stock plice moves (volatility). Some of tese positions are summarized on or graphically in Figure 3.17. We also categorize in Table 3.9 various strategies according to whether they reqect bullish or beatish views on the stock plice direction or volatility.? Netscape PEPS were equivalent to a bond coupled with an option spread, illustrating that the tools in this chapter have applicability eyond speculative investing.
.'
iliitiirli'irr''. j'y' ':')(:q:r?r:r::7!rt;' 7* ;r,-jjjjrrr#.-'Cl5'! q' iitr:tr;j'p)rrif 1!.:'. ':,1,.* .y'g jj.,j;jI'. qqqpf ((' )!' @' )' q)' prrf j'f(i' (@f' ittf r.iF' r' '#' g(' ;'t')'@#' !' p' t'(')7* y' j'.. yjf #' .j'r.tjyjyj'. ryf r' yyf r;pf (111:*. y' j')q)k-,p': ! ;i E E E i .. i q!E ! 'i i' E''' !:'E'ii..;:qt' ;rp ..;i.;. (. . j.r-i j r..r E.. (. . . ..E.. 5. . rl . r.q 7I.''i.Ei(:r-' . ,yy;,,,.(9).?.#;. . .!) . E..--k.).-yy-y-,y)j-:.E-q.E,yty-j.j,ji-#))g.t--:-.(q.y-)(:y..y-y)y).).4y)t)..y... .-)...rt ... . -.yj..y);.j.kEj-(-(j--t-j-,rk. ;-.-.(-tiy..y-..ty...-j,yy;q-tj-yr-kt.y. qjli E;.itqr, i.-..yE.. yj.tgr-))jjtj:y-y 4:::11 d:!!!r 'i-/-hk/,dt:::,jtr:;,, dl:y:),1::. j.yt.ty.y y jtl. yy.jyggy.yyy ( y g Ir..,. Ir..,lE!,l lr..il .:t').)-) ,!d; lr..,l jg::,jj;j!jj;;;. .1k::iie .1k:. ii.I:1I:: iE Ik..1' .IL:k ik ik . .r..;5.). :....jyy.yj.y j y,. yj,jjj. s d::::. sii:!;:,1:: e Ir..ll #-).; . . yj, . . . . . E E 'ii1t:qlii1''. -' -''1.1.:::2t!11:* IIiiEiiI'.X ;'-jjjjrrjjjjjjs'.f j,'j,)jj(' -ty.zjy'gjjyt-g-jjj'k:f -'y'. j'
.' .';
y'... jky'. (' :' () 781 11) t) jjjjjjjjj- (jk;;j))y, ')- tl.tlkqijt '1q.q;)...''jjjjjkk-k,. . - :q -
.
.
-
:
-
.
.
. . -. .
-
:... .-
--
71 .
. . . - -.
.
'
.
.
-
'
E
. .-..'.'.........;;............ ...
FURTHER
READING
In Chapter 4 we will see how flrms can use these stategies to manage zisk. We will further explore put-call parity in Chapter 9, in which we also will use bull, bear, and butterlly spreads to say more about what it means for an option to be fairly priced. Put-call parity was flrst demonsated in Stoll (1969).Merton (1973a)corrected the original analysis for the case of Amelican options, for which, because of early exercise, parity need not hold. Ronn and Ronn (1989)provide a detailed examinadon of price bounds and returns on box spreads. There are numetous practitioner books on option trading sategies. A classic practitioner reference is McMillan (1992).
7erable3.9 was suggested by David Shimko.
.;,:
......-.......--.....-
...,...,......,.
,
.
.
-
----
-
-
. -
.
.-
.
'ik;'-
;
VOIatlity
( ; y qjy ( (;y ) ((yt yj rgy'E.' r' (y ( E. ;. . :.(.E..y . -... r -. .i . : - q .. .. '-.-.'-..... q.............. '-E...i. ...'..;...;..;.....-............-...'.....-..... .....-..(.-... ...:.E....-..i-;.-i.r . .
'
-
'
-
. -.
-
'
-
.
'
-.
.
.
.
' -. -. .
'
. -. -. -' . -. - . . -. . -- ... . ..
-
'
. -
.
-
-
.
-
.
. ..
..
.
-. . .-..
. .
.
-.
.
-
-
Price will fall
-. . -. . . .
-
- . . - -. -
ut. .
.
. . .. . .. . . . . . .. . . . . . . .
-
.
--
q
direction.
'
i Elill iil''.ijwj.jjk q'. .. . .... ..... .
.
.
.
Buy puts
No price view
Buy saddle
Price will increase
Buy calls
PROBLEMS
'j-/-hk/,-
- - .. . . . . ......... .
.
u
F -'.tti',-.
..
' ' ' dEl,:' Ii..ll'
'.'L:. t!!;i 'Ir:::,tlr:.,:::,d::::. Ir..l. lr.' 11-:. iid::::. t!!;':::)p e a Ir..lk
dr::il
ju ujjjjjj wkjj
( ( jujrgqyy j j tjju r.;.. jry...E . (311:.r. ..# ... . .. . . . . . .. ...... ... Sell underlying Sell calls
-iik::r::.
-.. . - . -. .. . .
u
-...- - - . . ... .. .. ...
-
.
uk
. . . . . ..
-. ..-. .-. ..... . .
-
'
i
'
-
-
.-
-
.
.
y ..-...... . -
-
. ... ..... - - -. .. .
-
. . . .. . . . ..
-. . . ..
. .
. . . .
- . . .
.
.
-. .
-.
. . . . . .... .
. .........
. .
-
- .- .
.
.
...........
.
-. .....
-...
-.
Do noiing
Seltsddle
Buy underlying
Sell puts
.
. . . . . . . . . .. . . . ...
-
.
.
. .. .
.
- . . . . .. . . .. . . . .. .... ....... .. ..
.
.
.
.
3.1. Suppose that you buy the S&Rindex for $1000,buy a looo-strike put, and borrow $980.39. Pelbrm a payoffand prost calculation mirnicking Table 3.1. Graph the resulting payoff and prost diagrams for the combined position.
.
88
%.IlqsuRAxcE,
COLLARS,
AN D OTH ER ARATEGI
PROBLEMS
Es
.p/-t?pfc??l.5'
following 6-??7o?7r/7J5?a$?'J?-#>?-cc /c
assume
is $1020,t7?/#
6month
the cf/c/ve
r/7c.5'e#rc??)i'II???-./br
ttse
rate is 2%, the vS&k S&R options wr 6 ??7t???r-
nrcrEwr
3.12. Suppose you invest in the S&R index for $1000,buy a 950-stI4ke llW-strike call. Draw a prost diagram for zero-cost collar?
to expiration: Strike $950 1000 1020 1050 1107
Call
Put
$120.405
$51.777
93.809 84.470
74.201
71.802 51.873
101.214 137.167
a. loso-strike S&R straddle. b. Written 950-strike S&R straddle.
84.470
. ..
.
.
.
S&R straddle.
3.14. Suppose you buy a 950-strike S&R call, sell a looo-strile :-rike S&R put, and buy a looo-strike S&R put. a. Venfy' that there is no S&R price risk in this b. 'What is the irtitial ost of the position?
. '
mont
-
diagram by borrowing
'
s.
c. Consider buying n 950-stlike calls and selling ??7 1050-sike calls so that the premium of the position is zero. Considering your analysis in (a)and (b), what can you say about /1/??1? What exact ratio gives you a zero premium?
3.7. Vel'ify that you earn the same proft and payoff by (a) shorting the S&R index for $1000and (b)selling a loso-stn'ke S&R call, buying a 1050-stlike put, and borrowing $1029.41.
3.16. J.nthe previous problem we saw that a ratio spread can have zero irliti premium. Can a bull spread or bear spread have zero irlitial premium? A butterlly spread? Why Or Why not?
3.8. Suppose the premium on a 6-month S&R call is $109.20and the premium on a put wit the same strike price is $60.18.What is the jtrile price? .
'
3.9. Construct payoff and profit diagrams for the purchase of a 950-strike S&R call and sale of a looo-strike S&R call. Verify that you obtain exactly the samrm/r diagram for the purchase of a 950-strike S&R put and sale of a looo-strike S&R put. What is the difference in the payoff diagrams for the call and put spreads?
l
u
a. Buy 950-strike call, sell two 1050-strike calls. b. Buy two 950-stn'ke ealls, sell three loso-sfn'ke calls.
3.6. Verify that you enrn the same prolit and paypff by (a) buying the S&R index for $1000 and (b) buying a 950-strike S&R call, selling a 950-stlike S&R put, and lending $931.37.
3.10. Constnzct payoff and proft diagrams for the purchase of a 1050-stlike S&R call and sale of a 950-strike S&R call. Vel'ify that you obtain exactly the same proht diagram for the purchase of a loso-str1'ke S&R put aad sale of a 950-strike S&R put. What is the difference in the initial cost of these positions?
'ansaction.
3.15. Compute profit diagrams for the following ratio spreadj:
$1029.41and buying a 1050-strike put.
Why is there a difference?
S&R call, sell a 950-
c. What is the value of 4heposition after 6 months? d. Verify that the implicit interest rate in these cash :ows is 2% over 6
3.5. Suppose you s ort theS&R index for$1000 andbuy a loso-sfn'ke call. Construct diagrams forthis position. Verify thatyou ob,tain the same payoff payoffandproqt andprofit
.
c. Simultaneous purchase of a 1050-strike straddle and sale of a 950-strike
.
'
put, and sell a How position. close is this to a this
3.13. Draw profit diagrams for the following positions:
3.3. Suppose you buy the S&R index for $1000and buy a 950-strike put. Construct 'payoff and prost diagrals for this position. Verify that you obtain the same payoff nd prost diagram by inves ting $931 37 in zero-coupon bonds and buyig a 950-strike call. 3.4. Suppose you short the S&R index for $1000and buy a 950-strike call. Constnlct payoffand prostdiagrams for this position. Vel'ify tat you obtain the sam payoff and profit diagram by borrowing $931.37and buying a 950-strike put. .
89
3.11. Suppose you invest in the S&R index for $1000,buy a 950-sfrike put, and sell a loso-stn'ke call. Draw a profit diagram for this position. What is te net option premium? If you wanted to construct a zero-cost cpllar keeping the put stn'ke equal to $950,in what direction would you hay to change th call strike?
3.2. Suppose that you short the S&R index for $1000and sell a looo-strike put. Conseuct a table mimicking Table 3.1 which summarizes the payoffand profit of this position. Verify that your table matches Figure 3.5. For
%
3.17. Construct an asymmetric butterlly using the 950-, 1020-, and loso-stn'ke options. How many of each option do you hold? Draw a protit diagram for the position. .(
3.18. Verify tat th.e butterfly spread in Figure 3.14 can be duplicated by following eansactions (usethe option prices in Table 3.4): a. Buy 35 call, sell t'wo 40 calls, buy 45 call. b. Buy 35 put, sell two 40 puts, buy 45 put. c. Buy stock, buy 35 put, sell two 40 calls, buy 45 call.
i
;'''F' T' 'j'Jjd Ltd j)d y' ',' '')' Eyd i'.j::' F' ijd :'i'j'.pd ;L'.' tk' ?' ;'5(* -;,?rj)rE..' k;(jE...-,,,y)'k' (' iyd ;y5E:' f' !' !' t'-' F' jyyd 'r' k' j-' jjd ,(y' y-.ytlt'.' )E(' y' f' ;(' )' tlpE.ktd d?kp' ;i' ,'--.ji?#F' 'lltip':tr,d j,d Ed j't'jyrd ;;;)i,gj(;Eq@)p' 'td TE;' y' :'E' i'.;d ;'k' :r' k' .rd !'t' p' ,'j' 5* gd sjd ''tyd i'!;' ';' ?t';' ;;' f' ?' F' y' :'t't'(' r' j'4* jd krd j:':' pr'r' jtd i'jd :;,'))' ,j,' kd t'';' f' (td j-tjd ?' ,d (' iri;rd 7)* rtd jj,d E)' j'''$' ;)' jj-.jkd :;' ();q'.' ql'lrd 'k' #' jrittd )(;' yjy'.d y' pj'pj' pqqrd .j' @' !' ;'Cid 'j' fp;t'.' )')' g' ;k' j?d ))' 71* p' yg-l.td yjr'rd i't'F' ,,@' i..yppgjdyyyj:'yryytyy'.gyyj,rd j(' F' 'kttd :p'y(y' ;r).y' ;'rpE:);@t.-' ;'@' )' ,47*;r44@8)t* @'t' ,'(;' t'q' @!' $' d1* E' )' j'ijd y' ((' tld r' j:' y' r);d ;)(.4.j,)t.tjr)t)t'.' ;';' jtd jrd jjd qrd p' #(' lj'-' jhd 4* qryd ji')yd ()' lr'-kd )jj' ''T' ?p' iqid y)' pt.yjyjjjpyltijtl-..d @5j' #' C' t';r(';' .j' ild y(' ttd jjj-.d )t;' (qd (tj' ird j'4* lht!tkd :'k)d 1* ky'?p),;';' yy'd ,d j..'..--.,' jd )' .j.'y' 1/171'* jz' j..' ;(' j::-;i.-r' jld j;d j:'y,t'kt' yqd q' fl:;(p..' jkitt-.'.d fC''y'j$r(-i!.' LfjkqkLtl?i'-s'' jgd (j' ppd (t'-' jE'q' ''t''j' kkid y,d j(' f' ittt'-qd !yr'ktd qy'j)d (kd gtd ;-' !j' )kt' jpd ;'t't?d yy'-' ;)' yjd y' 4..*.* 1* sLqbiL'-ki'bffl:' 1;* y!qd jyd ;r' i'q,' jt-d y-.tjyjyyjyy',,j,;jgjd yryd (.td l.'p' ;@:.' j.:;d (jy' jj)d lp':jd jd ,L;b'.' f' (td j!d i-.' t)d 1,4q;j4:* @t(' jsd L:L;L;Lb,.'.' 1'* y?d p' (5* jrd i'.,d j'g' k' ('j')' 16 q' (' !!' ld p:)))-.' .(' .,' jLL'i.iqjL'.j.ikLtl'.' '(j' (jj' ttd ykd )t' td /1* )(' )j' .i?'?' J' .-' iiid )jjy' ;r.!)(r' :':' yrjd y'd k' t(' d' ,-).L'jL' (t' C' ,'ljd )'t' yyd ;d @' )?)' y' !' yt't' (hd g' jj$' )' ),..;j'-' 1* L,ik',' (jqjIJttIyIt'.X ii'-' ;LLb'.'.' ';' ).!' #' y-'-' k,' '''' i-'kpl'Frri;j'1' q;.' .'' q('(' .y..' ,',' i:d (r' ):' (' )j(' i'illd rqj@p;'.' pjrtyd pt',' .)t' p;td )-';' y:r)' y(' -L?'' );i'.' l-.' 1-* :;' q' t:' q.kd :'j;-.!$!7tp)'.-' '(.id k?qk.' !(');.,.g'' p' i?sd lpl'kd lid ?' ))' 1)*1* j))-'.' (s'-;' ;;' )':' prd ).?' t'!)d ikd 'td L:)'.' y' t:d 5* ;r' ;))'yytyt.:' .r' .(' jt))):j;.' (j' tq:yd (-' tcid q' ((' y;yd .p' lg'g' j)d rqd t.pd yjd ptd /?' y.jd (t;.$' !' f;3;.';j;t3LLk.' lrd tpd q.ti:r' .@' pfd J' #' '6jLLL'L-1Lf',' k)' )' (i-y' yy),,d )(' i'illi.' k' '(' p' Ip' rtyd t''j'1* j'E'5* tgr;jjy)y.' rtd st:td r)d r' rjt!rd t'ld t'tl't,d tktd r:'jj' r)' t';'lli' ,jL)' r');d t,:);d r))l77;:r'.' t.pd rrld t)d r?'.' tpqylyy-'d '' t.':' t)' likt;id (rr;)'-' tf'.j' t-;' )j;(-'-' r-' yjj.jyjjyjtyyqjkyk-ikjjg'-,-;-d td ry't;;d rpd ttd tjd tid sd .(' t@' r' 'r'jd ;'4* tld t't.(' r'rd ::' (''''''' '.','-',.-.' -'..' .'jt))).).q(t' -.j!' --';f;;'.b' -ir'.' ..' '.'-''.j' .lir:.ptd d-(' .'-'-' .'-..' .j' -;' -t' -i' .-'ykrllkiirljr,;d -'-' '''-.ky' .'.:rt' ...@:-;':i.t@!!$rr' '...(;r'..' -'.-')(jr' E' :':'.''*'x' t'..'-'-' t,)jL;'.L' L' --' :'-' ',';':;' --.-,.-(fbk..:;,,' -' -j' ')' ,;' ?' :yr'.' t'-r':y' -,' )?' 't'):' -'-' :t' -itt8i(.' j'IFX ,'-'r'kt).'-'))'(' td (' ,'p' :(' -y' i/.i'Ld -(.-'(.rj.'y' ;':'i.:jd -.' tyyjjytgj.iyjkjljtjlyd-lyyylgjjyy'yryyrs-yrjtyy.)ld -...' -' ?' '''.('d..'-!-,,)j)'..--' ;t' -t' ';' t'''Ej' -.jq.r';q(rj('.--,!)))yr.'(!rp?#))rrj?j'
rlq'yd r'rrd g' :;;qL3LL3jL(I;3L3L'.it?)b)$Lb'.L...jL:-L---LLLj..' ri-' rtd t).t'.' ry' 'z.ijrll-,jt'k-?d rd tjk)t:tg;..' rk-./tdrt.qlt;rrrd q)lll...d rrygj;)..' rij!qd
90
%
INSURANCE,
COLLARS,
AN D OTH ER STRATEGI ES
. . .'.' .'tp . .'! .'. .'(. . . . . . . . .''jj)) .'- . . ''.' . .'t):. '''''
.'.'.' .')qr(():'.' '.' ''.' .''i)'; .'....'. '''' .'.' '' ..' .'..'
-. :. .. i ! : -: - E.E - ! :E ' E E : i! - i 'E : ' :' :'.p);. '-'E '?i ! EE: :. . .;E!-. -!i . ii -. fE(! . : ! -'-lrI : .:. Ei,.i; j)iiE -' E : :: : !! - ); : 't.''' i' E. i - E :. i.i-.E: ir:. ;;;:-.i..;.!F:;!;k. (E E : r: E : i ;@; . qi .. ;:kjq; E: .r !! EEi. . . (. ..;ii7.1iEE.-.pE.;:''E.E:p.;i.. i. t' ...;)kjrtlq;'Ehjtpj));;)'....t,(!-.(j)pq;!(ppjji;/!)'' .; -3L3-'.i-')b33))3)L. ' (i E!Ey'ji. i. i IE '-.i ljki'rEjsp)p ..:k;;;;k!kq...---... ::j''!. ).iEr-.I:.-.-.E..:!E.I''';':I: -:ik.! ii.'E..:.'iE-..!.:.E-E!.!: -E.-'..!EkE;EEI:-:.EEE;;-..'iEii'--.)';!'E.tli!'E?, )(i?-'(:;'. ))! ;jiti.'PE ))'. ?p'' ' i!j.'''-E !E' '-;..!iEEi!'.E!;EEEEE-.iE.-.-i-EE.Ei'-t'.-:!iIt.:.;-.:-.EE'E-:E-.i2.EtE.!..E:';.. : r.(E. ,):t?L'. : .E.)'. i ji'fi i ; :,i. : ;!k;! . r-kqPE )'. -... );. : E E iEEE-E.E.. E.: : ;r;.('. ! ! E !' i ikji; ''.'Ei!.ii!jEj:f'(q. plji'E!i LqqkjfL.i?Litb.jklj3jj'.kstLLtLtiL'. E !i :. .-p-((E .! E E - E !i !; !; ::'.!!i (i'ikpr y'' i.- i-.E.: :.:!. : ;:-EI-E : .; 5*;*. E :. iE EE E g'EE!- .! i ;. . :.q jry'r). . r! L;'fLLb'LLb(33j..;jjL33?LLLqk)tfL(t3tLL;LL'. jL.jL$6L';L3Ltj'L)'. t.drr-r'.rsjq'yt.-tl .- i Ei : q . . ...i. .. .- . , . ; . E- -!-E!. E ; - (E : .: E . ,'. .i.E !EEE.E qk q. E -'- -y'. E Ej.' E -. i . . . . E !jkj-. ! (E E ; . ;. qj ..:.i .E..E...!; :.?'i.:..- . Eq.i- r -: rE !i i :qr ;pr'. . E -- tE- -'.E.g EEEEiE : :- p!. ..E . . -. . - iE Ei EE EEE.j.E E (.. E . :,,gjjjj2j/);*. @ ;.;! (( -! ECE( E-.EE-:i E : .! . 'iE -': . .! E q( .. jE(tq ...:. E- -i E, ! .- . ;E PEki. ' ...E... :. yj-.'. -. i ,E - : -. (E. E iE ; ! E j !.; E .. E.-- ;:.. ..E;!. .i EE .. . EE. : E.. E.. .EE. .:--...!i -EE : E - E . - : . - i ; ::- :: : E i E .:.. . ...; -!:EE ! ((jjE r : ; -E; (j,j'. . .. E : E :. E I'E ii - .! . ..,'.i.'.: (.iEijEE E ''. '-i' E -- ;. ;E j; -(Cyr.yr i ..E t'!.E IE k - -;)'. jpr i.tE . . : - ji(rj'i ; i-Lf'. EE E -.'E - E EEi :Fi -t'' i (t.. ' .E. : ,i . E!E y-yyy. ;. - yj'. . .- ..'i .E 'E ij. :; E ( . jqisk'''bblk. q );ktqqt,,. y :. . y lE. .E. E - E E;. . . ; j jt'tkyj y y ,3,43:. - . E' - i '''. -'. E;jj . - -: p i. E E :i ! ! E ( . E . !-y'E E . E :-'(;j!,p(rrrt(.'(kr)l;l:!r:'..... kE j. !. El; -. !: ; :EE - E . . E E.:E ! ! EE .. . i' .': i . E -:-. . jk.js'yE -.. t'--(. lE'E '. i ' E! iE ; --: - -jjjr'E j,jik,, E;.E:i!i.: ,!;'iEy :j(j)j)!tE E-j': :. . . ; E i E ! (q i.;. i.:.:(E . . -. . - tt','-;; ; ! E.j krE ,t Ey.-'.E... k. .EE- i - 't'- E: (. i. :.-(.. --!E! Ei.-.... , EE )j. -tt;ytjE. -. y j!i.E.E yjkIIy.E -E 2*1: . E i.. . E, E ; ;.. - -E fki i . -. ' i.'i.' E E LLLLI,LLLLjfL'-L,:;ILLj;LL. E. j..E E-5-. kjyyrl E ii - -E 'i i(-:t.. i . E; .-. E ,: .' ry':E . E (,; . .. E E E I i E E . . , j: j ). .. ; E LE . . i iyj'. ! . i ; . EE E kb'L.if'. ( j: E E E i E . . jt. ; E' ! !i E .. . .. . . qbq''. EEE. :. E ; E- ... .... . . -'. ! - : : ,. ; . jj))'.. E iE! EE ; -i .. ...E . . .. :. . .'i. '. :j- i . - i E .. .i ;k: -:j ! .. . .: . Fi.-; - -;. ! ;.ti-! : ' '. i E t;! E. E.E: -' E.-...;. !jy .. E. :i E( t.ll : ! .E : - i : y--;. i . . Ei q !(;:. . j'tl . . i ji (E . ... !E - --. . - E ?: :kt: )'';E);. : ! E . ; it: E . : : . ; . q . y yyr'qr E -t', . ! i q E ; E . . -. E! .. E E : .. . jj'yr qE. :. E y yiEE. -;(.. E ; E.-'. . E E E !i E E j rEE. E E -'-(j.. i k)'. E. ;- EE!-i -i!. g .!( , ..- E g ..(E-.... i: EE --(E. E..E.yy!( -q ..... q (.( E. ;: : -.. : . -. i -yyp; . j -.-yy;.rg ::,,(;', ;E-. -: (' (. E y; Cl )!: q . !: i E . (. -.. ; .-. j--'-: .. . .r E ; i - E- i k.E i . E:.- g: ;.i E ' -t.LL. ,j,. ?;);:,--'ii3/.. - :yy ! j- ,- yy. . -jy Ej . y q- :i-i : . -. L-f rtjjj':. E ; :. ; E . : i..!yt! -.---...-Eli(:; Elt. . E .;?LL.LL. E. E E .. E -E. E -'1)','. ,.!... hj'.. ,rkj'yykt'tjy'-)'. ifl';E !! E(rji: -'-. : ;... . ... .-jL. ,.. ; j'. E4.! E-E:E-. -, Eyy.E gj.jrjj q. jg. . ; !. t'. ,..-k.-. (-. (,'-.. - . y,yji'. j y-. . . '' E 'L. E@ . . . !2 y:jr'gjy .:'-LLLL;;L-'E ;. - y E )y'. E y . .jr'!,-j,,, . r - E Ejy. .. -.- - -. iEt. . qE ... . E!-kE.:. E. E ; . i. . Irtkttj;.. : k jt'. -.t! . -'-- - -'. : -. -- y . EyE.Ei.-; . .- .. r. -; !E -. . -.iii.( E. ;;y .. - -E j;.-.. !E ...L ; -.q y. j.... . ! - E. p. , -. ik. -.E ..!. -E-.-! .r,.- -(E-j......: .... E..y. -.. . i.-...k..- -,'.-)-, sy'.- @.. .- . -. .. E. 9'E .@)t$'. 5;-. p'.. E. .-- .E ..-,E .-. E -..E. . ' ,;:'.-. ; - 5 y-....;j5 y(.! .; E )- . ttlppE E E,- . E...--.. . ,,j... ; -. . . ?3*... . . .. . - ! i'.! -. . ,'. . ! . ''. y. .... . E .... .. .. 5-C-. . . ; ;. ! . ( ?l:;t':l'. .. .. - -- . . . . .. 'i ' . ....-'..E q i.. . . ' -. .. ... .. ....:y......E.- :.-.E..! ..t..E! . :.i,'i . .. . . E '. : .E-( -E . :.r:.:E .....('.i---IE . .. '- ;jE.k.....;---. - ... . ((E. . . k)'. )1 ?j3:..,.. tyr...t : -! E: ,()r;;t!. .. E- ... iE.:E-'-. E.E!- . EE.Ei - i!,E !;'. ; ; !@ E! ! ! . - E .-.- '.. E -. - . i i .E: E ! E!E ('k,F'. . @ :: : :gj,. - . i! ..E - yk(:'. ... - . : (!q'.;3)L-.k.'t;;'. E;.Ei!:.'-E i.! t'E ;:ry)j)?.. 51). '.-.-. .ii.- : E.yE. .E . -i i E jiE !'i :-. .;!i . Ei.E ....- E. .. i.-.i!. 'i iE;: !j;..;'.i..y'E L! .. . iEi EEi:....:. E .-.... .. k)E : :. E: . .. k-. ;;$.jkk);'. EE kq:, ! ; . EE ; ... ;'Sy;yy;)'): . : .E'::' Ei ItEPjI'?E!P)!)Ii rI(E. . E: ... .'qE k-('. E . : i.. .E- - ; .. . ..: . .. .-;'';):. -. ;j.- ! -. kt)E.: -. .EE. ' !!@E. E . )--. t.l',t-sl-' :'!q ; E E yt.((j'. i ! -tE IE .. E . . ilpyhdlllklhyljji-jlllptprlll'. ;E! E E E. -j)-. q!.-..j!. E ; --'. ; .-( tjq. E . :. . !: @ jlr2.5 -jt;': !.--'E ?'..,--..fiii--i.. 'r'fp'.': -'. !. E . ..E Eii : .. ;. -).tE. ;L(L1LL. E. y(i : :. ! E.. -.. !E E .E.jE. '. EEE . -'.. :.. ' .- i : E. E- ;)-..tjy'. . #'jj---E ;-.'-----t,. E . . E ;;: E ,'. Ffqrj;.1;i'y#)':yt';j@r.. ....j,k. . '.!' : ! Ei.:. !TE -(. . i 6L3LLL;LLL. k: ?;r44E''EI !'i' rrllkE;i i ; '.. (.;:i jp)y'. (: , (.,:jr-..'. r. ;):.;.'. -.- . , . ;iy))'. @ g(- ,i!.! E.. - . k .E jqr';rl - L1L. i-. ;p#'. .''.(.k:i E .. ;.; E.r i . .. .. .!..'.(.!E:'.. ?E ; -. . i. (i. )pL.t;r@?tt''. . .../)q*. '!;E 'I;;;,;'j. .: k).5!.:. .. ;;-. .lr;. -. t'.)'. ,t'iL'qL?'ib'jin'fq. . .. . '.-.2p 1)'t)'y .. ,j)'. .'.i!k'. . . y)h'. E.E:.:;..;;.. s..k(. . . . . L)t):'. ;)'pk',-'. . )'. :. L;;'. ;!;';,--(.. -j'. . ,8*. . a .. .s' . . .. .3*. ;..; LLL-. ; ;;'. . jbbq'. '. . aL:. k)-jj'. k'. . . j.Lb;'. j.;i-)r).-(t---;kr'. t.(t'. :C.'. :3;';'. ..E(..:gE::.;..j::yjj(.yjjj . . t.'. .. (r'. .jt'. ;,'(jIp(:'.. . ttl'. .T1;'.FC'j;'r'. . .i;..-;j'. . . . !il'. )'. :'kbL'. p);'. T)tIt,'. g?*?*tJ9*. :Ej Ellitjiil rqkgyyj , g : : a . '...:' . .' ..Eyjg( l . g .. . ' . .j..': yjr.. j.j.;., rj...yj. . .. T '
;.: :!:E ! -' -
''
-
'
One strategy investors are applying to the XYZ options is using thetic stock.'' A synthetic stock is created when an investor simultaneously purchases a call option and sells a put option on the same stock. The end result is that the synthetic stock has the same value, in terms of capital gain potential, as the underlying stock itself. Provided the premiums on the options are the same, they cancel each other out so the ansaction fees are a wash.
Etsyn-
Suppose, to be concrete, that the premium on the call you buy is the same as the premium on the put you sell, and both have the same strikes and times to '
expation.
.y..
-;
.
-
..
..
. .- .
.. .
..
.'.' r''
.
.
e Do you consider the Why
no t?
-
fees'' to really be
.y :
.
.
.,i..
.
.
.g
.
.
-.
-
.
..
... ....
,
...
.
.
;
.
.. .
.
.
..
-
.. ..
. ...
.
. ..
..
. . .
. ..
..'..
. .
.
t(k.. ..
.-
.
.
.
.
...
.
.
, . ..
. .
.........
.
.........
.. .
.
.
.
.
,
... .
.
.
.
.. .
t!'. .
-
.
.
--
.
.
-
-
-,'(!
;.
-y
.
-
..
-
.
. . -
-5-q.
.
. . . . . -. .
.. .
'
.
.
..
.
-
..
'.. .
-
.
..
. ..
.
.
-
.
'-
r..
' -.
.
.
-- . . .
..
.-b..
-'.
.
,
. ....
.
.
.-
'
..
t.
.
.
-
-.
.
-.
.
. .
..
. .
.
-
. .
-
-
-.
-
-
-
.
..
...
.
- . -.
.
-
.
.
- . . . . . .
.
.
-tt.. -
-
.
.
..
..
..
,'-
-
. .. . - .
.. -.,'.;t'.
j i'..' ' : .'' . ...' )(:.: . .. i..:.'jji)jj)j 5 (E : . ' ' . ( ' ' k! .ii : . ..j.....( .ii . .. : . :
.
'.'...
........
. .
::
.
;
!iEEE:
( . r El.
:.
:: ' ' Eg . E ...
.. .
r':..
-
'
-
. -
-
: .
:
. -
-
'
. .
. .
' '
- .
. . .
.
,'j(i
EE.. :
. ;
.
::
,'.y,!j,,j'.
,,,.tt
(g . . ......j..g(..E. .
Ej::j.:.'j ,,)
:.
. . :r .
..
tjtri,d',ltlj
,
,y't,)
i::
:E
: EE)l;t)E!
,
r.
,,i
'
.
: : '.
t.t'',,,Eii'.EE
,
Ei,itijiyjj
E
yy(,
k (g
..
.E !'
,L,l ')t.kl)..i .
.
.
EiE.Ei'tEi,.ji(;
.(
E
wash''? Why or
E(q
; :(: .ii ):E
r
:
.
tiitityti)
i! gj rE
l
E'.: . :i.':E:.'E:EIE
.,i).,
E E.';7,'),E';()
/)
.'...'..'..:.iE':'..f:E .'.:.'.'..
.:,E:,:)E)tEj'E :.
.....' . j.r..E..(:.j.: 'tE'' :j(':
).'.':.'EE:
:
..::.'.
E. . : .' . EtlE.,,lEltl)',E, ::j...i).i
..':
'.''
:.:.g:()
:
:( :
: . . i:.:
'
.
:
)
:.:1(').
il '..),,E())E ...
..(.E' .:'
f,
k.:g
.:i
::
.
:
..
.
:
l
..
: ' ' ' . .
:
.
.
' :
. .
.kEii
EEEEsqiii'iEEE jE:E::(;j :
E;,,E'(tt)t;,(,.ttp,
qE,:,:t,)qi,r
ttt
.IE:E
: j :i E: ::(j :.E. . ' .(..(.:. .j(q .' . . j.kj.j Ej ( :...E..g.. (j: : i (. : g .. . .E.;. .j.;j.k j . . . i.tj EE'E,'Et''t'i .
. .
:
:
.
.
-
.
.
.
:
: '
:
, 'l,yt,i).'..i,t. tti, ' ..E.EEE
..
. ..
'. ..
.,,)
E ' .: E' EjEiEE IE E ''.;(.. . ' . (..;(...lj.k . . E( .Et',Ertf,.,).i'.,, : E .! jyj:. : .E: : .;gi(!.(:kj ..yk . ... : . .. ( E :...-:EE;.:..i.!.....q.g....::...'.E(:.k. j ' :
.
.
:
' . . . : . .. .. . .
..
.
:
. .
. ).,)...(j....
.
1 ' E' j( ('j (;'EEE'. . . ':g..j. ..gi jE .( :i( ' .. :.( ' . . ' :. i'..E'!. (j! Ejyki 'yiy 'k !, : : iE. E E:. :y': .j yE. ! .( .
j. ;I ;. j':(. ;'r';. ii '' .
: g
'
'
'
' .
.
:
i j ( (j.;!rjlr .. jj., '. rtj.).'):..;( .. 'j'.l:.t..tjt ...t .
'
-.
.
. . :.j..i((
:.
...
(......g( jjjj.g g.jj'. ..... E E . j( kgjj .........g(E .. Ei jEijj;.('. .( Egiq q. ! k ( .'(...(.'(i('i(ji'ijI((.j.EE .. jijE(E;(g.. . i El.y:.iEE: !j.(. . ' . ;.''(:ji i 1 g . r'i. E.q ... ji;qj...z!I!y.jE 4-;! yj;t..'ry . :
.. y y.:. .
. yE : . . . :.. . y :: . iE . . :
:
.
: :'
-
.
:
:
5
:.
.
g
:
:
.::j:.' . :
,
. . E . . . - . . ..g.j gj.j.yj .. gg.
.
-))
- .
-
.
.
-
-
.
.
.. .
..
-
-
.
'- .
--
'
-;------..
rjk.,. )jt',--);y));;;---jjj'-. . . . LLL.LL..,:L-:LjLLLi'. 1 .
..j.
:
,
' '
'
.
,
.
.
.
-.
.
..
.
.
.
:. . : :.
'
: : ,:
:
.
-
:
.
j.
t
j'j
,i
:
:
.... ...
. ,
:
. .
:
: i .
: . .
jjjh
.g
).
.
y.
:
.
:
such as laborg raw matenals, Fmns convert .u. .m 1T tne COSt Ot Wnat lt Prodtlces IS ProntaDle n. Pnces can change, however, and what appears to be a exceeds the cost of the lnputs. are avmlproftable actwlty today may not be pro:table tomorrow. Many lnstnlments able that permlt tirms to hedGe = val-lous nsks, ranoolng rom commodlty pnces to Weather. jyn'fl tjjat actively uses delivatives and Other techm'ques to Zter its Iisk and protect its xprotablllty In thls chapter we tale a look at how ls engagmg ln nsk management. manaGe nsk. D derivatives such as forwards calls and puts are used ln practlce to ..
.
ls lnherently
ustness, llke hfe, and machines into
ns
lnputs
.
.
y.x
@
g
GOOdS nnd SCfVICeS. &
x,
.y
.,
.p,
.g.
@
.
*.
.
.
.
.
r
'
'
.
l
.
.
..
=
..,
.
rw ...
.,,,,..
.
x
.
.
'
x
x
.
.
o
.
a gold-mlnlng ll-11r webe-crmbv exnmlnlng tWO hypottletlcal III'IIIS Uolddlggers, what nsk.s they face andAunc Enterpnses, a mnnufacturer usmg gold as an mput to see nnd to demonstrate the use Of den'vatlves satecrjes to manatre those nsks. er lookmcr atthese examples we wlll explore somereasons fir'ms seekto managenskln the srstplace. .
.
.
.
..
.
:
.
.
'
..
*.
*.
.
&
&
.
.
'.
.
. ..
.
'
.
'
plannlng to mme nnd sell 100,000 ounces of gold Golddmgers ls a gold-mlnlng W11l slmphclty, We For assume that they sell a1l of the next year s over the next vear. 1Tom 1 tne Prlce IS that day. productlon Preclsely year totlay, recelvlng Wnatever l-trnore productlon beyond ls w11l te next year. We of $405/oz. todav The once the gold llke producer that hopes pnce w1ll nse Golddlggers Obvlously any .
*
.
.'''
.
1...,3
'
.
'
.
.
.
.
,.
..
'
.
..j oGol(1 .x
.j.
.1
,K
.
.p
.qold
1k.''''
'1k.
.
x
o
.
*
*
*
.
.
.
.
.
.
.
.
.
.
.
overthe nextyear. However, Golddlggers s mnnagementcomputes esttmatednetlncome net mcome calculatlon for a range of posslble pnces of gold tn 1 year (Table 4.1). .j,s j GolddlcGefs Oecteu Pr1CeS. uy IS S that Plrllt 1::::) Jr:::). ghu sjmws Gold pn'ces fnll enouch to make net GolddiGIVFS S1'mply Shut the mine JY sjjould neGau'veg cOStS On the extent to Which depends nnsWer The jncome are fixed. The irm & Only nOt it Produces GOld Whether Or Valiable j:i costs are incurred d XC tj.j C IIICWS cost o lt tf the mlne operates. Thus, for any gold pnce above the varlable cost of $50/oz., w11l mnke sense to Prodtlce gold. .
..po.
;2:::). k::::).
.
.
.
' .
z..j
...j
dk:::;l
;r:::1I
.
.
.
.
.
'
.
'
.
.
:
:
.
.. . .
-.
.-jt..
. .
.
r
E E: E i ' : ... :...(.': i :E.E !j (E. : ! ' .' ,',. ..j.i!...@r(') ' ' .Et,l
:
.
-
..
-
:
'
.
'
.
-..
.
-
' -
:
:'
.
.
.
-
.
...
*
jr ( jj )E:
:
.
:.....
. .
. :
.
.
,
( g . . ..(.(.:.:: .' ; ( j. j . .rg .(:.k .... . E(...:::?) . .. :E, :t ElE()j :: i E.EE . : ''..( i' ji EE. ! : ' '. EEiit:)::ji . .. . :::'!EE .
.
.
'
.
rfhe
,,','EE).,')$E) ',!.!)
.
:: . !
:
:
:
. : : .. .: .
:
.
,,iE.)jjy iytE,
,'
. ..
.
,,''i,l'.iti
E.'
:
:
: E. :
:
.
'',iq'g
E :
:
:
*
:'
.
-
'
-
.
-.
.
'.LL'.
..
.;..
.
-.
'
. .
-
,;
.
ANAGEMENT'. PERSPECTIVE THE PRODUCER'S
jjr'r . E.E.jL.'f.LLj.L E,.
.
.
.
.
.
-
-
.
.
.,j,
.
.
.
.
-
-
.
.
- -
.
.
.
-
.
.
'
-k:'.
.
.
..
.
.
'
'
--
.
.
-
4.1 BASIC RISK
) jlty yyE
'
.
.......'.'s..
.
''
-.
'-
. .
-
-
.
-'
'-
.
&
.
E yiE,iE
:
.
.
j. .
:.
-. . .
'
.
',i,i,'qsltlr'r'
.
..
-
.
.
'.
.
:' .
. j
. .
7
::
:.' E .:.:.' EL.L.L.:. .:E.': EE,
:
..
'-
.
.
. . .
.
' .
'
EEE1
'..'. .E
:
i
-
:
.
((E:(;'i.jj : :t:'.:(((E:.j)'('. .')'.:'.E.:
..
. . :
! E.
:
.
:
.
.
E(!:. . .. (: .!q: ... ;((..( E:. :. .:: ;(';!
...:(....E: ( '::l.
..
. ( ' gg(..g(gyr..Egg(jg(.jgj ...;j..j; E(; ( y(y yg;.(.j.g... (j(..'..(.j.j(. . . .. y E !E. EjE ! E.p E . ..!E E ( E. ... E... i !. i. i... p i. ...I(.E.;.; .Ey. . . .E. iE.. .(..Eq.;(.j. : .. iE g.(.iq ii. ' r. i (!EE.E .T.E . . . ( . (. .:...
:. ... . , : .
.
.
.
:. :
EE,'ri,il,tE
.
.-
-
-
.
'
.
.
: E( . : . :. .. . .. : : ' . q.E.E . : .E .... !.E.E: ... ! :. .
..''E,ltijt',)i
-
.
. .
:
''
E iki . . .. !i.. . iE... E .. il . '. . ; @ E. k ... . (j E,,!E' E
.
.
' -
:
E: .. iE' g. .
'
. :
:
-
.
-
'
. .
.
.
.
(. .
.
'
:
.
. -
.
.
-
: . .. .
Ei i ;: . .: ' .. .E .. iiEl. . iE.(: . . '. ' . .
-
.
. .
.
.
.
-
. .
. .
. -
;.'.
...
.
. .
i
:
:
-
. -.
.
' '.
. ..'
..
L
E : :' .'.' i . (E . ' ' .
-
-
-f33t;'. '
.
.
-
. . .
. -
.
. .
.
-.
. . ..'. -)j@y'j$'. .
.
,;.j?k
--
-.
.
.
.
.
.
-
.
-
.
..*;)1;71)/)..
.
-
.
..
-
.
-
. . . .
.
-
.
.
.
. .
.
-
. .
-
.';-t.. . .*5 ''. -
-..
.
-
.
.
... .
.
. . --..
. . .
.
.
.
-.
.
.
-
-
.'.
. . . . . .. . ..
.
,j,(.
.
.
.-.
. -.
-
.'i;
.
-... .. . .. . -. . . . . -
.
-
--.
.-
. .
.
-
.... .
...
-
.
-
.
.
.
. ltjssl'. ..
...'...........,....
.. -
'
.
..
.
.
. . . - -. . . . . -. . .
.
..
.
.'.
'.
.
. --.
-.
. . .
-
-
-
.
-
.
,
. . .
-i
.
.
.
-
.
.
-
. ..'. . . . . ....
-
.
.
.
-
.
-..
. .
-
-
.
-.
-
.
-
,r
.
.
.. .
'
-yj'.
-
-
.
,.
..
.
-
.1q -
-
. .
. .
.
.
.
.. - .
.
.
-
-
.
..
. .
...
.
.
-'- .
.. . -
-.
.
-. . .
-
.-:'.
-
.
.
. .
-
.
.
r'j. ..
..
.
.
- . . .
.q
-.
-
.
-
. ..
.
.
'
.
..
-
-
.
.
.
.
.
.
.
-
. - . -. -. - . .. . ' .
-. -: . . .-
,.
-
. . .
. -
.
.
.
.
.
'
..
...,.
.
.
.
.,..
.
.
.
-
. .
.
-
.
-
.
..
-.';ttr(j'.:!,y!ryt!-'. .
..
-
.
--
-
. -
-.
.
..
. .
,j- .
-
...
. .
..j52;:j(g'
.
-
.
.kr-
.
-
.
. . .. . -'. . .
-
.
.
. .
.
,;)r.
.
.
-
.
.
, ' t'il)t$), j.
3.20. Construct a spreadsheet for which you can input up to :ve stn'ke prices and quantities of put and call options bought or sold at those stdkes, and which will automatically construct the total expiration payoff diagram for that position. Modify the spreadsheet to permit you to choose whether to graph a payoff or proft ftlnction.
-
. . ..
. .
.'.j(k
'ta
.
. )r'--;--k)-.-)---I(irrt. t'.
.yr4k.,,........:...
.
.
.
..
.
-.
. . .
.
.
. .
.
-.-..
..
.
.-
r;'
. -
-
.. ...
' .
d. If you create a synthetic short stock with zero premium inclusive of the bid-ask spread, where will the strike pricebe relative ttj theforward price? 'ansaction
.
...
' -
.;j'
.
.
-
.
--.
.
.
. -
.
-
.
,;
..u
.
. . -
.
-j. .
.
.
.
' ' .'
.
..
--
.
.
.-'jjkk..
.
-
-
.
'
.
. .
.
.
..;r(-.
-
..
-
- . .
. . .. .
.
.. . .
.
.
-
-
.j)ppj'.
....drtqlyjyljylyj-r-.
-:-'.
- .
.
,
.
.
.
.
'
-
-
.
.
.
':
-
-
-'
.
.
-.
...
.
'..
---
' '.
'
.
'
.
.
.
-
-
'' .
-
.
-
-
-
-
.
. . .
-.
.
-
-
.
. .
..
. .
.
-
-
-
-
..
.
' .
-
-
'(.
.
-
.
.:
-.
' . ..ii.:llti,t.litrEi .
.-
. '
.
'':''i'
.
-
' '
r'
c. Suppose the options have a bid-ask spread. Ifyou ar: creating a synthetic purchased stock and the net premium is zero inclltqive of #7e bid-ask Jp?'rtz#, where will the strike price be relative to the forward price?
-
.
-t'-
-
.i..'' :.E ' ' E' )jj..:
a. What can you say about the strike plice? b. What term best describes the position you have created?
. . .
::r;2(4):k.
.. . .
.
-
';
-
'
'
'
:-'..;;fL(LL.. '
.'b'illqgilrlps-''sygyttjjjjyyygyy.s'-sirrttlprly...--:t,,;j;;.
..
.
-
using
'
. ' . -. .
E
3.19. Here is a quote from an investment Web site about an investment strategy options:
t!' -
'.
-
p. i@ . i . i; 'r.: )'-),'. lptsl'!i .; i . ;. . i i ..t.! )))-),'. ' : . .jrr'r'.'i t.)'.. ;. tijjj'.ir- )F:' rifjljtpktie's.-t?kjjiji.t,... i' r . 1.i. j i!t.!. (j'.ri.r;r j!y !;(jj 1 . i. .r.i (1 ' l. ))'.; jq-.ygdi t li.lp . :Ip:. r!L,lkp'k;j;j'r. $tt$...':(;4l$(tjkjlF4?;'.. . lir'i -)LL)LLL3):LL;;'. btk-iqsb:'. t)'))'. it!t)?)'. '. ;!;l.. !;j!.rliy-'! .j.:(. ... ri;..I.i yq'. .t33. . . k;t-'. . j'. jt! t)',LfLbt3L:'. .p. y'. . (.rty . .j'.i I 'r'. (-jr'. ;. t; ;yg. @jy'p,)!. i. )-. y .r..q'. r .;jq (jq).@@;. .; ;'.l!.yl'q rk'...(.j. ,.t.rt..i.i )r! .. . i k)'. .:. y.. ;t- . .;.!'.. I.''.. q..; '.. t'' ;. . ! . t?'j),'. ii'.,i j;LLq:tjjkLtL.. rq; . .j yl'ys .ttr )-.. ),Lk,'?LLL;)j3LL;jkk;. ..;; j,-gy:. . y)-'.l.; p! .r. j;;jjjr,'rrjjyy(j)jy'. . i ;. i .. (-.j.;Itik.q. ; ... !.i.'k! . rll . i')$,-'@ rr'. ):. ; . :'! ..j. j;'- ;' l.y'' .;ii8(''!?p1llt(I!I!!: . ..@.. . )tir.(jtj'q kt-tLLLj'. i . . k'. y,'. .. y;(.. y-. i''1:!7'.l Iti. ; j,':y.I:. .. ; .. i.!.)' jy';lr .$:j.:;j(yr'. 1-*, .;. j y. @'(. ;.l.)'. :!y( . I)'.;j..; ..pF'-.(.. ;-'L-'. '.'''.'.'..'l.!i:p'i.)h'f'jjjt)ry:y: y-')jjrjtjj.. ?'-).. . q.. .L:kLtjLL;,LLL-.'1L3'::tL)jL,).'jLL;f?LL))'. . rj j: @.. !-: i)'. ii jtL,,t,'; y-. 4t-yi!.t!t. tr.jtkt. .;yt,: y . ; . i.q.. t!.. . . r q'.)'! rtyj ..jy.j. ii.1, ;k!..i. li yy.yjjlkjyizyyyjr.jr. .:(!jj!t q, p'. j yy (:!' ;.;i t.. i yt. .-.;.i ()'. . ; r.;... ((.j:-.;. jy'j. .)y. f'li'll,kqLL.'qkqLL3LkL;.. . st.yg yyjyyjy.:y-j-..ryyjjyj.tyyjjjjjtj)jyy.-,.'yg$jjyyyy-'.yjj),,).. . . ty':ri . . rt;i. y'. .. !?'t7. ' . .ii q. . .. I r.t ;j)'.r't! . y'. y r y'. ): t pkiltr'ryllpyl'. :! . . (:'. . . . ).'t,2t'. . .. . . k)'. . ki i ii. ).. ; t,ii;.. .. ;--LLL'' ; . i;'LiLq63L'''1kLL3Lt31L'. q'pg'ps @,1 t'-'. jj.. . y. .:'. ,4,,,,tL,;jjL(;L-, ! (q. y i ?i.. . .(; iji)i);-i.'1*91-.. jyy yy.t.. g.j.--(k;'. . rqj'-tt-.(j'. t.. yj'. I ., . . . ;'. ;yyj.. )-.j-. )j( @'j y r; j j LL(L-'. r .t !.y)'. .-(-(-. !:'. . I lj'gq . .b-LLL. y;g. ; jy'. jyg r i :yj.rj;;i2(:;(;-. .jtrr.':. yyr. yt'! l . . y, j. .yty y,yy t '' . !. . L:1jL:LLL,..3.iiL(LL.-'-.bLLL'kL;'' . ii y.. .! t. .('. . r'jj)'. . r p. tj,. jLj''1L1?b'f', LjL'IL33tL.-..L3LLLbbqL . @ ) . . ! . k pk'. q5'i . yy (ljy; ( )'. t;-).j,'. . .t'. j. . . q! )! . . ri . k i p. r ; . (t(;('. ;. ! . i . . p'. q)). . tt.ql'lr't y LkLL, F'. j. k:'?;)pyr)'. t). . iy)C.. . . r! ;;Lj?--L,,j'. . jy . gr'l')-;'. i'.
[email protected],,j.. .! i . ($rg fi....)k. .i;. .. 3*. t'.; ( tpi . ;;;L;L;. yry)l . ( t! i ' kjy,. ! p.. p. !kp.;(j';jrr'yyy)):.. .j'i-. . l )2r.tg. rkr. . ;... . . r sq'-y ;t;r jj-. l Cr'. (. .(tjy': .. lg ..; (. . . r . ... )! y.q! q. . j; . ;.Lf':i ry-.()',)'. . yr'ti . y)r. j.r(k r jri. ;gr;(;. ( yj q. . p:str'li (t. j ?; ty y.)t.4q y ; p ! yjy.y.. i . . i #'yi ; ! yy q j j ))yjtjjy. 171.718. ..1 y'yyjly k))jjtt'.. (t; ( ti Li'. . y ,ijtLL;j.L''. $4),))): y, jik)7),i'!l;'.-.. j ; j ; . . . .y.. . k ( . j lj.y . ; p:'j)j k-'. (. :;'. r;. t: . r . . j g )?r;. . . . ;jrr:'y,k,'. . . g .y'-b'q.. j';,'j#'t')k y;. t'jr'Ci ti!i! (;'..'); I qi j.:jI rr q !.rjyy.y. l.lt).i. t:; k.j . . jy.y'i. .(,*1 ;. (!E. j..(; bqi,fk:,; . j: r; gjr . . j..i'f'. j rt. rt'-;yy'.. k; r'. . t!. ;ky-)'. !. (r-yj'. j.'ty'. rq.jjji )'t;! i. li11:1*. .r.ii! tt'. . . j!.j. . '''.'':.'i; il I..';II-. . . ;g'. . . ;g'. j'. . $ t. -LLL6qqL'. . . ! j)'( . j L 3;-)L'. qr.r;. ! j(ryyg-. ! . .. j y jj----. y'. j i. ( r y-.y(.t!i'-t-- t;si!pk. tlpll.yy)i y $k.jjj'. ), . ;-( (r. j:',tty': . ;'r'y, 'rk')-')-'' )'. it. jr));.-'.;p(r. r 7. t. . . i . y tyyyjk'!j. i. . -jL't,. yyj'. ( (q !.. iti(. . .. r. ). .1*- t.....l(. . tjqi. y'.t;y'. t i'. y;'. gy:'. .. .. .- ..r. .;;,)t';..ji'y k;-yyj.. fLL.. ri )!. ttpty,,. . f3LL:L.'.?3?bi;3L;.'. C-'. .; . .), .y; yy'. q. jy'j.'. k'. ;(;; . .. ;.'.. ;. ...s'. jjl ! .).y. j--! ;.(;'. .. .llr'r. p.L', q5.ji.(-tdy?kEs'. .. f'.E . I ty. .(,. . rr.l'. y)'. .r. .li. ()! )). r'..;1;';1;'. t.!;'. );;'jqjjj'. Iy'. .k'ttd-dii. )'.'. 1:.. .y)'. ...t'.'. j'.)'. tI...): ; j'. .i.)-..g .;j'. p'. )i:r.'.ji;.. y'j))j .. t.. jji yi tp'. rr . .. .': tkq.. jLj?1'. jt(:t).. r! r )j;)'. ..i('. -tq' tt(.. ; rt!rjT)...,-'(;) . jtb3$$;;;t.jbL;', . tf'. ptfs ..'.('. liq'iki'. t',pk-yi .i)(: :)j-''tqi.)jtjk)));. @'. .. . . ;;.. iji . )'. '...'...i.'! 3;;L. (7)lC;....:)i;,:!,p))pl#!-'' )j')jj'. .. .j(i'. ..'j$bL-,',. li ?. )'. . , . ;i;'tjj;''1(ri jjjjjjj:;-. .,y'. . (lli 'i .jr. ; ! l'i I yt'. r'. . ('. q . i r! !. l . ; ;'jt'. ( . qjl.tk.tj!;. . . .. ! k'. kj ; ((r'. !rr!.. ttjgjylpl rt; : .( y'. ! . jq jr j.Jr',..q)))j;. 31 ij)-:)l:. ! . ; (Ii t. ljs j).)(4p;rk)(';Ipjjj(jyq)T:.,... k'. i . ' rt'. .tk)'. yiit'!rr'- lij'y... LLLLjb;. 224jj44.. (;;;;;qri(. r(''. (r;;);;. . r)'i)'.. .).'. . . ( r;.. y.(j.: (jr);. )!!). . (y';k.trji. ))'.t ;('. ij-'jl. )'.; LLL3L-1. ((ii pjr,---t.. ( @-' . rrltlk r@j.. qk:. L..,.'j,Lkqjt . tt'.@(tl'-' @; '.r y'. iri. k!;j'. ; .:.-,-:(j(r-k:.. ii.. (. .'-.;1r. (qi-lkittj. p. :t*. ...t.. Li,-;'-L1.'. i.' (1't'-.t.' ('.)'! L''LqL. p'.l('-)tt);;. ;. t'.j'-. i. .. ( .jiik.q.j'Iii (i! .j. -LLLLLLLLL;L-L-,LLLLL.,L'. q., k r:.. (;jr;'. . rqt!. ;..fi. . L;);)LL).. :j)))t'.jjjLLj . . .k .'. k );jk;;rIT)'... v. tj/i; )k)'.Iq qi t)())'. ));'. .. )j-. .. r'. kil:2i)Ii t.kj'. . i'f.1...*:. t!)!' 1'. r)ylp. lir-#@ qi.pi i i r'-j'. i ..i t-1.. 2'. i'-l;..
-
-
-
.
-. y.
, :.
is $350/oz. If Golddiggers produces no gold, the firm loses its lixed cost, hlls hxed cost of $330/oz. and variable cost of $50/oz., produces gold, the 111111 If $330/oz. Golddiggers lt is better to lose only $30, so Golddiggers will produce and so loses $350- ($330+$50) = variable cost o $50, even when they have negative net income. If te gold price tvere to fall below the l
the gold price suppose
-$30/oz.
11.
1:111
d::)p ILCIL dz::: 1!:. 1!71k ii. 1!11; 1I:p 114:'
d:lk JZitEIPC:L 11:11721 d::zk 4k2:; 411,: d::s 1E71L 'l'.likF' 4:2), lrqlr 1!5; tEli14::11 1r: :: 1!9; 111; lkklt 1:71E:1: ii.10:
,.
'
1!!1j1111.
(
j q
'
E
E
% INTRODUCTION
92
.'.' '-'r' ''''''. .'.
BASIC RISK MANAGEM
MANAGEMENT
RIsK
'ro
;'. C' :''jj::ry:yf jy:ryy:j,f jjyygr!f !' Erillt::ird '11k:i!,')'' C' 4*. 7* yjf k)f ;;' j'yf k' j'(''''''''L !)' q' (t' 'ffr''tf L'jjjjjjjjjjjjjj)jjjjjjjjjjj... j;,,,.jj,,,,jjf 1* jjy.-.f llili)jjk;d $* yyyy);'E. ;y.)'jjt: ;;jj;jj'. ';1Iii5!i:'. :.iiIk)' .iiiii;''Ei,i:@:IiL' )' (:* ;'?r!' ))' qqf j'1* jjjyyy,j'. y' r' kif f' q' qrtf l'tiif (' jjjf 1*. @' (' jjyf ljjjjjgd : q .lIit:)'. r' r)' jf j.)' jy'i t')'f ii(ty j'-;;r,jjjjjjy'(.'.'('E'.EE. EEIE' .'.E. E... E.( ('!' i';'i TE i k 2r''E'i! .E''iE' (7E .i1Ii::' iyd j)f r5* .'?''@'?ff@@' y' y).jj)j);(rjy.' j'jjy'))jyyyjjyjyyt-. )' (ijiijk..;kjiq:',kkk;!i;;'. qr( EE q'ijt. (.2#E ; @q 1)q: (; .(;.'. (E qqf((q j(.ii . jjl . (. '.bkL. i .! i E C--yjl E :),i q.jj,,.,. . ! ; yj. E j i r!:.tiy-,jjjjj. jj:rrygjjj ...'..-....'.....r-?.).i..-..j-).).j . j j:y j. E;.E;t'-)tE.-.;-.-....-.,.......t ,j;gtjy,: jjr:yyyjj-jjrgjj!js jjyjgr!y .:::11 4:::11 yr.. . L.Lk....---L.Lbt-','-1..LL-$(2 E-..:E)t-.3 -8!@''--?Er .$jjyry y.E)j. yrjy..qj(jjry. jj(;(yyr.; .Gd:::)'11 '1tE. kijjyjjj jj,,,,yji .,j,yy. jj,,,,yjjjjy,,,jjy,,,)ji jj,.,,yjj . .:$k1t:))..:, 'Ei!t Ir''tf!;i t!;i'iii!t1!5; ii!E(2I' jyyjj,,,,, .it)r...-.-#?). lEzl
)': 1iiIi@E(!2t1ll;' kilrrii:tiijil'. ''yyyf yy'ytf y' l'y )1#* yj'jjj ;'yyjj(j'. q' jiik)d t'y'. )' ('. jjllyjtty-jjjjgrjty't;f jjljjkjjjj', p' .yii1ii;,;;i''''::ki)ik(y'. t)y.;(jjt'.' ttf .jjjii!)iij!,' . ): (. y. rt yyg tj t)t).. . jjjjjgjgrji,/pq
.'
''..' .'''. -' ,'i. .'..' -.' -'' ''.' ..,' '''IFY
-' -----' -kk)i--.' --' -.'t'. -'. ;'yyf t'f;f j.' -l:::::,,f -' --' y'. gyf (' j';' E (iI(E::I-' ;1* ''. :''Ej'. .jj'q yt'y r' j''
(1iiqii.' k,l:r.f !)'. (.' '-'. 1*1* ytf y' t''' t'yrjr q' i''q' jqf 1* j''i'! ('!'1! E (E(j' (ri ! !(EEEEliE'E 1.iiqlr /.; tlEqii)!;)rprrrsrlkq ,'t('. .11.::* :-:11111-*. 1IIl-:-:i1iE'. :Ep')' 1* ?T' ;:71*77757:7* iif F' p' (ltlr'i'tqf jtf jf (.':' i'$' ;.'E''EEE(.'(.'E'i.'!.1* ii;(. 'jq.. i.' '(EEqE ..';''i' j''jf yyjjg'. yj'.' jy'yjj'j' (' rrf t'irf .' ; ')-q..('. (''...(r.. i.j'g @ . . .ij,tttit. .q . j. j )'E ..L.. . . . t.lFELlplriltkljttiiFht'tj .Ey r . (.gygjjyy.yy;yggj y;yyyyg.;jj j
:iii (' ;'!iiiqr 7@ ( q' i')2tj;;t'-. .,' pyt)'-!'!II:::::;i)'. 1111:*. (' j'yyf yjjy'ryf jjjy'..tjyy' jtyj'y jyf.' !.' 1t*. t's-.' @' .. ;. . . tyjyty'jyyf (. t-tt--y---tjl-')p 1!!!iFFr-.'. .'. yj.. j.jj(.yyyjjj.. j.jj.jyj. . )). j j j .t i g . . tyjtg II't:iii'il. E. .;.y gjj y)!.. .E. !. ('. : i jyllrtkgrllt-kk... ! ..... . LLLbj//. ':'....,..-:-:11214-.r.;..;...:.i..;:,....r ;.4: l,rl t . . '....r-........ 1IliEEt-. .... .. . . ... . . j. ' . Prodr profit n one year, assuming hedging . with a short forward contract at a forward .
''''''''''. .jjj::' ;'t'''''''. y' ,''-' -' ;')'
E': :(:' : - ': : . .
-
.. . . . .
-
'
-
. .
.
'
:
:
!
'
: :. ..
-
.
. ---..-,')'..,----..-..-.......-...-...f.....'.''-.'....i........ E. .. .
E
' --.
'
:
.
.
'
:
.
- - . . -
.
.
!:
.ttt.pilri.
'
::
:
i :. -
-.
. .
-j,.
.
. .
t..-'!!;-y.i--.!..-...i)'t
;. E
tkkiL:,I;k. E' .. ' .
'
: : - q .. - . ... - .. . ' '' '' ' . S'.
.
... .
.jilrk-.
E - E'
E E.. ' :. j. . E:...' . ... . . . .
'. . .: '. . . .:
''''''''.
'
.
.
.:.E.' .
.
..
.
$350 $400 $450
''''''i . . ... . .. g: :
( E. : .
E-
.. .. . -.'
. .. . . ..
. .
.. . : .
: .. .
. :E .. ' :
-
.
. . - . ... ...:.... . . .. ..
E . .. . '
. y.
:
:
:
.. : . . . . . .
. . . ...... . -
.. .. ... .
.........
. -
E : i'. ' ' . . ... .
.
..
.
..
.
.
.
.
.
'
:
.
'.. :' ... . ..
:
( .:... :.: q: E. :.. . . g-:.iiij.i. .. . .-. r ' - . -. - . -..:..:.. . . .....
. . .
). ... -. . . .
. .... ..
'
. --
-
.: . g. . . . . -, ' '- ' -. - E- .. . . q-
-$330
$500
,kkkk!!,)'
41k:jijii().. ' ' E E.
-
. .. .. ... '' : E. ' . : '.E . E :.. .E g... .. . . .. - '. . . . -. . . -
. .
-. .
. .
.
. . .
:...
..
-$50
-$330
-$50
-$330
-$50
.
.
.
q. . . .. . ---
-
. .
-
.. . . .
.
-$50
-$330
.
.
.
i E- EE ; .. ! E .
y
-
..
-
. . . ' : :: E: : ... . y .
-
.
-
-
.
.
' -. .. ..
.
-
. g..
:
. . . . .
))
.
E y!
.
.
E : .. . .
'' -. . - -- . . . . . .. .......... . .. .
.
-$30
.
.
-
-
.
E i' g . .. .. E. - -. (:...: ...E....: ....... E.. . . E .
E
-
.. .......
.
.
...
$20 $70 $120
'
'
.
itrll.
::ij)jIl;.. tkIjjj!;.: i l i yijij tkiE:!ij. )j1I(. ;( .rI11E!iiikk! (....j('. (1k1qjk:.. .jIi(. . y;..(..' i..(.q(r..'...... j...(;..i..j....(.(ij..'''''''.( ..y.y'...'.j:g(....)y...y(..''ry.q'...!'........I.(. E..ri l.. E.. . (j1i(:jikk;i:jr. i.. . ... '''''''''. '''''''. k;))i::. ( j.. j j y. E.( .'. . j j'.... (.. '. ..((.g(. j j. .. j.j j. j... .j. .. .(.y jk ' .'..''((.. E.kjk:)ijiitl j . ( . '. . j j j. y y j .. j j . y . ' j ' y ( . . y y . . '. k ' . ': . y. y . y . . . t)II1E))IIky:ii1L.. E. i. . .' E . . .E. ... . -kkI:E:,k,-' . E .. . '' ''.'k'i!ji.i!'';' . ... ..' E-.. .q..'.. . -11Iiii!i.(11k:q1111:1@111.. ' -iI:E@)1,.. l '(1II.(III:qiIi...-'. '' i .E '. : ;.' ' '-kI:E:i,k-. )' !....l1Ii::qjE(ttr!. E ' ; i.E.. .qE..q.!;.'. l' 1 .r....r'. .i '.(..(.,..1III:EEq(2::!!( .. .',r .. (r.. r(.'r(. '....)...'y...((.....r'...!;..(-g.....(...r;. .Ii;:Ir).. . . . ' y .y .. ' ' .. . ry. . . .y i-.
'
!E
. .. jjy,,,;jy,,,yjj .. .
jjrjyyyyjp
:.
EE.
.
..
' ' ' ...
tE ' .
:
:
.
. -.
.
:
.
'.
. .. . ..
'
: E :' . .:
' '
:
:
:: .
,
:
:
' .
:
.
...
.
.
. . .. .
....
' '
:: .
- . ..
.
-. .
.
.
-
E' :
. .
i
---
:
.
.
:
:
:
y
-
y
.
-. .
y
(.
-.
:
.
-. .
.. . .
y k. :
y.-.
---
;ti5i;. dl:E (:),-
-
.
. .
x()
..+-
...-
150
Price Of 5420/oz.
$40 '
j
O
E E E
-150
l
250
300
.
.
Golddiggerp agrej to t yt a e sold today for $420/oz. jckldyroductonin 1 year. we will assume in atl exiilllj that th toixkard coneact its nanially. fonvaid is the si with phyjical noted earlier, the payoff : .
. . . .
.
.
. .
.
.
.
.
.
.
.
As jetlles sttlefnent. finapcial
to a
or
di calculations when Golddiggers is hedged are summarized ttaTable 4 2 This from Tabl . 4.1. Figttre 4.1 the profit on the fonvai d conact to net income adds table . .' . . . the showing following: three contains curves .
' Eultedged
'
'
.
'
.'' ;. ''.'..' -j;,kytj--'-' .:' .;' ;'';' :''-,'
! '' ! E: . :..
'qjj
'
i'
. :
(.
-
: :
- .
. ..
;.
..- .
-. . . .
.:
.7.
'
: :
.
-
.
.
-
-
.
-
: .
:
.
.
!
. . .. .
..:
1
. .
.. . .
. .. .
. .....
. . .. .
.
- .. .
.
...
:. .
..
.
ijltyt kttgt t r t y I yyri (y riltr 2% t'Vriiy
.
E
p $420
::
1
l
350 400 450 Gold Price in 1 Year
l
1
500
550
($)
'Eunhedged
EEshort
1. Added together vertically, the two graphs wi.ll have a slope of 0, so CWCSQOII iS ZC lvigllt 0f VIIC 0111Y linc. A Pfofit Calclllation at a Single point tells us tat it must be at $40/oz. SlOPe Of
-
ZC
..
-
- --
.
,
.....
.
'(q(
E
Insurance: Guaranteeing a Minimllm Price with a Put Option
-.' -;' -y' -' ..' --'
p'. yjf )' )$'-jtjj))'. j'try)@---'. t!ll!;f t'LjLlLL'j' jjiiykj,j'. ,'jjb;)kijLL;. )y'. )j.' ql't tjf t'g' ttf 'j((()'( r' rrfjt't:.j)y yj . .( ) ..72111:qr1:1!1)1t-.jyyyyjgy,yy-y---.. r j ()yy. .,jt)y tttlt) t;j...jyrly . t
.
'.
E
thedged
G
:'?' j,:yy::,,f -t' !' ff' yj)'Et' r' ;''''l:::::'''. il.'i'' ;',' :'kk,-bq' 4,:2:::4,* 4,:2::2,1* !)r' L' dl:::::.,f Il::::::f :.' ,j)'. ('. !' q;' F' q' t''(' j;' ,'('))i,,$,:jijjj;.'. kqf y' rrf (' Il....,l...'l.f (it.qf )1* (''l j'trd )cf i'qqf ' 1)* ik/tf 7* jj;f )' /;' q' t:d tld 1(* j;f !!qgggkjj' ,!((qgj,' t'jjgy'-jyjyg r' r)' tl)d tqf llr...!.:f rji-j---yyiyyy----kytt-tytk.,f j'.yr y'gy;r,jjjj;-rt,:t', ryf .!I--' k:!,h,--,.f iiif jl:kf rqf j't(' E IE Ci E E i'' ' E' E E..!(E. i.. EE.EE.(E((:pqq (ig E i E ii !('' . EE.. (!';! '( E Eqi iI ('' i 8* i 11 i i ( E.pi . . . t! . . (.. (. :;: E. . ty.r . ' ! .-;r ). .,-.tg, .-- .-.y : -.. -.-. ),-. -tjjjjjji!r:g .,-t- . r ;)!. - jjjjijjj(,. j jjjEjjjjk. y;j-...-yr.,ygqy.yr;.- (-y(j.;g.yy; . .. . . .EE. y y . -. y j , E ; E iEi; : .i ; ' . . ' .. ( ): . )):;:f..(i))kj)lI.-i. ... ' ;. ..... . . ' . .E(!i;((
i j
seller'' is the sum of te other two lines, @Hedged proht: The line labeled adding them vertically at every gold price. lt is flat at $40/oz.,as we wouldexpect fromrfable 4.2. Aquick way to add the lines togetheris to notice that the cold forward'' rrah has a seller'' graph has a positive slope of 1. and the
seller'' shows o Unhedged profit: Since costis $380/oz.,the line labeled lligher For exnmple, prices. and prot at zeto prost at $380,a loss at lower prices, ground, gold the Golddiggers has a long in at $4209profit is $40/oz. Since it has position in gold. ,* Prtt on the short fomrard position: The sh ort g old fomrd'' tte rpresents the profit from going short the gold forward contact at a fonvard pticeof $420/oz. :
I
.
We profit from locking in the price if prices are lower than $420 and we lose if prices are higher.
.
Pi'O
$380ii
.
i E !
...g.
-100
Hedging with a Fo-ard Contract t; olddiggis can lockin apriceforgoldin 1 yearby enteringinto ashortforFard contract, gtd for delivel'y in year. Suppose tht gpld t b delikred in ageeing tcy to sell its sell fofwfd al1 of and that
'
E
'
100 50
.
.. . . .
Unhedged seller shot.tgold fonprd Hedged seller
.+.
-50
. . ..
$
PERSPECTIVE
7; E E... E . i
r y qr.
.. . .
..' .'
:'
.. .
.
.
'.
-ji
. .
'
:
'
' :
'
.
.
.
-
'.
ENT: TH E PRODUCER'S
-
.
-
. . . ..
. .
.
: . ...
.
-
---.
jj::gyggg;;j
jl-,,.k:,!i;z Il,...:ll .11,2:. -j:E!!,,. l,....:lj -I,iE,,-
W ith
a forward sale of gold.
:
.:,j;k
y.q )E
y
'
-1::::2111 .1:::::11. -1:::::11( 119...117...)11 I,..'.:ll -1EE,,a Il'...Il'... .11t::.aL:,,b--''.-'-. e e
e.
-l:::::,,.
yjjyyjyytkyy jyjjjjyjj
ryii Jyyjtytiytyjyjt 2 yliy lktktty) tyqyqyrj(IC#y jt,o; o (yj oisy ry s zu u yqyyy zt-,o ys rosluoi'j l'bkt C/) Ci Hugeu yht mrt! @ trltyyttq tytiy gyiElyljjyyqjijryyygyjyypyjiyjylyylyyjgj#ygyglj
iltitki (
-l:iii,,.
'.:::::,,-
;q
..u..s
go
yj
yEuu
yyq
y(qyyrq!
t(E
,su
y
qy
$70 $20
-$50
$350
-$330
$400 $450 $500
-$330
-$50
-$330
-$50
-$30
-$330
-$50
-$80
qi
$40 $40 $40 $40
kytiylqlyylyyqyryyyyryyEiyyryy
j
yl
A possible objec:on to hedging with a forward contract is that if go'ld prices do rise, Golddiggers will still rceive only $420/oz',there is no prospect for greater profit. Gold instlrance-iae., a put option-provides a way to have higher profts at high gold prices while still being protected against low prices. Suppose that the market price for a 420strike put is $8.77/0z.2This put provides jloor on the price. Since the ppt premium is paid 1 year plior to the option payoff, we must tale into account interest cost when we compute prost in 1 year. 'I'he future value of the premium
z'rhisuses the Black-scholes formula for the c. 5.5%. J 4.879% and t 1 tyearl. =
=
=
put price with inputs S
=
420, K
=
420, r
=
4.879%,
93
% INTRODUCTION
94
BASIC RISK MANAGEMENT:
RISK MANAGEMENT
To
is $8.77 x 1.05 $9.21. As with the forward contract, we assume financial settlement, although physical settlement would yield the same net income. Table 4.3 shows te result of buying tlzis put. H te price is less than $420, the tlie put. This is put exercised and Golddiggers sells gold for $420/oz.less the cost of Golddiggers sells gold at price is the $420, incom of If net tan greater gives $30.79. market price. the the put-performs better than shorting the forThe insurance strategy-bpying than $429.21. Otherwise the short forward gold of 1 is if price in the more ward year outperforms insurance. Figure 4.2 shows the unhedged position, profit from the put by itself, and the result of hedging with the put. :
'
.
. .
....r.
..IE
' . ..
.
.''' ' -
....
--.
'' .
'
:' ' .
-. . -
..
;
'
..
-
.
:
:
:
:
:
S(
'
' '
:
.
.
. .
: :
-.. -.--,
. . .. . -.
-y
-
-
.
.,..;.....
......
. ...
......
...
.....
. ....
...
-.........
'' :
:
:
,,k,. -..
-
,
.
.. ,.
,,
.
-
- -. . .
.
-
--
:
..
'
'
'
E
:
!
-
.!,s;i
-
-
-q , ..
.tiitskfty, tt
yElqjEgy/itryriryy
tttgEl ( r)( q kt $350 $400
y
.g
yy
! yytkyo soy .yj
tt
cjtt
y
t
put option.
yi;
ct
(EtE#ilErtE
-$330
-$50
-$330
-$50
E
qyyyyq
tq
-$50
yE
yty
tppuonl
)
E!t
Insuring by Selling a Call
qyr
zrutLjLit ulcumvylyq
yyo.yyyyyyjyyyyjyjuu
yEq(E,rut
qu(
lyyty
With tlze sale of a call, Golddiggers receives a prernium, which reduces losses, but the written call limits possible frots. One can debte whether tlzis really constimtes insttrance, but our goal is to see how the sale of a call affects the potential prost and loss
$30.79 $30.79 $60.79 $110.79
$60.79 $10.79 -$9.21
-$9.21
-$50
-$330
e
r( trgyli ktytly); Willi )(r,;y y oryyxoy u
! yyy qyy ty y..us s j (
-$330
$450 $500
...111
---.
-
.
. .
:
.
- --
95
'
..1F.'
'''
'' :'
::
$
.
..';j '
y' jyt'-, jjl#'. 'j('. yy')' j'jjjjjjj;,'. )))q')' ;'tyf lif (' j(::kj!jjj;k'. )', t'iliitiii)f qqiiiiii77rr'-.'. ))'r(-ll-. y' ttyggt-gtkjyygyjyyyj'tjyy. ()' 1* yy'. )iiiiiiiiii;''' tjlyky)yf j)l j' ltjjjjjljjq liCt (( p )jy ; tytjyytkttjjtygi tl yr j. j y.)yjj.i ' ' ''' r.. .-. .j yj yjjyyyyy. .ytyjt; ;r. ).., wiyh a 420-strike . '
PERSPECTIVE
willbe in differentregions; thatis, how likely is it thatthe goldprice will exceed$429.21? The plice of the put option implicitly contains information about te likelihood that the gold prie will exceed $420,and by how much. The probability distribution of the gold price is a key factor determining the pricing of the put. We will see in later chapters how the distzibution affects the put price arld how to use information about te probability ' distribution to help us assess lisk. Figure 4.3 compares the profit from the two protectiv sategies we have exnmined: Selling a fonvard contract and buying a put. As you w. uld expect, neither strategy is cleady preferable', rather, there are trade-offs, with each contract outpedbrrning the other for some range of prices. The fact that no hedging sategy always outperforms the other will be true of all fairly priced strategies. J.npractice, considerations such as transaction costs and market views are likely to govern the choice of a sategy.
-(' dl:::::,,f jj::yy:,,f 77(7:!* -j'. Ey'. :':' t')' )' (' :.' g' tijyl-'. y' Il::::::f .1::::211* r'tl'l)f .4::::111* :,!s;;' 7* )' y' q' ,';' (' grkky.f rf l,r..':,:f .lr...f 1':* 11* )' ,gj' (q' Tl' ljgllggjjf jj;f 1* r',r !!((1I' jjf ;')' r' .-' '';''q'@' E(E EE' E i i :'- ytE r'jE( E'' ' i'('(! i I ;:(:'EE'.'q' .'.'i'i' ;.EE.i ('i'(EE'i 1;iriE@. j(iE(i' EEE'.. E'. ' i.(. 'E('r('('E;' 7( 77:74771:7q14r5 ' ((: ()- jjjj .j y .- - y E ' )(.. . - ,- !qq.,(.. . ! .' E j, - . -1:::::111 41:::2:,,. !!g(g(gjj.. g . .. j '.:::::,,ly j;.. (j . gj jy)yj-. d.:iis,,. jj:,,jj.jyjy.j.r.ky ,.. h) )f'' . tjIkt2,:.. y )j.ll::E::,,llr...llr...:.l dlr'.. ll'...'lr...ll. .tIt::. .y;yjjy)yy(j!( jj::yrryy;;k41::22,,,. iii .,It::.a(:!.,----.---. l,....l.i l. k(' Ii'...:ll p,r,.. Il:E,!r e e a .EI::' k:!,,--g-. ):.. ). . . y...,. .y- ... .. yy..qy.y..r.-..j r-..-. .,qy(. .. -jj)))-y'. -y'yf .-.j,4iiiikk2'''';'
PRODUCER'S
What this analysis does not address is the probability tat the gold price in 1 year
=
.
THE
for Golddiggers. Suppose that instead of buying a put, Golddiggers sells a 420-strike call and receives an $8.77premium. Golddiggers in this case would be said to have sold a cap.
jjyygg;jyd y.'j gtr'j j';jyy' (' 'j' y' t;j.j' j@!y.'j(jj)))y...yqjjjjjjj$))y'.?!(jjjjj)))'. ;'(' yrd ;rg' J' F' j'tj'yjy,. jygjd 1* ((' 14g* (' )' ll't'f jF'..7i 7* 'jj'ktd Jj)' .jjgyy' j'k' y' ay'j d4)* jgyj'yjtldizr/:'trr'l ,jj1):*1181,: (55)17;;* ((r7' if 1LLiq/;.11qiq21/'L1k'LL33k6/'Lj1f(L'''''13L'. (rff !' :f LLI'j6LL'LiL'k'q/?:'iLji'. lt'tti )' t'jjd (' y' rq;;:;rrrrq'r:r:!!?rrrr!q;srrri7qqql:sqr:!:q!f ll'frq (E !! (71 E !(E: 'fE1:(5 'k.' ilj' : i'Erii..,.7. Ei '(E ; '. :'.;15 E)Ji! 'iEl!' ' 1. E . . .. . p-. li i i L.)t E . EE i ! ' E; );t.E k! 4))4t924/!11):j! i. '. ( g . ;j)r ))!;! E ' .. )?plr llgjl';;i' . y! (E;!.iyg..i...!iE'E(.yyyjy '('.yj;tj . . 1IIiE!-Et':'r . . . y))r F:EEIII/E yr y y. yj. yj j.gjg II!:;I:L 'iii:lll-.t-k;/ trtlktrEi..j;.
t': t'kjjj(t)yttt)';' rjd t'g td ll'i'j. '),',' ' Ljjjj. kj(yrjjj).k-t'.rli.ty' it1.yygi--f)y.t111k,211. L.. Ii1q!h). --k)y,IIl-j.)41I::;;/(.;-)
.''.'
.
-' 'y'
')f )' jjf j' ''(' ,'(' y' q.---,('. )' )q'j(k(p'(jqj(j;((;)j)'jrrg'j(: Fff r' ,';t('. jjf t'(''j' q'. '11ik2--1iii'.11k::* q.-' .';l k:kti'f ij E ij(j.(( (j ' (' ;(! j l j... J(!(''y ; (: !. ij j. !.j.)tj ).'!j' 11111*.. kE'1iII::::;;k:'.-' (IIiqqjii.' r' LLLiL(LjLL'. qL:3i'33'. lf . . (:),' .j J(lj)(l! . ).:.y . ' . ' '. . . ( t'j';'gyf l.'(t..j.,j);.)).jk..L.L1;L,,L;).qi)jLL)Li13),LL.L. liiiiii: II1IEtlli:.. E. :ti15;k. (.:r. '. ). )-. ;. : (,1:F::,..: yj.. q-.. ( ) . t. .... . .. . t- (t. .I:I. .J(q . ...: . .. . . .-.' ,'. .;;'. -' .-' ;':' --' -.:z:r:,,f -,;r)f -.-;@-:-.'. -' -.' ''t')' S' -' f' -'. y'
'
.'.''. E j'
;
:
..@. .
i
E
:
,
'!: 7:E : rqE E ! i..IE ' E'' E'' i: E ' ' :. ' E ' E'' ': :EE : : . . ., . r...., .... : . . ' ... . . . t.t.t.)E.;..g'.(ljyy.j()E.gyg.!j(.,(y.;.,;,;..E.j.,ij,y.,.y....y..yq(..E.yj.....;E,,j..q( . .. '.Ey;.,'iL.E.yjljt:(jj(y.)ytyyy.)jjj'.tt.#. . . .E.,,E,.EEi,)..yyyj..y.ttk,,.'(''i#;;.;)' ..
' ' '' : !E:: E E E' . !.. ' .E ..E, . 'j ..
i
.. ,..(....).
.
-..
.
.
-.-
-
.
-
.
'
:
' :
'
'
. . .
5r
.
:
.
.
...
..
:
.
: . .
..
- -. .. . -,...;........, ......... '' .
:'
'
'
.' .
.. ..
-
-
:
.-, ::
.
,.
Comparison of
..
-
.
-
:
,
.
.
.
. -(! E-. ..
unhedged position, 420-strike put option, and unhedged position plus 420-strike put.
-
.
-r.. .
....
E .
.
-1-
200
..+-
....
15j
E j i E
-100 4(4g)4 kjjjk (jj..
250
1. ..............
300
.
-.-,''
:;
........,q..k.k
s,,
.
,. ...u.
..,c.
;k.;..
. .. ;@.;.u
i.
,j;.
:
. .
.,,ji;.p..yy?;kjj#ij))('.''..''...
.
;k:::
'
:y. ;;kjjk;:k,!i
,....':)j)!y.
...:ru,.g.;:..;
..
.
'
-
.
-
r
.y)j.y;
-y..
.
. j .yg y
y.k).: :;.y.js,
tjjgjytjj,s $2j4y4,, .yjrgy. jjjjy:;jj jgjj-s
200
+
senerhedged sellerhedged
M:.IA forward with put
k
150 100 50 0
'
-50
;
-
. -
for Golddiggers hedged with a forward contract and hedged with a put option.
$30.79
()
.
'
-
.
; : : ..
.
''.''
'y
Comparison of payoffs
Unhedted seller purchased put Hedged seller
100 50
j.
-.
i
.).tfyl
$380 iE
1L ..........................1L.. .........'................... .. ....... .
' i ! E E
$30.79
-100
5420 11
450 400 350 Gold Price in l.Year
I
500
($)
1.
550
-150
. : : : : : : : ::
-50
.:f
::
: $420
: :
300
450 350 400 Gold Price in 1 Year ($)
500
%
96
INTRooucTlox
To
Rlsx
;' Et,jj'. --kjj;;'--' -----jy' -' --' t'
.'.' '
-
.iiiE.i:' : ' ' E
:
,.
.
..;kE...4i.:..s. -
: :
.
y
.
'
.
.
. .
:.
.
-
:
:
.
.
-
-
!
:
.
.
.
'
:.
.
'
. :..
..
.....
' :
: .
'
E : .
'
tykjjrjjjjj..y t.yj y.y . .g ;,;jrjjjjjj,,, .
:
.
i
:
-
.
.y :
-
.
.
...................
,...
:' '
.
.
. .-... - .-.. -.. . .-., . -.---.. ... . .
tt.
-
-
-
.. -( - -
. .
.
-
y y
.
-
.
-
.
jtis
Golddiggers hedging with sale of 420-strike call versus unhedged.
.
-- -
. .
.+.
200
.+.
...
unlwdgedseller written call sellerhedged with
@ Sell some of the gain. Both of these sategies reduce the asymmetry between gains and losses, nd hence lower the cost of insurance. The first strategy, lowring the strike price, permits some additional loss while the second, selling some of tlie gain, puts a cap on the potential
written call
100
-100
!
250
.
.
. .
..
=
! E E i E E E$420
..
.
'l'he
=
..
I
350 400 450 Gold Price in 1 Year
300
I
I
500
550
The manager's view of the market and willingness to absorb risk will undoubtedly in:uence te choice among tese alternatives. Managers optimistic about the plice of gold will opt for low-strike-plice puts, whereas pessimistic manager; will more likely choose high-strike puts. While corporations per se may not be risk-averse, managers public's may b. Also, some managers may perceive losses to be costly in terms of the of perception boss's them. of f11-11: perception the or the This problem of choosing the appropriate strike price is not unique to corporate risk management. Safe drivers and more careful homeowners often reduce premiums
($) '
.
.
'
.
.
.
Figure 4.4 shows the payoff to this strategy. J.f we compute the acmal profit 1 year from today, we see tht if te golcl price in 1 year exceeds $420,Golddiggers will show Prots of
$420 + $9.21
-
$380
=
$49.21
.'.'..' .... . .----.-.@-' .''''. ' -..kki-'...'.
rr)'. ' i' i i
.)@
.
$9.21 $380
.
-' -;-1* .-t''. ' '5* '-)' -l:::::.-f -' -'' C'
t''(' ,'1111.---2111*. ;'qtd qrd 'qqd /)' (' y':)d f' t'1* (' T(' j(' ((F' t-'i''1::i:;;;i.--.' q' :k:;'. jlrd j'7* (' 1* q' )p'.f)q'ii(!r(!rrpj((;!q((p' 21(1* t'1)* !' ;'r' :!i1;;' .tIt::' i')')'(k' ?' LL--.-kfk.'.k.-kt-.LiL'. i;'. C'..k,-:)t,t2'.. jt'. ----'11iIF:pq'. ..' k' )' 11111*. (:;,''
'rhat is, Golddiggers sells gold for $420 (sincethe written call is exercised by the holder), receives the ftlttlre value of the premium, and has a cost of $380. If t.1+plice t gold is less than $420,Golddiggers will make Pgold+
'
.
Reducing the strike price lowers the amount of insurance; therefore the put option will have a lower premium. Figure 4.5 compares prost diagrams for Golddiggers's hedging using put options with strikes of $400 (premium $2.21), $420(premium 4oo-stlike, low-premium option yields = $8.77),and $440 (premium $21.54). lowest profit if needed the highest proft if insurance is not ttllepriee is high) and theoptipn yields the Oo-strike, high-premium is low). price The insuratlce is needed (the neded. profit is insurance and needed, the if highest lowest protit if insurance is not
$9.21
O -50
g ain
$49.21
50
(
97
* Reduce the insured amount by lowering the strik price of the put option.
jjryyy;jj grjjy;;s yjjty:,,j jgjj-
-
150
'
$
PERSPECTIVE
There are at least two ways to reduce the cost of insurance;
lyyjj.ikE't).tf ;'E jjry;f E' ':' ?' C' ')' ('. ,j-jjjjj(,jjjjy'. (' ''j' 1* ryf F' ';' lf EE E' iEC;!fE fj';E E gjjyy,f yjjgyyjjj.f E;' PIi.EqiiqT'. j':'jjjjyj;jjrj.'. f' ,kt' ytjrj-pt.-j;rttf )' 17* Eq'E!'E.(; E j,jjyjjjgy'jjf .jjgyy' .)' jr,'.;jj. 74E* @' t';j'f')'rtf )' (' i''q' tif ((' (lqii:i;llqf jrslf $J' q@ E ()'( ('E' (;7'7:777: ' . CE(.E(((.(E(;'7* r' 1. qip'E!qlF: ((;EE..i E.... qE i;E(;'(@( i E ;!' iliE i'.'@ . !' (l ri sjyf j;'. ;')j' jj'yj yyj'-. y' r.;tf r)'t?!f j')r yyf ('. 4141,* y'. y'y' jtl'tf r' j'y','py-jf 'jq'l).rq . . .'. q' . 'l .. . ' . ... ..E). . ; E. ! ;;; . E ! (. (:E (E;(E. (.E; .E.l:EikE.y E: ( !; j(i...E.)E I E : j (E1. .j.jy. ..!!:j;:::;:;j,. . yy(y ( . .jlji!!!:k ; j . . y (y, ; yi;j; . kj)jy. , fli .. ! i,......a.i-...:..........r... y yjy y.. yj j ; ., lj y;j jyj . yq j j .y.y yy ,( y . .!rj: y . . tyjjy--. y q y g y l-'qi.tq. . y yyyj , jg .yyy . . ..g. . ... . ... . jjj (jj C
)' lll;tf)... tyf t)f ty)y)y'.. t'pjf (j
ENT: TH E PRODUCER'S
BASIC RISK MANAGEM
MANAGEMENT
-
. ..
.
.
.
E
.
.
.
.
On the downside, Golddiggers has exposure to gold but keeps the option premium. By writing the call, Golddiggers keeps the $8.77 call premium and 1 year latef makes $9.21 more than an unhedged gold seller. On the other hand, if the gold price exceeds $420,the call is exercised and te plice Golddiggers receives is thus capped at $420. Thus, for gold plices above $429.21,an unhedged sategy has a higher payoff than that of writing a 420-strike call. Also, for prices below $410.79,being f'ully hedged is preferable to having sold the call.
Adjusting the Amount of lnsurance Consider again Golddiggers's strategy of obtaining insurance against a price decline by purchasinga put option. A common objection to the purchase of insurance is that it is expensive.Insurance has a prernium because it eliminates the risk of a large loss, while allowinga prost if prices increase. The cost of insurance reqects this asymmetry.
' ' : ! E' : qiE 777:q' E(E EE E.E7:EE:' :.'.E:''E''' 'iE' :(7'7q: !'':i E 7)17 .: ).@EE ; E. E( !!: :. E. E '.. . E ' E . I. . E)y4ktt();;@ip(1g;.t:jrk)p)!t,jE( , lttiytjyyyyyyjjiryy.Ejjjtsgqy,ljjsqgEyjygrEyjjyyjjlEyyjjyjy'yqsjsjlqEj,Ejjys,ijj ',lEjt EtEyijjtjyjyjjk,ElE !)E;E;),gjt)yy(#;(ig,:,jj,jqy.
8;1.( (. ( ( ' ! iqt'i;'i. tii ' (.' (. ( ), j
'' ' ..
ti . yttlk 7pi
' . .r. @11...'. ..
. -
...
i. .
.
.-..
- .. .. .
..
!. : -.. . . ... .
.
' '
.
.
.
'
t ilii!!!E:: 1111F:1111:..
..
. ..-....
...,
.....,..;.
-.
.
- -. . .
. . . . . . ... .
.
-. . ---,-
'i
E
rrl'l'!:!?)F!qqsl!q:r!q 1. ;' ,
(..r .t-. )..ti t;)k-;i -.
-.
-.-.. -. --. . .. .. ...
..
-
.
. .
.
-
..
.-i -; . i- -- .
Comparison of profit for Golddiggers using three different put strikes.
.
11::( (11(E::,r :1Iii-(ii..
160 l4o
-1-
.+.
w.
Hedged wit.h 420-strike put wjth zjlo-sjrrike put uedgedO:.II 440-strike put Hedged
'
120 100 80 60 40 20 jgj)j
250
I
300
I
1
.
I . ................ .......
I
400 450 350 Gold Price in 1 Year
500
($)
1 ............
550
98
k
INTRODUCTION
BASIC RISK MANAGEM
TO RISK MANAGEMENT ;';' !' y' 'Fr'; (' 59* )' (C' F' ?' qf (f ))' ltf li :E t'i'')#' (' . it?tljlji ''
:
' 'i E (E.'EEE:.! . !E ' ;!Ei'E '
:
:
:
'
' EiE ;
'
. ,:
.
-..
..
.'..'.'''-''-El-X''.''.',-..-. *17,.-$. . . today. ..-.. ..
:-l)--)-)). i.t).-.-.
-jjr,kkj
....;ij...;
'.
..
ae
,::rr,
..............,....
e
,:::l
e
e
,'.
-,.
,.11 . .
il:'
dr::)p lr..,r..,L lr,. lr,' (!j,/-. e tE:,l.
;y 4#2ijiyjjitiqjyjyyy yk jj lqyj tlttttlt yyy tztwkxyquujxoyyyyrj -w--sosotoyrt yyjj pltt ytt jjy ylrvy oitkujyy Rkenu yryyyjtgyjjylyytyyuqj-jyyyyyyy
;;
y jlEj:ryu;
l,''.,ti' ..
..
.
,,
yng
uyjy
.,..,..;i,.,r
.
.
,
E
,
,
jjjyttEjjjjytj
yj
yj gyy
.
y
$400 $450 $500
$800 $800 $800
;
yjygyjyyjyyyylyyjyiyrjEytjy
tj
tyyjy
$340 $340 $340
yyyjyy
$60 $10
$0 $0 $0
-$40
PERSPECTIVE
Auric Enterplises is a manufacturer of widgets, a product that uses gold as an input. We will suppose for' simplicity that th jrie of old is the only uncertainty Auric faces. In P articularwe assume that ,
* Aulic sells each widget for a fixed price of $800,a price lnown in advance. * The :xed cost per widget is $340. * The manufacture of each widget requires 1 oz. of gold as an input. @The nongold vadable cost per widget is zero. * The
'
:
.
.
....r
.' ,''--)-''.'i'..'...''-'.'.'.
4.2 BASIC RISK MANAGEMENT: BUYER'S
99
' E( .' .E iEE :i.' E : . . EE ....: E . (E . .
FilFE'i'E''E''' E' :.E''.:.:('i'' 'E'EE'( . .. . . .
.
$
BUYERYPERSPECTIVE
'!r': j' i. ' '!.(( (. ;' i;( (F' ';1. q''q.lq.?' @' . ('.ii)-C(. . !-)y.j.j.(:qqyy..y..-y.-.)y-qy.-.yj.g..;-y--y-yryyq.!yyt(.)tyq . . .r.(,(jj;.-.-ytyj$;jjjjj.rk-. )(. tt-)t'?rt).t))l)'-'''(q)t1' . '.1ti.(!t'.?-l7.)1'':.i!1iiiii' t.ytyt;j)yr)yjjt)y;gjyjyj.,?)..jy. tgyryq7r7q77-y;yjj.. i.. r.-.. y-yy jry-tj..)..t;;jjjjky-.d:::il 4:::j1 ,1:: j)k)))j;k2r... jj). L:):. y):---t)y. ... iklr..,r..,k jj ,kE,, dE!!,r lr..,r..,l Ik..,i lr..,k ,r..1l iiIr.., d:::,p ,k.,j; ,r,. ,jE; .(t-.j..).).-.).....-.).g..7 l''. I..,.d:::.lt(((Il .Ir:. .1r:. iii21''-. . i:..r.)..--..................--..---..-.-...)-k-.''.
by purchasing auto and homeowner's insurance with larger deductibles. This re:ects their proprietary view of the likelihood that the insurance will be used. One important difference between gold insurance and property insurance, however. is that poor drivers would lile smaller deductibles for their auto insurance; this differential demand by the quality of the isured is called adverse selection and is re:ected in te premipms for different deductibles. A driver known to be good would face a lower premium ?orany deductible t11a11 a driver known to be bad. With gold, however, the price of the put is of who is doing te buying. a independent
THE
ENT: TH E
quantity of widgets
to be lold is lnown in advance.
BecauseAuric makes a greaterprot if the price of gold falls, Auric's gold position lisk-management is implicitly short. As with Golddiggers, we will examine vadous strategies for Auric. The pro forma net income calculation for Auric is shown in Table 4.4.
(!,::.f E-' ;;' )' ').)' (' y' yj'lt';yd k)' J!;X f' q' (lf 1* (#1* E5!' t'2* '.': j'jjjyjjtjjtjtgyj.d j('. tl'l p' ';)' ;'?)' (r'i)' ?p' ;(' qqirf jjf ilf )i' l'itf p;;l' (' i:!ttf @?' j;': )r,' ))-'. i.f ;r':i!! (t(ky;'.' )' jj'jjd illlq:,'f ;1I7tiE.' j3iti$L6LL'---. 3-.k#(t'. ky't)y' pE;';' )'. p;' pIiIkt-:ilIr'. j'y' )jy'. j'yjyrrjjjjjjy'j.' tjd 11iIF:!?..' (' '-' (('. (y)' tjjl'py 'g' ltI$'--' !' t'r' tgy.jyd (:;! !qt) .(. (' '.i'.(i' :. ' .')ii.. ' ((!( -' --' -'. 7* -' -l:::::.,f y' (' qyj(y'. jjjjjgjjj;j''. !j' ;'(jd ''?f i.'E )' :.'(i ijd y'I :'tf r' (' I'L'LLI'L ;qlf ;'j'(E'
.'.'' lr! . -''j;yytj' .'-.' '
.'i ,. ---..iIII!.Illi!i!!zl)l. tyjtj tjyty).tj' ''. tti
' i E'E.E. E( (. (li!i i' ))l lt. .'(rq:qrrqql'7rr!?i . 5 . .. !.!tipr . . E Ei ! . ;.i jyyyj '.E jy. i j j jyj' j.j ;jjy j lly:.)tj .jj.j., yj. jjj .j. '.yj.! jj .j.j...gyg j j j j. -. . 1IIIFiilII:.. i..kj E . (.E . II.I::::CZEi-. ! -.. . i . i . ! ::tili't;.'. : ). : .. . ! ' I ....).!.. . ..r..... . . . t -. : . . . ... '
'
:
'
!:!
'
:
'!i E
:
iE r7E!((r11115jq15EE ! :
.
E
'
E!: ' ''!''' E E' ' IE : : !E ! .. E' E:. .E' E: E!(' E '
'
.
.
(t! j jjq lqr': (j j'.!i ( i(E('E' !E' ( ( i EE' 1; (1)! .)) ' g .r .'.. j ;j (j. j :
i
E
.
-..
-.......
-
''
E:
.
-..
-
.
:
'
'.
'
E:
. . . . .
:
. E :'!
i
. '
-.
. -
.-.
-- -..
.
: .
-----i' -.
: y
:
:
. E:j :
E
:
:
.
.
'
-
'
'
' : ' :.::. . . ..
-
.
Profit diagrams for
unhedged buyer, Iong forward nd buyer ' hedged with Iong forward.
:
- ..
:
.
'
E
.
.
=
y
11:: .11::: ' (2:,1
.
-:!1j4;-
unhedgedbuyer
-s
200
Lopg gold forwgrd Hedged buyer
-+-
...
1sO
100 $40
50
-150
E
..
0 . =50
lock in a proft by entering into a long prost of forward contract. Auric thereby guarantees a Profit
.
.
.100
Hedging with a Fprward Contract The forward pzice is $420as before. Auric can
.
.
.. . . . . .
. .. .
.
! . .. . E E
'' ''
..
,
.
j $,,.j:.0j $.,::60 g . :
300
.
35O 400 450 Gold Price in 1 Year (5)
500
$800 $340 $420 $40 -
=
-
Note that whereas Golddiggers was selling in the forward market, Auric is buying in the forward market. Thus, Golddiggers andAuric are natural countetpal-ties in an economic counterparties szce tey can enterinto forward sense. In practice tlzey need not be direct economic But exchanges. in an sense, one flrm's desire contracts through dealers or on fonvard. desire other's the to buy to sell folavard has a counterpart in
. along fonvard unhedged buyer and Figure 4.6 compares thepro tdiagramjforthe profit which shows the for hedged buyer, gold. It also is generated by the in position summing up the forward posiion and the unhedged payoff. We see graphically tat the buyer can lock irl a profit of $40/oz.
lnsurance: Guaranteeing Price with a Call Optioh 3.You might thirlk that a dealer would charge a higher price for a purchased option if the dealer knew thatan option buyerhad superiorinformation about the market forgold. Howe'ver,in genetal the dealer an ordinary investor will quiclcly hedge tlle risk from the option and therefore has less concern t.11m1 about future movements in the price of gold. '
Mnvimum
Rather than lock in a price unconditionally, Auric might like to pay $420/oz.if the gold price is greater than $420/oz.but pay the market plice if it is less. Auric can accomplish this by buying a call option. As a f'uture buyer, Auric is nattlrally short; hene, a call is
% INTRooucTloN
100
WHY
RISK MANAGEMENT
To
;'jj,,,,f (' y' 'E' y' $jjjkj;(y(jjjj:j!!'. r''j)iij)jFIii,' jr,,,,gj,f ;jjyyy' jjr.,,;jr,,,gjyf jjr,.,f jjr,,,jjyf y'y (' i'1* jj,,..f 'q' .y'( j'(y;j(. y'' tf j'y.yyyyjyyy'yy y'. r' )' lry!q:qryrqqqtl'qq's'f (f qjff qyf 41)t*(E i kjjygrjj,d j,ryjjjjjgj,;':;jjgjjjg,. jjjf .j' (' y' .y''j ''if;f jjjjjjjjj'. y'. t'j'yjj,. E kjjyyyjjj'. jjyf yf jjlkj:j,-'. tjgjjjjjjyf tjf .j:j,,,,,.-,,.' .;jj:y' jj'yyy,')(.jj.y'. !' E' ' ' E E' !;E i (! qjjjyjyjkj'. kjjjgrjjj'' rjjjjtjjjk.d ,ijIr)' jjgy'jy',jy'y,jjjyjjjj,yyyj. rrf (jj:jrj'. ijj,,,yjjf j'((.(g' j.gjy'. j', jtgty;(((' y' (' j'j'-. )' kjjkj!jr'. y)y'yjjjyj;t'. jy;yjytjyjf yyj'; yjyf y', kiij:jgqkgjj:;.d yj'-.f E ijEtr q'((. ' (!(.q'(!((; .y'yj )'. ('. y'. E J;(4)1 ;!!'!i'i E(' !'.' iE (((''..!' .: .Eq(.. '((.E !( :((.(.'((!.E'i'i'.. E.(('!'q('F'7:77:t:)!77.:7: . . yy. iilii:(yt)i E ; ( ... . ... (EEE EE E yggjjjjjjyys :-tjs ()(j:(. tjj ykj ( k y.jj!tyj'lj: jyyy(q. yj.j .j .y.,. g j.. q qj:. jjy j j y ...j y j y j j j j j jyyy (jg j . y .. y .. ;j. .yy yj-jy.( . (yjyjjjjjjjjjjjjggy.. jj .j. g j.. .. yjjyyyyjjjjjyyyya jjy.. jj.y.yy yyj (E)(EE.)Ji((..(..,....):.,..........:..............:....... y.yj .. t( (.. jj. @.'...q.. . ..
tjjjyj!r)d tjjijy(jjgjijj;'. jjjjjjygy'. jyyy'yyjyyyt(yyjjjy'. ljjjjd 4,j,y4yj;;2:jjj(.*. kjj:jjjljl.d t'ytj tjjjt.d tjjjjyrjjk(((jjj,': tjt!!kk.;tjjr.d ' ytyttttjt.yi jjjjjjjjjy;y. j j j.j. . tttyjjjjtjyjty . . 'jiljzg' -y' ,gyrj,,,,,' -y'. -;j;jjjjjj'. jk,j;jjjj,d t'-j!yyjyr,' y' yjryjjyr,f -y' 'jjjjj!jj;'. -;ygg;jj' .-t'(r'r' y.' jyryyygyf :'i'lE'jjjqjq')jjjjjjjjjttf yttjy'j ' yy'jjj
.'
i' . ::
'
r
:
;
.
E
.
: .
.
-.
. .
j
' ' .. . . . .
:
E
-
-
g s;..:li(ri:..( .....:r.:.........J....q..).......
,....
i :.
.
,
.
-
.
..
. .
,
:.
.
-
:
..
g
:
.
: y
,.
-
y
y
,
. ..
.
,j,,js:jjj-,
jjr:yyg:y
yjrggg;jjp
yjryyy;jjj
jjrrgg;,,, jj,.,,jj,,,,yjr .tjyyyy
jj:rrg;j,,
jkjggyjr jjzyjjg,-
jjyyjjgr,
jj:yyy;,,,
jj,,,,;jy
j . (').' .
,kjtrjjjj, '
'
-
'.
:;
..-t!!jikjjr,.;.
-
:
'o
,
)..,),r; .. ,t't'i.t1)'.. .
.
,
.
.
.
. ..
-.
. .
-
.-
-
-
$450 $500
:
:
-
:q
:
g
.
jkj,
.
.
i.
'yE EE.i y .. .
tj;jjg:jj;,.. t(jjjjty)jjjjrijkj.. ..( t . ..... t tjjjjjjyjjkryjjjj,. tjjjjjrjjjj.. tjjkk!kjjyjjjjj,. :
E
ji1jjjrj2'
,
:
-
-.
.
.: -
. :
:
-
.
y.. E
,j
-
-
-
-
-
:
.
-
:
.
-
-
..
,
:
-
-
y:E:
: .
,,
$20.79 $70.79
-.'
.jjjjrk jjjjjj!r:' ..
E:
.
-
... -
-$9.21
-$40
: -. .
:1 y
-$9.21
$110 $60 $10
$350 $400
' ' ''' .. .
jjk.tt'..
gjj,,,rjj, .rjyyy.
' 'yE i kjj:jrjk;(kkjjjljjjj, yljrj!k.,k;:jr;. '- (..(jjjjyly?i!?). '.((.. ( jq.(y..y..jj.y(..yy.(.(.jyj'.q.(:. j y.jj..y..g(jjj.y'ri . .y t .yjjjrjyjjjj,jjj,. jy kjjkjgj:jj jjjjjyrjjj,i ..... yj kk;jyjjkj,. k .. . . t,. '. .
..j. '. ). (jjjjjj-rjjjj;-j' . . . .. . . - .- . . ...)E E. . . . . - . . .
,,jk,,,-,j-jj,,
yjyygyyrr
420-strike call option.
:
''
.
yjrryyrjjj
.
jjjljjjjjj.. E E i E ' E 'y :.1jj(jj.j(jjr.-. '( E.. E. (.qqE(( j((y !jjt2:!j:' .(. E. . yjjjrgjjjjs, y jj qy(; ' 'kjikj!!jy kjjj2j!j:' ' jj;j:jl;,.' .j '( . ) y.k!;jyjjjjk. ...y. y!.jyE...Ej..!. ;q.y(( yjjjj:i-. .. y..... '..;( jyt'E . ..E. E:j :Eq ( '(lgy, q;.(..q... (jjjt, kjjigy j..yy..y.. .y.yy. y.j.E. . y(y y.y..Ey.. ( ( . (jjjjyrjjj,(. . . ....IE ('EEE ( ( yj.E.EjEE r i; E . q ( ; ,jjj:yjtj(yj:!. g E. . E j ..l j y . (. EE..@( q::gi: ; Eqr y j. yjtj E q y ( r . E'y. jjq .jjj2jj!j(E j( . EE . E . Ei ;! gjjj E)j y i ; q E E ( . ( . . ; ;Ey.(.!y.yj.;.y(E.y '. . q ; . .. .. .. tjjjjjyyjj;ys . .(( . .y ((.yi.y.j j.y.j.yy.yyj..1'....,. j ,,.))i). j (y '.''' .q..y. .(y y . .. y.. '. . ..j .y.y . . y ... .. y .jy,..!t... .q..jy,.,,,.,.r..,,r:,,.. . j . . y.j. yy . .,, . y . .. . . ., . . . . . .
' E' E. . E(
:-
.
g .
g.
....
'
:. .
:
.,,jjj
-
-
-
.
-
y
-
.
,
E
:y
.
y.
:,
.
..
-
.
r,
.
.
.-
FI RMS MANAGE
f1.$111*'!4.94:*''11:15/7r51(11)* ;)' 7r' ;'fjtt''jillqs!jf 1r' '.)q'yj$j.' yj'E j')' E' ' ''E /1* if T 1)* IjE.jt')X (' )!l;p)($' !j' f: iE E;)''''i rq jjf qff qrf qqf q' i'if 1* i!' ;'E' '!:q)))'(jyq. r' jjy'jf j'y'rf p' ;:.ECE ;i'. ''. i ' .17@7j: ! E E !'Jtf:ilsE.(Eql.. 5741j*'111* jj'. j'' ('.g(. qjjjj'.' tjjj)'.jjj)).' ('rf qE'.:('i'Ei(I1l;*( EEC'iiE'i(EE@(. 'if)l!' '.1: 1. E ' F::I::'::T:'T:F?T!TTFS'E:F!T:ES7:TIST7r)E .. l.' ()' 'qL'. (;' .:))jLj)j);...' y(jjyy'. . ;Etq ;ji. E.E-.))E. . ' . ; . EEjr 1)))4 pl.!)jl(CJCEE ! .il!:.!.; ii yjq r.q! . jj.jjljjjjtl-. t:tt. Etjtjri.sl jjqj.. (.t..;)E;.-Eti-i-.).-.E-k--E)j.( k .(.;jEj'j.E;'j;(-.).pEij...).-)..r,j . j.Ey'ty (((y 4 .. ITjj. . j..I2I.2g,.!p1::;:kF..... . jtj: ' . ' . . ,g)(@!)(. ..j;;;(. III:tIIf (jqtiy 4:g . .rr.!IIk-2II.. ,::2). ..!--y-ty,.,..Iir. ..t;)..)rtpj#y-riy)r):j)-y---. yjjl)r(.IIEq!q;. IIEEE.;.)r.-:y.y..!. .1r::!!4; :;;14!77:.E (I:y (1(F,, r g;l .j'i -'. L'L' ))'t#E7)@1)l :'''.'r;'@':E f' 7*.
.
-j-j
.
Do
$100.79 $50.79 $30.79 $30.79
' :- ::. i
:.
: .-
: i
E
i
:. E' E: -
i
--l-k;).y))'
' -
.
.
'
'
.
.
' -
:
-
:
y
-
'
-
:.
y
-.
Comparison of prfit for unhedged gold buyer, gold abuyer hedged wth call, and stand-alone call.
unhedgedseller purchased call Buyer hedged Mt.IA call
-s
200
.+-
.+.
150
50
530.79 i:
0
.
! E E ! ! ! (,460 E$420 i
E
-59.21
-50
!
'
'
.
-100
,
$420 since we have paid the call premium. Figure 4.7 shows the pro:t from tlze call by itself, along wit the results of hedging with the call. As before, the graph does not show the probability that the gold price in 1 year will be in different regions; hence, we cannot evaluate the likelihood of different Outcomes.
4.3
V AN AGE
RIsK?
The Golddiggers andAuric examples illustrate how the two compnnies carl use fozwards, calls, and puts to reduce losses in case of an adverse gold price move, essentially insuring their f'uture cash flows. Why would a flrm use these seategies? to hedge, In Chapter 1 we listed four reasons that firms might use derivaves: pracdce. to speculate, to reduce transaction costs, arld to effect regulatot'y arbirage. In discussed already one of these considerations may be important. We have more t.11a11 that fact market views-for example, opinions about te future price of gold--can te affect the choice of a hedging strategy. Thus, the choice of a hedging strategy can have cite the accounting eeatmnt o a tmnsaction a speculative component. Managers often obviously considertion. transaction and important, costs are as In this section we discuss why ficms might hedge, ignoring speculation, transactions costs, and regulaon (bt we do consider taxes). It seems obvious that managers would want to reduce risk. However, irl a world wit fairly priced derivatives, no trans-
'
250
300
350 400 450 Gold Price in 1 Year
500
550
($)
acion costs, and no otlwr market imperfections such as tues, derivativys change the distribution of cash flows but do not increase the valu of cash flows. Morever, large pubclyhld firms are owpedby diverse shareholders. Thes shareholders can, in theoty configtlretheir own portfolios to bear risk optimally, suiting their own Saste. ln order to credit hedge, the 51.1r1mvst pay commissions and bid-ask spreads, and bea.r coutiparty cosis? lcur these risk.Why There are sever at reasons th a t firms might seek to manage risk. Before cscssing about what derivatives accomplish. T be concrete, suppose that Goldthem, let's t111111 gold forward at $420/oz. As we saw, tltis guarantees a net income of diggers sells $40/oz
Do FIRMS
101
'
-
:
i
-..:111...
.
WHY
$
100
call has : prmit!p pf $8.77/9z.(recallthat this is the sap gs ipsuran,sppps the wlth . premium te same stnke price). The future value of tli premiumij the on the put $8.77 x 1.05 = $9.21. lfAufic bys the insurance contract, net income on the hedged position wi1lbe as d shown in Table 4.5. If the price is less tha!z $420,the call is worthless at expirtion Auric buys gold at the market price. tfthe price is greater than $420,the call is exercijd and Auric buys gold for $420/oz.,less the cost of tlle call. This gives a profit of $30.49. If the price of gold in 1 yar is less t.11%$410.79,iristtdng the price by buying the locldng in a price of $420. At 1ow prices, the option permits perfonus better t11:11 call of lower gold prices. If the price of gold in 1 year is greater t.11% advantage us to take insuring the plice buying te call perfonns worse than locldng in a price of by $410.79, .
RISK?
' .
i
E
E
Whe hedged ivith the fotwrd, Golddiggers will have a proft of $40 whatever price in 1 year. In effect, tlie value of the rduced pro/ts, should the gold price rise, the subsidizes the payment to Golddiggers should the gold price fall. If we use the term ttstat' to deriot apnrt iculagld price in 1 year, we can describe the hdging sategy as shifting dollars from more prostable states (whengold prices are high) to less profitable states (whengold prices are low). 'l'his shifting of dollars from lzigh gold price states to low gold price states will value fbr the s.rm have fthesnn valttes the dollar lntpr in a f/w goldprice state f/ltzn in goldprice state. 'Why might a 11n,nvalue a dollar differently in different states? a /yfg/7 '
E
-
-
-
''''''
.
An Example Where Hedging Adds Value Consider a 51%1that produces one unit per year of a good costing $10. Immediately after Production, the 51-mreceives a paypent of either $11.20or $9, with 50% pyobability. has an 'rhus, the 51-mhas eiter a $1.20prost or a $1 loss. On a pre-tax basis, the 51-111
.
%.Irq'rllooucn'lox
102
MANAGEMENT
Rlsx
'ro
wHv
.'''-''
-' 'kL'L'/L'i?L';1j1l;LkLILLLLL :jjj'. !' :'?' ''':''l:rf ;it' )(' ' ' : :. ::' ' '' E '' : : i: ii' E' E .. IE
Do
FIRMS
MAXACE
ltf )' jpif y' kf t'(' j';' tf !@' 1* fsf j)'ii:! jif #' q' y' r' )-.r!!l1:::1l1k.'('* qi(t)(. 5)57q7q:* '7(''J'1* jy)f )' j'(:)t;.y.jjk)' 1.l' y' )y' q' jtf (t)' E!' qi lt!.i !(@ qi i'iEE'' (i''i i.:.'!'(kri'(i (Ei itil i( '(''E.'((EEq EE (!lE)tqr'lq?l!;l:lrq qrrq'!tlq!rlr '15 (E(.j. j'Eit'E;;i((:1 . q( 'E ' . . ' .Eq'' .(..... .(....' '. ;q :: ; E. E (.:. !. ! ;! jj!E; qyqj E (5 E.qE ... (;E . .. F.i ;ra..,y,,j-y.y.jytq.j-:.jy.Ey;.q.jE(j.)..jt.jjjjyt.jq).yyjyrj..yy-.yjyjj q j(y !yjjs!jjj..l) ik. i.:jjjj;;.())j(jjgy;ygy-,jj. . '. iilTjqs ttrjjjk,jjI( . r r r . y y yj y .y j. r qEj. j.yy . . --j. gjgy;jyy jtyy;j gjjytjg E -. :.. E.::. jyyyj k. --. yjry, rjy.,..,,.yjgy,.;ggjjtgjk(:y .jjr .: .. . .r.. ... .
RlsK?
k
103
.'-',''
t'1iIkEii:r'. r' )'/' lf 1r' )7*:12 !1l);' ((;111,(*. .'1iI(((j,i,'. tqq;.,E;))2(kkkt)-))'.-ff Iiil,k-1'. ..' r)'-' y)qjIiL,)'. )-')' -'i$!!.:iiiI;''. 1111E1111..*. )'' 1IiIjiEE2...' ppf L-12-k-:-ttL',,-,,,-:,.,!-,.. @::j::!!:,'. t'. ), ) CE jljy). t :t- iE i -''3bb'L3.. rj--jtq7q7!j!!r77q:. E ' ; ii. .. Ei(!). '-) . : . ' k))r .r.:r!. .. .. . .. . .
--' -' -i--.''. -''' '''' ' ,')' EE : E k'. il..-tg-.--a'irilllt-t... ir)f;'ttf 9* jf i'@' ''7* :')'(' ' 'E jd )' ,''''-'-)'! 1Iik!!:'. kf q:' 7(r57777!'1717* !' qif ?)' E E : llEk .;' ;'. (2112:--.;411:* y'. .r' pf q' (ff ':'1* )' if ilq': Jlli':j)f j'(qf ip:: (!1( i!'E'E EEi!.;'l.' Eq': EE!'k('l!i'tii' i( ,(sEI.(E':('. E!.':i'tt(E.y)-(': k'E(. i'(EiE';(7E 'TE(:.E. E.E..EEE'E.(E ( 11 ; ( 11;1;111177111;8T$9 . . .jI . ..( : (,( . . oo. ss $ g g y ,, tyylyyoj q.,, :)jj:(.y.k is.E.E:y. Er..t!i:yjyr(!.:jj(: :' 'y((y..t(yg.j.jj; ; y=)L)p.j))j.;z)gj.j.;Lj;).jj.jjj.jL3j.L.;..LqjkjLjjL...L;L;),))..jjLL.;) Ljqjt(f1kb.b;L.1'j jj (yy()()jg. .
.' .'-
.'.j' '.'-(-' .'@' -' .',*,-66,33,'b3,3...*. '
'..' rrf '. -
:
'
:.
.
' .
.
..
.
'
.
. :
.
,
' . : .. .. ' . : . . .. .
:
,).,
.
.
:
.
.
iy.ygyyjrjjgl
. ,.. . .
. ...
-tt-j(t;);t-'
.
.
.
.
, -.. ..
... .
- - .. -. .
. . -. . -
.
.
.
-
-
-.
.
-
.
.
.
--
.
. . .. .
::
. .
.
jj( E(r g
L) Eg
.i
::
: '. . -. . .-. . ... . . . . -. - . . .
:
..
-
:
.
-
.
-
-
.
. .
(:Eq(r
)
.-
-
.
.g
q: k
E E( (j :j
jt(gjg
-
.
-. . . -
-
. .
-
-
jj jj jj
.4jy,
.jjg.
.4jr.
jj tjjytjyjj tjj;r jj..yj jj.,yj Expected pfterryax
jj
jtrgyj jtgg;jj jtyjjg j;..yj jr,.;j jg..j;,.;j
output price is 59.00 and $1 1 income is (0.5 x-$1) + (0.5 x $0.72) .20.
: E E. E.!
g -.
-
-
.
-
(.(. -.
.
::( :E ( (E( E (g (E.(Ei.(gy j jrljty( -. jjg E jq j)jE.@.
.
. . . - .. . - - - . . - . . . . . .
-
.
-
.. . . . -
.
-
- - . -.
y
( E(- :
Pre-tax operadng income
(1) (2) (3)
-.. . . . ..
:E
j'
:
E
.E. - . -. -
E.
:
k
:
.jgr
jkjyjrjgjgjryjjysjkygjg jr,, . .
.jjgj
jtjgjj
,jjjjjjjj,
.
..
-$0.14.
=
E (:E E: EE((:((g( (: (:(.:k:i:k:k:kkk)(E: ((E(::) ( !1jIE1)?iIk-. ( (E k E. . E E. E .
E: jEE/EEEEE E Ef
-
. -
.
-
.
-
-
-
(0.4x (2)) income ((2) (3)1
.
.
.
$1.20 $1.20 $0.48 $0.72
$0 0 -$1
-
-i--
-
. .
.
-$1
Tax @ 40%
(
-.
-
Tuable income
Aer-tax
pqf .' -
.
j
)).. -g .
-
.
'
.
-.
.. .. . .. .
.
-...
E: : . ..
.! ,-. - :.
.
.jj:.jjg,
jrygij..jj:r jgjjjr jtgggy jgyyjggy jkyjyy jtyg;j jr..yj kjjjjy jr.,...,kjj;js kjjjjs jj,,jjkyjyjy (ijjty jg..;j
E. E E. .. E i.t iE. i1Ii:t!:j. :liiljil;. Ey-,. . jillij',!II::i:. )y i i(...E;!. .y.. ( ( E :' (j; ; i-; i . Ei E.. i E E EEE. E.E....gE.Ey .Eq... E.j. E.Eq. .,q..;(. ..... E:1iIiI;.i!!()(liii q . qq ... y. ... ...y!!).q..EE.yr. ;. (..E.(. y y . .. . (.'..'E.....'i (;. ...' '.;.. .. .. ... r'.'i.'..y. ..'.q.;.......!EE...;.rE.... q......r....r.q.....q.....(. y....... (111:. .y;....E.. ...;..q. r.. q(.jy.((.!.... ....q!..-jy11I(Eq!i.i; j .y . . . . .. . . .. . . ... . . . . . .. ... ..yr....... . . . . . . ...y., .. .... . . .... . ... .... . .. ... . . . . ..... .... . .. . .... . .
ijy y E ' E.
(-' .y' jjjf jzfyyjjjqgjtf jj.ytyjryj;' p')' t:.' t'yyyf r' '- yy yy;y ( jjjgjj).;()..(.it)( -1)4(.rtjtyjjs j.(. yr t tjy . . .),-,k; jjji;jry 1lIi:!7-k.h'. t#))-44144. . . . -
,
,...
.
'-'.
.-.. .. .. .. . .
'
.
-
.
:
'
:
.
.
E
E
:
E ' : : .
: :
.. ..
:
. : ; !:-
-;y. -
'....... . . . . .. . .
, ..
.. .. . . .
-.. ..
- -. .....
----.----p;
..
. . .. .
. .
-.
.
.
. . ..
:
.
. .
(y. jjj.
'
-
:
.,jrjjk,
.
gjg:jj
.
-.
After-tax profit s a function of pre-tax profit. Poipt A is profit at a price of $9 (-$1 point B is profit at 1 ($0.72), point C is profit at 0.1 0 ($0.06),and point D is expec't.ed profit f price is or $11 20 wtttn one-half
Slope
:0.72
.20
-
TM
rate
c
$0.06 $9
.$9
probability,
1
---------------------------
0
.$1
=
B
.00),
,$1
' . '
:
. -- . .....--rtj(,y(yjt;(jj;-yyyrj..,-.
.,.
.
'
s11.2o
D : y$1O.1O
(jutput price
-$0.14
A
-$0.14.
expected prost of
($9 $10)) + (0.5x ($11.20 $10)q $0.10
(0.5 x
=
-
-
Slope
However, on an after-tax basis, the lil'm could have an expected loss. For exnmple, suppose that when the ftrm reports aprost, 40% of the proft is tued, but when the flrm reports a loss, it pays no taxes and receives no tax reftind. Table 4.6 computes expected after-tax proft under these circumstances. 'I'he taxation of prosts converts an expected $0.10pre-tax gnin into an after-tax $0.14loss.4 Because of tues, the 151-mvalues a dollar of prost at $0.60 ($0.40goeg to the government), but values a dollar of loss at $1. In this simatio, it is desirable for the f11-mto trade pre-tax prots for pre-tax losses. Suppose that there is a forward market for the filnn's output, and that the fomard price is $10.10. If the firm sells forward, prost is computed as in Table 4.7. lnstead of
.'llt'rti '
;')' '?' ))' )14@* ):pj'.q' )::qd @' )t''j@' f' (y' ll'ttjjd 1* '!'/'q17*:$* /j9*11.71(!1T* ill'l)tifqlljf (' ;'(;' )' (!;' prf q)' ljjyyd 11!*7)* jjjilljjjqlL'jljqjljll'. jz' ))' y' )k' j'. tjd (' )' t'j'yjjj'. g' k' t'j'y' i ' (' :;' ; ; i ): 11:111. q'E li !11544k1!(71 E ipj . ii ltpi yttlqi j.();(jjti '' ;:'qk(;i !:j; TErrr (lt: r)r!rrr::rqrqr?q;;r' rr . . . tji:k . . r . . ;Ji ! ' .ll7' j .( j'y ..jj.y ;qi)!jy r yj yj. .!. j'yjj. jyg 4::::: ,::r . ykjj.). 4::j1 ,::11 j(j((y r y -4tiI; tj)r .ii!!k iI1-.... ty . . . hl:1#.). . 1.LL.1Il166-.'......... . j . i . ipil IIEE,; ;.,-.....,.,..,k,.1,j.,,,..L..j.L,,,L.L.L.,;.;,.,j,,.L.,,,.,LL.,(.L,.,,jjk.,..L I u 1a 1t:id::)ln d::)p . . -.L:$',3. . !g(;I j). e aaeI.....tlra x ne ,l: iin d::r'::),me iE .............-....L.1.L.i-,.,....-q...-....-,., n stta ,l:es e . ..t#,tks. is and where 1 Expected output the price $9.00 ' after-tax income is $0.06. C' )' llrtjtEtl'llyytjyii:rd ;'$' 'Ff 7r(777771r!77:::7* E!:qjf ''
ipE
'--''
E:'
!
:
'
' '-''''
lj'-.L. '
..
:
..
.-..
' i E:
E' iiE :: .
.
.
'
:
I .': .
.
'q;. . i E E . ' -. .
.
-
,-;...;.-k
....
Ei
:
:' : i ' !E E.E. . . .E
:
;
' :
:
E
-
.
!' (
:
:
.
;ttr(gjjjj(yjj(y...
. Ejq EE E ;. )( 5 . -
:
..
-
: q EE:. E .
.
.@),.s.
:
.
(k.
i. EE qi : . y .
:
., ..........
.$1
(:.(.tt(;. (( ( .(! )-.( ;: .((:.: (E yt q-t.(...i)j )t :.(.'
E(
.'(...
:.
y.E;((.) ...
.....
qp'.
-
.20.
j7..Et (E E Ek(E. E. LLLL C..)( (.E. q'EE:!( C(t .'r i (:)j;k' EE F. .(.! E(.'..@ ElEi ' t.(tk!. i.E E(. .3@E(.. (. ..,.))':t..,,,i,(,.,,.,(#?,.r;,')''q,,,),,),).,,,.,'(,:,,..,,).'.i..k,.:,,..),.(,,,,..>,..k...,.!., . . E: .. r .( J. y...'E(;.ErqE;: (.:E(!!7 .:(E(( .. . . .
(:E..'.( ..E( )E:'iE.. .(E : . !
( .).'j
:.
((((.j( )
..
..
')( $.,.,,.,..''..r.,..',.,'..,.,k, 4.,.,,t,t,t.! -
...
..
..! .$,.
:
'EE
.
.E.
: i .-. ..!(E
(
=
1
an expcted loss of $0.14,we obtain a certain profit of $0.06. Hedging with a fomard transfers netincomefromaless-valued to amorelzighly valued stat, raisingteexpected, value of cash flows. Figure 4.8 depicts how the nondeductibility of losses affects after-tax cash Qows. First, observe that after-tax prost (lineACB) is a concave mction of the output price. (Aconcave ftmction is one shaped like the cross section of an upside-down bowl.) When prohts are concave, the expected value of prosts is increased by reducing uncertainty. We can see this in the graph. lfthe plice is certain to be $10.10,then prot will be given by point C. Howeyt, if plice ap be ither $9 or $11.20, xpected profit is at point D, onthe JineADB at the expected price of $10.10.Because ACB is concavq, pt/f?# D lies below #t?f/ll C, and /2#,j*W incrqases expected proAts-s Some of the hedging rationalej w discuss llinge on there being concav prosts, that value is increased by reducing unertainty. so q
.,,,',,..,,'ii.,........,k.,,,t,...),,i..,,,,).:',k).,#,t,,.,:;',..,!'..,.i''.',......')q,.i,..t',..,,.,..k,,.,.,.'.,,,',;', Pre-tax operaing income (1) $1.20 O ,,,!.,,..).
,...,,.
-$1.00
lncpme from short forward Taxable income ((1)+ (2)) Tax @ 40% (0.4x (3))
(2) (3) (4)
After-tax income
g(3) (4)) -
$1.10 $0.10 $0.04 $0.06
,
Y easons
-$1.10
$0.10 $0.04 $0.06
822/
There are in fact a number of reasons why losses might be more hnrmflll t.1)a1$ profts are beneficial. We now disculs some of those reasons.6
s'rhis is an illustration of Jensen t iltequality whih is discussed in Appendix C, and which we will often in this book. . encounter 6ef'hefollowing are discussed in Smith and Stulz (1985)and Froot et 1. (1994). .'
4.15 asks zlproblem
you to compute proht when losses are deductible.
,
(104
% llqTRooucTlolq
To
Rlsu
MANAGEMENT
WHY
Do
FI RMs MANAG E RISK?
Q
105
The previous exnmple illuseating the efect of tues was oversimplied in assurning that losses are completely untaxed, btg it is typically the case that governments tax prosts but do not give f'ull credits for losses. Tax systems usually permit a lps; tp be offset against a prost from a different year. However, in present value terms, ttle loss that applied to prots, which still geerates a will have a lower effective tax rate t.11:.11
Increase debt capacity Because of the deductibility of interest expense for tax purposes, irms may find debt to be a tax-advantaged way to raise f'unds.? However, lenders, fearful of banknlptcy, may be unwilling to lend to Erms wit.h lisky cash flows. The amoupt that a firm can borrow is its debt capacity.
income to anotherl, Capital gains taxation (derivativescan be used capital gains incolle as With collarsl, gnd differential taxation defef taxation ross Cafl be tlsed to SIO ifome Colm WCS ldelivatives Som O11e CotllllzlFto aliothefl.
Managerial risk aversion While large, public firms are owned by'well-diversised investors, fil'm managers are typically not well-diversied. Salary, bonus, arld compensation options are a1l tied to the performance pf the fil'm. An individual who is unwilling to take a fair bet (i.., one with an expected payoff equal to the money at stake) is said to be risk-averse. Risk-averse persons are hnrmed by a dollar of loss more tan they are helped by a dllar of gain. 'Fhus, thy benest from reducing uncertainty. The effect is analogous to that shown in Figure 4.8. and have wetll tat is tied to te company, we might lf managers are lisk-averse will they reduce uncertainty. However, matters are not this simple: try tat to expect Managers are often compensated in ways tat encourage them to take more lisk. For example, options given to managers as compensation, which we discuss in Chapter 16, lislier. Thus, a are more valuable, other things equal, when the Erm's stock price is manager's risk aversion may be offset by compesation that is more valuable if the firrli is rislder.
Taxes
motive to hedge. There are other aspects of te tax code that can encourage firms to shif't income using derivatives; such uses may or may not appear to be hedging and may or may not be approved of by tax authorities. TM ruls that may entice firms to use derivatives include the separate taxation of capital and ordinary income (derivatives can be ujed to COIWCXOI1e fol'm
Of
Of
to
and distress costs An unuspally large loss can threaten te survival of obvious that a money-losing 51-1':1 fu'm. The is most may be unable to meet reason a hxed obligations, such as debt payments and wages. If a tin'nappears to be in distress, customers may beless willing to purcase its goods. Aouldyoubuy acarorcomputerboth of which come with long-term warranties-from a company that alipears lilely to its warranties'?) would unable and be honor of business ten to out go Act'ual orthreatenedbanknzptcy canbe costly', adollarof loss can costthe company lirms to enter derivatives contracts moretllan a dollar. As with taxes, this is a reason for reducing the probability of loss profit thereby ansfer income states, f'rom states to that Bankruptcy
Afirm that credibly reduces the risliness of its cash flows should be able to borrow more, since for any given level of debt, bankruptcy is less likely. Such a flrm is said to will have raised its debt capacity. To the extent debt has a tax advantage, such a 51711 also be more valuable.
or distress. bnnknlptcy
Firms make lisk-management decisions when they organize and design abusiness. For example, suppose you plan to sell widgets in Europe. You can construct a plant in the United States and exprt t Erope, or you can nstruct the plant in Europe, in which case costs of consuction, labor, interest rates, and other' inputs'will be denominated in the snme cun-ency as the widgets yot! sell. Exchange rate hedging, to take one example, would be unnecessaty Of course, if you uild in a foreign countly you will encounter the cojts of doing business abroad, including dealing with different tax ctis qnd regulatory regimes. Risk can also be aseted by such decisions as lasing versus buyinj equipment, which determines the extent to which costs are Exed. Firmj can choose Qexile producbe reconfgured tion technologies thatmaybe more expensive atte outjet, butFhihcan at low cost. Risk is also affected by the decision to enter a particular line of business in the lirst place. Firms mnking computer mice and keyboards, for example, have to consider the possibility of lawsuits for repetitive stress injulies. Nonfinantial
coslly external financing Even if a loss is not large enough to threaten the survival ofa finn, the fil'mmust pay for the loss, either by using cash reserves or by raising funds externally(for example, by borrowing or issuing new securities). '
Raising funds externally can be costly. There are explicit costs, such s bnnk and undelwriting fees. Thre can also be implicit costs. tf you borrow money, the lender may worry that you need to bon'ow because you are in declze, which increases the Probability that you will not repay the loan. The lender's thinking tlzis way raises the interest rate on the loan. same problem arises even more severely with equity issues. rfhe
At the same time, cash reserves are valuable because they redce
a
srm's need
to
raise f'undsexternally in the f'uttlre. So if the firm uses cash to pay for aloss, thereduction in the in cash increases the probability that the flrm will need costly external snancing future. is costly can even lead the 111-1-r1 The fact that external snancing to forgo investment projects it would have taken had cash been available to use for financing. Thus, however the 51-111 pays for the loss, a dollar of loss may actually cost the firm Hedging dollar. t11a11 can safeguard cash reserves and. reduce the probability of a more costly external snancing.
risk management
106
k
Irq'rlkooucTlolq
To
RlsK
WHY
MANAGEMENT
The point is that risk management is not a simple matter of hedging or not hedging using:nancial derivatives, but rather a series of decisions tat start when the business is first conceived.
Reasons Not to Hedge There are also reasons why irms might elect not to hedge:
* Transacting in derivatives entails paying and the bid-ask spread. * The flrm must assess expertise.
costs and benests of a given strategy; this can require costly
* The fit'mmustmonitortransactions unauthorized trading. '
costs, such as commissions
'ansaction
andhave managerial coneols inplaceto prevent
* The lirm must be prepared for tax and accounting cmsequences tiO11S. ln Pal-tictllar, Yi S may Colr1 Plicate fekorting.
of teir
tansac-
Thus, while there are reasons to hedge, there are also costs. When thinking about lisk managecosts and benests, keep in mind that some of what srmsdo could be called derivatives. For example, supposeAuric but obviously involve enters into ment may not agreement with a supplier to buy gold at a fixed price. Will mangement tllink a z-yea1' this of as a derivative? In fact this is a derivative under urrent accounting standards (it is a swap, which we discuss in Chapter 8), but it is exempt from derivatives accountinps Finally, firms can face collateral requirements (the need to post exla cash with their counterparty) if their derivatives position loses money. The box on page 108 illustrates an attempt to manage risk that bacured. '
..
'
.
:
'
E
Empirical Evidence on Hedging d derivatives use of We know surprisinjly little about the risk-management practic titnns'in real life. It is dicult io tell, from publicly available infofmation, the extent to which fil'ms use derivatives. Beginning in 2000, Statement of Financial Accounting Standards (SFAS) 133 required fil'ms to recognize derikatiVes as assets (jr liabilities on the balance sheet, to measure them at fair value, and to report changes in their market va1ue.9 This reporting does not necessarily reveal finn's hedging psition (forward contracts have zero value, for example). Prior to 2000, fifms had to report notional exposure', hence much existing evidence rlies on dat f'rom the 1990s. Research tries to addrss two questions: How much do stzms use derivatives and ad oter why? Financial firms--commercial barlks, investment banks, brokr-dalers,
Scurrent
delivatves accounting rules contain a purchases and sales'' exemption. Firms need not use derivatives accounting for fonvard coneacts with physical delivery, for quantities likely to be used or sold over a reasonable period in the normal course of business. lenonnal
9See Gmstineauet al. (2001)for a discussion of SFAS l33 and previous accounting rules.
Do Fl RMS
MANAGE
RISK?
$
107
in derivatives frequently. ltisks are idenloable, and financial institutions-transact risk management. The more open question is the extent to which regulators encourage dedvatives. We can summarize research findings as follows: nfnancial til'msuse
* Roughly half of nonfnancial firms report usin dedvatives, with usage greater among large filnns (Bodnar et a1., 1998', Bartram et al., 2004). * Among firms that do use derivatives, less thn 25% of perceived risk is hedged, wi.thfin'ns likelier to hedge short-term lisks (Bodnar et a1., 1998). * Firms with more investment opporttlnities are likelier to hedge (Gdczy et al., 1997). * Firms that use derivatives have a higher market value (Allayanpis and Weston, 2001; Allaynnnis et a1., 2004., Bal-tram et a1., 2004) and more leverage (Grahnm and Rogers, 2002., Haushalter, 2000).10 Guay and Kothari (2003)verif'y many of these sndings but conclude that for most of minor economic signilicance. derivatives ln their sample of large tifns, firms, use is derivatives t.11%half Among derivatives slightly more report usage. users, the athrs median hedges only of that the about 3% estimate tirm to 6% exposure to interjt rates exchange and rates. Because data are hard to obtain, some studies have focused on particularindustries and even flrms. Tufano (1996),Petersen andThiagarajan (200), andBrown et al. (2003) have examined hedging behavior by gold-mining fmns. Using a uniquely detailed data set, Tufano found that most gold frms use some derivatives, with the median fil'min llis snmple (Nort.h American ftrms) selling folavard about 25% of 3-yearprodction. 'Fifteen' percent of ftrms used no delivatives. BroWn et al. found substantial variation oker time in the nmount hedging by gold finus. Firms tended to increase hedging as the plice rose, and managers reported that they adjusted hedges based on their views about gold prices. Thecurrency-hedging operations of a'U.s.-basedmanufacturingirm areexamined detail by Brown (2001),who finds tat foreign exchange hedging is an integral part in operations, but the company has no clear rationale for hedging. For example, of 51411 Brown reports one manager saying, W do not take speculative positions, but the extent we are hedged depends on our views.'' Faulkender (2005)sndsconsistent evidence for interest-rate hedging in the chemical industry. 'Fhese firms increase exposure to shortbut ocelations term interest rates as the yield curve becomes more upward-sloping,ll between cash Qows and interest rates do not affect behavior. The varied evidence suggests that some use of derivatives is common, especially large flrms, but the evidence is weak that economic t# eories explain hedgipg. at
loGraham and Smith (1999)5nd that after-tax profits are concave for a majority of srms,as in Figure 4.8. However, Graham and Rogers (2002)are unable to find a link between hedging and tax-induced concavity. 1lAn upward-sloping yield curve means that long-tenn bond yields are greater t.11m1 short-tenn bond less expensive., However, we will see in Chapters yields. Tltis appears to make short-term snancing costj, long-term and short-term 7 and 8 that if a company hedges :11 of its future short-term snancing linancing will cost the same.
%.Ihl'rRoouc'rlolq
108 -:iiiipr;f' ,' '''b''''. -1!14111;5* ,,)' -q;' ....j.yy...'. .''bb,6ik'i'.'' .',.((1;*. ,,jjij'y ,'' ,1...*: .li..ll.'y ''....','. ,;;'. .kk' ,;jy'. ,kr::jk'. .jr' ,3,t..3,,*. j'.....'. xqkj....,jp,'. .;j' 'rd'' .7,* jyjd ,k;' ,,jff)fj4,,)'. .;,' -'-;;' -jk:jj;;,k'. -;k' .jjjyj'. .;;' .''ii;,'6--'' .11:)*. .,.',;;' -..' .;' ,'.'-'.)'. '-.' ,'-'.',' j'..' '-1kkti!:;'. :'11ii:l'' g' ,'(jjjjjjjy'E r' (' 11....11;.*. l''....'':d ,11...:1..*. ''b...''''. 1;:::1,*. r' ,,,,,,,,'. ,,j,,,;,,'. ...t'. ytd jjjj:jj,,d !jyj;jggjj,'. ;'r' liiijirjl'r y' :tjyygjjjj.(gjj,': y'y ..),jj),j):'. j:jd :'.jjjjggj;'. '':::i1-'' -lt::r:y,-kii:iii''d -1ir:)' :';;,' -' )' C' Eyj'( ())S'yjjjrgjjjjjj':))' :';f 11::::1,,* -1:ii:-'' ?' :'-'E' '-,;,,y,k-'. .jj,...y.L'E :'y 11k,..:1k...11..*..* ry'. y' jj(yjjgjj'. j'yyd k' ,':'it::t,,tb...,3';'. s' Jlk.'y .11L.'. 1!(ii:!))'. ii::::::'. .iI...;1k.'. 11L..iI...;IE..',;;'. jjgggjgjj.d jygggg;jjd .36:..,'. Ejykj,yj,gij'g jjjkjggy;d jyggggggd yjggggyjj.d jjgggygjd jjggjj;yjd yjjt;'. jj;j.yjyd .1I:..11' 112..:$8*. ,4444.44*. ssgy'g ('. y' jjjjjjjjj,.'. ykgjjjjj;': jjkjjgjr;d (gjjjjyy;jj,'E ('....ar.,ysj,'jr gjjjgy(;jj'. 'jy'yjjggy;'. 1jkj!2rr)'. ...,.' Slk,.,gjj.'. :';'p' r'. jyjggyg;d y' .yjjggjjrj'E ;'jjj;;,,.;,yjjj'. 1311113,*. j'; ;,)' jjgjgg;jjd jkjyjjrj'y' jjyyjyjj,'. t'jyyjjjr;r'. jb:bbjt,ib'l 'i:ii:::' f'. -'. -'-' -k::i::i'. Frd )' j'-'. -'--' ''?' -.': -j#ty'. -'(t' :'-y)'. -')' '--' itljf -j' g' i',tf Iqj'yI:E Ftj'qtf 1* J'j irf )'jjjjjj((jjjjj': 7*
Gol-oolccERs
RIsK MANAGEMENT
To
.'.' ''''''' ''''''''' ;';' '''j' S' t' j'..'.(jjj(gtjjjgyyjjj.jjjjj,' ,'((jj1())ii1I,' Ejjjkjj!rrd :'yjf '-k':
.,jjjkjjk4Il.' )' f ..11.411:-11;..* 'lif i EEiF( ' i EE' ' '' 'E':; 7*E EE! ETE jjjjtyyjjjjjrjjjgjgjjj.d k' jjjgs';.,. r' 'j,' p' j;f :1jg.)jjji)' ;5:7* !1E;;;;;2!' )' !' E'iE E (.@ i!7EFE ' Fr E (' jjjjgyj,,d y)jjj'. fff q' ';'jq-' iiif )' ilprilipliliqijqiihlf t'tllf-'fqtt-l!l i'qrf @ y))gt))t)-'. 1.67*141: J' (!i'qt! p)r kjjjjjrjd rjjrj,;,d rttjf-tj'kl't .11Ijjj' j'y' (' tf t'j,)pf $' jll;i:kk-d ;':)'. r' ))*(1* tqlf 'Et2:1 .'E('' E'21: ' !' :'!'i'iE!i.Eq;'(. E!:EEl'iE'.(. .' r'(''';'. i '.' i. '. EEE...E 7! ('. Sjs.tlly:gyf ygy-jj'l-)!q . ( .J'ij' ! ! ;. !.E.yEyy. ;p! .. .-.gi-j'..- .)y' I -...r EE! E.EiE i ;(E.; Et.tk E E - ;.qr-, , j q y j ( t . 'C'tr ;:-(r.ylry.gj. , ; ) lljsyltj. jt)j y j . y j r ; j . z,....,jkttt)t,:,))iy...rE,,,,,..,ykt.,.;...,r..;.,).i...i..,.,.! . q . , j y j r ;..( .jy lj ..(r...jk y . y and
(:1j411jk.)j11()jj11t.*. ))' t'(,,jjjjjjj--'. y' tj',-ytf is!f tjjjjlkj:ji;kd '.1* r tj r
,jjr' ,jjj,..jjr,,)jjj' ..'; '
jtd i'k' 'ijd ,')' .1::4::11*. (' !1!::* ,1:::2:1,* $1:::111:.* ...y.,.'. asy...'y j'11!E:11..E111.* )'d1i:t:i,i'. ,'y' y'. :'Lj6L:..'. j'y' ..,,,....:,,*. yyyyy,gyyy'. .y...'. Ljjs,,,kkd ..Ij' jjjjgjjjd y'. .(4j;jg2:j44*. ;k;' y' yjjkjjyjj'q y'y '''!ii;;;;Iii1,;;'''5': j'..jj(k;.,.'. 11:jq):i;'. 112j4:j1j*. jjyggjjjj,'. y' jjtgygjjjjjjy,'. yjjjjyjgjj'y 1''11;131,1.*. 'l::drr',. g)''.::ii:'' 11::!::,k*. )' rrd j'tf ptf )' ,t' :'. (' ,'q' h)' y' :';LLL.'.--)'yjj'. j'j' th-'.k-i:' ' l5'jjjjjjyyj'' 1* (tj' j'('t'k' f'. )'. ,';!1'i.'. j''j')''''''y' :iit.' '1k..2:I''. '1L...1I.'. 'I::d!i' y' .1IEF2IL.'.jIk...II...j1l..'' ,'. k'':;g;ird .11:,(211.*. (IlrillLl'y d1:l:1I'' gjjjjjj'. yjjggd .yjj,ggyj.' .tyyj,,gj;gkjyjj'. y;yyyd .y' (' $j4jj,jjk* y' yjyd jjjtjjyjjjyjkjt'. gjjjsstjygygjyj.'.kj. .j' kI:j:::lp'. yy'y )' j;;d ...jjjj.j;jj()(jj,;.'. jjgjkjjjjd tgjjjjjgjj'jd j)jjj:'. iijkjjjj.jjjyj,,tjjjyytjjryrjkjyytjj..;d tlljr:ilL.d iI::jii:'. .I::i::'. :1!il1.kl'. (' ;'''t' tplrpi.ll-'.f --'E )' !' ''i'. !)y'i@);'rrrr q' (' ,tkytff ()y'jj(j:1j' ')''r)k'. )jj' yyjf @' g(' )' Elii.d IiE':)i.'' :.1::.,* .iI2,EiI(.' 1i!E:' ;I!!5iI..' '1ItE)' E?l:.d :1E!:11)(2)1:.f(11()* y' (' i1!:E'. -'!lkiE:.. ;pdiiijI;:t.I:#':'. (.Ii:E:I.):t)l.':i!EiI..'. E.i::7''. ()1:#7:1.:::11..*tEIIt(EiIi::1It.'. '1I2.tiI.' ti4(F':lt.'' :.1::.-** 11Ei5I...'' 11t:;' kk,d ;iE::1Ik.'(;,' ;....jj.j....jt'. :#qF:q)!!''':E''''EII..'. .ss,.ajsayis..j:;d jj,j.;y'-. .y..jy.-j..yj.'. y' '1EE@ilill' t.E2Ik..SII:EiIL.'. )' kiE!ll;)d pp;d 711:jFj11).((11).*. E11:q:)1Ly'. (jIr:q11.'. !1!2E' Lj6iLt.'' El1:::;It'' t'y' ;:iEEI1k.'(1rE::)1:qr!I!.'' t'y LL66''. (11:17:111':11t.*) (Jit:r'd Eli::r'd EiIii7:iIl71ikl'. 'lIi.:IEl', IiI:r!ii.'. gyjgygyj.d .yjjjjjj,.' g;jygyjd .,jyy..jjj'. yjjyj(jjjy'. kj;;;r!;d ;'k' jjj,kgjjj'. (jjrgrgjjyyyyjj..'. q' )' gjjgj,yj;jj,'. (jjgggjjjj'. j'j (ijjj'.d t''.tjjrgyjjjjjji.'.;lt.z'.. ggjjg;,.jjj,kyjj.'. Jjl:ld y'j gjjjgjyjjjjjjjg('. gjjyjjrdkjrjjjgjy,jtjtkkjy,'tjyjjjjj;' gjjjyjj'y gjjj.d g' jyjj,jgjy('. t'. y' j';kgjjjj,.'. gjjjj'. E!llqt:il):!!ld -i1,jjI(.' );j,jjjjj'. L'LiLj,,LLI;L''L yjjgg(jj(y'. )'lli?;;d :11j'#11E:::11)*. :Ii':ii)'' kl:Eir;d (I(:'11.'' (11r:.:11::.:11.** E?!.-d 1i:ii..'' -''EEIi(Et'''@11i:i)iL.'-1I.ii1E)'. t!IE:r-'. t1Eii!i-''. y' EE E ' ' ' ; ,j)'. ,')' k'. ?' ).2.:1)*. t'. j))', stf j(jj(': jyjyf r' ttt.!!!!5ii!!!!)-'-' jjly'jf (4(j:4*. y'i )jj' ((' )jf E ' Eg k' :'il:jir:;'. $' y' L..Lj,j..j.jj'. il:iii:d .(jr'. j'y'. r' .(4111j;E12:,.* jjjjgjygjg. (qIE tyjjgygjj.d ..yj,.' j(;,,gj'. 1* jjg;jjgjjd j'. t'ykj,jgjj,'. (;;' y' kjjk!gr;gd )' ,'qEIItE(iIIi:r.'. ;jtt')):)))p:---()((y):)):k'-' p' yyf (r)'. ))' j'jkq'. )y' ;'. ii''f 1* i(' .y('. '/'1* jqf q' t'jtf 'j': (' yy'tyf jjj'. yt:yf y' ( (j( i i !l ( ). 'q' E'. qj' ' q j' E.((y.' i lyj q ig(((j(Ei!r q i ( ''' jjl.i!FE ;E!(jIE.gE' jj E' .( ( . (.j jy (q . ' ' (1y !j. ( j(;(.gi ('k(.(.j j (!(j (ljjr(ij (yi j l@ j ( jy( (.(;(jjikiyy Ii!( j. j;'. E E(j' iEj( y I j r..y yj . ' 'jj jyF:E!i (jtl)j E j((jg .g g.j . ) r(E.'j.((iji t))-(Ejj 4 ;j (. E.jiyjj)(yikij(i' E.g'jj'jj!(Et''i E(;Eg j ( j . jjjjgj k-i-''i-' y.g' .jE. j. .jjj.j j j jyk.!yjt.E .. . j. j jj j --.jjji..Ej'gj -- j(( jj.j.j.g j . gE j ... .. )@t-. j j j jj ' . . i.E. g-. ' -. --; .E. .' :-.i ttt-y EE' . -.E . ;. E !. i. E E: E- ' . -;t('(-. i. ;..2:!:. E- E j j--. ;'p,; . -iEi! ;.' -.!-.. E-. . :i-(. :. -;:E. E . 8thp.. - -.. '. ; !..,':),.k(.. .- E :j; 7:J. r - - i;! ;. i .p );..y(:.-,:.qg..-.-.- - i.ijE! k-. .... ri . !E;k pt..i: . - -..i E - -. ;(,.-.-- E;- ErE-.: - .!- -.. E -.- -.. !. :;)-. -..-- i.' . (- -E ; ;:. : 2 . . .k). k:-:i . E! E . . . j -- y . . . . : . : . -. -y. . . . .. . . .. . . . . . E. E . . .. . E. EEF.. i;. ..y .. -.- j -. . ... E7. ... . - . E-.. -..E.... y-. ...E.Ey .. -.Ep. E.q. . (.. EE.E. .. ; ...... -. . ...-... . .. .... .'..r . - i . ... .E...!.j.) (..-E -. .. . . . . . . --. . . E. i. .. . . .. ;... ... .E.E...... q.yjyy.;. -..jE( q.-.q.y..E,, E. E....q.-. ;...y.q.....E :yy( ..- ;.......- . . ; g . - qy . ...q . y .-. (...y,)E .(('yy .. .LL - . (Et. y . .. . .t.... ( qg.y E ytt E y .. y .q .E....rq.. y(y . y..y... .( ... y.. E j. .. ( j y ! y .(jj;,. !g r. .. .q. . q .. . yy y.. E q.. . E E . ' . ....y y . .yj . . y g(,. . . ;,. i. ..... j. g E Ei jy,y,g . E ' : ( . . . .yy y y E .. E E . . E E . i . E ' y . . y E y, y . . .. E . . . . E . . E E E E . . . . ; ( g y . y . . .. ( ! . )( .. . (.y. (...(.qj.... (.. q(.. yq . j .. y g . g . . (y. .y..j.(..g .. 2.g . q).' . j; . . q... (. . (. . ( ..(E y(((... . ..(.......!...j .( Eq(. q... ... . ( .. .(. (..j((. . j. y(........ . .. . .. . . . . . . .. . .. . EyE E
.''. .'.... . r' i!!!!l;d :Ii:2'. ,L1tt''.'. ytd y' 4!5((;1,* r,,-'. gjijtjjtjy,jjjjjjygykjj'y j'r' jjjjj.d yyd ;j,,(:yjr' kj:...kl,d jjjjjyjjjd jjgjg(jjj.'. jjjjgjg'. gjyjjjy(g'. jyjjjygyyy'g ..y'g 'iI)2)'. y'g jjjjjjjjjjyd .jjjggg;yj' .y'g .jjjjgjjgj'FX tj;d jjjjgjjjj'. l!l'd j;;,t(jj'( jjjkjgj:jj;d jjjjjj(,yjjjgy(jj('. j'jjjjjy,gyjj,'. jgjjjjj,,,jj,'. j;(jyjkrjtjj(,(jjjj,'. kl::j!ill.d k;t:::jjd ijryjjjjt.d jyjjjj:jjjyjjjj.'. /11:tt11127111t.*. i!iil)d :1I:iq'II::' kiIti:'11..::11:.111:1-.*. y'. 11111,1*k,i!;d r' (' j'p' )' t'jqt'.rf t'. (kl).',(rjt)k)r;y;'. :(' g' ;'j'k).rl.'yltt.'.' q' t'-' y-,'!!!!1llI@'-' j)f jjk'.qjtjj k)'. jjf jjjf jj'(jj( j';'. ;'. y' )' (Iiqr':l..d (iil.d 2114!112:1(:11()::1,* 1.* i1:::iiI:.'. .L6q;''. t'j'r't'' LLt'''. (li::ll.d EllEiFi:;;'d (:!!i,''7y' illt''jltfd Elll;;:l'd Eii::''EiiE.'. (22!1,),''* !!(1(Ili$' ()II.)'. LL3b'i''.'; t1:):iII)'. jjLj)j'. ;LL,)L'. )' LL?3jii')'. .y' i:i)j1k,j(jj(.'. ((:jjj;/'' .yyj' jjjjjjjjjjyykjs' .y't' jjyjjjj;j'. jgj'gjj(.'. il.d (jIi(;i:i,' .',,,L. )'. (jj(jjr:j;'. t')' (11(.* (11)* (11.:':11.::111.* ((1IEiF::;7''. #'. kkEiiIi)''. di::iijIt.'; k1iii:.k'' (EIt:,t-...'-''''' tEiIitF::;i-'. k1:(:!II2.'. 1L:t'-i''';-,:1!5llk.(!I(.(1I:.'. Elld.d k' (' t')j' t.tq:()k;jj;'.. y' q);kf yyf fppr'. lp'-ktf j.;t' jjy'jjjj tjjtf Ljjjkljtjj' lyf jyf ))'jjjy(yjj(j(jjjj.. (yjtfttjtjjtjqkqk j'jjjf (,tf 44*4jj44j jyjkf 4* jqf ((j' j'j'5'. .'jjjj;yyjj'.' j'jf j'! jkf g' r' )'.. y'. y'jjjjyyjjjjd tjjyjj;j;d y.yd yyd 1* .gjjjg;;'. ..jj,jyjjj.'. tik:ii!:;'' t'. t7*t;f y'rf )' t)' (--' t'))'. k'-.jp))(. rjr'. ttf rr)')'. 't' t't' pl2'. qj'tgjyj qltfljq'tjl;y)lt)tjif ql'jjjjf gjjjyjyy.'. (y'j.. ('. (j' q' 4*. ((' qy( ' t')'. )jj();:jii'. tjijkyjj'j .jjjygj)j;* )j,)qjj,j'. jjjjyyd kjggjjjjj'pd 'yq?y'. jjjjjjygjj'yd jgjjjyy'. jjjjgjj;yd .,jy'. .tjjjjjggjj': jjjyjgjkyyyjjjj.jyjj.'. kjjyyjkjkjjyrgjjjjyjjj,d jyjjjy;'. .jjjgy'. ilirk'd k1::i::i;'. y .r'. tl'r. t-') t'(444g4j4j4*. .'jj'ygj'('. j'yj gjj. (' (:j j(.(' ( jy . iiE::lilt.d 11.1111.)j1.*. *11.::1:1;* 'ilt:liiiid 1111.*. (;jjggjjjj'. .tjjjgjjj.'. kjrjjjj;d tjrjjjjjj.d (jjjgjgjj('. ygjjtjjrrjj-'q t;gjjjjjj'. kjjjjgjjl.d k;2jjjr,'. (::21.* .,' jj.i ((j r' 't'((' tt'' ty'k tjjjjjjjjjj;jf jj.j 'tj kjjt. jq.tjjr yjj yjjj.( ( j k(ji(',7j'j 'k'(((j..(1 ' ( (.. ( kjj j ((j ( . ' j ' ', y( y(7jlyjgj . jtr.yi jyj''''. (*44 (jjjrkjjjjj, j(j jj.qj 'tjj;jjjjj.. .g( gj.. 'jjj.. 4j k ('j. (('jty . );. (k#i-. ..(. .j.j...'k...g (.. .-..p)l/333)333; . ri.. '.i..llt-. . tl. ..y (. r . t.. ;q j).!;kt;. ..lq . ....r.k(q. t. t--r (.p.. '.'(k.). ..r r);. (.r.l.'.. . . ;(;.. r..t.t-...; )r-.. .. . ).. . ; . f . y'. . . . r.,.... ...... ........ . (. y.(qk ....y.y.. .. . y,ijyr... r) ....j q( '')j'j)))il:(:ll . (j.,,.,,,y,' ) .. . .. . . . . . . t(.'t j y j . . y tjrj . . . . y . .. . t . t:jj. tyr jy y ; ;. j, :,, - rtjk ... . j); ....... .. j . stuwe . . ... dib!
tl:::i)d tz!!i,'!qd t1::i::lk'. .)jfj)44j,)'j t1(i:;'' tyjgjjji'g (jjjjj;jg)'. (gkjgggjjyjgykj.'. rf rilri'kl )j,jjjjjjjjt'. (jjjjjyjj'' (jj;jjjjjjjg('. tl:iiili.d .t;jjgjjjjjjjgjy,'. t:jjj;jd .gjygjjk,ygjyygj,' tjjjrr:jkg.'. r' .(gg;j;' .'.'. .*;!!!!777:)-*'.
.'.'. .' . .. ''.' .'
i
:
'
'.
: .
:
:''E'
-
.
-
.
.
.
.
-
..
'
-
.
.
. . . . .
. ,
I/IF. totf 4:1(11dlqllp :E;i:. iitrt :.
,q
.
.
. -. . -
-: ..
.
. . . . . . . .. . . . . . . , ,
.
.
.
. .
-
:
y.
.
g
'
.
IE
.. . . . :'
-
.
-
. .
-
.
.
.
.
:
'
'
.
-.. -. - . . -. -
--. -. .
.
.
.
- -.... . . .
'(!
E
: :
' tjjjjjjjjgg. .. - '. ..
. .. .
.
-
-
-
'
.
:
' . .
'
.
.
--
:
l jii
-
.. . '1k;$i.pr' ..
.
.
.
.
.. .
-
.
E.
.
...... .
,.--,k-..
.. -. .
.-.:-..
..
..-..;k...
...;.
.
.
.yjy
-
.
.
:
' :
rq.
-
'
'.
.
-.
. -
.
-
:
'
:
'
.
i
. :
.. .
-. .
'
.. -. .
-
.
.-
.
.
.
'
... --
.
--
.
,y
.
. .
-
. -. .
.
.
. ,; .
: .
.
-.
y
-. .
.
- .. -
:
.
-
.ui'
' :
' ' .
- --- -.-
-
.
-
:
---. .
-
'' :
.
. .
--:;jjjjj;;,
''
-
.
.
' '' ..
.
..
. .
.
.:,;. j )ry')tA.7.,. .A.,....,........ rt'.)t)
.
. -.
.-', . -
.
.,,
.y ..
.
'
-
-,
- -
!
'
: .. :' : . ...
-
.
.. . .
.-...r,,,......
.
: .
. :-
-
,
,
. -
.
.
-
. .
. ...
'
'
-
-
-.
-.. . ..
-..---
.
-.
'
E
:
'
-
-
-
.
-
-
:
.
.
.
--.
-
. ., r... tkti'Jkokiii tnlfkkr .jy-. -
.
.
'.
'
:
.
.
'
jttE
:
:
:.
:
-;.
-
,)
- .
.
tjzhk .
.
:
:
:
y
- . - -
-.
-
.
. . .
-
-.
. .
.
. .
.
.
..
.
.
..
.
.
.E. . .
.
.
.
.
......
...
.
- .
.;
...
.
:
.. . . ..
..
..
..1(:j:f.
. .
.
.. .
.
.
.
....
...
...
... . .
..
.t
....y
.
feit ytttinj/ikiljtgi,, #t
....
.-
-.
-.-
.
.
.- . .
. -.--..
..
..-j,;.;t-. . .
...
-.
..
call
.
.!1:::,.
...
IIEEI,E
.-
.
-
..
..
.. .
.. .. . . .. ..
.-. ....:jjjky.
..
- ..
...j.
.
..
..
...
-jkjjrjy. .
-
-
..
-.
y
.
..
.yy
.
-
:
.
.
.
.
.
. : . y.
. .
'
. ..
.
. ..
..
... . '
.
. .
....
.
.'
....
- .. .
. .
'
'
. -.
.
.
. . .
.
.
. - . . -. .
-. . . .
il::::::t'
-
.. ..
.
.
-.
-
.
.
--.
- . --
. .
.
.
:
.
.
.
.
-
.
.
- . . . . . .. . . . ,..
. - .. .
-
.
. . -. .
-
. -
-
.
-
.
,
..
tt
..t
-.-.yjjjjjy.
-
...y....y.,y.yjjg.
-.yjjjjjj.
..
-
y..-...y.j.y..yy-yj...y....-. -yy;jyjkjjjj,...-.... .
..
.
.
.
'
.
.
..
,.
' ' -. .
.
-
-
-
.
.. -.
..
.
...yjjjgjj.. ..
.
.
-.
-
. ,
yjjjjjjjjj,y y y
.
:
.
.
: , . .
.
...y.
: .. . .
g
:
. .
... .
.jjj.j:jjj
.
..
:
.
:
.
.
. ..
,.
.
. . .
:.
g:. ...
:. .
y. .
.
.;;..
:
.
.
.
. ..
:
. .
g
.
:
. . . y.y..;,.
:
E
.
. : y . :. . .. . : : y y .
.
.
:
..
,
!jtttj:::,
.
.
,
g :
:
. .. .
.
'
.
:.
:.,k,,jjjjy..
.
'
..
.
.
.
.
:
. y.
:
..
:
.
...
.. .
:
r
-j............
,,2;11k1'.*..
..
.
-'.
.
. .
....
-...... . ...3t.. '
-'itb
'
..
...
'
.-
..
....
''ikilk-''ii.
...
.
.-
.....
.....
....
.
..
.
.
.
..r
- .. .. . .-.,....,...
.
.
-;,gk.-
..,,,....,,..,. ..
-
-.
. -
.
- .
-.
..
..
.
-
.
.
..
,
. .
. . -. .: . . .. - - -- . 'E E. . . .. -. . . . -kjjgj:jjk. . . .. ..
E:. - ui'i .
..)
:- E
.
-
-
..
.
:-
- .
.
.
:.. .
.
i.
:
.
....
..
.
. :.
.. .:
.
.
:
: .: :
..
.,.
,
.:....2.
.
...
,
:
:
,
:.
. .
:.:
y : y. r, .y
y
.. y. . .
.
..
.
.
:
;.yy
gy . ..
:,.
r
:
.
.:
-
...
-:11!9k.
.
-. ..
... .
..
...
.
,.t.
g
, .
g
.
. .
. :
y.
g . .
.
, . :
:
.
-. . . . ..
-
.
.
q y . .. . .... . . II:EEE:IP'E ' :,i:
-
-
..
. .
.
.
. .. .
.
...
. .
. .
..
:
-.
.
'
E
-
.
.y. .
q .. .
..
.
:
.
put premiums for gold options.
y g.. g . . . .. g g . . g.. g .. . . ..
g.
y jjjjgy..g
j
. .
g
..,..g .g..g. g g sgy.. .g
,..,.
. g
.
:
q.:
g.:.. . . y :..
.
-
:
.
..
.
y
.
..
. y.
: .
-.
-(.k;..
.
.
-.(. .-, ..-..
.
,,,..,r..
-
.--.;;;.
ukil-y
..
.
.
.. . -
-
-
..--.-y.t
..--(... .
.
.-
....
.
-
....t..
-riy-..k.qg
..
..
...
....
-.r.
...
-
. .
... .
.
.:
.
.. -
..
. .
.
.
-
. .. E .
. . -
.
-
-
.
:
- . ..
..
.
....
....
. .
.y.y..E.. .
.
.
.
-
!.
..
... ..
q ..
:
.
-
.
- .
. .
'-iik;'';k-i''.
. . . . . '.
'. . .
.
'
dl::::::.
: .
'' ..
:' .
j
.
j y.' :
. . . .
-.
.
(. : . . .
...
'
'
.
...t.y.
.
.. .
-
.
qy. y .. .-y. .' E . E ! - i-E . -
.,rj.-.j.yys.r
'
:
.
.
.
'
'
'
.
( '.. .
:
(.... ' .' .. .
4.4 GOLDDIGGERS
:
.
-.
.
,
. ...4... gsy..y .yg. g . ggsgyjygg g g.. . g . . .
y
:
.-..
..-
g
.
-
.
.
.
. . -..g:jjj. . .
.
-
. .
.
..--
-k(yjjj.
-
.
E:
.
E
-
. .
.
.
.
.
.
. .
.
.
. ..-
-
..
E.
.
.
.
q . i . 'l...kl''y
. .-. .. -.
,
-
.-.'.
--
..E.ry......
.
...
.
.
' ' ' ' . ' : g. . . r . .: . .. . . . . . (. . .. . . . ..
.
. ...
.
q
!. .. -. . y 'ILLLt::'E.
.
. . ''bb''''
,.
.
...
-,E
.
.
.
-.
-
.
-
- -
..;
.
..
.
E- -. .
.
.
-
! q .q ,
E
.
.
.
q.q
. .
E.
!, ,.E
.
.
.
.
.
-.
E:
.y. .
l
..
-
..
...
..
.
'
'
. -
-
-
. . .
'
.
. . .
:
:
.
-
(' . y...
E.
...
-
. . .
-
:
.
.
,
...
-
...
E .
-E
;E
.
-
.
..-..
.
..
-
....-j...(jjjr,.t
y.y. -yy.y .q... . yy.... -...y.. y -. .. . y.. -. ..y .' ' E -. .. EE. . EEEE E. .E E.. . . E. E . . . .. . -
. .
...(112
-
.
.
.
.
'..
.
.
.
.
. . .-i...-. ... -.-.i-..-
...
,
.,;
.,
..
-
-
-
..
y. - . y .. . . -.. E -. .
-
.
.
.
i . . ;.
jjjjjjrjr
.
.jjj.
.
'
,
-. . . .
. .
.
.
i: Ej
-. . . ... . .
.
E
... : . . : . .. . . . .. . .. .
.
.
. .: , g : .. : .
-.
j. . .. . -
:
: . . . .
.u..)..
.
. ,
: . . : : '. . . .. .. ... ..E. E
-.
-.
-. -
. .
E:
- . . . -
. -. .
. .
.
.
.
.
.
.
. . .
:
. . .....
8.77
.
.
.
.
.
.
.
:
. . .
.
.
. : : . : : . ... . . ... .. : .... . . . . . . . :.. . ..
: . ..
: . .. .. ... .. ..
8.77
Suppose that Golddiggers buys for $8.77 for $2.49. Golddiggers $450/oz., price $440, $450. The $2.49 initially happen. tem Figure 4.9 depicts the combination of the purchased put ad written call, while Figttre 4.10 shs th two prot diagrams for Glddigers hdged with a 420-sfrike pui, hdged with a 420-stnke put plus Writinj zlklo-jfrlk cll. opposd t as
..jjyjj...
,,
-
E .
.
,.);.
-
.
y .
y .
y
'
.
E.,)
,.
:..)
'
.
.
.
'
.
.
..
.
.
.
.
.
. -
.
.
-.
.
.
-
'
.
.
.
:.
.
:
.
.
.
.
.
.
-
-
.. .
. .
.
.
. . . . . .
-
:
-
-
.
.
-j
.
-
. -
-
-
. -
.
.
.
-
.
-
.
:
.
.
-
.
.
.
.
.
.
-
-
.
.
.
-
-.
.
:
-
.
.
.
-
.
-..
-
.
-
...
'
. . . : .
.
.
.
-
-.
-
-
.
:. .
- .
-
:
.
.
.
.
.. .
.
..(.
.
.
:
. .
-.
'
.
-
-
-:k..-rjjjgrjj...t.y
- .
'
..
.
'
-
-
-t(
-.
- .
.
..
:y
: :
-
.
and year at
g
.
.
-
'
'
'
.
.
.
' .
y
-.
.
,
-
:
E
E .
..-:..
..
-
-
. ...
.j
y
.
.
...
- . (.....,.kE.....,(.......E.Er.i .. .E....
-.
i . 11;i::::.
.y
.. .
-. .. y
.. .
.
.
-
.
g .
g
g
....y!.(..g((jjj(y..
.
-
-.-
.
.
..-)
..-
.
-. . .
q.- ...
.
y
.
-y...,....(.r
.
..r.q-
........;..,). ..-.-(jjjjj.g.
. EE
EE
:g
-,kjjjj.-jjyjjk...
..
.
- .. . . ..
g
.
y : E ..:( . .. .. . .. .. . E:...-.E.:- ..E . :!. g.t E.. : . :y.E q!E .:. . . y E E.. .- .( . .(q(-.E - gE.. q . !...y . . . . ' )...;.. . . ( ' q ( . .: . . E-. -..g.. -.. - q g'' ' -. - E-. ' E '. EE- --E' - E -;-... . '' ' - ' ' E ' ' . ' ' ' E. - .. '. - E.. -.-... ..... -... E...- . .. -. .. .. ' ''E- ' - ' . . - .- . . -- '.. . i ... . ' E-- . -E'.. . . . - .. . E. -- . . i E E: - .E E. E (.E: - E.' -..E. :.'. EE.. ..E E . - E - . . E. . .. :. :E. .: . . . .(. . E . . - E i. -. . y q (( E E: : : E . -... (g .. i.E ( ( .. . . g - .. . q:( y .(.. ... :. : yg g : ..g . Ey r y(.g q . : . . yq q..y . qy . .. (.: .. .. . y .. .. . : g . y . . . j... y. . . --. -iy -y ' ' -. . .' . . -E - (E. -.E . .E -y- . E -. . ' E. . . . ' . .. .- E . -.E-.-. . . -...- ---. -E. E E. - --. E E. E. -.- -. E -. .. - .. - . .. . .. . .. .y -. E. .j .q .. . . -- -... - .y.- - -..- . i ----. . ..-. EE.-...-. -.- .- E - .. -. ..!- -. . - . . . . - g . . ; i -E.. .. . (- -.. . - .. .. -. . .. - - -.. . . . - - -. y . - - - . - - - - :. q . - . y. - . q . .. .y . .- .- - . .. y -..- E' -'' ' . Er . .... y .-.. .y- y - - -. -Er . .-. E E E-- E E y - -- y. - - E - - ..-. . . . ' ' '.'.. ! . . E.E E. . E. '. . - .. . . . . .- E. E. ... E. -.. ;.. .iE. E. . '.E -.. .. . .-... .. E E . - . - . .-. EEE-. . - ..- . . .-. E- .. E -- E . - .E E - i .. . -. E . .. . - - .. E.. ! -.. E ! . E . . . i . . . . - -. - ! . . ; - - . - . . . . E;- . .. . -.. . .. .. - . . --. q. . . q q ..-. - -. .. . -. q.... - .-. g ... . . . . . . .. . . . . . - ... - ..-. .. . . . . :' ' ' ' : . E . :. .. y :' E ' ' : ' ' :' .. ' .'' . . : . . . ( ( '' ( ' g '(. : ( : ( E EE : . :... (' :.. . ( . jj.' . . .E. 'E '.. . . ..gg.' .. .. .. . . . . .. : . j .r .: . . . : y . . . gj j g(g yy( : ( ..... . . y. . . . . . . j. . y. . . . .. j.'.j.y. ... .. ..g..(j. y. ( ...y.... .y(. .. y: j'..... g jr . gj'.....g..j..j.y... j. .y . r . y.. . .. . . y... y. ...yy . . .. ...y. . j. (g ..y . ... . . .y . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
... ..
-
-1i!2r:.-
-
.
'
.
-.
.-)'...r-.-'.-.
! .. .
:
g. . g . .. g.
.
..
-.
...
.
..
. -. , yygsy... ..y..jj..(. y .. ..12112::.' i E q y.. q 'lii -'. -dq;:,'.. kk'
-.
-
-
. . . .-
g
.
g
. g
.
:
y
:
:
.. . y y .. .; . y. ;
.
.
g
g
.. .
:
. .
-
...
-y-..
.- ,
-
'
' -
.
.
.
..
: . .. .
.... .
g.
. ..
...
-
.
-..
..
..
- ; . . ' :(!4:::. . . EE E :E . .E E . ..
.-'ki2d.
..
.
. . .... . y.. .. .
E.
.
..
-.
.
.p1kti#y'r(..r',.....,... ...(
. .
-.
...
.
.
.
..
.
..'.',iik..
-ik2,:i-.
-.::r:---:.::r:---:
.
!1i::i),-
.
.
.bil'.
.-1,,-;-r.
-,-.' .
.
.
.
.
' E
. .
-r::---k(i)k.'
.
.
..
..
-. .
.
.
' E ' .. ' . . . .. . . . . . .
... . . .- . .. . .
:
.
g g . .. : :y. : , . .r . : y y. y . . .
g
.
y
g
.
.
.
.
g
g
.
.
:
:j
.
y
g
,. yy yy . . .
.
-
.
.-..
..)).
; . .. 'iiii,:i.
..E
-.
.
.
.
.
y.
:
g
:
.
,
.
y
.
.
.
.
g
....tfq,3b..-,.;;...
.
..jkygjj..r..r
:
.
.
g
g
. .
-g.(.t
-'bi''i''-.
...-.y
-.
.
.. . .
-
:.
.
:y
y ,: . . :
:
-
..t
.
' :.
. : .. : :
g
yg .. ., . .yg... g ....... . . g g. g gssyyyyy g g. .
,
-
..
g
.
. gg
-
- - . . -. -. . . . .. .... ,
.
:. .
. . :. . ' :.
E
E(
-- . .
. . ..
y
.
.
y
. .. .
.
.y g
y. .y g
.
g . . , g.
.
E .. yy . -
. -
.
.
-y -
.
.
-y
.
-
:
:. : :.
-.
-()-'....-
...
.
y
...
: . .y
.
:
-.
..
.
-
.
-.1;'-
.
.. ..... ..... - . ..
.
,jj,..
-.
.
. .
-
:
.
.
.
...
... -
-
.
y
.
:.
y
g
,
-
..
-
y.
-
.
.
.. .. -
-
..,yyys,.s..(
.....,.y..
..,...
:
g g r:
.
.
:
.
. ..
-
-j
-.
--
.
-
-..-....y
-
-.
-
:
:
. .. :
y
:: :
:
:
-. ..
-
.
.
,y...j..
. .
..
.
'.
.
. .- . -
..
y
....
.
y y. . ..j .
.
.....
.
.
.
., . .. . .
.. .
..
.
:
:
.
:
E'
'.;
- ..
-
:
-
.
.
-
'I'''....
-. -
.
. . . .
-
.
.. ,
. ., .
.
,
y
.
.
.
.
.
g
-
-
-
.
,. .
...-..-.::2::..
...
....
.
-.
.-...-(.
..i
-
-. . .
, ,
.
-.
. ;.jj(jyygj,q . .
:
' ' .
:
:
.
.
.:
: .,
;. .
y .
y y.
: : :
........ .
:
. y. . . .
.
...
-
:
. .
.
g
.
y
.
. . y
.. . ..
.
-r - ..:g2....
. ..
-6,:.
.
.
.
.
..
. ,
,
.
.
:: . ..
.
E : : . . :
: . . :
:
'
.
-
.
-
-; .
y
. . :. .
: .
.
g
. .. : y. .
g :
:,jyy,,;yj,.:
..
.
:
.
:
'
'
.
.
..
.
: : .
.
. g
.
.
.
:
. .j,k...;,k., . .: y. : .. :. y . . ..
11::211...
.
y
..
-
..
. ..
.
:
y
66::.66.....
...
. ..
..
:
. .
.
g
:
.
.
.
.
.
.
... -
'
-'
.
.
.
.
:.
:. . .
. :,jjj.y,,: .
:
yg
y
.
. ' . .
.. . ..
!
'y
. : .. . . . . . .
.
.
.
.
'
-
.
:
y
:
,
.
.
.' ...
..
.
...
..
.
:
...:
.
y
. . .
. -
r
.
:
-
.
-
..
..j
-
. .
. .. .
-
:
-i!iiijl;2..
...
.p.k
-
- . .
.
,
. . . .
. ...c
.
..
.
..
.
. - . . .- . . -.
. .
: . .
:
:
-
.. . g .. :
)..
420 400
:(j:).
:.
.:
-
.
.
. .
..
...
-
.
y.
.
.
-
-
.
.
'
. .
-
.
.
E-
..
--
-
-.. .
E.
E .y
.. . -
.
-
-
,t
-
'
,
:
s..,.
,
.
.
j;gi
.
r
. . .
..
.
.
'
E
.
.
, ..
..
-
.
.
......yy
.... . -.(. ..-. . ...-.
'''i''....;''.
. . . . .
.
s.
-
.
..
-
..!jjjk;;jjjjjj;j,,jjjjjyggjj,g(jj,..
,jj;,,,.jjjj,;j,,. .
.
.:
'lrqiiqtl'y
.
...,
.....
.
. ''itk-;'-i
.
.
jjgjjgjg;E
.
.
.
w.'
'
..
.
.
:jjj;yjrr,.
..
IiiEEIiLEIIII.' i,ii.lk. ' 'i')5;.. ' ) ..y).-...... .... .... '.. . ..-.. .... '.) ..y)j..... .tiil. . .'. (111q::11:yt11:.. '. ' ) .. j . ( ' . j ' . .. j .. ( kjjyyyjj;.. .. .(jjjkgj:j. . . . .. . .. .. ... . . ......... . .'. . .yjjjjyyjjyygjjjj .... jjjj:,. . ....j k;yyjgjj,. jjjj,.. ....' .. t t;s!ikzl j!jj iltj!:it !(j))... . . . . . jjjy. ygjjjjg;j;.. $jjyjjjkj-. yjyjy;gjjj.. b? . )fif.j?j tjkyjjj;y. lygjytyjyjjrj;j fj,)j.. . .t.;jgjjj,jjjjyyjjjj.. . .. . ..tjgjjjjj(,.. . . kjjjjjgjjj . tkjjjjgji tj;gggyjj. gjjjjjj,(. (;jjjj;g..tyjjjyjj;r .. . tgjjjjj,.. ..'.tggijjjjjy . '.1.).. ... . . . .gjjjjjkjj4jjj.jjjjjj; '...' ' @* ;L66j(''. .. jgjjjjyrj. . tkjgyjjjjj. j jj.yyjjj.j.jj.jjyjjjj.jjjjj jj.jjjjj.jjj.. . jjjjs ....j j .. . (jjjjjj,.;jjgyjkj.jjjjj.. . . ... tjjgjgjjj,: j.jj.jjjjjj. jj..jjjjjjy j. j j.... .jj. j yjj....yj .j),jj)ffq4),).. . . ty . . .tjjjj(.. .(jgjjjjjj,. ' .. .. . . .. jjjjggjj.y jgjjyjj,s .tjjgggjyj,. 1i!i111ll. iil. . . . '. (. jkrjjjr;jryjj;j;! j jj .kgjjjgjj,.. j. . j.j...g j.. tkyggjjr;:. .. j.jj... ...'. j.. '. trj!j,;prri .g. r?!lr. . . . j . . .yiilEii'. j j . .. t1L,t.(.. j y kl)hk. *..1102. .. y . .g . 2t12'. iL y . .. ... .'.. .'. t )'. ii' j y .g '.....''..... t1... . . . iii:i-. .' ikII::(IE..f(f'.. .t1!E.Irq.. ili: ..:::1,-.....jjkj;( 4(El2i'. yj,jjkygjjj,. kjjjrr k::iil.. .gj:;,... .gjgrjjy..... j1L.. tj(rjky....j(ygrj., tlll tjjyj;,.(: jy(.-....yy.. '..' ' .jjjj;pj.,......y:.t.g:r:jj... jjy;j...). yjjj;j,..,.....t Jtik) .jjjjjjyjk(y. .jjjy.q pt iikjrill'r tks y y jj yq.jj ...yjj.jy.. qj.jjj. y y . y. yy j y y y..yk$j,j,. y... . y. y.y gjr(tiyyjy y. jg.jjg,j...jj.r..kg,. .. . t ry y.j.... j.L,Lj-. . .. ;1!ii!1.j:iI7''. ..gk,jy.. tl!Eiii.. . 1i1(E!iI'. .... yy., .. y lli:lit. . . . . iI:tj:!:;. yyy. . .jgy.(g ..yy( .y..C'.ry. '' .... y.. .. ilEEE21IL'.ElilEc7! 'i:::!iI.. ... .11:... li:ElE:ili ljIi:7lkl. .. :1.::11..:11E2.11:.li5E!):. l1Ii.. . . '1IL.)1I:. di::lilIl.. klliii!i;E 'I:ii!:'. .11:.... Eill...llEz.lt.. (iIE(iIIEC(1IL.. .'IE..ll..y qi-'k1:iEE:1li. .'.'.'..(111E111E!1:2 ;1i::1I.2:II:E''. EIIL'EII..' 11!511..'' ..!lil.1:EE112-.i.!.. ( ..(. . . . ..... .. . ; ! ... ' ( ) q . ( ' qy.( (y . ( . ; . ' . g : j . ! y ' . y . y. ; g.... ..211:.. (y.. yygjy.(y... .. .r...19' y. dii,' . y .)'..'.t. . .. .. . .. . . . ...j.... . '''ik;,i,;F'll::kLi.di::' ...-.'-..''.-'...-...E....'..--'.-;-'(. .. t!i!!i ;?L'b'ii;'.. . iI:;ii:iif(r'$iiii.. .d:EEl'...'i1::ii.. j....yj..i..-..... ..1iiil.. .. kk.kii . . .....-. dk:i;. 1/51.):1... k:!!It.. irEi.. :i::'.ki::EEi;l.tir,ht-;r1I1'. ii(iI.. . .Eii$i:i..1I:llIii.. --.rIr'1'''''......'Ir'l''.1..1i. tI!.. 'iki@i;. IE(r:1EiEi?7'::i,;.. 'lI2i, il'.i7ii..' ii:.11:11:51(7). EI7'.itiiill. '..- ' .i(:E;''. kk8:iik.'ii!i;-'.'.'..t1!iif...1!ii)?..)11.. .'. iiik.. dtEElp '.- ik,ii.. .l'.....-yt..l' .'..)- ..-.'t.'. ...' . .''-'ii,:.-ii. '' k:!(:'. ?..r.(,.y.r..,..,....k. ..L...L . . ). (.,....(...(.:. q..( .. tlllt?' j.( '' k ( . .. . ;...,.. (. . y.... t j(.. t r... .'. .).. nronq,,tkkjtt. ).tt,, .. . . .. t . .. . ti/, . rtlf t iectrfkkk .. yafs . . k#j#.,t.i.t.,. thktjl ..t gy .. ' .y.. y. ' ( E (..y( ; ( .(.y.'. .i..i. i '(j.; i@ ill::EltE:t)l.. ;' ' .'E. . . . . ki:::::''' kk:ii:r:k1i::rl.' (tIi.l 'Iti:(fii.. '' .:11:7:1!4. ; '. '.i.'' ;'. '' ilri:''Eil..' ' i'.i''''' '' ill:q'ii::li. ' '. tk::::!d. ''kl::i:ki' ' j.q.y.(j(y.. . ;!..jj yj(j y' ... ( ( y. ... (y.;r.....y..(..y(.y...( . k' y..jj(.(g'.l. ( (...:..j..gjr.j.( . ;.,. ('..' '1;iE::i'. . ...;(..'..j.j .r.j j'.'.'..-:'.....)'g;.j..y.. j.y(......ky)'(.jyj(. . j.....(;!;g..(...y.y(:.(;. y.y'jy.'j.y '. . . . .. . . j. .. j . .j.j..y... .. . . '1:t)t:li. . r . i1!E:!'.(:1.:@:1l.. ' . @ . i i il:EEt:)pd .y ; . ; ..... . . @. rtl;;!i:l ' ;. ;. '. . iq @ri't::)l:.. '. . .. IIEEEEI),..j y. . r.'t.t';i:i,.. q'. ''.) ..... . .'iq ..., .;;)iy..jr..yj..y...y.-. ( .y...y . .. ., ! .;.. r. (. .... (.. . .. . .qj .. j . .i.r.. !j... .-yyqy . . y.y ,.. . g. .. r.j . . r g..... y...j..( (y.. .. .. j y.r..... yr ' j ... ...... .. . .. . . . ' y..(.. y .yy... . . . .. .. .... . .. y y . . . . g y . . . . g ... j .;:!EiI1,.' y! . j . . . j . . . .g y. y . .. .. . . . y ., . E . . ' :!i!!i.@. .1dtEE ' ';EFE;. '. ' r:li'qlltiq?' .II2jr,) .11::: ' .r1It:t:)..' ...... '.. .i.. j... ... . . '. . . .; ;. ;.... .. '..'.(.. ..j. .....(. y.j;. ) . @. r i ' I1Ei::t,,..;. ' .'..' (.'.k .'.;.. (. ,.. '.... :.) ' '. . .. ' g.. .' ( ;. .. . ;(.. . q..r.... q ) .. . .. .. .. ( y ''. . r . r t. . r r . . . r . r ) . . '....'. r r r .. . .. r ' . . . . . . . r . . . 1.i l . . . . . . . . . l l l . . . ' ( . ; j ' . ( . .. ( j ( ( . ( y ' .. (... j. j(. j(...' ( . . g.... ' ' ( j (' '. . ( ... .. ... . ..j. . j..('. '..... j.. j. '. ..... j... j. j... ...( .. (.. .. ....... . .. . . . .... . .. '
-
'11!1(111:::1:..
..
E
...
...
....-..:yq
...
..
-'b;'bi;''..
.. .
..
-
-..y
.
.
.
E
.'6b;..
....tqq:q!!
...7r
t kjjjj;. .jkyjj;j(;g... .kjjjjy. .kjjjj:. .(kjjj,. . jjr(jj.. ..gjjjjky. . tlll;ii .gkgjj. jjjjjgjkkjjj;jj. kjyjjjy. . (jjjr.jjygjj,. jjjg;j-. tjt. (;,..yy tjj;j,y.jjj;;.y. . .(jj. yjgjjkgyjky..y sjkki.q) ..gjyy;jjt,. iy,.,y,j .....(j(.y........jjyjjj......y . jyj
.. .. .
....
.
... . . . .
-
.-...:E$..
-(
...4...
.
'.... . tik.llt. qiikk. ii!il. . . .. ljt:ji jiii;... . . . . . . . . tjgy:,i .tk:jr .. ii;.. 1!!l... y:j,.i!rq!.. . .
.
r).
109
EE E Er q ......-qqy..-r....-(...q...(...-....-.-...yyE..iy.. IIE.. ( yy.y.E ....,...-,g-j:,,E ikyikjy.. E.jy.(.....(y.--..,y ..E......;. y . . . . . . . . . .. q.. . 21..ELk.. .k.. t... . . .......... ..-..tjE-r..(.-,.;....)...;j .E.: . r........t;.. .-.4.rf. .. . ;. ;k: E( . .....tj.;-.. ( EEEi:' i (; (E: .!...,..-..... y...L-L... ...t. ..tjjt...ty.)..... ;).....j; )..y.......q.... Ell::g # klil:.f.kll.-,. . . . .E. ...-.. (!. E..E'.q.iE'.E. ( q;i( ;.(( (.)q.. E..istqtE( E.!Eq.!.' EEjy('.. I111t(!(q!2!!. rk....q......E.-,...E.y......qE...'.!y.'.y...EE.....Ey..E.(..y....j...E k. .;.y.....-..Lj.....LE.q . .,i;pk;i; . E . . . (!:E...)( . :il. ::511.11:t41. .. ..q.g.;.qi!i'..y (yjErg ..yEy Ii!1t). ... 7I::I'. 4(::j4 liit:411!9; . iIjt1l). yilzllr-'-y ..Ei.....-..y-.j..-.,...q...iyy....-....q(. . ..E...y-. dt::::y d(::)p'...11,.11,.11. .(:2::.. iE7q1.' ltt)i. 122.. rg(y(qy.E.q.....(yyy(..E...yy..,E...kEy...(..,.y..Ey.....E. :...'''''.y..r.......'*'*'*',.E itiip. ..;)..-.....(...k.-. '.. . .... g . 11::. ijj;. E-.).. -.E.-.;.E..(.-.-y..-Ey....E....r :112:11::. .*'..,L 4k:k. 4k:2l;-y.y . . Ey.y....(..q'...r',.()j...,..y(.y'...E.q(..iy........g....y.y(. .'.. ). '' !!i:. 'IL(!I.. -.:I:7)k.:IL.. ;!E1l.. kli:ip (1::21., 4::E1,. (1r:'4:EEIl.. .-:i!I-yy..-. ' 51:77j1. ...lEiIL.j... lil;y qIp:1I7q!:7r:!j,rr;.. ::;!:. ' l:ii.qig:1Ll' 21 . :!t14;.. il;ll. (11.)12.t1. y.y.- (ll.-1I:L1l::1l... y... 'Ir'71''11. 1It: . .... y..E......E...y...y...y.Eg...j,,..yEg......g.g.k.. .i:jiI. . . . .. . . . E)...'..,.'....y . . ).t.... . . ..........j.....L.j....LL).........L..j.L,..,. . . ..L. . r.. ..y... jy.. y . y. , .E y y kk. !Ejrjjkyjjjkjjj:.. . gjjy..E. . ( .. . . .( qt-. y .jjj(jj.. .r.jjjjj(,-.. j... .. .;.-.. ..y jijjl.lil iEi1li:kk.. ....q k; .)jjjyj(,. .. . .... E y . . . . ..... ..g.qy-y .. . . qj,j,;, .;. . -. : .);). ( (. . t. (i:#21i(ik. .. kj;j;j,. ;kjj;:.-. .. .1t k(jjjj.iil!ij(1Lriit:1.g....jjjyjjj. ..y.y :124,ilyE jjyjj(,.. kjjj:). .1k)iL)1I()' .y.j. ( tyy (jjyjjj. ji:r,i.:il;. . . 41::1: jjjyr;. kij.ji!zjs . .y j.. y.y.. .y iiil. L1giIyj(yjjyjjj.. . . . iz: jjjjj(.. ;!kdi, .jjjj.tjj,. y.. E g.. ... gy.. jjggyj.jqFyr?ryjjj......iti2:;.-;jjjj;,. ...y.E. ...q....; jl.)jt. 4::;)d::::: .,;. . . ir:.t jj: ,(:@y,, .. .jjr.,;jjr.,jjjj ljjjs llpdti:tjg;:li 422:, .j.'5.'.jyjjjgkgyjj!l .-y..y.g.y.yj.y..y...gy..yE...y.-.y..y..gqyy..-.-y...y.-.-......yyyy......yy..j..y-.y-.Eg.y... kl:iii ....j (12. ... il:E:i. (jt(:i.. :;. t.'.'(. . jkjyji ..y . ( .y((.. .. '''''. . . (' ' .. .. kjjj::,,. (l(... y E. yy.g y...j..jjjjjjj. ... .... .jy. E. . . i' (. E. ('' ..' y. y . (. . .. ysyjtggjj-gy ... yy. . .. .. .. .g y.(jjjjjj.-.y '(.. .. . ...i;iL. . . .....;jyg.y.-$jjj;. .. . .y ..... . . .. . .. ...j(kjrj.. .y :jgj;. .(. yjjj;yj1::j1t... )it-...y. . . .4tji:;. ,it2jr. . . ji@2:;. I:gtj:1). :rit,. . gjyjjj. tI:--..4(: . . .. jjj(yjjyrjgjtjg,. 4(1t), .iE#.1l:(. (j1.21:11.. jjygr;... .y. jl:r'd:jil. 4:::);.. iI2:... rIjrjI.. j;yjjgy,rgjy:jk.. kEjiI). . lI:gil.. ()IiF;i-. . 4:::: 4:2::. ji. ik:jf. :!(r'.jjj,. !ljq.:g!LyqIk.. :j;.4j2:: :j,t,. .... .. ...y.... ..qyj(jgjjj.......yy.gjjjyjy-..-y..... .....-..t..y ..L.....) ,j ..j. j;y. . y.y....rj :I40 21.54 y,. .... y 2.49 ...j(yjjjj.g . . . .yjrgjjj....y.j-..-j...-j. y yjyg.k,.yyjjjjj..y.y.y. ..y...y , . ....s..y.sjjjjgjj. . ...,,. . y....,y. .y...j y gyy.y.-yyy.y.yyy...jyy.-jy....y..y...yy..y...g-..y..y..-.j-.-.-.,g....y......y.yyy.y-y.y..-..-y..y..yy..-..y........yy.y.y... .. .ft' .jjjgjj,y ... .... ..y..y..k(yjj(jgjjy;.y y;r,'y,q.g..,,q.':i,.y,'j#f,.. . 111. . . ,,, E q ..L jjjj. ,,y E . .(; .y...j q. j.( (j.j((j. . ( ..... (. ; (..j .j.y(.:..... y, ( .) y(...(i.. ( y. j.(.y.. -11:1-. Eiiil(:ik:ii'.. . -1:::t:.,.. - E . . . '11::::.1::.'.11.::.11. . j .. .;( .j.... -.j - .k- ..i. -. .( j(.. ' . '.. EjE.jj - E--- -jjE. -. E ' r,I:i:)':.(iii).. i'i!ii.. . :1..E,1,.:111.::111F:111.. :'!i!;;. 'l::::::. - !. :hE!5.! . ' .E ' .. E 'i1iiE'iE1iitq1ii5T7'. -.t:EiiIE(. -.--. .y (tII::i'. .- . . . '.y. ' . .(41!:::1..' ,'E. U.E''EquE,' E(..t'Ej -il) .. ..ir-...'i.j-..g.....-..;.-!..;r-.. ,.;; ... . . . .; ;. (;-.r...y. 4,18.;.- -. h . . . . u .'i!,!;k .. '..'. - .. ., ..,-..- ; . .. .' E (.' r.tir::::.iq.j (.. E.....--....:. - q... .;..'.'.' (' i. i ( .i.( . , .).;....E (i . .L... ).) , )... . .w..:1 i .1..Ef )41::)::1k.. (.vr, ;-. . . ; y y ) ( . . q . . s'jE(E .v'r.. . . - .y yr . . i (. ( E. E . ( . E'' .1:Ei:iI;. . . .y . r. . . . ' ' '!'' !1(:::)i. E .. . . .. .. . (!!!i,iIt'' ' . . i . r t . E E. . .l2tj!r). ' E . i. E. E 'iiliq:ilL .;:!Eiii.. .II!E(EI1:''1IL..' . :!Iq. ' . .' .. . i!i!il. .. . '.' 'Ili!ii). . kiiEE!;. 1Ii!:!:;' y .E . y ' ...dIiiij!:.' ' 'E 2.21 21.26 ' (Iill;:iii '... 'iIi: ()I:':lI:.. '111511:t41::... . ;IEE!i;. ..''' ClE!ll,. ..y .yE. ..EE. .. .. '. .. TiE '11t:: '' ' '.E'-. '.;E E.E..;q.Eq . y..iq.q ..y..y '11:2(. ':iEijld.. i;:k:!::. '. kI::2jiI(..' iiil. . .. ( .r'.... .q..... y'y.(yEyy..' E... .....y.y... (. . y .ii.@lii.. . (.yy .rg.jqj.. (yj.y..yy . ;..(..E.y....j....y..y..yy. -;:jjjjjj,E . i .E E E;r . . . . .. E.. ... ....(jjkjgj::E E E. E . .. . . ;.. E E..(jIjy(. .... jLj,bii:ijb'.. ( ... .E... . kj;jry!::. ..... ( . ...... . . jj3LjjL.. . . .. .. E. q..y .g.y j. y 1:!jE11..(11((.(41(.. . j y.... .. ... rjjjjrjjjj! . j .. ...y. ygy.. y;..j..q.).yy. . ..E. ....yj. gEE..y...j..i.. y... . yj.... .j. . y . y. yjj;yjk,. . .. i q..E..( .. . .........y.... ..j. j E q.. .E.Ey. y.;.y...y ;i:EEilg. ..(jI(:r.:iii.. . y . . ... yy . . . y y. y (jlllt:zjlj j y . . y . E . E .E ... E. . y . . . y y y . y ;gjy(gjgjy,jjgjjgjgjj,.. . j y g . . . y. y y . y y :,E4;; y y y . .. . y . .. . ..yr. g.... ..y.y. . . y !. . . .. y. yy . y . . .y. ry q. g...y y. y...... .yy y. ..;. r . . .r. .q. y.. . .y . . y ... . j y . E . rE..... . . y. j . .y.y. jkgyjjjj(,.yg y ..y y. y y. .q.y.j..y .y.... y . . y..y... yy..jgjjjjyjjjjgjjjg.. y.y.. ...yyy ).. .y . .. . .y . . . . .. . . y... . . j. .. . y .yy ... . . . ....) ) . . .. L.... y ; y . .. . g yy y ... y. . y y.kk,,g;jj., y...- . y..y.. yyyg.yyy.j.yj.. . y.. .j. ji... . . ..j. yyy.r,qq:gyy,y .q . . . . ili-:tii'-: .. . jj-., gjgjj .. y.. .jjgyjjgjj.. ...E .fj;i. .. 1:i: . 4:j);:1t..r211 .kj . . jty tkjyjjjjjj., .. . .yyyy.. j.:.y. ..) ty Note: These prices are computed using the Black formula fpr opions on fumres, with a futures price of $420, j..y.y..j.. $j;jjyg,,.. .. . ... .. . . sj y. ... ., .. .... ,.. .. .. ....... y .. . gjjj... 11ql'1h. 1t:.i:t);i jp k:jr .y,.j.y...! ., .... .j..j..) kt:i ib. J. .,y . ,jys..j . . .,yj( .. . , .#q y y ..... ., , . , y. . j.,.y , j. jjgjgjjjjy . jjjygjjjt.s .r. y.... jyjjgjjjy ... .y.,.,....,,. .. ( . g. . y.. . q, Ey . ( q ( E.. .. y. jjjjjjj;,. y . .yy g.jjjgjjyjj ..(E.. . r .. ( (..(jj,E. y..jjygggj;,: . jjtjjjjjjj y;gjjjj((. j . y. . jjjjyjgj;. ;jgjjkj. j. .. yg . ;gkjgj,.. .(jjgy;j;;,. . ( ... .. .gjygy,,. yy jjjjt.j . . . .;kjgg;jjj(jjj.j(jj(j. . j;rj:jj,,. y .jjjjj .. (gjjyjjjj,. . . . jj;yjjyjy y. y . effective annual interest rate of 5%, volaulit'y of 5.5%, and 1 year to expiration. . gjjjjjgjjjjgjjjk(. illl'E .. ....(yjggygjjjyj;jj... ... .. y.. j...' jjrjjjjjjj. . y. .... j ..yj y jjjjjyt j jggjjjjjj .... . .. . . g .g . . . g . . . .. . . 5 . j . g . .' . (..!.k 'E j E(.yj. (.(. EE .( E...y. i'jjjjjjjj(jjjjj(.'. .j ..g. g .'iy'.. .E...ykjyk,gj;: .. (' r. .. . .. E.' ' q . (g. .. . .( E .( ... . .... .E. (. ...kj,.(jjk.jE . (. .g. E E (.. . (y ' ' ,1j1444;::. .. ..j.. ( j.( i jyj( j.( r.y.. Ei. (( (..E.. .E(. .(. (.y. .jjjgjg;j;q: k . . (.. .. .y y. k. ( (... yj(.(. (.. Eyg.. E... ... y... jg Eg.j..ytjk(ggj;y. .. ... :rj,.(.('..; ly jjjrj:ryjjj;j,,: . y. .. . y. ;. .y.;!jjj;;: . . . .. .. lli:lqqqjlql!r yy . .' y.; r . .. . y . 'y . . . .4,jj::jj;,,. . rt''r j .jjjjtjjjyjjjjjj. E. . . . .(4jjgjyj;j. (j1(:j'' . .. . .jrjg. . yj y j ( . Eg .. I:Sj,1,,. :jEj;;k. '11::24::..(ji.,. ' '(j1.. .1j!(j y. .jj.. ' ::(gj1... ' !j,,,. :jjjI;r. (pjj. ... .y .y..j.jj j . jji(:rF.. . .. yjy, j j j . gyg y .jj. yg.. j y. .y y. ty..y. .. jjj j j . . y . j..y.j. y. y. (jjj..E jyjjjjjj;y jj r ; r . yy y .. . . yjygggjjjjj.. g y . ... . j.j . . jjgjgg;jjy .j.y y.. j g y.y.y .jj. g..y. y..yjy.y .y..j.y... q. y y. gg.E q.yjgy.jgjjyjjj. j . .. .... . . .jg..jjggg. jjgjjjgjjj j j.y y jg.jgjj. y.yy j. j..jj..j...g.jj. jjj. j j j jj ..g . . y . . .. ;jgjjgj.jgjjgg..jgjjjj;y,... .. yj tyjjjjjjjy .,jjjgg. . . jjjgjjjjj.. .... . yj.. y. . . . . y.j g. . ....j. .gkjjjjkj,. jjggjjgjj. ......j)j)jjjj j.ggj.gjjyjjjyjjy .jjjjjjjjjj,j .yj g.. . . gggj. .j.gj jjgg. jgj jj.y . .. . 5. .. .y . j gjjjt.j. . y . .y j y ggg;gjjjgy y j . . . j . . j kjjggyjjy .g yjy j j.. . j . y j y y . . jy..g...yj..g . . y . . . j .. j ..Ej 'l 12,! .yy jgy.j j jy j. ....yj.yj. . jyjyy . . . . .. .. .q.. jq' E E ..gg E ( g y. y y..y .yyy. . y y.. .y.yy.jjgggjjjty E (.q( (.. .(. j (. y. j;,,,jjjjEqq .jjyE.. . (...(.. .y.y . i . .q..jjjjggjjj,. yy..yyjjjgjjjjj.. .. . yy..gy... y.y.j.y... . ;y E.......yyy..jjjjjg;j;,. g.y j. ..... y q :ggkjjj,y j y. y(yy.( . (y... y.. j.;yyyy..q..j.jygy q ;(jg..... (jjj;,yjjj(.,jkj.. . . .. y. j yyy.yyyyyy yyyy . . j ( .j j .y.g.. y j . ....y.y .. . (. ..(y y..(.... . .y . '( ''. .' . . . . r)j)jjj,. jjjjgyjri ( . y .. . j . . . ' j jr;:.!!kk ' $jkgjjjg,. . . j . . . . . . j y j . . jj:yygjjjj y j . .jjj,jjj(.. y .. . j . . y y . j. j . . gjy . . . yjyj . jj(jjjjjj,. . . gjyjj . . .... gggjgjjrj .. . . '. . jjjjgjjjjj k;:yk,. .......jjjjj .gjjjyjj . .. . $:gjj,4;,.jjj...jjj, .jj.gyjjj. .y..... ... ...j ... j..g . jjyjjjgjy .. yj.j.y.. jj .ygy...gg jL:'''; .j.yy;...;jj...jj,..jj .. ..1I... . g. . . ...j.yj.y.jg.j....gj.yy..yjjyy y....y.yjg.. j ..g..j...yjj.g.j.jj j ..g.j...jy.jj.jgj (g:jjj;y;.j. jjjjj . . .. y j . j. y. y. y .y g. g. .j. yj.k.ky.. . g yygy. y g g.. . gg. . . r r r . .... .y. . .j yy.. y. y.yyyyyy.y . . jg . .. .j;jy,.;jjj. g. . . j.. y y .. j ... . y .. k$jjjjgggjjk. . . . y y... . ... y..gygg.j.. y y j gj jjjjjygjjgjgjjjs j y.. ( y. yy .y.. gy..y.ggyy y.. y.yjjyjggyjy y.. .g...gg .yj.....yyj;j;gjr..,jjjyjj;;s ;k, y .. '1i!EE'. y.gg...j.y...y.gy .. .yjj.kyjjj, yg.j. .. y. y.... y,4...,. j... .y..y.j.y.. . .. j . y. .. j y. . . krgyjjjj. . .jggyyjyjj,..jjjs . . .. jg tjy,igk,. . . . ..gjy.. y...jyj.y..y.y.y.yyy..jyy. ...j jg.. ..y!:.j y' j'.yEj . gjjgg.yjj,ggjjy .. .. iliiil.. 'IIEiJ(11L.. .jjkjyygji 'E. '.'.''' ' . di2iij!).. . . .jjjjj .. ' ' jyjjj;yyjjy;j;s . .jjyygggji . g . 1 .j11,... 11.,,11;.. ' . ' (jj)l:ll' . yg..y..y.gg.y . E Iijj... 1I;;:1Ik... ' ( ' (ggjjjjjjj.. (11j(l;:lp. (y'qg.. y.'.j'y.k. qg jy g.y.g..y A 429 44@ tollae q. gy. g. g .11::r..,2. ..E. . .. . .....!.. yj. y.... .. y y y y.yj r r y . . g y y . . y . . . . . . y .jjgjgjyjgy . y y . .y: y . . y . j j j . a 420-sfrike put option . y y j . .... . y.. j. .. . . .. ..... g. . j j. . tyyjyyjyjjj. .. . . .. . y. j.j.j . y.j yy .y..y jy . yy .. . y . . ' E. g yy . j . .. .1kkr:r:;' j . .E. ..... . . g . g . . . Eq . kI:j::i,.. . 1122!22:. . ' ' .-iii;;.ik,;,-E y . . E . g ' jjgjjjjjjy. ' 'E . .' 'y.. ' '.I:@SirS.. .. .g ';iErI1... .I22!::.' ' . :,iEk:;. ;1:;11,..11t..,11.. . y . .. . ' ' ' kI222:r:.. . .y. jj j ... .jjj.!.. g.j..jyg ..g.. E 1I...IL.' '11::::,1.. (1IEr:1I. 1I;ii,!::' ' '::jE;: . ' :i!E!1i,.. '. i ;;!j51I,.'.iI.,!pIL.. .; .E.ygjj j.. ' . .r,;.. j!!Ej;;. j ( :EIIE(q. (,Ir:q.'..y ..y. '' ' ' ' . y . y y. . qy q. . . q. q. .y.. Vo-strike call y . 'y sells option premium of y. . y .. ..qq.-.:.-,..( y lf the price of gold in 1 . E...qr.k;.. . . y . .y E. . j j . ..'E.''.1!ii... ; ..-.y.... E. y y. y y j ,.-.. .y-.. ...... .... .y. ..... .. ..... q . E. . . a a g y.y...yyy.yjj. .. ... .. . .y . .y y y ; y. y y y . yy jyy. ..j. ... . . ....;..,.:. .;t .. E.d(::1l.y yyj. jyj ... . . .. .;y. . .C!iij;7iikik' . .. ... . ..it::1i7iI.. . .-...j...EE.j.yj...;-.g..y-g......y ....1Eiikli1)q'... :i-.'.').-)t. . y y... y .. . ( . 'i:'.. rg.gr-.y.... q-t-.'t7i:;'., ... di:t)'.. ...y . .. j.j..-l9!Ii.1iEE. .. ...gy.jjy .' iL....'lLl... y.j. yqk;.-i.E. ..jy...jjy.. yyyyjyy..j.y.....qy;y.......E...jg.-yE.....-y.......-y.--.q. .)-.EE . t(E't.... .tIE1;ii. ... ... (l!.E,(:Eil.ti.i1tf1I.. ....r-',i1:-2l.--qiL:1If. .. .y y..g..;E-t).(...... .(iE'i... -...'k:!L.-.. :1:.11::111. i.'fE::l'. 4k!:. ....y..yyy.yyyy-y $!Eil...., 1Eiil..IL.y --1!E1k...y.-...j..''L..-'.'L.L..,..L... 1ElL... ii::. (j(q:)'.-:E3lIL.lIL.:1E1; L''....LL. (1E41!1. -.(ElL:7-.y....y iE.. ,IL...y. ..r-.!tfEIpr',
[email protected]. (It:i' ''''(IE):)tii.ll.. j. .!.q yy y. yy ;.E.j j.. y.j..tjjygjj.).(..jqjjyj(.qt. (1(4:). 11:2... is the call will and ..' . exercise is obligated to sell gold ... . ..l ,y E)...,... E ....t,g '-.t y.r-E.. '.E...;. .?..-, owner E ..:. E .E. E. . E -.E... .L..... .y..:;.;.;;.. g.r(i.k.. (.(.....(f'LL .r .t...-E.... ..... . . . . .- . . .q . . .. ; . ::q...'ki::;yy.. . . q .E.. . . j ..( r..... ) ..E 112312y... (-..:;,Ey...y....(r r.. :IL:;.. ..) !i' 2.. .IE:'ii2.. .t .r. :iEIk.E. r E..- y..jy.-1iIi!:4-; ,di2::.E .';-...-;'.':...r.E(-;.E..-'...E..,.!..',.q--E..E.t' .d:i1'. . di:2:!. !IE:):lL.'t:t:''-'itEi;. 1117.-.... 11;:11E11(11t:!1. -.E1liqiI:i'iIE:1IrElk.' irEll.I::'. 'EIt:1)#il.. '.dCi:)..'.....qii' ..yE....:lk:r,q-y..--...-'-'Ir21I:E1l.EIL.y .d(::::. '. (IE!1::.. j'.q Ey..,i::r;... iksr). (1::-. 'lLkll.r.!y j.y..:i!I. .-'(k2rL.......E;r'IELy -iI:.t.-. ...-.'::,:,)....(.EI(27iIL-... E1::)l.--r.t(:j!I.. :I:.ii.lk ..).@ .IL. -....:EE;k.-. (1()4,1 (11.. :iE;;. (11842.-.. (...t-j.-.. (1(1:1,. (lE:?i.. of the .E '. ) strike price rather market ,....y-- ;-. the of premium ;.. g .L.;..t..-.-. .' r.... q... E.jjjjjj, tan .q .yy-...y.y;qj.yy..--.yy.y...-.-..jE.y.y-... E.gjyjjjjjjjj;E.. E .( .E q..y q.. ..jjjjr:.. y.......(.. . ,. .;;.. y. ...-. jjyj;;. E .. . . .. .. .. ..y kjrjj . y. q. . . . ... . .... .. . . . 'kiii:. ; y. .jjg;;.yjyygjjj.jgjjjy,(jjjj.. ...j... y. t;k;jyy;.. .jkgjr, y.... . . L''..14:):,:. '(:iil... . . . 21-11:111:1:.. .t-E,-...jgjjjj.y( j..-y .1i2:.... .(...gjjyj;.. .... . yjrjjjjtj;.. kgyrj;..(jjj:j. ... j..y.j gjgy.1it2. 1!11!) jj:.),j,;f.. .jjjj(j. 1E!'L5i:i1L.. .1t5il... !1:1ll '1E::tk. jy-y 1!!jt. ..'..qtE)l... ..-,g gjjy Ek. 'iEI.. . jjjjjyyjjjjjs 11:2. . ... gjjjjjj. jjjjj j ijjjrjkjEk.. -E.gyE..-...-.((y.-.yy......q.gy....y...r..-.-. ..y....,r,-gj-ygjy.g;..jj...y)t.(j. .,)..,L ... il::li' .' CIIE .. L.L..L..;.,;...;.j..(.-..;...;. .(-. Golddiggers received ...-jjgyj:.;t .' il:i)' for the ..(.y-...;jjj.. tl-.. possibility thatthis will Ey. ., Er. .yky(j(j.t.......jjyjrjy... compensates qjkgjj;...y y.).jyy.---.$j(;jjt;,j...-q.. rt yjjjjjyjjjgjjgrryq. ,--ygjjjygjg-..y. gjjy;.)..--.jj;;;..jjjj....,...--..y. y-qy,gjj,k; .-.yjjy;jj-.......y ..;....-.---..yyqy.-...y.....jy.g . ntkkjyt. .t.y... ttk' . lEirEiiizE I'E..q . E.
.....,
,(
.
. .
. g
.
.
'.
'-
.
-
.;,
-
-
: E .
:
:
-.
.
.
-
' . -
.
. .. - .. . . .
,t
i
. .
-
-
--
-
-
..
..
.
.
.. -.
investr .
-
:
:
y
-.
-.
.
-
-
.
. : ..
-
'5
:
.
:
;
-
.
. . - . .
.
g. -
'
'
'
'
.
:
-
-. .: .
:
:
:
: y
....
.
'
-
.
.
.
.
::
'
E'
.
,.
'
-
'
'
:
:
.
y.
.
%.
REVISITED
.
-
.
.
.
-
-
'
:
..
.
.
.
.
.
.
-
-.
... . :
.
. .
.
.. . . : .
..
...
. -
.
. -
-
'
,
.
' :
.
:
REVISITED
We have looked at simple hedging and insurance strategies for buyers and sellers. We now examine some additional strategies that permit tailoring the nmount and cost of insurance. For simplicity we will focus primarily on Golddiggers; however, in every case there are analogous seategies for Aulic. Table 4.8 lists premiums for three calls and puts on gold with 1 year to expiration examples use these values. and tlu'ee different strikes. rl'he
Selling the Gain: Collars
As discussed earlier, we can reduce the cost of insttrance by reducing potentif prot, i.e., by selling our right to profit from high gold plices. We can do tllis by selling a call. If the gold price is above the strike on the call, we are contractually obligated to sell at the strike. This caps our prohts, in exchange for anxirlitial premium payment.
i'(j('(j('.j;';' ').ff q(' ;:(1* k' .y' ;''k' a).::'f Elliqii.f p' )' (' t!!EE:2.I'i'-' IJX jjlljd t)d )-'. --.,.-..-L'' jt'lyd j.;' :,i1;;' ))'iihp'.;)lf jjd @:' ,:,'. !' j'tqd jjj'; ytjj'. (jI(q::.-' yjd j()'. (4)j'. IIIIq:piiI:..' jtky',' kkf 44* (' y' lI1iEiEEk'.. !111k-:-211:*. 1IIIiqp:''. 11111*. yyjtjttkgj:d kyjy'tjtd )s'. )(-'. t'. 4j': ttf t-)(''' ;(;;..'( .((ij'(). .'( @. :j)zi (j 1t@)! p .(.'..(..(i ( ( (!EE' (! y(jytjkr j(i! jj)): y j . ( jk )( . ....jit).-'(). jjjjj .j jj'jy kkyyj. . ! jgj.j.kij.y ):.;;..(j.''(q ! tk . : . :-r-; .i'. 'k.) '... ' ); !!! .).ii.,. '. . .
;'yJX ----' -' -'' ?' -' ')' i.jd -.' ---(kt---:-'--. .--' L(jLLq)$)kL6LjL.%)L.1'.L.LL..1LL)''.'..''LI''.'?LLL.q.q12.qLLL(.q$LjtLL',i,jLL'L1L'L''LL'1L'L''L'.L'.LL'L,L'/.L'LL.,LL. ..:Ef..1(.......:..: ....,.;....,..'.;'/Li.L.'LL.L2,L;kL1
''' ..'t ..' .'....' .:.)! '..kbfit ....
--'..' .-kq,:11iil2.-' .-'
iE!E
;:i:E 'E. :: EEEE ' !' :. ! :
:
-.).?!!!!p;;iiiis-..
.
.
:E
E .
-
-
:
rl
:
E 'EE .
.:jj((jj '
.
:
:
.
----.'. ,k.'i--.. ,,,...:q.-.-. ..
--:..:fkj:)'.
.. .
EE . EE ' : :
-
-
---
-
.
-
.. . . . .. . Net P at resulting from buying a 420-strike put with premium of $8.77 and selling a 440-strike call wth premium of 52.49. The profit for gold prices between $420 and $4402is (52.49 $8.77) x 1 .
. . ..
.
.. .
. ....
.
... .
--
..
. .
-
.
-
-
.
--r. -
11:2( .!Ir::. (::,1
Il::::),-
-
'xpiratioh
Gold collar
150 1O0
stj
..-
-$6.60
.
()
.05
i
-$8.60.
=
420-440
...s
200
-so
! -100 -15
5420
t)
250
I
300
1
.
I
!
j
:
j j
.@ : '
i
-
$440 I
.
.
I
350 400 45O 500 Gold Price in 1 Year ($)
l
550
%.lN'rRoouc'rlox
110
To
RlsK
MANAGEMENT
GouoolccERs '' :ltjil.f k'I:r.f gjf 5* r' i)' )' 'q f' y'tl';llf (q)f)' '' ' FE .11.::* -l::E::,'f ;'i'i' (' t'j,,' jjf y'lkjkyyy !' ;q' 1* i'i)f iftf ;$iL' jiiq't.ipl Jf qf' $9181 E i' '()' )' ,----iLLLLLLiL-'. r.'yyy y' j'j' 71*1)!7 )j5(q)' E q! ;E (('.i(( E ( (Ti g' ))'. ))' ryf j--gjjjy'kf y'. jjjjj'yyjf yyf j'. q(''I..(;:.li E:.'.(!E. ).EEiij'./((Eijii:)q; ('E.E''E'EE'E'EEE''(''!('E.E' i( (i. EE.(. E('(j7(.EI?( q' II1EEiEi,'. )' y' tj)((yy'. jjtjyj'yj (y', tlt-----.f t't--.)i-' yjf y'. '((' .i. . qq ; yy. '.j'. . '. .q.. E. .y..... .'. , E.y.;q g..y . i (.l() y. y .)y.)tjjijtjjy...E/;'7El(!ilii)!lt(! l )y . !y y y, yy ;j y .,.:jyyy j.(. jyyy..g yjj.y. jg((. . yyjy. gi I'IILLLt. . -..-. yjt. y .ttj y . y y j y tj . . . y. . .jjjjj. . g . ty. II1i@. -! .... (--:-j.-i.--..:.(;qlj. jlij- :!j:. '.t(.. .. '(. ..-. . i.. . . ....t.
.'..' qf. .'. .
-..' -.--khyf-rrrlllll.-'.x-x...axtliik;. -.'. t's' yjf :$f't' -' -..,!111:::::;---.* ,k--::111.1-*. lf ;'k' y' tlfgfytjjyy'j.y' !!' )' '7* (' -' '.'--)k);'-. )-''iilk''-1d1t'. j':'j';.jf C' E (: !E @ .
t.' J4441q511-*.
%.
REvlsl-rEo
.;' .''j'
E E : ' E: E :' plflff.tlllj7ftpqplEqlprlliplij'i j'E' y'( j.yf -.--kt:--.......'!!!4;;;)!!r...--;-))----------t()));;'--.:.-E--' @''r)' rC' : EltEl! E !::'E ': i''i E E'E : ((i' E:!EE (; !:' E ' IE : : l:''. EE' .E ; ! '' :. . l'E. ' E(''(E: E . .. .. . .. . . . '
---' @irft;;9E..' -7*j81(* -'' C' ;pqf(E E F' 5* y' :'((j' -' ';' '')' -' '-.'' .'-.'
f' /$)* 1* q' 'qj#'!q' jjyjyjjjyyjjjjjjjyjyy',yf krt'f ''''El' )' 'i'iif i)f ).;:' j''((;' q! (' q' (r' q:)f gjjj('-' ;':-.'t'ltf '1.* :i('( ( . ' ( )i ' (;@ i (''i ;'. :!ii;;' k' !i.)' jy', (l'yf j'. y'y (4'yy.yg )' itf y' yjf qrl:f q' j't' (1IiE-1ii.' .y'.jj'. ))' '.-'' ry.''lllLttk'lil'. 1l1lF:tIII:.'. ;'r'y)' r-'1iII;' ty'yyf t'(.;k(k)k)'....' rrf i'-.11p1iFFiqt.'..' 1111!*. !' IIItiii!,'. . qlllli' ( . . !' rqf i lli y. y ! jg ljlE')i i yi . jjj)..
''. '.''.-;. .'' '
TE i : t'l!'rE... E i.i( .!. !E(';.siI( . yj..jjj. y. jy gy , jjjj,j y;jyjyy; j.yy ; jy yj. (. y ; y y y . . . ! -i-.. . . j .qr yqj j y t y yj. j . j y y . y. '-j j- yi..j-.j pj, ,911::::;2k11i .k;i:!1IiI;'.' - . . ;-. .i-!p ky..,.: .... i.i. )#t.. '. -. .. -t.?!..!... . q..i. . . .
E
'
.
'
' .'
.
.
isili ttsE: E
'1i.p, t.
. .
. . ,. . .
..
i
.
:
''
.
: E. : . gi g.yt.y (E . .(!yyy.. EE E '.! i E E .. ' .'
. -
- .
.
.
..
.
.
.. .
-
.
.
. . . . . ..
' :
.
.
:
. . . .. .
.. .
-....
' . .. . . . . . .. . : .
:
:
.
,
g
--. . .
. --..
.
.
-
. -- -
.
..
. . ..
Comparison of Golddiggers hedged with 420-strike put Versus hedged with 420-strike put and written 440-strike call (420-440 collar).
-
.
-.
-s.
150
+
.
hedged sellerhedged
seller
wi:.IA Mt.IA
420-440 collar 420-strike put
100
$33.40
s0
! i !
$30.79
!
E
!
-50
E E
-100
i $420 !
-150 :
300
.
-
-
' ' E:. : : . - .
.:
,-..
.
41:2 .11::: y,.:r. ds::E:),(,1(q::1(ql,-
O
.
: '
.
.
.- ttj.y ,
-
-
-
. ..
:
:
.
.
'
677r:7t7:
:
.
.
:
-
-.'
.
.
..
-
-
'
E' . :.
'
.
E :
'
:.
.
' :.
'
.
:
::
:
'' i ' : !' :. E. : : . . :.
:!
...:
E
E
tjyyyyj. illiqi'ilii. . . .. . . - .
-
.
-
.
.
.
! :,
-
. .
.-
-
.
y:
-.
-
y
y
.. .
,
.
.
.-..
--
.
.. .. . .
y-
-
.
- -
-.
.
.
.
dl:r. :,!i5;;. (11E2::-(:),-
--
Compar ison Of unhedged profit for Golddggers versus .. zero-cost collar obtained by buying 400.78-strike put and selling 440.78-strike
-s
200
400.78-440.78
ollar
100
$60.78
stl
!
$20.78
q
' E
-100 .
j 5440
:
! !
-so
i
j
:
()
!
350 400 450 Gold Plice in 1 Year ($)
unhedgedseller sellerhedged M:.I
150
call.
E i ! i E E
+
$400.78 E !
-1
500
I
5
250
300
1.
I
w
j' !
j
! i E$440.78
..
'
I
350 400 q 450 Gold Price il 1 Year
I
1
500
550
@)
'
The 420 put costs Note tat the 420-440 colla.r still entails paying a of Thus, there premium only is a nd 440 call yields the $2.49. a net expenditure a $8.77, It is probably apparent, thokgh, that we can tinker with tlle strike plices and Of $6 2 pay a still lower net premium, including zero premium, if we wish. The ade-off is that te payoff on the collar becomes less atlractive as we lower the required premium.
.premium.
.
.
collar To construct a zero-cost collar, we could argue as follows: A400and put a uo-sttikecall are equally distant from the forward price of $420. 'This strike that the options should have approximately the snme premium. As suggests equivalence from the table of premiums for different stnke options, the 4oo-strike pui has we can see premium of wlle $2.21, te Oo-strike call has a premium of $2.49. The net premium a receive would buying this collar is thus $0.28.We can constnlct atrue zero-cost from we collar by slightly changing the strike prices, mnking the put more expensive (raisingthe strike) and the call less expensive (alsoraising the strike). With strikes of $400.78for the put and $440.78for the call, we obtain a premium of $2.355for both options. In reality this zero-cost collar of width 40 would be sold at lower strike pricej t.11a11 $400.78 and $440.78..'1'he reason is that there is a bid-ask spread: Dealers are willig to buy a given option at a low price and sell it at a high price. 'I'he ptlrchased put will be bought at the dealer's offer price and the call will be sold at the bid. The dealer can earn this spread in either of two ways: Selling the 400.78-440.78 collar and charging an explicit ansaction fee, or lowering the stlike prices appropriately and charging a zero transaction fee. Either Fay, the dealer enrns the fee. One of the triclk'y aspects of the more complicated derivatives is that it is relatively
A zero-tost
easy for dealers to epbed fees tlpt are invisible to the buyer. Of course a buyer can mitigate tllis problein y alpays sling quotes f'rol dlfferent dealers. t.tl b Wecan eynmlne ff ' epayo s y cons idering seprately the three interesting regions gold prices: of .
Prit
.
g
'
ofgold
<
..
..
'
.
o
$433.78: In this region, Golddiggers can
exercising the put option.
Price ofgold lae/wdepl$400.78 gold at tlm market price.
tw?#
.
jell gold for
$400.78by
$440.78: ln this region, Golddiggers cn sell
i
Price ofgold % $440.78.. In this region, Golddiggers sells gold at $*0.78. It has solda catl, so the owner of e call Svill exercil. This forces Golddiggers to sell gbldto t.1* calt owner for the strike price of $440.78. Figure 4.1 1 grap hs the zero-cost collar against the unhedge position. Notice that between $400.78 and $440.78, the zero-cost collar graph is coincident with the urlhedged prost. Above the 440.78-st11.1(:ethe collarprovldes prost of $60.78,and below the 400.78-stdke, the collar provides prost of $20.78.
The forward contrau Because the put and call with strike as a zero-cost collar prices of $420 have the same premiums, we could also construct a zero-cost collar by buying the $420-st*e put and selling the $420-st1411e call. lf we do this, here is what happens:
112
sEtEc-rlxc
Price ofgold < price of $420.
$420..Golddiggers will exercise the put option, selling gold at the
Price ofgold > $420.. Golddiggers has sold a 420-stIike call will exercise, obligating Golddiggers to sell gold for
call. The owner of that
$420.
ln either case, Golddiggers sells gold at $420. Thus, the collar'' is exactly like a fonvard conact. By buying the put and selling the call at the same strike price, Golddiggers has synthetically created a short position in a forward conact. Since a short forward and 420-420 collar have the same payoff, they must cost the same. This is 1W7ythe #?z??,I/??l- ol the 420-strike options t'lrc the Tllis exnmple is really just -anillustration of equation (3.1). ::420.-420
-rHE HEocE
Rvlo
%:
te call strike ptice below $440.78,in wllich case we would obtain a higher premitlm per call. To offset the higher premium, we could buy less than one call. The ade-off is that we cap the gold price on part of production at a lower level, but F maintain solpe participation at any plice above the stnke. lternatively we could raise the cap level (thesike price on the calll and sell would increase participation in gold price increases up to the than one call. more also effect of generating a riet short position in gold if prices rose the have level, but cap above the cap. rf'his
-twnn
synthetic i orwar d s a t p rices other 'than :420 We can easily extend this example to understand the relationship between option premiums at otlzer strike prices. In the previous example, Golddiggers created a synthetic forward sale at $420. You might think that you could benefit by creating a synthetic fonvard contract at a higher price such as $440. Other things being equal, you would rather sell at $440than $420. To accomplish this you buy the 440 put and sell the zl40 call. However, tere is a catch: 'Ihe zgo-strile put is in-the-money and the Mo-strile call is out-of-the-money. Since we would be buying the expensive option and selling the inexpensive option, we have to pay a premium. How much is it worth to Golddiggers to be able to'lock in a selling price of $440 instead of $42 ? Obviously,it is worjh $20 1 ye ar from toda# or $20+ (105) $19 05 in present value tenus. Since locking in a $410 price is f'ree, it should threfore be the case that we pay $19.05in net premium in order to lock in a $440price. In fct, looling at the plices of the Mo-strike put and call in Table 4.8, we hav premiums of $21.54for te put and $2.49for the call. This gives us ::2
,
Net premium
=
$21.54 $2.49 -
.
.
$19.05
=
Similarly, suppose Golddiggers explored the possibility of locking in a $400 price for gold in 1 year. Obviously, Golddiggers would require compensation to accept a lower price. In fact, they would heed to be paid $19.05-+e present value of $20-today. Again we compute the option premiums >nd we see that th 400-FtIike call sells for $21.26while the 400-stlike put sells for $2.21.Again we have Net premium
=
$2.21 $21.26 -
-$19.05
=
Golddiggers in this case receives the net premium for accepting a lower price.
Other Collar Strategies
Paylater Strategies A disadvantage to buying a put option is tat Golddiggers pays the premium even when the goldplice is high andinsurance was, afterthefact, unnecessary. One strategy to avoid this problem is a paylater sategy, where the premium is paid only when the insurance is needed. Wllile it is possible to constnlct exotic options in which the premium is paid only at expirain and only if the option is in ttze money, the sategy we discuss here is a ratio spread using ordinacy put options. The goal is to find a sategy where if the gold plice is high, tere is no net option prernium. lfthe gold plice is low, there is insurance, but the effecuve prernium is greater than with an ordinary inslzrance sategy. Ifthereis nopremiumwhen thegoldpriceis Mgh, wemusthaveno initialpremium. This means that we must sell at least one option. Consider the following sategy for Golddiggers: Sell a434.6-strikeput andbuy two4zo-strikeputs. Using ourassumptionsk the premium on the 434.6-strike put is $17.55,wllile the premium on the 420-strike put is $8.77. Thus, the net option premium from this strategy is $17.55z- (2 x $8.775)= 0. Figure 4.12 depicts the result of Golddiggers's hedging with a payter sategy. Whentheprice of goldis greaterthan$434.60, neitherputis exercised, and Golddiggers's profit is the same as if it were unhedged. When the price of gold is between $420 and $434.60, because of the written $434.60put, the firl'n loses $2 of profit fpr every $1 decline in the price of gold. Below $420 the purchased 420-.tri1e puts are exercised, and prost becomes constant. The nt result is an insurance policy that is not paid for unless it is needed. Also depicted in Figure 4.12 is the familiar rsult from a conventibnal insurance strategy of hedging by purchasing a single 420-sfrike pt. When the gold price is high, the pay 1a ter strategy wit.h a zero preinium outpebrms the sigte put. When the gold price is low, the paylater sategy does worse because it offets less insurance. Thus, the premium is paid later, if insurance is needed.
4.5 SELECTING
Collar-type strategies are quite :exible. We lave focused on the ease where the ftrm buys one put and sells one call. However, it is also possible to deal with fractional options. For exnmple, consider te 400.78-440.78 collar above. We could buy one put to obtain full downside protection, and we could vary the stn' price of the call by selling fractional calls at strike prices other than $440.78.For example, we could lower 'ke
THE HEDGE
RATIO
ln the Golddiggers and Auric exmuples, we pedbrmed all calculations in terms of one unit of gold, and made two important assumptions. First, we assumed perfectcorrelation between the plice of gold and the price of what each compahy wants to hedge. Second, we assumed that the companies lnew for certain the quantity of gold they would sell and buy. As a result of these assumptions, we effectively assumed that the hedge ratio is one,
:
k. IN-rRoouc'rlolq
.'
To
Rlsl
S ELECTI
MANAGEMENT
))E.' y' j'$'. E-j'i :'7rr* 5* f' jr.tjjf 'F' 'j1;)' @)*:. ITIStF'I;ICIEiqY j''llqi''prqf .'.' jlf r)f q' )' /'1* )ry' C' f' iif ))' ' ; J); jF; )-' t''('. (' (t' (l(5' )k':t' )' ;'tt'.)i iii';rf j'(1* jl'tti );' ji: iTE )1@. E 't!' :( ;;EE jtf r' $4jt* plftri'lpqrr'.)i t)';iji('(litkip!.l1( (;(i' i'Ir' i!i iE'')(l'' . (.:(E )' ;'y' 'l l''.(i.'..( ).7' lyf ltfpijq.'.' ('y' ;)'?r ')!)@' . .. i'..( yjylljjjij-''g ;'qif t'jjy't$r'. 1!1!qqq5q' (tf r' 'rjtyyf ):t'. i. i. iEy:. (. ! . ;@ ! (.;.ji.. jy.. ri.. 'rjj,yy;g--sijkj----(t(j)y--yk. :; g;.j.rj! lj)jg ' ( j . j j j j k -);.. . 8... jj. ). j ; yt:gg))j' ) (!11::::k2... y. ,-.jjjy-..qr,11I.).----(j--))))(.,;k.;.:;..j-):--L.L.L..,L;.).)-L;).;j-q..j-;.j.j;j).;,jb);-L,--L,j .:;;j2r;..-------;kyy.. .y,!(Ik:jIj. tj . .111:411: iyltt.-lgy. 187/3 !It!E:-tt.yj.;j1:tjI:y. i. . gjyyjj. jty 111; ,:22,, .jyy g,:s jg:jjgs ,kgg, jkygr jgyjjgjkjgr (j(y), jkjgr y;j .jgy.kjr.gjr.g,.s ...! rjj;i gjyrjt. . . rjjjjy:jg;jj.gj,pj:j E hedged with 4z-st'ri1 Dpiction of ,,pa#latk sener paqater 200 which in strategy, *. Seller hedged Mt.IA paylater Golddiggers sells a 150 434.6-strike put and buys two 420-strike 10() put's, compared to the conventional insurance 50 ao 70 E strategy of buying a E i 625.40 420-strike put.
,'.'' -' -.' fjy)jy;;')C.:' '(;;'@tq'..! :'E' j'
.'jt)f
:''' :( :
'
E
' ' '' ' E E!q. ii ; : E
Ei . .
-- . E -
:
.
.
--. ... ..
'. -
' E' . E
:
:
.
'
'
E . -. !. E ...E..E .
)E .
-
i
:
E.
'
:
'
.
IE (E ! E: :
..
'' .
E
:
. :.
.
.
E
' :E : .
:
.
-
'
:
.:
- : :
:
.
E
-.
. -
--' -E
.
. : -
.
E .
. ...
. . .... .
. .. .
.. . . .. . . .
. ..
-...-.
.
-L..-.... .
....jj.....
.
,,
.+.
put
.w.
Prot
''
E !
-50
j
-100 -150
250
1 300
I
5420 iq : l
Hedged profit
(
Nw Pw
=
=
$434.60 I
350 400 450 Gold Price in 1 Year
Nlwo'w2 + (H
2 chcdged
I
I
500
550
(5)
'
where te hedge ratio is defined as te ratio of the fonvard position to the underlying E
.
j ( g
assumption-may be valid. We ftrst exnmine the effect of widget ' ln practice, neither price uncertainty on hedging, and discuss cross-hedging. We then exnmine qttantity I/??c'?-lc?lfy
NwPw
=
in te price of widgets
-
(4.1)
N.g Pgold
Ng Pgold+ StPgold
-
-
F)
'
using an agriculmral exnmple.
gLH N g )20.2 g +
-
-
N
'g
(4.:)
lywpgwg.:
where o'm is the standard deviation of the widget price, o-g is the standard deviaion of the goldprice, and p is the price correlation between widgets and gold. Note that pavo'g is the covariance between widget and gold prices. rfhe variance-minimizing hedge position, S*, is13 H.
aSSC (
Ng
=
-
Nw
po'u)
c'p Equation (4.3)has a saighforward interpretation. 'Fhe flrst tet'm hedges costs-i.e., we second enter into a long position on the nmount of gold we use in production, Ng. term hedges revenue, prov iding the quantity of gold forWards we use to hedge widgets. iS 9om regressing the Widget ljlice ii tli gold p1ice.14 The telql pGwxg the Coecient times the number of widgets, #w, ives us the mber of This regression coecient result ij that the hedge gold conacts to use in hedging widget price risk. tuantity, 'l'he
'l'he
COSS-
gmg
In th Aul'i exampl we assumed that widget prices are sxed.However, since gold is used to produce widgets, widget prices might vary with gold prices. If widget and gold prices vat'y one-for-one, Auric's prots would be independent of the price of gold and Auric would haye tno need to hedge. 12 . . More realisttclly, the price of widgets ould chpnge wit.h the price of gold, but not one-for-one; othr factorq could affect widget plices as well. In this case, Auric might 5nd it helpf'ul to use gold derivaives to hedge the price of the widgts it sells as well as the price of the gold it buys. Using gold to hedge widgets would be an exnmple of crosshedging: the use of a derivative on one asset to hedge anotlzer asset. Cross-hedging adses in many different contexts.
lztrne-for-one''in tlzis context means tllat if the price of by$1 times the amount of gold used to make the widget.
115
The variance of te retllrn on the hedged position is given by
j.
:
%
Suppose that we go long H gold futures contracts, each coveling 1 oz. of gold. I.f F is the gold fonvard plice, the retttrn on the hedged posidon is
j
j
o
where Pw is the widget price.
.
()
RATI
and gold. Conceptually, we can think of hedging widgets and gold separately, and then combining those separate hedges into one net hedget To generalize the Auric example, suppose that buying Ng ounces of gold enables produce Nw widgets. Profit per ounce of gold, without hedging, is to us
.
.... .. . ... .
The hedging problem for Aul'ic is to hedge the dterence
E: 'i'(:.E: EE /73);:E . .
H EDGE
N G TH E
gold rises by $1, the.plice of a widget
lises
l3rrhis can be derived by differentiating equation (4.2)with respect to H*. 14ne term pcrmjo'g measures the comovement of gold and widgetplices, which is typically measured using a linear regression. A common approach' is to regress prce changes on price changes: P..t
-
#u,,J-l
=
+
ppg,l
-
#,g,,-1) +
lf/
'
where the subscript t denotes tlle value at time t. Other specilications, including ttle use of jercentage changes, or regressing levels on levels, are possible. The conect rejression to use dpendj (jn context. In general, regressions using clianges are more likely to give a correct hedge ceflicient since te goal of the hedge is (o haye changes in the price of the asset matched by changes in futures price. In Chapters 5 and 6, we will present examples of hedging stocks and jet fuel, and the appropriate regressions will be returns on returns (stocks)arld chang s on changes et fuell. Regressions of level on level are problematic in mafly contexa. Forexample, in the case of stoclcs, msset pricing models tell us that stock retttnts are related, but we would not expect a stable relauonship between twoprices. The appropriate regression is retul'ns on returns. A comprehensive discussion is Siegel and Siegel (1990, pp. 114-135).
116
k
Ilq'rltooucTlox
'ro
Rlsl
MANAGEMENT
sEt-Ec-rllqc
H*, is denominated in terms of units of gold. The net use of gold futtlres, H*, is the ditference in the number of coneacts needed to hedge gold inputs and widgt outputs. Note that if gold and widgets are uncorrelated-i.e., p = o-the hedge ratio is purchased), while fp > 0, the hedge of gold long fomard per ounce ounce one one (go ratio is less than one because the widget price itself provides an implicit long position in gold that is a partial hedge. When we hedge with H* f'uttlres, ctjgetj is obtained by substimting S* into
equation (4.2):
jg:yyy;jjf jjzjjq,!f j;:ryy:yf jj:rrgyyf E'. t)'kE ,'.y' ;'?' :.-j' ;?' '''J' E E Ei'E' jjygygg!f jjyjggr!f k'' ,'y' 11:::::)* jjr,,,,jjf 2:jj;;* yrjg;;f jjr,,,rjjf k;yjyyjjf jjr,,.f jjy.,.f (' r;f ;'' '5* ;'jj'. j'1* g' ,:;' 2!51;;* iiiil:f it' lr' itjlj' flff )q' j;' 5! 7j6: : #E !E'E E((i.:'EEi' ('.(: ! i .'E'E''' i i . EEE.'( (E'(( i E;E E i E' ;(!' jjj.,,yj;f .jjyyy' .jjyry' .$jjgry,,j,,;-!jjjj,,' jjy,,.yj;f p' ((:77777* )' Eqqr' ,'r' (' y' @' 1Ir'''' jj'. .IEEE!t' .IEiii!t' kf' 117.'':1:* .1:5E1,,* i''.''(s:'lf (!q(l' ;''r' (7( qr!q ill!lpp;lqqi jjglljyjjf jjyf jjjf jy);'tjjy'. )L)jLf',' jjt))y))))y-' jjjt'yjy rtk--?'.' ykk;t)jk(;(kk;j'-' (q' LiLk'jkji'. rr:' :j' q-' (p(' frili);lf 7f':)' . . ; ;.'' .5 jjyf jj:yyr:j,f 44!!,!7!!,4,14q*.* lllitj!j)f j'yjf rjf tf ;LLLjLL'. yk'. r' j)f t'li'l 1'*)j ('. jy'pj tjf lr' ()')j$j(r'i.rj)l1i(I 1)1!: . ((: . qr' t.i.'. q.' . . '. . . .. . q:' E E ..y ; !! Ey.pt'l ! jltt!ttji ;.!, E. .y j@ j ,pq y:.. ;Ey.E.iE.i E..E.j..yy. jqj!. i rrr)yp. .y . (,.yrr!E#:..y y l. y.y.(j ; y jy.,: y j . jjyyyjj g y ( E j j j ' ) 1lt.(qr7i''iIII7777:. y t-;4i:1IIIl2..; 1. IIIIEEEEII; 1111:)112.'!!EE;iI!!t.. i .. ( . @ . ! ljqjkk. ; . .LLIIIL . t.i((: . . .....:y;. . ('.r.r(rE... .. i. -'
tjyf y' y'. 'lll'f tj'y')q t'1171:* ttf . -;jjjjjj' -' ;''' -)'.
.' '..'-' '
-
'
E
:
.
-
-
:
-
'
.
-....,,k.
...........
-
-
:'
'
'' : .
.
:
.. ..
-
, -.......... .
....
.
:
-. -. -.
-. .. .-
. .
-
.,
E
.. .
. .
( -y
. .
-
-. .
-
-
(1 -
p2)
(4.4)
Te uncertainty remaining in the hedged position is due to basis risk, which is lisk due to the hedging instzument (gold)and the hedged price (widgets)not moving as predicted. The variance o f pro fits ultimately depends upon te ability to hedge the price of widgets, which, since we are using gold to hedge, depends on the correlation, p, between widgets and gold. The larger that p is, the less is basis lisk. tls section we have shown that the ability to cross-hedge depends upon the 1.11 coaelation between the hedging instrument and the asset being hedged, and that we can determine the hedging amount as a regression coefcient. We will see in Section 5.4 that the same analysis obtains when we use stock index futtlres conacts to cross-hedge a stock portfolio.
QuantityUncertainty
Thequantity afirmprociuces andsells mayvary withtheplices of inputs orottputs. When t'obvious'' hedge ratio (for example, by hedging te expected this happens, usina the quantity of an iput) can increase rather tan decrease lisk. ln this sectin we xamine quantity uncertinty. Agricultural producers commonly face quantity uncertainty because crop size is affected by factors such as weater and disease. Moreover, we expect there to be a correlation between quantity and price, because good weather gives rise to bountiful harvests. What q'uantity of fonvard onacts should a corn producer enter into to minimize the variability of revenue? We will look at three examples of different relationships between price and quantity: The benchmark case where quantity is certain, an example where quandty and price are neyatively correlated, and an example where quantity and price are positively correlated.lD
t
('')'! )ti r..Eii:rk r.. 'i:iii. d1t2f:.dt: lit'..E'1(EE22:l' 241. !;i1i,. tili!li. )El13'i'. :
. . .
.
.
.
..-
-
.
tRquanto''
-
..
i-
( .
.
- .
....
.
jjygj,r jjyygry
jjj jjj
jjjyyyrjj,
.jjygy.
jj:yyy:jj,
jj:yyy:jj-
jjr,,,
,jkyjyy!,
jj,,..
jjyjyj!y
and quantity for an agricultural producer. Each row is equally Iikely. In scenario A, there is no quantity uncertainty. In scenario B, quantity is negatively c rrelated with prce, and irl scenario C, quantity is positively correlated with price.
t')@,
.
..
E E'EE EE E ( tEEEE) E,;'E5E,.3!y(.Ei.'!q;'.,.'); E(
;(
(
(E ; El .E't',t''i
EEi
.).,r'E;,
;' 'E.;
-.
Li:4
JUOEE
V .Ewsk
..
lrry
' ! r. k @(E(EE( ( E( E t ( .,.,'(E ( x ( - y.tyyeyy nyay rlo; LL;,,ii! =(
tE (qEE
.r
,,
,
.j,k;j,..
.
--.)
.....E...... ....... -.......-...?.... .. ... ... ...-...-
........i...-.....-.-.-.y......-.....................-....-.......-.....-...-.-.-.......-... ..-...-..-..E.............
...-.
3
..-
.
.
..
jjjygjj,(y)j,Ej.. ...jj.y.y.j..
,jjrjjjj,
.-
y,,..!.t,;',t
...jjj,.,jjyyj;j,
! j(,Er, ( E E(
(E
.j
E
tl E ( ( r
-..-...--...E........-:-... ....E..-.-..........-.........-..i.......... ...-...........-E.....-....E................... ...-.............-.-.E...-................ -..........-.............Er..-r... i......-.......-.-...-.., .E--.-.- ....-.....-. -......-.... ... -...............
.
1.0m
1.5m
1.0m 1.0m
0.6m
0.8m
1.5m
1.0m
1.0m
0.8m
0.6m
-..y.......-.... .........
....-.....-..-.....
............
1.0m
);)E,EE..t !'i
.. (.. E .
..kjjgr:
....
,,j;;,,.
;E'.,
.
.
......-........ ......
....-....
.
addiion, for each ppssible price of corn there are two equally likely quantities, for a totat o f four possible price-quantity pairs. Table 4.9 illujtrates te three scenados. Note that in scenario B, average quantity is low when price is high (negativecorrelation), correlation). wherea i.nscenario C, average quantity is high whn price is high (positive
First considr ScenarioAwhere quantity iq certain: The producer always produces busels. t-etv and denotetheprice andquantityin 1 year. Revenueis SQ. Without 1m hedging, revenue will be either $3m(if the corn price is $3) or $2m (if the corn price is ..
$2). F
On the other hand, if the producer sells folavard lm bushels at the forward price 2.50 revenue is
=
Revenue
?' '''''''''. .'111r:;;12!,* y' :.' k;il..:ll,d L' !' y' f'jd ijlj::...d ill::,.-d .1i::ii@' 7* 1* ?' 2* 'ICY
CS x 1m)
=
-
g1mx CS 2.50))
2.5m
=
-
:uatanted rvenue in this case. 'I'he calculation is illustrated explicitly in
We hke
.'
'(' r' q' rf t'y' q' 'q' p)' 7' ;'((' j'('(' (' )' 9* 11* Cqf i'if i)'7872Eq7iEliqj))))' IEET:T/;?;:7!:T!I E' ' 77 )th))l' y' 'i'7* tilf (' E E t';'qtf ''1: il. i)E(i'!E i iEEE;!.E.(E'.. !IEE.E(.1( r))pf (EE...' E' i'(' :.i'(E.:E;!'!;( ';.;'qi.i.;!j.@ ji)).! .ltl rq .'E..E .'. tEy.j..::...F . . . ;Ejyj-)jtjy.r.jE:g.jy:iyi;.Er.j!.E..;-l--l 'yrq.. .ky)g).!E.g! .. r:!p y(!Ejjt.)));j.(jjtj(jjyyjkj' (...r.jrltsjjjljjl.y qj(j.j!.y(.gyqq )(j(( .Lj)jj))jILL;)(((;).j.jLLjL)jjjjj (. ...;6gE.lq-E)E;-'ri.;-.t-)j.!-E;;.-y-j.i).$.;.:-tr.r.;Ey.-.rEy.E-.jj tt;! j.gyjy .,-q)r-tyr.--;. .l.)9)tr. y.jjs(yy j y.jjjj. E..... .IiEE!lkk. . )( )i. .j 'ik;,'hk;/'' Ir..ll 4:::11d::::l 4:::j1d::::. . i.E'EE;(-)..4.7tL..)iu),-d.jz.>..dlEs!, 4::::). ..;k!47liik)d:ii!r d:::)p . 1k..iI . rlt:.4:::),..jr:::,dlE!!, 4E51,, Ir.' IE..II Ir,.i1E ,5;. t!;: Ir.' 1r,. 1..IId:Ei!qlr..tr..1i Ik..II Ii..ll Ik..1l I:::p, I..1k 1( .Ik:. 4:E5,: t!!i tf!;: p.. y)j)2y;... 14::),
jk-')l(#'-yj'. ''
'
i
-'
'
--
-
- . .
'
'
' :' ! E. . : : .
-
i
:
.(.(t.gj.j;!j(E : ..
- .
:
-
:
E'
.
-
- .(!yy;'t(-
'
.-
E: : :
.
. : .. . ..
' :
. . .
-
:
:
..
:
-
jt#i!:ii)'i!k--...' -'
-(j);:.'.-.))...l
-rr!j4rr))(....
t
ls-rhere are futures contracts intended to mitigate the problem of quantity uncertainty in an agriculttlral context. Corn yield futures, for example, traded at tlze Chicago Board of Trade, permit farmers to hedge variations in reoional production quantity, and provide alz aoericulmral eKample of a conact. We discuss quantos further in Chapter 22.
.jjy:;gjj::. jjy,,, .jjjgrj jjy
t t( t,! o.fluctl.oy t;y. ) j jjrj!y n..y j j ( j . ( ( . . y.. jj:jj,;(jjyrj(.. .. .y.( . jjrjjr.,j;jg;y. . gjt;j . .jjyjr. ..gjjjrjjy jjy2rj. .gjrj,;.kjIj,,. yjtytjjs :jgjjj.. :yjjj(. ;.j. ,jjgjj, jj;;. jjkgjytjjy j(yy(j:!rjjk4...j(ryyjj(gyrr,. ...jjtyyj. . gjjrgjjj. y.y . jjgyyyg:r. j;kgjjj. jjryjj,. jjyj!j. . jjjt,i .....-.....g jjgjj,.kjjyjr (jjy (jj, kjgjjjjs .kjrjj;; kjryjj)gjrj,, yyyyy.. .kjjyjj .gj;jj,..gjjji,.. yyjj.... . rjjkjj;;i (y)j. ..... -..-..................;..L.,k......L-.,...L.L.
i
. .
..11
--)-.
.
In all the examples we suppose that the col'n forward plice is $2.50*u and that there is a 50% probabili. ty tat in one year the corn price will be $2/b11or $3/bu. ln
jj:ygg;jjj
jj..,,jjj ;:jj;;y jjg jjy jj.
11'...
;Fl
i
:
=
jj,,..yji ''''rII'''''
) ,..
- -- . .. -. .
E EEEEIE rl (1 E( E C( E((E(E() E E C( E(EEE('llt (( EE )(l (()'E@..,' ('' E E ''' : @!tE'EC i)r)i)q E.,E,(). ,.:.(;.''y,'Eiy',Eyy'iEy(Er)qE.@ yy'E g!' E( E. E . :' j : . y -:.. ' . .g.- (g (j. ......r.. j... - .. iy.('. j E .:. . . .. . . -... . . .........$)..@p'-'.i;.-);-i.E.j........E-.......--......... .-..j(ry,jjygjj;;... . -,jryjj,. .y...;,rjjjk:y-. . . . ..-..
Nlo'l 2 trhedged u) u,
%.
RATlo
.
-
.
.
-
;
.........
:
. . . - - .
-
.. --. . E
.
. -. .
.........
' -
.
tq.kk-, ttj
THE HEDGE
.--
:
.
.
(Scenario A), revenuerof $2.5m is assured by sellinj forward 1 m bushels.
E : : E E E: j (.. ' :j E : :g . .. (.j('. g..(j. . g : (ii.(i(-.' 'g'. 1-.. (E..j.' . .- .('- i''-. EjE(. i.j(-..y...E.('...!EE...i.. ' E.(j'. . g..E... ..' E(j . :.' ' . @:- ... . .. . .. . ..:. ''. ' '. ' '(. . j-'g E.' ('.-.j.('..E.:.. . i. 1(.'.. '.j.'.'..E..'((..(. jE(.g-: ((...''. :. j..''-: :k...-(jj: kj'i..E'(': ... . i... .E . . .. . . . . . . . ..... .! ... E . .. . . .. . . E. .( . . . . . . . . . .. . . . . i.. . ' . : .. .. .. . . . . . ;.. . . . . . . E' . ' ' 1. . .' .: . ..E. ' '. '. ' .' . . ' ' '. ' . ' '' : ' : . .'''. ' E E1: . ... ' . . .' :... . i(' E:.. (('.'qi'.. : . ..;: . .'. . .. .'.. ('E.'' ('':E. '. .' . '.'.::.. ..E..' ' '.E' EE (' .('( ' ' ' ' '. . ' ' (' '.. .' ...( .. .. ' .('..E'E. . . . .... . . . . . . ( (. 'j:E (.' i . j..''..jj. (j .. (.. i'...:...'. ... .. . .jj. .... ''jg:''' ' ..i .. ....:y.. .. j..... . . . ' ..(.. ....j .. ..... E......... ... . .. .j.j.....'r g. .y ..'j.'jy j('.(.'.'..'...'.. . g .. . .. . ... .. j. . . . .. .. . .. . . .. . . . . . . . . .g... ..... .. ..... .
lig..... . .
( ... . ( .' . . ... j -
. .
. . .
.
. .
'
:
'
.
.
'
:
:
.
.
2
'.' .('.'...' .. .
. .
'
:
..
litildi 3 .
('..l
:
-
:
:
.
. .
.
:
'
.
:
q( .....
' '
:
.
'
.
'
.
:
. . . . .
.
.
. .
'
. '
.
.
'
:
.. . . .
.
1 0m 1.0m
( k
:
.
'
: .
g . . ; .y) Qidtr .
ty
'
.
:
:
Utidkd 3.0m 2.0m
:
.'
'
'
.
-
' '
' ' ' ' . . . . . . .
'
.
'
'''''.''''''''.
(' ''J' .. .(. Ej ( E( (' E: ..''. . . E : E(Ej(( :' .' ( ! : ' '(E' E. E'...EE ..E : j. ( (( (. - . . .. . -' .'. . ..'.. .'. . .E-. . !E . .. .E(..'.. : -. ... '... ...'.'. ..'E. ''.. .- . (E .. .-Iljk;,-. . ' ..'. '.' ...111,.111. .. . E E. . . - . . . .. E . . ( .. . . .: .. '''''''''''. '. . . . . . .. : ..... .' ' i . ' .E' (' ' '' E' ' . . . . .: ' . ''. .. .E'... E. E ' . ' ': . y.. g . . . . . . '. .. . .. . . . . :y.... . . . . . .. . (j '
'''
.
. .
..
'?!'''''
) ) #tt q
. .. .
'
''''''''i'.
.
)
'
.
.
2.5m
2.5m
.
-
. . ..
.
.
.. . .
:
ti
.
.
.
.
'
:
. .
.
'
.
Itwkd .
'
'
:
'
.
. .
.
.
bti .
j
.
.
i
k
118
Ilq'rstolluc'rlolq
CHAPTER
MANAGEMENT
Rlslt
To
ln general, if the producer enters into fonvard conacts Hedged revenue
RH4
=
(4.5),when there is uncertainty,
Using equation is
2 O'RCH)
2
y2
-1-
GSQ
=
:)
CS x
=
2 GS
+ (S x CS
0.25 x (1 + 0.6 + 1.5 + 0.8)
F)1
-
(4.5)
V gI.LI9SQ,S
(y.6)
SQ o's
g.
The standard deviation of total revenue, SQ, is o'sc, and the correlation of total revepue with price is psc,s. As in the preceding discussion of crossuhedging, the H tat minimizes the variance of hedged revenue will be H
.
=-
The variancecminimizing
calculation, we have psc,s have
pso,so'so Gs
H
This is the same as the second term in equation (4.3). The formula for the varianceminimizing hedge ratio in equation (4.7)is the negative of the coecient from a regression'of unhedged revenue on price. We can therefore determine the variance-minimizing B and C) in hedge ratios for the negative- and positive-conelation scenarios (scenarios Table 4.9 either by using equation (4.7)directly, or els we can nm a regression of revenue on p S ce. First, consider what happens in Scenario B if we hedge by shordng the expected quantity of production. As a benchmark, column 3 of Table 4.11 shows that unhedged revenue has vatiability of $0.654m.Fromrfable 4.9, expected production in the negative
ik'ik-'''. j')'. jjyjjyjjf tsjyyf t'r' ;i' :'(1* tiitiiiiL'. :ii'.)i1rl'. j''iiii!,''t'' ,iltqllipi'. jjy,,,yjjf jj,,,,jjf iyyjy;jjf jjy.,,:jjf 'p' (' ;'ffjd iE;f i.jj' f' -'I'.(iIIt.-' E!,ij::II.)')' 'iIIt!i:II.l' EiiI::II'.'. likirlikklrikkll'' kkI5k)!k'.IjItE)ii,,'. ('1!' Ci.EEEi''. (Iik:rikik)'. (111j#:Iii.'. .kIk)(Ik))'. kII:!ii1i()'. jjy,,.yjjf y'y t'.rr@'r q' qr't' si'fl.tjllt'sf q)' lq'k.yqf 1* (' :11.1:1111t1.111*. iliEiiil:ld rrjjj;f jy'. 1.1.1111..t111t.*. i'.' :2!ijii;:'. !jjiij,r;2'';'C'jjyf jjj,,,:jjf jjryyg:j,f kjjf jjf j;f yj'tttf ))' rf t'''E:iiIIEEqi,'-' ''ii:EEt1Ek;l):'. :i1Ii:f':(jjII(()'' k1k)!iiIi)'. tE1IlEE):''i''. tIiI())ii()'. tk:jIjI()''' 11iijjI1(y'. (llillitii'i-'. j'jy'tjf ))j' jj'yt(j(r'-. i'1t:E)El;1l:.'. (:1I1tEEi'-'.' y' .yy;gjjjj'y 4* ''C'y k11:111:*. ('. tr'. )' yjy'. jjy.jjyyjtj':.1211,1-*. 'iii.iii!:.-' jqFft!l )i(((l((tl5ltql;' iq : ': '.E('E ( (' i.(!(!!( !lq !' (((( q i' (((' i.'' i('' ( ;q (:'i' !'i((iq.''' '1(!;'ik'i'f!q:(iE(;'r!1* jjjj . 1115 . lt llqq ktl . j.j.....jy.. j . j .j .i' i.. Ej.. !'.y . E )))-.tti))t-tl4:-Ijipj!. li;:.(--pq . .). j ...t.j q ( (y. . j ;(.yyj. j.(y! .jyg..j q( j j ..;-( y(.jjjjjjjg(-. jjyl . (q y . y .j jj). ,j-t.).yjjij;j,jy,,.,y.j.k..... .. .-j..($;krjjjjj;-y. .....y..j j..j...yjj.j.jyj. lj jj:rj;rj . yy , .jty.-.---;j)j. . . .gyjyr..jj;jjjjj. tr .LklLLskk;. ijI!!1;;. .,,36?qbb;t. y jjjjjgjjkj. . g, j.;.. .jy.t . ... ....L1............,.... .y .. '
'
'
:
.
lttilttlrq. !
...
.yj.E),...
.......,.
.......
......
....)
-
-
E '' i: E E El' . . ! ! :. ;
' '
:
:
:
E 5' EE! . ttt., . .
. . .
.
E
:: ..E .
:
:' ...
E
'
:
..
: : : . . E?;E .. E : . .i E
i y-.jj E y y .
.
......;.........-.........!:
-.....
,
. .
.
'
'''.
.-
--
..
.
-
-
.
.
.
' .
-'.''.. . . .
-.
. ..
--
lE
.
. q
-.
E .
'
.q . .-
IEEC tE,..
-.
.E -. EE. EE..E....EE' -.. EE.. - .. .E E - . - .E . . . . -; -. .. iEE. . .. .. q'E : . . i: . i:. .
.jjyyy. jj,,,. .jjyyyy,jk,,,,jjj;j,, .jjjrg, jj,,,, jj, jjjyyyrjj, jj,,,, kkjjyjjj rrjjjj;i of production) from shorting and 975,000 th quantity corn forwards (columns4 and 5) and from selling fprward corn 100,000 bushels (columns6 and 7). Each rice-quntity combinaton is equally Iikely,with a probability of 0.25. Standard deviations are computed using the population estimate of standard deviation. jjyjyy,!y jjr,,, jjjjjjy;j jj. kjgjgjjj jg,.,,jjj jj,,,.yjj gjgjjjjjj.
-
E
.
'
'
' -...
-
,,jk,;,,,
.. q. !: . ..
.. . . -. . .
.
.
,
..
:
-
E(
(
g
:
::
,
,
$3
0.6m
$2 $2
1.5m
0.8m
revenue o'total
.
.
.
.
.
Et
-
-
.
.
-
-
-. .
.
.
$3.0m $1.8m $3.0m $1.m
$0.654m
g
-
'
; -.. ..... .
.
-
. E(E
.
- . .
4ki2?.. .
-
E
-
.
.
-. . .
0.07647 x $0.654m $0.5
-0.100m
=
,jjygg.
jjygyj,,r
'.1IIt:(,.(:iE'E.
-
'
-.
-
-y
. y,y--.yy
...
jj:yyy:jjs jjyjjjrs
jj:jjgrjs
jj:yyy:jj,
tIIiiE),kk-. i;Ik(iqkkk)qikkk).. !
jjrggy,r
jjygjyyr, jj:gyygjjr
>. .... .-. ....
.
.
(
. - -- .. . . .
.kkti;j,!ik;;,. .
(
.
-.
y
.r
. .. .
-.
-
-
-
-
-$0.488m
-$0.488m
$0.488m $0.488m
.
. . .
-
-i1ti:ii'..
:
'111t:'.
'
'
-
'
-
.. -
-.. .-
-
..
'
.. . . .
E-
.
.
-.
.
.
-
.
.
-
--,.' '
-
-
-
-. .-
.
'E E. ' ' E'E. - '- -. -. - . i -. ' . . . -.. . . .....
$1.312m $3.488m $2.088m $0.8l4m
. .
t(ijitjjjikk-. .. -
E
.
l:jiikikL).
-' .
..
.1i(:(l1,-
--k:. '
' .
-
-
-
.
:
.
E ( ili'i!'-E. ,
y( . .
-
.
.
.
,
.
-$0.050m
-$0.050m
$0.050m $0.050m
-
'-.EE.- ' . . . -. -
..E
((
.
.
:
-'1ii!i.,k.
'
. -.
-.
..
. E:' -
- .. . . -
-
-.
(
,
,
z, ..
..=. tM q1... 1HAZVuj;:',..,.',.?.3i
CHAPTER
$kI:i:jk-. ;i,i!1ii'. ' i!!!l;Ili-. i. (:,tk;),t1;:k;:. i1i()(li,-. .1IiiIi,.. ' iiLj;'i''. (i1Ii)'. '' ()jIt(. ..'. . . ! i. ( k:!!iiikiI:. .. .$111111,:. .'''.' ikk))pk.. . ..Ej-p....g.. ..'... . .. y. . (.j... '. ( '. ... ..'.'.-;.......'-.'....'...).'.'......(!r'y..'....'..q ..t:!. (... '. ' . . .('.(. '.... (iik,ri-t-. .('.'..' .. :iIi:;,.. .. .y.....(l .(.iiI.:I-k.. . .. ( ) ( .. ((( (...'. (j,l.r:i ( (( ( .( ('..(; ..( . . ( (. . ... ( ) . . . ), .( .(... .. r.)( . (ij t .tjt. r . . .)( .)..!( r(:t (.( . j (j..(j. . . :ii!!ih:. . y . y . k ) . . . . . :1iE::iiiilEiiii' j ''tEE1iiEEE-i-. .. . .. . '.iEEI'iL.. Eiii::'''. i E . 1.1E1111..:1111.. iiiL.:11il. 'il(':1'i:.. ' '. -iIiEiii.. ' t ' t: -iIi!!,i-. :111:.:11.:::11,... . (( . tEi1IEE;.1I2i1.' ' . . ; (IIi::7''. E!1iIEEp''''iiI!:! E '. '.. y.- ' '..'. ; ;' ;.' tilili. ..;' '.. r'.'-' ; '-. . .'. k'li:E::;;!d!!. :... . .. ; ' p'.) . . k'':r!.' ('. '(:. rq . .....q. . . . .. . '.. . .). . . .. . .. .. . . .. .. . . .'. .. . 1.0m $3 $2.512m $2.95m . ' i.'- ! E.. . .. -. - . . iE .. - . E( .. . E.. .
i --
l
=
hedge can be obtained using equation (4.7). By direct o's = $0.5, and o'sc = $0.654m.16Thus, we
0.07647,
i.
-
-y
-
-
=
'l'h
.'. .'.' .'-.!!ir!''.' '. ' ' ' : : '' ! EE E: E E. . E E . E! EE E j.
:IE
0.975
=
Column (7) of Table 4.11 shows that variability is reducd to $0.652mwhen hedging optimal hedge quantity is closer to no hedging t.1)a11 hedging. tis amount. to R111 In fact, we gain little by hedging optimally, but we increas: the standard deviation of revenue by 25% if we adopt the plausible but incorrect hedging strategy of shorting 975,040 bushels. Problem 4.21 asks you to venfy' that you obtain the same nswer by On ning a regression of revenue on price. Yo right guess by now that when correladon is positive (Scenario C), the optimal hedge quantity exceeds expected quantity. The fact that quantity goes up when price goes up inakes revenue tat much more varibl than when pric alciil: vaties, and a corrspondigly larger hedge position is reqired. Problem 4.23 askj #o to compute 2 millio bushelseven . j hedge in scenario C. The answer j s to jjmrt mmost th ptlm though production is never tat large.
,'-ifj!!;;''. -'' ,jjjyy.:!jj;;' jjkj'y ''' -'-' '''.jjy::' jj:ryygyf -j' :y': S' 11i(j)li-'. jj:rygg)f jjyjjyjf jj:rgy;jjf jjrryyf jjyjy,!f jj::yggyf jjygjjgjf t't' )' ;'
rllli:::i!j;;ii'd tlit)ljil;d .'
119
Ifwe short tltis quantity of corn, column 5 of Table 4.11 shows that there is sti.llvariability irl hedged revenue. Perhaps more surprising, the variability of total revenue acmally itcreases. The reason is that price decreases when quantity increases, so nature already provides a degree of hedging: The increase in qaan:ty pnrfially offsets the decrease in pric. Hedging by shorting the f'UIIexpected quantity leaves us overhedged, with a commensurate increase in variability.
tr2s(s),
the variability of hedged revenue,
%.
cotnelation scenario, B, is
on H units, hedged rev-
will be
enue, RH),
SUMMARV
.
g
y ' --- -. - -' . . ' . -y. E. . . - .. . . . .
.
.
.
$1.75m $3.05m $1.65m
$0.652m
-. .
.. . .. . .. .
-
':.
77:F... .
.:....4.
..
..
.
b.. .i..:q7r. ..'.L.jjL.j3LL.Lk?L'LzjkA.g'a5. >. 'T. -N...'..t'..iiIii'!'.'fy?i6,?.:1 ..XN.... '3i;'q6qLl'?3.... . . . ... ... .c.
..
,
.
..
.
.
......
. . .
q(E4irE;;):(1Un..!.>g= ytL'>'Z
?..7...i.i.>. :::T7..1. LL'LLjLLLLLLLL133. qjl ; .
SUMMARY
A producer sellig a rislty commodity, such as gold, has ll inherent long position in the commodity Assuming costs are :xed, the firm's profit increases when the price of the commodity increases. Such a f11-mcan hedge profit with a variety of sateocries, including slling forward, buying puts, and buying collars. A 51-r11tat faces price l'isk on inputs hs an inherent short position in te commodity, with prost that decreases when the price of the input increases. Hedocringstrateocriesfor such a 51-minclude buying forward, buying calls, and selling collars. Al1 of the sategies hwolving options can be customized by chanocing the option strike prices. Sateoeies such as a paylater can
l6Because Table 4.11 presents the complete population of outcomes, which are equally likely, it is appropriate to use the populauon estimate of the standard devition. In Excel, this is STDEVP as opposed to STDEV. ne calculation for o'sc is obtained as STDEVPO,1.8, 3, 1.6) 0.6538. =
120
%.I?q-rftoouc'rlolq
To
RlsK
PROB LEMS
MANAGEMENT
provide insurance with no initial premium, but on which the company has greater losses should the insurance be needed. Hedging can be optimal for a company when an extra dollar of income received in times of high prosts is worth less t1::,1:an exa dollar of income received in times of low prosts. Prosts for such a fil'm are concave, irl which case hedging can increase expected cash iow. Concave prosts can arise from taxes, bankrptcy costs, costly external nance, preservation of debt capacity, and managerial risk aversion. Such a 51-mcan increase expected cash flow by hedging. Nevertheless, firms may elect not to hedge for reasons including transaction costs of dealing irl derivatives, the requirement for expertise, the need to monitor and con'ol the hedging process, and complications iom tax and accounting considerations. FURTHER
'''''
:
'
.
.
'
'
.
'
.
.
.
.
'
.
.
.
l-year continuously compounded 'Fhe l-year forward price of copper is $1/1b. option prices for interest rate is 6%. One-year copper are shown in the table below.l? 'rhe
Strike 0.9500 0.9750 1.0000 1.0250 1.0340 1.0500 In your answers, at a $1.00, $1.10,and $1.20.
,
.
.
minimum
Call
Put
$0.0649
$0.0178 0.0285 0.0376 0.0509 0.0563 0.0665
0.0500 0.0376 0.0274 0.0243
0.0194
consider copper prices in 1 year of
$0.80,$0.90,
4.1. lf XYZ does nothing to manage copper pri risk, what is its prot 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copperproduction, what is its estimated prot 1 yearfrom now? Cnstnlct graphs illusating both unhedged ynd hedged profit.
'
. .
121
@ Telco installs telecommunications equipment and uses copper wire from Wirco as an input. For planning purposes, Telco assigns a sxedrevenue of $6.20for each unit of wire it uses.
READING
In this and earlier' chapters we have examined uses of forwards and options, tnking for g'rantedihe pricing of tlpse contracts. Two big unanswered questiops afei How are prices determlned? How does the market for them work? those ' In chapters 5 tlaiough 8, we will explore forwartt and ftures coniacts cticussing Capters 10 througti 13 we will Pricingas well as how market-makers function. 1.11 answer the same questions for optlons. Chapter 14 will discuss how exotic options can be used in risk-management sategies in place of the ordinary puis nd clls discussed in this ch>pter. Wlkarton and irms to assess their hedging. survey noafinanlal cmc regularlyBodnar A recent survey is summ arized in et al (1998) Bnrtram et al (2004)exanune hedgsng bhavior in an internadonal sample of over 7000 firms. Tufano (196, 1998), Petersen andrfhiagarajan (2000),andBrown ei a1. (2003)have smdied hedgingpractices in thegold-mlnlngindustly Otherpapers exnmlnlghedogincludeGczyeti. (1997), Mlayannis and Weston (2001),Maynnnis et a1. (2003), and Allaynnnis et a1. (2004). Guay and Kothari (2003) attempt to quantifyv derivatives usage using information in 51-mnnnual reports from 1997. Brown (2001)provides an interesting and dtailed by one (anonymous)'Iirm and Faulkender (2001) description of the hedging exnmines imerest rate hedging in the chemical industry. Gastineau et a1. (2001)discuss Statement of Financial Accouting Standards 133, wlzichcurrently governs accounting for derivatives. Finally, Reming (1997)relates some of the history of ttheficdtous) Auric Enterprises.
%.
4.2. Suppose the l-year copper forward price were $0.80instead of $1. If XYZ wer to srll folavard its expected copper production,whpt is its estimated profit on'e year from now? Should XYZ produce copper? What if ie forward copper plice is $0 45?
.
4.3. Compute estimated profit in 1 IIIIaI'it'XYZ buys a put opon with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case. 4.4. Compt: estimated prost in 1 IaTiI'XYZ sells acall option Fith a sfrike of $0.95? $1.00, or $1.05. Draw a graph of prost i each case.
.decisions
4.5. Compute estimated profit itl 1 yearif XYZ buys collars with the following s-rikes: a. $0.95for the put and $1.00for tlkecall. b. $0.975for the put and $1.025for the call. c. $1.05for te put and $1.05for the call. Draw a graph of protit in each case.
4.6. Compute estimated profit in 1 year if XYZ buys paylaterpf/l. premium may not be exactly zero):
PROBLEMS For the following problems consider the following three ftrms:
@A'I''Z mines copper, with xed costs of $0.50/1band variable cost of $0.40/lb. * Wirco produces wire. lt buys copper and manufactures wir. One pound of copper can be used to produce one unit of wire, which sells for the price of copper plus $5. Fixed cost per unit is $3 and noncopper variable cost is $1.50.
as follows tthenet
a. Sell one 1.025-strike put and buy t'wo 0.975-strike puts. j
g These
are opion prices from the Black formula assuming that the risk-free rate is 0.06, volatility is 0.l and time to ekpiration is one year. ,
122
%.llq-rRoouc-rloN
'ro
RlsK
P Ro
MANAGEMENT
%.
BLEMS
123
probability of a $600 loss each year. Firm B has a 50% probability of a $300 proft and a 50% probability of a $100 prot each year.
b. Sell two 1.034-strike puts and buy three l.oo-strike puts. Draw a graph of profit itl each case.
4.7. If Telco does nothing to manage copper plice tisk, what is its profh 1 year from now, per pound of copper tat it buys? If it hedges the price of wire by buying copper forward, what is its estimated prost 1 year from now? Construct graphs illustrating both unhedged and hedged prost. 4.8. Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95, $1.00,or $1.05.Draw a graph of profit in each case. 4.9. Compute esmatedprofit in 1 year if Telco sells aput option with a strike of $0.95, $1.00, or $1.05. Draw a graph of prost in each case. 4.10. Compute estimated prost in 1 year if Telco slls collars with the following sikes: a $0'.95for the put and $1.00for the call. b. $0.975for the put and $1.025for the call. c. $0.95for the put and $0.95for the call.
a. What is the expected pre-tax prost nekt year for ftrms A and B? b. What is the expected after-tax profit next year for fil'msA and B? 4.17. Suppose thattirms face a40% income tax rate on positive prosts and that netlosses reeive no credit. (Thus, if profits are positi' ve, after-tax income is (1 0.4) x proht, while if there is a loss, after-tu income is the amount lost.) Firms A and B have the snme cash flow distzibution as in the previous problem. Suppose the appropriate effective annual discount rate for both firms is 10%. -
a. What is the expected pre-tax profit for A and B? b. Whgt is the expected after-tax prot forA and B? c. What would Firms A and B pay today to receive next year's expected cash flow for sure, instead of the variable cash Qows described above? For the following problems use the BSCall option pricing function with a stock price of $420 (theforward plice), voladlity of 5.5%, continuously compounded interest rate of 4.879%, dividend yield of 4.879%, and time to expiration of 1 year. The problems require you to vary the stlike prices.
Draw a graph of profit in each case.
4.11, Compute estimated prost in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero): '
one 0.975-strike dall and buy t'wo 1.034-st!ike calls.
a. Sell b. Sell t'wo l.oo-sfrike calls and buy three 1.034-sike
calls.
a. Suppose that Auric insures against a price increase by purchasing a +1.0strike call. Vel'ify by drawing aprost diagrnm that simultaneously selling 400-stIike put will generate a collar. What is the cost of yhis collar to a
'
Auric?
Draw a paph of profit in each case.
4.12. Suppose that Wirco does nothing to manage te risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose thatWirco buys copper fomard at $1. What is its prost 1 yeay from now? 4.13. What happens to the vatiability of Wirco's prost if Wirco undertakes any strategy (buying ealls, selling puts, collars, etc.) to lock in the price of copper next year? You cn use your answer to the previous question to illusate your response. 4.14. Golddiggers has zero net income if it sells gold for a price of $380. However, by shorting a fonvard contract it is possible to guarantee a profit of $40/oz. Suppose a manager decides not to hedge and the gold price irl 1 year is $390/oz. Did the 511.11 em'n $10 in prost (relativeto accounting break-even) or lose $30 in prot (relative to the prost that could be obtained by hedging)? Would your answer be different if the manager did hedge and te gold price had been $450:7 4.15. Consider te exaple in Table 4.6. Suppose that losses are f'ully tu-deductible. What is the expected after-tu profit in this case?
4.16. Suppose that srmsface a 40% income tax rate on a11prohts. ln particular, receivefull credit. Firm A has a 50% probability of a $1000 proft and
4.18. Consider the exnmple of Auric.
losses a 50%
b. Find the strike prices for a zero-cost collar (buyhigh-sike low-strike put) for which the sikes differ by $30.
call, sell
4.19. Suppose that LA1N lnvestment Bnnk wishes to sell Auric a zero-cost collar of width 30 without explicitpremium (i.e.,there will be no cash payment fromAulic to LMN). Also suppose tat on every option the bid price is $0.25 below the lack-scholes price and the offer price is $0.25 above the Black-scholes price. LMN wishes to enrn their spread ($0.25per option) without any explicit charge to Auzic. What should the stlike plices on the collar be? Note Since the cpllar involves two options, LMN' is looking to make $0.50 on the deal. You ned. to 5nd strike prices tat differ by 30 such tat LNIN makes $0.50.)
4.20. Use the snme assumptions as in thepreceding problem, withoutthe bid-askspread.
Suppose that we want to construct a paylater strategy using a ratio spread. Instead
of buying aco-strike call, Auric will sell one Mo-sfrike call, and use theprerium to buy two higher-sfn'ke calls, such that the net option premium is zero.
a. What higher stn'ke for the purchased calls will generate a zero net option premium? b. Graph the proft for Auric resulting from this strategy. h
124
k
Ilq'rlkochucn'loN
To
RIsK MANAGEMENT
4.21. Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coecient of about 100,000. 4.22. Using the information in Table 4.9 about Scenario C: a. Compute ctotal revenue positive.
When
correlation between price and quantity is
b. What is the correlation between price and revenue? 4.23. Using the infonuation in Table 4.9 about Scenario C: a. Using youranswerto theprevious queson, useequatipn (4.7)to compute the variance-minimizing hedge ratio. b. Run aregression of revenue on price to compute the variance-mirlimizing hedge ratio. c. What is the variability of optimally hedged revenue? 424. Using the infonnation in Table 4.9 about Scenazio C: a. What is the expected quantity of production? b. Suppose you short the expected quantity of cor'n. What is tlze standard deviation of hedged revenue? 4.25. Suppose that plice and quantity are positively correlated as in this table: Price
Quantity
Revenpe
$2 $3
0.6m bu 0.934m bu
$1.2m $2.8m
There is a 50% chance of either price. The f'utures price is the effect of hedging if we do the following'.
$2.50. Demonstrate
a. Short the expected quantity. b. Short the minimumquantity. c. Short the maximum quantity. d. What is the hedge position that eliminates variability in revenue? Why?
)..' 'j'jtd (' jpl'ltd j'1' g' y,@' q(' jyd jjd 'yd y,.y' I.' k' )' y' (jj' q' #' h' j;'.' ;'18 j''g' 18 $.)' (' 1* ;'!' )' l'11* j'(jd !d jjd @)' @' :'p' q' (' ..' j'.'.-.'jLL;LLL((LL.-'.-' k' j'j' Lt',' q;jr..'.' q(' )(' j'jd jkd $' ktd 1* .)j(;.-' i'k' j.td j;d .jj;-' ;;rgtrryjjry,r'' yyd ..yj' !?/81pf:::111* .k' .-' j-d kr'r@' y.jyt-$' ..jyy(r;;g'.,.,yjy;y'.' $!:* rt!--' .1' t);d (jd )'!.p' 5* i.:d t)?d p.qjd Lkt:.' pd /)F'.' fih',' g' );' jtd t'(-.)@.t...' ?' g(' kkyd 11--.-4;jj4r*.* j-kjy'y' $k' '''jd q')d t;' ..1;j'y--'j;p' j'-' (t..' @)-' ):;;)..@p' !' kkk;--' 'X i;d )d :j)))'-.' q;d ljd j.jtd ld ('(' ))' !!yd 1*1* .-jf(-' qyd k'k' ;t';;(..' frd f:d g)d jqd 1* )'' @k' y)I!rpj;'(r!ptqp!!)';' iq'-' 1#* !)t' y' .)' y'.' .p' )Ek'i.))..' p'pd (* ;),(.*-4j44,ry*.,* 'j' t'1.* @' .11* F;' /t)d (ryldyity'pyz's)qqqtEpd ;))t' p.d '';11E12*-* td ltd ..)' q' :tl' j'rid :1:1,-*....-9;;-.jqj!/42tt*,'11@:)* ..j(' yjjd jy'jd (.4* 1(* $' t'(kj' 1* ttd rjjjd r' 16 ).' ;')' ::;i!''''' y' (11r:11L.' (11::..* q' t'4* i;d )' p' j-.,' ;,-'(!..j' tj/'-kd -jtT' .y' jrk'!d (' q't'jd t))I)' ,'?'?' 1'';jjy)':$' y' f-Ltjt'.' ..'' itd pdyyyd yyd jri'rd ;'(;,* 1;ppp))'.,' )d )@' ;kjjtl!.' j)'(j.#!@rIjT)r'..' fq'tkj;qj'..gt' kd $' l.t;d rrd )y' (r' t(' t:r';r;..?);'(y' (jjjq'';:jjlt)d y:r)d )lp':' (t/9k:2);*.* g' tjd ;)rrt(I;-.;:)Ippqj;(k;k.'....' :'q' j))k-.' 'J.'rp')t(tqrjf'@)'jd!' ttd ,jr..' h' jjr(-'-' #' ..-(-(lI!r:.' @' )' tltd ti':qd /'!'id (Jj.-' t.-' !' )4* 1* .(' y;,d ;L3L-.' )l:d .(t' lqp'-?d ,)*6:* lt' td pjd i-':' 'r'J' -Jj'.' ltj.--rs'rd t'','qid r)' '.' rjd .t' r';?;ip' t'.jd t.jj':t.tid r'pd r.rd 1* ,;.:' r'yd t','t' r)' tr,ji'-)d rjd rgtj'.' sd yjtty)y.' rll'.pd r!rld .,' rd ttd il#lfl:':' t'j.j;((' i';j).' tyd ll' t;tjd tpd tld k)j:('.' r'lj' -;-J''' ytj'jq;jsd :':' ,'':F' 'j' ?;'tjj' :'Eg.td' 'L' :':' :'L' .''rq(11!((r'''' ,j,,' S' :;' e' *' 'F' t'?' :'-s-jjly'lt'-E)d -' ..,;.' :$5* :ri' y' ''E.r' itlr'-d rs.E-d .-,' .'.F'-i'i' )' -y' ;'yjyjjyyrd 'ijd -:)' pjrttktiL' -il/ytl-k:d --' 'jEjd t'.j(' -' ;'-j' :i' ;j;,,,.ty,,.'.' t'(' j''t' E.r' yL.;rd !y' 'i.' i':r' f' ,;E-it' -*-,jjjt4j4y1'* j'jrd (' y' tl'd p' E' ''jyd )' 'j.')' k' t'18 ('E' ,,,' 111::(((4(2::1,,* r' .!' 'j' (' !' tqd 5* lqd ;'r'!' i'L' ilj'd (gt' ,-' r' ..' ,t' (' -.(!)@pp)7'.:8jjp'.;)jijyj)rrtr;hqF'.' ''.' F' '''yd jjsrjjd it:?ltyrd t'',,jy.y,*5* g'td ,..' J' .,..*.11!qtt4:j:F....* ryyl'ri'tgd 5* 'Er' '.' ..L:j;.--...' .'.;$(j,'.' #' 5:j(2j*.* ;'-'rtd ,,'..t' jy'.p);d ttk.'---ijjkt'i,i' F;prr' Epj' Ey'r;' i'(;4442*.* ''E' (;y' ?' r)E.' diyjd d(sd ,'jd js:d :'F':'f' '(' qd j'y' !' j)d )' y,;'::)':,jq-.' jtd rEE-'i' j')':':t';'C' r'jd jyjd j'jd )' j':j,' (11::((((q(::' t'j''kjd ;'y' (' )' ?' 1(* #' ...--.#' ,'!tpj2!!)rrrkr'.' j'j' )p..' @' 'kl''td i'j$f,.' ktd ,):*.t* fd yjyyyj':.'-f6' q' j-jd Cjd jlpj-trls'lyyjyd )' 1* j';.t(t(..yjj;y'-' 1*.*r:.* ;:' qtd 74* ls'-;d p' .jii,'..' ))'E;'.' j.-,' :':'ti'-(?:' yrd rd p' jp)d ?-';()('q!' #' (kd y' j.td 5* pjrk)d (gtj'..' f' ?' .pr' jgd jjrd q55'.' )1* ((' j''''lltdtjk'.-iL..'d /'k.ptd py'kiyg'jd tytd ;j' yg-jjErd k(' ;'g.(' .jj' k' ki' ##' (jd ?k' 7)* ;'yd iE..)t' p)' ;,' j(' k' jytd (' ?d (jjd :()y' td i' klkd #j)' .',k' .'' (rd ........L33bqb..' t':';,..-()' j(;..'.' itfd .'L;'.' Cis'jd l'$l5$li!'i')' jjyd !' jyyd .'''h.ijd '.;.'yg;;;p!'' 'tjd ,':''''' .;,.''' '...' ('.'.' -ty(' ;'('.,.j!r!!!!1ik,.' .,j,' .j,' ,,.' .,,d .,;j!!!!!!1.,,.' '''''' ,,,' .';''.'..,-.-,,,--!r:' +' ,-ylt;ktd ,'..j)jj'.' .-' -' .j()t'-' .''j' ...,.';f'6'3' .'.y' ,-;$j' -r' -'...' .$--' -' -p.' '--yr'kjyd ..'qpfj$' '.p' .?' .ktl,..' ..' ,r''' ....' '-,d .f' ,ykr(y'.' ''@rk!t/' -y' ,',.' ,(' .j' PJX ...' .'-:' 't' -;-' -.' .-kl'ptd .'-' -' --' ..)' .-' .--' '-t' ..)'
.' .' r'rjp))d rtd '..)-'-' ''1Ij((...jjj;''' ((j1Iqgg;I::.,.' (lllljjjil:,d ((111q)* '-'rq(jI1(('''-' rllqlll..,d
.'..'.',;q;f'. . . . . .'.. . .
''.'$ly'y' .'.' ''''*-)5:-* .'.*9)*)* .'.' .';' .'. ' . ' ..'. ...'. .. . ..'. ..'-. .. ....'...
li. )'! . !! i.r-../ ft! .. k ;. ! jljj)t. ;j;tyr'gj.i. .t:.i.! ... j. ;. jjqjL. jiE i.: i i. ;):r'. ; (.. .. jjy.f;.. p; .fkjL($3L;j3;,L. ()jjjp@-;. ;'r.rk. I!(k.'f;LL1. E ityllljf'-.---il)jyj)r i. t:i'rjrj7: .!.''iIEi-!i; f!?;1. '.' @lj'. ;.p ; E I.grj.. jr:. !ilti;5*'. :j'. ..!t. . j ! pr 1: i p'E! !tl'ttt: .gE .;'! . 'kI; )'.@. ' ' q)')!; l'tir!!.;t!; @!r'jj.i..)!.;...y'. t@jiLLttq3L:'. .;.jj'I;-; tkr'. rli.jl .,j(j'.(jjjr'r;rqj!t;Ir((:.-$t:''. !)!kdtrldlt.yit:rltps'.-lilr'i. ..yj'. i . t('. t-.'pj'. i@ . .'.,i!(jI'.ryrplpiqdjsjledkprllklplird..--.lllt!. ; EC! ti)!:;ir'js .,..LL);k;3;3LL)-...,j,,..,-L;k(. . .'Iii t; .tk'. .yi r; i)'.-,;):,,--'t2t-. k'. . 14(!. .k:s'ti ;j( .t':$.. q'.'. .(-j!... r..y.!.;',j . j.l (I. ..tE3L;. ! r.,;))-. r.(i t(,... .rjj/t44,1 r;.E t. . qjbbqslL'' li..):1,-*. 1.(js!.-kk;(.. i )r.'. :!y; . ; . ti !.:!))'. ;(.i'ji-y. rt ! r)t;-.! ; ..9*jj.. ...' ' . . i.' ?-';r)-'. t...(;'. ;-j)t5:. ..' ;'ji' tr'j;fqLL6LL;,-'. (4'. ;i. iip)rii t'jr',);.. .; I. k Ei'.?.q' ;i( j..i .)r'. '.ij' .l kf?: k' t'. !);g- 5*-)*12 '- @.' qlsdEqr.rt'!Eq iii.2@. . ;;?E. ; t.' L/LL' (q('(j,y'y)p.j)tj(rj)1p1)tyk.. ' (tb.-f'' ..'.).-.. jj-'g;)'. (;.j'i)';; y!. kj);...)'ii:. .jk'. );-y. ' ..''..'. ri(.;rg)p))r;l:'. .y. !' ii.;j)(. :rq''C yjtjjyjgjyrkjjjrr (;ryyy'jrjy)y;', ty'i tj... t'ttjgrjy. jj,-tgt;'. .ty..;jjj!k)jj;. i)r... iti. tp . '.iiEt)''')j@!Ip)9;;E.!'q@);'-)!)r;i7@??:;:,fll)t'2lql!!plE'' -L'j1k;?qLLLLLLk'. . ji. (i...( j'-i rLi'. . t!)rj't'. ::'. (jjk)p'..q(k'::g))ty(((j'.. ..ktr'. . g.; .. ('jy;r.. t:-fi--. L;'.k-L'. . -iC-'. iL.'. t,.,.(jkr. ti l 17.. r.tit-'.:.ir.. LLLb';kLk', ;l($';r'.(4k;(),.' .:;; y.j;..jz'jl )jj)'.lll'qit!-;i .t.. l'113f'36133i'q'. . .. )jr);)),..' )(j'. )';;'. :.; 11:. i;(:jr h'.t.- :.i :j)'.(;2*-: yt4;jt'. . . :!r'.F'. j!.).j: .@'. .j. . . L'-L'..j;.tl. jy': . jkLq. ;rj'. ih'. ; r;;:)(!';y'! (i tjy.-'yi jr;(;. :jfjkkkltpjrljtjryr'. ft'. .E;. (i j. IE; )f:. . i' .;!t.. yj.i j)i ;,.L3jL'. (i;1)('! LLLLLj;:t4VLL'C'LL;LIL'LLILLI'. .. ('(t'...;:;;(!(;,:.ilit'lq.'E ikbklLkk. t.jjjt'.jl: ..(! !. t)''-.-.t)'' .@ j-ji )1 !jy).. . )'(q.;ll.. q. :-)j.; ... ?). .j..g : y$.. tj!'. .t.. . pt.)4jy),'. jlt.jl (.yt. ;i, )(.('.i,5 ): t(y.'y.')@rtjj))'. J,'rI (..; )(LL. jt'... . )'jr'ri yi qj 9E )jpjt'. jt; k ' r .yr'. . ,jjjL..;LLLLq3'j$L)3L;LLLLL.. )..r.. ).. Sj-i.i ! )! ;;!'. .tlEl)!p y -ki : . . tb. i ; .(1. ( . (sg rjjijirj-rp--i-'sjyj, ljtEjltt.ii y'.. 1i)l.' . . t .jq y:y'. r'. .. ; . (-k-j'i '' . ' ..'' '..i.C-!pE'l'.'. :!:2j:)*;;)11111111.,5,,;44tjr. rt':lll.ji 'b.(LL' (!;'. illkj. tyt'!-.rjk.. ; (.(yj;sryi!. ki 11k2. l.. . j .ty'; j i ..k'. :r2t)j))'. ).i jjl p: jyyj,k.. j('. ; jpt'jr . . ri1j;r;.. j ' . j-. !. j .. y'it'i .(,j'. ' i' 1:1)..24-:tt1;.-!5:112,*,j4, j.;.jy'(-. ... j.-.. E ; ykg'. gjlyy ; . , .y i.y jil'ikLLl-Lkj.LbkLLkL'tb'qjlqktfjsL'kikLLLs'jbL'. ).).jy j,.-krrjydyy'yj.tt j.(y'. yCjjEP';. j.,)L'.',L.. i j:. ..:;i ,(pr)'. p;'. . k'. t; t'(.. (j t.j t.t..jj)y.. L;);-j-'LjjL.. ..kppr; j.t; . 3,,j3. j );( ry. (ttjjjt'))jt,(-r2. j;;)'..'. jjkqjifjkff. ';3Lf'1'-'' . 111*.(r)';Il.7'. . jk!. j-j.:.jjj-';j: .trty.yy r ; 1. . . jjjr.. (yjr(j$rj);.. ;. j.-j.t.ii..);.. .i;.(;. k j.(j. ttl;;'' :;)'. ..!. yr .;''' ..( ilr-. jg.j..;.j'!;li 1...;!p jkr: )jj'. :;E ))-y'.jL-k;;;LL ):j'.-;..'.-L.-1'. , jj-jj 433;3433j3. yy. !'. g. j).....k. .j tq.'. 16;. . $!$! . . ;jjyk., . j'.p;:. ;.. ili))'. j LjjLlLj'j '@. . ,!t:! .jj.jlj'ktjgpy-jl y.. ': )1-. ''' 1E. yj, tyjl';yjy. $'. : i). ; . j.. 1l)'' rll @-'(j'q(. :! ; )'.ty'y 11tL'-''LL1Lq'1:'LL11'.,3jLL;(?LL,. ilk )j'. fi. ..).t.'. tLLL'Lf,;?'L6;i, jp!;.)! !t....; '.i i' $. #??!! gty'. q( rrykj,'j('yjy'tr$j);'-. ' .' . ' '..' '-'.'i I'l )y'E'tll''':(((!,L6t1'. j: :tttkkbqlkql'... . ti )hl'' '.$1!. yj;'. ii i. (!!! (.p!'. ' .('.' . ,'. j'. 1.rpj(.'. .j',yj'))'-. ' :7k1;. ;1:.. tf'yj'.. t!.i:.. : . jj.;jjj'yjyj'-(.:;'.! j. t.. qrI; 1*. ..!! .i; t-1. t!;r',yj.'. . :)'.tj'.yyj,y . q'. . .y-. .. . ';.t-.. l.iii ;jyy'., iiqllk!i . $hkj(jy'j,. . . Ii .f?p.(r. . . .(r-tt rp (-.ptj'. .pi L)'.?q'. ;;r!)p!t;; . .. ;'. !;''. . ;(; l--r'. . y-.. f'.: i .'... . k'. ..i('. ..kjj)j;L' . q.. .kq-.. k'. (@'.. .r.g.'.j.r'. 'r1?p))):. j y'. .(;, .yjy.g,j'jt.y,g. ...j..g'. . j'. . . y;p('. IE l)k.)))!... ' .. g .....t'. ;. ..g.rj'. . ?::.1141145*59. . @)': . t'. 115*'. . j;;,j. yjyy'.yy yy '.!!Ii k.j'. !. . . t'.i y.'. yy'. q,*. ry,. j.. . i . . ty'. . LLLLLj3. $ltltr-;;))rjt'.))l,!rt'rjr,rr)lrr:jk)tj)p:'. p! kr'! j . . . j pi.-jifr k ; !y( t'tkqr j '. ( p. k. .. ' ., .. . ;LL(',.:4,L(33L3L. )$'. i. j:. . @;. p i . . . j )Ii! L? t! p -'i j !-qh';t! ;)1kt'. k;'. . . :jiij?e'ljkkllrpllppl'.. ; bl'. t; (LjjLL.. ): t' rr)E();ptr.. ('jL.jqqj;Lji. r)q'.;'J-. ( /. f@t'-'jfj)p2yr',;j:-r. j q((;'-.-'jtLL;.;3Lj-qq33L'jLkjL.f3LL..-f3L3LbL3.16',6 jytll r jittt;jq'yyy,'yyy',yyjy'tkI@.. j ).. i . ;. . @)t!tty:@));))'. r. ;'; k.-'. .. @p. f(;:tj(y)'.y . t ; :'...i-ij'yr )'. . . rii)ky....-,,.,. )jr.'. . /*-*3111!!. . .r. -j,j,L);4jj;.r.(;'. 'iqfqbjL'j)tt')t')L;b3.. tkft'..;;.. ..'j3ktLiL..,. ... jp'. f'qsE r-jLtL. . ?LLLLLLL'. (j';j ... !. $! .,yy . .!'. . kf.j;. r; . kj'jy'. q@jj)jgjrt'. ..);;Lbt'. . . ,j,')4(',,j..,;j,.,;j,. qlrjt'.Lf'f'L;. . ;)qI.....'?. ikLL'. ..kk3t4)t31;;L ....j- ;; ;q,)L. ijry tl.jjj);.!.. . . . i'';-.r(tp))))' ' (r',-...;I-;'j;'.)I. jyitrqf.l'i-'',. rq;i k. ! S-;'E !.'t)I. jy..(...,--,,., .j).;f;ytj .yy j.yy j rj(--t;'.. )y;j.,p y))jr:t'. r)'. .;-j.'-j.'1;';--tyr. . ((j.. yy, . jr;.j'j;---t.. .lk-j,i I..?'' .ti. y,. ......'. iftLq. (ty'#1. ..),)j.. . ' . ' . .' ' ..'..'.E.'..
[email protected]:. I... kyjj. . .y(..jy,.,. ;,.LLL'L;L;L;;. .yr t.:;)... I()pIktj'.j''.;irljji;.. j(k'''')'''',;Eytij@':I)Il);'Il);'kr)k7ll)lp;. '?.(?'.)-. ... ..r'..'Lt.. )..-.'qI(Ijj:,;-. j'..t-.g)'.. jI i'-.--.kjjr. k'..i ))')':i t. j .,ty. (-;.Lf)k;. ;: L1.'. pIjr$q:t?'. . ..1-.(i. lb. . ..ji. g;-k. . .. . ..'p; . .. . j)y'.;L)L'. .;t:LLLqf.'. . tiijj/tjst.'r... . (j----).. (k))-)'. .,-yy. 336*. . . tjy.'. ll-s'. . . i)'. . LL?LLqqLb336)'. . .. . . .. . (i);'..j', . . . t'tjf'. . ('. ;'. p'. . . . . . . y'. . )ykrr'.rt))ypkj;'..,'. . kf'-k ty'. (:) . . ''. . r))--'. . .y-)'. L'b'', $(;'.t.'. . . t'. .:k:l#)''. 3;';363'. .. j;)@jjI.;#p)t'. ;j3;j;'jLq(LLb31$q'. . .r'. . . . .: .. . )t:'. t:( EE E. E ' iEE . . ' .. . - .' . .. ' ' ' - ' .-.'. ' - .' E. ''-E.' . E.E @111711:'' ' '- ! :. . ' E' '' ' . ' '.' . - . . . .. . : . . .. . . . -. .. . '.
'
'
:
.
(jyg'(r)rj'. i .,.-:,5/));*-
'
rpl$rl!).-;:ik@;'f)ttE.. ' ,):.-. i - l---';... -
..
.
i.. . EE)
.
.rppl:liyi).. (L. ' . .
.
: E i .ykEjli:jljj ji: : : (i:( .. E EE ; gj :: '. .: (( i. g . i !:. :.. . .. . ' . . .E'..E'.. i . !i:'..ii;'E:i: : . E: :E ( 'E.E . ' .. .......... ..
. -.
,.
..)-iji...''.
'i. -.33*.-
t.-,E
rI. ! .
'..
. .
!L-:-
:-
:
-
-
.
-i
'ti
.
...
-
:
. . .
-!!//)2),*.
..
-
-..i.' -E
.
. .. ...
y
y'.
.
:.;.
.
--
. .
i
-
........'..
-?,..
. ..:j;j'.
!
.
!.
! ! i.
-
. .
-f-
..(j
: . .
'
''
.'-'.;((jy;..j..
. . . ... . ..'.. .
i ..
' .
.
.
. .
.-
.
-
--
-
'
. . -.
-
.
-,
.-
.-
. '. .-
.
.
--
. .
.
-
.
.
.
.
.
-
.
.
.
..
.
...
.
-
.
-. .
,.
. ... . . -y..
..
.
-
.
-
. .
.
.
- .
.
.
..
..
.. . -.
.-'y
. -j-... ,-'jij,;)j..
-ty'. .
.
--
,t).y.,
--
.
.'.. .
.. .
.. . . .
.
.
,.
..
-.
-
. . -
. .-
-
.. .,
...
,
.
. . . . . .. . .. . . .
.. .. .
- . . .. -. . . . -'. . -..'. . . .
-.;r.
. .
..
. -
.
-:
.
. .
.
.
-
-t(jyt'. -'.
.
. .
'
.
,- -.
. .
...
-
-
.
-
.
-
-
..
-'. .
-.
.
. .
. .. -. . ..
-
. ...
.
.
.
-
- . . -j'.
..... -
.
.
-'.
--
-
. .
.
-
.
-. .-1
.
'
(. i.E.(. .
.
'
.. .
.-
: . .
:
.
:! ' .E '
:
..
' :
:
.:. .
.
:
.
:
.
.
.
. .. :
:
:. . .
:
:
.
. .
E(
:
. ..
. ..
.
.
E ,E
:
:
i:i'.E..jili).'.jtjj. . .:g.:.
..1
'.
:
E ::'E,'i'E(';,'EytlEiyEy,i'y!E('y ji', E. ( EiEEi ; i . . .. . .. ..E..g Ei ..((.. .. ( k ,l!'.!t)Ji, :: !: : ::(:j:: .
:
:
. .
.
..
..
.
.' :
E
! jjjjj
'
.
:
:
:
.
.
jjy
. .
'::
.'.i.',,'Ei.,i.':pj,.t!@'@, '.)..
.
,
::.tilj'j'
..:) Ejji:! '.:j :. :.' ' L: : .EEE,,,,E';!tE,i..!)),,)!.(.' .
.
.
''
'
.
.
.
.
:!
:r)
:
j:jyjjjjyjij: :Eg:j
E: : . . . . . . .!j jjj .).iE (. g j .: : E . .:g . :. . (.: : ''i :(.'r !(i'.iiElj E. . ' . ... 'gi ! (.EE'; !: l : !i : . .:.E..: . E.EE.E..(i;-.-E.;.gk q'. g--- ..... E. ' ' .- .il 'E . E. i-1. i i. k'! E .!i. : ti@ ..-E.i. ..E. - 'E. g.... I'Ei-'i . p'r i. . '.. g.E i j ' j . j ( .. y . g .. y j . j E . . . .. . (...: . ..(. .. . ... : . E( : .!.k .Ij!.Ii j;EEj.. . .: .(r.. : Ei i . Iigl'qj .. .. . E'EE:E'E,E;i'g.)I'E!:2i!yir)gr'r,,'k . tE.
t
:
: ..
.
.'
.
. :
.:.
' E
l
E
.
*
*'
s.j
*
,u.
.
o
*
'
.
.
.j,
.,,
a
. .
:
:
:
.
.
-
.
-
.
. . -.. --.
.
. ..
E
,'
'
:
. .
.
.
. .
:
(t,
.i.ij!i.tl'j..rfj
.
( ..
.
4t
!!. E.Ei .E . . E :.E : !: i ' E E ! E. g ....: q. ::ijj i ( qi . .
: .
.
,t;,:,
;.i.;.rjkkjyj;j jj
'. ' . . i : -.: . E.
' . .
.
(:
..L.'...L .
,qitikg/fjtt
E E
E
Z. stocl has The purchase of a share of ee components'. (1) nxlng tne pnce, (2) ownerslup the buyer mnklncr & payment 47 to the seller atld (3)the seller trnnsfelq-tncrshare that oavment nnd physlcal recelpt cnn occur to the buver. If we nllow for the posslblhtv 1's lliizr). illi!!l,.g' xed thefe are four t (11:::51r:::11.. es say time 0 and time F then Once tlze plice lopcally posslble purchaslng arrnngements: Payment can occur at bme 0 or F, nnd yeceipt can occur at time 0 Or F Table 5 1 depicts these four possiblh' Pjj y' al 1 along wlt.h thetr custom nnmes. Let s dlscuss these d erent arrnngements.
........L.L.j.L..,;. Etili,,,.pi,iutlllll. Eiit,,i' ryEiljij'
. . .
-.
.
-. ..
.. .
.
.
r
.. g. . ..E . ..y. ...'.'!'i'iij'.ii!I'Ii.jjr(., E
..
;i,i,.(
,
.!:
E.E..':.i.:.(::)j.E:..E..:;... .. 'iEiEtti,lt),. 'Er'i,..
!.yjjt
' . '.
E
.
.
.
. ..
.
'tl,yy ,El',,;:E)'i,i,iyi.jy4$.iig
':i ..' : .. '.g(E;,,.,ti;r..yrjqt.y,ti.y'!4'!..g,y;y .g..g.'...j....j..Ig(gt.!.'jj (i,.yyiyi,,y,.j.,, r jj jg , ' ::' E
:
.
, .
'':
'
i E
'
.
:
IE
( j.i j. gg.!( y . ,EiEEE'E'''r',i;'E':',?E;';,q'j')';'yij '
. .. E' . q i E. .. .. : . !i jj. .
..
.
:
.
:. y
.
:
:
: :
. .
.
. .
:
:
:
'
' @ E: . (jiEk E: E . : E EEiE@!' i ;I jI ij((:: EE itiy/ijjljjjyl LLLLLLLLL,LL E
:
: . :.... j ( jgg ; y . . . .. ....E Ek!:.i ;.ii!jjjj .j jj.: . ..E (: ;g I j ..E!'E.',i..;!j2;j.i....)j;'rrty,jjrjr ..'. r . kj. .j.ry j :
. .
'
.
..
:
:
. . ,,...,g,i,,yy,, : .
.
. . I i .'.'. i.( ; g l.j ( 2 . j ! ! 1lI( . . . ..... E . . ... r .
.
E: : : i(g. '. .!: : .: i : i : : . . . ' ' . ' . i : .. g E. ' E . .' ... E:(... E: i ':: : . E E '. .... .( . (g: . g .g. : : .. :i (..... : qgE:,E;iE!;iIj;E1ijEj';E$y Etli'ii''E
.' .
:
.
.
.
rrllil;
....:.
.
.
.. .
.
:
Eil...tljili
: . .
,
E
. .
: .
.
: :.: ;
:
' .
E
'
E
..
:
.
y
.
:
::'. ' ..
.
.
IEE . . E
!':.:
;E
:.
.
:@
: . i ' ' . ' i . :' . . . .. .. .. . .:i :. : . .. : .
'r...j.tllj,,
: : . . ' . .
.
,.q,ii
.., :
:' : . !
.
'. .
: : : .. ;'. . . :'''rtljjtytl::j . S(
.,i..
:
.
: :.: .. ..
:
' :' .' . ' '
:
i
: : '. . i-. . :.: .: . t;jj?(E::i .
:
:
..... : .. .
: ' E : : .: :. . . . :. ' ' . ' : . E .. E: : : : :j'.. :. g E . .. : :, ,. :: : ..: :. . . . '. 'Eri . E ' :. . E' : : i iy ::y..g':: . : : . . . : .. : : . . : .: . y :. . . . . : , y :. : :.. ,. ,, . .. y.: : . y t'E '' .: . ! '' : ' . ''.. . : . . . . . . i. E :: . . ' : . :i . . tlEz.y.j'jl.:tig.r.. :. . :..(....j(( .. ..(...::.(.gj!...;.g.i.j(ijy. ... : .:. .. E.
.
.
:
. : . .
:. ::
...
...
...
.
.
*
,
.
.
.
Eii(:: .'..
t
...l,jjjqi
'
:
.
.
...'.
,
E,
..
j(LEE.i ..
.
EE .
E .. .
:
.
. j . '.' ... :.E:t t E. ....E.).!gyii.!) yr.;:... .:)jE.. @!( E j.t....... . .. E.. !.(. rjljj... ..gE .@'ilrltiit j!.j.. i E ).Ei ! ! i iji. !(. . .((gE.j k . ( Ei.iyj'hl., . .....';E::E!E;Ef;'(),.;(.EyrlJI, I hk 'I :' 'E.'''..Ei ;. !i.E;. '.l (.I.jlp L/)i,i: EE E:g:E.:,:'j!JEE.''ipt.ilE'1Jli i ( : .''.i.. E.jr' ijj!1'jE1 i ;@ l! .I;; ?i('yj'. E ; EEE.' . !i E..!.1. ; . I E'EEE,::ilrriEEtij'jfj.l'ijky''qj . . ....... .'!.i: . . . (! '. :((s'iEl'yilrltlp$j' ! ' E i @ l ' j.t! gE ; : '.' E@ r' jj'. ttliyli'' fEi.('('E'Ei !;ii $).. ..... E Ii !q'' ti '! F'. . :. . E....i).!ijj)y ' ;j q;'i)'. j.)!:. jE .. . ..... EEE(.EE..!ii(..jIrijp!.)y(y. . r;j ;li jk i I . ! ! ( . . kj'. !j jE . .....g...(.j.( .;qg!i k(.jyjy. yg.2.. t!.Li. .j g . . . q.... . .. ... Ej E(tj'. . . E i !Ii! i. E . iE... p ! Iij4gEj.. g jI jy( ... . . E.g. g. ..y.. j.ijjjyg .. r . j: 4. j .jj . . . . . .. . . . ...gy.y g..!.y .j . tr.. . . . . . EE... ' ji jj jyh'1'. . . . ' ' .....E; .;. i . i.i . t!;'. ..l 1 ii ( ' . .(. . (j'. . .;. .( . . . . .. .. . . .. .. . . '. . '......L . ,J'. f:r>.)7.:)jt. k..y(.y.jj(. jylj .. . . .. E 'yjt )ytj . j .. . y .jy:.j.. . j.j.j)jy.j:y:..... . . . . .. . . .. . . . ''.j'.':E!..E((!.'lE!C1tEj.Ej.i tjt.i: :
'.
.
:.i
...
:
.
.'
tll' WIVCIA yotl PV fOF the Stock today 2f&11 &Gemellt Prepaid fomvard CO11tTaCt*. f'utllre date. 2 and recelve the Stock at an atrreed-upon = e d erence between a
:qrj::'ljj.j i.E.ljlqiE .jj; itt.k :.. : .:::: :E . ..,:E':'iiiEt;E'.ii,i!i)rg!) E i)jijjijjjl j . jjjyiyti E ((E.'.liqliijqj! j::.i)(. .: . . . Ei!t!...(. :.:EE:..::..E::!'i:/j:l)l). . ! : i i;(q(i.q. ( . j''
.
.
'ties
.
'.,
@
,..',EEt,,,ii.i,..(E.'.;'.'4;k i i i C( r .' ..(.'
.,
.
A purchase m Wmch yotl borl'tm ttle enttre purchase purclmse: . of the Suppose secunty. you borrow tlle share pnce, %, arld aotrreeto repay pnce the bo1ToWe (j Rmotln t at me F If the continuously compounded intefest rate iS r ipr.'' ::i!''''' at tlme F you would owe e per dollar borrowed, or Sze
E: '
'y'y'jtg . . .ijjjj
'
.
.
y'uny jeveraged
E:. . E ' : ' ' Ei q (E E i.1. ( g jj j y ( : ..' i:.'j. ......E(.j.:jj) i.( ;.'. !j(jg . '..j. . E. . . : g((j ;tl;'t,) . .. jl ..
.
...
:
,Eltil Et'
E
,,.,j,
purchase: e typlcal way to nk about buymro stock. You slmultaoutm*ght the stock neously pay pnce m cash and recelve ownerslup of the stock.
y'jjit:
E E 'Eilj.siijl..
,.,..,,,
'
.
,,
..
.
.!.,
,;,.
11221!:).
.
!((E E .gy.jj...j. .jrjtjjjssjj E.;iE .:g . . .(: .Ey . . .. . :... .. . '
:
..,,
.
: .:t:..)t1I (
E
.,,,,...j,,.
.
...
. j!r .',,.,;i,;r),i,.y,y;yyt;jgjE.,ig y,(jt,!,)j :
111::(4q(q::))41
.
. .
.
.
''-'!!j1,(j!''''
..jlljrllljii,.'''''!!!lltj!,'''
:
kri' !
'
,..11111,,;;.-..
-.'(4(11(((.'--
. : . . . . . . ..
(.: .. E ;(!!i E: ' .g5E. . . . . .).,j, llli'jlqi E : .(' E. . EEE:t:!(( :::... .
. . .
.
.
.. .
(lliiiij;;l '''(((I1((q'''' qqllEilt::zll ((411141111211,.
((jII(q)..-,. ((j1:!!''.'k2;1r'' (qjlliiiig:,.:lliiij;i
.
.
.' .. .
,;:j111;7-
.
:
. . .
-
'....'.' '-.'. j j ..... .
-
.
*
.
.
.
.
.
.
w
x
*.
..
.
.
*
.
.'
.
.
.
.
.
.
.
.
A.
.
.
.
.
.
.
#'
jl,
A11of these arrangements can be reversed in the case of the seller. Problem 3.1 asks you to desclibe them t'rom that PCFSPCCUVe. .,.,,.
zTj)e term prepaid contract or prepay 1-s Widely used in Practice and Stlch cont-ac? are common The Enron tansaction discussed in Chapter 8 and a hedging transaction by Roy Disney lfld FOGV1tFYS. (jj cussetj jyj Gjjagur jg used jlrejljll'd SWITS s ./brljptzrtf
,
,
'''
rt'
:
,.
.
:
'
. ... . .
' ..
... .
. '. .
.
:
..
.
.
'
.
.
..
.
..
. . ..
.
.
.
. . ..
' . ..
. . . .. '
.
'.
.
(
. . : . ...' . .. ' . .
..
.
.
.
.
..
.
..'
.
'
E
. .
:
''.' Et. :::...:::.:..t4ij.?.;j.jj)r;.E
i:. s....j.)l.r)...:...... E (-)'; t.(LL:., !.:y': jrjj k )jlE.. .E jql ,.;(;. ... ( r k! (strj'ljjjjtpllp'.s yjr.. r r..
. ( 'j ( ! . ( '. ':..E . . E' E '. .. . .. .: .. ( i k . tttj'. : . . :.,;. ..(j;'. ! :
..
.
. ..
' '
..
''
'
'''.. .16:
'
EE . .
E
.
...
. . . . ..
:
,
.. . .
-
.
: E- E.;':
;
j
!
('.
.,,
:
.
'
@
),4'
' i : ii !@ ' E IEElit'. .i.:g E ' '. .''..1E.E.':Elirijjjjtljljj . r.'. ; EEl(g '
. E
. . . .
. :
.
. .l g.j(g .
,
:
.
:
fOr omard contracts Which permit ST'ITIS nnd 1'nvestors to oGtlarantee a Price a futtlre mnnaGement PWC j) RSC OF Sale tool ln this chapter we continue are a basic finnncialrisk o to explore these conacts nnd sttldy in detail folavard and fumres contracts on Ennncial lnstruments, such as stocks, mdexes, currencles, nnd mterest rates. Our oblectlves are m .y. nOW ttley are Pnced, and hOW to understand mofe abotlt tlle Use 0f Enese Contracts kasmakem gcyg= tem Questlonsto keep m mmd oughout the chapter lnclude. o mlght buy or sell speclfc contracts? at klnds of firms mlght use te contract for nsk mnnagement? ZS lt 1S9 Y js tjw coutmct deslcllcd o
:
(jfljjl !jj .LL.... ijiryttttt, ';)Ei,
:.f.:
k
.
: ' : :
:
:
'
lj
.
:
. :
. .
:.
' '
... .. . ..
:
.
.
: ' : ' . .
:
E. . .: . ..
:
tit,iiqiil,i j:
(
!:
.
.E,, riii tiEiil ' .. .g .(.::E(.' : . ijEi (EE.: ... .: :. 'EE),i)',t!tj!!r, .. . .E. . . (EE . g. : j y jj E ' j .g . E. : . : E .. . (( (ii i g gj . Eg g .. . g...: ....EE
'.
j
E
..
. ,: : .. : :. : ' ' .:.
. :
:
.
..
. :
. .. ..
.
t: '
:.
.
: .
:
. .
:
:.: .
:
:
.
E
E E
. .:
:.
.
... . .. r'Eiilii:,itsi)i('
' . .. :: E.:.E'j i (iii.
. . :
:
:
.
Ig
:: . . . .
:!
r:. .... r: ..
'
..
.. :' .
.
.
. . .
:
. j(.j.: (.y: (Ej!. j(E(j j.:.. ' :... (. yj.E.(y.: i. yE! E( i!j. .. gy! i. . (E . . . : .j. 5 E ( :(. . ..El.E... : LL. :. ;; j: ii. j. i.Eg( jE..(:.EEg!44.. Ejj.:y. jr.. y.j E r. .E:.:E:.!E....: .(t. :.( (i .: . 2:. r! jj. i . Ei (.j. i :...j.y.j.: ..( y..i. .E.;yE;Eg;'.. ...g.g:. 5 '(.E.:...(. E..;(.j.'E':.i .2. . .. :k E :jq' .;r'. . : :.: E). : ;' !(' . .E(EE:E; t::!Ej. . ..EE: E jEE.: .iEliE:E('..:iE(EEjj.E.gy. .jyjE . 5r.Egij: !.. :':'E!.jgg ijj'jjj.. .i il. . l',' . 1. . E j. j E ; .. . g..: . . t.;:. .;!.....E.... .
:
'
.
j
jj
11
:
: . .
. . . . : . : ..
'
:
.
.
,;i;--,)..r'-p,-,-..
: . ..
' :
. .
.
.
E
..
.
,,*.
#
:
'
..
- '.
.
.
..
.
'
.
:
y
-
:
5: :
'
:
:
.'
. -
j' j j.' ..E: k.!g.!i .. i
. ' :.. '. ' .EE:'E.:. jj'j'jEiE;((ig:i !r' . .. . . ........g.gE.q.)E. ' :: : E . '.. .i j:. El... 2l:q .. .E.. Ei .E... E! qg. . . .. .. .....E:.'.EEE :;.;gg . : .' . E;i.j(.I: . .E:.EE.. . E........ .:EE. :...rj(:. EE: Ei .!... ' '' .E E. Ei ... . . E..j:..Ei.!I .j:. i . E ' j E'E.i: ' i ki Ej' E 1. ' E. E @E: .
.
. -
-
. -
. .
.
..
..
.
-
..
.,
-
...
.
.
,
.
.
.
.
.
.
-.
-.
.
.
-
-
. . ..: .
.
..
..
.
-
.
.
,
-
..
.. .
.
,)333k).
'
.
..
-
.
.
.
-
'y-j'.
-
.
-
. .
-
'
-
.
.
-
''1't11111911111'-j,4-*.
.y;
. -
-
'
.
'
'.
-
..
.
..
.
' .
.
. - .-
-
- ..
.
,..
.-k(-.:
'
. -
.(
.
. . -
-
. .
.
'.
.
.
.
. .
.
.
.
.y
.
,
.
. . . -.- .. . - . -
.
-
'
.
-.
--'j.
-gj)..
-
-
-.
.
.
. - . .-.)'.
'.r$-.''
. .
-k
- .
-
.
.
.
.. . .
.
-
.-
.
. .
-.
--
..
-((..
.
' -
.
-
'
. . .
.
..i
,*11.1!.
.
-'.
-. ' ' ..
'
.
-l;
.
-
.
.
-(.
..
..
.
.
-
-g
...
.
.
: ' . ... ' . . . ... '.': (. (. ' . . ' ' ' '. : : '... E' j ('' E. . .. : .E . : : . : Ei : rk.. :
'...
-
...
:
'. .
. .
' .
'.
rE
jE
..
.'
'.
' .
:
-j -
-
......
E
.
?!Ef'i t': -
:
j ( ('. ( l
. .
-
:
:
...
.
'
' ..
(E i
! :.
'
-
.
.
s:!)'.
..-j.)q))$$)jp(,qi'.. .
.5($i;.,,...-ipjp)!;'. ..
-.
.
,-t',.'
...(:..
;tj,'pk(q.F)'. t'tqEj- --'-;iit. .- r! ! .. rE. . . E iE -,.'jj'pjp''tI)iy());)))@jF.'q)tlj();' -' -'(tt.' i i - : E.i . .:. . -i-. E -. --g': jllE:! -.... E .- !jE; - : . E-. -ji.-. ! ----lli.. jj-).----.. i! i :. E i g,.-!- E.E! E ! ii-.jk.))E -. . ! ll'ss :qE.! E EE. iEEE.. . !. '.. E.E ! E;;. . :! . E jE - EEEE,. . . .4,jj,(;;.;-,;Fjj:j,4jj*. E E. . E:. : E :. i i Ei -jp$);E-'-. 'E (E 'j! ! ...'. i. i i' . ;-/: E .-. i i i --).'. E: - E . E. E EEi : ; EE E !.. E- - E-E ,?)''. ! E E E jp). i ! E .. . @t':k'j' E E.. : LLL3tfCC,,t,,,'. . .. .. li!tj :;(i!'.-'-)$-'-'' ' ' ;. : . qg . ! ;'' :EE (ty'E g EE . ' ' E-- EE ' t.E E '(q'-j-'k)); : . .5r:. . E IE tiEE-; jE : -: ,tt'tt.Ej!. ('. : i i. : : . tE!. :.Fr. E ';.. . i . . E - Eq'. i i EEi .j;'E -.-. E j'i:- E . i -. i - - :!.: : - E,)'.E! ;E i - . . :. i EE E y tE ,': : :jj(()).'t'. - (Ety':)g.. . si y'.. !- E . : E -Ei . .jE!iI'E (Ey-jE. ;rp):. . (E ,. . --Ej - IE : -11: -ktty'. E ; -. y E . @ : krEjyy'. ''. . :(. . -. i . g E :t:(k E E ''. ( -E E ,1';1.. E tj'. yjjtyyj ''1''Eji7!iil'tg. . E. g (',yE E y r i ''E'''. ('i!;l';f'' . E'.E Ei i i iE. i -. L': q. '?t. -. -'E . j-q'. t'iji':'3333*. Fi. y ,....jpi FE. E ! ' E'. -j:. ! . E j!(E; j: E -ti ' .: : .i E . y .E i E ! ': . -iE -. Jg.-sq-tjl EE E E . . ;! rE . ir'.i E ! .:ig.ji;. tE-. : E-. - : '. )tj''i ! ?,'. !. ?.E jirfi i ... E.i. :. .. .E :t; ----?3. l:.:,,:-5:,,1. -;ij)--'... : - E ... ?. . .. -. f@'. , . .-ii E-j-.E' -: ,...... -..!i i . :)'. . ... .. .E .. .. . E.- (.. . E-. i .. ..'.. . . . .jf-'.,, :. . . . ... . . .. riq'ljlj)E :.. ttttttttjr..rllssi.etptkltd'lkdtjkdlkdlplj--jlkrdl'jpdisrljidlyjktlEpjrk.tktllp'stt!trr!l y'--,;;ii;,!j(!!E. :pp):tt$p;p;;.. (E'. !)'ttk?E-p:. TjjE ! E - -. . g. !! - -. - r E- 5*.-. E :-. . : iE : Eii. Ei E g g ;. ggg y.: - ; yE.. (;g y... ( . i i yE - y .y.. - yyg.- r ! y-gg-y .. E q.'; . y. g. .. y E y -'- . E .. . ! E E . q .'(;jp))j;F('.-ih)j(pq:.. . .-pp'E i E -;:E (....;49gjr)418:y.. jsjjt. ,;j4(j4...*: E.-'i ! E y ,)j. y (j-y. E; i yjy .. .yyj. jj,'tkt,. jk.. -j,,.'jj,j;-),,j'. i'. !, . ; E -......-..,-..,,.,.. p,y'. ,*::1.*. ijgj., :yE(:(kt'.. . . E . . j.:;'. . E . . .. -;E .. ; :!-...'(:..;rjj'. , .-,. . .. .. ... -'. .... . . . .. ... .. . . g . .. j.'. y..ry;'. . . r'r.'jij'. . E
-
.
.
lr'ltEj'
--r-i'.
.
E6..
..
,
-
' -
-
.. .
.
FE .
:'
:
.
:.jq;,.jgyjr'. . . -!r;'. :;ryy!jyyj'--(. . . 6!.2y!. !
.
..
. .
-(I'E ..
!E...E@ .. . . :.
-.
. .
.
'.
...
-.
i '! . J!;. ). .E
-.-
tE
f-' '
.
.
y-
. .
'
;
.
..*:q
--
, j . : qsI . E: ;:'.. . j;li.. -.lfi. -.
.
'
-
yjt'ri
.
-tt.;rjtyi
:1141(*'.. '''
''. -. , .-h.
..
.-''
'
!
.
i:
..
..:).j)',tj:;.iif'. . '
-.
'
.
....
. .
:
?(J:))p,E E jE . '- .i' .
rF;;q!rrjk$jyrj(!(r(r..... ...
.
jygkkkg!!r;, jjjjjjj,,j j:;j
128
k. FINANCIAL
PREPAI D FO RWARD CO NTRACT.S ON STOCK
AN D FUTURES
FORWARDS
.'y .'.'.'. -' -y' ;'y'i :.' -jf -''('E
.'
'
-'
'
. r((!. ' ) (-) ( ..)(. j j .'(,-ljki r.r. li Four different ways to buy a share of stock that .has ..jjjj)-;jjj))k. qL1.b1.L3'6.'..-,. y . 11 (t.r..,#t.)r,!y.,t.u.,))z-;.:, . .)qt...tr. price S at tme 0. At time 0 you agree to a price, which is paid either today or at time T.The shares are received either at 0 or T. The interest rate is n E
'i'4''' .
-
'
' :
.
-
'
EE
-
-q- -
..
.
..;...-l/i$'ii-....--.-..-.-..t.!1pi!!q#.
-
-
. -
.
-
.
! E : '' ' . i
i
.
.
'' :.
'
' : .-
-
.
'. .
.
.
:-
E
:
.
. .
.
-
. . -q -. --,,.,))?,,,,.,r,y!,.,.,.y,,.t(,t.,.t,rg.t,,.),.,.,,),jj,.,,,,.j,t,,.) -
.. -
( E E. .' E . r jjtjjrtjjykk. '(.'j.'E ( (Ej(' .,jk/,.E E'( E Er . ( 7 E (.) j i.(.E(:( (.(j. ( . (. ''(' ' ;'...:.!. (E.(.(.' .Ey (' (E. .: .E.'( ' ' E. y''gjj.til jjj,j . . . E.j '(( . .. . ' E. ('.-'. .j.y...'....E (... .. (....:( .jE.. .EE. ..L. .. .:.E . . .....(....E jjtyjy 'jjI:;, q.. . E..Ej ;j..gj j..E;.. .t:).(: ..E .q.L. $;jjjj.. . ( E. .y .y:( jjjjjyi)k jjjj!j.qjkjj(.' kjtjjj .E ( i.q. E.y.j.t . .y.$;yjjj, tjjjgjjI:k.jjgjj. yy.. .j!jjj;j )( y:.. ii.- (.. ; ii t ( i ) i@ . E E ( ( . (. . ' . . j. . ' E . j ( . j g. E E' . . ' ' E j ( j ( ..( ( .. .(.y. .. .j. ...g....... j . .gj j... (' ' . ...(jy((EE( j' E kgE ........jj.. j.j. j. . .g.jy......j.. . jj..y jj....... . . jryyyq .. ... . 'r '.. j .gj.'j.g.gg.. ...j jj.. (.. g j( ( jj ('j...ygj j..'j j g.. ....(.j.. (.g j gj..j. jj.. .j;. g..j.(yyrj j j....'''. ''''. (.''E('' jEE . . .. . :jE j.E.' .' ( jg'. .( . . '' . . j . .. . . j .. ( .gi (. .j . . E(j( ( j gg . y EEj . . . . E( .j ) . . ( . y E g g ( j yyy . .... ; . q(. . . j ( . .EE.Eg. ..j:: ... ( .( .. ((jj .j.Ej jj r jE'.yE .. EE.EE.yj E.j . E y ... .... .E.jjj. ;. ;E.y -:I!EklIi:. yk :111:1: ; 111:::,19: t.iIr''7I1r.'iIIr. y.; qj .. ( jj 'E E . '. i.. E ; ' Ey. t '' . ' . E E. i y.E. i . (. .i41rE. .. y . . q E.E. ..... . y(E. y . E. . E . 111kE!ij: .'. E. (.. '.k . .y q ..' .yq'. .y illi:jr!i . . E E y ' . q .. . y . . y ( ) . y . . j q r y y E!ll. y q ; . y . . ; y (jjlql)il,l (.. . . j... . . ..... j j .,.....y( y .. . .. . .( y..j.y.. . ; y . .. . y .. . . . . j.yyy ... ....y ..j.. . j..... y. . y.;. ....j;....j. j....j...yyy... yy .. .. . . . . . .. r. ry ... yj...,.... y...j..y ........ y..... ............ .. .y.y j..... . ..... . ...,..y..y.y..,..jy....... ry... . . . .. . .(....-(: j.:( '... .: . . .
(
...
.
.
E ' ..
'
E.
:
:
. ..E . ..
'. ' g .
.'.
'.
..
.
:
'
. .
. . . . . .
: .. : . . ..
( .(....''.':'( (E(' j. j..' ':'E..k. ).(....g .(...
((q.E.... .. (.. EEEE(E. gi'.(.y.. ( ; t; (. ' . '' (
: .' ' ' '. . . : . . .. . :
....(
...:
.. ..
.
' :
: .
.'
' : . . . . .' y . .
'
:
.
'
-
.
:
'
.
.
' . ' . : . . - . .. . .
. . .
E :
.
-.
.
-.
:
'.... .,... -.
-
-.
.
Fully leveraged purchase Prepaid forward contract Forward contract
-
:
:
'. . : ....: ....g.
# ''''''.
. ' . . .
: .. .. .. . .
j.... .0 -
F 0 F
.
, .. .. .
.
. . . .
'
: . .j .
:. ..
:
.
.'
:
:
.jjr)..
:..:
E
: ' . '. . .
'
:
-
- '..'. .Outright plzrchase -
g..
:.
.
:
'.
-. .
: .'
:E'...E :' .
:.: .
:
: .. .
. g y.
. . .. .
y -y
.
:
:
:
i E(
E
: .. :y
E
-.
E ' '' '
EE
.
. .
:
''
.
.jjy(yjjjr:.'
':
''
..:.
:.
.
. .
:yj y
E .
.
j .
'''
: :.)..'-;
.
.
:
.
.
-
-
-.
..
. : . .
. -. - ... .
. .
' ..'
E.
:
. . -
E .' ' : ' '. ' ' . :-.. ...
. .. y . ..
')' .:
. .
'.
'.'.
'E(!'
y y .
.
. .
'
E.'
:
:
:
.
y yyuj :
E''(. E' (E'. .' ...
. . . . y g.. .. : -
.
'.j.(.E
':'('g''g
: .. ..
-.y
-
.
:
:
.
y
.
'
'
;
0 0
- -. .
.
- .. .
. .
.
-.
y. ... ... . . . .
.
. .
.
. .
.
.
-. y .
. - . ., . .. . -.
-. ; .
E
-
..
Sv at time 0 Szer- at time F ?
?x
erF
From Table 5.1 it is clear that you pay interest when you defer payment. 'I'he interesting question is how deferrinj the physical receipt of the stock affects the ptiee; this d e fen'al occurs with both te folavard and prepaid fonvard conacts. What should cases'p you pay for the stock in those
CONTRACTS
ON STOCK
a foreign Aprepaid forward contract entails paying today to receive something-stocks, permits the prepaid of contracts forward The sale future. a currency, bonds-in the of period possession time. for retaining physical while a owner to sell an asset We will derive the prepaid forward price using three different methods: pricing pricing by present value, and pricing by arbiag. analogy, by
Pricing the Prepaid Forward by Analogy Suppose you buy a prepaid forward contract on XYZ. By delaying physical possession of the stock, you do not receive dividends and have no voting or conol rights. (We ignore here the valpe of voting and conol.)
the possibilility that the arrangements also differ with respect to credit risk. which arises f'rom his or her end of the deal. (And of course the 11511 of will side not transaction other the the person on wonied about yott f'ulfillingyour obligation.) person on the other side of the deal may be
S'rhe
w
=
y0
(5 j) .
-
prepaid forward contract and an outzight purchase is tht wit.h the former, you price you pay is not necessarily the receive the stock at time F. We will see that the ttock price. Al1 arrangement in which you both pay for the stock and Forward contract: with te time F price specised at time 0. receive it at time F,
5.2 PREPAID FORWARD
FnP
.y
. . -
129
In the absence of dividends, whether you receive physical possession today or at time F is irrelevant: In eiter case you own the stock, and at time T it will be exactly as if you had owned the stock the whole time.4 Therefore, w//cn there c?'e no dividends, the #?-fcc ofthe prepaidfonvard ctmrmcl is r/7cstockprice today. Denoting the prepaid fonvard jrice for arl asset bought at time 0 and delivered at time F as FpPw, the prepaid forward price for delivety at time F is
;''(' .y'. (' ?' f(' k;f jtf :q)' ;'y' y' y''''';IIIC'''t.lIi.. 'y' jd r' f' qif r' :5* ''i'(F' )' pif q' .y' y' ;:ijjdIi.'(111:2111111:111:*. .iIiiiit'(''''i(y' :liij:fd glil:j-'rEllr.d ltf 'f:tfjf j't)y-ytyt;))y'p;yf F' il'ii)tf iq'i( i'ilf lijf q' /'(1* pqllf .jyg.yy't't..E)';' t';'-jtt' lq(q'jqil1j!llqliEI7''piirE.Ei ?' EE' j.' !kIEEi!,:'. )' (llljqilll)d tr.yjf ;r':;.jy'rj,p' q;f r' ()' .' EiEjq' ! Ei E. 'Ei;i' ( !qE.EE(t(5!@ : E(ql!' E': (777q777)77771:: ip( ! EiS)'E i'i ql @i.i''.liE7: ((l: li . EEE !l! J (i jlt.j!r . (i' . );,. . ryt)))E. .!yE .i E. E ! ! E.jgE...E. !;j; El. jjr. .Ek. ; yj)jy .qg. Eq!.q , ! ;p().. i.pt)..k'. t:j; ;-.E.....'.'......,..-t..E..E)......-...--..)..r. . j ( j y
jj?3(;;'''. :';iI!::j y'. )'; y' :(L'.j; :;t'p' t;f j't' )'(' jjf r'((;1IIq1r'''.. (liljlqElEijjpr'. )' (' .-j'y' ttf 'i' (i
$
.
Pricing the Prepaid Forward by Discounted Present Value We can also derive the price of the prepaid fonvard using present value: We calculate the expected value of the stock at time F and yhendiscount that value at an appropriate stockprice at time F, 5'w,is uncertain. in computing te present rate of rempz. risk-adjusted price, eed appropdate of the stok value to use ap rpte. we If the expected stock price at tlme F based on information we liave at time 0 is EoSv4, then the prepaid forward ljrice is given by 'fhus
'l'he
FnP
= E 0 sw)c-W (52) w where a, the expected return on the stock, is determined using the CAPM or some other model of expected retllrns. of the expected How do we compuie th expected stock price? By desniti the stock will worth F be retum, we expec t that irl years ''''
.
'
.
E Thus, equaon
()(uT )
=
zbhe'T
(5.2)gives
eEWe-T = = FpP = E 0 sv s() s0 w For a nondividend-paying stock, the prepaid forward price is the stock price.
)e-T
Pricing the Prepaid Forward by Arbitrage Classical arbitrage describes a sitation in which we can generate a positive cash flow either today or in the future by simultaneously buying and selling related assets, with no net investment of funds and wit.h no risk. Arbiage, ill other words, is free money. A.n extremely important pricing principl, which we will use often, is tat the prfcd ofa thqt nt7 arbitrage is possible. be deriyative Hre is an exnmple of arbiiage. Suppose that the prepaid fomard price exceeds P J arbitraoeur w ill uy 1ow anci sell high by buying te stock price-i.e., ' Fo,w > %. 'qlfltf
'lfc/;
rfhe
lsuppose that someone secretly removed shares of stockfrom your safe arld remrned them 1 year later. From a purely linalzcialpohlt of view you would never notice tlzestock to be missing.
'
130
%.FINANCIAL
'.'.' '''.'. .'y -' -(' -'.,-p,,'.,-,..*. -'.-','-' -..' ,k2k);;))(:;:@2);)l)'' ..' -;'
;,,,-k,,;k,,k,,.,k-.'. ;'.,' .j' -' ....!.-,;(l;)...,'. ,yjj;j,-jj-;,,f ' Ekk,t..'. 'r' )';'-'Ili-,.jllk.'.d j--:--j'. ,,k;,,k,,;'. ('7* t'-'y' ;'',p)ygjy'qjE j,::yy:yf jjygg,,f -'E ;'y ,,i!I!i'-.:jIIj::.IIr'j':-'Ejd y' kii:i,llk.'. ;')'. j-,-ij.;d ,';':'qrr'r-lr''?E f' ,*5 T'E i'.';''1*
PREPAI D FO RWARD CO NTRACTS
AND FUTURES
FORWARDS
;(' (E E E 1IIk;;:::'.' )'r' )' 1117*. i',i1I...III,..'. E(' q' y'. '(-,,;j'jj. r.:;k;jjgrr'. :.-)j' tyltftt'y)))f E' )'' F' kijgyjjf ,!jj;;' jjr.,,,jjf kjyjyjjrf t)f iijjyjjj' ,rjg;;' jjy,.,,j;f jjr,,..jjjyyf (' ;'@' f:ii' E'i E'.E'EE i(E@ E''EE . E ?. ! E'(E 'i11'':I1r''1I1r'iIIii!!!;'. y' .111::':111)* (!IIty:1Ii.' ;i!!iIk'. j:iEll,d qlf k,if y'' #' ;;' r' 1* ill'iii;'lif t)f ).tf !j' i'';' q' qqqqf qf j'y' jjy,,,rjjf t')' E i lJ)llil (( ( 1: j.' EE E(;tEi.tt: ,L'--.'L )lilj:.lld (' .ktlir!.:.il;;i;d t'' jtyj'j ).;(':qJpl(pEir(r pjf ljgljjgjjf jjlf y'. .!jj:r' '5.'.. k( 'l (. .'iiEi;i . .k . . Si . . ((..;!.. .q.... '' .(. .i.;...@(.; !. .. i.j j k: :. :j- !t:.! .k..:..EEE.r !. EEE , .E.;.y yE! !q : y y . . L-k(jii- (( .. ; jyty. j;):Er yj jyy y y E E. . 5ti):j(;;:. F ! t!;!iiE.ik; E;).. :.E)i,. . ).!! ; ' .... .. .. ......
l')'g' j't' k' yy,' jt'yjjjy)jjjjjk. jyjjjy',).yjjf ;'r' y)'' jjyy' j'. .jk'. jjj,..rjjf (' jjj,::,f r )j', 7#'. )yjy'j)'l')' 45. (*q y' flf j'! (J#' rd ygrf )' )'jr')y .t' jjjjjjj;;;'; t'1(111t1E12-1:1111)*. r' t')f kpl:i k..jjjjj(k;j, y. jjj . ..jy . j,).j.. . yjjjjjyyjjj' rj.j(. ..jj( y(j. jt. jj.)r-yy kjyygjj, jjs liti. ltpll 1111E15!11;-. .y. (. iy. r( . '..' .. ... when the
.'..' .'@ .'.*.-6633333b6b11*.-. rrq7p!!!q77ql'' '' ... . .. ' -
!
-
'
i' : ' :.
'
.-
:
. .
'
-y
-jjj;gjjj;j
-
.
-
-
'
:'' : :
: '' ' : :
E E
. -
-. . .
-
-
'E . :..
E
. .
,yy)j-jyr.
.. .
-
-
-
'
.
...
.
-
. . .. . . .
. . . .. . .
.
...
..
..
.
.
-.
.
. .
-.. .-. -
. ..
-.
-.
.
.
-
-
' '
-.
..
-
--
-
..
.
,j$2yyyyyyyy
. ..
jj:rry:j,, jjyjj,..,
jj::yyyjji
jj:ryyrjjj
kyyjgjj,
jj:gyy:,,s
-
prepaid forward price, stock pricez
jjzg,,r k;jjyjji ;;yjy,j; jjr,..
%Pp
jjykyg,y
When a stock pays a dividend, the prepaid forward price is less than the stock price. The owner of stock receives dividends, but the owner of a prepaid forward contract does not. This difference creates a linan cial distinction between owping the stock and holing the prepaid forward. lt is necessary to adjust the prepaid fonvard price to re/ect dividends that arereceived by the shareholder, but notby the hplderof the prepaidforward contract.
exceeds the
.%.
E E E: : E : : : E E : : ' E E : (((.( E(i EE. .'' (. E(j. E'(( (' . (i.((.qEl.( ' E. . : g' .. ' :.. : ..( :: . . . . : E ( ' ( . (.('.j.E. E(j 'E.' -. . E. dklriijlkk ( ' 'g ' . (( . ( : . ' E .y-E. ('j. . j. :. (r''E.( i. : '.: E j .1111:!1(:2::2:. .. . ( : . .r : . EE E E . .( . . . g g .. r ..( ' ' E jy . j... .E.('j '.-. '.g... .( . . .('.. : :. . (g((. '.:'''. .. (''EE. '( ' . E. . . ' j' ..... . i. . . ' ;- . EE . . .-. ;. .' E.. (- E . j.. . ..- . .. ..' . '. . j. (' g . - . (.. . E . .. . . IidililI E - . E. .. .- . . E EE E -E.-- . .. .. ..;- . . - .. E... E... . . ..-. . . E. i . .. .. . ... .E.. .. - -. ;..j - ..-. - q!.i EiE - - E. !..E- !E...:: ,. : j'yi..--!. - :. . qq- E- . . E... y.. qg E.. E . .E.. . - - .. . .. ;;-. .. y.E:--. !-.. .. . . .. . . . . . . . ... . . . . . . . . . ' E . .. . . . q E . . . ,,i.:,,';.q,k' ' E. .. . . . . . . .. ' E' ' . E E. ' . . . ' ' . . .. ' 'E .. . . .. '. E. . E. ' . ' ' E . 'E . E. E. EE ' . ' .;kf,k;k,:;;,' . ... -. ; E..iqE.y. .. E E ...E -..... . E... ... .... E '.- E..EE. E E'.... E... : tkjjj,.k:rj:' E'..E'- . l E. 'E'';.-''. E... .. '.. ' E'. E - .. -.. . E... qE ....... . E.. . E. - - . . E. '. . ...... . - ..E.... ..y qq. y.y.. . ; .. .g.- q.q,,,..,,jr,,,,....,. . . .. . . - . -,i. . . -E- ... . . ... E. . .. . . .... .ICCEi!IIL. . .. E y 'kl:i:llpk .. y . (.(.-. (.q. . y.j E.. . ( .. ...y . ( q . ( .q . y g . g( . . . . . . . .. . . . y . ( y , . ,. . . .- . y E 1111..311,.. .:iiliill .. . .:. y..-. ...- y . I1k--... d1dI:::,:rlli.qllp.. apllk.ajllk..q iIIr''1I1r''1I1'. :ilEE!11i(' . . k'.''' ' ' .1I::d!:. . - . ' E: . i... .E' : . .'E...' . E..E. !... .: '. .... ... Eq.E-.. . qEqE.! E... .E.:j;.y..-.i..- ..E.. .t....- i. - i.(E... ii . .. E ..y...y.q . :.. . .. - - IIIr1II. .y . . y . -y !. Eq. yy q- .q . - q . yy . -. -- y . . j. E y ' .E:. . ... ..... . .E. ' : .... . .. ..... ... .... .. . .. E.. E. .. :.. EiE.E. .EE'. l @ . . .. . .... . ....... .... .. ... . .... ... .. ;..y. . ... . . .. . . .. ..:... E.E : . .. ..E ......'......: :. : ( ( E.. ...Ei.. . . . .. y . y- q.2.y. y. y......... .... y.... . ...... y.-........ ;.....-. ..y. . y- ...y... y..q. ;.....y- y . . .. .......... . .... . .. . ..... ... .. ... . . . . .. . . :
:
k ('(i. l . rr..( ..rq.. ..y... ).'..r... '.'' . . ...$,.,-i.L .(. .. . j. :
:
:
'
.
-.
)
-
' '. .'. ...,.... '
. .
. .
'
.......
.
-
-
. .
- . .....
-. . ..
..:..--
. .
.
-
:
.
.
.
. . . . . . .
.
'
-
.
.
.
.
.
. . . .
.
.
.
-
.
-
.
.
-
-
.
.
.
-
-
.(. ..'. ) .. ;. . .. r.) .... .. ... . '.-ky.. ' '.' . . .....',i.-,1ji,..-j; ---L. ''. . .. .. .. r....'-'. . . r.... ... .j. ...j ..'..366...'. ... ..... y....r. .r.....,y.... . .. .
.
-.
(.' '....... .....' .-... .......r..r........ .i.... ... i' ... g.''y' .. ) '. .......r) (r r'!. r' r '.. '. ''E(iII!q(''!1I1::-''' lllrf:lllt. '... .... l. (. . .... .j.. . ' '. ' . .. .. .. .. '
-
. .
'
-.
.
. .
.
-
.
.
.
.
-
'
.
. .
-
.
. . .
Buy stock @ So
.
-
. . ..
.
.
.
.
. . ..
.
-
.
. . .
-
.
. -
.
-
-
-
.
.
..
'
-
-
.. .
.
- .
-
.
. . ..
.
..
.
.
.
.
.
-
-.
.
.
.
,
.
-
.
.
-&
+FoPp
Flv
-
--
.
.
. - . .
-.
Sell prepaid forward @ Fopp
Total
.
.
-
'... . -
-
..
-
.
. ,
.
.
- .
-
-
.
..
.
..
.
. -
.
-
.j,.j--.. .
.
.,-illr.....y
. . . . . .... . . , .
. ,..
.
.
.
.
.
.
-. .
.
.
. . . .
,
. . .. .
. . . ..
. . . .
,.
.
.
.
-.
+Sv -Sp 0
Sz
To understand the effects of dividends, we will compare prepaid dividends StockA pays no dividend, and othenvise identical stock B pays stocks: forwards on two days dividend from today, just before the expiration of te prepaid forwards. 364 a $5 We know that the prepaid forward price for stock A is the current stock price. What is the prepaid fonvard plice for stock B? Since the $5 dividend is paidjust before the delivery date for the stock 1 year from today, on the delivel'y date stock B will be priced $5 less than ock A. Thus, the price we pay today for stock B should be lower than that for stock A by the present value of Discrete
,
the stock for Sz and selling the prepaid forward for &%.This ansaction mkes money arld it is also risk-free: Selling the prepaid forward reqires tat we deliver th stock at time T and buying the stock today nslzres that we have the stock to delivef. Thus, we the buyf of the prepaid earn Fopv s'atoday and at expiration we supply the stock to offset all and futilre fisk. Table 5.2 profts positiv earned today have forwar. We simation. this summadzes
$5.
-
< &. Now suppose on the other hand that Xpv we can engage in arbiage ' eprning So shorting stock, Oe yar from and the by buying the prepaid forward stock folward arld via tp clse te short we us tat te prepaid now we acquire te stock reversed. simply table abo-ve the afe P osition. Te cash flows in Throughoutthebookwe will assume thatprices are atlevels thatpreelte rbitrage. This raises the question: Ifprices are such that arbitrage is not prohtable, who can alord to become an arbitrageur, watching out for arbiage opporttlnides? We can resolve rbitrage to earn livig, this paradox with the insight that in order for arbitrageurs ttan of degree equilibrium be there from must time; time to opportunities must occur disequilibzium.''s However, you would not expect arbiage to be obvious or easy to undrtAe. The transactions in Table 5.2 are the same as those of a market-maker who is hedging a position. A market-maker would sell a prepaid forward if a cujtomer wished to buy it. The market-maker then has an obligation to deliver the stock t a sxedprice and, in order to offset tis risk, can buy the stock. 'T'he market-maker thus 4ngages in the same ansactions as an arbitragetm except the purpose is risk management, not twptf arbitrage. Thus, the rl't//kmclDn described ?7 Table 5.2-sellilq theprepaidfonvard ??7t7r/l-?nckc?: buying l/kc stock--also describes the acfsn, ofa The no-arbitrage arguments we will make thus srve two funcions: 'Fhey tell us how to take gdvantage of rnispricings, and they desc/be the behavior f market-makers
In general, the price for a prepaid folavard contract will be te stock price less the present value of dividends to be paid over the life of the contract. Suppose there are multiple dividend payments made throughout the life of the forward contract: A stock n. A prepaid is expected to make dividend payments of Dti at times ti i = 1, receiving the receive time but without will entitle the stock F at fomard contract you to prepaid price is the dividends. 'Fhus, forward interim
'fhen
-'
-
FJw.
managing risk.
sThe phrase is from Grossman and
stiglitz(1980),in which this
131
.jjjyy,
jj:ryy:,,, j,:yyy:j,. .jjjry, .4j4rr, ,j..,,jjj jjy,,.yjj yrjj;;i
.!jjyy, jjy.,,
$
Pricing Prepaid Forwards with Dividends
.
-
.
- -
-
-
''
'
.
i
-y -- .
.
ON STOCK
idea was first proposed.
.
,
.
.
.
,
'
11
F-o.w p
=
S()
PV0,Jf
-
lhi
)
(5.3)
i= l
where PV() ff denotes the time 0 present vzue of a time ti payment.
)' )y'. rj Example 5.1 Suppose XYZ stock costs $100 today and is expected to pay a $1.25 /1 quarterly dividend, w ith the srst oming 3 months from today and the lastjust prior to ytr tE) the delivery of the stock. Suppose the annual continuously compounded lisk-free rate is )'!qi 2.5%. prepaid therefore continuously compounded is A l-year quarterly 10%. The rate )q 'i) forward contract for the stock would cost tti 'j @ (qyj: k E i
rj
. i) ,)(
ijtr)
%
YU
The calculation in this example implicitly assumes that the dividends are certnin. Over a short horizon tlzis might be reasonable. Over a long horizon we would expect dividend risk to be greater, and we would need to account for this in computing the present vale of dividends.
132
%.FINANCIAL
FORWARDS
AND FUTURES
FORWARD
.'y .'.' .'.'.
CONTRACTS
$
ON STOCK
133
klkzzj!:;d jjjljl::jpd kyjjjjj;j'. jjjjrjjjrgg'. jjjj,jjjj.'. .,j(jjgjg;'. jjjjj.d jjjyd j,jjj;;d jjjjjjjjjjg:,d rjlt.d j'.jjyjy;jjjyrjjj.'. j'. ;'.yjjy.yyjj.;'. rlj:.d kli:::pd .1It)ji1i(:72' !lp2ir:' jjjjjjt;jjd jj'(' r' rjd jjjjjjy:jyd jjjlgl:i,d y'( jyd kl!!i,.d yy'jjj' ()j' p'. t''t'. ?' ).'. trtf .p' p' 'l''t'tkf )'-(().. i'@'--' t't'. 'r' j'q' ('. )-'III:'-''-'-)'..---;;...-''. hf )'-; jjd (' ('@.-' jjtj'jjgtjj jjj(jj'. ;)k';j*(y'j):'r':-'. 1* y' .jjjgggjjj.'. 'tIE(1(Iil.'. jjrgyjjjj.'. ..' '1Ir-k::r.'. yjjll;::prd L(3fLLt''. rjlrjr:;j.d 11::jjI(.' ((jIqEFEpi,', (r-'iik.,;;iq;''-' tqlIiq)..-,'. pliitizii.d 1i::ii1I.' -(!iiEii!i:-.(qlliEEi:'''. (jltll::ltd )'. 1Ei!IiL'' t'('.(' ,j::2jjI(.' t1i!2r:;'i:!j!il.d y'y (ty' (4*4j yqjd yytj'y iil.d jijijjjg;;d ki:itjiI(.'. (j11.* tlt::ilirj,rjlr.d )')-t'. ;i',' )',.-()-----plpk-. )'. y' )'-' ;;;'. ji!jji;'. .g(;jjj;yg'. tjltq::li.d r' (jjj.'. ggjjjjyd tjjjjjy:jjd t'..;'''j'. '..))(;'-' t;'-'. t'. tlf trf r;'. '-;'. t'r' tt'-'. trj!j,;p;.d .;:gjjjj.'. kl::jlll.d (4kj4jg,4*. .)jj)qj;)jj)j))'. .gyjjyjjjjgggyyj;gj.'. (jgyjjd tjjjj.'. (jjj.'. trfrp'. '-'L.'bk'b''lLk. .:'. zkr':: .'jjj(.(4'ry'' .,' r-'. tjjjjyg:jjd tjjjy,jjj.'. .)jfjj4j4),'. rf .:'tjjjjjy.j..j,'. t)lig:ril.d .;jggj;'. tjjjjy;d (jtj yj:tj y. jj g (. (; lllir':'lli' ' (r(; . j' y(.j j j (. r..g((.'':'g' (i... )!J;...)!p)t).. '. :j)-.- r. g'- iIII::''',r.. . .. -!!!!!EE!!!r:)-. pi.. ' '.'- ( ..ip bib,. '..t.. r. t2i))))t)l):t);.r)))r(;t:. .(.. I; .tt.. ( r i .. ..-)').. . . i . (.- .-;-..-tt-.;:ryi .-... . ): ...-j ..-..i i. !r)'i ... . ;...'.. '(.. .. .. r . . . r . . .( r . . .) . . . ..(' .). . (. .. . . . . .. llltr . . . )y. '..- ;' (,-.'. .) j. i' '... '. .. .: ii2g;(. ( j. ... j . ... ... ''..'. j:!i,l,lllt;:llj. iiiik;i . . . .' . y.... j j.. y. .. .. ..y .. j. .j ..j. j . ... .,r.j .. ..jj .. . ....j .............-....j........j. j..... .jjjrjj;;jjjj;; .. . .. rjjjj;, jjfjj,4,f).. . j . . . . ..j y (.. . . . .. . j.jj j. . j .jj ..j.. ... j . j...j . j... .; jjjjjjyjj,. . .. .; ..(j......... ktj::: 'Ii:gr ij!r . iiyyjlq1ky. ))t. . ( jglrqk j j . )jg:j::j;gq.. . j . t::rj)k . .. . .. . ... j y.jj. .(. ... .g. y t; . j . . . y. .. iki2r;. kt:j:r iijjjr jlj;:ll n' .jj4jjfj.). ' 'Ir$:r'. jtjgyjy.jykjykjj,y. ikjr;. .-. jjjykgjjy ( yasssyyyyyasyy... ' 'ijj; t:jyj,i .. i(2i;. t. it@!i. (' ( .. .)j'.....j.)....)..j t-q .,r:$,...(. .gjjj;j...jjy (.j. .jjjj;q ''' .(. )jI(:1; '(jjjjy ...jj;...-...j..j..;..j.;L.j.....L. ' . ... jjgjjgj;.. jjjjjggjj,.. jjjjjjg;jj. . . ..(. j j .jjjjjggjjy. ... tgjjjyyjji j jj.j .;j j. jjj j. .. .j )y.(j... t yy y x . ... . . tt-is-tkp y j. j. jj.j opb lLlr,rtpl. lld ) ph tik . ygyqj. t . k j t y 'j' lr:piki;:.lj;i. jj,i::'. '. ' ' .jjygy' ).'.. ' '... '.....j '...... .... ..' (.'gjjjyjjjj,' . ... '. j j .' r j '. .kjyyyjgj;.. ' . . . ' -). il!i::)il:/k' j . . . . . . :jI:j. . . . ... . . . . . . rjjyy . . r ki:::t:ipl ... . .. il!::qiiil. (:1!5,.,7'. '. . .'..j j '.'.(. )'. r...j . ... .- .. .. ... j .... .... . . ... j. .. ... j.. ... j.. jj.-r.jj...(..g j..jj j j r... .. j j.. ........j.. .....,),q,qq,. . . . . .. . 'iI@::. tl::i:i,; . ':Ii:i Iil:lEl!!::li-:!jii!I;: i!ii;:. . .. .i::jEIIC.. . . . . . .,Lbfb4i:f:L'1.. ..i .- j . .-.. ...)'. ..... ....-. j. . ( . j.. j..j..- ...j .. .(. j ..... . .... ( . . ........ .j . .. . .. . . . . . . .. . . j . . . . 1k(::iIIj . 1I:ijiI(.. . . j...j.. . . . .. tl:jzjill. L Ltff33b3ii-L.. ( . . '. ' . . ..t:jjiEj;::. ... ........ r... . ...... jj ... ... r ' kl::jjrrq.. . .. . r (!1j::jig. ) . ' kj:jyy;k;-. . . .. '...... ....... . . (yjjgggj:j,.. .. . yjjj(;:jj,. .. .... .. ..... jjjjjjjjjjg;, . ....r.. ). .. . j,jjj:;. . . . ... . ... . . .tjjjy, ..j. . .yjjjkjjq?)l ... j.... .g jjyj;j;j,. '11t2:1111. i i!ii!l: ;k;.. . (1144::)11 . ( 'tjj:::;T!gjj4.tgjjyg,,. . .'. .. jj)Ljj,i.. ... .. .g jjjrjy,r;jjjj;jyk,;j,,. . . ... . .kjjjgjjj,. . yjjgjjjjj... '.j...j j.... j ........(.:...j jj..(....j......jj. .g..'. ...j..g .... .. . j. j.. ....:..g..g tjjjjjggjy..)jjjj44,:.. j. j......j......jj.yj j .'j..'j.j.j..j.:.. .. .gjjyjyjjjj. . . ..... j . . j. jj. . j j.. j.... jy .yy. j .. j . j.jjjj klzjq::lp. . . ...... . (jjjjpr;jp. ... .. . . tjiijrjri . ...... (41j.,jyz),i ...... . .. ....g j j.jj j.. j .j . . ...
!jkjg!g' jjIyjjjk;'E .jjjggjgjj'. jjgjgjgy'. jyrjggjy'y jyjggy;'( jkgjjjjjj'y jjyggg;jj'. .JIk..'. .jIk.'. 11.....11* 111:::;:.*. jjj:jjjjjyjjjj'. (jjjjjyjjj'E jjjjjgj:;d ;'. ;'kj:yrjj:'E .7*. :'Ijyqrjjd ;-,' r'. jj:!j!:j;d @' #jj' 4* j(j;;;'. jj:jjyjjj.d .jj.,.jjj' E.' j;),,b'. .jjr:j,j...' jj;,..;jj.'. t'7* j)' fl:::jrr.':j:::y;r,r :'E' ,1kj:2!:r'. kli,rjli.d -'. -.-' j';'t' L' -' --'-!!'!r--.---'''::'. -'-''i'::::'i'.!. -);)'.(.. -'--'i. ,';' ..y..;;',,;'E '',': --'' ''''l:::''i''r ...j,r'. -))'' g' (?' -''-$-'.. ILX 41::2j::* ilki,.,,,.-;l!d .jjj:2:::!k,' (': 11....11...11.*. 11....J1:.*. ,11...21L.*. il:ztiid 4j2!gj:g'. 4i2j:r:l)'. 11:2!:,4* Eyd j'. jj(:gygy)g:jj;'. k-i'. .k!Ik.,,,.' 11::::::*. (:1jjiq'''I12i!::'. :'j'F' k' jjgjjjggjj'g .r' )' 15* !!r' jjjd t'. t'.,,lllljli,,t'fd -'t'-.()-' ;')r'. y..j.'. (' --' ,':-'j'bilii-f;f. --'. -'' y)' 'p' j'qkjj;j)j))'. ((jjg(4)'. jjjj)j))'..'jjjjj)jj))))j.' ).;)r.' t,d y' :1!!511;.* 41::j)* jgjjy.,,d gyjjj;;'. y'. ky,'y (iIE::1ILl' gijjyjj;;d ijjjgjjt,d jjjryyjjj.d :11...:1r..,..11*. 1Ik@gri:'. .yjj'. ;;gjjjjj.'. .bqbj,kblti,t' ;jgjjjj;j'.. Ilid:;d .ii$ib)''. 11::4)* I1iEE::Ik' iliiiilkl''l::!,,;lr' il:t::iI;'. $:85511(.1115141*. .)j(.j:jj.' q' y' ;jjjjj;'. y'. gjjjjyjd jjjj.d gjjjjjj'. j'jjjt.d jjd j.))jj* jjt.d iljjjjjd jd k' 'j!d (j' j,yd jjjrg'.d jj:yjjrjjd gjjjjgfd .jj(kjyjj,' jrd ).-' Iil!iksd .jj' y' .1Ii:::j:' j'r' tjd jjtjjd 444*jk4444* (' )' :-',',''i'.t')#,-' ,-' qkL-''''-'. :!!iii;', (tf ,';t)'.@' jkd )'. (jlr:r.lt.d ..II::r.!'-' .j,ik;;..' j'.yjjyjjjjjgkgjj,' gjj(,j;jj.': gjjjsjgjjyjk,s,;d .yjj,jgjg.'. yjd t'jjjgy.. ki::Eiill.d Ell.d II1:E2j1p'. 'i1:j(iI2lj$l!I1L.' ijtgyd gjjryj,d gjgjjjj,.gjjj.d ,1iEq:il.'. .'k1:E:q:,i.. .'(jIiq':rii.'. yyjr.';-i jj:ry::;'7!::F:;'!-jjjjr,(jj)'. yjgjrtjjj'. jjtyyg;j,': yjjy'. .jjyjjrj(.'. :ii;;'. (iljrt)lt.d iiEj;sd (lir:r:lt.d 1i::E2:1,' ::;j,.-y'. i!li::d r!1i:;Ii:t'. ii::::::d (iI(i7:I..'. y' CitEs!l.d (.1(.* kIlii!i:'. gpl:tjll.d l1i4i;)'. ;1!EEIi.'' IIE#'t11('. yjjrrrjjjgrr,jt.d iiij2)'. ii!EE'. )' gjjj,-;,d y.t,j(,;;,y' i!!E4:;'. .2111.y(11:j4,1:.*' (ji(:r'' (qllliltil!j)d yyd jjd (' .jjjjygggy'. yyjjjjj,.d fjjj4).,)'. j1LLj)'j('. .jjjy)' gjjg.d )jj' jgd ygjjgy(yyjjygjj.'. t,d (i:ji!;::.d :!iij;;'. .j' iijjjjik'. (jIrqr:il.'. Eii.;ilrrr,itd .11()j)I(.' 1:2E:ji' j;jd y't-ty-y--'-. j,f !'. lilli-'r'. )t)----i.iiEi'','-?gf.--ttt:.'j't' t'. 1..,)*. t(' i''Ir'''''''. jd 'tf j,kd )' r..-tk#)''. kjkggjgjj'. j'g j)jry.,d jj:rrr:;d kj:jgyjjjd :It:'. j,;.,jyjj'. ;'. jf)jj,,f.'. -l:::F::r'rjj,jgjjyytjjjj;;': :!IL' :,!E4;)* ,1::::::* r' .1r::' y' j,jj)j')Ljj'. kjjd j()':' jj2j)q:jj'. jj't'tyt. jjkjjrjr.d jy'. jjjd jjjijjt'jjjj ,').'. )'4*. :):;'. :,'-' jjd j.-'-,.t' '-'.;' r' p' 4* yjjtll''jjy ;t'. :'. :'1*, 7'f'---'. plf'E #' y' !r!!d!'rr!!!!!r7s'-------.'.------'. q'. ('p-'. E 1:. i ji lj i i. '.E ' i. :E''j'E(' ' jjl ' ;.plijj!' (E :Ejj ;j'' 'j.; ; . i 'jy jyj:E. j;;kz E!ttyyt. EE jjll jj4.. j' i'5jjjjyty j(j. j jj E( j ( E (E . . j. (';j. ' jk E E E )j,(!j'j(j.'jj i j'( 't.! .ig().jj;l j('E)((y . 'jj.y rjt.:lyjl ' ( .i!:. .!Ely#Ey. (jjjjyj .r. ykt( jjj(E.E.ljjg .i. . 2!'.Ii2:E; . j. ''k'i'iE',''j ' kjr' ('. (;!jj ji j 'y;(.(-( y.j j'; jj.;j .y.jj ((yj.E.( kggjjggj(g!';gij.'.j'yjjj('' ji ErEj( 1..('yjj.jE-j .(..y.('.Ejjy.'( 'jE.j.'.(j j. :g'E.j'jgr,-;j(.( . . )!kjj''j j---lllllitl:.q j . .yj j . (4(' jj---yj jjg j ' .....k gj(.j'jj;jj.jEEj.j;j(jj(k'.j''j(.j(.gjg( . .. .)). . . jjtjj .jjjgrj.g'-j jj. j;jy . jj! ;jjjtltj j'.. jjj!( j jjj)ytgyttt)y .j jj jy j. .g..j-'..j .j.(... j.g . :.- -ti. -k;i; - . . -;;-.-. -- ' E -.- E--. i-:.i ;. kii.- -. . - --'.. . ;-:-- .E. -- -f -.. . -. - '-- ; .-.. -. -.'. ;-.- - -. j.!.; . - i: . t. - .-.-t--- - .-.--. . -. -.. q. !.;. i i j ( -. k--:. .-..y6..:.E..E'.. .. ;.k.. .-. -....:.--...;;;- ;..r: -..-.-...... :. pi. .. .! . . -.. - ;. i. .--.......(. ,!-.- !;:-.-i-.!. ... .. -. krp-. ...-..-.-.i- -...;.!.. ; .-E . . i. .. .,.- , !... .! .. s; . : .- . . .-. -... .. - r - ;; : ; . @ : .F . . y (E .. . . . . . '..E. . EE. E )E.. .....-(.(..Eqi...E.i'( .( .... . .. .E E E .E:.. .. ...E.. (... . .q(,E.. . !.. r7..: . .. ...E J. . .'... .y.E. E. E.E ' .. .E. ')'... .. .... .. , .j) . . (.'. . . E... ... E. .E. .. t-r'yj)Er(j .... . . E. .( . . .. .L .. .. E.. .( Eq..:..E .() . ) .... .E. E.E ) j).. ,,-L).. . .....(.. .' .-E. ........E ... ...q... E..y. (..... E q '( y... E. ....Ei.:.q(...-. . ..(E. ..(.q(...E. . .. E.. .... ..- ...,. . q . . .' . ..E E..( 'tE. ' .E. ... ('(...). ( q. . ... .. yr.ll . .. . . . .. E .. y. ( i.:. E .L...E..: .. . ... . .... . .: E ... .... E.E. E.!. Eq . E ... .. . .y Eqq.... Eq.... . . . E'. . . ' E. g .. Ej... ..' ... '.' E. . .. . E E't . .. . E. . .. . ).. E E .... ...' ' . . :.. j. r:j:( E i . '.. E -.........E ;,y.E y...(.(. E . E . . . '''1IrIII.''': . E E . ... . . . . ( E . . . . . . . . . . . . E (. . .. q. . 1Iij:2!:E. . . . .. . ... ..y q.(..; . q(q y..'..y .. qy . '. j.. (.y.; r .. y.q.y '... . . .. q. . yg... .. 1ij2jr:: q rq.. . q. q . . .. '. .. .. (.. yy.. . ,. . . . y .....y.( y.y(y.j. .(...y. (.y... (j.q..(..q...y.....;y.y.... (y.......y.E..;i....(.(q r .11,)!Ii, . '. y . kII:r:)I(,. i..... . . .. . . . q. ......j. y. . ' ..11::2:11(.. . y .ilzjjrri ..... . .k,,! yy j.... yy y.. ..y..yy ..jj...... .. .. .E E. . . . .yjj . .. ....iiljl!))ql!:j:)jt, .'.('..j. , . . q. E.E.. q .. . . . . ;1E211,.. (jI(:q:Ip,. (EII.2II:.:jIL... rj,gyr1IEr,1t,. . y.. (214... .. j .-. . . y ... . .. yq. q .y.. . Eqqj... E-.( . .. . .. .....gy.y.. ; yy .y.... . .; y. y .. y. y . .. .y.. . . y. . . gy. .. j . j:jjjjj;;k . y. . ....ygy. ....y y. y..y.... j...y.y... .. . j j.y . . .. j . . . y .. .. y . j y . y jjjrjj:;, . $:4jjjjjjtjjjy44j(,. y y . . . .. i . . .. . . . jjj,..kjj.jjj.,jjjl $jkj2j4)j . . jj.., jjjjs j.jj.g. . .. y yy.g... yy . . y. . g ..( . . j.j y( j r jjyyyr)jj g ..jj(, jj,,, jj. g (jjjj . ..... ... . . . .. .jj. . . .. . ...ygy.j . y j. .... j.j .j.y..j...j..g... g..j. j. ...... . .y.. j... yjjgjj.. . j. .j j. y.j ... j . j . jjggjggjj.y . g .(..y .. jy. ..... . j. .. g . g. . . . j. . .r . gy;jjjyjj,. . .j ... . g. y g.. .y. . y k!kjik;,. . .. . .. . . y. . . .. .y .. .. .. . . .... ... .. . . . .. y... . . . .(igr:jig. E.y . . yjtgjg;yyg yy . . gj. g....jg....g jj ygjj.j. .. . j.. . . tjjj.y jjj . .gy.y jy. j . j. jgg . gj j . jg j jj .yjjj;,. . .jjjgg. . . . .tjjg 4jjj:: .. g .. j..... .g .. j . j.. .. g gg.j... j.....;y .j.. g.....jj... . j..g t!jji 11.: g y j .yyyy.j yj. .y.yj . jj.y . j . yjjjy . j y.yj.....y . y g j. . g rjr..ql j gyy j g 422,,, yyj . j . y y . . . . .. . . . j .. .......)j . .;-j. ....y yy.. j yyy. y. y ..yy... y.. ... .y y..y. y . .y y ...y . . ...j.-. . . y.j.....L.. .. .. .. y..y yq. j..y y . ... .( . . ..y-... yy .y ...- . .y y . . . . y. ..yy . y j y y yy . .g.. . . y.. y. . . . .. . (E E E. . j j y yy j... ( E .m.-z . . . E' .;.;.; (.(( (. . .i.. y. . .j.. .y..y. . . ..y. ..y.$yjy;.. ).. g. .: . . y . yy..y y.. y.. .y.y. j . yy.y.j. (y. .j E...'....y .... y ....y E( j . w. . .. . k!t,/,. yy .r .(. .t...( ...''..L'j'..'j...j.'''..;..-..).j'.L.jj...'..-.' t. . .. . :12:1L(41)(j1) .y ' .. (.. E y... . lirr:p, zsj.;t. .. y.. .(q.: ('':EE' .y u.41.2j: .yjy..;, y. y. j. . .. l:iz:l . iIi:9IjrC-yI.E .yy j-t..y... .. jjj.j...,j. .y.'y.....yt .Et... ... j. . k.-v s:.u. (.'.'y 4k12:9:k$1;:gg144:)8 (.l . . :1q ' (1Ir(Ir7'1i:;.. )I(:)i... qI'.'jh''1r' ...yj.jgggjj :127-112: dkj!r :r:1@ '4t:E)). 41!. ql!jlq:lL '..q' qIk)jIL.. . (j((:i@)I(g-ji(.. !'.'ii;;jk/h'(jl..IL..IL.. .jk..jl...IL.j1.gl.. 11.. 11..'1rik2r:. ysysj-ys.. .E ....':.'.... .4(kg;).. .. y .. '!r. y. i,rj.j E .q.... ( ;-(.q...L..d1k)117ii.1(i!; -.j,,.,,Ej (: E ........... ..:j;. ..... .;.. q.-...kj;..,;E...j,.jj. Ekj:jy. y,,,r,jry.EEyjj ..,,;jj.( k;.. .... E ;.tr. ..,;.t;;.;,. .. . . .... (..... . . . . .. iiiygkj; . .E.. )..... .(.. !:;di2. . y:.jj.kk,E ...;;..j;,.. .E. .jE yy .g. E. ....j(yjr;. Ej(jjjy.. qE.yjk(gr,:(.y ... ..,jk; )j(yjrj.. ;. yjj,,k... jjk;y;. jj.k,;. jjj;.jk,... (. ...y..jj...j;,kjy .- yy . .jjr...gjjyyjj,jk.gjj... . .L.j).. q;..yy(-(...( . ( y ...:...j. . lkij);t:jj)l' . .. jj., t:y . jjr,kj .-... .., .. ;(.j .t.. t- yy litEli.tlErtir 4(1:1... . .j.j.... iEjk. .11E?1 . kjy. kjj,j, ,k..jl.. ;...,k... 2.. (lL.. jrj :1k.:li71l. '1E''lE'q1 .'!..'!Ik... 1kkkk;: 1i..1(7!iy. (1(j:1k71k). .Ik.klL... 11..12.. 'k.ik..l:y .ILit:ll. 11.01.. Ik... Ik.. ....I.. . ii:!1..IL.. .dC:'k.. .It.,k.. i;. .Ik.. .........tj..y sk..lk. ') .g.: .j.( j y. ( '': ( .(. .. (' .jE:.,r yy.y ( L:.( . (. . ( Ekj, .: qj.r...... ( . ( . j: . y y.. r r jjgjjggjj;. . jjjjggjj;. ( ..... y,.... ..jjjjjgy;s . (' . . jy .. .. (.. . (. .. . ... j ; .. g g yjgggjgg,.( .y...y.y. . (. . .jjgggg;jj.. ;jjjgjj.jgjjgggjjt,. . j . . . gjjjj;j .. .y..y... .y.. ....... . j . .jjjjjgjj, . . . ..jy. .y. g g . g . (.gjjjj;;. .yy .......g j.j j g g . g . . gg(. ....... ..j. ... . .j . j .j ... .. . .j . yg .. . ..y ..t . .jkjj;ggjj.t(jjgygjjj,. y ... ... . . .jjy...jj,,,..yjy. ggjjjyj . . j. tyjjgj,jjjjja .... . . .jjjg.gjjjgggjj(.. j jy. . . jjjj,s gj. . ... .j.. . j. g .j. . . j . . j . j j . g . . . g g . .y . E.. ... . . . . j . j j . g . . . j . . . . j . j j.yg qqE j g . j j . . jEy,t . .. qy. yj . . . . . . . . y. .jjy.. . j. y ..y .. . j. y j jj.. ... . j j . . y g .... g .. gyy . . .. y... . .. .j y . j . .j j . . y. j jj yj.jy.. y . E y .. ... y j . yy ( E .y iku Eq : q ( y . j . j .. . . . j g y. y q yty q . jj . qg. jyj. jyjy jj.yjyj yj.. . . . ( ..( . ; y y y y; y . .). .. . .yy.;. . y yq qy.yEy ( ( y.E. .. E . yg tg y . .jy. yy . . y g . yy . y y y j jyy y. yyyEj,jjytj (E) y,EE y y...y yy g . .yyyy . yyy y y,j.y(E,jyyjy,yj,jyyyyyj;(jy,yyyyryy,(,g.yy,jy,qq,y 'jj.y . tE,yj.jyjy.,(g ; . . y y . (,.y y,.yjjq,y((l ,( . E jq ;y.y yq y; E. '11ki2!:' yy ' j y ' E' g E' 1i2j!:!j' 11::):1,)-.E ' 411j:::2 E' E j (jlfqt,jg. !' ' '':!ij:; kl;E::,;. ' :1:111,)j:j,k4:2. 1122.:::. '' E . E. '.' 11::j:jp-. 4k2ir.' 1it:E2i1,( 'y )jI!tr!. ' . 41:::t:1,. E' E. . E ' '.' E. E.q' '....'y(.. E.. 'yE.q'E.E.'q 'E'j.. . .( . E (jI(..E lltrq :EEjiii' i .yEE. ...'.. E.y.. yiygjyjj..q . .E q.. .E.;( y.. . .. (.q.(.'...E.....! i..kk. q...(g . '.g. r... .( . .. . qy'. . E. .. .. ( y.y. jkjgjj;jj.. y . . .y. j y..j . r.. E.. . E. . y . .. .....q E i. . E.y . E. r . . ... .. E.q..k-;. . . y . j . y ... r E.. E.iy . .. .E t.,,.. yy . $jgygjj(j.. y . y . E . . y i . E . . . y . j q . g .E... . ' .. . .. E !!.E..Ey... y g . y .. E . . y y . . i . . y y . i q . gjjgj;gjyyj. . .. y . ;. .. . (:jjjjjyyjjt.. .. '-1t1-.':1E1)0',. . . . . . .. I..... .... . ( .. .. (. . .. . . ....y.. y(j:qqj:jji . jyj... .. y. . t-.;. iq.. . ... ( q E . ...(. qj. ..q g . j g. .. y . . j;j!j,lj, 'Iii!!r::. . .. . .... jjkjygjrj .. . . 11:::2511.. . .;L3jj?-. . ' )''E'. ' (''' E E' ' .(..y...y. yr.. y..y;..(yyyyy...y. .(. y .q.yy. 11::t::).. . .i'kksF'. ' '114::.11!7::11... iI:::E1IL.;iI(..:IIE7-'. ' '. ..' ..q.'yE E.E 'E :!pE!l; E E. y . . ( y .,. y .. . ..j g. y y.yq.;y....E.yqyq. ry(.. g(. ... ....y y..,.y.. E. (. .yq.. y yygj. .g.;..... yq..y yj.E.yg jj. y.y;y...:.gy...yy;. Eqy.yqyy y.Eyy (. E.y..ygq.qj i.(..y..q. ...y;y. ..y. ..E..yqq;Eji....yy..jgE.. . ;!.gy.yy . yy ...j...jr y . .. y j y..yy.. .. . . . . . . yy y. .. . y. yyy yy . . . . y. y. jy . y y.yy ... .y....yy j.y . y j. '1t:i::.,. .. .. j . j y y y :11f.(:111q:11.(. . . . y . .. . . y y . . E. . . . . y . . . ' . kl!Eli,. y y. .. . . . kI2!:r);. . .. . . y . . . j.. yy . . . . .(qr)1I. E. . E .... E E E. . y. y . .... . . .q. . y.E. . ..-ikfLiii-., . ..y .. (: .j jt (.. . .j ;. r.( ;. .j ;..:!Ep;l;. . ... ..y.. . y. .g . . . j q..i. yE iq. . . .. .. E ( ; .( j... k( j. . . .......yy ...... g.E.... ..,.y. q....q qq.jgy... Ey..i. ......y...y..yy..j...q...E..... .y,yqy..y! yg.. y.........qy q... ;. y . ..y. y.y.. y. .. g . j .g ;:1EEEii,) . ...( . .! j. . j: .j. .; E,1;rri,k. .)(11(1-,211(). ... g;. q. . ;-y..y.. y. y.y.. ; yyy. y.yy.y.y.. y. y....yq. !....y.y.. j yy...y yyyEg. . -i;::jI8.. . y t 1Ikij:j. j . yy . . . . .j .. . . E :1Ii2q(jI(.. . y E . y 21!j.:11., ' . E E y ... . ' .. . . E. . . j .. i1:j::)p' . j . . . :1E:411j.j11:,k:.r:j;..111.,;,;.. .;i-.. E.. . . y .:11). . .. E. .... . .. E . i .11:22. .. . .. iy . j. . .EjqE.. E.. .E. ...... yq. y. ..E. .... .E ...(. E....q.j . . ..y. .i i. y.j...j .. .. ... . . .. .. . . ... y q. !1i22yj1rr:),. . ..' y . .... q .. . . .. . .. . . y .j q ....... ..q. q . .. ..y.yjjj((;rj,I. .i . ' (jjjjrjjj,E E.. E . .. 4j2@j,rrj. . . . . ki:Ejj:j). . ... ,i::::i:.. 1ik2k2r:4.... yy.. .... .y.y...yy.j.y.y..... .(.y ;y.y.yy... .y.. . .y . . . . E.. . E .:yjjjggjjj-E .j . . jjjyyjjjj,.. ;. ..g yj... j.q.E y.' (j1.yr11:#:,1t,. .. . . . .. . q . y..'.(. . .. .. j.E;.. .. E .. (.. '!.. . .. ..,,j,.y:.. :!Eg;k. . y ..:,jj;;r: . . .. y. !y ( y . .. .y.. q.y..q... . .E..y.yjyEy.qE.....q.y.. r.(.. q :!!k;:. .. yq... ..jjgjgjjjj. y. ... .E . .. . . y (...y.. yyy y j ( .yy . E E y ... . . E .. E E . .E. y (. .. E. .. ... .(.. t .(...((.. :rj,... ( .. y j .. ..y.yj .yj..yE. ..y...j..k.. . . .. g..t . )(( jjkjjrrji (.. . ....jjjgyg:j;. ( .E.. .. . . ( . . .. E . E...(.. . . ! . .) (. ( E.(.. '. . . (j.(. ( ( E gq... . . ..... y k.. . kjj,.,.jjj,j,j(. . E ... y.. . i . . .. y .. jj:ygy:j;. . . . . . q . . . y . k,, . ,),,)),L. . y(. . ' .. . . . . .. .... . .y j;zgggrri . .y .. . . . . ... . ..g . .y j . . . ... .. .. jjjsk,jjj. .kjggyjj.jjjjtj .. . :jjjj,r. jjjyg. y:;kjjr;i . .jjkjjyr;i $,j44,j . (. .y(g..j.Ej(...j..i... . iljjj;i . . .. . il!:z. . . . .,jj;.j. .. ..... '..... .yy.g... .. y y .j y 11E1111).511j::11). 'Ik..J1'. . E ... ' ( r''(' '1Il((1i(1 j jy.. .. .y .. 'Ik.,..;p'. .' gjjys . j.g ( 11.... ... ' 41k::::;: 4I1ii!ii.. E '' . E. . . . .jjktyggjj(. jjjgjgyjjjj,,. . ( . .g . . .y. (' ( Ey . ..... . ( .... .j . .q . . . ( .. g.j. .. y . ... j ... y .... ...(... .. Cjjjgjj;,E.jjyjjgrj,.. . . . j . j. y( ;j(.. E.y.(:.... j E . g...y. j.. E .. r q ( .. .. .g. .y. .. (jjjj,yjj,.. . ... . . .;;jggjj,.. . . .y. yq.jjjgj. . . (y j.jy.j.. ... j.y.yyy. ..y.....j yg..j.yy...jg..y.y.gg g yy.ygj.y.j.q .q.yj.y'..j..gyE...g yE LLLjIj,).L i jjjggj. . g.E. q .. ... q. .q..qirgjyjjjj. .. .. . . . .(..jjggtgy;,. . .y j y . . .. .. . gjj;gg. y... y . . y . yj.. . g . .. .kjygj(jj,. . y . g g.. .. . . . . . . . . . g . . j . ' .. . y. gj y..gg .j.gj g. y gyj . y . y.. .. . ....... .. y .. g . j j... g . .y..g .. . .. y gyj .. .. j.('.('g. . . yjjr::ryqj.. .g . . . . . g . y.g y. g... ...y... . ..gg g. .. yg g .. . . g g y . . . g ..gy j . ...j...j.g g. g g .g g.g y.g. . . . j... ..... .. y.....y. .... .y..y.....y.....y .y.. y ...yy... . y yj.......y.j...y..y. yy . ....y. .y...g. . ....... ........y.y...g. j... j...yg..y..... .. y.y. . ... y. g g j.yyyj .y....y.y.yj..ggyy..y.y..j I:tjll, ..,jjg..,,.,;! q..... . . E. E qg .. (gjjrrr. E . :jIr(:qqjIy:y. q .. . y . q . y.. .. . y .y . j y y y. . .yy.y. . . r .. . . yy...E y. g . y.. . g. . g y.r. .. yy.. . qr. .. . . y... . E E i' .' .y. . . ; y. . .. . y y y .. .. y '. ... .. ... '.. .. .. ... ..' .. . .. .. ''. .' . .. .. '.' . ( . E ' . . ' ' . '' y ' ' ' ' . l . . . . . y .. . . . . . . .. .. . j... ..j ..y.. .y. ..yy. y. y . jg.. y..yy.y.g....y y ....y...gy.. j y. y...yg.j y......... y. gj. .y....y.y...g..... ...j y.. y; y...y...g...ygyjyj... y.y...y..y..y.y..y.g .. . . .. . . . ....... .. . .... . . . .. . . .. .. . . . .... ' .. .... .. ... ... .....''..' ..' . . .. . . .y . g. . . ...j..g.. . g . .. ... . .. ...g.y. .. . -,,': ,gjjggjjjj': .'.$1...11:.*. ,,k,,j,*. j,;,.,k,.'. ,k,' ,jj,,,j'. ,;,' .;;' -,,' .11.*. ..,,j'. ,;,' .y'. ,,gjjj,.I.' ,jjgjy' jky'. ,),jjj,jjj),j)'. .'.112:.241:.* .,,j.,,j,j,,,jj,,,,,,'. ,)jjjjqj4j,.'. '',k:p):k:j.t:-' .,kt!!f!!!j,k).' '' .'-'. ,',...'..-, -.'
k-!ik,,'. :jjjjjjg;yj'g y'.. :'''''-''''.'(jjj(gkj.,kjy'; ''1::;k..' -' -.' jizjgrjgd yyd :yjjj.'. 1,.42)11(41,:(j1:41,1)*. Eti:::jijl.d y'. i' p' r'yygjjjjj;;d kl!jj;;i;j;jp,d ,k2:,,4* ;'t'..11:::*. ''tjilq:z,i.d -111!t)1::4:11)*. ')'-@)t.!!rppt:'. -''' :'.j'. :':.' --jrtr)).'. -' '-;t','y' ..-..yq;' -.ii:i:::'-r:' :(' Ejd -'t'. -'.
For stock indexes containing many stocks, it is common to contlnuous dividends dividend being paid condnuously at a rate that is proportional to the level the model as of the index', i.e., the dividend yield (the annualized dividend payment divided by the stock price) is constant. This is an approximation, but in a large stock index there can be dividend payments on a large proportion of days.o The dividend yield is rioi likely to be sxedin the short nm: When stock prices tise, the dividend yield falls, at least temporarillt Nevertheless, we will assume a constant proportional dividend fieldfor purposes of this discussion. and the anTo model a continuous dividend, suppose that the index price is the dollar dividend over one day nualized daily compounded dividend yield is is xa
'rhen
.
) Daily dlvidend
=
365
x Sz
& 365x7' jr 1+ e 365
=
At the end of F years we have approximately e&T mom shares t.11:11 initially. Now suppose we wish to invest today in order to have one share at time T. We can buye-T shares today. Becaus of dividend reinvestment, at time F, we will have e&T we started wit, so we end up with exactly one shar. Adjusting the more shares t.11:.11 initial quantity in this way in order to offset the effect of income f'rom the asset is called tailing the position. Tailing enables us to offset the effect of continuous dividends. We will encounter the concept of tniling frequently. Since an investment of eMTSZ gives us one share at time F, this is tlze time 0 prepaid forward price for delivery at time F:
/-0z'T
=
s0
.' '
g-JF
' E'
:
.
:
'
i
,
''
.
:
:
'
.
: .
':
y
..--kl)-)::)-'
.
- -
. ..
.)/().
-..
..
- .
:-;
-
. ..
:
'''
''--
... .
.
: .
$ :.. .. . .
.
...
. -
.
..
:
i5.jgkjj :.
'
..
..-
.
..
-
-
:
.
. : . .. . ..
. . . .
:
..... . . ..
-.
. g
)
. ..
.
:
.jj..
:
.. . . ..-- - - . -. . ....
. -
:.
.
-)7t:-. .-.
-
..
;
t( '
:
.' :.
.
.
.
.
.,,
.
. y. .
.
.
.
'
.
.
.
.
.
.
......
. .
:
,jj,.,;;;.
k'i
.. :.. :.y:
.s.;
..
..
.:. y
.
.: .
..
..w
.
.
:
'.
.
.
.
.
:
..
..
....-
..
.jj...4;.
..
.
..
. .
..
-
.
...
..:
..
:
..
...
:: ' .
.
.
..
-..q.
....
.
:..,ji;.;...
..
,;..k;. .
,,kj,
.
.
'
: ' .. .. . jjjt,gjjjj. .
.
. .
:
..
.
.
. . :.
: ' .. . . .
'
.11):2
: . ... . . . . .. . -.-. .. .
:. . . .
.
E
'
-
'
-
E'
.
.
.
-
.
.
.
. . - --.ji....
. ...,,-...
.
: . .. . .
.
-
. : . - . . . :
.
..
. .
.:j
..)
.
:(
;
.
:..
.:
.11:2(
.
.
:: . .
:
,
. .
.
,,,jjjry
.
g.
't
.
.
.
.
.
.
:.
:
(jjgjjjg;,E .. : . : .
. . .
:
: .. . . . :.
y
r
''11k(:::...
.11...:1r..-..;1..
,1I:::j))I
.. .
E
:
.
.
)' )-1* )-*1: )1*
1) ))j. or a
.
.
. .
-
'
y .
.
.
-.
y
y . . ..
,..
..
y. -
.. .
g
.....
..
-
' : ..
.
.
jj,,
'lk,'!k/,.
... .
. .
-
.
,
.
g
. .
:
.
. .
.
..
.
.
g
..jj;......
.
g .g
. . .
,
y
,1../,...
. . .
..
.
:
,.:,..
.
:
.
y
g
:
.
'k..d.. qk..-.
..
.
.
y
g
.. . . . : . -.
:
.
.
.
.
j,.gjjjtyyky,i . . . . . .
. :
. y.
.
. .
.
.
'. . . . .: -. E . . y . .
-
..
dk.,k.:i...
..
.
g
. .
.
g
.y
.
.
.
.
. . g y. .. .
y .
.
-
.
.
.
.
-.
:
-
.
. .
-
. ..
y
.
g
. . y.
-
.
.
,
-
-
.
. . .
.
---
.
.
. . .
-
-.
-
-
-
,
.. y
,
-. ..
-
. -
E. : . .
-y. :
.
.
. . :. :. . . .. . . .. . :y . . .. . . .
. . .
:
.
.
E
.
,jjk...j,,. . .
.
.
.
:
.
.
.
: . .:
.
11.....4,..
E. .
'
..
.
:
EE . .
:
.
..
. . :
E -E
y
.
-.- . . . . .. . :.
:
,,,j..,,,.
,jjr.,kj,
.
' - .-. . . . . . - - .. .
,,,. .
-
.
g
y
g
.
.
.
.
..
.
:
y
:
:
..
.
.
.
.
.
-
.
t..
.
.
. .
.
.
-
'
-
-
.
. . -
..
y : . . .
.
. .
-
.
-
. . .
.
.
. .
.
. . . . . .
.
.
. .
.
..
-
:
.
.. . . .
.
: .. . .
-
.
...
.
.
.
.:
...-
-
E''
'
. .. .
.
: ' .
E.
:
. . -
....
. .
g g. . . . . .
.. g. . . .. . .
:
g
.
.g .
.
:
.
g
.
g
. g. . . .
g
.
.
. .
' .
:
.
'
. . . .. . ' :
E .
' . . .
:
g . . g .. . g. . g. .. . ..
.
g
E. . . . . . ... ..
g
.
.
.
.
.
. g
.
.
;
r
.
.
. r .
..
.. g
.
.) ti /Ei
i'
!(
)? shares. t) ')
0.03
=
lt shares. 'I'he prepaid forward price is lp I)t) $125/ ltt
() q-ytjzjaz
=
y!a) alo
.
- -. . .
-
.
-
. : .
.
: .: . - . .
..
y
:. .
. .. ..
.
.-
-
g
:.
. .
-. . . -
...
.
:
.
-
:
:
.
.
:.
:
y
. . . -
- .
-
..
.
:
.
-
-
:
: ..
.
-.
.
- . . .
:.
.
:
.y.. y
:
:.
:.
y
g
.
.
g
. y. .. r . .
.. . ..
.
.
. . . g
:y .. . .
. g . .
g
.
.
-.
-.
. . g g
(
.. . y . .
. g
g g g . . .
. . . g
.
.
.
,
.
.
gg .
.
, . . . . .
.
y
-
-
.
.
g
.
.
.
.
:
Suppose that the index is $125and the armualized daily compounded p Example 5.2 jj ' d ividend yield is 3%. The daily dollar dividend is y)'t lj'il .l( Dividend tt (0.03 365) x $125 $0.01027 ->
=
6nere is significant seasonality in dividend payments, which can be important in practice. A large numberof U.S. firms pay qumerly dividends in Februmy May,August, and November. Gennan firms, by coneast, pay mmual dividends concentrated in May, June, and July.
CONTRACTS
y .y. y r y yg , .y. y g ..
q.
ON STOCK
Now that we have analyzed prepaid forward contracts, it is easy to derive fomard prices. The only difference between tlle prepaid forward and the forward is the timing of the payment for the stock. Thus, the .(??'u'cI?'# price is just 1/7chttttrevaltte of the prepaid
fonvard. Here are forward prices for the cases we have
considered:
No dividends: Taldng the fumre value of equation (5.1),for the time 0 forward price of a stock tat is delivered at time F, we have F 0.F
=
P'MCFP
0sr
)
=
SverT
:
g.
.
y
-0.a3
5.3 FORWARD
..
: : . .
:.. .
-
-
:
..
: : .
: .
.g g. . .
-
-
-
.... .:y
jjzjjgr:y
g
g
. -
..
y
-
.
-. -. .
-
.
.'
y
.
g
. : . y - .. -
. .
-.
:
.. .. y . y .
.
-
-.
:
.
--
-
.
.. .
-
. . . - -.
. .:. . . . .E - y . .. ..
-
.
-
y
-.
. --
-.
.
. y
y
.
.
-
-. . . . . : y.
;
-
.
.
y
1.0 3:4,,55
=
. .
y. .
,
-0.(s
e
.
g
y .
Thusa if we wish to end the year holding one share, we must invest in
Et
y
.
g
:
-, . .
.y -
-
..
.
e
. y
.
little more than one penny per unit of the indek. jf we start by holding one unit of tj index, the at the end of 1 year we will have t.':
' IE
.
. . .; y . .. .
.
.
.
.
:.
.
:. . : :. : . . . . : .. .. . : y g .. . . .. . . y . . . g .g . y. ; -
-y
;
,
. .
g
. . . ; y.
y.
. . . E - . - . . . . .. . . .
.
:
'.
. . .
.
. : . . ' . : ' '.. . . ' . . .
.
.
-...
y
-,
-.
.
: . . . . . - . .. . . .
.
. y
y
.
. .
. . y y.y
.
-
-
.
-. :.
-
-
..
:
;.
.
.
.
.
.
.
:jjjjgjj,jjjjggj,. . .. .
. .
.
.,
y. .
:
g
y
;
.
: . .
g . . .
.
y
y. .
g
.........y.
.
.
.
. .
.
-
. .. .. .
-
.
y
.
.
.
.
.
-
..
:.
.
.
:. .
.
y
-. .
. y. y
g
g .
.. ..
. .
y
. .
g
.
-
. .
.
.
;
'
. . E. .
. . .
.
.
. -.
.
.
g
..
E . -. .. ..
-
-. .
.
.
g
y. y
...
y.
. .
-
.
-.
y
g
y
g
-. .
E
:. . : ! . . .
:
-
. .
..j(yjjk.,
... .
-
:
-
y.
:
.
.
....
..
y.
; . y g. .y . . . .; ;
.,
.
y
.y. .
g y
.
... .. .. .g .g ...... ..; .
-
,
-
. .
.
.jjrrj)j,
. .
.
Sk.-,.y
y .. . . . . . . . . . . . y.
. y
y
-
.
-
. y . .g gg .
.
...
g
-
..q...j(gjjg.-
.
. .. y
.
.......
y.
. . . . ..g y y y . y ..y. . .y
..
..yy
..
.g...
.
. -. . . .
.
-
. .
.
.
. .....
..
. yy y
-
y
.
Ik..'b..t-j
...
..
...q..y
..:
:
..
-
.
:
.
.
-. - .
-
.. .
-
-
-
.
.
.
.
-
-
. . . . .
,...c
.
,j..,k,.... .
'l..;,.
. .. . :
.. . . . . .
y
.
.
--
y
lji?,.
:.
.
jj.jj.
..
.
.
-
'
.
-
: .
.
. -. .
'...
..g
...
,t)
'
-
,
.
.
.
. . . ..
.
.
.y
.
.
y, : .
.
.
...
.E'
:. . . . . . .
.
. :
'
:
:
.
-
..
....
-
.
. .
.
. .
,
.....
....
:' : :
.
g
-
..
.
-
. .
.
.:. E...EE
. : . . .....: .. ,.....,.. . . ..,3633... . r .
:
.
.
,
.
- -
-
-
-.
..... :
,k,.
.
.
-
..
. -
.
.
.
--
.
..'
'
. ,.. .
,
:jjkty2rr.. .
.
jl;:yr:,)l
.1 '
: .'
:
:
-
-.
. ..
.
..11:7::1k,.
. .
.
y
'
.
:
'''tljlljjy'sljllgrrjl..y
:
:
,k-,i.y
.....-
.
..
llk...llk.. 11.......
.
:
- -
.. ,-..
yjE
::
'
--
. .. .
,;,j.,,j,.
,,;,,$j,..
:
.
' ' . .
:
..
'
.
y
.
.
: . :.
.
..:
..
.
. E
.
: :
: ..
:
.
.
.i
-
-
..;
.
.
. . . E: . . . . :. . : .. . .
.
.
.
-
.
.
..
: . . . . .. . .
E'
.-111,;,..
,k;.
.
. .
411:::.
:
.
..'.
'
:
.
.
,.
-
.....
.(:u.
....
..
:
.
.. . .
-
.
'
.
.
.
..
. : .
:
g. ..
.
y'
,
'
..
.
...q
,:j,k.
..(.
.
.
,
-
-
. .,
.. .
..;tt'I.
.
-,-' .
.
. .. .
.. .' .. . . . . . : y'.(-'...''.'.:y'.('E.'-.y.'.'y.(
.
..
.1k:7'.k..
:.
-
-.;.
.
..
.
.
..
.
.
-.
:..q ' . . : .: : . .. . .. . .. . . .
E . .
. g . . . : .. . . .r . .. . . . g . y. .
yy
:
.
'
.
. .
-
.
:
,jjjjgj
y
.
. . -..
:' .,
:
-. . ' . . ..'
.
'
.
.
..' . -
....
.
:
:
:
-.
-.
-
.
.
..
..
.1!:tjirjr'.
: : :'j
:
.
-
-
-
.
' . . g . y . .
14k...,4,.
.
.
. .
. .
-
. :
.
.
....
....
'j
'. . . j
...
.
.
-
-
..'
..
.
:
.
.
-
.
.. .
...
.
E.
. . . . . :. : . .... . . .
!,
: :
.
.
.
--.
-
..
....
:
:
.
.
. -.
--
...
.
:
.. . . .. , . -..
.
. ..
...
.k.
.
.. ...
-
.
'.'
..
..i
-
...
,;,y.
..
.1r
..
...
. . .
...
'.
. -
.
.
. .
;E
-.'
-
. '!-. - .
.
y . . .. .
. .
,,q,;,E
.itipqkr.ljl,
: -. .
...
.
-
.
: .
:
. -
' '
. .
:
y
.
:
''-k!j!jj:
'
....
.E. ..
..
.
.
.
.
.
-
where is the dividend yield and F te time to maturity of the prepaid folavard ontract.
=
' ' :'
-)
.
.
.
Now suppose t.1). at we reinvest dividends in the index. Because of reinvejtment, after F will have more shares than we started with. Using continuous compounding years we approximate dily compounding, we get to Number of shares
''.' .' .'g )' .'. .''.')k'. .'r .'''''.'-
g
g
. .
134
% FINANCIAL
FORWARDS
Fo RWARD
AN D FUTURES
CONTRACTS
$
O N STOCK
135
Tls formula shows thattheforward coneactis apurchase of the stock, with deferred P ayment. The hlterest adjustment compensates for tat deferral.
For the case of continuous dividends, equation (5.7),the annualized forward premium is simply the difference between the risk-free rate and the divideljd yield.
Discrete dividends: To obtain the forward price for a stock that pays discrete dividendj, we tale the f'uture value of equation (5.3). The forward price is the fumre value of the prepaid forward.
Creating a Synthetic Forward Contract
lt
F 0, T
=
SoerT
-
J''lerT-tbD
ti
i= 1
Whereas for the prepaid fonvard we subtract the present value of dividends frorrt the current stock price, for the fonvardwe subact the fumre value of dividds from the future value of the stock price.
Continuous dividends: future value of equation
A market-maker or arbitragelzr must be able to olset the Iisk of a fonvard contract. It is possible to do this by creating a synthetic fonvard conact to offset a position in the act'ual forward onact. In this discussion we will assume tat dividends are continuous and paid at the rate and hence that equation (5.7)is the appropriate fonvard price. We can then create a , ' synthetic long forward contract by buying the stock and bolwwing to ftmd the position. To see how the synthe:c position works, recall tlzatthe payoff at expiraiion for a long fonvard position on the index is
When the stock pays continuous dividendsj we tale the
Payoff at expiration
(5.4): e rr &e
=
Or
Fo,r
=
,Sze(r-)W
.
zquation
as Forward premium
(5.8)
uv
have forward premium
=
-
1 T
E
Fonvard
.'
gjjyd r' y' yjjy'.lll III;.I.'. r,,1j,,'Ei.ii;,,.' 'y )' k' (tf if 5* ''ii.!if 'if ;r' r'' (4-j.*: rlljjltrqtpjyjy'lqkrq'.ktf :;(ry' t'?'qf )4* .t'. jjjkjjrjjd j': j',')' ,'. y' ::ltIiI'. jjjjygjjr'. )'' :!IiIit)ili.'. )'yq ;'j'. .j' .;j,,.,' g'. .Lt...'. q';jf @if if hif @)' 1* )' (t' 'j'. ;r' IE ' :'' y' y.'. r' 'j' .j' )p' ))' )f jrf t)tf yjf tsf tljly'iljlpfiplEtif'lppiilslrf'lpll'/ii'q (iEi i);i iE j'j';'. ;'jjjjrygjjjjyd y' (' 'tld )'. ii!ji;;yd g'. ('. $' yyf jjrtjjjjiyjjyjf .t'((jyyj)'., rjf !'; q: EifE ' lE k'. jjjjyjy,'. t'.-' i1Ir::'. (111)* E 7! ( EE E' !!' E'li.;' lliE.E E''.E(.E' (.' E.. C.. ;liiE (:ii !q! '7!qiqi . .r'. !! i! j-). iE i. .yjy'E E E .y EE!i E E E ()(.. tk !.tt-;.it))-p)!),.k:t))i,..'.,g));k.!-? j.. .. . :. j.;t!ij;;. .(jrrtrjji ;.!;!.4.jj.k.!.)iy:(.y,(.yjy(. .))y.
'y,' -' -';'F' :'11d1::((q(q:!r'. .t' -y' jj;;f jjj;f '..'(()rj)jl;it!;(ii;ki!!)TrrIE ;'( y' -' ;','y' '':'E )' f' E' ''j'E Et'g y'E (' '$' E' .k' )' y'i ''r'. j.(jj' k' jjf r'
t't'. 't' tjyyt:.' r'EEiIII(E2'!)i1i:F';'. f)Lj?f).)ff'. ';' tt..jjjt'tqlp!j:'. t);j('si',3'.'.(tjjjjjj.j( (()lh..'
ln
Fn w '
Sz
Stock
=
(5.9)
zerp-coupon bond
-
.''ll'r''l'''. .'-'-'q .,..,...' --(jy'j!
!
!' .
'
:
:
' !
(E
' : : E- : ..
.
- ''
-
-
:.
.
i:
'
' -.
''
-
.
-
.
' ';. : ..
. . .
i :. : ..
.
:
: .q7IiE:7q-.. ty ltJ.t-''. . .. ':kt4r,1ik:'. . qLs.k.;..--;.-.L).,--.-L.-;L-...,Lj.k-. . ..--p-,).-.-.:E!!ppk-. .I1EE!2...k-.-T.. 121:k,.. lr...r..,l Ir..,k Ir.. iii'. IEIE::), I:::'lr..Ir..dr:::,. iiIr..ll Ii..1. IiiEl.k .Ir:) .Ik:. ii Ir..ll.Ir:( t!!;: ').br..'.ql.; lgl -
f- -,
.
-
-.
-
-y
..
:
.
.
'!lEE.E.!.-.r
E
.
...,......-..:-....
...,..
-
-t-;.-f.--.k;k-t.t-.--,y..k).i.--..-ir..
..-
,)...-.....--......)....-...--..............-,-
..
-.,-
...
.
d:::)p kjyyyjjj jj,,., Ik..li Ik:::,(!$,/-' jjj,.,yjj a l::::,forward index replicates the payoff to a contract, of the S.t- Foy. .lldi',::)-
.1k:(
:
-. - '
...'.i.. !''. '
E
. .:..
.
E ' E
'
-. .
E.E . E?'... . . .
i
E. E i E E' . E' ': : E. EE E.q.. . .. .( i .. E .E: . . . : .. . .
t@ .. .. -
(:
-
.
.
. .
.
i
-
'
.
-
'
.
-
.
. .
-
.
.-
.
-
orrow
Total
.
:
-
..
-.
.
-
.
(
'.
:
. .
-
.
-
. y.. ..tk. j. ...:y'qjlljr:jll;, .. ....(q!II((Ei(-'...j. (' . .. . ... ...jjjjy j .. .;,jjjjjjj,. . . .. q.;; t. jt; y. tjj:jjjrj. '
.
-
.
.
-
.
- . .jjj,,rjjj,,yjjj. . .
. .
-
.
.
-
.
.
.
.
-
'
'
-
-
units of the index
soe-T
.
-
-
-
. y j . j.. jyy..jy. ..j... . (-' . yq. .r y. y...j. . .j. q. ..r. r.(y. . -.%e-T .. . .
.
e-&T..
.
-
. -.
'
kyjj;;y
:: E . - EE. E- E -. E -,.. -.' E . EE.. -- E :. E. ' . . . . .E . . .. . .y .. E..E. . :: E. . i:rli;:k ; E.. EEE : E.. .. E (I1r(!)Ip.y E EE- E. . . (.yy .. I:tlEjllt. E q . . . .. . q. . y- g yq , : ' q ' .. . r: y. y y : g.. y. : y . i. .. EE. . E.'' :y.:''Ey.y. y ;.y y y.y-yy.y.y . q.y. . jjy . : yEE; : !. . Ey.E: .y. :y.. . . . Ey. . . q.,,,k. y' .. . ' . i .. . .. ... .. .. .. . ( :. E. .j . . . . . .g .. . . .. . . ( y.E'. . j,ryyjjj(yyjI:. .. : : .y i ;.. :.,,ryyjjj(yy!,:. . . . E. . . . : ::.E. : .. . E..: . .: .. . . .... .. . jjj(yjjj;,. ( q.q. :. ... . .. . . E . q. jjgyjj.. . (444,,j:4y4j4444j4,4,,. . .j kjjjgtjjjjyj kjjjj,. ....-tE.. . k y...y:.: y..-;..y..: -y....g.jj.gyy yy. g... .. ... . j. gj..y .y(..gy. .y . . . y. .. j .. . . j . -. .. . . . .y . . . ,.. q;.-..q.- -.. -..-. ...q.!- q....gq-yy.-- yy-qq:.. . . -qy- ..q- yq.. q-.. : . .-.-q. - . y... .-.. .-q-. - qyy..-qy.q.;y. - -qL. .. -..y . - --. . q; q- . -- --. . qg--.g. -qq- --.- . y ... -.. .. ... - ----q, y.. jg.. . .. . .. . ...... . r..... . ... .. .gqy j.y. -.-.- qq.t.. . .. .. . .. .... .... . .. . . . .. . ..... ...... .. . ...... . . . . . .. .. .. .. . . . ... . . . ....... ......... .. .. ....... .. -
-.
.
.
.
.'. ...' ..iii;@)ir.. .,iIii:)ik-. -:'i::pI'''. ' ' i' .. ..r r. Ell,;. .. l .. ,... ,.y . .
Buy
B
.
-
..
d!!!,-
:
-
:. ' E E E: ' : : ' ' iy : '. 'E: . i. . ; 'yE. .. :. :. . i. @ : iE . E . .. y-- E.E.. : y. . !y. q . q g . y- - . yyEy. . -y yy.. ! . .. : . : E.. . ! EE E : .. . .. i (g...g. .: . . . ...g . E..... . . . E .. :. . : . . q . . j . q q E' E i. -- i ..y- ;y. j: ..yy. . Ey . . . . - - - qi - . - (E L?.L..': iL'i '.'. :'' .. il :: '.i .(1. L'j'.. q : .- ... E i.' . ) '!..'.j:..E.l((.(::. L..:... . .; .... k . . . . .. .....
E
.1r:.
d!,'.....'
-i-/-hkd'-
fr::;,.
dt:::,-
e
.. jy. .'. . ..( ( (( ... .... ..g., '.'(; ..(!' '..' ; (.. IEy'..y..(k.y'..(.yq.'..( y..'...jr '. '.. ..... qyjj..j.(..;y..iyy j...y g..yy :'i..'('.;(l'.;E'..1. . ..yyjgj..yj.. y.'.y..(rj. ..(...j. ..'..y-. .; y.'.... ..yjjjryyjjjj. . )...(.. .jy.i(.j.. .q(. .y...( .... .ty ..'. . .i..(. .i .;.. .. .. kkk. . . . (.. q... ! . . . .
We can annualize the forward premium and express it as a percentage, irl which case we Annualized
,synthtically
'
F0,z
=
toEa
(5 g;
It is important to distinguisk between the fonvard #?-cd and the p?z/nfa/l, for a fonvard coneact. Because te fonvard coneact has deferred payment, its nfrftzlp?-'?/7rll?i is zero; it is initially costless. The forward price, however, is the future value of the prepaid forward price. This difference between te fonvard price and premium is in conast with the prepaid forward, for wltich plice and the premium are the same: 'I'he prepaid forward price is the amount you pay today to acquire the asset in the future. Occasionally it is possible to observe the forward price but not the price of the underlying stock or index. For example, the ftltures coneact foi t S&P 50 index ades at times when tlw NYSE is not open, so it is possible to observe the f'utures price but not the stock price. The asset price imptied by the forward pricing fonuulas above is said to dene fair value for the underlying stock or index. (5.7)is used irl tlzis case to infer the value of the index. 'l'he forward preminm is te ratio of the forward price to the spot price, defined
F0 w'
-
In order to obtain this same payoff, we buy a tailed posiion in the stock, investing SOe-&T. This gives us one share at time F. We borrow this amount so that we are not arlything additional at time 0. At fime F we must repay Szer-&sT and we required to pay Table 5.3 demonsates that bon-owing to btty the stock replicates stock Sp. for sell the payoff expiration forwrd contract. te create a forward, we Just as we can use the stock apd borrowing to can also use the fonvard to create synthetic stocks and bonds. Tabl 5.4 demonseates that we can go long a folavard contract and lend te piesent vale ju gol'war tj p yoe to synthetically create the stock. The expiration payoff in this table assumes that equation (5.7) holds. Table 5.5 demonstrates that if we buy the stock and short the folward, we create cash flows like those of a risk-free bond. The rate of rettllnA on tltis synteic constnlction of which is summarized in Table 5.5-is called the implied bond-te repo rate. To summarize, we have shown that
-5z'
Fc, r
S
=
-
'. .
-
.
.
..
-
-
.
.
.
. .. .
.
. .
-
-
.
.
.
-
.
.. ...
.
' . . . . .
.
..-
..
.
.
.
-
-.
.
..
. .
.
' .
.. . . . ' . . .
'
.
-
. ..
-
.
.
. . .
+&c-:T
0
.
.
-
.
-
.
-
.
.
.. ..
... . . . . . . . .
.
. -
.
.
.
-
. . . . . .jjj,,!jjj,,yjj;. . . .
-
.
.
.
. . . .. .
.
. . . . .,
.
.
+&
.. . .
. . . . . .. . .
, ...
.
E
. . .
,.
'
,-vzer-stT ST
-
Sber-ntT
. . .
. ...
.
,.
.
,:gjy;;
jjyyjjyry
.
% FI NANCIAL
136
AN D Fu-ruREs
FORwARDS
FORWARD
.'. '';j;jyk.,;.'. .'..' . . ..'..
CO NTRACTS
$
ON STOCK
137
..-6'3*. .'..-'l,k.......d .q'''k.q'. .--81.*. .k,,.-.2' .--;.-k.* .:11f.:k;,;:-k*. ...,'k'',-,,t-.' -' -.' .,' .'-33b'. '' l.lii.lli.'. -' (),'. ;'..-:,'b..j-' 1.'1..:...*. -;,,.;k;1*. .-11j,;*:111;;;;.*. .-;i.!)?!l..'':'.'' -i.d --;1*. -')t()!.-' -' '-:' ;y.' ''I...lr-'.d 1.,1.....1*. '''' E-' ''I-.'p'''. ;lk;kp,;;d j.-,--.d y' ;'.'E I'E ';'j'5*
()'. iiii;:ll!.d 111.-..111.* 'i,;..11Ir.k-!$..,,kk2' ii'k:llirz'iid y' jr.'E )' (:/' !f !' r' y' .y' j'3''..'':'. ,iIr..1l1i' :ijjji;;d i-d 'f' iIrCiEISX J)IiICZ;PjiX ).;' (' ;'j'q!r?l?r'f il)f )' F' t'i'((' .;' .1111::-!.,*, ''s'. );' :','.'E )' )qf 7*. llEt' )q' 1* j'(p' J' (' (rf
)' p' t'r' ;'j't'. 'y' )' (111.* lqjljlEjk'!;d 'rjII(rqj1j.'. (' y' :kk1:111112.-* 1111E11111.*. 9':*. )y' tttd )' llil!iikd t/'y t'y '' !jly);p!E'lpiir1: @(.(E(. i' ;;.'jil;!'.!ii.qsji.Ei'.. Ei. ((i.'(. (' .('.;E('::4..'( ; E.....('!.E..i'; . j!..q; k' ..k.jy;!(,;j.(gi.:Eyj('jj(jyqjjj(j(!yyyqr L,j)),... . r . .L,L1(bq)2/$Ljq'L j!. (')kt(E :! )). . 7
[email protected](.Iy.E.. ' )j. . )y...t)j()(j))tjt(j l.jlqttj yjj ..(' '!kjjjltrj j.,jggjj ttt .)tqyjy.j jz js.u jgg;jj,.j yjy jggg jjyjjjyj kk).-k.. jgg;j yjgjjyjjjy yjy jjy..jjjjyjjjtg;jj..yjjjjyjkgyjgj:ojg.wkjyj jjjgg;jj..j jjyjj.;j .jggjtg;jj..jjyjg kjjj jjjg;j . .. j.,kjgj -;---q---..C$jk))-. tji j,.kjyj '.ii '.tg). tyx. i...q. ..)!.. ;' i ti ?: ii '.r7':Ii1I7777't(.(;--;::jk(k(q... .. . . . . .y.. r.. the price /kr Szer-nT and lending the ' .. . . .. value present of the forward price creates a synthetic share of the C index at time T.
.*1111:.1--.. .'..' ... . i
/)E E ' !! jpE E ; . E : :'
'
E
-
.
-..'
E
''-
.
E
-
' :' ! E! : EE . .
. . . ..
' :
:
.
.
-
ipll,'L
-.
. . :
'i ' ' '. . EE
:E : : ' . i.
: :jE :E
.
:
.E E ijtt EE .:;. FE
EF
-
-
-.
.
. . -..
'
' i' ..
:
-.
-.
''
. . . .
...
.
:'
.
-
:
:
-
.
.
E. E E.. .
....
-
-.. ..
.
IE
E E(.::E:: : E E ' ' . .. .. . . .- . . E
:
'
. - . . . ... .
.
:
y.j.(.j
-
-
.
i
'
- .E ..
'.-
. .
.
. - - -. .
=
E'E' E' jf (t#' :')'f 'j' ))' ''.'(' f' ' E(E'E kjf kkjjjjjrf jj,,..yjjf kkjjyjjf j;,.,jyyjj,;,,,f k;yjjjjjf jj,,..f (;' pjf .'(' 7''jit'.fi tr* ))t' ;'jjjL'. )') y' i'J''qq5:47:* )' lif l'!i;f 1751'7577777'77:7:'57* lf 17T!2F444,1)* (:E . E jijjtjjt-f );',.jjj-' .gjj::' g' t'y' )1* j'j'(')')jq' C' ).'. l:(! li(!'! ;1! i 'iqiE' i:'(' i (Eli'.(..E('EEi):.EE(.'ji.:iE. iq.:ti ;'':!('(''C'I jjf jjj,.,gjjf jy-yf yrf rrjj;;f j'. qtf y'. jf j'y'r'(jy'k,s :r-rjjjj;rt:'. ;'lf'if .j.))'ty r' )' j-,jj'. jjjjj:'. '.(':': .....E.(.;E.EEEEr.'.'(.' '' E.:.(k-. t't'-jj' yjjy':. y')17* tjf ..( tiE . E!C!E @! iy kjky. ;yj E.. ij.; tyl E E ! . :E q .;)-!(..jjjjj,jkE. ! l.!1y! ... E.....E. (; !;.q .. E Ej j .j E gk jE . . .si . !.jy j ; ; y . y ( . y y yy-. . y.yjjy.. .. y ( r j y yy . y j j . . jjE . .., y j rtyj y. y l . jj;j!j!,jk. .. ;(g, p j j . j.. . i . t)k y y y . . y E .yyjjy;yj y .(y..y....jy ...y. yyyyy jy.yyyk. .E. .ki... ...... y. TE.. . .y. ... y. .jjjy.. .. . . . ... .,
ttyygyy-ytj.;f ttf .'tj);y;;-'-.
.'.'- ..-36tf:3kL,j;. . ... ... .... .',,,,,' ak..jjj..,..jjy,,,f ,';y' .-' -;jgjgjj' jj::rrryf -' --' -q'k;jtj,jj ,;yjjyjj' ,;jjg;jj' ' jj:yyyyyf jj:yyyrj,f j'-.' jj:ryr;j,f II'E jj:yyy:jjf jt:yrgyyf :'.jkjjjjjj;;j,y' ''; tjf f' ' .! -
:.!
E'(E
.'
-
)
-
. .
'))
,r ..(
,
. : .
')..
.
...
: .. .
.
.
..
.
. . : . .
.
y
,.;i::..k--.
'
-. -
.
E.'
.. ..
:
;
:
-
E E. . . . -
.
. . . .. . E. .--
.
-
' !
. .. E: .
...... .
y -.
,
-
.
.
-
,
. . .
-
''ikkiii:.
..z'li....
-
:....
..........
.. .
' . . . . . . . .. .
.. .. -. . '
: .. .. . . .
. E ..
.
,..,...:....
......:..............
'
-
- .
-;'!:,.-
. . . . .
.
. .
: .- . . .. . . . .. ..
..
.
.
: x'Il-,.j.'.,tpll. .' . ..
'
'
: : :.. .:..,................:..:... .......:...'....
.
.
:. .
-
-
..
.
.
. . . .
EE. .
.
-
... .
., . . ; . . . .... .
y -'i1
'
. -.
. . . .. . . . .
.: ..
. ,,
.
--
''
'
-
-
..
E -.-
Fo,r
-
.
.- . .
. . . . ..
.... ....
!
.
'
t).tjjj
.. -
-.y
-
,kyykjjj..
,
- . -. . .
.
. .
..
-
.
.
,-;
,, .
:. .
-.
.
.
-
..
-
(:jj;;y
jj,,,,yjj
,:gjj;; ,jjj,,,,jk-j,,.
..
tE, ( ( '.: : E ' E . . . .. : . ( . .. - ... . .. .. ) .. . - . E.. .
E
..i
y E :E .-
irli
..jj:::
jj,,,,jj,,,,,.
:
....
.
.
..
:
..
..'
:
.
.
.(. tyj
.)-
..
...... .
.
..... -. ..
..
....
-..
..
. .
..
..
.
j.
. .. . . .. . . .... .
-
,
).E
(
E
gyy..jsj.j.
...j
-..y.......,.y,(.y -.!.y. ...y......
-....
tj
:
E
t ..g..... gk
...
.....
.
.... ... . . . .
..
j
. ....r.(.
.
-.
..
ur
.. ... ..
....
. .
+Sv
-JF Borfow s ()d
+&dtr-d)F
..
+s
e-&T
,
-s
()
0
..
tr;j!y-,. .
;
.
.. - .. . . . . . .... . . . . . . ... . .. . .
t ),
,
.
y.
-..(....y...
...
Buy tailed position in stock, paying
.
jj,,., ::: tjjjjr!jjjjj
kyjjgjjj kkyjjjjj jj..,,yjj
jjr:rgg)j-
market-maker is short a fofward contract and Iong a synthetic forward contract.
.,
))
.tt p ilisiilk1:ii:' tl'::l' .-).1* ''
jj::yyyjjr
jjj trjjj;y .jjyyy. jj,,..yjj
,
y
,
--,
-.
E
-
E E EEl) E il (EE E E iE yrgqtgE E) EE j ) q E y q ytg j y r E( y ( Ey E( E yg qE j,q iq!g jtjyu yyjqyy y(g ;yj( )y q (( q ; j! i jqEr q ! i ( yq y;y ) E y ( E i i.y..j.y.gj...yg.......j.yk.y......j.q.j..g.. y E ); q) Et y ..(... ( . 'y ( .' .'..'...' .: ...j ..g yy . j ) y . y jl y .. . . .. . g''. . .. E... . . jj...y.jj.sg . . . -..-..).(-. y... ..jr jj .j..y j ....).....y.. g. . . g ...' jj jys.sysa..j )- j.)..)...; . .j,jjr ..-.-...-. (.y.-,.t.. ,.....yt. ..,jrj .gjirjlg.y;)j .-.,. 412j,: . .-. ..y yj ...q .. ..ks.j ...-.......y ja... . y.,jt(j)-. 41y2j, :1,44:)2j14yj, . . .-...)... . . .jji.,ir-kl!iii' 4(:y il!:tiL . j,gg,i .-...' ''ElIIE??''i7' ...-..y..-.;. ;...- ,rq,j(q,-j;ky)j:rjjr#,,, j;.j,t:jliyj,t ,-rj1Iyy.yy .. i't:l'k ;;.... kjty,j .- . .y, ...q.. yy.. .y. ...-... .y.y (;k. '... ii!7l''ii;: .....t't .'.-.-.i i t... ......-... .....y.. .......-..... .).. - ... ......-..... y.y .-. .... ..... .........y ... ... y .... . ..-. ..... ... . ... ... . . .... . . .. . . ...y .. . -%e-&T %e-&T
(E
ryy
E
.
j::!iik:jg.jjj;:rliyd jjy'ttjjjjf j'(' tff ;'.j' g)'. yy'y' )y-'. )' y' ytyf t:ijijijj)d t'jy)yqtf rjf tjjjjrrjjjjy'. (r)' t)).li.li/t'i-')-77Iirt7( yjj r t.. .. j! .jrjjttj. rii!L2L--. . .!.;#-.-)....ktj-.-;-)i-;t.. j. . . ryy-)t-r).. yj . )j):t?i.ljjty g.2..(.-.r-......--.-..t.......--r.. tj tjyyyj . d!!!. '.r'' I..,. ijtil:-. Ir..,. Ir..,l.1k:( Ik..l. lr.. IEEE::), lr,. s d:::), iit'.'jt.....-,;-....-..-.,. i,i,. Ir..,. Ii..1l iidr::)p iilr..,l .Ig:' 1/..1. -Ir:. ls -..--(L(LL,-(L. tp. . a.Ir:I::),. e 1r..1:..1. s.1r:. (!j-,-. a .Ir:. IEII s iEi,l e iiIr..l e :)-4:: !t).-. 4444444# .
.
. .
Lt )(t()
(y l ( t
i
-y -y -. . y
:-.
'7q:t'
' :
.
-
., .
. ..
1. .t.
.,l'k..l''...
:
-
.. . .
.
.
E '' :
-
-. .. . - .,, . . . . ...
.
.. .,.,..
-
.
'
.
l''-'.'-',
. . . . . . , . . . . . . .. . . . , . .
Sv
..
.
'
, E- -i,;bi.': . . .. .. '''k'''''.
'''-.
: : .
.
.
.
.
.
.
:.(y.y
:(..::
.
'
'
-SZe-&T
Llijjrtilllyd .111r..111,..11,*. I'rIIIIIIIT'tjIII:j-;'E y' r' ((' j'jIyy4j.'E g' jyjf qqf q' j)' k' y)' j':f 111It:(gIli.' j:;jjiI(y(j11rj:iI1.'. .t'( y'. y' )f r' ($' 'r' (' (r(' t'j;f )' jtf q' lijilqtfljqkqjyrrjjjjljplrjg't'iri jf i E):(('tqjE j';'i'yt'. jy,k('.y(t'. (y'. rqqrl'rqqrql':'lq:jrrrr:rrq!rrrlqqprq')!qj)rf ' ! ! yjl iI.('y:E'.;(qi E' !Ej(E'' E (trjjj ( '' 'Et:t)i! E@ ;!l('j('gj'k ).E.)! 'jI(i;!Eg'.@ ' . ((!g(.;j.qli((' j. '(' .E (q..i'q(!' ( ( .j((()! ; jE E i; ! g (; .!!! . :E i.; ''y E;.i ; .E i E i.. g; iyqy j jjE-jgg ...ryj. yy y y.y .j y.. . y! rt( . ktj.! lr,-))--,-)t?...-j y .y j .jjyyyyj ; y i..7j..r@t-qr).-y-.i.ptj..
Sj:
-
-,%d-:?-
Total
'y
-
0
Long one folavard Lend Sze-tT
EqE :((.
E
..
.
:
-
. .
''
.
-jj(.
.
)E (iE EEjE':(l .Fi .E. .!i.y.. .E .!.)( E .. ;' '!. E E(E(L'l E 11ij::j,;-. (..:(j( .(.( E: .!((:E( .;.' .j:. .'.. .;(( q.. j'EE Er (. .() E(.1.) (:(. .E : .!!.;i ;:,iriiI);jijjj;: j:' p ) (E7i(((E.EEC(E(g.r E (. .((j . i. .) E(: E 'E(. j' EE.r .E(E( Ey E j..:j:.!) ..E .' . E.ig .j...(( E EE E.. E (E-; ( t t E. ..... . . . .(.EE..((g.j .. q tI1I:()q((q:!). . i . ...j E.. '.y...qy.r.'E.... .... .q'y.('!EE . .y..y..yi.. (E.. q)l.E q. yj .. y tEi g ...'E.' ! .'r.y.q.. ..(E.((.qtEq...(y...y.(j.j......j.yj. y.....g..yy..j.y..y...yE.....y....y-....yE...j....yjyyyj y.. ... q. y.r. .(.. yy.... ..y.. y. ... E' ..yy.E. y.. ... y Er'( ...;'Ej'y. qy.. . .E.y . . y.qit ,E . . . . j . .. g . ' . . .( .E . . ( . . E.y . . ( . ()!(E) . ) l 'q ',...iII;..i,.'E'' . kkkj(,k;... ; . .'t ((. . ; )L---,,,-k.E. EE ) . . . EEE'EE.i.. '..(y ). '.,1;,,,41,-. l r E...yE'..!iy . x. E E .lll'..lpr..rppl. .k-k,,L-k.(kL,.k;. . . ; '(E. '.E ) . .. . t,j14,.kk,.' j;k';i. i' ' (..... illk..kll' ....E.. .( .. ... ... .. . . . . .Eii;:lll'. '' -.E. ..qq! . EE q.......q. .. E..q.EE.E... . . . .,3''. lllk.:z:E .'Ilq..illt..: ll'..'.pl'... .111...111.::2!!:1.111...111.. E . E...E.,.:...E'EtE.. .. . . E.'......E... ...:..............:.................:.E(.',E . . q.qq. q q. .k. ... .y..q 'E').''...'' 'E ..1111....1112.'.. q q. (...r .. . i ; ...;'dk.,.2g;. . ' )..()('. .. .. .. . r.. .r.. ..... .. . . ...... ... ... .. .. . ... .. . ... . . -'
:
'
' ' : :. ' :
E
.
-. .
. . . .
. .. .. . . ...
..
.
-
..
.... . . . ..
.-
........
.....-.-...-
.. .
.-
..
'
e(r-J)r
Short forward
0
F(),w
Sv
TOtal
0
F0 z.
Szec-ntT
=
-
,
,'!kitjjjj:' ;''',ij:j:ir)' 1,:4:111((4!.* jjjrjyjjj;'. y' F' 'j(' j':'lrf E' (' ;' 111j22,,,* .t' y' t')' (j'E q'
'.(' .111r..111..:114* qtkjkyk.,;jj;j;,d
.'yj'
'
'
:
:
.
-
E
.
:
-
-
....-147111-..
:
'
E
E
''
.
g
: -. -
.
E . :'
i
:.
:
:.--'y.t)-....)'-..-,''-,)-t?.?,.
.
-
-
-
-
,
: .
.
.-).
....
.
y
.
.........
.,
.
-..
..
''-
......
,t:::,-
......,....;
-.
,(:::11
dlsi,t
4:::j1
. .
a forward creates a synthetic bond.
shorting
E ' i E .E .E .E (y ( .E. ( q ! gEq (qqE E.q grjjjj;rjjjjjjtjjjgk!l ( .y (..i'.'.E.) j ... (. ..(.. ((E!. q..;.. .iq. .)'.)'E j(E ..y(.jryy.(yq.(y.jE.jy.....Eyyy.j.y;yjE(.y)yj. ;. ...(ygj .jyE(.jj. . j .jy.'y.. ). . j'; q q. . E . ... . E jq (.y i(iE.'...E.'...,. . . q.E#i)(..E,!..... '.r..(.)..)(..E, g :;jjy;jjj.. jjjjjjj;. j .. (;Ei...... ..yt.tEE..ti....r ;.. ...( jgq.. (i... ; . .. j . , g. y ..)E.... j. ... ......).... jjg).),... . jk ... . .. .yy .j...(..iE.(......E.E.;.',
[email protected]., . j..jr y ....)..:; .. y . . ..... . i.. .j.r.... . .EiEi ..j.,'..t... EIt E.jg ... ..;. j.'E.E.) j,F.y..) y;E y.t.y jy y. . .. . .. ... jjjjj, . EEyr: .)i..)) . yyE.g.g.). .. j.y..j.yjjg EEIE. . (.,..' (IE (.'i (E.E . ...j..(..y.gj..g.. . E( . Ei .(g . .. . Ey ( E . . yl yi ( y.(g . q ( ;( . Iilii;;j iy r . E '. y ; ( y E ( i .jl( g y ; lillqlj))l jyjgtjyygj j ( j y j y E qjiit. : y i . y y 414:r.. i. t kjiiji!jl ! yjy kilz:j!:.rjllrjrlil). .. rjjirgrijj,. yjljrjj,. . qE. ( E . . .( . . .. ( (;y ! g r .111, ....?fjk6??;',. ( tjj:j)i;l y...:..E .. !..(j..y.-)..(E.'yy.'(:.E) ...... .k::jjill, .r:....:............j...(. . . L......x.... ..r'....'..k..(......'....E.y....'L....l. (..'. .).. E.(.....'....';. . .().... )(.':..'E j.'....' .......E ...E.''....(y..y....y...... .111- ..y....(..: .... ..r ..... .:E.).. ..L r:.. .i(.. r.. ( . .. . .. . .... ......... .-soe-T e-&T uriits of the index :
:
.
:-
.
. .
.
-
-
..
EE :
:
-. '
'
-
. .
i
.. ..: :
:.
..
..
.
,q.i:
:. . .
.. :
:):t
E
:
.
. ;. . ..)) . . ,
.. .
..'
'.
.
.
.
.
; :
: . ..
.
: ..
. .. ..
:
.
:
:.
: . ...
...
....
. .
.. .
..
: ..
.
'E
..':
y((
. .
. . .. . .
:
.
:
-
-
-
'
'.
:
.
: .
.
-
.
.
.
: E :.
: ..
: .kjjj:yjjjjg:!E. .
:
.........,........)... -
:
.
. .
:'
..
'..
-
.
: ... .
,
.
..
.
.
... ' :.
..
........
.
.: ..E... .. . ..
. ...:
....
:..
E
.
....
. .
.
-
:
.
:
.
E .. .
:
:
-
.
:
.
:
:g
-
-
:
:
..
'
.....: .... . ...: . . .:. ...... . . .
...
...
.
..
..
..
:.y)
..'...
..;.:: ....... :.................... ...
....
.
...
.
(j
shor t one forward Total
. y
.
.
. yy . . :y
-
.
:
-
.
,y
'q:jpi
.
,...''..
..........L..L.L..L... :: ..r
Fa v
-%e-T
-
:
: .,.
....
,...
...:..... .....
.
......,..
....r...
..
...
.:.
. -. -.
-
....
'
+sv
Btzy
E
-
-
tgj.
E y . .. . . .... . . .. .L..j.L... :. .. .;.
.....
,
-
t'-';'jjtryjd ,,yyjjjjjjy,.' ,,,yyjjj(y),,' '':)' jjjtytjjj;d )'?''''t!rr'q?:;q:'q!f iiEp7f !f pf E E E E' 5*
)' :,:1111:)(111(r1111)*. .y' C' )t)' ' EE li 1. FE EE'E''''' E 7 E ' ' i' E''E ;ii;r'' yy:jjkkj:yjjjjtj;jjjjd .yjjy,,jjj,.;jjy'. yjjr'---. y' EE7 i iqE r' p' If )' k!if lC' i'if (,' lllkjriij)d jjjjgjjjj'. $jjrjyrj,'. kjjyjj!,jd 'j' qqqq!rl'ltr::r'l q'E''7' 'E;!i! Jl (q( jjjj.d r' E E l' .q!'il' ;@' E EE i!i 7)771! 1):1@ . . 'E. E... '. !'E. Ey;-!-(..j..)j:.jq-.;y)7rit.!.jyytyyjjEy.j-yyyyty,yt.'jyy ..' rjjjyjjjjjyrjjjjjd ::j)?:;..tt'. i.i1jjgjjjd t'q'j'('t)'?'qq!f rjtf . tj.jjyy.y..E.jiy.,Ey:yry.).-y..;-...-...-j,..-E.g.y(yj.jy.E.tEy.g-jy.yj;j..jygy.y.jy.y.y).jyyj l ('!iii .i i ':'(q. .r 7r:' ..' ) E.EjqEt..E( .)itEjt.j) LIIZ'LLLLL i ;!lt( (y,I )...ly )i'EjI;ICEIjIEi .)(jt(').. ..t .. .'1.--.;'6b.....-........ jtygjjy .....-..Iri!iC,.......... -'...i1iEdtjgy:gy ('..'.--t'..:!!Eii;...-..-r-'.. .'....iiI-. jyjjjs gjj;ikgjjjj jr.,;j jtyyyy )'..-'4C'lkk..-.,.-. jj..;j .jg: jjjryg;j .jyr jr.,;j kjg;j jjjgyy;j jr..yj tjj;y L-...k-',.;k'... tjj;s y-(--. i,-...LL-L-. );-.:.(k....)kjgjj ..2kr.(22---)..
.'.' ..' ,''-'
llpiE
:
'
'
:
.
..r
.
'-c
ijjE ' E' . .
'
:
' '
'
. :
.
:: :
-
'
........i..
-
=
:
. E E. .: .
'
:
' :
:
:
.
.
11r. . .
E
y..
E
g
.
..
..
..
p.
. .
...
........
.
...
..
.... ... ......
..
...
Zero-coupon bond
=
Stock
-
.. : : y. y . . g . E -. .. .. ..
:
.
...z. :. .
Lend Sze
Total
Forward + zero-coupon bond forward
A1l of these synthetic positions can be reversed to create syntheic short positions.
.:..
...
.
:
y :
.
Now we will see how market-makers and arbitrageurs use these seategies. Suppose a customer wishes to enter into a long forward posidon. The market-maler, as the counterparty, is lef4 holding a short forward position. He can offset this risk by creating a synthetic long fonvard position. Speci:cally, consider the ansactions and cash Eows i.nTable 5.6. The marketmaker is short a forward conact and lng a synthetic forward contract, constnlcted as Sfjer-&bin Table 5.3. There is no tisk because the total cash llow at time T is F0,r A1l of the components of this cash flow-te forward price, the stock price, the interest rate, and the dividend yield-are known at time 0. The result is a risk-f'ree position. -
yjjsjjjj
jjjjj-
,,,,
:
-
......
:
('
E-
.
-
-
.
E
...
.
.
.
.
.
.
..
.. .....
. .
-JT
,..,
'
:
