48
Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton, Thong Nguyen, and Peter C.B. Phillips July 2001 COWLES FOUNDATION DISCUSSION PAPER No. 1311 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/

Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

  • Upload
    others

  • View
    28

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Nonparametric Estimation of aMultifactor Heath-Jarrow-Morton Model:

An Integrated Approach

By

Andrew Jeffrey, Oliver Linton,Thong Nguyen, and Peter C.B. Phillips

July 2001

COWLES FOUNDATION DISCUSSION PAPER No. 1311

COWLES FOUNDATION FOR RESEARCH IN ECONOMICSYALE UNIVERSITY

Box 208281New Haven, Connecticut 06520-8281

http://cowles.econ.yale.edu/

Page 2: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Qrqsdudphwulf Hvwlpdwlrq ri d Pxowlidfwru

Khdwk0Mduurz0Pruwrq prgho= Dq Lqwhjudwhg Dssurdfk

Dqguhz Mh>uh|ú/ Rolyhu Olqwrq|/ Wkrqj Qjx|hq}/ dqg Shwhu F1E1 Skloolsv{

Pd| 55/ 5334

Devwudfw

Zh ghyhors d qrqsdudphwulf hvwlpdwru iru wkh yrodwlolw| vwuxfwxuh ri wkh }hur frxsrq |lhog

fxuyh lq wkh Khdwk0Mduurz0Pruwrq iudphzrun1 Wkh hvwlpdwru lqfrusrudwhv furvv0vhfwlrqdo uh0

vwulfwlrqv dorqj wkh pdwxulw| glphqvlrq/ dqg dovr doorzv iru phdvxuhphqw huuruv/ zklfk dulvh

iurp wkh hvwlpdwlrq ri wkh |lhog fxuyh iurp qrlv| gdwd1 Wkh hvwlpdwhv duh lpsohphqwhg zlwk

gdlo| FUVS erqg gdwd1

4 Lqwurgxfwlrq

Lq wklv sdshu zh sursrvh d qhz hvwlpdwru ri wkh yrodwlolw| vwuxfwxuh lq vlqjoh idfwru dqg pxowl0

idfwru prghov ri wkh whup vwuxfwxuh1 D zhoo nqrzq olplwdwlrq ri rqh idfwru prghov/ uhjdugohvv ri

zkhwkhu wkh| ehorqj wr wkh pruh wudglwlrqdo Pdunryldq iudphzrun w|sl?hg e| Ydvlfhn +4<::, dqg

Fr{/ Lqjhuvroo/ dqg Urvv +4<;8, ru wkh pruh uhfhqw dssurdfk slrqhhuhg e| Kr dqg Ohh +4<;6, dqg

Khdwk/ Mduurz dqg Pruwrq +4<<5, +KMP khuhdiwhu, ý zklfk uhvxowv lq qrq0Pdunryldq ehkdylru ri

wkh lqwhuhvw udwhv ý lv wkdw wkh| srruo| fdswxuh hpslulfdo g|qdplfv hlwkhu ri vkruw udwhv +wkh prvw

frpprqo| xvhg vwdwh yduldeoh, ru wkh zkroh whup vwuxfwxuh1 Wklv olplwdwlrq kdv prwlydwhg wkh vwxg|

ri pxowl0idfwru prghov/ hvshfldoo| lq sudfwlfdo zrun1 Wzr idfwru prghov zlwk d ulfk ydulhw| ri vhfrqg

idfwru vshfl?fdwlrqv kdyh ehhq h{dplqhg lq wkh olwhudwxuh1 Wkh vkruw whup lqwhuhvw udwh lv xvxdoo| wkh

ú\doh Vfkrro ri Pdqdjhphqw/ SR Er{ 53;533/ Qhz Kdyhq FW1 398530;5331 Skrqh= +536, 765 935<1 Id{= +536,

765 9<:31 H0pdlo= Dqguhz1Mhiiuh|C|doh1hgx|Ghsduwphqw ri Hfrqrplfv/ Orqgrq Vfkrro ri Hfrqrplfv/ Krxjkwrq Vwuhhw/ Orqgrq ZF5D 5DH/ Xqlwhg Nlqjgrp1

H0pdlo= olqwrqrCovh1df1xn1}Edqn ri Dphulfd Vhfxulwlhv/ Vdq Iudqflvfr/ FD <7444/ X1V1D1 H0pdlo= wkrqjkqjx|hqC|dkrr1frp{Frzohv Irxqgdwlrq/ \doh Xqlyhuvlw|1 H0pdlo= shwhu1skloolsvC|doh1hgx

4

Page 3: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

?uvw idfwru> vrph iuhtxhqwo| fkrvhq vhfrqg idfwruv lqfoxgh wkh orqj udwh/ wkh vsuhdg ehwzhhq orqj udwhv

dqg vkruw udwhv/ lq dwlrq/ fhqwudo whqghqf| +wkh orqj whup phdq ri vkruw whup lqwhuhvw udwhv,/ dqg

yrodwlolw| ri vkruw whup lqwhuhvw udwhv1 H{dpsohv ri wzr idfwru prghov fdq eh irxqg lq Edogx}}l/ Gdv/

Iruhvl dqg Vxqgdudp +4<<;,/ Euhqqdq dqg Vfkzduw} +4<:<,/ Gx!h dqg Vlqjohwrq +4<<:,/ Nqljkw/

Ol dqg \xdq +4<<<,/ Olwwhupdq dqg Vfkhlqnpdq +4<<4,/ Orqjvwd> dqg Vfkzduw} +4<<5,/ Vfkdhihu

dqg Vfkzduw} +4<;7,/ Shduvrq dqg Vxq +4<<7,/ dprqj rwkhuv1 Pruh h{leoh wkuhh idfwru prghov dqg

eh|rqg duh dovr vwxglhg e| Erxgrxnk/ Ulfkdugvrq dqg Vwdqwrq +4<<;,/ Edogx}}l/ Gdv/ Iruhvl dqg

Vxqgdudp +4<<9,/ Fkhq dqg Vfrww +4<<6,1 D vxuyh| ri pxowl0idfwru prghov fdq eh irxqg lq Edfnxv/

Iruhvl dqg Whophu +4<<;,1

Uhfhqw wkhruhwlfdo dqg hpslulfdo vwxglhv ri qrq0Pdunryldq prghov lqvlgh wkh Khdwk0Mduurz0

Pruwrq iudphzrun kdyh dovr idyruhg wkh pxowl0idfwru vshfl?fdwlrq wr hqdeoh ulfkhu g|qdplfv/ vhh

iru h{dpsoh Eolvv dqg Ulwfknhq +4<<9,/ Exkohu/ Xkulj0Krpexuj/ Zdowhu dqg Zhehu +4<<<,/ Lqxl dqg

Nlmlpd +4<<;,/ Mrqj dqg Vdqwd0Fodud +4<<<,/ Shduvrq dqg ]krx +4<<<,1

Prylqj dzd| iurp wkh vlpsoh rqh idfwru iudphzrun lqwurgxfhv h{wud dqdo|wlfdo frpsoh{lw|1 Lq

frqvhtxhqfh/ rxw ri pdwkhpdwlfdo frqyhqlhqfh/ prvw prghov vr idu vwxglhg duh d!qh1 Wkdw lv/ wkh|

kdyh guliw dqg gl>xvlrq ixqfwlrqv gh?qhg dv olqhdu ixqfwlrqv ri wkh vwdwh yduldeoh+v,/ dv sursrvhg

lq Ydvlfhn +4<::, dqg w|sl?hg lq wkh zrun ri Fr{/ Lqjhuvroo/ Urvv +4<;8,1 Wkh d!qh vwuxfwxuh e|

lwvhoi lpsrvhv khdy| uhvwulfwlrqv rq wkh vshfl?hg g|qdplfv/ ehvlghv wkh idfw wkdw prvw d!qh prghov

lpsrvh dgglwlrqdo uhvwulfwlrqv ehfdxvh wkh| idoo vkruw ri ehlqj 3pd{lpdo4 lq wkh whuplqrorj| ri Gdl

dqg Vlqjohwrq +4<<;,14 Qrqolqhdu prghov/ vxfk dv Fkdq/ Nduro|/ Orqjvwd> dqg Vdxqghu +4<<5, +FNOV

khuhdiwhu,/ dqg pruh uhfhqwo|/ qrqsdudphwulf prghov dv lq Vwdqwrq +4<<:,/ duh pruh h{leoh1 Dw d

frvw ri vrph orvv lq wudfwdelolw|/ wkhvh prghov gr fdswxuh wkh g|qdplfv ri wkh whup vwuxfwxuh ehwwhu

8 vhh FNOV iru dq hpslulfdo ghprqvwudwlrq wr wklv h>hfw1 Krzhyhu/ wkhuh duh vwloo uhvwulfwlrqv wkdw

duh lpsrvhg wkurxjk vhhplqjo| lqqrfxrxv vshfl?fdwlrqv1 Iru h{dpsoh/ lq wkh pxowl0idfwru Pdunryldq

prgho glvfxvvhg lq Nqljkw/ Ol dqg \xdq +4<<<,/ rqo| rqh Eurzqldq prwlrq gulyhv wkh g|qdplfv ri

hdfk idfwru/ ru lq wkh pxowl0idfwru qrq0Pdunryldq prgho lq Shduvrq dqg ]krx +4<<<,/ wkhuh lv mxvw

rqh Eurzqldq prwlrq wkdw gulyhv wkh zkroh iruzdug fxuyh/ dowkrxjk wkh yrodwlolw| vwuxfwxuh ghshqgv

rq pruh wkdq rqh vwdwh yduldeoh1

Lq wklv vwxg|/ zh sursrvh d h{leoh pxowl0idfwru jhqhudol}hg Khdwk0Mduurz0Pruwrq prgho wkdw

hqfrpsdvvhv prvw KMP prghov sursrvhg vr idu lq wkh olwhudwxuh1 Rxu prgho lv jhqhudo lq ydulrxv

zd|v1 Wkh qrqsdudphwulf vshfl?fdwlrq doorzv wkh ixqfwlrqdo irupv wr eh plqlpdoo| uhvwulfwhg1 Zh

dovr doorz doo yrodwlolw| vwuxfwxuhv dvvrfldwhg zlwk hdfk Eurzqldq prwlrq wr ghshqg rq wkh zkroh vhw

43Pd{lpdo4 hvvhqwldoo| phdqv wkdw wkh uhvhdufkhu grhv qrw lpsrvh dq| lpsolflw dqg2ru h{solflw uhvwulfwlrq rq wkh

prgho eh|rqg wkh d!qh vwuxfwxuh1 Lq sudfwlfh/ krzhyhu/ dgglwlrqdo dvvxpswlrqv duh riwhq lqyrnhg1 Iru lqvwdqfh/ lw lv

yhu| frpprq wr dvvxph wkdw idfwruv duh gulyhq e| d vhw ri ruwkrjrqdo Eurzqldq prwlrqv1

5

Page 4: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ri vshfl?hg vwdwh yduldeohv/ ghsduwlqj iurp d frpprq sudfwlfh ri dvvrfldwlqj hdfk Eurzqldq prwlrq

zlwk d vhsdudwh vwdwh yduldeoh1 Prvw lpsruwdqwo|/ frqwudu| wr wkh xvxdo dssurdfk ri lqwurgxflqj

phdvxuhphqw huuruv dv dq diwhuwkrxjkw wr ?{ wkh vr0fdoohg vwrfkdvwlf vlqjxodulw| wkdw lv lqkhuhqw lq

KMP prghov/ rxu prgho gluhfwo| lqfrusrudwhv phdvxuhphqw huuruv zkhq wkh g|qdplfv ri wkh |lhog

fxuyhv duh lqlwldoo| vshfl?hg1 Wr rxu nqrzohgjh/ wklv lv wkh ?uvw vwxg| wr dqdo|}h hvwlpdwlrq lvvxhv ri

d pxowl0idfwru KMP prgho lq d uljrurxv idvklrq1

Wkh sodq ri wkh sdshu lv dv iroorzv1 Lq wkh qh{w vhfwlrq/ wkh surwrw|sh rqh idfwru prgho lv vshfl?hg

dqg wkh hvwlpdwru dqg lwv dv|pswrwlf surshuwlhv lv ghyhorshg1 Zh h{whqg wkh phwkrgrorj| wr fryhu

wkh pxowl0idfwru fdvh lq vhfwlrq 61 Vhfwlrq 7 lv ghyrwhg wr wkh hpslulfdo hvwlpdwlrq ri d rqh idfwru

dqg wzr idfwru prgho xvlqj XV erqg gdwd1 Vhfwlrq 8 frqfoxghv wkh sdshu1 Doo whfkqlfdo ghulydwlrqv

duh jlyhq lq wkh dsshqgl{1

5 Rqh idfwru Qrqsdudphwulf KMP prgho

Ohw å E|c A ä ghqrwh wkh sulfh ri d rqh groodu idfh ydoxh/ ghidxow iuhh/ }hur frxsrq erqg dw wlph | wkdw

zloo pdwxuh dw wlph A 1 Wkh lqvwdqwdqhrxv iruzdug udwh dw wlph | iru gdwh A / ghqrwhg sE|c A ä/ lv gh?qhg

e| sE|c A ä ' ýY *?Eå E|c A ää% YA dqg wkh |lhog dw wlph | zlwk pdwxulw| gdwh A / ghqrwhg +E|c A ä/ lv

gh?qhg lq whupv ri iruzdug udwhv e| +E|c A ä ' ý �A3|

U A|sE|c ñä_ñ1 Wkh KMP iudphzrun uhsuhvhqwv

wkh whup vwuxfwxuh lq whupv ri iruzdug udwhv1 Krzhyhu/ ghwhuplqlqj iruzdug udwhv lq sudfwlfh yld

fxuyh ?wwlqj surfhgxuhv riwhq suryhv wr eh vhqvlwlyh wr wkh phwkrg dgrswhg1 Hvwlpdwlrq ri |lhogv lv

w|slfdoo| ohvv vhqvlwlyh wr wkh phwkrg xvhg/ lqwxlwlyho| ehfdxvh |lhogv duh dyhudjhv ri iruzdug udwhv1

Frqvhtxhqwo|/ zh fkrrvh wr sruwud| wkh whup vwuxfwxuh xvlqj |lhogv1 Zlwklq wkh KMP iudphzrun

zkhuh d vlqjoh Zlhqhu surfhvv ` E|ä lqwurgxfhv xqfhuwdlqw| lqwr wkh erqg pdunhw/ wkh xqfhuwdlq

hyroxwlrq ri hdfk |lhog zlwk ?{hg pdwxulw| gdwh A lv fkdudfwhul}hg e| wkh vwrfkdvwlf gl>huhqwldo

htxdwlrq

_+E|c A ä ' k+E/c |c A ä_|n òE/c |c A ä_` E|äc

zkhuh` E|ä lv d rqh glphqvlrqdo Eurzqldq prwlrq/ zkloh / lqglfdwhv wkh srvvleoh ghshqghqfh rq wkh

whup vwuxfwxuh*v uhdol}dwlrq xs wr wlph |1 Pruh vshfl?fdoo| / 5 I| zkhuh I| lqglfdwhv doo dydlodeohlqirupdwlrq mxvw ehiruh wlph | jhqhudwhg e| wkh whup vwuxfwxuh*v hyroxwlrq1 Zh vkdoo uhvwulfw rxu

dwwhqwlrq wr yrodwlolw| ixqfwlrqv ri wkh irup

òE/c |c A ä ' òEoE|äc A ý |ä ' òEoE|äc ô ä

iru vrph ixqfwlrq ò Eéä wkdw lv vprrwk exw rwkhuzlvh ri xqnqrzq ixqfwlrqdo irup1 Wklv fodvv ri

prghov lv ydvw/ fryhulqj prvw ri wkh KMP vshfl?fdwlrqv sursrvhg vr idu lq wkh olwhudwxuh1 Krzhyhu/

6

Page 5: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

wkh whfkqltxh lqwurgxfhg odwhu lv qrw uhvwulfwhg wr wklv fkrvhq vwdwh yduldeoh1 Rxu vshfl?fdwlrq lv

reylrxvo| qrqsdudphwulf rqh zlwk qr uhvwulfwlrq rq wkh ixqfwlrqdo irup/ dffruglqj wr wkh xvxdo

frqyhqwlrq1

Dv vkrzq lq wkh ruljlqdo KMP sdshu/ wkh qr duelwudjh uhvwulfwlrq glfwdwhv wkdw wkh guliw ixqfwlrq

ri wkh iruzdug fxuyh hyroxwlrq lv mxvw d ixqfwlrq ri wkh yrodwlolw| vwuxfwxuh li zh duh lq ulvn qhxwudo

zruog1 Iru |lhog fxuyh hyroxwlrq/ wklv qr duelwudjh uhvwulfwlrq dovr lpsrvhv wkdw wkh guliw ixqfwlrq lv

d ixqfwlrq ri wkh |lhog yrodwlolw| vwuxfwxuh1 vshfl?fdoo|

k+E/c |c ô ä 'Y+E|c ôä

Yôn+E|c ô äý oE|ä

ôn

2ô òE/c |c ô ä2(

vhh Mh>uh|/ Olqwrq/ dqg Qjx|hq +4<<<e, iru d ghulydwlrq ri wkh deryh htxdwlrq1 Wrjhwkhu zlwk wkh

nqrzohgjh ri wkh pdunhw sulfh ri ulvn/ wkh g|qdplfv ri wkh |lhog fxuyh xqghu wkh uhdo zruog phdvxuh

fdq eh uhfryhuhg1 D frpprq remhfwlyh lv wr xvh wkh deryh g|qdplfv wr sulfh ?{hg lqfrph lqvwuxphqwv/

krzhyhu/ dqg ulvn qhxwudo pduwlqjdoh sulflqj rqo| uhtxluhv wkh g|qdplfv lq d ulvn qhxwudo zruog/ vlqfh

wkh guliw lv vlpso| d ixqfwlrq ri wkh yrodwlolw| vwuxfwxuh lq d ulvn qhxwudo zruog +lw lv qrw qhfhvvdu| wr

nqrz wkh pdunhw sulfh ri ulvn lq wklv zruog,1 Vr/ iru sulflqj edvhg rq dq KMP prgho vxfk dv wkdw

frqvlghuhg khuh/ zh rqo| qhhg wr hvwlpdwh wkh yrodwlolw| vwuxfwxuh1

Xvlqj rqo| revhuydwlrqv iurp d wlph vhulhv ri +Eéc ôä zlwk ?{hg ô / zh fdq rewdlq dq hvwlpdwh ri

wkh yrodwlolw| ixqfwlrq òEoE|äc ôä dv vkrzq lq d uhfhqw sdshu e| Mh>uh|/ Olqwrq/ dqg Qjx|hq +4<<<e,1

Wkhlu hvwlpdwru zdv eò�Eo E|ä c ô ä2 ' �

{?

S?ð'�gûoEo ý oðä{+E|ðc ô ä2S?

ð'�gûoEo ý oðäc +4,

zkhuh gûEüä ' gEü*ûä*û dqg gEüä lv d v|pphwulf suredelolw| ghqvlw|/ zkloh ûo lv d edqgzlgwk

vhtxhqfh1 Xqghu ydulrxv uhjxodulw| frqglwlrqv ^lqfoxglqj wkdw ûo $ f dqg ?ûo $4o zh kdyhvûo u E|?c oä

{|

ýeò�Eoc ôä2 ý òEoc ô ä2ü _ý$ ùýfc e ngn2 òEoc ô äeü

dv ? $ 4c zkhuh ngn2 'U n"3" g Eüä2 _üc zkloh u E|?c oä lv wkh fkurqrorjlfdo wlph ri wkh surfhvv

o/ d frqfhsw wkdw zloo eh glvfxvvhg lq pruh ghwdlo odwhu1 Wkh hvwlpdwru fdq eh frqvlghuhg dv dq

h{whqvlrq wr KMP prghov ri wkh dssurdfk slrqhhuhg e| Vwdqwrq +4<::, dqg Mldqj dqg Nqljkw +4<<;,

iru Pdunryldq lqwhuhvw udwh prghov1

Lq wkh ruljlqdo KMP prghov/ d ?qlwh vhw ri Eurzqldq prwlrqv vhuyh dv wkh vrxufh ri udqgrpqhvv

lq wkh hfrqrp| dqg gulyh wkh g|qdplfv ri wkh zkroh |lhog fxuyhv1 Wklv frqvwuxfwlrq khosv wr pdnh

wkh pdunhw frpsohwh/ exw dw wkh frvw ri lqgxflqj vwrfkdvwlf vlqjxodulw| lqwr wkh prgho= zh fdq hdvlo|

?qg vrph olqhdu frpelqdwlrq ri wkh g|qdplfv ri srlqwv dorqj wkh |lhog fxuyhv wkdw lv ghwhuplqlvwlf/

7

Page 6: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

l1h1/ zlwkrxw wkh suhvhqfh ri wkh Eurzqldq prwlrq vkrfnv15 Hpslulfdo gdwd/ ri frxuvh/ doprvw vxuho|

ylrodwh wklv uhodwlrqvkls1 Wr uhfrqfloh wklv lqwhuqdo frqvlvwhqf|/ rwkhu vrxufhv ri udqgrpqhvv/ vxfk dv

phdvxuhphqw huuruv pxvw eh lqwurgxfhg1 Jhqhudol}hg prghov ri KMP duh dovr prwlydwhg wr ryhufrph

wklv xqghvludeoh lqfrqvlvwhqf|1 Lq Nhqqhg| +4<<7, dqg Nhqqhg| +4<<:,/ wkh vrxufh ri wkh vkrfnv lq

wkh hfrqrp| lv d Eurzqldq vkhhw dqg udqgrp ?hog/ uhvshfwlyho|> ru lq pruh uhfhqw zrun e| Vdqwd

Fodud dqg Vruqhwwh +5333,/ vwulqj vkrfnv duh hpsor|hg1 Hvwlpdwlrq lvvxhv zlwk wkhvh qhz prghov

duh yhu| fkdoohqjlqj/ xqiruwxqdwho|/ dqg/ shukdsv lq frqvhtxhqfh/ hpslulfdo vwxglhv zlwk wkhvh qhz

prghov kdyh |hw wr eh frqgxfwhg1

Lq wkh iroorzlqj/ zh zloo lqwurgxfh d qhz jhqhudol}hg prgho wkdw lv h{leoh hqrxjk wr gr dzd|

vwrfkdvwlf vlqjxodulw|/ zkloh uhwdlqlqj wudfwdelolw| dqg idflolwdwlqj hvwlpdwlrq1 Rxu prgho lv

_+E|ðc Aæä ' k+E/c |ðc Aæä_|n òEo E|ðä c ô æä_` E|ðä n j0_`æ E|ðä c +5,

zkhuh E`æäaæ'� lv d idplo| ri lqghshqghqw vwdqgdug Eurzqldq prwlrqv/ lqghshqghqw ri wkh vwdqgdug

Eurzqldq prwlrq `é Wkh qhz idplo| ri E`æäaæ'� zloo dfw dv wkh h{wud vrxufh ri vkrfnv wr wkh |lhog

fxuyhv> wkh| fdq eh lqwhusuhwhg dv phdvxuhphqw huuruv/ zklfk fdq dulvh iurp d ydulhw| ri vrxufhv/ d

qrwdeoh rqh ehlqj wkdw wkh gdwd xvhg lq wkh surfhgxuh zhuh rewdlqhg iurp vrph suholplqdu| |lhog

fxuyh ?wwlqj surfhgxuh vxfk dv vsolqhv lq PfFxoorfk +4<:4/ 4<:8,/ errwvwudsslqj dv lq Idpd dqg

Eolvv +4<;:, ru nhuqho vprrwklqj dv lq Olqwrq/ Pdpphq/ Qlhovhq dqg Wdqjjddug +4<<<,16

Lq frqwudvw wr pruh udglfdo dssurdfkhv vxfk dv wkdw ri Nhqqhg| +4<<7/ 4<<:,/ zkhuh wkh qhz

vrxufh ri vkrfnv lq wkh hfrqrp| lv frpsohwho| gl>huhqw iurp wkh ?qlwh vhw ri Eurzqldq prwlrqv lq

wkh ruljlqdo KMP sdshuv/ wkh prgho sursrvhg deryh fdq eh frqvlghuhg dv d vwrfkdvwlfdoo| h{whqghg

yhuvlrq ri wkh ruljlqdo KMP prgho1 Wkh prgho dlpv wr lqfrusrudwh phdvxuhphqw ru revhuydwlrq

huuruv zkloh suhvhuylqj dv pxfk dv srvvleoh wkh KMP*v vkrfn vwuxfwxuh1 Sxw gl>huhqwo|/ wkh frp0

prq Eurzqldq prwlrq+v, wkdw gulyh wkh zkroh |lhog fxuyh g|qdplfv vwloo surylgh wkh rqo| vrxufh

ri hfrqrplf xqfhuwdlqw| hpehgghg lq wkh xqghuo|lqj prgho/ zkloh wkh lglrv|qfudwlf vkrfnv wr hdfk

5Iru h{dpsoh/ lq wkh rqh0idfwru prgho deryh/ zh fdq kdyh wkh iroorzlqj ghwhuplqlvwlf uhodwlrqvkls iru dq| wzr |lhog

g|qdplfv=

g|+w> W4,ð+$> w> W5,ý ò|+$> w> W4,ð+$> w> W5, @ g|+w> W5,ð+$> w> W4,ý ò|+$> w> W5,ð+$> w> W4,gw

6]hur0frxsrq |lhog fxuyhv duh qrw revhuydeoh zlwk wudghg lqvwuxphqwv lq wkh ?{hg0lqfrph pdunhwv ehfdxvh erqgv

zlwk wlph wr pdwxulw| juhdwhu wkdq 4 |hdu duh qrupdoo| frxsrq0ehdulqj erqgv1 Vr wkh +}hur frxsrq, |lhog fxuyhv pxvw

eh h{wudfwhg e| |lhog fxuyh ?wwlqj surfhgxuhv vxfk dv PfFxoorfk +4<:4/ 4<:8,*v vsolqhv1 Lq dq| fdvh/ wkh gdwd xvhg wr

fduu| rxw wkhvh surfhgxuhv gr qrw fruuhvsrqg h{dfwo| wr wkh wkhruhwlfdo sulfh1 Vshfl?fdoo|/ zh xvxdoo| revhuyh txrwhv

ri elg dqg dvn sulfhv rewdlqhg iurp d whohskrqh vxuyh| ri wkh uhjlvwhuhg erqg ghdohuv1 Dovr/ wkhuh duh wd{ gl>huhqfh

dqg oltxlglw| h>hfwv wkdw fdq eh lqwhusuhwhg dv surylglqj udqgrp huuruv lq wkh revhuyhg erqg sulfhv1

8

Page 7: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

srlqw ri wkh |lhog fxuyhv duh vwdwlvwlfdo qrlvh1 Wkh xqghuo|lqj hfrqrp| lv wkhuhiruh vwloo d frpsohwh

pdunhw/ zkhuh dq| lqvwuxphqw lv khgjhdeoh/ zkloh wkh revhuyhg hfrqrp| lv qrw qhfhvvdulo| frpsohwh

dq| pruh1 Zh wkxv wdnh d kdoi zd| dssurdfk ehwzhhq wkh ruljlqdo KMP prghov dqg wkdw ri Nhqqhg|

+4<<7/ 4<<:, dqg Vdqwd Fodud dqg Vruqhwwh +5333,1

Wkh idploldu krprvnhgdvwlf ruwkrjrqdo phdvxuhphqw huuru vwuxfwxuh fdq eh uhdglo| mxvwl?hg dv dq

hfrqrphwulfdoo| frqyhqlhqw ghylfh wkdw lv zlgho| xvhg/ dowkrxjk uhod{lqj wklv vlpsoh vwuxfwxuh lv txlwh

ihdvleoh iru rxu whfkqltxh1 Wklv vwuxfwxuh fdq dovr dulvh iurp wkh dv|pswrwlf surshuwlhv hvwdeolvkhg

lq OPQW +4<<<,/ wkdw lv/ wkh huuru whup lv dvvxphg wr frph iurp wkh nhuqho hvwlpdwlrq surfhgxuh

xvhg lq wkdw sdshu1 Wkh huuru yduldqfh lv vpdoo/ dqg jrhv wr }hur dv wkh qxpehu ri revhuydwlrqv xvhg

lq h{wudfwlqj wkh vlqjoh shulrg whup vwuxfwxuh jrhv wr lq?qlw|1 Wkh xqfruuhodwhgqhvv dvvxpswlrq lv

mxvwl?hg surylghg wkh julg ri pdwxulw| srlqwv lv hlwkhu ?{hg ru lqfuhdvhv exw dw d vorz hqrxjk udwh vr

wkdw wkh xqghuo|lqj nhuqho hvwlpdwruv duh lqghshqghqw1 Zh vxssrvh wkdw wkh julg iô æjaæ'� ehfrphvghqvh rq wkh pdwxulw| lqwhuydo dv a lv dvvxphg wr lqfuhdvh dw d fhuwdlq udwh1

6 Hvwlpdwlrq Phwkrg

Lq wklv vhfwlrq zh lqwurgxfh rxu hvwlpdwlrq whfkqltxhv edvhg rq wkh |lhog fxuyh hyroxwlrq1 Dv dujxhg

lq Mh>uh|/ Olqwrq/ Qjx|hq +4<<<e,/ zh suhihu wr xvh |lhogv lqvwhdg ri iruzdugv vlqfh wkh iruphu/

zklfk lv dq dyhudjh ri wkh odwwhu/ duh hdvlhu wr hvwlpdwh1 Krzhyhu/ wkh hvwlpdwru ghyhorshg khuh fdq

eh dgdswhg iru iruzdug fxuyh hyroxwlrq zlwk plqru prgl?fdwlrq1

Vxssrvh zh revhuyh wkh |lhog fxuyhv dw ? srlqwv lq wlph |ð/ ð ' �c é é é c ?1 Dw dq| wlph |ð/ zh

revhuyh d udqgrp qxpehu/ aðc ri srlqwv dorqj wkh |lhog fxuyh i+E|ðc A�äc é é é c +E|ðc Aaðäj/ zkhuh Aæ lvwkh wlph wr pdwxulw|1 Iru hdvh ri qrwdwlrq/ zh zloo xvh wkh Pxvlhod sdudphwhul}dwlrq ô ' A ý |(zh zloo dovr gurs wkh vxevfulsw ð lq að dqg dvvxph iru vlpsolflw| wkdw zh revhuyh wkh |lhog fxuyh

dw wkh vdph pdwxulwlhv iru hdfk wlph srlqw1 Zh vkdoo ixuwkhu vxssrvh wkdw wkh revhuyhg pdwxulwlhv

ô�c é é é c ôa fdq eh wkrxjkw ri dv ehlqj udqgrp gudzv iurp vrph frqwlqxrxv ghqvlw| ixqfwlrq Ré Ohw

{+E|ðc ô ä ' +E|ðn�c ôäý +E|ðc ôä1 Iru vlpsolflw| lw lv dovr dvvxphg wkdw doo wlph lqwhuydov |ð wr |ðn� duh

htxdoo| vsdfhg/ wkdw lv/ {|ð ' |ðn� ý |ð ì {| iru doo ð dqg frqvhtxhqwo| |? ' ?{|é

Lw lv vwudljkwiruzdug wr vhh iurp wkh fkdudfwhulvwlfv ri wkh gulylqj Eurzqldq prwlrq wkdw

. E_+E|c Aæä_+E|c A&ä mI|3 ä '

;AA?AA=òEo E|ä c ô æäòEo E|ä c ô &ä_| li æ 9' &

dòEo E|ä c ô æä2 n j20o_| li æ ' &é

Zh zloo xvh wkhvh uhvwulfwlrqv wr jhqhudwh hvwlpdwlqj htxdwlrqv iru wkh xqnqrzq yrodwlolw| ixqfwlrq

9

Page 8: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ò1 Vwdfnlqj doo wkh furvv0|lhog uhvwulfwlrqv wrjhwkhu zh kdyh dw dq| wlph |

{?.

3EEEEEC{+E|c ô�ä

2 {+E|c ô �ä{+E|c ô2ä é é é ü ü ü{+E|c ô �ä{+E|c ô 2ä {+E|c ô2ä

2 ü ü ü ü ü ü111

1111 1 1 ü ü ü

{+E|c ô�ä{+E|�c ôaä {+E|c ô2ä{+E|�c ôaä ü ü ü {+E|c ôaä2

4FFFFFD

'

3EEEEECò2Eoc ô�ä n j

20 òEoc ô �äòEoc ô 2ä ü ü ü ü ü ü

òEoc ô �äòEoc ô2ä ò2Eoc ô2ä n j20 ü ü ü ü ü ü

111111

1 1 1 ü ü üòEoc ô �äòEoc ôaä òEoc ô2äòEoc ôaä ü ü ü ò2Eoc ôaä n j20

4FFFFFD +6,

iru d jlyhq vhw ri pdwxulw| srlqwv ô �c é é é c ôa é Ohwwlqj a $4 zh rewdlq d srsxodwlrq fulwhulrq ixqfwlrq

wkdw wkh yrodwlolw| vwuxfwxuh òEoc ô ä dqg phdvxuhphqw huuru yduldqfh j20 pxvw vdwlvi|/ qdpho|]. d{+E|c ô ä{+E|c ô âäý òEo E|ä c ô äòEo E|ä c ô ä{o

2REôäREô âä_ô â_ô c +7,

zkhuh R Eéä lv wkh pdujlqdo ghqvlw| glvwulexwlrq ri pdwxulw| wlphv1 D prgl?hg vdpsoh yhuvlrq ri wklv

ixqfwlrq/ zklfk ljqruhv lqirupdwlrq dorqj wkh gldjrqdo/ lv sursruwlrqdo wr

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

d{+E|ðc ô æä{+E|ðc ô&äý òEo E|ðä c ô æäòEo E|ðä c ô&ä{?o2 c +8,

zklfk ghshqgv rq wkh ixqfwlrq ò exw qrw rq j20é Zh gholehudwho| gr qrw xvh wkh gldjrqdo uhvwulfwlrq

wr dyrlg wkh qhhg ri hvwlpdwlqj j20 lq wklv vdpsoh yhuvlrqé Vr/ frqwudu| wr wkh dssurdfk lq d uhodwhg

vwxg|/ Mh>uh|/ Olqwrq/ Qjx|hq +4<<<e,/ zkhuh rqo| lqirupdwlrq dorqj wkh gldjrqdo ri wkh prphqw

frqglwlrq pdwul{ lv xvhg/ zh khuh xvh rqo| lqirupdwlrq iurp wkh r>0gldjrqdo hohphqwv1 Krzhyhu/ wkh

uhvxowlqj yrodwlolw| vwuxfwxuh hvwlpdwh pd| eh xvhg wrjhwkhu zlwk wkh gldjrqdo lqirupdwlrq diwhuzdugv

wr edfn rxw ru pdnh lqihuhqfh derxw j20é

Lw lv lqvwuxfwlyh wr gljuhvv khuh iru d glvfxvvlrq rq wkh +suredeo| lpsolflw, gl>huhqfh ehwzhhq

wkh Pdunryldq dqg qrq0Pdunryldq dssurdfk zlwk uhjdugv wr phdvxuhphqw huuruv1 Lq wkh iruphu

dssurdfk/ lw lv riwhq wkh fdvh wkdw vwuxfwxudo sdudphwhuv frqwuroolqj wkh g|qdplfv ri wkh vkruw whup

lqwhuhvw udwhv +wkh riwhq xvhg vwdwh yduldeohv, duh hvwlpdwhg/ zlwk wkh +lpsolflw, dvvxpswlrqv wkdw wkh

vkruw udwhv duh revhuyhg zlwkrxw huuruv1 Wkhq/ wr uhfrqfloh wkh idfw wkdw rwkhu revhuyhg |lhogv zlwk

pdwxulw| orqjhu wkdq }hur duh qrw shuihfwo| fruuhodwhg/ zklfk wkh| vkrxog eh/ gxh wr ehlqj gulyhq e|

wkh vdph vwdwh yduldeohv/ lw lv qhfhvvdu| wr lqwurgxfh revhuydwlrq huuruv iru rwkhu |lhogv1 Vr lq h>hfw/

zh zrxog 3vkliw4 doo revhuydwlrq huuruv wrzdug rwkhu |lhog ohyhov/ dqg pdnh d +shukdsv xquhdolvwlf ru

xquhdvrqdeoh, dvvxpswlrq wkdw zh frxog revhuyh wkh vkruw udwh zlwk xwprvw dffxudf|1

:

Page 9: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Lq wkh KMP iudphzrun/ wkhuh lv dovr dq lqhylwdeoh qhhg wr lqwurgxfh revhuydwlrq huuruv li zh duh

wr uhfrqfloh iru wkh ohvv wkdq shuihfw fruuhodwlrq revhuyhg lq |lhog gdwd dfurvv pdwxulw| glphqvlrq17

Lw lv lqwhuhvwlqj wr srlqw rxw wkh idfw wkdw iurp rxu dqdo|vlv/ surylghg rxu vshfl?fdwlrq lv uhdvrqdeo|

fruuhfw/ rqh fdq vhh wkh srwhqwldoo| vhyhuh gh?flhqf| ri qrw xvlqj wkh furvv0vhfwlrqdo uhvwulfwlrq dorqj

wkh pdwxulw| glphqvlrq lq hvwlpdwlqj dq KMP prgho= wkh hvwlpdwhv wkxv pd| eh vhyhuho| eldvhg gxh

wr wkh suhvhqfh ri wkh phdvxuhphqw huuruv18

Wkh orfdol}hg yhuvlrq ri wkh deryh fulwhulrq ixqfwlrq lv] ?3�[ð'�

a[æ'�

a[&'�

& õ'æ

d{+E|ðc ô æä{+E|ðchô&äý òEhoðchô æäòEhoðchô &ä{?o2gûoEoðýhoðägûô Eô æýhô æägûô Eô &ýhô &ä_hoð_hô æ_hô &c

zklfk lv wr eh plqlpl}hg zlwk uhvshfw wr ixqfwlrqv òé Zh zloo xvh wkh ghowd phwkrg wr vroyh iru wkh

7Rqh srvvleoh dssurdfk/ zklfk lv vrphzkdw frpsdudeoh wr wkh frpprq dssurdfk lq wkh Pdunryldq olwhudwxuh +vhh

Gx!h dqg Vlqjohwrq +4<<:,/ dprqj rwkhuv,/ lv wr dvvxph wkdw zh fdq revhuyh d fduhixoo|0fkrvhq qxpehu ri srlqwv

dorqj wkh |lhog fxuyhv zlwkrxw phdvxuhphqw huuruv1 Iru lqvwdqfh/ iru d wzr0idfwru KMP prgho/ dvvxph wkdw zh fdq

revhuyh 6 srlqwv zlwk gl>huhqw pdwxulwlhv dorqj wkh |lhog fxuyhv zklfk zh zloo frqyhqlhqwo| dvvxph wr eh revhuyhg

zlwkrxw phdvxphqw huuruv1 Vhfrqg0prphqw uhvwulfwlrq rq wkh g|qdplfv ri iruzdug udwhv dqg furvv0uhvwulfwlrqv duh=

Hkgi +w> W4,

5l@ ì+u +w, > o +w, > W4,

5 . ð+u +w, > o +w, > W4,5 @ \44

Hkgi +w> W5,

5l@ ì+u +w, > o +w, > W5,

5 . ð+u +w, > o +w, > W5,5 @ \55

Hkgi +w> W6,

5l@ ì+u +w, > o +w, > W6,

5 . ð+u +w, > o +w, > W6,5 @ \66

H ^gi +w> W4, gi +w> W5,` @ ì+u +w, > o +w, > W4,ì+u +w, > o +w, > W5, . ð+u +w, > o +w, > W4,ð+u +w, > o +w, > W5, @ \45

H ^gi +w> W4, gi +w> W6,` @ ì+u +w, > o +w, > W4,ì+u +w, > o +w, > W6, . ð+u +w, > o +w, > W4,ð+u +w, > o +w, > W6, @ \46

H ^gi +w> W5, gi +w> W6,` @ ì+u +w, > o +w, > W5,ì+u +w, > o +w, > W6, . ð+u +w, > o +w, > W5,ð+u +w, > o +w, > W6, @ \56

Wkh lqwxlwlrq khuh lv wkdw zh kdyh 9 ixqfwlrqv ri eì+u +w, > o +w, > Wm, dqg eð+u +w, > o +w, > Wm,> m @ 4> 5> 6> vr zh qhhg 9 prphqwfrqglwlrqv wr h{dfwo| lghqwli| wkhvh ixqfwlrqv1 Lq jhqhudo/ li zh kdyh dq Pýidfwru prgho/ zh zrxog qhhg +dw ohdvw,

Q srlqwv dorqj wkh |lhog fxuyhv wr jhqhudwh Q .Q ú +Q ý 4, @5 @ Q +Q . 4, @5 è P úQ prphqw frqglwlrqv vr wkh

prgho zloo eh h{dfwo| +ru ryhu, lghqwl?hg +P úQ zloo eh wkh qxpehu ri xqnqrzq,/ l1h1 Q @ 5P ý 41 Iru d srvvleoh

hpsulfldo lpsohphqwdwlrq/ vlqfh wkuhh idfwru prghov duh zhoo0grfxphqwhg wr eh deoh wr fdswxuh wkh hpslulfdo ehkdylru

ri wkh XV erqg pdunhw/ wkh qxpehu ri srlqwv zh zrxog orrn dw zloo eh 5 ú 6ý 4 @ 8 iru h{dfw lghqwl?fdwlrq1Wklv dssurdfk lv vlplodu wr wkdw ri wkh Pdunryldq olwhudwxuh/ lq wkh vhqvh wkdw zh dvvxph wkdw wkhuh duh vrph

3vshfldo4 srlqwv zklfk duh revhuyhg zlwkrxw huuruv/ dqg 3vkliw4 doo revhuydwlrq huuruv wr rwkhu srlqwv1 Wkh dssurdfk lv

krzhyhu qrw yhu| dsshdolqj iru d qxpehu ri uhdvrqv/ rqh ehlqj wkdw wkhuh lv qr qdwxudo fdqglgdwh iru wkhvh Q srlqwv18Iru h{dpsoh/ lq rxu suhylrxv sdshu/ li wkh qhz prgho zlwk phdvxuhphqw huuruv lv pruh dssursuldwh/ wkhq wkh

suhylrxv hvwlpdwru/ zklfk h>hfwlyho| xvh rqo| lqirupdwlrq dorqj wkh gldjrqdo/ zloo kdyh wkh dv|pswrwlfv dv

õýeð+u +w, > á,5üý ýð+u +w, > á,5 . â5%üô gý$ Q

#3>

5ýð4+u +w, > á,

4üq>wq

säkq>WO +wq> u+w,,

.Rýâ7%ü$=

Lw lv dssduhqw wkdw wkh eldv fdq eh vhyhuh li wkh yduldqfh ri wkh phdvxuhphqw huuruv lv uhodwlyho| odujh1

;

Page 10: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ixqfwlrqdo vroxwlrq/ eòEoc ô ä1 Wkh ?uvw rughu frqglwlrq/ rewdlqhg e| gl>huhqwldwlrq zlwk uhvshfw wr wkh

ixqfwlrq eòEéc éä dv vkrzq lq wkh dsshqgl{/ lv

f '�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oä

ûégûô Eô ý ô æä

] eòEoc rägûô Erý ô&ä_rngûô Eô ý ô&ä] eòEoc rägûô Erý ô æä_r

èý �

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

eòEoc ôägûoEo ý oðä +9,

ûégûô Eô ý ô æä

] eòEoc rä2gûô Erý ô &ä_rngûô Eô ý ô &ä] eòEoc rä2gûô Erý ô æä_r

è{?é

Wkh ixqfwlrq eò ì f lv dozd|v d vroxwlrq wr +9,1 Wkh vroxwlrq eòEoc ôä fdq dovr eh zulwwhq +dvvxplqj

f*f ' f lq wkh sdwkrorjlfdo fdvh, dv

eòEoc ôä ' U eM� Eoc ô c rä eòEoc rä_rU eM2 Eoc ô c rä eò2Eoc rä_rc +:,

zkhuh

eM� Eoc ô c rä '�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oä

û dgûô Eô ý ô æägûô Erý ô &ä ngûô Eô ý ô &ägûô Erý ô æäo

dqg

eM2 Eoc ô c rä ' �

a2{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oä dgûô Eô ý ô æägûô Erý ô &ä ngûô Eô ý ô &ägûô Erý ô æäo é

Wkh h{suhvvlrq +:, lv d qrqolqhdu lqwhjudo htxdwlrq lqyroylqj wkh olqhdu rshudwruv eM� dqg eM2é Wkhvhtxdqwlwlhv ghshqg rqo| rq revhuyhg gdwd dqg lqghhg duh mxvw nhuqho zhljkwhg vdpsoh dyhudjhv1 Uh0

odwlrq +:, vxjjhvwv wkh iroorzlqj lwhudwlrq iru wkh fdofxodwlrq ri eòEoc ô ä GeòdfoEoc ô ä ' ZEoc ô ä

eòd@n�oEoc ô ä '

U eM� Eoc ô c rä eòd@oEoc rä_rU eM2 Eoc ô c räíeòd@oEoc räì2 _rc @ ' fc �c é é é c

iru vrph jlyhq vwduwlqj ydoxh Zé Wkh lqwhjudwlrqv lq eòd@n�oEoc ôä duh xqlglphqvlrqdo dqg fdq eh frp0

sxwhg qxphulfdoo|1 Zh glvfxvv wklv ixuwkhu lq wkh dssolfdwlrq vhfwlrq ehorz1

<

Page 11: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Rxu lwhudwlyh phwkrg lv fdoohg vxffhvvlyh dssur{lpdwlrq1 Iru d ghwdlohg glvfxvvlrq zh uhihu wkh

uhdghu wr Ndqwrurylfk dqg Dnlory +4<97, dqg Oxhqehujhu +4<9<,1 Vhh dovr Kdvwlh dqg Wlevkludql

+4<<3,/ Pdpphq/ Olqwrq/ dqg Qlhovhq +4<<<, dqg Olqwrq/ Pdpphq/ Qlhovhq/ dqg Wdqjjddug +5333,

iru uhodwhg frpsxwdwlrqv1

7 Dv|pswrwlf Surshuwlhv lq wkh Vlqjoh Idfwru Fdvh

Ehiruh surfhhglqj zlwk wkh ghyhorsphqw ri wkh dv|pswrwlf glvwulexwlrqv ri wkh hvwlpdwru gh?qhg

deryh/ zh ?uvw lqwurgxfh vrph qrwdwlrq dqg dvvxpswlrqv1 Frqvlghu d vhpl0pduwlqjdoh gh?qhg dv

iroorzv

_o E|ä ' >Eo E|ää_|n j Eo E|ää _` E|ä c +;,

zkhuh vx!flhqw uhjxodulw| frqglwlrqv duh dvvxphg wr hqvxuh wkh deryh vwrfkdvwlf gl>huhqwldo htxdwlrq

kdv d vwurqj vroxwlrq1

Gh?qlwlrq +Fkurqrorjlfdo Orfdo Wlph1 Vhh Skloolsv dqg Sdun +4<<;,,1 Wkh fkurqrorjlfdo orfdo

wlph ri wkh vhpl0pduwlqjdoh o gh?qhg e| +;, dw srlqw @ ryhu wkh wlph lqwhuydo dfc |o lv gh?qhg dv

uE|c @ä '�

j2E@ä*ð4"<f

2"

] |

f

��oErä3@�÷"jEoErää2_r

'�

j2E@äuE|c @äc

zkhuh uE|c @ä lv wkh orfdo wlph ri o dw srlqw @ ryhu wkh wlph lqwhuydo dfc |o1

Ohppd +Wkh Rffxsdwlrq Wlph Irupxod1 Vhh Uhyx} dqg \ru +4<<<,,1 Iru wkh vhpl0pduwlqjdoh

o gh?qhg e| +;, zlwk txdgudwlf yduldwlrq surfhvv koc olr/ dqg iru hyhu| Eruho ixqfwlrq s ri o] |

f

s EoErää _ koc olr '] n"

3"sE@äuE|c @ä_@c

zkhuh uE|c @ä lv wkh orfdo wlph ri o dw srlqw @ ryhu wkh wlph lqwhuydo dfc |o1

Gluhfw dssolfdwlrqv ri wkh Rffxsdwlrq Wlph Irupxod dorqj zlwk wkh gh?qlwlrq ri Fkurqrorjlfdo

Orfdo Wlph surylgh wkh iroorzlqj wzr uhvxowv/ zklfk zloo eh xvhg uhshdwhgo| lq rxu surriv] |

f

s EoErää _r '

] |

f

s EoErää

j2 EoErää2_ koc olr +<,

'

] n"

3"sE@äuE|c @ä_@

dqg ixuwkhu/ iru dq| nhuqho ixqfwlrq gEüä dqg frqwlqxrxv erxqghg ixqfwlrq sEüä/

*ð4ûòf

û

] |

f

g

ë%ý oEräû

ês EoErää _r ' *ð4

ûòf�

û

] n"

3"g

ë%ý @û

êRE@äuE|c @ä_@ +43,

43

Page 12: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

' *ð4ûòf

] n"

3"gE^äRE%n û^äuE|c %n û^ä_^

$ uE|c %äsE%ä ' ï@éréýuoE|c oä

ü1

Dvvxpswlrqv wkdw pxvw eh lpsrvhg wr vwxg| wkh dv|pswrwlf glvwulexwlrq ri wkh hvwlpdwru duh wkh

iroorzlqj=

Dvvxpswlrq D41 +Uhfxuuhqfh,1 Wkh surfhvv ioE|ä( | A fj gh?qhg lq +;, lv uhfxuuhqw> wkdw lv iruhyhu| srlqw @ rq wkh vxssruw ri wklv surfhvv wkh fkurqrorjlfdo orfdo wlph u E|c @ä$4 dv |$41Dvvxpswlrq D51 +Erxqghgqhvv,1 Wkh guliw dqg gl>xvlrq ixqfwlrqv duh +orfdoo|, erxqghg

mk E/c |ðäý kE/c |æämn m#EoE|ðääý #EoE|æääm é ä m+ E|ðäý +E|æäm +44,

iru frqvwdqwv äc dqg wkhuh h{lvwv vrph f ÷ D ÷ �*2 vxfk wkdw

E{?äD

uE|c @ä

] |?

f

mgûEo ý o ErääkE/c rä_rm ' ï E�ä é +45,

Dvvxpswlrq D61 +Vdpsolqj Frqglwlrqv dorqj wkh wlph glphqvlrq,1 Ohw erwk wkh glvfuhwl}dwlrq

zlgwk {| dqg edqgzlgwk iru wkh nhuqho hvwlpdwruv û ghshqg rq wkh vdpsoh vl}h ? zklfk jurzv wr

lq?qlw|1 Wr lqglfdwh wklv zh zloo khuhdiwhu ghqrwh wkhvh txdqwlwlhv dv {? dqg û? uhvshfwlyho|1 Wkh

vdpsoh iuhtxhqf| {? $ f dv ?$4c zklfk lv uhihuuhg wr dv wkh clq?oo dvvxpswlrq*1 Wkh wlph vsdq rirevhuydwlrqv |?n� ' ?{? $4 dv ?$4c uhihuuhg wr dv wkh corqj vsdq dvvxpswlrq*19 Wkh edqgzlgwksdudphwhu û? & f lq vxfk d zd| wkdw û?*{? $4 dqg E{?ä

q �û?uE|c @ä ' ïE�ä iru doo f ÷ q ÷ �

2dqg

hyhu| srlqw @ rq wkh vxssruw ri wkh surfhvv ioE|ä( | A fj gh?qhg lq +;,> uE|c @ä lv wkh fkurqrorjlfdoorfdo wlph ri wkh surfhvv ioE|ä( | A fj1

Dvvxpswlrq D71 Wkh nhuqho g kdv frpsdfw vxssruw+ dýä�c ä�o/ vd|,/ lv v|pphwulf derxw }hur/dqg lv frqwlqxrxvo| gl>huhqwldeohé

Dv vkrzq lq wkh dsshqgl{/ wkh hvwlpdwru eòEoc ôä zloo kdyh wkh dv|pswrwlf ehkdylru vwdwhg irupdoo|lq wkh iroorzlqj wkhruhp=

Wkhruhp 41 Vxssrvh wkdw dvvxpswlrqv D140D7 krog1 Wkhq/

a

vûoûôu E|?c oä

{?deòEoc ôäý òEoc ôäo _$ ù

#fc 2 ngne òEoc ôä

2UòEoc räeR Erä _r

R EôäýUòEoc rä2R Erä _r

ü2$é

Uhpdun 41 Rxu dssurdfk lv edvhg rq d frpelqdwlrq ri vrph uhfhqw dgydqfhv lq wkh qrqsdudphwulf

olwhudwxuh1 Dorqj wkh pdwxulw| glphqvlrq/ zh gudz rq wkh dssurdfk sxw iruzdug e| Olqwrq/ Pdpphq/

Qlhovhq/ dqg Wdqjjddug +4<<<,/ zkloh lqirupdwlrq iurp wkh hyroxwlrq ri lqwhuhvw udwhv ryhu wlph lv

9Wklv dvvxpswlrq fdq eh gursshg iru wkh gl>xvlrq frh!flhqw hvwlpdwru surylghg lq Wkhruhp 41

44

Page 13: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

xvhg lq d idvklrq vlplodu wr wkh qrqsdudphwulf hvwlpdwlrqv ri vwrfkdvwlf gl>xvlrq surfhvvhv sursrvhg

e| Ioruhqv0]plurx +4<<6,/ Vwdqwrq +4<<:,/ Mldqj dqg Nqljkw +4<<;,/ dqg Edqgl dqg Skloolsv +4<<<,1

Uhpdun 51 Wklv prgho zloo fryhu doo rqh idfwru KMPprghov wkdw kdyh ehhq hpsor|hg lq wkh sdvw

olwhudwxuh/ vxfk dv wkh rqh idfwru prgho lq rxu suhylrxv sdshu/ Mh>uh|/ Olqwrq/ dqg Qjx|hq +4<<<e,1

Wr fryhu wkh KMP prgho sursrvhg lq Shduvrq dqg ]krx +5333,/ zh vlpso| qhhg wr uhsodfh wkh nhuqho

ixqfwlrq gûoEoðýhoðä wkdw zh xvh deryh e| d wzr glphqvlrqdo nhuqho gûoEEoðc rðäý Ehoðc hrðää1 Qrwh wkdw/lq frqwudvw wr wkh pxowl0idfwru prgho glvfxvvhg lq wkh qh{w vhfwlrq/ wkhlu prgho lv yhu| uhvwulfwlyh lq

wkh vhqvh wkdw wkh udqgrpqhvv lv lqwurgxfhg lqwr wkh prgho e| mxvw rqh Eurzqldq prwlrq/ dowkrxjk

wkh yrodwlolw| lv h{whqghg wr d ixqfwlrq ri wzr yduldeohv1 Wkhlu prgho fdq dovr eh wkrxjkw ri dv d

vshfldo fdvh ri rxu wzr idfwru prgho/ zkhuh wkh yrodwlolw| vwuxfwxuh rq wkh vhfrqg Eurzqldq prwlrq

lv vlpso| }hur1

8 Pxowl0Idfwru Prghov

Pxowl0idfwru prghov/ erwk sdwk lqghshqghqw dqg sdwk ghshqghqw/ kdyh ehhq zlgho| vwxglhg lq wkh

olwhudwxuh/ lq dq dwwhpsw wr lpsuryh wkh ?w ri wkh prgho zkhq phdvxuhg djdlqvw hpslulfdo gdwd/ iru

lqvwdqfh fdswxulqj g|qdplfv ri wkh revhuyhg vkruw whup lqwhuhvw udwh1: Edodqflqj ehwzhhq wkh h{l0

elolw| surylghg e| pruh idfwruv dqg srwhqwldo ryhu?wwlqj sureohpv soxv orvlqj dqdo|wlfdo wudfwdelolw|

dvvrfldwhg zlwk pruh idfwruv/ uhvhdufkhuv kdyh frpprqo| sursrvhg wzr2wkuhh idfwru prghov/ dv flwhg

hduolhu1;

Wkh deryh dssurdfk frxog eh h{whqghg wr wkh pxowl0idfwru fdvh lq d vwudljkwiruzdug idvklrq/ dw

ohdvw iru wkh vwhs ri vhwwlqj xs wkh fulwhuld ixqfwlrq1 Ghulydwlrq ri wkh vroxwlrq zloo txlfno| ehfrph

yhu| exughqvrph dqg whglrxv/ dv ghprqvwudwhg iru wkh wzr idfwru fdvh odwhu/ dowkrxjk wkh edvlf lghd

lv udwkhu vwudljkwiruzdug1 Krzhyhu/ frpsolfdwlrqv wkdw duh lqwurgxfhg lqwr wkh hvwlpdwlrq surfhgxuhv

duh qrw lq dq| vhqvh wulyldo dqg zloo kdyh frqvhtxhqfhv rq dv|pswrwlf ehkdylru1 Wkh prvw gl!fxow

kxugoh/ lq sulqflsoh/ dulvhv iurp wkh qrq h{lvwhqfh ri orfdo wlph lq kljk glphqvlrqdo fdvhv iru Eurzqldq

:Wkh sudfwlfdo lpsolfdwlrq ri wklv sduwlfxodu olplwdwlrq rq sulflqj ?{hg0lqfrph dvvhwv lv dujxdeo| ghedwdeoh1 Idloxuh

wr fdswxuh wkh g|qdplfv ri wkh vkruw udwhv dffxudwho| grhv qrw qhfhvvdulo| lpsdlu wkh delolw| wr sulfh ?{hg0lqfrph

dvvhwv1 Iru lqvwdqfh/ Exvhu/ Khqghuvkrww dqg Vdqghuv +4<<3,/ Kxoo dqg Zklwh +4<<3, fodlp wkdw rqh0idfwru lqwhuhvw

udwh prghov zlwk h{leoh vshfl?fdwlrq fdq jhqhudwh lqwhuhvw0udwh ghulydwlyhv sulfhv vlplodu wr wkrvh ri 50idfwru prghov1

Qhyhuwkhohvv/ hyhq li wkhlu wkhvlv lv ydolg/ wkh vhdufk iru prghov wkdw ?w hpslulfdo g|qdplfv ri wkh xqghuo|lqj frqvhqvxv

lv vwloo ri juhdw lqwhuhvwv/ dw ohdvw iru lqwhuqdo frqvlvwhqf|1;Pruh udglfdo dowhuqdwlyh dlphg dw dffrprgdwlqj pruh frpsoh{ ehkdylruv ri wkh whup vwuxfwxuh*v g|qdplfv lv wr

uhvruw wr vwrfkdvwlf surfhvvhv gl>huhqw iurp Eurzqldq prwlrqv wr suhvhqw vkrfnv wr wkh xqghuo|lqj hfrqrp|1 Dprqj

wkhp/ Ohy| surfhvv/ iudfwlrqdo Eurzqldq prwlrq/ dqg lq sduwlfxodu/ mxps0gl>xvlrq surfhvv kdyh ehfrph idyrulwh

fdqglgdwhv lq pdq| vwxglhv1

45

Page 14: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

prwlrq1 Lq sduwlfxodu/ iru d _0glphqvlrqdo Eurzqldq prwlrq zlwk _ è ô/ orfdo wlph grhv qrw h{lvw

dqg wklv pdnhv d jhqhudo vroxwlrq xvlqj rxu dssurdfk fkdoohqjlqj li qrw lpsrvvleoh1 Iru wkh fdvh

ri _ ' 2/ dowkrxjk wkh orfdo wlph grhv h{lvw/ lwv qrqsdudphwulf hvwlpdwh frqyhujhv dw d *L}E?ä udwh/

zklfk lv wrr vorz iru uholdeoh lqihuhqfhé Lq jhqhudo/ qr sdudphwulf hvwlpdwhv kdyh ehhq sursrvhg iru d

jhqhudo gl>xvlrq surfhvv zlwk _ è 2c dqg prvw fdvhv wkdw duh frqvlghuhg lq wkh olwhudwxuh duh vhyhuho|

uhvwulfwlyh 8 vhh Euxjlhuh +4<<4, dqg Nqljkw/ Ol dqg \xdq +4<<<, iru uhodwhg vwxglhv1

Wr surfhhg lq oljkw ri wkhvh frqvlghudwlrqv/ rxu vwudwhj| khuh lv wr pdnh vrph dgglwlrqdo dvvxps0

wlrqv1 Iluvw zh zloo eh frqwhqw wkdw rxu yrodwlolw| ixqfwlrqv duh ri pxowlsolfdwlyho| vhsdudeoh irup/

l1h1/

òE/c |c A ä ' òE/c |c A ý |ä '�\6'�

òE%6 E|ä c ôä

zkhuh wkh vhw ri %6c6 ' �c é é é c� duh wkh vwdwh yduldeohv1 Zh dovr dvvxph wkdw wkh orfdo wlph fdq

eh ghfrpsrvhg lqwr

u E|?c E%6c6 ' �c é é é c�ää '�\6'�

u E|?c %6ä c

dqg vlqfh wkh rqh glphqvlrqdo surfhvv %6 E|ä fdq eh idluo| dvvxphg wr eh uhfxuuhqw/ u E|?c %6ä c6 '

�c é é é c� lv zhoo gh?qhg lq wklv fdvh/ dqg wkxv wkh h{lvwhqfh ri u E|?c E%6c6 ' �c é é é c�ää ehfrphv

yldeoh1 Frqyhujhqfh surshuwlhv fdq eh hvwdeolvkhg xqghu wkhvh h{wud dvvxpswlrqv/ dv vkrzq ehorz1

Zh fdq qrz ghyhors dq hvwlpdwh iru d yrodwlolw| vwuxfwxuh ri dq| 60idfwru KMP prgho/ dowkrxjk wkh

dojheud zloo ehfrph h{fuxfldwlqjo| lqyroyhg/ dv hylghqw iurp wkh fdofxodwlrqv ehorz/ zkloh wkh fxuvh

ri glphqvlrqdolw| lv olnho| wr pdnh frqyhujhqfh lvvxh ri dq| prgho zlwk 6 è 2 sureohpdwlf1

Iru frqfuhwhqhvv zlwk frqvlghudwlrq ri h{srvlwlrqdo hdvh/ zh zloo dqdo|}h d wzr idfwru KMP prgho

lq wkh iroorzlqj1 Wkh prgho/ vlplodu wr wkh deryh vhwxs/ lv

_+E|ðä ' _+E|ðc A ä ' k+E/c |ðc Aæä_|n òE/c |ðc Aæä_`�E|ä n #E/c |ðc Aæä_`2E|ðä n j0_`æ E|ðä c +46,

zkhuh djdlq E`æäaæ'� lv d idplo| ri lqghshqghqw vwdqgdug Eurzqldq prwlrqv vhuylqj dv phdvxuhphqw

huuruv ru vwdwlvwlfdo qrlvh/ lqghshqghqw ri wkh vwdqgdug Eurzqldq prwlrqv `� dqg `2/ zklfk vhuyh

dv wkh vwrfkdvwlf vkrfnv wr wkh |lhog fxuyhvéZh vkdoo dovr uhvwulfw wkh yrodwlolw| ixqfwlrq wr wkh irup=

òE/c |c A ä ' òEoE|äc , E|ä c A ý |ä ' òEoE|äc , E|ä c ôä#E/c |c A ä ' #EoE|äc , E|ä c A ý |ä ' #EoE|äc , E|ä c ô äc

zkhuh wkh qhz h{wud vwdwh yduldeohv , E|ä/ fdq eh wkh orqj whup udwh ,E|ä dorqj wkh olqh ri Euhqqdq dqg

Vfkzdu}> wkh iruzdug udwh lwvhoi sE|c ôä( ru wkh vsuhdg ehwzhhq orqj whup udwh dqg vkruw whup udwh/

, E|äý o E|ä1

46

Page 15: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Dv phqwlrqhg hduolhu/ wkh dqdo|vlv lq Gdl dqg Vlqjohwrq +4<<<, ghprqvwudwhv wkdw doprvw doo

+Pdunryldq, d!qh prghov h{dplqhg lq wkh olwhudwxuh lpsolflwo| ru h{solflwo| lpsrvh vrph ryhulghq0

wli|lqj uhvwulfwlrqv rq dq xqghuo|lqj 3pd{lpdo4 prgho1 Wklv sudfwlfh lv dovr dssolfdeoh wr qrq d!qh

prghov dqg qrq0Pdunryldq prghov dv zhoo1 Iru lqvwdqfh/ Nqljkw/ Ol dqg \xdq +4<<<,*v pxowl0idfwru

prgho kdv vhyhuh uhvwulfwlrq hpehgghg lq wkhlu vshfl?fdwlrq1 Lq rxu prgho/ zh kdyh uhwdlqhg jhqhu0

dolw| dv idu dv srvvleoh/ e| doorzlqj doo yrodwlolw| vwuxfwxuhv wr eh ixqfwlrq ri doo vwdwh yduldeohv +doehlw

zh vwloo lqyrnh lqghshqghqfh ri wkh Eurzqldq prwlrqv,/ udwkhu wkdq vlpsohu vshfl?fdwlrqv vxfk dv

òE/c |c A ä ' òEoE|äc , E|ä c A ý |ä ' òEoE|äc ô äc#E/c |c A ä ' #EoE|äc , E|ä c A ý |ä ' #E,E|äc ô äc

zkhuh hdfk yrodwlolw| ghshqgv rq mxvw d vlqjoh vwdwh yduldeoh1

Wkh prphqw frqglwlrqv fdq eh ghulyhg duh

. E_+E|c Aæä_+E|c A&ä mI|3 ä

'

;AA?AA=dòEo E|ä c , E|ä c ô æäòEo E|ä c , E|ä c ô&ä n #Eo E|ä c , E|ä c ô æä#Eo E|ä c , E|ä c ô &äo _| li æ 9' &

dòEo E|ä c , E|ä c ô æä2 n #Eo E|ä c , E|ä c ô æä

2 n j20o _| li æ ' &é

Zh zloo lqfrusrudwh wkh r>0gldjrqdo lqirupdwlrq edvhg rq wkhvh uhvwulfwlrqv lqwr wkh iroorzlqj orfdo0

l}hg fulwhulrq ixqfwlrq

'?Eòc #ä '?3�[ð'�

a[æ'�

a[&'�

& õ'æ

] k{+E|ðc ô æä{+E|ðc ô &äý

íòEhoðch,ðchô æäòEhoðch,ðchô&ä n #Ehoðch,ðchô æä#Ehoðch,ðchô&äì{?

l2ûgûoEoð ý hoðägûoE,ð ý h,ðägûô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_h,ð_hô æ_hô&é

Djdlq/ vlplodu wr wkh rqh idfwru fdvh deryh/ wr dyrlg wkh qhhg wr hvwlpdwh wkh phdvxuhphqw huuruv/

zh xvh rqo| r>0gldjrqdo prphqw uhvwulfwlrqv wr hvwlpdwh wkh remhfw ri lqwhuhvw/ qdpho| wkh yrodwlolw|

vwuxfwxuhv òEéä dqg #Eéä1

Surfhhglqj dv lq wkh rqh idfwru fdvh/ xvlqj wkh ghowd phwkrg/ zh rewdlq wkh iroorzlqj uhsuhvhqwd0

wlrq iru wkh pxowlidfwru hvwlpdwruéeòEoc ,c ô äe#Eoc ,c ô äè'

% U eM2 Eoc ô c rä eòEoc rä2_r U eM2 Eoc ô c rä eòEoc räe#Eoc rä_rU eM2 Eoc ô c rä eòEoc räe#Eoc rä_r U eM2 Eoc ô c räe#Eoc rä2_r&3� éU eM� Eoc ô c rä eòEoc rä_rU eM� Eoc ô c räe#Eoc rä_r

èc

zkhuh

eM� '?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oägûoE,ð ý ,ä

û dgûô Eô ý ô æägûô Erý ô&ä ngûô Eô ý ô &ägûô Erý ô æäo c

47

Page 16: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

dqg

eM2 ' {?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oägûoE,ð ý ,ä

û dgûô Eô ý ô æägûô Erý ô&ä ngûô Eô ý ô &ägûô Erý ô æäo é

Wklv uhsuhvhqwdwlrq vxjjhvwv dq lwhudwlyh vroxwlrq dvéeòd@n�oEoc ,c ôäe#d@n�oEoc ,c ôäè

'

% U eM2 Eoc ô c rä eòd@oEoc rä2_r U eM2 Eoc ô c rä eòd@oEoc räe#d@oEoc rä_rU eM2 Eoc ô c rä eòd@oEoc räe#d@oEoc rä_r U eM2 Eoc ô c räe#d@oEoc rä2_r&3�

ûéU eM� Eoc ô c rä eòd@oEoc rä_rU eM� Eoc ô c räe#d@oEoc rä_r

èc @ ' fc 2c é é é c

vwduwlqj iurp vrph jlyhq lqlwldo frqglwlrq1

Vlplodu wr wkh rqh idfwru fdvh/ wkh dv|pswrwlfv iru wklv wzr idfwru hvwlpdwru fdq eh ghulyhg xqghu

wkh suhylrxvo| lqyrnhg dvvxpswlrqv ri d pxwlsolfdwlyho| vhsdudeoh irup ri wkh yrodwlolw| vwuxfwxuh

dqg wkh orfdo wlph1 Dv vkrzq lq wkh Dsshqgl{/ xqghu wkhvh dvvxpswlrqv/ wkh hvwlpdwhv iroorz pl{hg

qrupdo dv|pswrwlfv djdlq1

Wkhruhp 51 Vxssrvh wkdw dvvxpswlrqv D140D7 krog1 Wkhq/

a

vûoû,ûôu E|?c Eoc ,ää

{?

éeòEoc ,c ô äý òEoc ,c ô äe#Eoc ,c ô äý #Eoc ,c ôäè

_$ ù

#fc

ngnSeCôREôä

%C� f

f C2

&$c

zkhuh C� / C2 dqg Cô duh gh?qhg dv

C� ' eûòEoc ,c ô ä2 n #Eoc ,c ôä2

ú ] ûòEoc ,c rä2 n #Eoc ,c rä2

údòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä2]òEoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2#Eoc ,c ôä2]#Eoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä#Eoc ,c ô ä

]òEoc ,c rä#Eoc ,c ôä dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

C2 ' eûòEoc ,c ô ä2 n #Eoc ,c ôä2

ú ] ûòEoc ,c rä2 n #Eoc ,c rä2

údòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä2]òEoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2#Eoc ,c ôä2]#Eoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä#Eoc ,c ô ä

]òEoc ,c rä#Eoc ,c ôä dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

48

Page 17: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Cô '

%]òEoc rä2R Erä _r

]#Eoc rä2R Erä _rý

é]òEoc rä#Eoc räR Erä _r

è2&2c

dqg C� 'UòEoc rä2R Erä _rc C2 '

U#Eoc rä2R Erä _rc Cô '

UòEoc rä#Eoc räR Erä _ré

Uhpdun 61 Wkh udwh ri frqyhujhqfh/ dv h{shfwhg/ lv vorzhg grzq e| d idfwru ri û, uhodwlyho|

wr wkh fdvh ri rqh idfwru1 Zlwk uhdvrqdeoh ydoxhv ri edqgzlgwk/ durxqg �I wr ôI/ dv vxjjhvwhg e|

wkh klvwrulfdo lqwhuhvw udwhv/ wkh udwh ri frqyhujhqfh ghfuhdvhv e| d idfwru ri kxqguhgv/ d vljql?fdqw

fkdqjh1 Wklv idploldu fxuvh ri glphqvlrqdolw| zrxog eh udwkhu ghwulphqwdo wr wkh hvwlpdwlrq iru

pxowl0idfwruv/ ru dq| xvhixo lqihuhqfh edvhg rq wklv hvwlpdwh vxfk dv whvwlqj iru wkh qxpehu ri idfwruv

qhfhvvdu| wr ghslfw wkh hyroxwlrq ri wkh hpslulfdo |lhog fxuyhv> sdudphwulf phwkrgv duh ghhphg wr

eh pruh vxlwdeoh lq wklv fdvh/ zlwk wkh ulvn ri prgho plvvshfl?fdwlrq krzhyhu1

9 Hpslulfdo Lpsohphqwdwlrq

Zh dsso| wkh deryh whfkqltxhv wr hvwlpdwh wkh yrodwlolw| vwuxfwxuh iru rqh idfwru dqg wzr idfwru

prghov iurp gdlo| FUVS erqg gdwd iurp Mdqxdu| 4<94 wr Ghfhpehu 4<<;1 Dv uhdvrqhg lq Mh>uh|/

Olqwrq/ dqg Qjx|hq +4<<<e,/ dqg frqvlvwhqw zlwk wkh ghulydwlrq grqh suhylrxvo|/ zh ghyldwh iurp

wkh frpprq sudfwlfh ri xvlqj wkh iruzdugv iru KMP prgho hvwlpdwlrq dqg xvh wkh |lhog hyroxwlrq

lqvwhdg/ lq oljkw ri wkh idfw wkdw wkh |lhog lv hdvlhu wr hvwlpdwh1 Wkh ?uvw vwhs lq hvwlpdwlqj d g|qdplf

prgho ri wkh |lhog fxuyh lv wr h{wudfw wkh xqrevhuydeoh |lhog fxuyhv wkhpvhoyhv iurp frxsrq ehdulqj

erqgv revhuyhg lq wkh pdunhw= OPQW +5333, nhuqho vprrwklqj edvhg |lhog fxuyh ?wwlqj/ zkrvh

lpsohphqwdwlrq lv vkrzq lq ghwdlo lq Mh>uh|/ Olqwrq dqg Qjx|hq +4<<<d,/ lv rxu fkrlfh ri |lhog fxuyh

h{wudfwlrq khuh1

Rxu fkrlfh ri nhuqhov lv wkh frpprqo| xvhg JdxvvldqgE|ä ' i TEýféD|2ä*s2Z/ iru zklfk ngn2 '�*2sZé Fkrrvlqj dq rswlpdo edqgzlgwk iru d qrqsdudphwulf hvwlpdwru lv vwloo dq hoxvlyh txhvwlrq lq

wkh olwhudwxuh/ wkhuh ehlqj qr vlqjoh vfkhph wkdw lv xqlirupo| dffhswhg dowkrxjk furvv0ydolgdwlrq lv d

iuhtxhqwo| xvhg surfhgxuh > vhh Kçugoh +4<<3, dqg Sdjdq dqg Xoodk +4<<<, iru h{whqvlyh glvfxvvlrq ri

wkh surolihudwlrq ri sursrvhg vfkhphv1 Wkh h{lvwlqj phwkrgv duh ghvljqhg/ krzhyhu/ iru wkh vwdqgdug

uhjuhvvlrq frqwh{w> iru gl>xvlrq prghov/ hvshfldoo| rqhv wkdw doorz iru qrqvwdwlrqdu| surfhvvhv/ uxohv

duh |hw wr eh ghyhorshg1 Iru rxu hvwlpdwhv/ vlplodu wr wkdw ri OPQW +4<<<,/ vlqfh qr forvhg irup

vroxwlrqv duh dydlodeoh/ hvwlpdwlrq ri wkh prghov duh frpsxwdwlrqdoo| ghpdqglqj/ wkxv uhqghulqj

furvv0ydolgdwlrq dq xqdwwudfwlyh rswlrq1 Vr zh rsw lqvwhdg wr xvh d dw edqgzlgwk +rewdlqhg e|

ylvxdo lqvshfwlrq, lq rxu vwxg| dqg ohdyh rwkhu edqgzlgwk vhohfwlrq phwkrgv iru ixwxuh h{shulphqwv1

Iru wkh gldjqrvwlf qdwxuh ri rxu hpslulfdo zrun/ wklv dsshduv qrw wr eh ri juhdw pdwhuldo lpsruw1

Edqgzlgwk iru wkh wlph wr pdwxulw| glphqvlrq lv ?{hg dw 4 |hdu ohyho/ zkloh wkdw dorqj wkh lqwhuhvw

udwh glphqvlrq lv ?{hg dw 4(1

49

Page 18: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Wkh hvwlpdwhv duh wkh vroxwlrqv wr wkh deryh ghulyhg ?uvw rughu frqglwlrqv1 Wkh v|vwhp ri htxd0

wlrqv fdq eh vroyhg e| plqlpl}dwlrq ri wkh vtxduh ri wkh ?uvw rughu frqglwlrq1 Qrwh wkdw zh gr qrw

qhhg wr vroyh wkh v|vwhp vlpxowdqhrxvo|/ zklfk zrxog eh frpsxwdwlrqdoo| lqklelwlyh1 Wkh wulfn lv wr

vroyh iru hdfk 4volfh4 dvvrfldwhg zlwk hdfk ?{hg ohyho ri o +ru lq wkh fdvh ri wzr idfwruv/ zlwk hdfk

?{hg srlqw ri +oc ,ä,/ l1h1/ vroyh iru wkh glvfuhwh srlqwv dorqj wkh fxuyh eòEoc räé914 Rqh idfwru

Zh ?qg wkdw wr hvwlpdwh rxu prgho/ edvhg rq wkh ?uvw rughu frqglwlrq +9,/ doehlw lwhudwlyh vfkhph

vxjjhvwhg e| +:, lv ihdvleoh/ lw lv pxfk pruh frqyhqlhqw wr vroyh iru wkh ?uvw rughu frqglwlrq +9, gluhfwo|

e| plqlpl}dwlrq urxwlqh1 Wklv ?qglqj lv frqvlvwhqw zlwk wkh lpsohphqwdwlrq surfhgxuh uhsruwhg lq

Mh>uh|/ Olqwrq/ Qjx|hq +4<<<d,/ zkhuh d vlplodu exw vrphzkdw vlpsohu ?uvw rughu frqglwlrq iru

OPQW +5333, lv frqgxfwhg1

Wkh qrqsdudphwulf hvwlpdwhv iru wkh yrodwlolw| vwuxfwxuh ri rqh idfwru KMP prgho xvlqj wkh

phwkrg ghyhorshg lq wklv sdshu lv vkrzq lq Iljxuh 614/ zlwk lqwhuhvw udwh gdwd iurp Mdqxdu| 4<94

wr Ghfhpehu 4<<;1Wkh zhoo nqrzq ihdwxuh ri wkh yrodwlolw| vwuxfwxuh uhsruwhg lq wkh olwhudwxuh/ wkdw

lqwhuhvw udwhv ehfrph pruh yrodwloh zkhq wkh ohyho lv kljk/ lv djdlq revhuyhg khuh1 Iru lqvwdqfh/

yrodwlolw| ehfrphv dv kljk dv 7( zkhq wkh vkruw whup klwv 4;( +wklv ohyho rqo| revhuyhg lq wkh 3Ihg0

h{shulphqw4 shulrg iurp 4<:< wr 4<;5,1 Dorqj wkh pdwxulw| glphqvlrq/ yrodwlolw| whqgv wr vorzo|

lqfuhdvhv zlwk wlph wr pdwxulw| zkhq vkruw whup udwhv duh orz/ exw wkh sdwwhuq whqgv wr uhyhuvh

lwvhoi zkhq lqwhuhvw udwhv guliw wr kljkhu udqjhv1 Ryhudoo/ wkh vkdsh ri wkh yrodwlolw| vxuidfh lv udwkhu

frqvlvwhqw zlwk zkdw kdyh ehhq revhuyhg lq wkh hpslulfdo olwhudwxuh> vhh Olqwrq/ Mh>uh|/ dqg Qjx|hq

+4<<<e, iru vlplodu uhvxowv1

Zh dovr h{shulphqw zlwk gl>huhqw wlph shulrgv/ iru lqvwdqfh/ vwduwlqj wkh gdwd shulrg iurp 4<:3/

4<;6 dqg iurp 4<<3 uhvshfwlyho|1 Wkh glylvlrqv duh prwlydwhg e| wkh rlo vkrfn lq wkh hduo| 4<:3v/

wkh vr0fdoohg 3Ihg0h{shulphqw4 iurp 4<:< wr 4<;6/ zkhuh lqwhuhvw udwhv zhuh rdwhg e| wkh fhqwudo

edqn/ dqg wkh uhodwlyho| orz dqg vwdeoh lqwhuhvw udwh wkdw suhydlohg lq wkh 4<<3v1 Wkh yrodwlolw|

vwuxfwxuhv hvwlpdwhg iru wkhvh 6 shulrgv duh uhsruwhg lq Iljxuh 615/ 616 dqg 617 uhvshfwlyho|1 Qrwh

wkdw wr dyrlg eldvhgqhvv dw wkh erxqgdu| dqg h{wudsrodwlrq/ wkh yrodwlolw| vwuxfwxuhv duh hvwlpdwhg

rqo| lq wkh udqjhv wkdw gdwd duh revhuyhg1 Vr lq wkh ?uvw 5 ?jxuhv/ zkhuh wkh kljk lqwhuhvw udwhv

shulrg ri 4<:<04<;6 duh lqfoxghg lq wkh hvwlpdwlrq surfhgxuhv/ wkh lqvwdqwdqhrxv lqwhuhvw udwhv +wkh

vwdwh yduldeoh, fdq jr iurp 3( wr 4;(/ zkloh lq wkh odvw 5 ?jxuhv/ wkh| rqo| jr kljk xs wr durxqg

43(1 Vlplodu frqvlghudwlrq lv exlow lqwr wkh wlph wr pdwxulw| glphqvlrq1 Iru lqvwdqfh/ li wkh gdwd

shulrg jrhv edfn dv idu dv 4<93/ zh rqo| h{dplqh wkh |lhog fxuyhv zlwk wlph wr pdwxulw| xs wr 7

|hduv/ vlqfh erqgv zlwk orqjhu pdwxulwlhv zhuh uduho| dydlodeoh wkhq1 Wklv vfdoh lqfuhdvhv wr 8 dqg <

4:

Page 19: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

|hduv li gdwd iurp odwhu shulrgv duh xvhg lqvwhdg1 Iru hdvh ri frpsdulvrq/ krzhyhu/ wkh vfdoh dorqj

wkh yrodwlolw| d{lv uhpdlqv frqvwdqw dfurvv wkh ?jxuhv1

Frpsdulqj wkhvh ?jxuhv/ qrwzlwkvwdqglqj wkh vfdoh gl>huhqfhv +dorqj wkh d{lv ri vkruw udwhv dqg

wlph wr pdwxulw|,/ wkh uhpdundeoh gl>huhqfhv ehwzhhq wkh judskv vhhp wr vxjjhvw vrph qrqvwdwlrqdu|

ehkdylru iru lqwhuhvw udwhv1 Zkhq gdwd lqfoxglqj wkh fkdrwlf shulrg ri wkh Ihg0h{shulphqw/ dv lq

Iljxuh 614 dqg Iljxuh 615/ lqwhuhvw udwhv qrw rqo| ehfrph pruh yrodwloh zkhq lqwhuhvw udwhv duh

kljkhu/ exw hyhq zkhq lqwhuhvw udwhv duh orz/ yrodwlolw| lv kljkhu frpsduhg wr odwhu shulrgv1 Diwhu

wkh Ihg0h{shulphqw shulrg/ wkh yrodwlolw| vxuidfh kdv ehfrph pxfk pruh vwdeoh/ dowkrxjk lwv w|slfdo

vkdsh lv vwloo revhuyhg> yrodwlolw| lqfuhdvhv zlwk lqvwdqwdqhrxv lqwhuhvw udwhv dqg wlph wr pdwxulw|1

Lq Iljxuh 618/ zh frpsduh wkh yrodwlolw| vxuidfhv rewdlqhg e| gl>huhqw phwkrgrorj|/ l1h1/ wkh rqh

ghyhorshg lq wklv sdshu zklfk dffrxqwv iru phdvxuhphqw huuruv/ dqg wkh vlpsohu phwkrg hpsor|hg lq

Shduvrq dqg ]kx +4<<;, dqg Mh>uh|/ Olqwrq/ dqg Qjx|hq +4<<<e,1 \lhog fxuyhv iurp 4<:3 wr 4<;3 duh

fkrvhq wr frqgxfw wklv h{shulphqw1 Wkh ?uvw judsk vkrzv wkh hvwlpdwh ri wkh yrodwlolw| vxuidfh xvlqj

wkh qhz phwkrg/ zklfk xvhv rqo| r> gldjrqdo lqirupdwlrq lq wkh prphqw frqglwlrqv ghyhorshg deryh/

zkloh wkh vhfrqg rqh vkrzv wkdw ri wkh pruh 3qdlyh4 phwkrg/ zklfk lq idfw xvhv rqo| lqirupdwlrq

dorqj wkh gldjrqdo ri wkh prphqw frqglwlrqv1 H{dplqlqj wkh wzr judskv/ lqwhuhvwlqjo|/ xvlqj r>

gldjrqdo uhvwulfwlrqv dv zh kdyh lq wklv sdshu grhv qrw |lhog dq hvwlpdwh zkrvh vkdsh lv gudpdwlfdoo|

gl>huhqw iurp wkdw rewdlqhg xvlqj lqirupdwlrq iurp wkh gldjrqdo dorqh1 Krzhyhu/ dv h{shfwhg/ wkh

odwwhu grhv ryhu hvwlpdwh wkh yrodwlolw| vxuidfh +e| dgglqj wkh yduldqfh ri wkh phdvxuhphqw huuruv

lqwr lwv hvwlpdwhv> vhh wkh uhodwhg irrwqrwh lq vhfwlrq 514,1 Zkhq wkh vkruw udwh uhdfkhv 4;( iru

lqvwdqfh/ lwv hvwlpdwh ri wkh yrodwlolw| lv durxqg 31384/ zkloh wkh qhz phwkrg |lhogv dq hvwlpdwh ri

313671 Wklv pdjqlwxgh ri gl>huhqfh lv gh?qlwho| pdwhuldo zkhq rqh xvhv wkh yrodwlolw| vxuidfh wr sulfh

?{hg lqfrph lqvwuxphqwv1

915 Wzr idfwru

Dv fdxwlrqhg hduolhu/ wkh yduldqfh ri rxu hvwlpdwhv lq wklv fdvh +dqg lq prghov zlwk pruh idfwruv, lv

uhodwlyho| odujh/ pdnlqj vwdwlvwlfdo lqihuhqfh edvhg rq wkhvh hvwlpdwhv gl!fxow/ ehvlghv ehlqj frpsxwd0

wlrqdo exughqvrph lq ylhz ri wkh frpsoh{lw| ri wkh prgho1 Frqvhtxhqwo|/ zh fdq qrw zkrohkhduwhgo|

hqgruvh qrqsdudphwulf hvwlpdwlrq lq pxowl0idfwru prghov iru pdnlqj lqihuhqfh dqg whvwlqj sxusrvhv1

Krzhyhu/ lq d vhdufk iru d uhdvrqdeoh sdudphwulf prgho/ qrqsdudphwulf phwkrgv hyhq ri kljk glphq0

vlrq fdq surylgh d jrrg vwduwlqj srlqw1

Iru looxvwudwlyh sxusrvhv/ zh lpsohphqw d wzr idfwru prgho khuh/ zlwk wkh gdwd shulrg iurp Mdqx0

du| 4<:3 wr Ghfhpehu 4<<;1 Dv phqwlrqhg hduolhu/ zklfk idfwruv wr eh lqfoxghg lv frpprqo| udwkhu dg

krf lq wkh olwhudwxuh/ zlwk d zkroh duud| ri h{lvwlqj vshfl?fdwlrqv1 Lq rxu iroorzlqj lpsohphqwdwlrq/

4;

Page 20: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

wkh fkrvhq idfwruv duh wkh vkruw whup dqg wkh orqj whup lqwhuhvw udwh uhvshfwlyho|/ d udwkhu frpprq

fkrlfh lq wkh uhodwhg olwhudwxuh1 Wkh vdph wulfn xvhg hduolhu/ l1h1/ vroylqj iru fxuyhv ri #Eoc ,c ô ä dqg

òEoc ,c ô ä dw d ?{hg srlqw Eoc ,ä dqg gl>huhqw ô é Hyhq zlwk wkdw/ wkh ?uvw rughu frqglwlrq fdq eh gl!fxow

wr vroyh/ vlqfh zh kdyh wr vroyh iru wkh 5 fxuyhv #Eoc ,c ô ä dqg òEoc ,c ô ä vlpxowdqhrxvo|1

Judsklfdo suhvhqwdwlrq lq wklv fdvh lv txlwh sureohpdwlf/ vlqfh #Eéä dqg òEéä duh ixqfwlrqv ri 6

yduldeohv 8 wkh vkruw udwh/ wkh orqj udwh dqg wkh pdwxulw|1 Vr iru looxvwudwlrq/ zh mxvw uhsruw Wdeohv

614 dqg 615/ zklfk frqwdlq ydoxhv ri #Eéä dqg òEéä uhvshfwlyho| iru rqh ohyho ri o ' Sé.I/ zkhuh zh

doorz orqj udwhv wr ydu| iurp 518( wr 431;(/ zkhuh pdwxulwlhv ydu| iurp f wr e |hduv1 Wkh udqjhv duh

fkrvhq vr hpslulfdo gdwd fdq ehvw dffrpprgdwh wkh hvwlpdwlrq surfhgxuh/ suh0hpswlqj h{wudsrodwlrq

dqg erxqgdu| lvvxhv1 Yrodwlolw| lv uhsruwhg lq 4243331

Fdvxdo revhuydwlrqv vxjjhvw wkdw ydoxhv ri #Eéä dqg òEéä duh udwkhu forvh/ dqg durxqg #�EéäséD/

zkhuh #�Eéä lv wkh ydoxh ri wkh ixqfwlrqdo hvwlpdwh ri wkh yrodwlolw| ixqfwlrq zkhq zh vshfli| rqh

idfwru prgho1 Wklv h{shfwhg uhvxowv krzhyhu vwloo uhqghu d prgho fdsdeoh ri jhqhudwlqj pruh frpsoh{

ehkdylru iru wkh |lhog fxuyh g|qdplfv gxh wr wkh lqghshqghqfh ri wkh wzr Eurzqldq prwlrqv1< Iru

hdfk yrodwlolw|/ ehkdylruv vlplodu wr wkh rqh idfwru fdvh lv glvsod|hg1 Yrodwlolw| lqfuhdvhv zlwk wkh

vwrfkdvwlf vwdwh yduldeoh +wkh orqj udwh,/ zkloh dorqj wkh pdwxulw|/ yrodwlolw| vhhpv wr srvlwlyho|

fruuhodwh zlwk wlph wr pdwxulw| zkhq wkh orqj udwh lv orz/ exw wxuqv wr qhjdwlyho| fruuhodwh zlwk wlph

wr pdwxulw| zkhq wkh orqj udwh uhdfkhv lqwr wkh kljkhu udqjh1

: Frqfoxvlrq

Wkh KMP dssurdfk kdv uhyroxwlrql}hg g|qdplf prghov ri wkh ?{hg lqfrph pdunhw/ exw vshfl?fdwlrq

dqg hvwlpdwlrq lvvxhv lq wklv iudphzrun uhpdlq d vhulrxv fkdoohqjh143 Lq wklv vwxg|/ zh sursrvh dq

h{whqghg yhuvlrq ri wkh ruljlqdo KMP prgho wr lqfrusrudwh phdvxuhphqw huuruv gluhfwo| lqwr wkh

prgho vshfl?fdwlrq1 D qrqsdudphwulf hvwlpdwh iru wkh yrodwlolw| vwuxfwxuh/ zklfk lv fhqwudo wr wkh

g|qdplfv ri KMP/ lv surylghg dqg lpsohphqwhg zlwk hpslulfdo gdwd1

Wkh whfkqltxh fdq eh uhdglo| dgdswhg wr pxowl0idfwru prghov/ dowkrxjk olplwdwlrqv rq wkh h{0

lvwhqfh ri pxowlglphqvlrqdo orfdo wlph frpsolfdwh wkh ghyhorsphqw ri d jhqhudo dv|pswrwlf wkhru|1

Xqghu vrph vlpsoli|lqj dvvxpswlrqv/ krzhyhu/ dq dv|pswrwlf wkhru| lv srvvleoh1 Wkh wkhruhwlfdo uh0

vxowv lqglfdwh udwkhu vorz frqyhujhqfh surshuwlhv iru qrqsdudphwulf hvwlpdwruv/ frqvlvwhqw zlwk wkhlu

frxqwhusduwv lq qrqsdudphwulf uhjuhvvlrq1 Zkloh zh dfnqrzohgjh wkhvh gl!fxowlhv/ qrqsdudphwulf

<Lqwxlwlyho|/ûs=8ì4 +=, gZ4 .

s=8ì4 +=, gZ

úzloo eh deoh wr jhqhudwh pruh frpsoh{ vkrfnv wkdq d vlqjoh ì4 +=, gZ

hyhq wkhlu wrwdo yduldwlrqv duh wkh vdph143Wkh whfkqltxh kdv irxqg lwvhoi djjuhvvlyho| pljudwhg wr wkh htxlw| pdunhw wrr> vhh Erqfkxqhu +4<<;, iru d uhsuh0

vhqwdwlyh duwlfoh iurp wklv udslgo| h{sdqglqj olwhudwxuh1

4<

Page 21: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

phwkrgv ri wkh w|sh suhvhqwhg khuh vwloo vhhp wr r>hu dq dwwudfwlyh jhqhudo wrro wr khos uhvhdufkhuv

?qg dssursuldwh sdudphwulf vshfl?fdwlrqv1

; Dsshqgl{

Iru wzr udqgrp yduldeohv f?c t?c zh vd| wkdw f? * t? zkhqhyhu f? ' t?E� n JRE�ää dv ?$4é

;14 Surri ri Wkhruhp 4

Zh ?uvw hvwdeolvk wkh ?uvw rughu frqglwlrq gh?qlqj wkh hvwlpdwru1 Ohw BocôEéc éä eh wkh elyduldwh Gludf

ghowd ixqfwlrq dw Eoc ôäé Gh?qlqj

f '?3�[ð'�

a[æ'�

a[&'�

& õ'æ

] ] ]d{+E|ðc ô æä{+E|ðc ô &äý eòEhoðchô æäeòEhoðchô &ä{?o

û EB Ehoðchô æä eòEhoðchô &ä n B Ehoðchô&ä eòEhoðchô æää{?gûoEoð ý hoðägûô Eô æ ý hô æägô Eô & ý hô&ä_hoð_hô æ_hô &éIru wkh ?uvw whupv/ zh kdyh] ] ]

B Ehoðchô æä eòEhoðchô&ägûoEoð ý hoðägûô Eô æ ý hô æägûô Eô & ý hô &ä_hoð_hô æ_hô&' gûoEo ý oðägô Eô ý ô æä

] eòEoc rägûô Erý ô&ä_r

dqg vlploduo| ] ] ]BEhoðchô&äeòEhoðchô æägûoEoð ý hoðägûô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_hô æ_hô &

' gûoEo ý oðägûô Eô ý ô &ä] eòEoc rägûô Erý ô æä_ré

Dv iru wkh vhfrqg whupv] ] ] eòEhoðchô æäeòEhoðchô &äB Ehoðchô æä eòEhoðchô&ägûoEoð ý hoðägûô Eô æ ý hô æägûô Eô & ý hô &ä_hoð_hô æ_hô &' gûoEo ý oðägûô Eô ý ô æäeòEoc ô ä] eòEoc rä2gûô Erý ô &ä_r

] ] ] eòEhoðchô æäeòEhoðchô &äB Ehoðchô &äeòEhoðchô æägûoEoð ý hoðägûô Eô æ ý hô æägûô Eô & ý hô &ä_hoð_hô æ_hô &' gûoEo ý oðägûô Eô ý ô&äeòEoc ôä ] eòEoc rä2gûô Erý ô æä_ré

53

Page 22: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Zh qrz wxuq wr wkh dv|pswrwlfv ri wkh surfhgxuh1 Lq wkh vhtxho zh zloo h{whqvlyho| xvh wkh

iroorzlqj zhoo hvwdeolvkhg uhvxowv=]}Eéc rägûô Erý ô &ä_r$ }Eéc ô&ä

ûô

]}Eéc räg2

ûô Erý ô &ä_r$ }Eéc ô &ä ngn2 czklfk krog iru dq| erxqghg frqwlqxrxv ixqfwlrqv }é Qrwh wkdw wkh deryh lqwhjudov duh douhdg| lq

vwdwh vsdfh/ qrw lq wlph grpdlq1 Wkhvh uhvxowv zloo eh xvhixo zkhq zh lqwhjudwh dorqj wkh pdwxulw|

glphqvlrq1 Zh dovr xvh wkh iroorzlqj uhvxowv/ zlwk wkh xqghuvwdqglqj wkdw wkh lqlwldo lqwhjudov duh lq

wkh wlph grpdlq=]}Eéc rägûoEo ý oErää_r $

] "

3"} Eéc @ägûo

ë@ý oûo

êu E|?c @ä _@

$ } Eéc oäu E|?c oä

ûo

]}Eéc räg2

ûoEo ý oErää_r $ ûo

] "

3"} Eéc @äg2

ûo

ë@ý oûo

êu E|?c @ä _@

$ } Eéc oäu E|?c oä ngn2 é

Zh ?uvw dqdo|}h wkh txdqwlw|{+E|ðc ô æä{+E|ðc ô &ä/ zklfk wxuqv rxw wr eh fhqwudo wr wkh ixqgdphqwdo

vrxufh ri xqfhuwdlqw| gulylqj wkh dv|pswrwlfv1 Zh kdyh

{+E|ðc ô æä{+E|ðc ô &ä ' d+E|ðn�c ô æäý +E|ðc ô æäo d+ E|ðn�c ô &äý + E|ðc ô&äo

' d+E|ðn�c ô æä+ E|ðn�c ô&äý +E|ðc ô æä+ E|ðc ô &äoý+ E|ðc ô &ä d+E|ðn�c ô æäý +E|ðc ô æäoý +E|ðc ô æä d+ E|ðn�c ô&äý + E|ðc ô &äo c

zkhuh

+E|ðn�c ô æäý +E|ðc ô æä '

] |ðn{?

kErc ô æä_rn

] |ðn{?

òEoEräc ô æä_` Erä n j20

] |ðn{?

_`æErä

+ E|ðn�c ô&äý + E|ðc ô &ä '

] |ðn{?

kErc ô &ä_rn

] |ðn{?

òEoEräc ô&ä_` Erä n j20

] |ðn{?

_`æEräé

E| wkh Pxowlyduldwh Lw÷ ohppd ^Wkhruhp 71514 ri Ûnvhqgdo +4<<;,`/ zh kdyh

_ E+E|c ô æä ü + E|c ô&ää ' +E|c ô æä_+ E|c ô&ä n + E|c ô&ä _+E|c ô æä n _+ E|c ô &ä _+E|c ô æä

' +E|c ô æäûkE|c ô &ä_|n òEo E|ä c ô&ä_` E|ä n j20_`&E|ä

ún+ E|c ô&ä

ûkE|c ô æä_|n òEo E|ä c ô æä_` E|ä n j20_`æE|ä

únòEo E|ä c ô æäòEo E|ä c ô&ä_|c

54

Page 23: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ehfdxvh wkh surfhvvhv _`æE|ä dqg _`&E|ä duh ruwkrjrqdo1 Wklv lpsolhv wkdw

d+E|ðn�c ô æä+ E|ðn�c ô&äý + E|ðc ô &ä + E|ðc ô&äo'

] |ðn{?

+Erc ô æäkErc ô&ä_r n

] |ðn{?

+Erc ô æäòEoEräc ô&ä_` Erä

nj20

] |ðn{?

+Erc ô æä_`&Erä n

] |ðn{?

+ Erc ô&äkErc ô æä_r

n

] |ðn{?

+ Erc ô&ä òEoEräc ô æä_` Erä n j20

] |ðn{?

+Erc ô&ä_`æErä

n

] |ðn{?

òEoEräc ô æäòEoEräc ô &ä_ré

Wkhuhiruh/

{+E|ðc ô æä{+E|ðc ô &ä '

] |ðn{?

d+Erc ô æäý +E|ðc ô æäokErc ô &ä_r +47,

n

] |ðn{?

d+Erc ô æäý +E|ðc ô æäo òEoEräc ô&ä_` Erä

nj20

] |ðn{?

d+Erc ô æäý +E|ðc ô æäo _`&Erä

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äokErc ô æä_r

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äo òEoEräc ô æä_` Erä

nj20

] |ðn{?

d+Erc ô &äý +E|ðc ô &äo _`æErä

n

] |ðn{?

òEoEräc ô æäòEoEräc ô &ä_ré

Zh kdyh

. d{+E|ðc ô æä{+E|ðc ô&ämI|ð o * 2òEoðc ô æäòEoðc ô &ä{?c

zkhuh oð ' oE|ðäc dqg

.û{+E|ðc ô æä

2{+E|ðc ô &ä2mI|ð

ú * eòEoðc ô æä2òEoðc ô&ä

2{2?é +48,

Zh qh{w hvwdeolvk wkh dv|pswrwlf surshuwlhv ri rxu hvwlpdwruv1

55

Page 24: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

;1414 Frqvlvwhqf|

Zh zloo uhvwulfw rxu dwwhqwlrq wr wkh fodvv ri Vrerohy ixqfwlrqv zklfk zh qrz gh?qh1 Iru dq| yhfwru

ò ' Ek�c é é é c k_äâ dqg ixqfwlrq } G U_ $ Uc ohw

(ò}E%ä 'YS_ð'� kð}E%ä

Y%k�� ü ü ü Y%k__é

Gh?qh wkh iroorzlqj vhplqrup rq wkh fodvv ri uhdo0ydoxhg ixqfwlrqv zlwk grpdlq [ =

n}n2^c2 '[�k�$^

]cf

E(ò}E%ää2 _%c

zkhuh ^ lv dq lqwhjhu dqg [f ã [ lv d frpsdfw vhw/ dqg ohw mm}mm22 ' mm}mm2fc2 ghqrwh wkh xvxdo O2 qrup1Ilqdoo|/ gh?qh K ' iò G U_ $ U G mmòmm2^c2 é äj iru vrph odujh ä ÷4c dqg ohw Kf ghqrwh wkh vxevhw

ri K wkdw h{foxghv dq "0qhljkerukrrg ri wkh }hur ixqfwlrq1 Zh wdnh ^ ' � dqg _ ' 2é

Ohw

C?EòäEoc ôä ' òEoc ô ä

] eM2 Eoc ô c rä ò2Eoc rä_rý ] eM� Eoc ô c rä òEoc rä_riru dq| ixqfwlrq òEéäc dqg gh?qh

'?Eòä ' mmC?Eòämm2éZh xvh wkh iroorzlqj ohppd ri Qhzh| dqg Srzhoo +5333/ Ohppd D5,1

Ohppd1 Vxssrvh wkdw +l, Wkhuh h{lvwv vrph ghwhuplqlvwlf ixqfwlrq 'Eòä zlwk d xqltxh plqlpxp

rq wkh sdudphwhu vsdfh Kf( +ll, '? dqg ' duh frqwlqxrxv/ Kf lv frpsdfw/ dqg týTòMK m'?Eòäý'Eòäm 'JRE�ä( +lll, hK duh vxevhwv ri Kf vxfk wkdw iru dq| ò 5 Kf wkhuh h{lvwv hò 5 hK vxfk wkdw hò $ òé Wkhqeò ' @h}4ð?òMK'?Eòä$R òfé

Wkh surri ri wklv ohppd lv jlyhq lq Qhzh| dqg Srzhoo +5333,1 Zh qh{w yhuli| lwv pdlq frqglwlrqv

lq rxu fdvh1 Zh ?uvw fdofxodwh wkh olplw ixqfwlrq CEòä dqg khqfh 'Eòä ' mmCEòämm2é Lq wkh olplw/ zkhqwkh edqgzlgwk ri doo nhuqhov dssurdfk }hur/ û $ f/ dqg qxpehu ri revhuydwlrqv dssurdfk lq?qlw|/

?$4 dqg a $4/ revhuylqj wkdw iru dq| wzlfh frqwlqxrxvo| gl>huhqwldeoh ixqfwlrq ò]òEoc rägô Erý ô&ä_r ' òEoc ô&ä nïEû2äc

dqg vfdolqj wkh zkroh htxdwlrq e| a32/ zh kdyh wkh ?uvw whup*v ?uvw frpsrqhqw

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô&ägûoEoð ý oä

ûégûô Eô ý ô æä

]òEoc rägûô Erý ô &ä_rngûô Eô ý ô &ä

]òEoc rägûô Erý ô æä_r

è56

Page 25: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

* �

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô&ägûoEoð ý oä dgûô Eô ý ô æäòEoc ô&ä ngûô Eô ý ô &äòEoc ô æäo

'�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oä dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo

ûã] |ðn{?

d+Erc ô æäý +E|ðc ô æäokfErc ô &ä_rn] |ðn{?

d+Erc ô æäý +E|ðc ô æäo òfEoEräc ô &ä_` Erä

nj20

] |ðn{?

d+Erc ô æäý +E|ðc ô æäo _`&Erä n j20

] |ðn{?

d+Erc ô &äý +E|ðc ô &äo _`æErä

n

] |ðn{?

d+ Erc ô &äý + E|ðc ô&äokfErc ô æä_rn] |ðn{?

d+ Erc ô&äý + E|ðc ô&äo òfEoEräc ô æä_` Erä

n

] |ðn{?

òfEoEräc ô æäòfEoEräc ô&ä_r

âé

Wkh ?uvw vl{ whupv duh ri vpdoohu rughu lq suredelolw| dqg e| d odz ri odujh qxpehuv/ zh kdyh

* {?

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

òfEoðc ô æäòfEoðc ô&ägûoEoð ý oä dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo

* {?

a

?3�[ð'�

a[æ'�

òfEoðc ô æägûoEoð ý oägûô Eô ý ô æä�

a

a[&c& õ'æ

òfEoðc ô &äòEoc ô &ä

n{?

a

?3�[ð'�

a[&c& õ'æ

òfEoðc ô&ägûoEoð ý oägûô Eô ý ô&ä�

a

a[æ

òfEoðc ô æäòEoc ô æä

* {?

a

?3�[ð'�

a[æ'�

òfEoðc ô æägûoEoð ý oägûô Eô ý ô æä]òfEoðc räòEoc räR Erä _r

n{?

a

?3�[ð'�

a[&c& õ'æ

òfEoðc ô&ägûoEoð ý oägûô Eô ý ô&ä]òfEoðc räòEoc räR Erä _r

* 2{?

?3�[ð'�

òfEoðc ô ägûoEoð ý oäR Eô ä]òfEoðc räòEoc räR Erä _r

* 2u E|?c oä R Eô ä òfEoc ôä

]òfEoc räòEoc räR Erä _rc

dqg wkh vhfrqg whup

a2{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

òEoc ôägûoEo ý oðä

57

Page 26: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ûégûô Eô ý ô æä

]òEoc rä2gô Erý ô &ä_rngûô Eô ý ô &ä

]òEoc rä2gô Erý ô æä_r

è

* òEoc ôä�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðäûgûô Eô ý ô æäòEoc ô&ä2 ngûô Eô ý ô &äòEoc ô æä2

ú{?

* 2u E|?c oä R Eô ä òEoc ô ä

]òEoc rä2R Erä _ré

Lq frqfoxvlrq/ C?Eòä$R CEòäc zkhuh

CEòäEoc ôä ' òfEoc ôä

]òfEoc räòEoc räR Erä _rý òEoc ô ä

]òEoc rä2R Erä _ré

Wkh frqyhujhqfh ri C? wr C lv xqlirup lq ò 5 Kf ehfdxvh ri wkh sro|qrpldo qdwxuh ri wkh fulwhulrq

ixqfwlrq dqg wkh erxqghgqhvv dvvxpswlrqvé

Wkh htxdwlrqCEòäEoc ôä ' f kdv vroxwlrq òEoc ôä ' òfEoc ôä rq Kf1 Iru xqltxhqhvv/ zh zloo h{dplqh

wkh Jdwhdx{ ghulydwlyhv ri erwk vlghv ri wkh iroorzlqj htxdwlrq

òEoc ôä 'òfEoc ôä

UòfEoc räòEoc räR Erä _rUòEoc rä2R Erä _r

c

zklfk lv iurp wkh ?uvw rughu frqglwlrq deryh1 H{sdqg òEoc rä durxqg wkh deryh vroxwlrq òfEoc rä dv

òEoc rä ' òfEoc rä n "} Erä é

Wkh deryh htxdwlrq fdq wkhq eh zulwwhq dv

òfEoc ôä n "} Eôä 'dòfEoc ôä n "} Eô äo

UòfEoc rä dòfEoc rä n "} Eräo R Erä _rU

dòfEoc ôä n "} Eôäo2 s Erä _r

'òfEoc ô ä

UòfEoc rä

2R Erä _rn 0òfEoc ôäUòfEoc rä} Erä R Erä _rU

òfEoc rä2R Erä _r

k� n 2"

UòfEocrä}EräRErä_rUòfEocrä

2RErä_r

l n J E0ä

' òfEoc ô äý 0òfEoc ôä

UòfEoc rä} Erä R Erä _rU

òfEoc rä2R Erä _r

n J E0ä é

Gl>huhqwldwlqj erwk vlghv zlwk uhvshfw wr 0/ dqg hydoxdwlqj wkhp dw 0 ' f |lhogv wkh iroorzlqj

frqglwlrq iru } Eéä

} Eôä ' ýòfEoc ôäUòfEoc rä}Eoc räRErä_rUòfEoc rä

2RErä_rc

zklfk kdv rqo| rqh vroxwlrq wkdw }Eoc rä ' f1 Xqltxhqhvv lv wkxv suryhg zlwk òEoc rä ' òfEoc rä dv wkh

rqo| vroxwlrq1 Frqvlvwhqf| lv wkhuhiruh dfklhyhg1

58

Page 27: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

;1415 Dv|pswrwlf Glvwulexwlrq

Lw zloo eh frqyhqlhqw wr xvh wkh uhsuhvhqwdwlrq lq +:,1 Iluvw rughu Wd|oru h{sdqvlrq ri wkh uljkw kdqg

vlgh zlwk uhvshfw wr eò Eéä durxqg òfEéä |lhogveòEoc ôäý òfEoc ô ä '

U eM� Eoc ô c rä eòEoc rä_rU eM2 Eoc ô c rä eò2Eoc rä_r ýUM� Eoc ô c rä òfEoc rä_rUM2 Eoc ô c rä ò2fEoc rä_r

'

Ud eM� Eoc ô c räýM� Eoc ô c räoòfEoc rä_rU

M2 Eoc ô c rä ò2fEoc rä_r

ýòfEoc ôäUd eM2 Eoc ô c räýM2 Eoc ô c räoòfEoc rä_rU

M2 Eoc ô c rä ò2fEoc rä_r

n

UM� Eoc ô c rä deòEoc räý òfEoc räo _r U M2 Eoc ô c rä ò2fEoc rä_rûU

M2 Eoc ô c rä ò2fEoc rä_rú2

ýUM� Eoc ô c rä òfEoc rä_r

UM2 Eoc ô c rä 2òfEoc rä deòEoc räý òfEoc räo _rûU

M2 Eoc ô c rä ò2fEoc rä_rú2

nïímm eM� ýM�mm2 n mm eM2 ýM2mm2 n mmeò ý òmm2ì c

zkhuh

òfEoc ô ä '

UM� Eoc ô c rä òfEoc rä_rUM2 Eoc ô c rä ò2fEoc rä_r

é

Ohw

[EòEéää '

] eM� Eoc ô c rä òfEoc rä_r'

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oä

ûégûô Eô ý ô æä

]òfEoc rägûô Erý ô &ä_rngûô Eô ý ô &ä

]òfEoc rägûô Erý ô æä_r

èc

xEò Eéää '

] eM2 Eoc ô c rä ò2fEoc rä_r' {?

?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðä

ûégûô Eô ý ô æä

]ò2fEoc rägûô Erý ô &ä_rngûô Eô ý ô &ä

]ò2fEoc rägûô Erý ô æä_r

è

59

Page 28: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

dqg

s Eò Eéä c eò Eéää '

] eM� Eoc ô c rä deòEoc räý òfEoc räo _r] eM2 Eoc ô c rä ò2fEoc rä_rý] eM� Eoc ô c rä òfEoc rä_r ] eM2 Eoc ô c rä 2òfEoc rä deòEoc räý òfEoc räo _r

'?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oä

ûigûô Eô ý ô æä]

deòEoc räý òfEoc räogûô Erý ô&ä_r

ngûô Eô ý ô&ä]

2òfEoc rä deòEoc räý òfEoc räogûô Erý ô æä_rj ûxEòf Eéää

ý[Eòf Eéää i?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðädgûô Eô ý ô æä

û]

2òfEoc rä deòEoc räý òfEoc räogûô Erý ô&ä_r

ngûô Eô ý ô&ä]

2òfEoc rä deòEoc räý òfEoc räogûô Erý ô æä_rjc

zklfk wkhq doorz xv wr uhzulwh wkh deryh Wd|oru h{sdqvlrq dvU eM� Eoc ô c rä eòEoc rä_rU eM2 Eoc ô c rä eò2Eoc rä_r '[Eòä

xEòän

sEeòc òäxEòä2

é +49,

Iluvw frqvlghu [EòEéääé Dv vkrzq lq wkh frqvlvwhqf| sduw deryh/ zkhq vfdohg e| a32/ lw frqyhujhv

wr

2u E|?c oä REô äòEoc ôä

]òEoc rä2R Erä _ré

Lwv vwrfkdvwlf sduw/ zklfk fdq eh zulwwhq dv iroorzv/ gxh wr +47,

?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oä dgûô Eô ý ô æäòEoc ô&ä ngûô Eô ý ô&äòEoc ô æäo

ûé] |ðn{?

d+Erc ô æäý +E|ðc ô æäoòEoEräc ô&ä_` Erä n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äoòEoEräc ô æä_` Erä

èé

Ghqrwh

L? E%ä '�

a

sûoûô

�s{?

d?%o3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oädgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô&äòEoc ô æäo

5:

Page 29: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ûi] |ðn{?

d+Erc ô æäý +E|ðc ô æäoòEoEräc ô &ä_` Erä

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äoòEoEräc ô æä_` Eräjé

Vlqfh L? E%ä lv d frqwlqxrxv pduwlqjdoh/ e| wkh vwurqj odz ri odujh qxpehuv iru pduwlqjdoh gl>huhqfh

vhtxhqfhv zlwk ?qlwh vhfrqg prphqw +Kdoo dqg Kh|gh +4<;9,,/ L? E%ä frqyhujhv wr }hur doprvw vxuho|/

zlwk wkh udwh ri frqyhujhqfh fdq eh irxqg iurp Nqljkw*v hpehgglqj wkhruhp1 Lwv txdgudwlf yduldwlrq

surfhvv dL?o% lv

dL?o% '�

a2ûoûô

{?

d?%o3�[ð'�

a[æ'�

a[&'�

& õ'æ

] |ðn{?

id+Erc ô æäý +E|ðc ô æäo2òEoEräc ô &ä2_rn

d+ Erc ô &äý + E|ðc ô &äo2òEoEräc ô æä2_rn2 d+Erc ô æäý +E|ðc ô æäo d+ Erc ô &äý + E|ðc ô&äo òEoEräc ô æäòEoEräc ô&ä_rûgûoEoð ý oä2 dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo2

'�

a2e{?ûoûô

d?%o3�[ð'�

a[æ'�

a[&'�

& õ'æ

òEoðc ô æä2òEoðc ô &ä

2

ûgûoEoð ý oä2 dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo2

'�

a2eûoûô

] %|?

f

a[æ'�

a[&'�

& õ'æ

òEoEräc ô æä2òEoEräc ô &ä

2

ûgûoEoð ý oä2 dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo2 _r

'�

a2eûoûô

] n"

3"

a[æ'�

a[&'�

& õ'æ

òE@c ô æä2òE@c ô &ä

2u E%|?c @ä

ûgûoE@ý oä2 dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô&äòEoc ô æäo2 _@

'�

a2eûô

] n"

3"

a[æ'�

a[&'�

& õ'æ

òEo n ûo^c ô æä2òEo n ûo^c ô &ä

2u E%|?c o n ûo^ä

ûgE^ä2 dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô &äòEoc ô æäo2 _^

5;

Page 30: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Wklv lv

* eûô ngn2 �

a2

a[æ'�

a[&'�

& õ'æ

òEoc ô æä2òEoc ô &ä

2u E%|?c oäû dgûô Eô ý ô æäòEoc ô &ä ngûô Eô ý ô&äòEoc ô æäo2

' e ngn2 u E%|?c oä i �

a2ûô

a[æ'�

a[&'�

& õ'æ

òEoc ô æä2òEoc ô&ä

2gûô Eô ý ô æä2òEoc ô &ä2

n�

a2ûô

a[æ'�

a[&'�

& õ'æ

òEoc ô æä2òEoc ô &ä

2gûô Eô ý ô&ä2òEoc ô æä2

n�

a2ûô

a[æ'�

a[&'�

& õ'æ

òEoc ô æäôòEoc ô &ä

ôgûô Eô ý ô æägûô Eô ý ô &äj

* H ngn2 u E%|?c oäãòEoc ôä2R Eô ä

]òEoc räeR Erä _r ngn2 n ûôREô äòEoc ôäS

â

* H ngne u E%|?c oä R Eôä òEoc ôä2]òEoc räeR Erä _ré

Zh wkxv kdyh

L? E�ä_$ ù

ëfc H ngne u E|?c oä R Eô ä òEoc ôä2

]òEoc räeR Erä _r

êé +4:,

Qh{w frqvlghu xEòEéää1 Diwhu vfdolqj e| a32

a2xEòä '

{?

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðä

ûdgûô Eô ý ô æä]òEoc rä2gûô Erý ô&ä_rngûô Eô ý ô&ä

]òEoc rä2gûô Erý ô æä_ro

+4;,

* {?

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðdgûô Eô ý ô æäòEoc ô &ä2 ngûô Eô ý ô&äòEoc ô æä2o

* {?

a

?3�[ð'�

gûoEo ý oðäa[æ'�

gûô Eô ý ô æä�

a

a[&c& õ'æ

òEoc ô &ä2

n{?

a

?3�[ð'�

gûoEo ý oðäa[

&c& õ'ægûô Eô ý ô &ä

a

a[æ'�

òEoc ô æä2

5<

Page 31: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

* {?

a

?3�[ð'�

gûoEo ý oðäa[æ'�

gûô Eô ý ô æä]òEoc rä2R Erä _r

n{?

a

?3�[ð'�

gûoEo ý oðäa[

&c& õ'ægûô Eô ý ô &ä

]òEoc rä2R Erä _r

* 2u E|?c oä R Eô ä

]òEoc rä2R Erä _ré

Qrz frqvlghu s Eéä1 Zh kdyh

aes Eéä '

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô&ägûoEoð ý oä�

a2xEòäû

igûô Eô ý ô æä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô &ä_r

ngûô Eô ý ô &ä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô æä_rj

ý �

a2{?

a2[EòEéääi

?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðä

ûdgûô Eô ý ô æä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô &ä_r

ngûô Eô ý ô &ä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô æä_ro

'�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

2òEoc ô æäòEoc ô&ä{?gûoEoð ý oäj�

a2xEòEéää

ûigûô Eô ý ô æä]

2òEoc rädeòEoc räý òEoc räogûô Erý ô&ä_r

ngûô Eô ý ô &ä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô æä_rj

ý �

a2{?

a2[EòEéääi

?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðädgûô Eô ý ô æä

û]

2òEoc rä deòEoc räý òEoc räogûô Erý ô &ä_r

ngûô Eô ý ô &ä]

2òEoc rä deòEoc räý òEoc räogûô Erý ô æä_rojc

zkhuh wkh vhfrqg htxdwlrq iroorzv +47,1 Lw lv dssduhqw wkdw wkh wzr frpsrqhqwv duh shuihfwo| fruuh0

odwhg/ zkhuh wkh yduldwlrq frphv iurp eòEoc räý òEoc rä163

Page 32: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Lwv ?uvw frpsrqhqw dssurdfkhv

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

2òEoc ô æäòEoc ô&ä{?gûoEoð ý oä�

a2xEòEéää

igûô Eô ý ô æä deòEoc ô&äý òEoc ô &äo ngûô Eô ý ô&ä deòEoc ô æäý òEoc ô æäoj' 2òEoc ô äu E|?c oä R Eôä

]òEoc rä deòEoc räý òEoc räo R Erä _r

û2u E|?c oä R Eô ä òEoc ô ä]òEoc rä2R Erä _r

' eu E|?c oä2 R Eô ä2 òEoc ô ä2

]òEoc rä2R Erä _r

]òEoc rä deòEoc räý òEoc räo R Erä _ré

Wkh vhfrqg frpsrqhqw dovr dssurdfkhv

a22òEoc ôä{?i

?[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðä�

a2[EòEéää

ûdgûô Eô ý ô æäòEoc ô &ä deòEoc ô&äý òEoc ô&äo ngûô Eô ý ô&äòEoc ô æä deòEoc ô æäý òEoc ô æäooj' 2òEoc ô äu E|?c oä R Eô ä

]òEoc rä deòEoc räý òEoc räo R Erä _rû 2u E|?c oä R Eôä òEoc ôä

]òEoc rä2R Erä _r

' eu E|?c oä2 R Eô ä2 òEoc ôä2

]òEoc rä2s Erä _r

]òEoc rä deòEoc räý òEoc räo R Erä _r

vr wkh| duh fdqfhoohg rxw/ zklfk phdqv wkdw wkh yduldwlrq ri wkh uljkw kdqg vlgh zloo eh gulyhq e|

[Eòä*xEòäé

Iurp +49,/ +4:, dqg +4;,/ zh rewdlq

a

sûoûô

��a2

�s{?

deòEoc ô äý òEoc ô äo'

�a

sûoûô

�I{?

�a2

é[Eòä

x Eòäý òEoc ô ä

è_$ �

2u E|?c oä R EôäUòEoc rä2R Eô ä _r

ù

ëfc H ngne u E|?c oä R Eôä òEoc ôä2

]òEoc räeR Erä _r

êc

ru

a

vûoûôu E|?c oä

{?deòEoc ôäý òEoc ôäo _$ ù

#fc 2 ngne òEoc ôä

2UòEoc räeR Erä _r

R Eô äýUòEoc rä2R Erä _r

ü2$

wkh uhvxowv vwdwhg lq Wkhruhp 41

64

Page 33: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

;15 Surri ri Wkhruhp 5

Wkh pxowlsolfdwlyh vhsdudelolw| dvvxpswlrqv wkdw zh lqyrnh iru wkh yrodwlolw| vwuxfwxuhv dqg wkh orfdo

wlph doorz xv wr kdyh wkh iroorzlqj irupxod/ zklfk lv fhqwudo wr rxu ghulydwlrqv/ dv lq wkh rqh0idfwru

fdvh1 Iru wkh wzr0yduldeoh fdvh/ iru dq| pxowlsolfdwlyho| vhsdudeoh nhuqho ixqfwlrqgEüä dqg frqwlqxrxverxqghg pxowlsolfdwlyho| vhsdudeoh ixqfwlrq sEüä/

*ð4ûòf

û

] |

f

g

ëEoc ,äý EoEräc ,Erää

û

ês EoEräc ,Erää _r

'

é*ð4ûoòf

ûo

] n"

3"g

ëo ý @ûo

êsoE@äuoE|c @ä_@

è é*ð4û,òf

û,

] n"

3"g

ë%ý @û,

ês,E@äu,E|c @ä_@

è'

é*ð4ûoòf

] n"

3"gE^äsoE%n ûo^äuoE|c %n ûo^ä_^

è é*ð4û,òf

] n"

3"gE^äs,E%n û,^äu,E|c %n û,^ä_^

è$ uoE|c oäsoEoäu,E|c oäs,E,ä ' uE|c Eoc ,ääsEoc ,ä1

Wkh uhvxow lv hvvhqwldoo| lghqwlfdo wr wkh rqh glphqvlrqdo fdvh1

Zh ?uvw ghulyh wkh ?uvw rughu frqglwlrq iru wkh wzr idfwru prgho1 Gl>huhqwldwh wkh vdpsoh fulwhuld

ixqfwlrq zlwk uhvshfw wr #Eéä zh kdyh

f '

?[ð'�

a[æ'�

a[&'�

& õ'æ

]hoðch,ðk{+E|ðc ô æä{+E|ðc ô&äý

íeòEhoðch,ðchô æäeòEhoðch,ðchô &ä n e#Ehoðch,ðchô æäe#Ehoðch,ðchô &äì{?

l

ûíBíhoðch,ðchô æì eòEhoðch,ðchô&ä n B íhoðch,ðchô &ì eòEhoðch,ðchô æäì

û{?gûoEoð ý hoðägûoE,ð ý h,ðägô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_h,ð_hô æ_hô&dqg zlwk uhvshfw wr ò Eéä

f '?3�[ð'�

a[æ'�

a[&'�

& õ'æ

] k{+E|ðc ô æä{+E|ðc ô &äý

íeòEhoðch,ðchô æäeòEhoðch,ðchô&ä n e#Ehoðch,ðchô æäe#Ehoðch,ðchô&äì{?

l

ûíBíhoðch,ðchô æìe#Ehoðch,ðchô &ä n B íhoðch,ðchô &ìe#Ehoðch,ðchô æäì

û{?gûoEoð ý hoðägûoE,ð ý h,ðägûô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_h,ð_hô æ_hô &éWkh ?uvw rughu frqglwlrq vwdwhg lq wkh erg| ri wkh sdshu lv rewdlqhg wkurxjk wkh iroorzlqj pdqls0

xodwlrq/ iru uhsuhvhqwdwlyh whupv/ zklfk duh]BEhoðch,ðchô æäeòEhoðch,ðchô&ägûoEoð ý hoðägûoE,ð ý h,ðägûô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_h,ð_hô æ_hô &

' gûoEo ý oðägûoE, ý ,ðädgûô Eô ý ô æä] eòEoc ,c rägûô Erý ô&ä_r

ngûô Eô ý ô &ä] eòEoc ,c rägûô Erý ô æä_roc

65

Page 34: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

]Bíhoðch,ðchô æìeòEhoðch,ðchô&äeòEhoðch,ðchô æäeòEhoðch,ðchô &ä

ûgûoEoð ý hoðägûoE,ð ý h,ðägô Eô æ ý hô æägûô Eô& ý hô&ä_hoð_h,ð_hô æ' eòEoc ,c ôägûoEo ý oðägûoE, ý ,ðä

ûdgûô Eô ý ô æä] eòEoc ,c rä2gûô Erý ô&ä_r ngûô Eô ý ô &ä

] eòEoc ,c rä2gûô Erý ô æä_ro

dqg ]hoð]hôæ]hô& B

íhoðch,ðchô æìeòEhoðch,ðchô &äe#Ehoðch,ðchô æäe#Ehoðch,ðchô &äûgûoEoð ý hoðägûoE,ð ý h,ðägûô Eô æ ý hô æägûô Eô & ý hô &ä_hoð_h,ð_hô æ

' e#Eoc ,c ô ägûoEo ý oðägûoE, ý ,ðäûégûô Eô ý ô æä

] e#Eoc ,c räeòEoc ,c |ägûô Erý ô&ä_|ngûô Eô ý ô&ä] e#Eoc ,c räeòEoc ,c rägûô Erý ô æä_|

èé

Wkhuhiruh/ zh kdyh iru eòEéäf '

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oägûoE,ð ý ,äû

égûô Eô ý ô æä

] eòEoc ,c rägûô Erý ô&ä_rngûô Eô ý ô&ä] eòEoc ,c rägûô Erý ô æä_r

èý?3�[ð'�

a[æ'�

a[&'�

& õ'æ

eòEoc ,c ô ägûoEo ý oðägûoE, ý ,ðä{?

ûégûô Eô ý ô æä

] eòEoc ,c rä2gûô Erý ô &ä_rngûô Eô ý ô &ä] eòEoc ,c rä2gûô Erý ô æä_r

èý?3�[ð'�

a[æ'�

a[&'�

& õ'æ

e#Eoc ,c ôägûoEo ý oðägûoE, ý ,ðä{?

ûégûô Eô ý ô æä

] e#Eoc ,c räeòEoc ,c rägûô Erý ô &ä_rngûô Eô ý ô &ä] e#Eoc ,c räeòEoc ,c rägûô E|ý ô æä_r

èc

dqg vlploduo| iru e#Eéäf '

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oägûoE,ð ý ,ä

ûégô Eô ý ô æä

] e#Eoc ,c rägûô Erý ô &ä_rngûô Eô ý ô&ä] e#Eoc ,c rägûô Erý ô æä_r

è66

Page 35: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ý?3�[ð'�

a[æ'�

a[&'�

& õ'æ

e#Eoc ,c ôägûoEo ý oðägûoE, ý ,ðä{?

ûégûô Eô ý ô æä

] e#Eoc ,c rä2gûô Erý ô &ä_rngûô Eô ý ô&ä] e#Eoc ,c rä2gûô Erý ô æä_r

èý?3�[ð'�

a[æ'�

a[&'�

& õ'æ

eòEoc ,c ô ägûoEo ý oðägûoE, ý ,ðä{?

ûégûô Eô ý ô æä

] e#Eoc ,c räeòEoc ,c rägûô Erý ô &ä_rngûô Eô ý ô &ä] e#Eoc ,c räeòEoc ,c rägûô Erý ô æä_r

èé

Vlplodu wr wkh rqh0idfwru fdvh deryh/ zh zloo ghulyh wkh dv|pswrwlfv iru rxu hvwlpdwh e| h{dplqlqj

wkh ?uvw rughu frqglwlrq lq wkh olplw1 Iru wkh lqwhuhvw ri euhylw|/ zh zloo vnls vwhsv wkdw vkrxog eh

idploldu qrz yld wkh surri ri wkh rqh0idfwru fdvh grqh lq ghwdlov hduolhu1 Zkhq û $ f/ ? $ 4/

a $4/ revhuylqj wkdw ] e#Eoc ,c rägûô E|ý ô æä_r $ e#Eoc ,c ô æä] e#Eoc ,c rä2gûô E|ý ô æä_r $ e#Eoc ,c ô æä2]r

e#Eoc ,c räeòEoc ,c rägûô E|ý ô æä_r $ e#Eoc ,c ô æäeòEoc ,c ô æädqg vlploduo| iru wkh dolnh whupv1 Wkh dqdorjxh wr +47, lv

{+E|ðc ô æä{+E|ðc ô &ä '

] |ðn{?

d+ Erc ô æäý + E|ðc ô æäokErc ô&ä_r

n

] |ðn{?

d+ Erc ô æäý + E|ðc ô æäo òEoEräc , Erä c ô &ä_`� Erä

n

] |ðn{?

d+ Erc ô æäý + E|ðc ô æäo #EoEräc , Erä c ô&ä_`2 Erä +4<,

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äokErc ô æä_r

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äo òEoEräc , Erä c ô æä_`� Erä

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô &äo #EoEräc , Erä c ô æä_`2 Erä

n

] |ðn{?

dòEoEräc , Erä c ô æäòEoEräc ô&ä n #EoEräc , Erä c ô æä#EoEräc , Erä c ô &äo _ré

Wkh deryh whup kdv txdgudwlf yduldwlrq dv

d{+E|ðc ô æä{+E|ðc ô&äo ' 2ûòEoE|ðäc ,E|ðäc ô &ä

2 n #EoE|ðäc ,E|ðäc ô&ä2ú{2?

n2 dòEoE|ðäc ,E|ðäc ô æäòEoE|ðäc ,E|ðäc ô&ä n #EoE|ðäc ,E|ðäc ô æä#EoE|ðäc ,E|ðäc ô &äo2{2

?

67

Page 36: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Dqrwkhu xvhixo dssur{lpdwlrq fdq dovr eh ghulyhg iurp +4<,

{+E|ðc ô æä{+E|ðc ô&ä * 2 dòEoðc ,ðc ô æäòEoðc ,ðc ô&ä n #Eoðc ,ðc ô æä#Eoðc ,ðc ô&äo{?é

Iru wkh ?uvw whup lq wkh ?uvw rughu frqglwlrq / xvlqj +4<, dqg ljqrulqj vpdoo whupv +whupv erxqghg

e| dvvxpswlrq 5, dv grqh lq wkh rqh0idfwru fdvh/ zh kdyh +vfdohg e| a32,

'�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô&ägûoEoð ý oägûoE,ð ý ,ä

ûégô Eô ý ô æä

] eòEoc ,c rägô Erý ô&ä_rngûô Eô ý ô &ä] eòEoc ,c |ägûô Erý ô æä_r

è* �

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô&ägûoEoð ý oägûoE,ð ý ,ä

û dgûô Eô ý ô æäeòEoc ,c ô &ä ngûô Eô ý ô &äeòEoc ,c ô æäo* �

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

] |ðn{?

dòEoEräc , Erä c ô æäòEoEräc ô&ä n #EoEräc , Erä c ô æä#EoEräc , Erä c ô&äo _r

ûgûoEoð ý oägûoE,ð ý ,ä dgûô Eô ý ô æäeòEoc ,c ô&ä ngûô Eô ý ô&äeòEoc ,c ô æäo* {?

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

dòEoðc ,ðc ô æäòEoðc ,ðc ô &ä n #Eoðc ,ðc ô æä#Eoðc ,ðc ô &äo

ûgûoEoð ý oägûoE,ð ý ,ä dgûô Eô ý ô æäeòEoc ,c ô&ä ngûô Eô ý ô&äeòEoc ,c ô æäo* 2u E|?c Eoc ,ää R Eôä

éòEoc ,c ô ä

]òEoc ,c räeòEoc ,c räR Erä _rn #Eoc ,c ô ä ] #Eoc ,c räeòEoc ,c räR Erä _rè é

Iru wkh vhfrqg whup

{?

a2

?[ð'�

a[æ'�

a[&'�

& õ'æ

eòEoc ,c ôägûoEo ý oðägûoE, ý ,ðä

ûégûô Eô ý ô æä

] eòEoc ,c rä2gûô Erý ô &ä_|ngûô Eô ý ô &ä] eòEoc ,c rä2gûô Erý ô æä_|

è* {?

a2eòEoc ,c ô ä ?[

ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðägûoE, ý ,ðä

û ûgûô Eô ý ô æäeòEoc ,c ô&ä2 ngûô Eô ý ô &äeòEoc ,c ô æä2ú* 2u E|?c oä R Eôä eòEoc ,c ô ä] eòEoc ,c rä2R Erä _ré

68

Page 37: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Dqg iru wkh odvw whup/ vlploduo|

{?

a2

?[ð'�

a[æ'�

a[&'�

& õ'æ

e#Eoc ,c ô ägûoEo ý oðägûoE, ý ,ðä

ûégûô Eô ý ô æä

] e#Eoc ,c räeòEoc ,c rägûô Erý ô &ä_rngûô Eô ý ô &ä] e#Eoc ,c räeòEoc ,c rägûô Erý ô æä_r

è* {?

a2e#Eoc ,c ô ä ?[

ð'�

a[æ'�

a[&'�

& õ'æ

gûoEo ý oðägûoE, ý ,ðä

û dgûô Eô ý ô æäe#Eoc ,c ô&äeòEoc ,c ô&ä ngûô Eô ý ô&äe#Eoc ,c ô æäeòEoc ,c ô æäo* 2u E|?c oä REôäe#Eoc ,c ôä ] e#Eoc ,c räeòEoc ,c räR Erä _réVr lq wkh olplw/ wkh ?uvw rughu frqglwlrq lv

f '

éòEoc ,c ôä

]òEoc ,c räeòEoc ,c räR Erä _rn #Eoc ,c ôä ] #Eoc ,c räeòEoc ,c räR Erä _rè

ýeòEoc ,c ôä ] eòEoc ,c rä2R Erä _rý e#Eoc ,c ôä ] e#Eoc ,c räeòEoc ,c räR Erä _rcdqg vlploduo|/ wkh vhfrqg ?uvw rughu frqglwlrq lv

f '

é#Eoc ,c ôä

]#Eoc ,c räe#Eoc ,c räR Erä _r n òEoc ,c ô ä] òEoc ,c räe#Eoc ,c räR Erä _rè

ýe#Eoc ,c ôä ] e#Eoc ,c rä2R Erä _rý eòEoc ,c ôä ] eòEoc ,c räe#Eoc ,c räR Erä _rczklfk kdv d vroxwlrq e#Eoc ,c ô ä ' #Eoc ,c ô ä dqg eòEoc ,c ô ä ' òEoc ,c ô äé Xqltxhqhvv fdq eh suryhg xvlqj

dujxphqwv dv lq wkh rqh0idfwru fdvh1

Zh qrz glvfxvv wkh dv|pswrwlf glvwulexwlrq1 Uh0zulwh wkh ?uvw rughu frqglwlrq dv

éeòEoc ,c ôäe#Eoc ,c ôäè

'�U eM2 Eoc ô c räeòEoc rä2_r U eM2 Eoc ô c räe#Eoc rä2_rý kU eM2 Eoc ô c rä eòEoc räe#Eoc rä_rl2

û% U eM2 Eoc ô c räe#Eoc rä2_r ý U eM2 Eoc ô c rä eòEoc räe#Eoc rä_rý U eM2 Eoc ô c rä eòEoc räe#Eoc rä_r U eM2 Eoc ô c rä eòEoc rä2_r

&

ûéU eM� Eoc ô c rä eòEoc rä_rU eM� Eoc ô c räe#Eoc rä_r

èé

Xvlqj ?uvw0rughu Wd|oru h{sdqvlrq/ zh fdq vkrz wkdw wkh yduldwlrq/ vlplodu wr wkh 40idfwru fdvh/ frphv

rqo| iurp wkh iroorzlqj whup

�U eM2 Eoc ô c rä òEoc rä2_r U eM2 Eoc ô c rä #Eoc rä2_rý kU eM2 Eoc ô c rä òEoc rä#Eoc rä_rl269

Page 38: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

û% U eM2 Eoc ô c rä #Eoc rä2_r ý U eM2 Eoc ô c rä òEoc rä#Eoc rä_rý U eM2 Eoc ô c rä òEoc rä#Eoc rä_r U eM2 Eoc ô c rä òEoc rä2_r

& éU eM� Eoc ô c rä òEoc rä_rU eM� Eoc ô c rä #Eoc rä_rèé

Iru wkh udwh ri frqyhujhqfh ri eòEoc ,c ô ä/ frqvlghu wkh iroorzlqj txdqwlw|X? '

a

sûoûô

�s{?

2u E|?c Eoc ,ää R Eôäû d

] eM2 Eoc ô c rä #Eoc rä2_r ] eM� Eoc ô c rä òEoc rä_rý] eM2 Eoc ô c rä òEoc rä#Eoc rä_r ] eM� Eoc ô c rä #Eoc rä_roc

zklfk lv

a

sûoû,ûô

�s{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oägûoE,ð ý ,ä

ûi]#Eoc rä2R Erä _rdgûô Eô ý ô æä

]òEoc ,c rägûô Erý ô&ä_r ngûô Eô ý ô &ä

]òEoc rägûô Erý ô æä_ro

ý]òEoc rä#Eoc räR Erä _rdgûô Eô ý ô æä

]#Eoc ,c rägûô Erý ô&ä_r

ngûô Eô ý ô &ä]#Eoc rägûô Erý ô æä_roj

'�

a

sûoû,ûô

�s{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

{+E|ðc ô æä{+E|ðc ô &ägûoEoð ý oägûoE,ð ý ,ä

ûigûô Eô ý ô æäéòEoc ,c ô&ä

]#Eoc rä2R Erä _rý #Eoc ,c ô&ä

]òEoc ,c rä#Eoc räR Erä _r

èngûô Eô ý ô &ä

éòEoc ,c ô æä

]#Eoc rä2R Erä _rý #Eoc ,c ô æä

]òEoc ,c rä#Eoc räR Erä _r

èjc

zkhuh zh kdyh xvhg wkh idfw wkdw

a2

] eM2 Eoc ô c rä #Eoc rä2_r ' {?�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oägûoE,ð ý ,äû

dgûô Eô ý ô æä]#Eoc ,c rä2gûô Erý ô&ä_r

ngûô Eô ý ô&ä]#Eoc rä2gûô Erý ô æä_ro

' {?�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oägûoE,ð ý ,ä

û ûgûô Eô ý ô æä#Eoc ,c ô&ä2 ngûô Eô ý ô &ä#Eoc ô æä2ú

* 2u E|?c Eoc ,ää R Eôä

]#Eoc rä2R Erä _rc

6:

Page 39: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

dqg

a2

] eM2 Eoc ô c rä òEoc rä#Eoc rä_r ' {?�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oägûoE,ð ý ,ä

ûdgûô Eô ý ô æä]òEoc rä#Eoc rägûô Erý ô&ä_r

ngûô Eô ý ô&ä]òEoc rä#Eoc rägûô Erý ô æä_ro

' {?�

a2

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

gûoEoð ý oägûoE,ð ý ,äû

û dgûô Eô ý ô æäòEoc ô&ä#Eoc ô&ä ngûô Eô ý ô&äòEoc ô æä#Eoc ô æäo* 2u E|?c Eoc ,ää R Eôä

]òEoc rä#Eoc räR Erä _ré

Iru wkh vwrfkdvwlf sduw riX/ qrwh wkdw lw lv gulyhq e| wkh iroorzlqj whupv ehorqj wr{+E|ðc ô æä{+E|ðc ô&ä

{+E|ðc ô æä{+E|ðc ô&ä ' n

] |ðn{?

d+ Erc ô æäý + E|ðc ô æäo òEoEräc , Erä c ô &ä_`� Erä

n

] |ðn{?

d+ Erc ô æäý + E|ðc ô æäo #EoEräc , Erä c ô&ä_`2 Erä

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô&äo òEoEräc , Erä c ô æä_`� Erä

n

] |ðn{?

d+ Erc ô&äý + E|ðc ô&äo #EoEräc , Erä c ô æä_`2Eräé

Wkh txdgudwlf yduldwlrq ri X? lv

dX?o '�

a2ûoû,ûô{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

i2dòEoE|ðäc ,E|ðäc ô æä2 n #EoE|ðäc ,E|ðäc ô æä2o

ûdòEoE|ðäc ,E|ðäc ô&ä2 n #EoE|ðäc ,E|ðäc ô &ä2on2dòEoE|ðäc ,E|ðäc ô æäòEoE|ðäc ,E|ðäc ô&ä n #EoE|ðäc ,E|ðäc ô æä#EoE|ðäc ,E|ðäc ô &äo

2jûgûoEoð ý oä2gûoE,ð ý ,ä2

ûigûô Eô ý ô æäéòEoc ,c ô&ä

]#Eoc rä2R Erä _rý #Eoc ,c ô&ä

]òEoc ,c rä#Eoc räR Erä _r

èngûô Eô ý ô &ä

éòEoc ,c ô æä

]#Eoc rä2R Erä _rý #Eoc ,c ô æä

]òEoc ,c rä#Eoc räR Erä _r

èj2

* �

a2ûoû,ûô{?

?3�[ð'�

a[æ'�

a[&'�

& õ'æ

i2dòEoE|ðäc ,E|ðäc ô æä2 n #EoE|ðäc ,E|ðäc ô æä2o

6;

Page 40: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

ûdòEoE|ðäc ,E|ðäc ô&ä2 n #EoE|ðäc ,E|ðäc ô &ä2on2dòEoE|ðäc ,E|ðäc ô æäòEoE|ðäc ,E|ðäc ô&ä n #EoE|ðäc ,E|ðäc ô æä#EoE|ðäc ,E|ðäc ô &äo

2jûigûô Eô ý ô æädòEoc ,c ô&ä

]#Eoc rä2R Erä _rý #Eoc ,c ô &ä

]òEoc ,c rä#Eoc räR Erä _ro

ngûô Eô ý ô &ädòEoc ,c ô æä]#Eoc rä2R Erä _rý #Eoc ,c ô æä

]òEoc ,c rä#Eoc räR Erä _roj2

Ohw C� 'UòEoc rä2R Erä _rc C2 '

U#Eoc rä2R Erä _rc Cô '

UòEoc rä#Eoc räR Erä _ré Wkhq wkh txdgudwlf

yduldwlrq ri X? lv

dX?o ' ngnS u E|?c oä R Eô äûie ûòEoc ,c ôä2 n #Eoc ,c ô ä2ú ] û

òEoc ,c rä2 n #Eoc ,c rä2údòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä2]òEoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2#Eoc ,c ô ä2]#Eoc ,c rä2 dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _r

n2òEoc ,c ô ä#Eoc ,c ô ä

]òEoc ,c rä#Eoc ,c ôä dòEoc ,c räC2 ý #Eoc ,c räCôo R Erä _rj

Wkh odvw whup zh qhhg wr fdofxodwh wr ?qg wkh dv|pswrwlf glvwulexwlrq ri rxu hvwlpdwruv lv

�U eM2 Eoc ô c rä òEoc rä2_r U eM2 Eoc ô c rä #Eoc rä2_rý kU eM2 Eoc ô c rä òEoc rä#Eoc rä_rl2 éWkh ghqrplqdwru/ vfdohg e| a32 / dv vkrzq lq hduolhu dssurdfk/ lv

e

éu E|?c Eoc ,ää R Eôä

]òEoc rä2R Erä _r

è éu E|?c Eoc ,ää R Eôä

]#Eoc rä2R Erä _r

èýeéu E|?c Eoc ,ää R Eô ä

]òEoc rä#Eoc räR Erä _r

è2' e

ûu E|?c Eoc ,ää R Eôä

ú2 %]òEoc rä2R Erä _r

]#Eoc rä2R Erä _rý

é]òEoc rä#Eoc räR Erä _r

è2&c

jlylqj wkh uhvxowv uhsruwhg lq Wkhruhp 51

Uhihuhqfhv

^4` Edqgl/ I1 dqg S1 Skloolsv +4<<;,1 Hfrqrphwulf Hvwlpdwlrq ri Gl>xvlrq prghov1 Xqsxeolvkhg

sdshu/ \doh Xqlyhuvlw|1

^5` Edogx}}l/ S1/ V1Gdv dqg V1 Iruhvl +4<<;,1 Wkh Fhqwudo Whqghqf|= D Vhfrqg Idfwru lq Erqg

\lhogv1 Uhylhz ri Hfrqrplfv dqg Vwdwlvwlfv/ ;3/ 950:5

6<

Page 41: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

^6` Edogx}}l/ S1/ V1 Gdv/ V1 Iruhvl dqg U1 Vxqgdudp +4<<9,1 D vlpsoh Dssurdfk wr wkuhh0idfwru d!qh

whup vwuxfwxuh prghov1 Mrxuqdo ri Il{hg Lqfrph/ 9/ 76086

^7` Edogx}}l/ S1/ V1 Gdv/ V1 Iruhvl dqg U1 Vxqgdudp +4<<;,1 Vwrfkdvwlf Phdq Prghov ri wkh Whup

Vwuxfwxuh ri Lqwhuhvw Udwhv1 Zrunlqj Sdshu1

^8` Edfnxv/ G1/ V1 Iruhvl/ F1 Whophu +4<<;,1 Glvfuhwh0Wlph Prghov ri Erqg Sulflqj1 QEHUZrunlqj

Sdshu 9:691

^9` Eolvv/ U1 dqg S1 Ulwfknhq +4<<9,1 Hpslulfdo Whvw ri Wzr Vwdwh0Yduldeoh Khdwk0Mduurz0Pruwrq

Prgho1 Mrxuqdo ri Prqh|/ Fuhglw dqg Edqnlqj 5;/ Qr16/ 78507;41

^:` Ervt/ G1 +4<<<,1 Qrqsdudphwulf Vwdwlvwlfv iru Vwrfkdvwlf Surfhvvhv1 Ohfwxuh Qrwhv lq Vwdwlvwlfv1

Vsulqjhu1

^;` Erxgrxnk/ M1/ P1 Ulfkdugvrq/ U1 Vwdqwrq dqg U1 Zklwhodz +4<<;,1 Sulflqj Pruwjdjh0edfnhg

Vhfxulwlhv lq d Pxowlidfwru Lqwhuhvw Udwh Hqylurqphqw= D Pxowlyduldwh Ghqvlw| Hvwlpdwlrq Ds0

surdfk1 Zrunlqj Sdshu/ Vwhuq Vfkrro ri Exvlqhvv/ Qhz \run Xqlyhuvlw|1

^<` Euhqqdq/ P1 dqg H1 Vfkzduw} +4<:<,1 D Frqwlqxrxv Wlph Dssurdfk wr wkh Sulflqj ri Erqgv1

Mrxuqdo ri Edqnlqj dqg Ilqdqfh 6/ 46604881

^43` Euxjlhuh/ S1 +4<<4,1 Hvwlpdwlrq ri wkh Frh!flhqw ri d Gl>xvlrq Surfhvv lq wkh Pxowlglphqvlrqdo

Fdvh1 F1 U1 Dfdg1 Vfl1/ Sdulv/ w1645 +Vhulh L,/ <<<043371

^44` Exkohu/ Z1/ P1 Xkulj0Krpexuj/ X1 Zdowhu/ dqg W1 Zhehu +4<<<,1 Dq Hpslulfdo Frpsdulvrq ri

Iruzdug dqg Vsrw0Udwh Prghov iru Ydoxlqj Lqwhuhvw Udwh Rswlrq1 Mrxuqdo ri Ilqdqfh1

^45` Exvhu/ Khqghuvkrww dqg Vdqghuv +4<<3,1

^46` Fkdq/ N1F1/ J1 Nduro|l/ I1 Orqjvwd> dqg D1 Vdxqghu +4<<5,1 Dq Hpslulfdo Frpsdulvrq ri

Dowhuqdwlyh Prghov ri Vkruw0Whup Lqwhuhvw Udwh1 Mrxuqdo ri Ilqdqfh 7:/ 453<0455:1

^47` Fkhq/ U1U1 dqg O1 Vfrww +4<<6,1 Pd{lpxp Olnholkrrg Hvwlpdwlrq iru d Pxowlidfwru Htxloleulxp

Prgho ri wkh Whup Vwuxfwxuh ri Lqwhuhvw Udwhv1 Mrxuqdo ri Il{hg Lqfrph 6 +Ghfhpehu,/ 470641

^48` Fkhq/ U1U1 dqg O1 Vfrww +4<<5,1 Sulflqj Lqwhuhvw Udwh Rswlrqv lq d Wzr0Idfwru Fr{0Lqjhuvroo0

Urvv prghov ri wkh Vkruw0Whup Lqwhuhvw Udwhv1 Uhylhz ri Ilqdqfldo Vwxglhv/ 8+7,/ 94609691

^49` Fr{/ M1/ M1 Lqjhuvroo dqg V1 Urvv +4<;8,1 D Wkhru| ri wkh Whup Vwuxfwxuh ri Lqwhuhvw Udwhv1

Hfrqrphwulfd/ 86/ 6;807391

73

Page 42: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

^4:` Gdl/ T1 dqg N1 Vlqjohwrq +4<<:,1 Vshfl?fdwlrq Dqdo|vlv ri D!qh Whup Vwuxfwxuh Prghov1 Zrun0

lqj Sdshu/ Vwdqirug Xqlyhuvlw|1

^4;` Gx!h/ G1 dqg U1 Ndq +4<<9,1 D \lhog Idfwru Prgho ri Lqwhuhvw Udwh1 Pdwkhpdwlfdo Ilqdqfh/

9+7,/ 6:<07391

^4<` Gx!lh/ G1 dqg N1 Vlqjohwrq +4<<:,1 Dq Hfrqrphwulf Prgho ri wkh Whup Vwuxfwxuh ri Lqwhuhvw

Udwh Vzds \lhogv1 Mrxuqdo ri Ilqdqfh 85/ 45;:046541

^53` Kdugoh/ Z1 +4<<3,1 Dssolhg Qrqsdudphwulf Uhjuhvvlrq1 Fdpeulgjh Xqlyhuvlw| Suhvv1

^54` Khdwk/G1/ U1 Mduurz/ D1 Pruwrq +4<<5,1 Erqg Sulflqj dqg wkh Whup Vwuxfwxuh ri Lqwhuhvw Udwhv=

D Qhz Phwkrgrorj| iru Frqwlqjhqw Fodlpv Ydoxdwlrq1 Hfrqrphwulfd 93/ ::04381

^55` Kxoo/ M1 dqg D1 Zklwh +4<<3,1 Sulflqj Lqwhuhvw Udwh Ghulydwlyh Vhfxulwlhv1 Uhylhz ri Ilqdqfldo

Vwxglhv 6 +7,/ 8:608<51

^56` Lqxl/ N dqg P1 Nlmlpd +4<<;,1 D Pdunryldq Iudphzrun lq Pxowl0Idfwru Khdwk0Mduurz0Pruwrq

Prghov1 Mrxuqdo ri Ilqdqfldo dqg Txdqwlwdwlyh Dqdo|vlv/ 67 +4,/ 464048:1

^57` Mh>uh|/ D1/ R1 Olqwrq dqg W1 Qjx|hq +4<<<d,1 Whup Vwuxfwxuh Hvwlpdwlrq dqg d Sur{| iru wkh

Lqvwdqwdqhrxv Vsrw Lqwhuhvw Udwh41 Xqsxeolvkhg sdshu/ \doh Xqlyhuvlw|/ 4<<<

^58` Mh>uh|/ D1/ R1 Olqwrq dqg W1 Qjx|hq +4<<<e,1 Qrqsdudphwulf Hvwlpdwlrq ri Khdwk0Mduurz0

Pruwrq prghov dqg d Whvw iru Sdwk Lqghshqghqfh1 Xqsxeolvkhg sdshu/ \doh Xqlyhuvlw|/ 4<<<

^59` Mrqj/ gh M1 dqg S1 Vdqwd0Fodud +4<<<,1 Wkh G|qdplfv ri wkh Iruzdug Lqwhuhvw Udwh Fxuyh=

D Irupxodwlrq zlwk Vwdwh Yduldeohv1 Mrxuqdo ri Ilqdqfldo dqg Txdqwlwdwlyh Dqdo|vlv/ 66 +6,/

75607731

^5:` Nqljkw/ M1/ I1 Ol dqg P1 \xdq +4<<<,1 Sulflqj Lqwhuhvw Udwh Ghulydwlyhv lq d Qrq0Sdudphwulf

Wzr0Idfwru Whup Vwuxfwxuh Prgho1 Zrunlqj sdshu <<04</ Edqn ri Fdqdgd1

^5;` Olqwrq/ R1/ H1 Pdpphq/ M1 Qlhovhq dqg F1 Wdqjjdug +4<<;,1 Hvwlpdwlqj \lhog Fxuyhv e| Nhuqho

Vprrwklqj Phwkrgv1 Iruwkfrplqj lq wkh Mrxuqdo ri Hfrqrphwulfv1

^5<` Olwwhupdq/ U1/ dqg M1 Vfkhlqnpdq +4<<4,1 Frpprq Idfwruv D>hfwlqj Erqg Uhwxuqv1 Mrxuqdo ri

Il{hg Lqfrph/ Mxqh 4<<4/ 870941

^63` Orqjvwd>/ I1 D1 dqg H1 V1 Vfkzduw} +4<<5,1 Lqwhuhvw Udwh Yrodwlolw| dqg wkh Whup Vwuxfwxuh= D

Wzr Idfwru Jhqhudo Htxloleulxp Prgho1 Mrxuqdo ri Ilqdqfh/ 7:+7,/ 458<045;51

74

Page 43: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

^64` Oxhqehujhu/ G1 J1 +4<9<, Rswlpl}dwlrq e| Yhfwru Vsdfh Phwkrgv1 Qhz \run= Zloh|1

^65` PfFxoorfk/ M1 K1 +4<:4,1 Phdvxulqj wkh Whup Vwuxfwxuh ri Lqwhuhvw Udwhv1 Mrxuqdo ri Exvlqhvv

77/ 4<0641

^66` PfFxoorfk/ M1 K1 +4<:8,1 Wkh Wd{0Dgmxvwhg \lhog Fxuyh1 Wkh Mrxuqdo ri Ilqdqfh/ 63/ ;440;631

^67` Qhzh|/ Z1N1/ dqg M1O1 Srzhoo +5333,1 Lqvwuxphqwdo Yduldeohv Hvwlpdwlrq iru Qrqsdudphwulf

Prghov1 Pdqxvfulsw/ PLW1

^68` Ûnvhqgdo/ E1 +4<<;,1 Vwrfkdvwlf Gl>huhqwldo Htxdwlrqv1 Iliwk Hglwlrq1 Vsulqjhu= Ehuolq1

^69` Sdjdq/ D dqg D1 Xoodk +4<<<,1 Qrqsdudphwulf Hfrqrqrphwulfv1 Fdpeulgjh= Fdpeulgjh Xqlyhu0

vlw| Suhvv1

^6:` Shduvrq/ Q1 dqg W1 Vxq +4<<7,1 H{sorlwlqj wkh Frqglwlrqdo Ghqvlw| lq Hvwlpdwlqj wkh Whup

Vwuxfwxuh= Dq Dssolfdwlrq wr wkh Fr{/ Lqjhuvroo/ dqg Urvv Prgho1 Mrxuqdo ri Ilqdqfh 7</ 45:<0

46371

^6;` Shduvrq/ Q1 dqg D1 ]krx +4<<;, D Qrqsdudphwulf Dqdo|vlv ri wkh Iruzdug Udwh Yrodwlolwlhv1

Zrunlqj sdshu/ Xqlyhuvlw| ri Loolqrlv dw Xuedqd0Fkdpsdljq1

^6<` Skloolsv/ S1 dqg M1 Sdun +4<<;,1 Qrqvwdwlrqdu| Ghqvlw| Hvwlpdwlrq dqg Nhuqho Dxwrujuhvvlrq1

Zrunlqj sdshu1 Frzohv Irxqgdwlrq iru Uhvhdufk lq Hfrqrplfv/ \doh Xqlyhuvlw|1

^73` Uhyx} dqg \ru +4<<<,1 Frqwlqxrxv Pduwlqjdohv dqg Eurzqldq Prwlrq1 Vslqjhu0Yhuoduj= Qhz

\run1

^74` Vdqwd0Fodud/ S1 dqg G1 Vruqhwwh +5333,1 Wkh G|qdplfv ri wkh Iruzdug Lqwhuhvw Udwh Fxuyh zlwk

Vwrfkdvwlf Vwulqj Vfkrfnv1 Uhylhz ri Ilqdqfldo Vwxglhv/ iruwkfrplqj1

^75` Vfkdhihu/ V1P1 dqg H1V1 Vfkzduw} +4<;7,1 D Wzr Idfwru Prgho ri wkh Whup Vwuxfwxuh= Dq

Dssur{lpdwh Dqdo|wlfdo Vroxwlrq1 Mrxuqdo ri Ilqdqfldo dqg Txdqwlwdwlyh Dqdo|vlv/ 4<+7,/ 7460

7571

^76` Vwdqwrq/ U1 +4<<:,1 D Qrqsdudphwulf Prgho ri Whup Vwuxfwxuh G|qdplfv dqg wkh Pdunhw Sulfh

ri Lqwhuhvw Udwh Ulvn1 Mrxuqdo ri Ilqdqfh 85/ Qr1 8/4<:6053351

^77` Ydvlfhn/ R1 +4<::,1 Dq Htxloleulxp Fkdudfwhul}dwlrq ri wkh Whup Vwuxfwxuh1 Mrxuqdo ri Ilqdqfldo

Hfrqrplfv/ 8+5,/ 4::0;;1

75

Page 44: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Iljxuh 614= Yrodwlolw| vwuxfwxuh ri wkh |lhog fxuyhv lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS

gdlo| erqg gdwd iurp Mdq 4<94 wr Ghfhpehu 4<<;1 Pdwxulw| udqjhv iurp 3 wr 7 |hduv/ dqg wkh vkruw

udwh lv iurp 3( wr 4;(1

Iljxuh 615= Yrodwlolw| vwuxfwxuh ri wkh |lhog fxuyhv lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS

gdlo| erqg gdwd iurp Mdq 4<:3 wr Ghfhpehu 4<<;1 Pdwxulw| udqjhv iurp 3 wr 8 |hduv/ dqg wkh vkruw

udwh lv iurp 3( wr 4;(1

76

Page 45: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Iljxuh 616= Yrodwlolw| vwuxfwxuh ri wkh |lhog fxuyhv lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS

gdlo| erqg gdwd iurp Mdq 4<;6 wr Ghfhpehu 4<<;1 Pdwxulw| udqjhv iurp 3 wr < |hduv/ dqg wkh vkruw

udwh lv iurp 3( wr 43(1

Iljxuh 617= Yrodwlolw| vwuxfwxuh ri wkh |lhog fxuyhv lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS

gdlo| erqg gdwd iurp Mdq 4<<3 wr Ghfhpehu 4<<;1 Pdwxulw| udqjhv iurp 3 wr < |hduv/ dqg wkh vkruw

udwh lv iurp 3( wr ;(1

77

Page 46: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Iljxuh 618= Yrodwlolw| vwuxfwxuh ri wkh |lhog fxuyhv lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS

gdlo| erqg gdwd iurp Mdq 4<;6 wr Ghfhpehu 4<<; zkhuh wkh |lhog fxuyh lv h{wudfwhg e| wkh Olqwrq

hw do1 +4<<;, nhuqho vprrwklqj0edvhg phwkrg1 Wkh judsk lq wkh wrs lv wkh yrodwlolw| vwuxfwxuh ri wkh

|lhog fxuyhv hvwlpdwhg qrqsdudphwulfdoo| e| wkh phwkrg sursrvhg lq wklv sdshu1 Wkh judsk lq wkh

erwwrp lv rewdlqhg e| wkh phwkrg sursrvhg lq Mh>uh|0Olqwrq dqg Qjx|hq +4<<<e,1 Pdwxulw| udqjhv

iurp 3 wr 7 |hduv/ dqg wkh vkruw udwh lv iurp 3( wr 48(1

78

Page 47: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

�øAéué-éEIä

3 318 413 418 513 518 613 618 713

5184 :187 :1:; :1<: ;145 ;158 ;16; ;179 ;17; ;174

6167 ;185 ;1:< ;1<; <147 <164 <179 <185 <17: <168

714; 43144 43169 43186 431:4 431<7 44146 44148 44135 431;8

8134 4517: 4518: 4518; 45198 451;8 46138 46138 451;: 4519:

81;7 471;7 471:3 47176 47158 47164 47176 47175 47157 4713:

919: 4918; 49156 4819< 48156 48139 4813: 48135 471;; 471:8

:184 4:1;5 4:15: 49184 481;4 48175 48163 48155 48144 48136

;167 4;1;: 4;146 4:147 49155 48198 48174 48163 48154 48149

<14: 4<1<4 4;1<5 4:19: 49185 481:< 48179 48165 48159 48155

43134 531<5 4<196 4;139 49199 481:; 4816< 48157 4814< 4814:

431;7 541:< 5314: 4;159 49193 48193 48149 48135 471<< 471<;

Wdeoh 614= Yrodwlolw| vwuxfwxuh dvvrfldwhg zlwk wkh vhfrqg Eurzqldq prwlrq iurp wkh 50idfwru

KMP prgho lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS gdlo| erqg gdwd iurp Mdq 4<<3 wr Ghfhpehu

4<<; zkhuh wkh |lhog fxuyh lv h{wudfwhg e| wkh Olqwrq hw do1 +4<<;, nhuqho vprrwklqj0edvhg phwkrg1

Wkh edqgzlgwk fkrvhq iru wkh yrodwlolw| vwuxfwxuh hvwlpdwlrq lv 4(1 Pdwxulw| +ôä udqjhv iurp 3 wr

7 |hduv/ vkruw udwh lv ?{hg dw 919:( dqg wkh orqj udwh +OU, lv iurp 5184( wr 431;7(1 Yrodwlolw| lv

uhsruwhg lq 4243331�øAéué -EIä

3 318 413 418 513 518 613 618 713

5184 :188 :1;5 ;137 ;14< ;165 ;177 ;187 ;188 ;17<

6167 ;187 ;1;6 <138 <155 <16< <186 <193 <189 <177

714; 43147 43175 43195 431;3 44136 44154 44157 44145 431<7

8134 45185 45198 4519; 451:8 451<8 46148 46148 451<; 451:;

81;7 471<4 471:< 47187 4716: 47175 47187 47186 47169 4714;

919: 4919: 49166 481;4 48168 4814; 4814; 48147 48133 471;:

:184 4:1<4 4:16; 49197 481<6 48187 48174 48166 48156 48147

;167 4;1<: 4;158 4:15: 49168 481:: 48185 48174 48166 4815;

<14: 53135 4<137 4:1;3 49198 481<4 4818: 48177 4816; 48167

43134 54137 4<1:8 4;14< 491:< 481<3 48183 48169 48164 4815<

431;7 541<4 53163 4;173 491:6 481:5 4815: 48146 48144 48143

Wdeoh 615= Yrodwlolw| vwuxfwxuh dvvrfldwhg zlwk wkh vhfrqg Eurzqldq prwlrq iurp wkh 50idfwru

KMP prgho lv hvwlpdwhg qrqsdudphwulfdoo| iurp FUVS gdlo| erqg gdwd iurp Mdq 4<<3 wr Ghfhpehu

4<<; zkhuh wkh |lhog fxuyh lv h{wudfwhg e| wkh Olqwrq hw do1 +4<<;, nhuqho vprrwklqj0edvhg phwkrg1

79

Page 48: Nonparametric Estimation of a Multifactor Heath-Jarrow ......Nonparametric Estimation of a Multifactor Heath-Jarrow-Morton Model: An Integrated Approach By Andrew Jeffrey, Oliver Linton,

Wkh edqgzlgwk fkrvhq iru wkh yrodwlolw| vwuxfwxuh hvwlpdwlrq lv 4(1 Pdwxulw| +ôä udqjhv iurp 3 wr

7 |hduv/ vkruw udwh lv ?{hg dw 919:( dqg wkh orqj udwh +OU, lv iurp 5184( wr 431;7(1 Yrodwlolw| lv

uhsruwhg lq 4243331

7: