Zklfk Vkdsh Iru Wkh Frvw Fxuyh ri UlvnB
Orxlv Hhfnkrxgw
Fdwkrolf Xqlyhuvlw| ri Prqv
Fkulvwldq Jroolhu
JUHPDT dqg LGHL/ Xqlyhuvlw| ri Wrxorxvh/ dqg LXI1
Dsulo 4<<;
Devwudfw
Lw lv riwhq vxjjhvwhg wkdw wkh odujhu wkh vl}h ri ulvn/ wkh odujhu rxu
zloolqjqhvv wr sd| +ZWS, iru d jlyhq uhgxfwlrq ri wklv ulvn1 Zh vkrz
wkdw wklv lv qrw wuxh lq jhqhudo lq wkh h{shfwhg xwlolw| prgho1 Zh h{0
dplqh xqghu zklfk frqglwlrqv wkh ZWS iru d pdujlqdo uhgxfwlrq lq
wkh vl}h ri ulvn lv lqfuhdvlqj1 Zh dovr h{dplqh wkh forvho| uhodwhg txhv0
wlrq ri zkhwkhu wkh ulvn suhplxp lv vxshudgglwlyh lq wkh vl}h ri ulvn1
Glhuhqw qhfhvvdu| dqg2ru vx!flhqw frqglwlrqv duh surylghg zkhq wkh
glhuhqw xqlwv ri ulvnv eruqh e| wkh djhqw duh hlwkhu shuihfwo| fruuh0
odwhg ru lqghshqghqw1
Nh|zrugv= zloolqjqhvv wr sd|/ ulvn suhplxp/ elg sulfh/ ulvn yxoqhu0
delolw|/ surshuqhvv1
4
4
Lqwurgxfwlrq
Lq plfurhfrqrplf wkhru| xqghu fhuwdlqw|/ d orw ri dwwhqwlrq kdv ehhq sdlg
wr wkh vkdsh ri wkh frvw fxuyhv1 Iru lqvwdqfh/ wkh surshuwlhv ri wkhvh fxuyhv
duh xvhg wr ghwhfw wkh suhvhqfh ri lqfuhdvlqj ru ghfuhdvlqj uhwxuqv wr vfdoh
lq surgxfwlrq1 Ehvlghv/ wkh dqdo|vlv ri d frpshwlwlyh htxloleulxp dqg lwv
surshuwlhv lq whupv ri zhoiduh dovr fulwlfdoo| ghshqg xsrq wkh vkdsh ri wkh
dyhudjh dqg pdujlqdo frvw fxuyhv1
Lq dq xqfhuwdlq hqylurqphqw zlwk ulvn0dyhuvh ghflvlrq pdnhuv/ ulvn uhsuh0
vhqwv d frvw dqg pdq| dfwlylwlhv duh ghyhorshg wr wudqvihu dqg2ru wudqvirup
wkhvh ulvnv1 Iru lqvwdqfh/ lqvxudqfh frpsdqlhv dffxpxodwh ulvnv lqlwldoo| khog
e| lqglylgxdov zkr ghfuhdvh wkh txdqwlw| ri ulvn wkh| idfh1 Wr hydoxdwh wkh
jdlq lq zhoiduh dvvrfldwhg zlwk vxfk d wudqvihu rqh qhhgv wr nqrz krz wkh
frvw ri ulvn fkdqjhv iru hdfk ghflvlrq pdnhu dv d ixqfwlrq ri wkh qxpehu ri
ulvn h{srvxuhv kh srvvhvvhv 1
Vxusulvlqjo| hqrxjk/ qrw pxfk dwwhqwlrq kdv ehhq sdlg wr vxfk txhvwlrqv1
Lq wkhlu slrqhhulqj sdshuv N1 Duurz +4<:4,/ M1 Sudww +4<97, dqg L1 Od Ydooìh
+4<9;, ghqhg wkh frvw ri ulvn iru rqh xqlw ri d orwwhu| wr eh sod|hg rqo| rqfh
dqg wkh| frpsduhg wklv ulvn| vlwxdwlrq wr d vxuh rqh lq rughu wr phdvxuh
wkh frvw ri ulvn1 Krzhyhu prvw riwhq shrsoh wudgh d ulvn| vlwxdwlrq djdlqvw
dqrwkhu rqh dqg lw lv wkhq lqwhuhvwlqj wr nqrz krz wkh frvw ri ulvn lv prglhg
e| wklv h{fkdqjh1 Zkloh wkh ulvn suhplxp lv ghqhg iru joredo fkdqjhv lq
ulvn +ulvn yhuvxv fhuwdlqw|, dwwhqwlrq lq frvw ehqhw dqdo|vlv lv riwhq sdlg wr
wkh ydoxh ru frvw ri vpdoo fkdqjhv lq ulvn1 Lqwxlwlrq vxjjhvwv wkdw iru ulvn0
dyhuvh ghflvlrq pdnhuv wkh odujhu wkh lqlwldo ulvn wkh odujhu vkrxog eh wkh
zloolqjqhvv wr sd| +ZWS, iru d pdujlqdo uhgxfwlrq ri wkh ulvn1 Zh vkrz
wkdw ulvn dyhuvlrq lv qrw vx!flhqw wr jhqhudwh wkh uhvxow dqg wkdw dgglwlrqdo
frqglwlrqv pxvw eh lpsrvhg hlwkhu rq suhihuhqfhv ru rq wkh glvwulexwlrq ri
wkh ulvn1
Wkh lpsdfw ri ulvn wudqvirupdwlrqv rq zhoiduh lv ri frxuvh ghshqghqw xsrq
wkh lqwhuuhodwlrqvkls ehwzhhq wkh ulvnv lq wkh ghflvlrq pdnhu*v sruwirolr1 Wklv
lv wkh uhdvrq zk| zh zloo dozd|v frqwudvw wzr vlwxdwlrqv1 Lq wkh uvw rqh +wkh
pxowlsolfdwlyh prgho, wkh ulvn| h{srvxuhv lqlwldoo| khog zloo eh dvvxphg
shuihfwo| fruuhodwhg zkloh lq wkh vhfrqg rqh +wkh dgglwlyh prgho, wkh| zloo
eh wuhdwhg dv lqghshqghqw1 Wkh uvw vlwxdwlrq lv jhqhudwhg h1j1 e| rzqlqj q
xqlwv ri d ulvn| vwrfn zkloh wkh vhfrqg rqh fdq eh looxvwudwhg e| wkh fdvh ri
dq lqvxudqfh frpsdq| slolqj xs lqghshqghqw dfflghqwv ulvnv1
Wr looxvwudwh/ frqvlghu dq djhqw zkr krogv ? xqlwv ri shuihfwo| fruuhodwhg
5
ulvnv1 Ohw xv phdvxuh wkh wrwdo frvw ri wklv ulvn0ehdulqj e| wkh ulvn suhplxp
dwwdfkhg wr wkh ulvn| vlwxdwlrq1 Lv wklv wrwdo frvw frqfdyh ru frqyh{ zlwk ?B Li
zh wdnh wkh Duurz0Sudww dssur{lpdwlrq dw idfh ydoxh/ zh zrxog vd| wkdw wkh
ulvn suhplxp d prqhwdu| hydoxdwlrq ri wkh frvw ri ulvn lv d ixqfwlrq ri ?2 /
zklfk lv frqyh{ Lq wklv vhqvh/ pxowlsoh ulvnv duh vhoi0djjudydwlqj1 Wkh ulvn
suhplxp iru wzlfh wkh ulvn htxdov irxu wlphv wkh suhplxp iru wkh lqlwldo ulvn1
Vhjdo dqg Vslydn +4<<3, uhihu wr wkh qrwlrq ri vhfrqg0rughu ulvn dyhuvlrq
zkhq wkh ulvn suhplxp jrhv wr }hur dv wkh vtxduh ri wkh vl}h ri wkh ulvn1
Vhfrqg0rughu ulvn dyhuvlrq/ d ihdwxuh ri wkh h{shfwhg xwlolw| prgho zlwk d
glhuhqwldeoh xwlolw| ixqfwlrq/ lv dq looxvwudwlrq ri wkh lghd wkdw wkh frvw ri
ulvn vkrxog eh vxshudgglwlyh= wkh frvw ri wkh ulvn ri vl}h ? n ?2 vkrxog eh
odujhu wkdq wkh vxp ri wkh frvw ri ulvn ri vl}h ? dqg wkh frvw ri ulvn ri vl}h ?2 1
Krzhyhu/ zh vkrz lq wklv sdshu wkdw wkh vxshudgglwlylw| ri wkh ulvn suhplxp
uhtxluhv vrph dgglwlrqdo dvvxpswlrqv rq suhihuhqfhv wr krog1
Wkh sureohp lv glhuhqw li zh frqvlghu wkh dffxpxodwlrq ri lqghshqghqw
ulvnv1 Li zh wdnh djdlq wkh Duurz0Sudww dssur{lpdwlrq dw idfh ydoxh/ wkh ulvn
suhplxp iru wzr lqghshqghqw ulvnv zrxog htxdo wkh vxp ri wkh ulvn suhpld
ri wkh wzr ulvnv wdnhq lq lvrodwlrq1 Wkhuh zrxog eh qr hfrqrp| ri vfdoh
iru lqghshqghqw ulvnv/ dv h{sodlqhg e| Vdpxhovrq +4<96,1 Djdlq/ wklv lv qrw
qhfhvvdulo| wuxh iru odujh lqghshqghqw ulvnv1
Rxu sdshu lv rujdql}hg dv iroorzv1 Wkh edvlf frqfhswv dqg wkh qrwdwlrq
duh lqwurgxfhg lq wkh qh{w vhfwlrq1 Lq wkh iroorzlqj wzr vhfwlrqv/ zh glvfxvv
wkh surshuwlhv ri wkh ulvn suhplxp dqg ri wkh frpshqvdwlqj suhplxp zkhq
wkh qxpehu ri ulvn h{srvxuh fkdqjhv1 Wkh iwk vhfwlrq lv ghyrwhg wr d vlplodu
dqdo|vlv dssolhg wr wkh zloolqjqhvv wr sd| iru pdujlqdo fkdqjhv lq ulvn1 Lq wkh
odvw vhfwlrq zh zrqghu li wkh ulvn suhplxp sdlg iru holplqdwlqj d ulvn fdq eh
h{wudsrodwhg iurp wkh zloolqjqhvv wr sd| wr pdujlqdoo| prgli| lw1 Wkh pdlq
uhvxowv duh wkhq vxppdul}hg lq d vkruw frqfoxvlrq1
5
Wkh frqfhswv
Zh frqvlghu d vhtxhqfh ri udqgrp yduldeohvih
+jMl / zkhuh l lv vrph lqgh{
vhw1 Djhqwv duh h{shfwhg xwlolw| pd{lpl}huv zlwk d wzlfh glhuhqwldeoh/ lq0
fuhdvlqj dqg frqfdyh xwlolw| ixqfwlrq 1 Wkh| kdyh dq lqlwldo zhdowk f Iru
dq| frxsoh E?c ? 5 l2 c zh ghqh Ef c ?c ? dv wkh zloolqjqhvv wr sd| ri wkh
djhqw zlwk zhdowk f wr uhsodfh ulvn +h? e| ulvn +h? G
6
.Ef n +h? ' .Ef n +h? Ef c ?c ?
Wzr ihdwxuhv ri wkhvh ixqfwlrqv kdyh ehhq h{dplqhg e| Sudww +4<97,1 Wkh
frpshqvdwlqj suhplxp iru +h? lv ghqhg dv REf c ? ' Ef c fc ?/ l1h1/
Ef ' .Ef n h
+? n REf c ? +4,
Wkh frpshqvdwlqj suhplxp phdvxuhv wkh sulfh wr eh sdlg wr frpshqvdwh iru
wkh ehdulqj ri ulvn h
+? zkhq lqlwldoo| wkh djhqw lv xqghu fhuwdlqw|1
Dqrwkhu frqfhsw lqwurgxfhg e| Sudww lv wkh Fhuwdlqw| Htxlydohqw iru ulvn
+? ' f/ zh rewdlq ZE? '
+h? zklfk lv ghqhg e| Ef c ?c f1 Zkhq .h
Ef c ?c f zklfk lv wkh ulvn suhplxp iru ulvn +h? Lw lv ghqhg dv
.Ef n +h? ' Ef ZEf c ?
+5,
ZE? lv wkh dprxqw wkdw wkh djhqw lv uhdg| wr sd| wr jhw ulg ri ulvn +h? 1 Lq
jhqhudo/ ZEf c ? glhuv iurp REf c ?/ dv vkrzq e| OdYdooìh +4<9;,1
Zh qrz lqwurgxfh pruh vwuxfwxuh wr wkh vhtxhqfh ih
+ jMl Zh zdqw ?
wr uhsuhvhqw wkh vl}h ri wkh ulvn h
+? 1 Wzr qdwxudo rughuhg vwuxfwxuhv duh
frqvlghuhg lq wklv sdshu1 Wkh vlpsohvw rqh lv d pxowlsolfdwlyh prgho zkhuh
+h?6 ' ?h
% iru vrph udqgrp yduldeoh %
h dqg ? : f Wkh rwkhu prgho lv dgglwlyh1
Pruh vshflfdoo|/ frqvlghu d vhtxhqfh ri udqgrp yduldeohv %
h c %
h2 c wkdw duh
lqghshqghqw dqg lghqwlfdoo| glvwulexwhg dv udqgrp yduldeoh %
h Zh ghqh +h@?
dv wkh vxp ri wkh ? uvw udqgrp yduldeohv ri wklv qhz vhtxhqfh=
+h?@
'
?
[
%
h iru ? ' c 2c '
Zh ghqh E6 c R6 c Z6 dqg E@ c R@ c Z@ dv deryh uhvshfwlyho| iru wkh pxowl0
solfdwlyh prgho dqg wkh dgglwlyh prgho1 Lq wkh uhpdlqlqj ri wkh sdshu/ wkh
devhqfh ri d vxshuvfulsw @ ru 6 wr d ixqfwlrq phdqv wkdw zh uhihu wr wkh wzr
prghov dw wkh vdph wlph1
Zlwk rxu wzr vshflfdwlrqv ri vhtxhqfh ih
+ jMl c zh fdq dovr h{dplqh
wkh zloolqjqhvv wr sd| iru d pdujlqdo uhgxfwlrq lq wkh vl}h ri wkh ulvn1 Lq
wkh pxowlsolfdwlyh prgho/ ehfdxvh l lv wkh vhw ri uhdo qxpehuv/ zh fdq xvh
glhuhqwldo fdofxoxv wr ghqh wkh zloolqjqhvv wr sd| +ZWS, iru d pdujlqdo
uhgxfwlrq lq wkh vl}h ri wkh ulvn dv
7
Y6 .h
%Ef n ?h
%
`A Ef c ? '
'
Y? ?'?
. Ef n ?h
%
6
+6,
Lq wkh dgglwlyh prgho/ wkh zloolqjqhvv wr sd| iru d uhgxfwlrq ri wkh vl}h ri
wkh ulvn e| 4 lv ghqhg dv `A @ Ef c ? ' @ E?c ? / ru
@
.Ef n +h?@ ' .Ef n +h?3
` A @ Ef c ?
Wkhuh duh pdq| uhodwlrqvklsv dprqj wkhvh glhuhqw frqfhswv1 Rxu sxu0
srvh lq wklv sdshu lv qrw wr h{dplqh wkhp lq ghwdlov/ exw udwkhu wr fkdudfwhul}h
krz wkhvh ixqfwlrqv ehkdyh dv ? lv lqfuhdvhg1 Wkh frpshqvdwlqj suhplxp
dqg wkh ulvn suhplxp duh wzr dowhuqdwlyh zd|v wr ghqh wkh frvw ri ulvn1 Lqwx0
lwlrq vxjjhvwv wkdw wkh| duh frqyh{/ ru dw ohdvw vxshudgglwlyh1 D uhdo0ydoxhg
ixqfwlrq s lv vxshudgglwlyh li sE? n ?2 lv odujhu wkdq sE? n sE?2 iru doo
? : f dqg ?2 : f Iru wkh fdvh ri wkh zloolqjqhvv wr sd| iru d pdujlqdo
uhgxfwlrq ri wkh vl}h ri wkh ulvn/ zh duh lqwhuhvwhg lq zkhwkhu lw lv lqfuhdvlqj
lq wkh vl}h ri wkh ulvn1 Lq wkhvh wkuhh fdvhv/ wkh lghd lv lqghhg wkdw ulvnv duh
vhoi0djjudydwlqj1
6
Wkh frpshqvdwlqj suhplxp
Lv wkh frpshqvdwlqj suhplxp vxshudgglwlyhB Li dq hpsor|hu kdv wr frpshq0
vdwh wkh ulvnv wkdw zrunhuv ehdu rq wkh zrunsodfh/ dqg li surgxfwlrq qhfhvvl0
wdwhv dq h{srvxuh wr wzr lqglylvleoh lqghshqghqw ulvnv/ vkrxog kh dvn d vlqjoh
hpsor|hh wr eh h{srvhg wr wkh wzr ulvnv/ ru vkrxog kh dvn wzr hpsor|hhv wr
eh h{srvhg wr d vlqjoh ulvn hdfkB
Rxu lqwxlwlrq lv wkdw zkhq d sodqqhu kdv wr frpshqvdwh wkh djhqwv zkr
ehdu ulvn/ wkh frvw0plqlpl}lqj vwudwhj| lv wr vsolw wkh djjuhjdwh ulvn lqwr ulvnv
ri vpdoohu vl}hv wr eh doorfdwhg wr glhuhqw djhqwv/ udwkhu wkdq wr doorfdwh wkh
djjuhjdwh ulvn wr d vlqjoh djhqw1 Wklv zrxog phdq wkdw wkh frpshqvdwlqj
suhplxp lv vxshudgglwlyh1 Wklv lv dozd|v wuxh lq wkh pxowlsolfdwlyh fdvh/ dv
vwdwhg lq wkh iroorzlqj Sursrvlwlrq1
Sursrvlwlrq 4 Lq wkh pxowlsolfdwlyh prgho/ wkh frpshqvdwlqj suhplxp lv
vxshudgglwlyh lq wkh vl}h ri wkh ulvn h{srvxuh=
R6 Ef c ? n ?2 R6 Ef c ? n R6 Ef c ?2 8
Surri= Wdnlqj wkh uvw dqg vhfrqg ghulydwlyhv ri wkh frqglwlrq=
.Ef n ?h
% n R6 Ef c ? ' Ef zlwk uhvshfw wr ? |lhogv=
.Eh
%n
dqg
YR6
% n R6 Ef c ? ' f
Ef c ?Ef n ?h
Y?
.Eh
%n
' +7,
YR6
Ef c ?2 Ef n ?h
% n R6 Ef c ?
Y?
Y2 R6
Ef c ?. Ef n ?h
% n R6 Ef c ?
Y?2
Ehfdxvh wkh ohiw0kdqg vlgh ri wkh deryh lqhtxdolw| lv qhjdwlyh e| ulvn dyhuvlrq/
dqg ehfdxvh . lv srvlwlyh/ zh rewdlq wkdw R6 lv frqyh{ lq ?1 Revhuyh
dovr wkdw R6 Ef c f ' f Dv lv zhoo0nqrzq/ vxfk d ixqfwlrq lv qhfhvvdulo|
vxshudgglwlyh1
Wkh surri ri wklv uhvxow lqglfdwhv wkdw wkh frpshqvdwlqj suhplxp lv frq0
yh{ lq wkh pxowlsolfdwlyh prgho1 Wklv lv pruh ghpdqglqj wkdq vxshudgglwlylw|
iru ixqfwlrqv s vxfk wkdw sEf ' f1
Zh qrz wxuq wr wkh dgglwlyh prgho1 Zh vwduw zlwk d frxqwhuh{dpsoh1
Frqvlghu dq djhqw zlwk d slhfhzlvh olqhdu xwlolw| ixqfwlrq=
li 5 ff
5
+8,
E5 '
Df n 52 li 5 ff
Wklv ixqfwlrq lv lqfuhdvlqj dqg frqfdyh1 Lw lv qrw wzlfh glhuhqwldeoh/ exw fdq
eh dssur{lpdwhg dv pxfk dv zh zdqw e| d vprrwk ixqfwlrq1 Vxssrvh wkdw
f ' ff1 Frqvlghu ulvn h
% wdnlqj ydoxh 093 dqg .<3 zlwk htxdo suredelolwlhv1
Lw lv hdv| wr fkhfn wkdw wkh djhqw kdv d frpshqvdwlqj suhplxp R@ Ef c  ' f
iru wklv vlqjoh ulvn1 Li R@ lv vxshudgglwlyh lq ?/ lw zrxog lpso| wkdw R@ Ef c 2 2R@ Ef c  ' 2f1 Wklv lv qrw wuxh krzhyhu/ vlqfh zh fdq yhuli| wkdw R@ Ef c 2 '
f Lw dsshduv wkdw lq vxfk dq hfrqrp|/ lw lv ohvv frvwo| iru d frpshqvdwlqj
sduw| wr doorfdwh wkh wzr lqghshqghqw ulvnv %
h dqg h
%2 wr d vlqjoh djhqw udwkhu
wkdq wr doorfdwh hdfk ri wkhp wr wzr glhuhqw djhqwv1
Zh kdyh wzr lqghshqghqw ulvnv/ +h? dqg +h?2 / zkrvh frpshqvdwlqj suhpld
duh ghqhg dv
9
.Ef n +h?@ n R@ Ef c ? ' Ef +9,
.Ef n +h?@ 2 n R@ Ef c ?2 ' Ef +:,
dqg
Wkh frpshqvdwlqj suhplxp lv vxshudgglwlyh lq wkh dgglwlyh prgho li
.Ef n +h?@ n +h?@2 n R@ Ef c ? n R@ Ef c ?2 Ef +;,
Lq zrugv/ zh kdyh wzr lqghshqghqw ulvnv/ +h@? nR@ Ef c ? dqg +h?@ 2 nR@ Ef c ?2 c
zklfk wkh djhqw lv lqglhuhqw wr dffhsw ru uhmhfw li rhuhg lq dq h{foxvlyh zd|1
Wkh lqhtxdolw| deryh phdqv wkdw wkh djhqw uhmhfwv wkh exqgoh ri wkhvh wzr
lqghshqghqw ulvnv1 Wkh txhvwlrq ri wkh vxshudgglwlylw| ri wkh frpshqvdwlqj
suhplxp lq wkh dgglwlyh prgho lv wkxv uhodwhg wr wkh iroorzlqj sureohp= duh
wzr lqghshqghqw/ lqglylgxdoo| xqghvludeoh ulvnv mrlqwo| xqghvludeohB Lq wkh
whuplqrorj| xvhg lq wklv sdshu/ wklv phdqv wkdw R@ Ef c 2 vkrxog eh srvlwlyh
zkhqhyhu R@ Ef c  lv srvlwlyh1 Sudww dqg ]hfnkdxvhu +4<;:, h{klelwhg wkh qhf0
hvvdu| dqg vx!flhqw frqglwlrq rq suhihuhqfhv wr jhw vxfk d surshuw|1 Wkh|
fdoohg lw surshu ulvn dyhuvlrq/ ru surshuqhvv1 Doo idploldu xwlolw| ixqfwlrqv olnh
wkh h{srqhqwldo/ wkh orjdulwkplf dqg wkh srzhu ixqfwlrqv duh surshu1 D qhf0
hvvdu| frqglwlrq iru surshuqhvv lv ghfuhdvlqj devroxwh ulvn dyhuvlrq +GDUD,1
Dv vkrzq e| Nlpedoo +4<<6,/ d vx!flhqw frqglwlrq lv GDUD soxv ghfuhdvlqj
devroxwh suxghqfh1
Qrwlfh wkdw rxu sureohp lv d olwwoh glhuhqw iurp wkh Sudww dqg ]hfnkdxvhu
sureohp/ vlqfh zh rqo| uhtxluh wkdw wzr lqghshqghqw ulvnv iru zklfk wkh djhqw
lv mxvw lqglhuhqw zkhq wdnhq lq lvrodwlrq duh mrlqwo| xqghvludeoh11 Zh gr
qrw uhtxluh wkh surshuw| wr krog iru dq| sdlu ri xqghvludeoh ulvnv1 Wkxv/
surshuqhvv lv vx!flhqw/ exw qrw qhfhvvdu|1 Exw lw lv hdv| wr fkhfn iurp
wkh Sudww dqg ]hfnkdxvhu*v zrun wkdw surshu xwlolw| ixqfwlrqv duh wkh rqo|
GDUD ixqfwlrqv vdwlvi|lqj frqglwlrqv +9,/ +:, dqg +;,1 Krzhyhu/ vrph xwlolw|
ixqfwlrqv zlwk dq lqfuhdvlqj devroxwh ulvn dyhuvlrq dovr vdwlvi| wklv vhw ri
frqglwlrqv14
Sursrvlwlrq 5 Lq wkh dgglwlyh prgho/ wkh frpshqvdwlqj suhplxp lv vxshu0
dgglwlyh lq wkh vl}h ri wkh ulvn li devroxwh ulvn dyhuvlrq lv surshu1
4
D ixoo fkdudfwhul}dwlrq ri wklv zhdnhu qrwlrq ri surshuqhvv lv dydlodeoh iurp wkh dxwkruv1
:
7
Wkh ulvn suhplxp
Zh qrz wxuq wr wkh vxshudgglylw| ri wkh ulvn suhplxp1 Wr looxvwudwh/ wkh
vxshudgglwlylw| ri ulvn suhplxp lpsolhv wkdw d prqrsro| lqvxuhu zloo h{wudfw
pruh uhqw li wkh djjuhjdwh ulvn lq wkh hfrqrp| lv xqhtxdoo| glvwulexwhg lq
wkh hfrqrp|1 Wr h{dplqh wklv sureohp/ zh zloo uhvwulfw rxu dqdo|vlv lq wkh
uhpdlqghu ri wkh sdshu wr wkh fdvh ri sxuh ulvnv= .h
% ' f15 Wkhuh lv d
uhodwlrq ehwzhhq wkh frpshqvdwlqj suhplxp dqg wkh ulvn suhplxp1 Lqghhg/
frpelqlqj htxdwlrqv +4, dqg +5,/ zh rewdlq wkdw
.Ef n +h? ' Ef ZEf c ?
' .Ef ZEf c ? n +h? n REf ZEf c ?c ?
Lw lpsolhv wkdw
ZEf c ? n REf ZEf c ?c ? ' fc
ru/ htxlydohqwo|/
ZEf c ? ' REf ZEf c ?c ?
+<,
Wkh ulvn suhplxp dw zhdowk f iru d ulvn ri vl}h ? htxdov wkh frpshqvdwlqj
suhplxp hydoxdwhg dw zhdowk f ZEf c ?1 Wklv surshuw| krogv lqghshq0
ghqwo| ri wkh prgho frqvlghuhg1 Iurp wklv surshuw|/ zh vhh wkdw zh fdqqrw
gluhfwo| lqihu wkh frqyh{lw| ru vxshudgglwlylw| ri Z zlwk uhvshfw wr ? iurp
wkh fruuhvsrqglqj gxdo surshuw| iru wkh frpshqvdwlqj suhplxp R1
Lq Sursrvlwlrq 4/ zh vkrzhg wkdw/ lq wkh pxowlsolfdwlyh prgho/ wkh frp0
shqvdwlqj suhplxp lv vxshudgglwlyh zlwk uhvshfw wr ? iru dq| frqfdyh xwlolw|
ixqfwlrq1 Wklv lv qrw wuxh iru wkh ulvn suhplxp1 Wr suryh wklv/ ohw xv frq0
vlghu djdlq wkh slhfhzlvh olqhdu xwlolw| ixqfwlrq ghvfulehg lq +8,/ zlwk d nlqn
dw zhdowk 4331 Frqvlghu dq lqlwldo zhdowk ohyho f odujhu wkdq 433/ dqg d ulvn
% zklfk wdnhv ydoxhv 04 dqg .4 zlwk htxdo suredelolwlhv1 Reylrxvo|/ dv orqj
h
dv ? lv ohvv wkdq f ff/ wkh vxssruw ri srwhqwldo zhdowk ohyhov lv lq d uhjlrq
zkhuh lv olqhdu/ |lhoglqj ZEf c ? ' f Iru vl}hv ? odujhu wkdq f ff/
5
Wklv dvvxpswlrq fdq eh mxvwlhg e| wkh dvvxpswlrq wkdq rqh fdq frpshqvdwh ehduhuv
ri ulvn iru wkh h{shfwhg frvw/ qrw iru lwv yduldelolw|1 Wklv dvvxpswlrq lqgxfhv d orvv ri
{@h
{ . zlwk Hh
{ @ 3= Wkhq/ lw lv hdv| wr fkhfn wkdw jhqhudolw|1 Lqghhg/ ghqh h
e
e+z3 > q, @
+z3 . q> q, q1 Wkh vxedgglwlylw| ri e zlwk uhvshfw wr q lv wkxv d pruh frpsoh{
sureohp wkdq wkh rqh h{dplqhg lq wklv sdshu/ vlqfh d qrlv| zhdowk hhfw hqwhuv lqwr wkh
slfwxuh1
;
wkh ulvn suhplxp ghshqgv xsrq zkhwkhu f ZEf c ? lv deryh ru ehorz wkh
nlqn1 Zh rewdlq wkh iroorzlqj vroxwlrq=
;
s ? f ff
?f
?3Ef 3ff
s f ff ? Ef ff
ZEf c ? '
= ?nEf23ff
s ? : Ef ff
e
ZEf c ?
a
a
a
a
f ff
Iljxuh 4 =
a
a
a
aZ '
a
a
a
a
a
a
a
a
ZZ
ZZZ
ZZZZ
ZZZZ
Z'
ZZ
?nEf 3ff
e
?3Ef 3ff
2
!
Ef ff
?
Wkh ulvn suhplxp iru wkh slhfhzlvh olqhdu ixqfwlrq zlwk f : ff
dqg % ' Ec ( *2( *2/ pxowlsolfdwlyh prgho1
Wklv ixqfwlrq lv gudzq lq Iljxuh 41 Lw fohduo| dsshduv wkdw wklv ixqfwlrq
lv qrw vxshudgglwlyh zlwk uhvshfw wr ?1 Lq sduwlfxodu/
Z6 Ef c eEf ff Z6 Ef c Ef ff n Z6 Ef c f ff
D vx!flhqw frqglwlrq lv suhvhqwhg lq wkh iroorzlqj Sursrvlwlrq1
Sursrvlwlrq 6 Lq wkh pxowlsolfdwlyh prgho/ wkh ulvn suhplxp lv vxshudggl0
wlyh li ulvn dyhuvlrq lv ghfuhdvlqj1
<
Surri= Ohw ghqrwh Z6 Ef c ? n ?2 Iurp frqglwlrq +<, dqg Sursrvlwlrq
4/ zh kdyh
Z6 Ef c ? n ?2 ' R6 Ef c ? n ?2 R6 Ef c ? n R6 Ef c ?2 +43,
Revhuyh wkdw ' Z 6 Ef c ? n ?2 lv lqfuhdvlqj lq lwv vhfrqg dujxphqw1 Wklv
lv gxh wr wkh idfw wkdw
YZ6
.h
% Ef n ?h
%
%
.h
%.Ef n ?h
Ef c ? ' ' f
Y?
Ef ZEf c ?
Ef ZEf c ?
Lw lpsolhv wkdw ZEf c ? n ?2 ZEf c ? / iru ' c 21 Zh dovr nqrz iurp
Sudww +4<97, wkdw R6 lv ghfuhdvlqj lq lwv uvw dujxphqw xqghu GDUD1 Frp0
elqlqj wkhvh wzr revhuydwlrqv/ frqglwlrq +43, ehfrphv
Z6 Ef c ? n ?2 R6 Ef Z 6 Ef c ? c ? n R6 Ef Z6 Ef c ?2 c ?2 c
zklfk lv uhzulwwhq dv
Z6 Ef c ? n ?2 Z 6 Ef c ? n Z6 Ef c ?2 Wklv frqfoxghv wkh surri1 Zh wxuq wr wkh dgglwlyh fdvh1 Wkh vdph dpeljxlw| dv deryh dulvhv iru
vxpplqj l1l1g1 ulvnv xqghu ulvn dyhuvlrq dorqh1 Exw/ frqwudu| wr wkh pxowl0
solfdwlyh fdvh/ GDUD lv qrw vx!flhqw wr uhvroyh wkh dpeljxlw|16 Lq wkh qh{w
Sursrvlwlrq/ zh vkrz wkdw ulvn yxoqhudelolw| lv vx!flhqw1 Wkh frqfhsw ri ulvn
yxoqhudelolw| kdv ehhq lqwurgxfhg e| Jroolhu dqg Sudww +4<<9,1 Ulvn dyhuvlrq
lv yxoqhudeoh wr ulvn li lw lv lqfuhdvhg e| dgglqj d }hur phdq ulvn wr zhdowk1
Xqghu ulvn yxoqhudelolw|/ d }hur0phdq edfnjurxqg ulvn uhgxfhv wkh ghpdqg
iru ulvn| dvvhwv1 GDUD lv qhfhvvdu|/ exw qrw vx!flhqw iru ulvn yxoqhudelo0
lw|1 Lw lv nqrzq wkdw surshuqhvv lv d pruh ghpdqglqj frqglwlrq1 Dqrwkhu
vx!flhqw frqglwlrq iru ulvn yxoqhudelolw| lv wkdw devroxwh ulvn dyhuvlrq eh gh0
fuhdvlqj dqg frqyh{/ ru wkdw devroxwh ulvn dyhuvlrq dqg devroxwh suxghqfh eh
ghfuhdvlqj1
Sursrvlwlrq 7 Lq wkh dgglwlyh prgho/ wkh ulvn suhplxp lv vxshudgglwlyh zlwk
uhvshfw wr wkh vl}h ri ulvn li ulvn dyhuvlrq lv ulvn yxoqhudeoh1
6
D frxqwhuh{dpsoh lv dydlodeoh iurp wkh dxwkruv1
43
Surri= Zh kdyh wzr ulvnv/ +h?@ dqg +h?@2 / zkrvh ulvn suhpld duh ghqhg dv
+?@ ' Ef Z@ Ef c ? .Ef n h
+44,
E| ulvn yxoqhudelolw|/ dgglqj wkh lqghshqghqw }hur0phdq ulvn +h?@ 2 wr zhdowk
pdnhv wkh djhqw pruh ulvn0dyhuvh1 Iurp Sudww +4<97,/ zh nqrz wkdw lw pdnhv
wkh ulvn suhplxp odujhu1 Wkxv/
.Ef n +h@? n +h?@ 2 .Ef Z@ Ef c ? n +h?@ 2 +45,
Exw zh kdyh
.Ef Z@ Ef c ? nh
+?@2 ' Ef Z@ Ef c ? Z@ Ef Z@ Ef c ? c ?2 +46,
Pruhryhu/ zh nqrz wkdw Z@ Ef c ? lv srvlwlyh e| ulvn dyhuvlrq1 Xqghu GDUD/
lw lpsolhv wkdw Z@ Ef Z@ Ef c ? c ?2 Z@ Ef c ?2 / zklfk lpsolhv lq wxuq
wkdw
.Ef Z@ Ef c ? n h
+?@2 Ef Z @ Ef c ? Z@ Ef c ?2 +47,
Frpelqlqj frqglwlrqv +45, dqg +47, gluhfwo| lpsolhv wkdw
Z@ Ef c ? n ?2 Z @ Ef c ? n Z@ Ef c ?2 Wklv frqfoxghv wkh surri1 8
Wkh zloolqjqhvv wr sd| iru d pdujlqdo uh0
gxfwlrq ri ulvn
Wkh lghd wkdw ulvnv duh vhoi0djjudydwlqj/ zkhq dssolhg wr wkh qrwlrq ri wkh
zloolqjqhvv wr sd| iru d pdujlqdo uhgxfwlrq ri wkh vl}h ? ri ulvn/ lv wudqvodwhg
lqwr wkh srvlwlyh uhodwlrqvkls ehwzhhq ZWS dqg ?1 Lq wkh pxowlsolfdwlyh
prgho/ wkh zloolqjqhvv wr sd| lv zulwwhq dv
%
.h
% Ef n ?h
+48,
. Ef n ?h
%
lv lqfuhdvlqj lq ? li wkh iroorzlqj lqhtxdolw| lv vdwlvhg=
`A 6 Ef c ? '
Wkxv/ `A 6
44
.h
% Ef n ?h
%.h
% Ef n ?h
% .h
%2 Ef n ?h
%.Ef n ?h
%
+49,
Zkhq orrnlqj dw wkh irxu h{shfwdwlrqv lq wkh deryh lqhtxdolw|/ lw kdsshqv
wkdw wkh uvw dqg wkh wklug duh qhjdwlyh e| ulvn dyhuvlrq/ dqg wkh irxuwk lv
srvlwlyh1 Dq| frqglwlrq wkdw pdnhv .h
% Ef n ?h
% qhjdwlyh lv wkxv vx!flhqw
iru wkh uhvxow1 Krzhyhu/ wklv whup lv qhjdwlyh rqo| li lv ghfuhdvlqj/ l1h1/
rqo| li suxghqfh lv qhjdwlyh1 Wklv fdq eh pdgh pruh dssduhqw e| orrnlqj
dw htxdwlrq +48, gluhfwo|= zkhuhdv wkh qxphudwru lv dozd|v lqfuhdvlqj/ wkh
ghqrplqdwru lv hlwkhu ghfuhdvlqj ru lqfuhdvlqj ghshqglqj xsrq zkhwkhu lv frqyh{ +suxghqfh, ru frqfdyh1 Wkh lqwxlwlrq iru wkh dpeljxlw| xqghu
suxghqfh lv wkdw dv ? lqfuhdvhv/ wkh lqfuhdvhg ulvn lqgxfhv wkh djhqw wr olplw
klv sd|phqw wr uhgxfh lw/ lq rughu wr vdyh pruh17 Ehfdxvh f lv qrw d
uhdolvwlf dvvxpswlrq/8 zh wxuq wr pruh uhdolvwlf vx!flhqw frqglwlrqv1
Li zh olplw wkh dqdo|vlv wr wkh fdvh ri }hur0phdq ulvn %
h zlwk rqo| wzr
dwrpv/ l1h1/ %
h lv glvwulexwhg dv E% c R( %2 c  R zlwk R% n E R%2 ' f/ qr
rwkhu uhvwulfwlrq lv qhfhvvdu| wr rewdlq wkh uhvxow/ dv vkrzq lq wkh iroorzlqj
Sursrvlwlrq1
Sursrvlwlrq 8 Lq wkh pxowlsolfdwlyh prgho/ wkh zloolqjqhvv wr sd| iru d pdujlqdo
uhgxfwlrq ri wkh vl}h ? ri ulvn lv qrqghfuhdvlqj lq ? li wkh vxssruw ri %
h lv frp0
srvhg ri wzr dwrpv1
Surri = Xvlqj frqglwlrq .h
% ' f iru h
% y E% c R( %2 c R/ zlwk % f %2 /
zh rewdlq
R'
%2
%2 %
Iru vxfk d glvwulexwlrq ri %
h/ frqglwlrq +49, ehfrphv=
E% %2  % %2 2 E% %2  % %2 2 E%2 %2  % %22 2 E%2  % 2 c
zkhuh E&
' E&Ef n ?% 1 Diwhu vrph whglrxv pdqlsxodwlrqv/ wklv lv htxly0
dohqw wr
7
Dv ghqhg e| Nlpedoo +4<<3,/ d suxghqw djhqw lv zloolqj wr vdyh pruh lq iurqw ri dq
lqfuhdvhg ixwxuh lqfrph ulvn1
8
Lw lpsolhv lq sduwlfxodu wkdw devroxwh ulvn dyhuvlrq eh lqfuhdvlqj1
45
%2
Ef n ?%2 Ef n ?% %

Ef n ?%2 Ef n ?% Wklv lv dozd|v wuxh xqghu ulvn dyhuvlrq/ vlqfh % f %2 1 Zh pd| zrqghu zkhwkhu zkdw lv wuxh iru elqdu| udqgrp yduldeohv lv dovr
wuxh iru udqgrp yduldeohv zlwk wkuhh dwrpv1 Wklv h{whqvlrq lv qrw srvvleoh/ dv
vkrzq e| wkh iroorzlqj frxqwhu0h{dpsoh1 Frqvlghu djdlq wkh slhfhzlvh olqhdu
xwlolw| ixqfwlrq +8, zlwk d nlqn dw zhdowk ohyho 4331 Frqvlghu dq djhqw zlwk
dq lqlwldo zhdowk f ' f mxvw deryh wkh nlqn1 Ilqdoo|/ wdnh wkh wkuhh0dwrp
udqgrp yduldeoh %
h wkdw wdnhv ydoxh 2c *e ru b*e zlwk htxdo suredelolwlhv1
Lw lv hdvlo| fkhfnhg wkdw
;
li ? *2
? f
` A 6 Ef c ? '
fD li *2 ? e
=
feD li ? e
Iru d yhu| odujh h{srvxuh wr ulvn/ wkh djhqw lv uhdg| wr sd| rqo| 3178 wr
uhgxfh wkh ulvn pdujlqdoo|/ zkhuhdv kh zrxog eh uhdg| wr sd| 318 iru wkh
vdph pdujlqdo uhgxfwlrq li klv h{srvxuh lv vpdoohu1
Lq wkh qh{w Sursrvlwlrq/ zh surylgh d vx!flhqw frqglwlrq wr jxdudqwhh
wkdw shrsoh zlwk d odujhu h{srvxuh wr ulvn duh zloolqj wr sd| pruh iru d
pdujlqdo uhgxfwlrq ri wkh h{srvxuh wr ulvn1 E ' E* E ghqrwhv
wkh devroxwh ulvn dyhuvlrq phdvxuhg dw 1 -E ' E lv uhodwlyh ulvn
dyhuvlrq19:
Sursrvlwlrq 9 Lq wkh pxowlsolfdwlyh prgho/ wkh zloolqjqhvv wr sd| iru d pdujlqdo
uhgxfwlrq ri wkh h{srvxuh ? wr wkh ulvn lv qrqghfuhdvlqj lq ? li 5Ef n 5
lv qrqghfuhdvlqj lq 51 D vx!flhqw frqglwlrq lv wkdw devroxwh ulvn dyhuvlrq eh
ghfuhdvlqj dqg uhodwlyh ulvn dyhuvlrq eh lqfuhdvlqj1
Surri = Zlwkrxw orvv ri jhqhudolw|/ ohw ? ' 1 Ohw 8 ghqrwh wkh fxpxodwlyh
glvwulexwlrq ixqfwlrq ri %
h1 Ohw C ghqrwh wkh fxpxodwlyh glvwulexwlrq ixqfwlrq
ri 5h/ zlwk
9
Reylrxvo|/ uhodwlyh ulvn dyhuvlrq lv ghqhg rqo| iru srvlwlyh ydoxhv ri z1 Dv zh xvh lw
{ lv lq U .1
ehorz/ zh lpsolflwo| dvvxph wkdw wkh vxssruw ri wkh glvwulexwlrq ri z 3 . qh
:
Lq wkh surri ri wklv Sursrvlwlrq/ zh gr qrw xvh frqglwlrq H {
h @ 31 Wkxv/ wklv uh0
vxow krogv iru dq| glvwulexwlrq ri ulvn {
h1 Dovr/ lw lv dssduhqw lq wkh surri wkdw frqglwlrq
^}D+z3 . }, qrqghfuhdvlqj lq }` lv qhfhvvdu| dqg vx!flhqw iru wkh surshuw| wr krog iru dq|
glvwulexwlrq ri {
h1
46
Ef n %_8E%
U
_CE% '
Ef n %_8 E%
Frqglwlrq +49, lv htxlydohqw wr=
Ef n 5h
h
2 Ef n 5
.h
5 . 5h . h
5 c
Ef n 5h
Ef n h
5
ru/
.h
52 Ef n h
5 .h
5 .h
5Ef n 5h
+4:,
Exw zh kdyh
.h
5 2 Ef n 5h ' .h
5 .h
5Ef n 5h n SJEh
5 c 5hEf n h
5 Li 5Ef n 5 lv qrqghfuhdvlqj lq 5/ wkhq wkh fryduldqfh lv srvlwlyh/ lqghshq0
ghqw ri wkh glvwulexwlrq ri h
5 1 Lw lpsolhv wkdw frqglwlrq +4:, lv yhulhg1 Wklv
frqfoxghv wkh surri ri wkh vx!flhqf| ri 5Ef n 5 ehlqj qrqghfuhdvlqj lq
51 Li -E ' E ghqrwhv uhodwlyh ulvn dyhuvlrq/ zh kdyh
Y5Ef n 5
' Ef n 5 n 5 Ef n 5 ' -Ef n 5 f Ef n 5
Y5
Wklv lv qrqqhjdwlyh li - f dqg f
Xqghu GDUD/ lw lv qrw fohdu zkhwkhu 5Ef n 5 lv lqfuhdvlqj lq 51 Lq0
fuhdvlqj uhodwlyh ulvn dyhuvlrq uhpryhv wklv dpeljxlw|1 Zkhwkhu uhodwlyh ulvn
dyhuvlrq lv lqfuhdvlqj ru ghfuhdvlqj lv vwloo dq rshq txhvwlrq1 Prvw prghov lq
qdqfh dvvxph frqvwdqw uhodwlyh ulvn dyhuvlrq/ zklfk lv dovr vx!flhqw khuh1
Exw wklv dvvxpswlrq lv pdgh iru whfkqlfdo frqyhqlhqfh/ zlwk qr dwwhpsw wr
mxvwli| lw1 Duurz +4<:4, eholhyhv wkdw uhodwlyh ulvn dyhuvlrq lv lqghhg lqfuhdv0
lqj/ ehfdxvh lw lv frpsdwleoh wkdw wkh uhodwlyh vkduh ri zhdowk lqyhvwhg lq
ulvn| dvvhwv lv ghfuhdvlqj zlwk zhdowk1 Krzhyhu/ prvw hpslulfdo whvwv whqg wr
lqglfdwh wkh uhyhuvhg uhodwlrqvkls ehwzhhq zhdowk dqg wkh uhodwlyh vkduh ri
zhdowk lqyhvwhg lq ulvn| dvvhwv1
Wr hqg xs zlwk wkh hhfw ri wkh lqwhqvlw| ri wkh h{srvxuh wr ulvn rq wkh
ZWS/ zh wxuq wr wkh dgglwlyh fdvh1 Lq wklv prgho/ wkh h{srvxuh lv lqfuhdvhg
vwhs e| vwhs e| dgglqj dq lqghshqghqw }hur0phdq ulvn wr zhdowk1 Li zh dvvxph
ulvn yxoqhudelolw|/ wkh lpsolflw ulvn dyhuvlrq lv lqfuhdvhg dw hdfk vwhs/ wkhuhe|
lqfuhdvlqj wkh zloolqjqhvv wr sd| wr hvfdsh wkh dgglwlrqdo ulvn1 Wklv lv wkh
lqwxlwlrq ehklqg wkh iroorzlqj uhvxow1
47
Sursrvlwlrq : Lq wkh dgglwlyh prgho/ wkh zloolqjqhvv wr sd| iru d pdujlqdo
uhgxfwlrq lq wkh h{srvxuh ? wr ulvn lv qrqghfuhdvlqj lq ? li ulvn dyhuvlrq lv
ulvn yxoqhudeoh1
@
Zh kdyh wkdw
Surri= Ohw h ghqrwh f n +h?3
.E
hn%
h? ' .E h ` A @ E?
Qrz/ ghqh wkh lqgluhfw xwlolw| ixqfwlrq E5 ' .E5 n %
%?n
h?n Ehfdxvh h
kdv d }hur phdq/ ulvn yxoqhudelolw| lv vx!flhqw wr jxdudqwhh wkdw lv pruh
frqfdyh wkdq lq wkh vhqvh ri Duurz0Sudww1 G|eylj dqg Olsspdq +4<;6,
kdyh vkrzq wkdw doo ulvnv iru zklfk dq djhqw lv lqglhuhqw duh glvolnhg e|
pruh ulvn0dyhuvh djhqwv1 Wkxv/ zh nqrz wkdw
.E
hnh
%? .E
h `A @ Ef c ?c
ru/ uhsodflqj /
h?n .E
hnh
%?n `A @ Ef c ?
.E
hnh
%? n %
Vlqfh h
%?n duh l1l1g1 yduldeohv/ wklv fdq eh uhzulwwhq dv
%? dqg h
@
.Ef n h
+?@ ` A @ Ef c ?
.Ef n +h?n
E| ghqlwlrq/ wkh ohiw0kdqg vlgh ri wklv lqhtxdolw| htxdov .Ef n +h?@ `A @ Ef c ? n  Wkxv/ lw pxvw eh wkdw `A @ Ef c ? n  `A @ Ef c ?
Iurp wklv surri/ lw lv dssduhqw wkdw ulvn yxoqhudelolw| zrxog dovr eh qhf0
hvvdu| li %
h lv doorzhg wr eh dq| udqgrp yduldeoh zlwk d qrqsrvlwlyh phdq1
9
H{wudsrodwlqj wkh ZWS wr fdofxodwh wkh
ulvn suhplxp
Lw kdv ehhq qrwlfhg wkdw ZWS lv zkdw rqh lv uhdg| wr sd| iru d pdujlqdo
uhgxfwlrq lq wkh h{srvxuh wr ulvn zkhuhdv Z/ wkh ulvn suhplxp/ lv zkdw rqh lv
uhdg| wr sd| iru d ixoo holplqdwlrq ri wkh ulvn1 Fdq zh lqihu dq|wklqj derxw Z
e| mxvw nqrzlqj wkh ZWS iru d pdujlqdo uhgxfwlrq ri wkh h{srvxuh wr ulvnB
Wkh dqvzhu wr wklv txhvwlrq lv vlpsoh iru wkh pxowlsolfdwlyh prgho1
48
Sursrvlwlrq ; Lq wkh pxowlsolfdwlyh prgho/ wkh ulvn suhplxp iru wkh holpl0
qdwlrq ri wkh h{srvxuh ? wr ulvn lv dozd|v vpdoohu wkdq ? wlphv wkh zloolqjqhvv
wr sd| iru d pdujlqdo uhgxfwlrq ri wklv h{srvxuh= Z6 Ef c ? ?` A 6 Ef c ?1
Surri= Zlwkrxw orvv ri jhqhudolw|/ ohw ? htxdo 41 Ohw 5 ' f Z6 Ef c 
dqg +h ' h
+ E51Reylrxvo|/
% n Z6 Ef c  Ghqh ixqfwlrq MEk ' .E5 n kh
MEf ' f E| ghqlwlrq ri Z6 Ef c / zh nqrz wkdw ME ' f1 Zh zdqw wr
suryh wkh iroorzlqj lqhtxdolw|=
Z6 Ef c  `A 6 Ef c  '
%
.h
%Ef n h
. Ef n h
%
Wklv frqglwlrq fdq eh uhzulwwhq dv
.Eh
% n Z6 Ef c  Ef Z6 Ef c  n h
% n Z6 Ef c  fc
ru/ htxlydohqwo|/
.h
+ E5 n +h f
Wklv lv htxlydohqw wr M E f1
Xqghu ulvn dyhuvlrq/ M lv frqfdyh1 Vlqfh MEf ' ME ' f/ lw pxvw eh
wkdw M E f1
Qrwlfh wkdw wklv uhvxow lv lq idfw lqghshqghqw ri zkhwkhu h
% kdv d }hur phdq
ru qrw1 Wkxv/ lq wkh pxowlsolfdwlyh prgho/ xvlqj wkh ZWS iru d pdujlqdo
uhgxfwlrq ri ulvn wr h{wudsrodwh iru wkh ZWS iru d ixoo holplqdwlrq ri ulvn
dozd|v jhqhudwhv dq ryhuhvwlpdwlrq ri wkh odwwhu1
Lq wkh fdvh ri dgglwlyh l1l1g1 ulvnv/ wkh vroxwlrq lv ohvv wulyldo1
Sursrvlwlrq < Lq wkh dgglwlyh prgho/ wkh ulvn suhplxp iru h{srvxuh ? lv
odujhu +uhvs1 vpdoohu, wkdq wkh vxp ri wkh ZWS iru vxffhvvlyh pdujlqdo
uhgxfwlrqv lq ulvn li devroxwh ulvn dyhuvlrq lv ghfuhdvlqj +uhvs1 lqfuhdvlqj,=
S?
S' ` A @ Ef c li devroxwh ulvn dyhuvlrq lv lqfuhdvlqj>
@
Z Ef c ?
?
@
' ` A Ef c li devroxwh ulvn dyhuvlrq lv ghfuhdvlqj1
Surri= Zh nqrz wkdw
@
.Ef n h
+?@ ' .Ef n +h?3
`A @ Ef c ?
49
dqg
@
.Ef n +h?3
` A @ Ef c ?
@
' .Ef n +h?32
` A @ Ef c ? `A @ Ef ` A @ Ef c ?c ? 
Zh frqfoxgh wkdw
@ Ef c ?c ? 2 ' `A @ Ef c ? n ` A @ Ef `A @ Ef c ?c ? 
Vlqfh .h
% ' f/ ` A @ Ef c ? f/ dqg GDUD lpsolhv wkdw;
` A @ Ef ` A @ Ef c ?c ?  ` A @ Ef c ? 
Wkxv/ zh kdyh
@ Ef c ?c ? 2 `A @ Ef c ? n `A @ Ef c ? 
E| uhfxuuhqfh/ zh rewdlq
@
Ef c ?c ? & &
[
` A @ Ef c ? n 
'
Wkh idfw wkdw Z@ Ef c ? ' @ Ef c ?c f frqfoxghv wkh surri iru GDUD1 Wkh
fdvh ri lqfuhdvlqj devroxwh ulvn dyhuvlrq lv rewdlqhg e| v|pphwu|1
Wkh frpelqdwlrq ri sursrvlwlrqv :dqg < gluhfwo| |lhogv wkh iroorzlqj fruro0
odu|=
Fruroodu| 43 Lq wkh dgglwlyh prgho/ ulvn yxoqhudelolw| lpsolhv wkdw wkh zloo0
lqjqhvv wr sd| iru d ixoo holplqdwlrq ri wkh h{srvxuh wr ? l1l1g1 ulvnv lv odujhu
wkdq ? wlphv wkh zloolqjqhvv wr sd| iru wkh ixoo holplqdwlrq wr wkh h{srvxuh ri
d vlqjoh ulvn=
Z@ Ef c ? ?`A @ Ef c 
Surri= Uhphpehu wkdw GDUD lv qhfhvvdu| iru ulvn yxoqhudelolw|1 Wkxv/
e| Sursrvlwlrq </ zh kdyh
;
Wkh surri wkdw GDUD pdnhv Z W S +}> q, ghfuhdvlqj lq } lv ohiw wr wkh uhdghu1
4:
Z@ Ef c ? ?
[
`A @ Ef c '
E| Sursrvlwlrq :/ lw lpsolhv wkdw
@
Z Ef c ? ?
[
`A @ Ef c  ' ?` A @ Ef c 
'
Wklv frqfoxghv wkh surri1
:
Frqfoxvlrq
Lqwxlwlrq vxjjhvwv wkdw ulvnv duh vhoi0djjudydwlqj iru ulvn0dyhuvh djhqwv/ l1h1
wkdw wkh frvw ri ulvn lv frqyh{ lq wkh qxpehu ri +lghqwlfdo, ulvn| h{srvxuhv1 Lq
idfw/ lqwxlwlrq lv fohduo| frquphg rqo| lq wkh pxowlsolfdwlyh prgho zkhq wkh
frvw ri ulvn lv phdvxuhg e| wkh frpshqvdwhg suhplxp1 Lq doo rwkhu fdvhv/ pruh
uhvwulfwlrqv wkdq ulvn dyhuvlrq kdyh wr eh sxw rq wkh xwlolw| ixqfwlrq wr uhfryhu
wkh lqwxlwlyh uhvxow1 Wkh frqglwlrqv uhtxluhg duh dozd|v ohvv ghpdqglqj iru wkh
pxowlsolfdwlyh prgho wkdq iru lwv dgglwlyh frxqwhusduw1 Wkh| duh vxppdul}hg
lq wkh iroorzlqj wdeoh1
Surshuw|
Pxowlsolfdwlyh prgho
Frpshqvdwlqj suhplxp vxshudgg1 dozd|v wuxh
Ulvn suhplxp vxshudgglwlyh
li GDUD
ZWS lqfuhdvlqj
li LUUD dqg GDUD
Wdeoh 4= Surshuwlhv uhodwlqj wkh glhuhqw frqfhswv ri ZWS wr
ulvn1
Dgglwlyh prgho
li surshuqhvv
li ulvn yxoqhudelolw|
li ulvn yxoqhudelolw|
wkh vl}h ri
Li rqh uhphpehuv wkdw wkh vxshudgglwlylw| ri wkh frvw ixqfwlrq grhv qrw
qhfhvvdulo| lpso| dq hyhu|zkhuh lqfuhdvlqj pdujlqdo frvw/ wkh lqwxlwlyh lghd
wkdw wkh pdujlqdo frvw ri ulvn vkrxog eh lqfuhdvlqj zlwk wkh qxpehu ri h{0
srvxuhv douhdg| dffxpxodwhg lq idfw lqgxfhv udwkhu vhyhuh uhvwulfwlrqv rq wkh
vkdsh ri wkh xwlolw| ixqfwlrq1
4;
Uhihuhqfhv
Duurz/ N1/ +4<:4,/ Hvvd|v lq wkh Wkhru| ri Ulvn Ehdulqj/ Pdunkdp Sxe0
olvklqj Fr/ Fklfdjr1
G|eylj/ S1K1/ dqg V1D1 Olsspdq/ +4<;6,/ Dq dowhuqdwlyh fkdudfwhul}dwlrq
ri ghfuhdvlqj devroxwh ulvn dyhuvlrq/ Hfrqrphwulfd/ 84/ 55605571
Jroolhu/ F1 dqg M1Z1 Sudww/ +4<<9,/ Ulvn Yxoqhudelolw| dqg wkh Whpshulqj
Hhfw ri Edfnjurxqg Ulvn/ Hfrqrphwulfd/ 97/ 443<044571
Od Ydooìh/ L1/ +4<9;,/ Rq fdvk htxlydohqfh dqg lqirupdwlrq hydoxdwlrq lq
ghflvlrqv xqghu xqfhuwdlqw|/ sduw L= Edvlf wkhru|/ Mrxuqdo ri wkh Dphulfdq
Vwdwlvwlfdo Dvvrfldwlrq/ 96/ 58505:81
Nlpedoo/ P1V1/ +4<<3,/ Suhfdxwlrqdu| Vdylqj lq wkh Vpdoo dqg lq wkh
Odujh/ Hfrqrphwulfd/ 8;/ 860:61
Nlpedoo/ P1V1/ +4<<6,/ Vwdqgdug Ulvn Dyhuvlrq/ Hfrqrphwulfd/ 94/ 8;<0
9441
Sudww/ M1Z1/ +4<97,/ Ulvn Dyhuvlrq lq wkh Vpdoo dqg lq wkh Odujh/ Hfrqr0
phwulfd/ 65/ 45504691
Sudww/ M1Z1 dqg U1 ]hfnkdxvhu/ +4<;:,/ Surshu Ulvn Dyhuvlrq/ Hfrqrphw0
ulfd/ 88/ 47604871
Vdpxhovrq/ S1D1/ +4<96,/ Ulvn dqg xqfhuwdlqw|= wkh idoodf| ri wkh Odz ri
Odujh Qxpehuv/ Vflhqwld/ <;/ 43;04461
Vhjdo/ X1 dqg D1 Vslydn/ +4<<3,/ Iluvw Rughu Yhuvxv Vhfrqg Rughu Ulvn
Dyhuvlrq/ Mrxuqdo ri Hfrqrplf Wkhru|/ 84/ 44404581
4<