Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
)DVW *UHHNV LQ )RUZDUG /LERU 0RGHOV
3DXO *ODVVHUPDQ
*UDGXDWH 6FKRRO RI %XVLQHVV
&ROXPELD 8QLYHUVLW\
1HZ <RUN� 1< �����
;LDROLDQJ =KDR
'HSDUWPHQW RI 6WDWLVWLFV
&ROXPELD 8QLYHUVLW\
1HZ <RUN� 1< �����
-DQXDU\ ����
$EVWUDFW
7KLV SDSHU GHYHORSV PHWKRGV IRU IDVW HVWLPDWLRQ RI RSWLRQ SULFH VHQVLWLYLWLHV LQ 0RQWH &DUOR
VLPXODWLRQ RI WHUP VWUXFWXUH PRGHOV� 7KH PRGHOV FRQVLGHUHG DUH EDVHG RQ GLVFUHWHO\ FRPSRXQGHG
IRUZDUG UDWHV ZLWK SURSRUWLRQDO YRODWLOLWLHV� :H LQYHVWLJDWH WKH HpFLHQW HVWLPDWLRQ RI RSWLRQ
GHOWDV� JDPPDV� DQG YHJDV LQ WKLV VHWWLQJ� 9DULRXV JHQHUDO PHWKRGV DUH DYDLODEOH LQ WKH 0RQWH
&DUOR OLWHUDWXUH IRU FRPSXWLQJ VXFK HVWLPDWHV� ZH WDLORU WKHVH PHWKRGV WR WKH WHUP VWUXFWXUH
PRGHOV DQG GHYHORS DSSUR[LPDWLRQV VSHFLnF WR WKLV VHWWLQJ LQ RUGHU WR HLWKHU DFFHOHUDWH WKH
PHWKRGV RU H[SDQG WKHLU DSSOLFDELOLW\� :H SURYLGH VRPH WKHRUHWLFDO VXSSRUW IRU WKH DSSOLFDWLRQ
RI WKH EDVLF PHWKRGV DQG HYDOXDWH WKH DSSUR[LPDWLRQV WKURXJK QXPHULFDO H[SHULPHQWV� 7KH
UHVXOWV LQGLFDWH WKDW WKH SURSRVHG DOJRULWKPV FDQ VXEVWDQWLDOO\ LPSURYH RYHU VWDQGDUG nQLWH
GLmHUHQFH HVWLPDWHV RI VHQVLWLYLWLHV�
� ,QWURGXFWLRQ
7KLV SDSHU GHYHORSV PHWKRGV IRU IDVW HVWLPDWLRQ RI RSWLRQ SULFH VHQVLWLYLWLHV EDVHG RQ 0RQWH &DUOR
VLPXODWLRQ RI IRUZDUG /LERU PRGHOV RI WKH W\SH GHYHORSHG E\ %UDFH� *DWDUHN� DQG 0XVLHOD >�@�
-DPVKLGLDQ >��@� 0LOWHUVHQ� 6DQGPDQQ� DQG 6RQGHUPDQQ >��@� DQG 0XVLHOD DQG 5XWNRZVNL >��@�
7KHVH PRGHOV DUH VLPLODU LQ VSLULW WR WKH JHQHUDO IUDPHZRUN RI +HDWK� -DUURZ� DQG 0RUWRQ >��@� EXW
GLmHU LQ WKDW WKH\ PRGHO WKH G\QDPLFV RI GLVFUHWHO\ FRPSRXQGHG IRUZDUG UDWHV �GLUHFWO\ REVHUYDEOH
LQ WKH PDUNHW� UDWKHU WKDQ LQVWDQWDQHRXV FRQWLQXRXVO\ FRPSRXQGHG IRUZDUG UDWHV� $V LQ WKH +-0
VHWWLQJ� DUELWUDJH UHVWULFWLRQV GHWHUPLQH WKH G\QDPLFV RI WKH IRUZDUG FXUYH �QRZ UHSUHVHQWHG E\ D
YHFWRU RI GLVFUHWH UDWHV� RQFH WKH YRODWLOLW\ VWUXFWXUH DQG QXPHUDLUH KDYH EHHQ FKRVHQ� 7KH UHVXOWLQJ
G\QDPLFV DUH W\SLFDOO\ FRPSOH[ HQRXJK WR PDNH 0RQWH &DUOR VLPXODWLRQ WKH SULPDU\ FRPSXWDWLRQDO
WRRO IRU XVH ZLWK WKHVH PRGHOV�
3ULFH VHQVLWLYLWLHV DUH� RI FRXUVH� RI FHQWUDO LPSRUWDQFH LQ DQ\ PRGHO IRU SULFLQJ GHULYDWLYH
VHFXULWLHV EHFDXVH WKH VHQVLWLYLWLHV GHWHUPLQH WKH WUDGLQJ VWUDWHJ\ WKDW KHGJHV WKH GHULYDWLYH VHFXULW\�
$ FRPPRQ FULWLFLVP RI 0RQWH &DUOR VLPXODWLRQ LV WKDW LW SURGXFHV SRRU HVWLPDWHV RI *UHHNV� ,QGHHG�
XVLQJ VWUDLJKWIRUZDUG VLPXODWLRQ� HVWLPDWLQJ GHOWDV ZLWK UHVSHFW WR 1 XQGHUO\LQJ DVVHWV RU UDWHV
UHTXLUHV VLPXODWLQJ D PLQLPXP RI 1 � � WLPHV DV PDQ\ SDWKV DV HVWLPDWLQJ D SULFH DORQH� DQG LQ
�
VSLWH RI WKLV WKH GHOWD HVWLPDWHV REWDLQHG ZLOO RIWHQ EH PXFK OHVV DFFXUDWH WKDQ WKH HVWLPDWHG SULFH�
7KHUH DUH� KRZHYHU� 0RQWH &DUOR PHWKRGV VSHFLnFDOO\ GHVLJQHG IRU WKH HVWLPDWLRQ RI VHQVLWLYLWLHV�
6RPH RI WKHVH DUH WUHDWHG LQ� H�J�� *ODVVHUPDQ >�@� *O\QQ DQG /(FX\HU >�@� +R DQG &DR >��@� 5HLPDQ
DQG :HLVV >��@� 5XELQVWHLQ DQG 6KDSLUR >��@� DQG WKH DSSOLFDWLRQ RI WKHVH DQG UHODWHG PHWKRGV WR
RSWLRQ SULFLQJ KDV EHHQ FRQVLGHUHG LQ %URDGLH DQG *ODVVHUPDQ >�@� )RXUQLuH� /DVU\� /HEXFKRX[�
/LRQV� DQG 7RX]L >�@� )X DQG +X >�@ DQG 3LNRYVN\ >�@� VHH DOVR WKH RYHUYLHZ LQ %R\OH HW DO� >�@� %XW
WKH FODVV RI PRGHOV IRU LQ ZKLFK ZH LPSOHPHQW WKH PHWKRGV KHUH LV VRPHZKDW PRUH FRPSOH[ WKDQ
SUHYLRXV nQDQFLDO DSSOLFDWLRQV RI WKHVH PHWKRGV DQG LW UDLVHV ERWK SUDFWLFDO DQG WKHRUHWLFDO LVVXHV�
7KH SUREOHP RI HVWLPDWLQJ VHQVLWLYLWLHV E\ VLPXODWLRQ PD\ EH IRUPXODWHG TXLWH JHQHUDOO\ DV RQH RI
HVWLPDWLQJ WKH GHULYDWLYH RI DQ H[SHFWDWLRQ ZLWK UHVSHFW WR D SDUDPHWHU� ,Q WKH FDVH RI HVWLPDWLQJ
D GHOWD� IRU H[DPSOH� WKH UHOHYDQW SDUDPHWHU LV WKH LQLWLDO YDOXH RI D SULFH RU UDWH� 0HWKRGV IRU
HVWLPDWLQJ VHQVLWLYLWLHV PD\ EH EURDGO\ FODVVLnHG E\ ZKHWKHU WKH\ SXW WKH GHSHQGHQFH RQ WKH
SDUDPHWHU LQ WKH XQGHUO\LQJ VWRFKDVWLF SURFHVV RU LQ WKH SUREDELOLW\ PHDVXUH� %RWK SHUVSHFWLYHV
DUH JHQHUDOO\ SRVVLEOH� DQG WKLV oH[LELOLW\ LV DQDORJRXV WR WZR ZD\V RI DGGLQJ D GULIW x WR %URZQLDQ
PRWLRQ� ZH PD\ DGG xW DW WLPH W WR HDFK %URZQLDQ SDWK� RU ZH FDQ OHDYH WKH SDWKV XQFKDQJHG
DQG XVH *LUVDQRYV 7KHRUHP WR DGG D GULIW WKURXJK D FKDQJH RI SUREDELOLW\ PHDVXUH� 3XWWLQJ WKH
GHSHQGHQFH RQ WKH SDUDPHWHU LQ WKH VDPSOH SDWKV RI WKH VWRFKDVWLF SURFHVV OHDGV WR HVWLPDWRUV WKDW
GLmHUHQWLDWH WKH SDWKV RI WKH SURFHVV _ ZH FDOO WKHVH SDWKZLVH GHULYDWLYHV� 3XWWLQJ WKH GHSHQGHQFH
LQ WKH PHDVXUH OHDGV WR HVWLPDWRUV EDVHG RQ GLmHUHQWLDWLQJ SUREDELOLW\ GHQVLWLHV� WKLV LV RIWHQ UHIHUUHG
WR DV WKH OLNHOLKRRG UDWLR PHWKRG �/50��
:H LQYHVWLJDWH WKH XVH RI ERWK SDWKZLVH GHULYDWLYHV DQG /50 LQ HVWLPDWLQJ GHOWDV DQG JDPPDV
DQG WKH XVH RI SDWKZLVH HVWLPDWRUV IRU ?YHJD� �VHQVLWLYLW\ WR FKDQJHV LQ YRODWLOLW\�� 2XU SULPDU\
FRQWULEXWLRQ WR WKH OLWHUDWXUH RQ IRUZDUG /LERU PRGHOV OLHV LQ GHULYLQJ DQG FRPSDULQJ D YDULHW\ RI
PHWKRGV DQG LGHQWLI\LQJ ZKLFK DUH PRVW SUDFWLFDO DQG HmHFWLYH LQ WKLV FRQWH[W� ,Q WKLV UHJDUG RXU
FRQFOXVLRQV DUH DV IROORZV�
p )RU HVWLPDWLQJ GHOWDV ZKHQ WKH RSWLRQ SD\Rm LV D �/LSVFKLW]� FRQWLQXRXV IXQFWLRQ RI WKH
IRUZDUG UDWHV� XVH WKH SDWKZLVH PHWKRG ZLWK D IRUZDUG�GULIW DSSUR[LPDWLRQ�
p )RU HVWLPDWLQJ GHOWDV ZKHQ WKH SD\Rm LV GLVFRQWLQXRXV �H�J�� D GLJLWDO RU NQRFNRXW SD\Rm��
XVH /50 ZLWK WKH IRUZDUG�GULIW DSSUR[LPDWLRQ�
p 1R PHWKRG LV HQWLUHO\ VDWLVIDFWRU\ IRU HVWLPDWLQJ JDPPDV� &RQYHQWLRQDO FHQWUDO GLmHUHQFH DS�
SUR[LPDWLRQV DUH YHU\ VHQVLWLYH WR WKH VL]H RI WKH SHUWXUEDWLRQ LQWURGXFHG� $ PL[HG SDWKZLVH�
/50 PHWKRG DSSHDUV SUHIHUDEOH�
p $ SDWKZLVH HVWLPDWRU XVLQJ D IRUZDUG�GULIW DSSUR[LPDWLRQ LV IDVW DQG HmHFWLYH LQ HVWLPDWLQJ
YHJD ZKHQ WKH SD\Rm LV FRQWLQXRXV�
�
5HODWLYH WR WKH JHQHUDO OLWHUDWXUH RQ HVWLPDWLQJ VHQVLWLYLWLHV WKURXJK VLPXODWLRQ� WKLV SDSHU
PDNHV WKUHH SULQFLSDO FRQWULEXWLRQV�
�L� LW SURSRVHV DQG HYDOXDWHV IDVW DSSUR[LPDWLRQV WR DQ H[DFW SDWKZLVH DOJRULWKP VSHFLnF WR WKH
IRUZDUG /LERU VHWWLQJ�
�LL� LW DQDO\]HV WKH FRQYHUJHQFH WR WKH FRQWLQXRXV�WLPH OLPLW RI SDWKZLVH HVWLPDWRUV EDVHG RQ
GLVFUHWH�WLPH VLPXODWLRQ�
�LLL� LW XVHV DQ DSSUR[LPDWH /50 HVWLPDWRU LQ D VHWWLQJ ZKHUH WKH UHOHYDQW SUREDELOLW\ GHQVLW\ LV
XQNQRZQ DQG GHYHORSV D PHWKRG IRU DSSO\LQJ /50 LQ D VLQJXODU VHWWLQJ ZKHUH QR GHQVLW\
H[LVWV�
:H FRPPHQW EULHo\ RQ HDFK RI WKHVH SRLQWV� �L� ,Q LWV H[DFW YHUVLRQ� WKH SDWKZLVH PHWKRG HQWDLOV
VLPXODWLQJ D VWRFKDVWLF SURFHVV RI GHULYDWLYHV RI VWDWH YDULDEOHV LQ DGGLWLRQ WR WKH RULJLQDO VWDWH
YDULDEOHV� ,Q D PRGHO ZLWK WKH FRPSOH[LW\ RI WKH IRUZDUG /LERU PRGHOV� WKH HmRUW LQYROYHG LQ
VLPXODWLQJ WKH GHULYDWLYHV SURFHVV FDQ EH FRPSDUDEOH WR WKDW UHTXLUHG WR VLPXODWH D SHUWXUEHG FRS\
RI WKH RULJLQDO SURFHVV� VR WKH SDWKZLVH PHWKRG PD\ QRW RmHU D ODUJH DGYDQWDJH RYHU D VWDQGDUG
nQLWH GLmHUHQFH DSSUR[LPDWLRQ WR D GHULYDWLYH EDVHG RQ UHVLPXODWLQJ WKH RULJLQDO SURFHVV� 7KH
DSSUR[LPDWLRQV ZH GHYHORS DGGUHVV WKLV LVVXH� �LL� 7KH SDWKZLVH PHWKRG FDQ EH IRUPXODWHG LQ
FRQWLQXRXV WLPH �GLmHUHQWLDWLQJ D GLmXVLRQ SURFHVV ZLWK UHVSHFW WR D SDUDPHWHU� RU LQ GLVFUHWH
WLPH �GLmHUHQWLDWLQJ WKH GLVFUHWL]HG SURFHVV LQ WKH VLPXODWLRQ�� :H JLYH FRQGLWLRQV XQGHU ZKLFK
WKH GLVFUHWH�WLPH HVWLPDWRU JLYHV XQELDVHG GHULYDWLYH HVWLPDWHV IRU WKH VLPXODWHG SURFHVV DQG DOVR
XQGHU ZKLFK LW FRQYHUJHV WR WKH FRUUHFW FRQWLQXRXV�WLPH OLPLW� �LLL� 7KH DSSOLFDWLRQ RI /50 WR
HVWLPDWLQJ GHOWD HQWDLOV NQRZOHGJH RI WKH WUDQVLWLRQ GHQVLW\ RI WKH XQGHUO\LQJ VWDWH YDULDEOHV� 1R
VXFK GHQVLW\ LV DYDLODEOH LQ IRUZDUG /LERU PRGHOV� VR ZH XVH D *DXVVLDQ DSSUR[LPDWLRQ� 7KLV GRHV
QRW HQWLUHO\ UHVROYH WKH SUREOHP EHFDXVH LQ D PRGHO ZLWK IHZHU IDFWRUV WKDQ VWDWH YDULDEOHV �L�H�� D
PRGHO LQ ZKLFK WKH GLPHQVLRQ RI WKH GULYLQJ %URZQLDQ PRWLRQ LV VPDOOHU WKDQ WKH GLPHQVLRQ RI WKH
VWDWH YHFWRU� WKH GLVWULEXWLRQ RI WKH LQFUHPHQWV RI WKH VWDWH YDULDEOHV RYHU RQH VLPXODWHG WLPH VWHS
LV VLQJXODU DQG IDLOV WR KDYH D GHQVLW\ _ HYHQ LQ WKH *DXVVLDQ FDVH� 7KH LQFUHPHQW RYHU PXOWLSOH
WLPH VWHSV PD\ QHYHUWKHOHVV KDYH D GHQVLW\ DQG ZH XVH WKLV REVHUYDWLRQ WR DSSO\ /50�
7KH UHVW RI WKLV SDSHU LV RUJDQL]HG DV IROORZV� 6HFWLRQ � UHYLHZV WKH G\QDPLFV RI IRUZDUG
/LERU PRGHOV� 6HFWLRQ � GHYHORSV SDWKZLVH GHOWD HVWLPDWRUV� nUVW GHULYLQJ DQ H[DFW PHWKRG DQG
WKHQ SURSRVLQJ DQG HYDOXDWLQJ DSSUR[LPDWLRQV� 6HFWLRQ � GHYHORSV /50 GHOWD HVWLPDWRUV� nUVW
UHYLHZLQJ WKH PHWKRG LQ D SXUHO\ *DXVVLDQ VHWWLQJ WKHQ WDLORULQJ LWV DSSOLFDWLRQ WR IRUZDUG /LERU
PRGHOV� 6HFWLRQ � DGGUHVVHV WKH VRPHZKDW KDUGHU SUREOHP RI HVWLPDWLQJ JDPPD DQG 6HFWLRQ �
GHDOV ZLWK YHJD� 7KHRUHWLFDO DQDO\VLV RI WKH SDWKZLVH HVWLPDWRUV LV JLYHQ LQ 6HFWLRQ ��
�
� 3UHOLPLQDULHV RQ WKH 0RGHO
:H EHJLQ ZLWK D EULHI UHYLHZ RI /LERU PDUNHW PRGHOV EDVHG RQ D nQLWH VHW RI PDWXULWLHV� DV GHYHORSHG
E\ -DPVKLGLDQ >��@� 7KH WHQRU VWUXFWXUH LV D nQLWH VHW RI GDWHV
� 7� � 7� � c c c � 71 � 71��
UHSUHVHQWLQJ PDWXULWLHV VSDFHG� H�J�� WKUHH PRQWKV RU VL[ PRQWKV DSDUW� )RU VLPSOLFLW\� ZH DVVXPH
WKDW WKH GD\�FRXQW IUDFWLRQV pL� 7L�� b 7L� L �� � � � � 1 � DUH DOO HTXDO WR D n[HG p �H�J�� p
���� \HDUV�� ,Q SUDFWLFH� GD\�FRXQW FRQYHQWLRQV ZRXOG PDNH WKH OHQJWKV RI WKHVH LQWHUYDOV VOLJKWO\
GLmHUHQW� 7KH OHIW�FRQWLQXRXV IXQFWLRQ s � ��� 71��@� I�� � � � � 1 � �J GHnQHG E\ WDNLQJ s�W� WR EH
WKH XQLTXH LQWHJHU VDWLVI\LQJ
7s�W�b� � W � 7s�W�
JLYHV WKH LQGH[ RI WKH QH[W WHQRU GDWH DW WLPH W� $VVRFLDWHG ZLWK HDFK WHQRU GDWH 7L LV D ]HUR�FRXSRQ
ERQG PDWXULQJ DW WKDW GDWH� %L�W� LV WKH SULFH RI WKDW ERQG DW WLPH W � >�� 7L@ DQG %L�7L� ��
7KH IRUZDUG /LERU UDWH DW WLPH W IRU WKH DFFUXDO SHULRG >7L� 7L��@� W � 7L LV
/L�W� �
p
t%L�W�
%L���W�b �
u� L �� � � � � 1� ���
,W LV DW WLPHV QRWDWLRQDOO\ FRQYHQLHQW WR H[WHQG WKH GHnQLWLRQ RI /L EH\RQG WKH LWK WHQRU GDWH� ZH
GR VR E\ VHWWLQJ /L�W� /L�7L� IRU W ! 7L� $W D WHQRU GDWH 7L WKH SULFH RI DQ\ ERQG %Q� Q ! L� WKDW
KDV QRW \HW PDWXUHG LV JLYHQ E\
%Q�7L� Qb�<M L
�
� � p/M�7L��
PRUH JHQHUDOO\� DW DQ DUELWUDU\ WLPH W � 7Q ZH KDYH
%Q�W� %s�W��W�Qb�<
M s�W�
�
� � p/M�W�� ���
7KH G\QDPLFV RI WKH IRUZDUG /LERU UDWHV GHSHQG RQ WKH IRUP DVVXPHG IRU WKHLU YRODWLOLWLHV DQG
RQ WKH PHDVXUH XQGHU ZKLFK WKH PRGHO LV VSHFLnHG� 7KURXJKRXW� ZH DVVXPH WKH /LERU UDWHV KDYH
GHWHUPLQLVWLF YRODWLOLWLHV �VR WKDW FDSOHWV DUH SULFHG E\ %ODFNV IRUPXOD� DV LQ >�@� DQG ZH ZRUN LQ
WKH VSRW /LERU PHDVXUH LQWURGXFHG E\ -DPVKLGLDQ >��@� 7KLV LV WKH HTXLYDOHQW PDUWLQJDOH PHDVXUH
DVVRFLDWHG ZLWK WKH QXPHUDLUH
%e�W� %s�W��W�
%����
s�W�b�<M �
%M�7M�
%M���7M��
ZKLFK PD\ EH LQWHUSUHWHG DV WKH UHVXOW RI EX\LQJ � %���� ERQGV DW WLPH � PDWXULQJ DW 7�� DQG
WKHQ DW HDFK WHQRU GDWH VHOOLQJ WKH ERQGV WKDW PDWXUHG DQG LQYHVWLQJ WKH SURFHHGV LQ WKH ERQG
�
WKDW PDWXUHV QH[W� 7KLV LV WKXV D GLVFUHWHO\ FRPSRXQGHG DQDORJ RI WKH PRQH\ PDUNHW DFFRXQW WKDW
JLYHV ULVH WR WKH XVXDO ULVN�QHXWUDO PHDVXUH� $ SDUWLFXODU FDVH RI -DPVKLGLDQV FRQVWUXFWLRQ LV WKH
VSHFLnFDWLRQG/Q�W�
/Q�W�
Q;L s�W�
pwQ�W�wL�W��/L�W�
� � p/L�W�GW� wQ�W� G:W� Q �� � � � � 1� ���
LQ ZKLFK :W LV D VWDQGDUG G�GLPHQVLRQDO %URZQLDQ PRWLRQ XQGHU WKH VSRW /LERU PHDVXUH DQG HDFK
wQ LV GHWHUPLQLVWLF� ERXQGHG� DQG SRVVLEO\ WLPH�YDU\LQJ� ZLWK wQ�W� D G�GLPHQVLRQDO URZ YHFWRU�
�7DNH wQ�W� s � IRU W w 7Q WR NHHS /Q�W� /Q�7Q� RQ >7Q� 71��@�� 7KH IRUP RI WKH GULIW LQ ���
LV QHFHVVLWDWHG E\ WKH DEVHQFH RI DUELWUDJH RQFH WKH YRODWLOLWLHV �DQG WKH QXPHUDLUH� DUH VSHFLnHG�
,Q SDUWLFXODU� ZLWK WKLV FKRLFH RI GULIW GHoDWHG DVVHW SULFHV �UDWLRV RI DVVHW SULFHV WR %e�W�� DUH
PDUWLQJDOHV� ,W IROORZV WKDW WKH WLPH�W YDOXH &�W� DQG WLPH�7 YDOXH &�7 � RI D GHULYDWLYH VHFXULW\
�WKDW FDQ EH UHSOLFDWHG E\ WUDGLQJ LQ WKH EDVLF ERQGV� DUH UHODWHG E\ WKH SULFLQJ UXOH
&�W� %e�W�(
�&�7 �
%e�7 �M)W
w� ���
ZKHUH I) � W w �J LV WKH nOWUDWLRQ JHQHUDWHG E\ WKH %URZQLDQ PRWLRQ�
,Q RUGHU WR GHOWD KHGJH D GHULYDWLYH ZLWK SRVLWLRQV LQ WKH XQGHUO\LQJ ERQGV� ZH QHHG WR FDOFXODWH�
H�J��#&���
#%Q���
#
#%Q���
t%e���(
�&�7 �
%e�7 �
wu� Q �� � � � � 1 � ��
,Q OLJKW RI WKH GHWHUPLQLVWLF UHODWLRQV ��� DQG ���� WKLV LV HTXLYDOHQW �WKURXJK WKH FKDLQ UXOH RI
RUGLQDU\ FDOFXOXV� WR FRPSXWLQJ VHQVLWLYLWLHV
#&���
#/N���
#
#/N���
t%e���(
�&�7 �
%e�7 �
wu� N �� � � � � 1�
VHWWLQJ /� �>� %����@b�� p� 7KLV PD\ EH YLHZHG DV D W\SH RI EXFNHW KHGJLQJ LQ ZKLFK D VHSDUDWH
GHOWD LV FRPSXWHG ZLWK UHVSHFW WR HDFK FRPSRQHQW RI WKH IRUZDUG UDWH YHFWRU� *LYHQ HLWKHU EXFNHW
GHOWDV RU GHOWDV ZLWK UHVSHFW WR ]HUR�FRXSRQ ERQGV RQH FDQ LQ WXUQ FRPSXWH GHOWDV ZLWK UHVSHFW
WR WKH EDVLF LQVWUXPHQWV XVHG WR EXLOG D IRUZDUG FXUYH DJDLQ XVLQJ MXVW WKH RUGLQDU\ FKDLQ UXOH�
EHFDXVH WKH ERQG SULFHV DQG IRUZDUG UDWHV DUH GHWHUPLQLVWLFDOO\ UHODWHG WR WKH EDVLF LQVWUXPHQWV _
WKH GHWHUPLQLVWLF UHODWLRQ EHLQJ HPERGLHG LQ WKH FXUYH EXLOGLQJ DOJRULWKP�
� 3DWKZLVH 'HOWDV
��� 3UHOLPLQDULHV
7R PRWLYDWH WKH nUVW PHWKRG ZH GHYHORS� FRQVLGHU D FDSOHW ZLWK VWULNH . SD\LQJ p�/Q�7Q�b.��
DW 7Q��� )URP ��� DQG WKH GHnQLWLRQ RI %e LW IROORZV WKDW LWV SULFH DW WLPH � LV
&Q��� %����p(
��/Q�7Q�b.��
Q<L �
�
� � p/L�7Q�
�� ���
�
0XFK DV LQ %UDFH� *DWDUHN� DQG 0XVLHOD >�@� WKLV H[SHFWDWLRQ LV HYDOXDWHG E\ WKH %ODFN IRUPXOD
&Q��� &%ODFN�xwQ� .� /Q���� %Q������ 7Q�
ZKHUH
&%ODFN�}�.� U� E� 7 � pE
�Uj
�ORJ�U .� � �
�}�7
}S7
�b.j
�ORJ�U .�b �
�}�7
}S7
��� ���
ZKHUH j LV WKH FXPXODWLYH QRUPDO GLVWULEXWLRQ DQG
xwQ
V=7Q
�
NwQ�W�N� GW�
2I FRXUVH� VLQFH ZH ZDQW WR GHYHORS D JHQHUDO PHWKRG ZH ZLOO QRW XVH WKH IDFW WKDW D FDSOHW FDQ EH
SULFHG LQ FORVHG IRUP H[FHSW WR FRPSDUH QXPHULFDO UHVXOWV�
7R FRPSXWH� H�J�� #&Q��� #/N��� ZH QHHG WR FRPSXWH WKH VHQVLWLYLW\ RI WKH H[SHFWDWLRQ LQ ���
ZLWK UHVSHFW WR /N���� 3URYLGHG GHULYDWLYH DQG H[SHFWDWLRQ FDQ EH LQWHUFKDQJHG� ZH KDYH
#
#/N���(
��/Q�7Q�b.��
Q<L �
�
� � p/L�7Q�
� (
�#
#/N���
��/Q�7Q�b.��
Q<L �
�
� � p/L�7Q�
���
���
,I ZH FDQ HYDOXDWH WKH GHULYDWLYH LQVLGH WKH H[SHFWDWLRQ RQ WKH ULJKW IRU HDFK VLPXODWHG SDWK� WKHQ
E\ DYHUDJLQJ RYHU SDWKV ZH REWDLQ DQ HVWLPDWH RI WKH H[SHFWDWLRQ DQG WKXV RI WKH GHULYDWLYH RQ WKH
OHIW� 7KH FKDLQ UXOH VXJJHVWV
#
#/N���
��/Q�7Q�b.��
Q<L �
�
� � p/L�7Q�
�
Q;L �
#
#/L�7Q�
��/Q�7Q�b.��
Q<L �
�
� � p/L�7Q�
�#/L�7Q�
#/N����
2QH PD\ TXHVWLRQ ZKHWKHU WKH H[SUHVVLRQ LQ FXUO\ EUDFHV FDQ EH GLmHUHQWLDWHG DV LQGLFDWHG LQ OLJKW
RI WKH SUHVHQFH RI WKH SRVLWLYH�SDUW RSHUDWRU� +RZHYHU� WKH PDSSLQJ [ �� �[ b .�� LV /LSVFKLW]
FRQWLQXRXV DQG WKXV GLmHUHQWLDEOH DOPRVW HYHU\ZKHUH DQG HTXDO WR WKH LQGHnQLWH LQWHJUDO RI LWV
D�H��GHnQHG GHULYDWLYH� :LWK SUREDELOLW\ RQH� ZH KDYH
#
#/N����/Q�7Q�b.�� �I/Q�7Q� ! .J#/Q�7Q�
#/N����
�7KH H[SUHVVLRQ �IcJ WDNHV WKH YDOXH � ZKHQ WKH HYHQW LQ EUDFHV RFFXUV DQG � RWKHUZLVH��*HQHUDOL]LQJ WKH VHWWLQJ� WR HVWLPDWH
#
#/N���(>J�/��W��� � � � � /1 �W1 ��@
IRU VRPH /LSVFKLW] FRQWLQXRXV J DQG DUELWUDU\ GDWHV WL� ZH EULQJ WKH GHULYDWLYH LQVLGH WKH H[SHFWDWLRQ
WR DUULYH DW WKH �FRQWLQXRXV�WLPH� SDWKZLVH GHOWD HVWLPDWRU
1;Q �
|#
#/Q�WQ�J�/��W��� � � � � /1 �W1 ��
}cQN�WQ��
�
ZLWK
cQN�W� #/Q�W�
#/N���� Q� N �� � � � � 1�
,Q SUDFWLFH� ZH FDQ DW EHVW VLPXODWH GLVFUHWH�WLPH DSSUR[LPDWLRQV A/Q DQG AcQN WR WKHVH FRQWLQXRXV�
WLPH YDULDEOHV� :H WKXV DUULYH DW
3DWKZLVH 'HOWD (VWLPDWRU
1;Q �
|#
#/Q�WQ�J�A/��W��� � � � � A/1 �W1 ��
}AcQN�WQ�� ���
,W VKRXOG QRZ EH FOHDU WKDW WKH NH\ WR WKLV PHWKRG LV WKH HYDOXDWLRQ RI WKH /LERU VHQVLWLYLWLHV cQN
DQG WKHLU GLVFUHWL]HG FRXQWHUSDUWV AcQN�
��� ([DFW 3DWKZLVH 0HWKRG
5HFDOO WKDW WKH HYROXWLRQ RI WKH IRUZDUG /LERU UDWHV LV GHWHUPLQHG E\
G/Q�W� xQ�W�/Q�W� GW� wQ�W�/Q�W� G:W� Q �� � � � � 1� ���
ZKHUH
xQ�W� Q;
L s�W�
pwQ�W�wL�W��/L�W�
� � p/L�W��
+HXULVWLFDOO\ GLmHUHQWLDWLQJ ERWK VLGHV ZLWK UHVSHFW WR /N��� VXJJHVWV
GcQN�W� cQN�W� >xQ�W�GW� wQ�W�G:W@ � /Q�W�1;M �
#xQ�W�
#/M�W�cMN�W�GW
cQN�W�G/Q�W�
/Q�W�� /Q�W�
1;M �
#xQ�W�
#/M�W�cMN�W�GW� ����
Q �� � � � � 1� N �� � � � � 1�
,Q 6HFWLRQ �� ZH MXVWLI\ WKLV HTXDWLRQ E\ VKRZLQJ WKDW WKH V\VWHP RI 6'(V �������� KDV D VROXWLRQ
IRU ZKLFK LQGHHG cQN�W� #/Q�W� #/N����
)URP WKH SHUVSHFWLYH RI VLPXODWLRQ� WKH SUREOHP KDV QRZ EHHQ UHGXFHG WR RQH RI VLPXODWLQJ
GLVFUHWH�WLPH DSSUR[LPDWLRQV WR WKH V\VWHP RI 6'(V ��� DQG ����� 7KH TXHVWLRQ RI GLVFUHWL]DWLRQ
RI ��� LV LQYHVWLJDWHG LQ *ODVVHUPDQ DQG =KDR >�@� ZKHUH ZH VKRZ WKDW WKHUH DUH DGYDQWDJHV WR
GLVFUHWL]LQJ 6'(V IRU GHoDWHG ERQG SULFHV �RU WKHLU LQFUHPHQWV� UDWKHU WKDQ WKH IRUZDUG /LERU
UDWHV WKHPVHOYHV� 'LmHUHQWLDWLQJ WKHVH 6'(V OHDGV WR D VHW RI GHULYDWLYH 6'(V DQDORJRXV WR ����
DQG LW LV SRVVLEOH WR VLPXODWH D GLVFUHWH�WLPH DSSUR[LPDWLRQ WR WKRVH� ,QGHHG� RQH FRXOG HYHQ FKRRVH
�
WR VLPXODWH WKH GHoDWHG ERQG SULFH 6'(V WRJHWKHU ZLWK WKH GHULYDWLYH 6'(V ����� ,Q FRQWLQXRXV�
WLPH� DOO VXFK YDULDWLRQV DUH XOWLPDWHO\ HTXLYDOHQW� 7KH SRVVLEOH GLVFUHWH�WLPH DSSUR[LPDWLRQV DUH
OLPLWOHVV� 7R PDNH WKH JHQHUDO PHWKRG DV WUDQVSDUHQW DV SRVVLEOH� ZH UHVWULFW DWWHQWLRQ WR ���^�����
$PRQJ WKH PHWKRGV IRU VLPXODWLQJ ��� FRQVLGHUHG LQ >�@ LV WKH UHFXUVLRQ
A/Q��L� ��K� A/Q�LK� H[SK�AxQ�LK�b �
�wQ�LK�wQ�LK���K� wQ�LK�
SK=L��
L����
LQ ZKLFK K LV WKH WLPH LQFUHPHQW� =�� =�� � � � DUH LQGHSHQGHQW G�GLPHQVLRQDO VWDQGDUG QRUPDO YHFWRUV�
AxQ�LK� Q;
M s�LK�
pwQ�LK�wM�LK�� A/M�LK�
� � p A/M�LK�� ����
DQG A/Q��� /Q���� �$ KDW LQGLFDWHV D GLVFUHWH�WLPH DSSUR[LPDWLRQ WR D FRQWLQXRXV�WLPH YDULDEOH��
(TXDWLRQ ���� PD\ EH LQWHUSUHWHG DV DQ (XOHU VFKHPH IRU ORJ/Q� $PRQJ DOO ZD\V RI GLVFUHWL]LQJ
����� WKH RQH WKDW GLmHUHQWLDWHV ���� VHHPV PRVW QDWXUDO DQG \LHOGV
([DFW 3DWKZLVH $OJRULWKP
AcQN��L� ��K� AcQN�LK�A/Q��L� ��K�A/Q�LK�
� A/Q��L� ��K�1;M �
#AxQ�LK�
# A/M�LK�AcMN�LK�K� ����
ZLWK LQLWLDO FRQGLWLRQ AcQN��� �IQ NJ�
,W LV HDV\ WR VHH WKDW� LQGHHG�
AcQN�LK� # A/Q�LK�
#/N���� ����
IRU HYHU\ RXWFRPH RI =�� � � � � =L� DQG WKLV LV WKH VHQVH LQ ZKLFK WKH DOJRULWKP LV H[DFW� 0RUH�
RYHU� WKH VDPH DOJRULWKP HYDOXDWHV # A/Q�LK� #%N��� LI ZH FKDQJH WKH LQLWLDO FRQGLWLRQ AcQN��� WR
#/Q��� #%N����
0RUH VLJQLnFDQW WKDQ WKH VDPSOH SDWK SURSHUW\ ���� LV WKH IDFW WKDW IRU DQ\ /LSVFKLW] FRQWLQXRXV
J � 51 � 5 DQG DQ\ n[HG WLPHV L�K� � � � � L1K�
#
#/N���(>J�A/��L�K�� � � � � A/1 �L1K��@ (
�� 1;M N
#J
# A/M�LMK�AcMN�LMK�
�� � ����
DV VKRZQ LQ 6HFWLRQ �� �7KH UDQJH RI VXPPDWLRQ VWDUWV DW N EHFDXVH AcMN s � LI M � N�� )RU
H[DPSOH� WR XVH WKLV WR HVWLPDWH WKH VHQVLWLYLW\ RI D FDSOHW SULFH WR DQ LQLWLDO IRUZDUG UDWH� OHW J EH
WKH GLVFRXQWHG FDSOHW SD\Rm �WKH ULJKW VLGH RI ��� EXW ZLWKRXW WKH H[SHFWDWLRQ� DQG HYDOXDWH
1;M N
#J
# A/M�7Q�AcMN�7Q�
�
ZLWK
#J
# A/M�7Q� %����p
Q<L �
�
� � p A/L�7Q�
��IA/Q�7Q� ! .J b �A/Q�7Q�b.��
p
� � p A/M�7Q�
��
(TXDWLRQ ���� LQGLFDWHV WKDW WKLV JLYHV DQ XQELDVHG HVWLPDWH RI WKH GHOWD LQ WKH GLVFUHWH�WLPH PRGHO
�DVVXPLQJ WKH WLPH JULG IK� �K� � � �J LQFOXGHV WKH WHQRU GDWHV 7Q��
��� $SSUR[LPDWLRQV
)RU RSWLRQV ZLWK /LSVFKLW] FRQWLQXRXV SD\RmV� WKH SDWKZLVH PHWKRG PDNHV LW SRVVLEOH WR HVWLPDWH
GHOWDV IURP D VLQJOH VLPXODWLRQ SDWK _ L�H�� ZLWKRXW DFWXDOO\ FKDQJLQJ DQ\ LQLWLDO YDOXHV LQ WKH
PRGHO� +RZHYHU� LPSOHPHQWLQJ WKH H[DFW SDWKZLVH PHWKRG VLPXODWLQJ ���� WRJHWKHU ZLWK �����
7KH FRPSXWDWLRQDO HmRUW UHTXLUHG E\ ���� �IRU DOO N �� � � � � 1 DQG DOO Q N� � � � � 1� LV FRPSDUDEOH
WR WKH HmRUW LQYROYHG LQ UHVLPXODWLQJ DOO �A/�� � � � � A/1 � DQG DGGLWLRQDO 1 WLPHV� VOLJKWO\ FKDQJLQJ WKH
YDOXH /N��� RQ WKH NWK RI WKHVH� +HQFH� WKH H[DFW SDWKZLVH PHWKRG PD\ QRW RmHU DQ RYHUZKHOPLQJ
DGYDQWDJH FRPSDUHG ZLWK D VWDQGDUG nQLWH GLmHUHQFH HVWLPDWRU� :H SURSRVH DSSUR[LPDWLRQV WR
WKH H[DFW DOJRULWKP WKDW DUH PXFK IDVWHU WR VLPXODWH DQG DSSHDU WR JLYH YHU\ JRRG DFFXUDF\�
2QH RI WKH PRVW WLPH�FRQVXPLQJ VWHSV LQ ���� LV WKH UHFRPSXWDWLRQ RI DOO #AxQ # A/M DW HYHU\ WLPH
VWHS� )RU W\SLFDO SDUDPHWHU YDOXHV� HDFK xQ ZLOO EH TXLWH VPDOO �WKH\ GLmHU IURP � MXVW HQRXJK WR
NHHS WKH IRUZDUG UDWH G\QDPLFV DUELWUDJH�IUHH� VR RXU nUVW DSSUR[LPDWLRQ VLPSO\ VHWV #AxQ # A/M s �
LQ WKH GHULYDWLYH UHFXUVLRQV� &OHDUO\� ���� WKHQ FROODSVHV WR
=HUR 'ULIW 3DWKZLVH $SSUR[LPDWLRQ
AcQN��L� ��K� A/Q��L� ��K�
/N����IQ NJ� ����
7KLV ZRXOG JLYH WKH H[DFW SDWKZLVH GHULYDWLYH LI WKH IRUZDUG UDWHV ZHUH GULIWOHVV PXOWLYDULDWH JHR�
PHWULF %URZQLDQ PRWLRQ _ L�H�� LI DOO WKH xQ ZHUH LQGHHG �� ,W PXVW EH HPSKDVL]HG WKDW DOWKRXJK
ZH PDNH WKLV DSSUR[LPDWLRQ LQ WKH DOJRULWKP IRU Ac� ZH FRQWLQXH WR XVH WKH RULJLQDO AxQ IRU WKH
VLPXODWLRQ RI WKH A/Q� DV LQ �����
(YDOXDWLQJ WKH AcQN XQGHU WKH ]HUR�GULIW DSSUR[LPDWLRQ UHTXLUHV YLUWXDOO\ QR HmRUW EH\RQG WKDW
LQYROYHG LQ VLPXODWLQJ WKH IRUZDUG /LERU UDWHV WKHPVHOYHV� +RZHYHU� WKH DSSUR[LPDWLRQ VHHPV
UDWKHU FUXGH� 2XU QH[W DSSUR[LPDWLRQ OLHV EHWZHHQ WKH H[DFW DQG ]HUR�GULIW PHWKRGV ERWK LQ WHUPV
RI FRPSXWLQJ WLPH DQG WKH DFFXUDF\ ZLWK ZKLFK LW HVWLPDWHV # A/Q #/N� ,Q WKLV DSSUR[LPDWLRQ� ZH
GLmHUHQWLDWH A/Q DV WKURXJK WKH RUGLQDU\ GLVFUHWL]HG GULIW AxQ�LK� LQ ���� ZHUH LQVWHDG
AxRQ�LK� Q;
M s�LK�
pwQ�LK�wM�LK��/M���
� � p/M����
�
,Q RWKHU ZRUGV� ZH UHSODFH WKH A/M�LK� ZLWK WKHLU WLPH�� IRUZDUG YDOXHV /M���� LQ WKH VSLULW RI WKH
DSSUR[LPDWLRQV LQWURGXFHG E\ %UDFH HW DO� >�@ WR GHULYH SULFLQJ IRUPXODV� 7KH VHQVLWLYLW\ RI WKH
DSSUR[LPDWH GULIW WR /N��� VLPSOLnHV WR
#AxRQ�LK�
#/N���
pwQ�LK�w�
N�LK�
�� � p/N������Is�LK� � N � QJ�
2EVHUYH WKDW WKHVH YDOXHV DUH WLPH�YDU\LQJ EXW GHWHUPLQLVWLF� ,I AxR ZHUH WKH WUXH GULIW� ZH ZRXOG
EH DEOH WR VROYH WKH 6'( IRU WKH IRUZDUG /LERU UDWHV DQG GLmHUHQWLDWH WKLV VROXWLRQ ZLWK UHVSHFW LWV
LQLWLDO FRQGLWLRQ� 'RLQJ VR \LHOGV
)RUZDUG 'ULIW $SSUR[LPDWLRQ
AcQN�LK� A/Q�LK�
/N����IQ NJ� A/Q�LK�
Lb�;U �
#AxRQ�UK�
#/N���� ����
7KH GHULYDWLYHV RI AxR XVHG LQ WKLV H[SUHVVLRQ FDQ EH SUHFRPSXWHG VR WKLV DSSUR[LPDWLRQ LV RQO\
VOLJKWO\ PRUH HmRUW WR LPSOHPHQW WKDQ WKH ]HUR�GULIW DSSUR[LPDWLRQ� ,Q SDUWLFXODU� XQOLNH WKH H[DFW
DOJRULWKP� LW GRHV QRW HQWDLO VLPXODWLRQ RI DQ DGGLWLRQDO UHFXUVLRQ�
��� 1XPHULFDO &RPSDULVRQV
:H FRPSDUH WKH VSHHG DQG DFFXUDF\ RI WKH H[DFW DQG DSSUR[LPDWH SDWKZLVH DOJRULWKPV WKURXJK
QXPHULFDO UHVXOWV� $OO RXU UHVXOWV DUH EDVHG RQ p ���� �TXDUWHUO\ UDWHV�� K p �VLPXODWLRQ
WLPH VWHS HTXDO WR OHQJWK RI DFFUXDO LQWHUYDOV� DQG 1 � � �� �D nYH�\HDU KRUL]RQ�� 7KH LQLWLDO
WHUP VWUXFWXUH WDNHV WKH IRUP /Q��� ORJ�D � EQ�� ZLWK D DQG E FKRVHQ VR WKDW /���� ��� DQG
/����� ���� 7KH YRODWLOLWLHV DUH FRQVWDQW RYHU WKH LQWHUYDOV >7L� 7L � ��� ZLWK
wQ�7L� w�Qb L�� L �� c c c � Qb �� Q �� c c c � ���
DQG HDFK w�M� GUDZQ UDQGRPO\ IURP WKH XQLIRUP GLVWULEXWLRQ RQ >����� ����@� 7KH VSHFLnF YDOXHV RI
WKH w�M� XVHG DUH
������� ������� ������� ������� ������� ������� ������� ������� ������� ������
������� ������� ������� ������� ������� ������� ������� ������� ������� ������ �
:H FRPSDUH WKH SHUIRUPDQFH RI WKH H[DFW SDWKZLVH DOJRULWKP� WKH ]HUR GULIW DSSUR[LPDWLRQ DQG WKH
IRUZDUG GULIW DSSUR[LPDWLRQ LQ HVWLPDWLQJ #&Q��� #/N��� ZLWK &Q D FDSOHW SULFH DV LQ ���� $OWKRXJK
LQ SUDFWLFH RQH ZRXOG EH LQWHUHVWHG LQ GHOWDV IRU PRUH FRPSOLFDWHG LQVWUXPHQWV RU SRUWIROLRV RI
LQVWUXPHQWV� XVLQJ FDSOHWV DOORZV XV WR FRPSDUH ZLWK H[DFW �FRQWLQXRXV�WLPH� YDOXHV IURP %ODFNV
IRUPXOD�
��
,Q SULQFLSOH� WKHUH DUH 1� YDOXHV RI #&Q��� #/N��� WR EH HVWLPDWHG FRUUHVSRQGLQJ WR WKH SRVVLEOH
FRPELQDWLRQV RI Q DQG N� WKRXJK WKH GHOWD LV FOHDUO\ � IRU N ! Q DQG WKH PRVW LQWHUHVWLQJ FDVH LV Q
N� (VWLPDWLQJ DOO WKHVH GHOWDV XVLQJ nQLWH GLmHUHQFHV �L�H�� FKDQJLQJ HDFK /N��� DQG UHVLPXODWLQJ�
UHTXLUHV 1 �� VLPXODWHG SDWKV SHU REVHUYDWLRQ _ RQH IRU WKH RULJLQDO VFHQDULR DQG DGGLWLRQDO SDWK
IRU HDFK SHUWXUEHG /N���� 8VLQJ FHQWUDO GLmHUHQFHV WKH QXPEHU LQFUHDVHV WR �1��� $W WKH H[SHQVH
RI VRPH RYHUKHDG SHU SDWK� DOO WKH GHOWDV FDQ EH HVWLPDWHG IURP WKH VDPH VLPXODWHG SDWKV XVLQJ
DQ\ RI WKH SDWKZLVH DOJRULWKPV� ,Q RXU H[SHULPHQWV� HVWLPDWLQJ DOO GHOWDV XVLQJ WKH H[DFW SDWKZLVH
DOJRULWKP LV DERXW IRXU WLPHV DV IDVW DV HVWLPDWLQJ DOO GHOWDV XVLQJ nQLWH GLmHUHQFHV� WKH IRUZDUG
GULIW DSSUR[LPDWLRQ LV DERXW WKUHH WLPHV DV IDVW DV WKH H[DFW SDWKZLVH DOJRULWKP� DQG WKH ]HUR GULIW
DSSUR[LPDWLRQ LV IDVWHU E\ DQRWKHU IDFWRU RI WZR�
5DWKHU WKDQ DWWHPSW WR UHSRUW QXPHULFDO UHVXOWV IRU DOO 1� GHOWDV� ZH IRFXV RQ WKH PRVW LQWHU�
HVWLQJ DQG PRVW GLpFXOW FDVHV� 7KH PRVW LQWHUHVWLQJ GHOWDV DUH WKH GLDJRQDO FDVHV� Q N� 7KH
PRVW FRPSXWDWLRQDOO\ GHPDQGLQJ FDVHV n[ Q 1 DQG OHW N UDQJH IURP � WR 1 � ,Q FRPSDULQJ
PHWKRGV� WKHUH DUH WZR VWDQGDUGV RQH PLJKW UHDVRQDEO\ DSSO\ LQ JDXJLQJ DFFXUDF\� SUR[LPLW\ WR
WKH GLVFUHWH�WLPH GHOWD REWDLQHG E\ GLmHUHQWLDWLQJ ZLWK UHVSHFW WR /N��� ZKLOH NHHSLQJ WKH WLPH
VWHS K n[HG� RU SUR[LPLW\ WR WKH FRQWLQXRXV�WLPH GHOWD� 7KH H[DFW SDWKZLVH DOJRULWKP LV XQELDVHG
IRU WKH IRUPHU EXW _ OLNH DQ\ VLPXODWLRQ PHWKRG _ LV VXEMHFW WR GLVFUHWL]DWLRQ HUURU LQ HVWLPDWLQJ
WKH ODWWHU� ,Q RUGHU WR JLYH DV FRPSOHWH D SLFWXUH DV SRVVLEOH� ZH LQFOXGH LQIRUPDWLRQ RQ ERWK W\SHV
RI HUURU�
)LJXUH � VKRZV HVWLPDWHG ELDVHV �LQ SHUFHQW� IRU WKH GLDJRQDO GHOWDV #&N #/N FRPSDUHG ZLWK
WKH GHOWDV REWDLQHG IURP %ODFNV IRUPXOD� 7KH H[DFW YDOXHV UDQJH IURP ���� WR ����� :H FDQ VHH
WKDW DOO WKUHH PHWKRGV DUH FORVH WR HDFK RWKHU DQG SHUIRUP ZHOO� 7KH VWDQGDUG HUURUV RI WKHVH
HVWLPDWHV DUH DERXW ����� VR PRVW RI WKH HVWLPDWHG ELDVHV IRU WKH H[DFW DQG IRUZDUG GULIW PHWKRGV
IDLO WR EH VWDWLVWLFDOO\ VLJQLnFDQW� ,W VKRXOG EH VWUHVVHG WKDW WKH UHVXOWV IRU WKH H[DFW SDWKZLVH
HVWLPDWH UHSUHVHQW WKH EHVW RQH FRXOG KRSH WR DFKLHYH XVLQJ RUGLQDU\ nQLWH GLmHUHQFH HVWLPDWHV�
,I ZH ZDQWHG WR FRPSDUH ZLWK WKH GLVFUHWH�WLPH GHOWD UDWKHU WKDQ WKH FRQWLQXRXV�WLPH OLPLW� ZH
FRXOG XVH WKH H[DFW SDWKZLVH HVWLPDWH DV WKH VWDQGDUG� VLQFH LW LV XQELDVHG IRU WKH GLVFUHWH�WLPH
GHOWD� 7KH IRUZDUG GULIW DSSUR[LPDWLRQ GRHV D SDUWLFXODUO\ HmHFWLYH MRE RI DSSUR[LPDWLQJ WKH H[DFW
PHWKRG� ZLWK VRPH JUDGXDO GHJUDGDWLRQ DW ORQJHU PDWXULWLHV�
)LJXUH � VKRZV WKH UHODWLYH ELDV LQ GHOWDV RI WKH ODVW FDSOHW� #&�� #/N���� N �� � � � � ��� 7KH
DEVROXWH YDOXHV RI WKHVH GHOWDV IRU N �� � � � � �� DUH DURXQG ������� ZKLFK LV DERXW ��� WLPHV VPDOOHU
WKDQ #&�� #/������ ,Q UHODWLYH WHUPV� WKH ]HUR GULIW DSSUR[LPDWLRQ PHWKRG SURGXFHV D ODUJH ELDV
�XS WR ���� IRU WKH Rm�GLDJRQDO GHOWDV� WKH IRUZDUG GULIW DSSUR[LPDWLRQ SURGXFHV D VXEVWDQWLDOO\
VPDOOHU ELDV� DQG WKH H[DFW PHWKRG SURGXFHV QR GLVFHUQLEOH ELDV DW DOO� ,W VKRXOG EH HPSKDVL]HG
WKDW LQ DOO FDVHV LQ )LJXUH � WKH DEVROXWH HUURUV DUH YHU\ VPDOO DQG WKH ODUJH UHODWLYH HUURUV DUH GXH
��
WR WKH IDFW WKDW ZH HVWLPDWLQJ YDOXHV VR FORVH WR �� %DVHG RQ WKHVH DQG RWKHU FRQVLVWHQW QXPHULFDO
UHVXOWV� WDNLQJ LQWR DFFRXQW ERWK DFFXUDF\ DQG FRPSXWLQJ WLPH� WKH IRUZDUG GULIW DSSUR[LPDWLRQ
DSSHDUV WR EH WKH PRVW HmHFWLYH PHWKRG�
1 2 3 4
-0.4
-0.2
0.0
0.2
0.4 Exact Pathwise
Zero DirftForward Drift
Caplet Expiration (in years)
Est
imat
ed R
elat
ive
Bia
s (p
erce
nt)
)LJXUH �� (VWLPDWHG ELDV RI #&N #/N���
5 10 15
-60
-40
-20
0
Libor Rate
Est
imat
ed R
elat
ive
Bia
s (p
erce
nt)
Exact PathwiseZero DirftForward Drift
)LJXUH �� (VWLPDWHG ELDV RI #&�� #/N���
� /LNHOLKRRG 5DWLR 'HOWDV
7KH RQO\ VLJQLnFDQW OLPLWDWLRQ RI WKH PHWKRG GHYHORSHG LQ WKH SUHYLRXV VHFWLRQ LV WKDW LW LV UHVWULFWHG
WR SD\RmV WKDW DUH DW OHDVW FRQWLQXRXV� 7KLV SUHFOXGHV DSSOLFDWLRQ RI WKH PHWKRG WR� H�J�� D FDSOHW
ZLWK D GLJLWDO SD\Rm
�I/1 �71 � ! .J
RU D NQRFNRXW FDSOHW ZLWK SD\Rm
�/1 �71 �b.���I PLQL ������1
/L�7L� ! EJ� ����
,Q ERWK FDVHV� WKH SDWKZLVH GHULYDWLYH ZLWK UHVSHFW WR VRPH /N��� DFWXDOO\ H[LVWV ZLWK SUREDELOLW\
�� EXW IDLOV WR UHoHFW WKH GLVFRQWLQXLW\ LQ WKH LQGLFDWRU IXQFWLRQ DQG WKXV SURYLGHV DQ XQLQIRUPDWLYH
HVWLPDWH� �)RU WKH GLJLWDO FDSOHW� WKH SDWKZLVH GHULYDWLYH LV LGHQWLFDOO\ ]HUR ZKHUHYHU LW H[LVWV�� 3XW
PRUH SUHFLVHO\� WKHVH DUH H[DPSOHV LQ ZKLFK WKH LQWHUFKDQJH RI GHULYDWLYH DQG H[SHFWDWLRQ UHTXLUHG
LQ ��� GRHV QRW KROG� :H QRZ SUHVHQW DQ DOWHUQDWLYH PHWKRG IRU HVWLPDWLQJ GHOWDV EDVHG RQ PRYLQJ
WKH GHSHQGHQFH RQ /N��� IURP WKH VDPSOH SDWKV WR WKH PHDVXUH� WKHUHE\ HOLPLQDWLQJ WKH QHHG IRU
VPRRWKQHVV LQ WKH RSWLRQ SD\Rm� $V QRWHG LQ 6HFWLRQ �� WKH GLVWLQFWLRQ LV DQDORJRXV WR WZR ZD\V RI
DGGLQJ D GULIW WR %URZQLDQ PRWLRQ� ZH FDQ DGG xW DW WLPH W WR HDFK %URZQLDQ SDWK� RU ZH FDQ OHDYH
WKH SDWKV XQFKDQJHG DQG XVH *LUVDQRYV 7KHRUHP WR DGG D GULIW WKURXJK D FKDQJH RI SUREDELOLW\
PHDVXUH�
��
��� /50 LQ WKH *DXVVLDQ 6HWWLQJ
:H EHJLQ RXU GLVFXVVLRQ RI WKH OLNHOLKRRG UDWLR PHWKRG �/50� E\ FRQVLGHULQJ WKH VRPHZKDW VLPSOHU
VHWWLQJ RI HVWLPDWLQJ VHQVLWLYLWLHV ZLWK UHVSHFW WR D SDUDPHWHU RI WKH PHDQ RI D *DXVVLDQ YHFWRU�
:H WKHQ H[WHQG WKLV WR DVVHWV GHVFULEHG E\ JHRPHWULF %URZQLDQ PRWLRQ DQG XOWLPDWHO\ VKRZ KRZ
WKH PHWKRG FDQ EH DSSOLHG �ZLWK VRPH QHFHVVDU\ PRGLnFDWLRQV� WR /LERU PRGHOV�
6XSSRVH� WKHQ� WKDW WKH UDQGRP Q�YHFWRU ; LV PXOWLYDULDWH QRUPDO ZLWK PHDQ YHFWRU P�t�
DQG FRYDULDQFH PDWUL[ h� +HUH� t LV D VFDODU SDUDPHWHU DQG ZH DUH LQWHUHVWHG LQ VHQVLWLYLWLHV ZLWK
UHVSHFW WR t� :H VXSSRVH h KDV IXOO UDQN DQG GHQRWH E\
��[�P�t��h� H[S�b�
��[bP�t���hb��[bP�t���
��{�Q �MhM� �
WKH GHQVLW\ RI ;� )RU DQ\ J � 5Q � 5�
(t>J�;�@
=5Q
J�[���[�P�t��h� G[� ����
ZKHUH ZH KDYH VXEVFULSWHG WKH H[SHFWDWLRQ WR HPSKDVL]H WKH GHSHQGHQFH RI WKH PHDVXUH RQ t�
'LmHUHQWLDWLQJ DQG WKHQ LQWHUFKDQJLQJ GHULYDWLYH DQG LQWHJUDO \LHOGV
G
Gt(t>J�;�@
=J�[� B��[�P�t��h� G[
=J�[�
B��[�P�t��h�
��[�P�t��h���[�P�t��h� G[� ����
WKH GRW RQ � LQGLFDWLQJ GLmHUHQWLDWLRQ ZLWK UHVSHFW WR t� 6RPH DOJHEUD VKRZV WKDW
B��[�P�t��h�
��[�P�t��h� �[bP�t���hb� BP�t��
0DNLQJ WKLV VXEVWLWXWLRQ LQ ���� DQG LQWHUSUHWLQJ WKH LQWHJUDO WKHUH DV DQ H[SHFWDWLRQ� ZH DUULYH DW
G
Gt(t>J�;�@ (t>J�;��; bP�t���hb� BP�t�@� ����
+HQFH� WKH H[SUHVVLRQ LQVLGH WKH H[SHFWDWLRQ RQ WKH ULJKW SURYLGHV DQ XQELDVHG HVWLPDWRU RI WKH
GHULYDWLYH RQ WKH OHIW� 0RUHRYHU� WKLV GHULYDWLRQ UHTXLUHV VPRRWKQHVV LQ WKH GHSHQGHQFH RI � RQ
t� EXW QR VPRRWKQHVV DW DOO LQ J� 7KH NH\ TXDQWLW\ B� � LV WKH GHULYDWLYH ZLWK UHVSHFW WR q RI WKH
OLNHOLKRRG UDWLR ��[�P�t � q��h� ��[�P�t��h�� KHQFH WKH QDPH OLNHOLKRRG UDWLR PHWKRG�
,Q D VLPXODWLRQ� ZH ZRXOG W\SLFDOO\ VDPSOH ; E\ VHWWLQJ ; P�t� �$= ZKHUH $ LV DQ Qd Q
PDWUL[ VDWLVI\LQJ $$� h DQG = LV D YHFWRU RI LQGHSHQGHQW VWDQGDUG QRUPDO UDQGRP YDULDEOHV�
0DNLQJ WKLV VXEVWLWXWLRQ� ZH JHW
G
Gt(t>J�;�@ (>J�P�t� �$=�= �$b� BP�t�@� ����
��
7KH H[SHFWDWLRQ RQ WKH ULJKW LV ZLWK UHVSHFW WR WKH Q�GLPHQVLRQDO VWDQGDUG QRUPDO GLVWULEXWLRQ�
KHQFH QRW VXEVFULSWHG E\ t�
7KLV GHULYDWLRQ DSSOLHV GLUHFWO\ WR WKH HVWLPDWLRQ RI GHOWD IRU SDWK�GHSHQGHQW RSWLRQV RQ JHR�
PHWULF %URZQLDQ PRWLRQ� /HW
6W 6� H[S�xW� }:W�� W w ��
ZLWK :W D RQH�GLPHQVLRQDO %URZQLDQ PRWLRQ DQG x DQG } FRQVWDQWV� 6XSSRVH ZH ZDQW WR HVWLPDWH
G
G6�(>I�6W� � � � � � 6WQ�@
IRU VRPH GDWHV � � W� � c c c � WQ DQG VRPH I WR EH LQWHUSUHWHG DV WKH GLVFRXQWHG SD\Rm RI D
SDWK�GHSHQGHQW RSWLRQ� :LWK J FKRVHQ DSSURSULDWHO\� ZH FDQ UH�H[SUHVV WKH SD\Rm LQ WKH IROORZLQJ
IRUP�
I�6W�� � � � � 6WQ� J�ORJ6W�� ORJ6W� b ORJ6W� � � � � � ORJ6WQ b ORJ6WQb���
1RZ PDNH WKH FRUUHVSRQGHQFHV t� 6��
; �
�%%%%#
ORJ6W�
ORJ6W� b ORJ6W�
���ORJ6WQ b ORJ6WQb�
�&&&&$ � P�
�%%%%#ORJ6� � xW�x�W� b W�����x�WQ b WQb��
�&&&&$ �
DQG
h�
�%%%%#
}�W� � c c c �� }��W� b W�� c c c ����
���� � �
���� � � }��WQ b WQb��
�&&&&$ �
:H PD\ FOHDUO\ WDNH $ WR EH GLDJRQDO LQ VROYLQJ $$� h� /HW = � �=�� � � � � =Q� EH LQGHSHQGHQW
VWDQGDUG QRUPDOV XVHG WR VLPXODWH WKH SURFHVV� LQ WKH VHQVH WKDW
6WL�� 6WL H[S�x�WL�� b WL� � }SWL�� b WL=L���� L �� �� � � � � Qb ��
�VR WKDW ; P�$=�� :H QRZ nQG WKDW
= �$b� BP =�
6�}SW�
DQG ���� EHFRPHVG
G6�(>I�6W� � � � � � 6WQ�@ (
�I�6W� � � � � � 6WQ�
=�
6�}SW�
w�
:H PD\ WKHUHIRUH XVH
I�6W� � � � � � 6WQ�=�
6�}SW�
��
WR HVWLPDWH GHOWD� $ VLPLODU H[SUHVVLRQ ZDV GHULYHG LQ %URDGLH DQG *ODVVHUPDQ >�@� 1RWLFH WKDW
ZH XVHG WKH IXQFWLRQ J WR PDNH D GLUHFW FRUUHVSRQGHQFH ZLWK WKH SUHYLRXV H[DPSOH EXW J SOD\V QR
UROH LQ WKH nQDO HVWLPDWRU� 0RUHRYHU� I FRXOG EH JHQHUDOL]HG WR DQ\ IXQFWLRQ RI WKH SDWK RI WKH
XQGHUO\LQJ DVVHW WKDW GHSHQGV RQO\ RQ YDOXHV RI WKH XQGHUO\LQJ DIWHU VRPH WLPH W� ! ��
7KH FDVH RI PXOWLGLPHQVLRQDO JHRPHWULF %URZQLDQ PRWLRQ ZRUNV VLPLODUO\ DQG ZLOO EULQJ XV RQH
VWHS FORVHU WR WKH /LERU PRGHO� 6XSSRVH ZH KDYH G DVVHWV 6�L�W � L �� � � � � G� VDWLVI\LQJ
6�L�W 6
�L�� H[S�xLW� }L:
�L�W ��
ZLWK (>:�L�W :
�M�W @ |LMW� 6XSSRVH WKH Gd G PDWUL[ h ZLWK HQWULHV hLM }L}M|LM KDV IXOO UDQN DQG
OHW $ VDWLVI\ $$� h� :ULWH I�6� IRU WKH YDOXH RI VRPH IXQFWLRQ RI WKH G DVVHWV WKDW GHSHQGV RQ
WKHLU YDOXHV RQO\ DIWHU VRPH WLPH W� ! �� 6XSSRVH ZH VLPXODWH WKH G DVVHWV E\ VHWWLQJ
6�L�W� 6
�L�� H[S�xLW� �
SW��$=�L��
ZLWK = D YHFWRU RI G LQGHSHQGHQW VWDQGDUG QRUPDOV� 3URFHHGLQJ DV EHIRUH� ZH DUULYH DW
G
G6�N��
(>I�6�@ (
�I�6�
�= �$b��N
6�N��
SW�
��
7KH H[SUHVVLRQ LQVLGH WKH H[SHFWDWLRQ RQ WKH ULJKW WKXV SURYLGHV DQ XQELDVHG HVWLPDWRU RI WKH GHOWD
ZLWK UHVSHFW WR WKH NWK DVVHW�
��� /50 LQ /LERU 0RGHOV
,Q RUGHU WR VHH ERWK WKH SRVVLELOLWLHV DQG GLpFXOWLHV LQ DSSO\LQJ /50 LQ WKH /LERU PRGHO� LW LV
FRQYHQLHQW WR WDNH ORJDULWKPV LQ ���� WR JHW
ORJ A/Q��L� ��K� ORJ A/Q�LK� � >AxQ�LK�b �
�NwQ�LK�N�@K�
SKwQ�LK�=L��� Q �� � � � � 1� ����
7ZR LVVXHV QRZ DULVH� 7KH nUVW LV WKDW AxQ LV D IXQFWLRQ RI WKH IRUZDUG /LERU UDWHV WKHPVHOYHV DQG
KHQFH LPSOLFLWO\ RI WKH /N���V� 7KLV PDNHV LW GLpFXOW WR PRYH DOO WKH GHSHQGHQFH RQ WKH /N���V RXW
RI WKH VDPSOH SDWKV DQG LQWR WKH SUREDELOLW\ PHDVXUH� :H DGGUHVV WKLV LVVXH DV ZH GLG LQ 6HFWLRQ ���
E\ GLmHUHQWLDWLQJ DV WKRXJK WKH GULIW ZHUH GHWHUPLQLVWLF �ZKLOH VLPXODWLQJ WKH IRUZDUG /LERU UDWHV
ZLWK WKH RULJLQDO GULIW�� ,I ZH XVH WKH ]HUR�GULIW DSSUR[LPDWLRQ� WKH SUREOHP UHGXFHV WR DSSO\LQJ
/50 WR FRQVWDQW�GULIW� PXOWLGLPHQVLRQDO JHRPHWULF %URZQLDQ PRWLRQ� MXVW DV LQ 6HFWLRQ ���� %XW
ZH ZRUN SULPDULO\ ZLWK WKH IRUZDUG�GULIW DSSUR[LPDWLRQ� ZKLFK LV RQO\ VOLJKWO\ PRUH FRPSOLFDWHG�
8QGHU WKH IRUZDUG�GULIW DSSUR[LPDWLRQ� ���� GHVFULEHV WKH HYROXWLRQ RI D *DXVVLDQ SURFHVV VR
WKH GHYHORSPHQW RI WKH SUHYLRXV VHFWLRQ SRWHQWLDOO\ DSSOLHV _ LI RQO\ DV DQ DSSUR[LPDWLRQ� %XW ZH
VWLOO IDFH D VHFRQG LVVXH QRW GHDOW ZLWK SUHYLRXVO\� HTXDWLRQ ���� GHVFULEHV WKH HYROXWLRQ RI D YHFWRU
��
RI 1 UDWHV GULYHQ E\ �VD\� G�GLPHQVLRQDO YHFWRUV RI QRUPDO UDQGRP YDULDEOHV� ZKHUH G LV VLPSO\ WKH
QXPEHU RI IDFWRUV LQ WKH RULJLQDO IRUPXODWLRQ RI WKH PRGHO� 2YHU D VLQJOH WLPH VWHS� WKH FRYDULDQFH
PDWUL[ RI WKH LQFUHPHQWV LQ ���� KDV UDQN G� ,I G � 1 �DQG ZH XVXDOO\ KDYH G �� 1�� WKH PDWUL[
LV VLQJXODU VR WKH GHYHORSPHQW LQ 6HFWLRQ ��� _ ZKLFK LQFOXGHV LQYHUWLQJ WKH FRYDULDQFH PDWUL[ _
LV QRW DSSOLFDEOH� ,QGHHG� HYHQ WKH VWDUWLQJ SRLQW RI WKH GHULYDWLRQ ���� LV SUREOHPDWLF EHFDXVH WKH
1 �YHFWRU RI LQFUHPHQWV IDLOV WR KDYH D GHQVLW\ LQ 51 � 7KLV LVVXH LV QRW VSHFLnF WR WKH /LERU VHWWLQJ�
+DG ZH QRW DVVXPHG WKDW WKH FRYDULDQFH PDWUL[ h RI WKH PXOWLGLPHQVLRQDO %URZQLDQ PRWLRQ LQ
6HFWLRQ ��� LV QRQVLQJXODU� SUHFLVHO\ WKH VDPH LVVXH ZRXOG KDYH DULVHQ WKHUH�
7R DGGUHVV WKLV LVVXH� ZH FRQVLGHU WKH GLVWULEXWLRQ RI WKH LQFUHPHQWV RYHU PXOWLSOH WLPH VWHSV
UDWKHU WKDQ MXVW RQH� 8QUDYHOLQJ ���� \LHOGV
ORJ A/Q�LK� ORJ/Q��� � K
Lb�;M �
>AxQ�MK�b �
�NwQ�MK�N�@ �
SK>wQ���MwQ�K�M c c c MwQ��Lb ��K�@
������
=�=����=L
�������
ZKHUH WKH URZ YHFWRUV wQ�MK� KDYH EHHQ FRQFDWHQDWHG LQWR D VLQJOH YHFWRU RI OHQJWK L c G DQG WKHFROXPQ YHFWRUV =M KDYH EHHQ VWDFNHG LQWR D FROXPQ YHFWRU RI WKH VDPH OHQJWK� )RU VXpFLHQWO\ ODUJH
Le� WKH 1 d LeG PDWUL[
eK�Le�
�%%%%#
w���� M w��K� M c c c M w���Le b ��K�
w���� M w��K� M c c c M w���Le b ��K�
������
������
w1 ��� M w1 �K� M c c c M w1 ��Le b ��K�
�&&&&$
PD\ KDYH UDQN 1 � HYHQ LI G � 1 � 7KLV PHDQV WKDW WKH FRYDULDQFH PDWUL[ eK�Le�eK�L
e�� RI WKH
ORJ/Q�LeK�� Q �� � � � � 1 � LV LQYHUWLEOH DQG _ XVLQJ D GHWHUPLQLVWLF DSSUR[LPDWLRQ WR WKH GULIW _
WKH GHULYDWLRQ RI WKH SUHYLRXV VHFWLRQ DSSOLHV�
6XSSRVH� WKHQ� WKDW eK�Le� KDV IXOO UDQN� 7R DSSO\ WKH PHWKRG RI 6HFWLRQ ��� LQ WKH IRUP JLYHQ
LQ ����� PDNH WKH IROORZLQJ FRUUHVSRQGHQFHV� t � A/N����
; � �ORJ A/��LeK�� � � � � ORJ A/1 �L
eK���� ����
PQ�t�� ORJ/Q��� � KLeb�;U �
>AxRQ�UK�b �
�NwQ�UK�N�@� Q �� � � � � 1� ����
�AxR LV WKH IRUZDUG�GULIW DSSUR[LPDWLRQ RI 6HFWLRQ ����� h � SKhK�L
e� s SKeK�L
e�eK�Le��� DQG
$�SK$K�L
e� IRU DQ\ LeGd LeG PDWUL[ $K�Le� VDWLVI\LQJ $K�L
e�$K�Le�� eK�L
e�eK�Le��� DQG
BPQ � �IQ NJ/N���
� KLeb�;
U �
#AxRQ�UK�
#/N���� Q �� � � � � 1� ����
��
:LWK WKHVH VXEVWLWXWLRQV ZH DUULYH DW �VHH ����� WKH IROORZLQJ GHOWD HVWLPDWRU IRU DQ DUELWUDU\
GLVFRXQWHG SD\Rm J�A/��W��� � � � � A/1 �W1 ���
/50 'HOWD (VWLPDWRU
J�A/��W��� � � � � A/1 �W1 ���; bP��hb� BP� ����
ZLWK ; � P� h� DQG BP DV LQ ����������
3UHFRPSXWLQJ WKH YHFWRUhb� BP UHGXFHV WKH FRPSXWDWLRQDO HmRUW SHU VLPXODWHG SDWK WR HYDOXDWH
WKH TXDGUDWLF IRUP LQ ���� IURP 2�1�� WR 2�1�� ,I ZH FDQ WDNH Le 1 G VR WKDW eK�Le� LV VTXDUH�
WKHQ ZH FDQ UHZULWH WKH TXDGUDWLF IRUP WR JHW WKH HVWLPDWRU
J�A/��W��� � � � � A/1 �W1 ��Kb� �= �
eK�Le�b� BP�
ZKHUH = LV WKH FROXPQ YHFWRU REWDLQHG E\ VWDFNLQJ WKH Le G�YHFWRUV RI LQGHSHQGHQW QRUPDOV XVHG
WR VLPXODWH WKH G�IDFWRU PRGHO IRU Le VWHSV� 7KLV SXWV WKH HVWLPDWRU LQ WKH IRUP RI ����� 0XFK DV
LQ ����� ZH FDQ SUHFRPSXWH eK�Le�b� BP�
7R LOOXVWUDWH WKH XVH RI WKLV PHWKRG� ZH UHWXUQ WR WKH H[DPSOHV ZLWK ZKLFK ZH EHJDQ WKLV VHFWLRQ�
&RQVLGHU WKH HVWLPDWLRQ RI
%���� cG
G/N���(
��I/1 �71 � ! .J
1<L �
�
� � p/L�7L�
��
WKH GHOWD RI WKH GLJLWDO FDSOHW ZLWK UHVSHFW WR WKH NWK IRUZDUG UDWH� )RU WKH /50 PHWKRG WR EH
DSSOLFDEOH� ZH QHHG WKH TXDQWLW\ LQVLGH WKH H[SHFWDWLRQ WR EH D IXQFWLRQ RI WKH A/L�W� IRU W w LeK EXW
QRW W � LeK� 2QH ZD\ WR DFKLHYH WKLV FKRRVHV WKH WLPH VWHS K VPDOO HQRXJK WKDW 7� w LeK� %XW WKH
PHWKRG FDQ DFWXDOO\ EH DSSOLHG ZLWK DQ\ K � p LI ZH UHFDOO WKDW /L�W� s /L�7L� IRU DOO W w 7L� 7KH
GLVFRXQWHG SD\Rm RQ WKH GLJLWDO FDSOHW FDQ WKXV EH UH�H[SUHVVHG DV
%�����I/1 �71 � ! .J1<L �
�
� � p/L�71 �
DQG WKH GHOWD HVWLPDWHG XVLQJ �ZLWK WKH QRWDWLRQ LQ ����������
�%�����I/1 �71 � ! .J
1<L �
�
� � p/L�71 �
�Kb� ��; bP��hK�L
e�b� BP�
$ VLPLODU UHZULWLQJ RI WKH GLVFRXQWHG SD\Rm RQ WKH NQRFNRXW FDSOHW OHDGV WR WKH HVWLPDWRU�%����p�/1 �71 �b.���I PLQ
L ������1/L�71 � ! EJ
1<L �
�
� � p/L�71 �
�Kb� ��; bP��hK�L
e�b� BP�
IRU LWV GHOWD ZLWK UHVSHFW WR /N����
��
7KH GHULYDWLRQ DERYH VLPSOLnHV VRPHZKDW LQ WKH LPSRUWDQW VSHFLDO FDVH WKDW K p _ L�H�� ZKHQ
WKH VLPXODWLRQ WLPH VWHS FRLQFLGHV ZLWK WKH VSDFLQJ EHWZHHQ WHQRU GDWHV� ,I� LQ SDUWLFXODU� ZH KDYH
G � �D VLQJOH�IDFWRU PRGHO�� WKHQ WKH NH\ PDWUL[ WR FKHFN IRU QRQVLQJXODULW\ LV
ep�1�
�%%%%#
w���� M w��7�� M c c c M w��71b��w���� M w��7�� M c c c M w��71b�����
������
���w1 ��� M w1 �7�� M c c c M w1 �71b��
�&&&&$ �
8QGHU RXU FRQYHQWLRQ WKDW wQ�W� � IRU W w 7Q �VR WKDW /Q�W� s /Q�7Q� IRU W w 7Q�� WKLV PDWUL[ LV
EORFN ORZHU WULDQJXODU� 6XSSRVH WKH IRUZDUG UDWH YRODWLOLWLHV GHSHQG VROHO\ RQ WLPH�WR�PDWXULW\ LQ
WKH VHQVH WKDW
wQ�W� w�7Q b W�
IRU VRPH IXQFWLRQ w�c� DQG DOO Q DQG W� �:H UHTXLUH WKDW w DVVLJQ D YDOXH RI � WR QHJDWLYH DUJXPHQWV��
,Q WKLV FDVH� ep�1� DQG LWV LQYHUVH KDYH WKH JHQHUDO �7RHSOLW]� IRUP
ep�1�
�%%%%%%#
D�D� D�D� D� D����
������
� � �
D1 D1b� D1b� c c c D�
�&&&&&&$� ep�1�
b�
�%%%%%%#
E�E� E�E� E� E����
������
� � �
E1 E1b� E1b� c c c E�
�&&&&&&$�
7KH LQYHUVH LV SDUWLFXODUO\ HDV\ WR FRPSXWH EHFDXVH E� � D��
E� b�D�E��E�� E� b�D�E� � D�E��E�� � � � � E1 b�D1E� � c c c� D�E1b��E��
)DVW FRPSXWDWLRQ RI eb�p�1� PD\ EH HVSHFLDOO\ LPSRUWDQW ZKHQ WKH VLPXODWLRQ LV HPEHGGHG LQ DQ
LWHUDWLYH SURFHGXUH WR FDOLEUDWH D PRGHO WKURXJK FKRLFH RI w�c��
��� 1XPHULFDO 5HVXOWV
&RQVLVWHQW ZLWK REVHUYDWLRQV LQ >�@ DQG WKH EURDGHU OLWHUDWXUH RQ VHQVLWLYLW\ HVWLPDWLRQ� ZH nQG WKDW
ZKHQ WKH SDWKZLVH PHWKRG LV DSSOLFDEOH _ LQ WKH SUHVHQW FRQWH[W PHDQLQJ WKDW WKH RSWLRQ SD\Rm
LV /LSVFKLW] FRQWLQXRXV _ LW SURYLGHV PRUH SUHFLVH HVWLPDWHV WKDQ /50� :H WKHUHIRUH HYDOXDWH WKH
/50 HVWLPDWRU LQ HVWLPDWLQJ GHOWDV IRU WZR GLVFRQWLQXRXV SD\RmV� D FDSOHW ZLWK D GLJLWDO SD\Rm
�I/Q�7Q� ! .J� DQG D NQRFNRXW FDSOHW ZLWK SD\Rm
�/Q�7Q�b.��Q<L �
�I/L�7L� � ELJ� EL ���/L���� L �� c c c � 1�
7KH PRGHO SDUDPHWHUV DUH DV LQ 6HFWLRQ ���� %HFDXVH WKH SDWKZLVH PHWKRG LV LQDSSOLFDEOH� WKH
DOWHUQDWLYH DJDLQVW ZKLFK ZH FRPSDUH LV D nQLWH GLmHUHQFH HVWLPDWRU� )RU WKH GLJLWDO FDSOHWV ZH FDQ
��
0HWKRG 'HOWD (VWLPDWRU 6WG� (UU� 506(
([DFW #&� #%����� ������� b b
9DOXH #&�� #%����� ������ b b
/LNHOLKRRG #&� #%����� ������� ����� �����5DWLR 0HWKRG #&�� #%����� ������ ����� �����
cE #&� #%����� ������� ����� �����)LQLWH ������ #&�� #%����� ������ ����� �����
cE #&� #%����� ������� ����� �����'LmHUHQFH
����� #&�� #%����� ������ ����� �����0HWKRG cE #&� #%����� ������� ����� �����
����� #&�� #%����� ������ ����� �����
7DEOH �� 'HOWDV IRU GLJLWDO FDSOHWV XVLQJ /50 DQG nQLWH GLmHUHQFHV� 7R EDODQFH WKH FRPSXWLQJ WLPHUHTXLUHG WR HVWLPDWH DOO GHOWDV� ZH XVH ��������� UHSOLFDWLRQV IRU /50 DQG ������� IRU WKH nQLWHGLmHUHQFH HVWLPDWRUV� cE LV WKH LQFUHPHQW LQ %����� XVHG LQ WKH nQLWH GLmHUHQFH HVWLPDWLRQ�
nQG WKH H[DFW �FRQWLQXRXV�WLPH� GHOWD IURP D VWUDLJKWIRUZDUG YDULDQW RI WKH %ODFN IRUPXOD� IRU WKH
NQRFN�RXW FDSOHW RXU ?DFFXUDWH� YDOXH LV REWDLQHG IURP D ODUJH QXPEHU RI VLPXODWLRQV RI D nQLWH
GLmHUHQFH HVWLPDWRU XVLQJ D VPDOO LQFUHPHQW�
$V LQ 6HFWLRQ ��� WKH ODUJH QXPEHU RI GHOWDV RQH FRXOG FRQVLGHU PDNHV LW QHFHVVDU\ WR IRFXV WKH
QXPHULFDO FRPSDULVRQ RQ LQIRUPDWLYH FDVHV� (DFK FDSOHW &Q �ZKHWKHU GLJLWDO RU VWDQGDUG� FDQ EH
SHUIHFWO\ KHGJHG XVLQJ WKH ERQGV %Q DQG %Q��� VR KHGJH UDWLRV ZLWK UHVSHFW WR WKHVH XQGHUO\LQJ
DVVHWV DUH SDUWLFXODUO\ LQWHUHVWLQJ� 2XU QXPHULFDO UHVXOWV IRFXV RQ WKHVH GHOWDV�
$SSO\LQJ WKH OLNHOLKRRG UDWLR PHWKRG� ZH FDQ FRPSXWH WKH HVWLPDWRUV RI DOO GHOWDV #&Q #%L���
IRU DOO Q �� c c c � ��� L �� c c c � ��� IURP HDFK VLPXODWLRQ SDWK� +RZHYHU� XVLQJ D nQLWH GLmHUHQFH
PHWKRG� HDFK SDLU RI SDWKV \LHOGV DQ HVWLPDWH RI #&Q #%L��� IRU DOO Q �� c c c � �� EXW ZLWK L n[HG�
,W IROORZV WKDW WKH FRPSXWLQJ HmRUW UHTXLUHG WR HVWLPDWH DOO GHOWDV XVLQJ nQLWH GLmHUHQFH HVWLPDWLRQ
LV DSSUR[LPDWHO\ �� WLPHV JUHDWHU WKDQ XVLQJ /50� ,Q RXU QXPHULFDO H[SHULPHQWV� ZH EDODQFH WKH
QXPEHU RI SDWKV VLPXODWHG XVLQJ HDFK PHWKRG VR WKDW WKH FRPSXWLQJ WLPH XVHG WR HVWLPDWH DOO
#&Q #%L��� LV WKH VDPH DFURVV PHWKRGV�
7DEOH � VKRZV WKH QXPHULFDO FRPSDULVRQ RI VHOHFWHG GHOWDV IRU WKH GLJLWDO RSWLRQ� �:H FKRRVH
L �� DV D W\SLFDO FDVH�� %HFDXVH (>= �eK�L
e�b� BP@ �� ZH XVH = �eK�L
e�b� BP DV FRQWURO YDULDWH LQ
WKH LPSOHPHQWDWLRQ RI /50� DV LV RIWHQ GRQH� 7KLV UHGXFHV WKH VWDQGDUG HUURU E\ DERXW ���� 7KH
nQLWH GLmHUHQFH PHWKRGV DUH EDVHG RQ VLPXODWLQJ IURP WKH LQLWLDO YDOXHV %����� DQG %����� � cE
IRU WKUHH YDOXHV RI cE� 6PDOOHU YDOXHV WHQG WR UHGXFH ELDV EXW LQFUHDVH YDULDQFH� WKH WZR HmHFWV
DUH FDSWXUHG E\ WKH URRW PHDQ VTXDUH HUURU �506(�� 7KH UHVXOWV LQGLFDWH WKDW WKH /50 HVWLPDWRU
RXWSHUIRUPV WKH nQLWH GLmHUHQFH HVWLPDWRUV�
7DEOH � VXPPDUL]HV D VLPLODU QXPHULFDO FRPSDULVRQ IRU NQRFNRXW FDSOHWV� 7KH ?DFFXUDWH� YDOXHV
DUH HVWLPDWHV FDOFXODWHG XVLQJ WKH nQLWH GLmHUHQFH PHWKRG ZLWK cE ������� DQG ��� PLOOLRQ
��
0HWKRG 'HOWD (VWLPDWRU 6WG� (UU� 506(
$FFXUDWH #&� #%����� �������� ������� b
9DOXH #&�� #%����� ������� ������� b
/LNHOLKRRG #&� #%����� �������� ������� �������5DWLR 0HWKRG #&�� #%����� ������� ������� �������
cE #&� #%����� �������� ������� �������)LQLWH ������ #&�� #%����� ������� ������� �������
cE #&� #%����� �������� ������� �������'LmHUHQFH
������ #&�� #%����� ������� ������� �������0HWKRG cE #&� #%����� �������� ������� �������
����� #&�� #%����� ������� ������� �������
7DEOH �� 'HOWDV IRU NQRFNRXW FDSOHWV XVLQJ /50 DQG nQLWH GLmHUHQFHV� 7R EDODQFH WKH FRPSXWLQJWLPH UHTXLUHG WR HVWLPDWH DOO GHOWDV� ZH XVH ��������� UHSOLFDWLRQV IRU /50 DQG ������� IRU WKHnQLWH GLmHUHQFH HVWLPDWRUV� cE LV WKH LQFUHPHQW LQ %����� XVHG LQ WKH nQLWH GLmHUHQFH HVWLPDWLRQ�
UHSOLFDWLRQV� 5RRW PHDQ VTXDUH HUURUV �506(� DUH HVWLPDWHG UHODWLYH WR WKH DFFXUDWH YDOXHV� 7KH
/50 PHWKRG VXEVWDQWLDOO\ RXWSHUIRUPV WKH nQLWH GLmHUHQFH HVWLPDWRUV ZKHQ WKH FRPSXWLQJ HmRUW
UHTXLUHG WR HVWLPDWH DOO GHOWDV LV KHOG n[HG�
� *DPPD
6HFRQG GHULYDWLYHV DUH W\SLFDOO\ VRPHZKDW KDUGHU WR HVWLPDWH WKDQ nUVW GHULYDWLYHV� ,Q WKLV VHFWLRQ�
ZH SUHVHQW DQG FRPSDUH WKUHH PHWKRGV IRU HVWLPDWLQJ VHFRQG GHULYDWLYHV RI RSWLRQ SULFHV ZLWK
UHVSHFW WR LQLWLDO YDOXHV RI IRUZDUG UDWHV �RU HTXLYDOHQWO\ RI LQLWLDO ERQG SULFHV�� WKH VWDQGDUG
FHQWUDO GLmHUHQFH HVWLPDWRU� D FRPELQDWLRQ RI WKH SDWKZLVH DQG OLNHOLKRRG UDWLR PHWKRGV� DQG D
SXUH OLNHOLKRRG UDWLR PHWKRG�
$V LQ 6HFWLRQ �� GHQRWH E\ J�A/��W��� � � � � A/1 �WQ�� WKH GLVFRXQWHG SD\Rm RI VRPH GHULYDWLYH VHFXULW\
DQG FRQVLGHU WKH JHQHULF SUREOHP RI HVWLPDWLQJ
#�
#/N����(>J�A/��W��� � � � � A/1 �WQ��@ ����
RU#�
#%N����(>J�A/��W��� � � � � A/1 �WQ��@� ����
8VLQJ WKH GHWHUPLQLVWLF UHODWLRQ EHWZHHQ WKH LQLWLDO IRUZDUG UDWHV DQG WKH LQLWLDO ERQG SULFHV ZH FDQ
FRQYHUW DQ HVWLPDWRU RI HLWKHU JDPPD LQWR DQ HVWLPDWRU RI WKH RWKHU� 7R HPSKDVL]H WKH GHSHQGHQFH
RI WKH H[SHFWHG YDOXH RQ WKH LQLWLDO WHUP VWUXFWXUH ZH ZULWH
*�/����� /����� � � � � /1 ���� (>J�A/��W��� � � � � A/1 �WQ��@
RU *�%����� � � � � %1������ IRU WKH VDPH TXDQWLW\� 7R HPSKDVL]H WKH UROH RI D VLQJOH IRUZDUG UDWH
/N��� ZLWK DOO RWKHUV KHOG n[HG� ZH ZULWH *�/N�����
��
)RU D FHQWUDO GLmHUHQFH HVWLPDWRU� ZH FKRRVH DQ q ! � DQG PDNH WKH DSSUR[LPDWLRQ
#�
#/N����(>J�A/��W��� � � � � A/1 �WQ��@ {
�
q�>*�/N��� � q�b �*�/N���� �*�/N���b q�@ � ����
7KH WHUPV *�/N���h q� DUH HVWLPDWHG LQ VHSDUDWH VLPXODWLRQV LQ ZKLFK WKH LQLWLDO YDOXH RI WKH NWK
IRUZDUG /LERU UDWH LV VHW WR /N��� h q� 7KH DFFXUDF\ RI WKLV PHWKRG FDQ EH YHU\ VHQVLWLYH WR WKH
FKRLFH RI q� VPDOOHU YDOXHV ZLOO OHDG WR ODUJHU YDULDQFH LQ WKH GLmHUHQFH HVWLPDWRU EHFDXVH RI WKH q
LQ WKH GHQRPLQDWRU RI ����� ODUJHU YDOXHV ZLOO OHDG WR ODUJHU ELDV GXH WR WKH DSSUR[LPDWLRQ LQ �����
,GHDOO\� RQH ZRXOG OLNH WR FKRRVH q WR EDODQFH WKHVH FRQVLGHUDWLRQV E\� H�J�� PLQLPL]LQJ PHDQ VTXDUH
HUURU� EXW WKH RSWLPDO q PD\ EH TXLWH VHQVLWLYH WR WKH IRUP RI WKH GLVFRXQWHG SD\Rm J DQG WR PRGHO
SDUDPHWHUV�
$ SXUH H[WHQVLRQ RI WKH SDWKZLVH PHWKRG LQ 6HFWLRQ � WR JDPPD LV UDUHO\ SRVVLEOH EHFDXVH
RSWLRQ SD\RmV DUH VHOGRP WZLFH GLmHUHQWLDEOH� 7KH H[DPSOH RI D FDSOHW VKRXOG PDNH WKLV FOHDU� :H
DUJXHG LQ 6HFWLRQ � WKDW WKH PDSSLQJ /Q �� �/Q b.�� FRXOG EH GLmHUHQWLDWHG DOPRVW HYHU\ZKHUH
WR \LHOGG
G/Q
�/Q b.�� �I/Q ! .J� ����
:KHQ ZH WU\ WR GLmHUHQWLDWH D VHFRQG WLPH WR SURGXFH D JDPPD HVWLPDWRU ZH IDFH H[DFWO\ WKH
VDPH REVWDFOH DV LQ HVWLPDWLQJ GHOWD IRU D GLJLWDO SD\Rm� 7KH LQGLFDWRU FDQ EH GLmHUHQWLDWHG DOPRVW
HYHU\ZKHUH� EXW LWV GHULYDWLYH LV ]HUR ZKHUHYHU LW H[LVWV DQG LV WKXV FRPSOHWHO\ XQLQIRUPDWLYH�
,W LV� KRZHYHU� SRVVLEOH WR FRPELQH D SDWKZLVH HVWLPDWH RI GHOWD ZLWK DQ /50 WHUP WR DUULYH DW
D PL[HG HVWLPDWH RI JDPPD� WKH /50 WHUP KDV WKH HmHFW RI ?GLmHUHQWLDWLQJ� WKH SDWKZLVH GHOWD
HVWLPDWH� &RQVLGHU� WKHQ� D SDWKZLVH GHOWD HVWLPDWH RI WKH IRUP
1;
L �
#J
# A/L�WL�AcLN�WL�� ����
:H UHVWULFW DWWHQWLRQ WR WKH FDVH LQ ZKLFK Ac LV FDOFXODWHG EDVHG RQ WKH IRUZDUG�GULIW DSSUR[LPDWLRQ
����� ,I ZH QRZ ZDQW WR DSSO\ WKH /50 PHWKRG� ZH PXVW QRWH WKDW WKH LQLWLDO YDOXHV /N��� DmHFW
WKH H[SHFWHG YDOXH RI ���� LQ WZR ZD\V� LPSOLFLWO\ WKURXJK WKH GLVWULEXWLRQ RI WKH A/L�WL� �MXVW DV LQ
6HFWLRQ ���� EXW DOVR H[SOLFLWO\ WKURXJK WKH IXQFWLRQDO GHSHQGHQFH RI WKH AxRLRQ WKHVH YDOXHV� 7KH
ODWWHU GHSHQGHQFH HQWHUV ���� WKURXJK WKH G\QDPLFV RI AcLN� 7KLV GHSHQGHQFH LV GLmHUHQWLDEOH D
VHFRQG WLPH �HYHQ WKRXJK WKH GHULYDWLYH RI J PD\ QRW EH� VR ZH XVH D VHFRQG DSSOLFDWLRQ RI WKH
��
SDWKZLVH PHWKRG IRU WKLV WHUP WRJHWKHU ZLWK /50� 7KH UHVXOWLQJ HVWLPDWRU LV
0L[HG 3DWKZLVH�/50 *DPPD (VWLPDWRU�1;Q �
#J
# A/Q�WQ�AcQN�WQ�
��; bP�hb� BP�
1;Q �
#J
# A/Q�WQ�AcQNN�WQ� ����
ZLWK
AcQNM�LK� # AcQN�LK�
#/M���
A/Q�LK�
/N���
L;U �
#AxRQ�UK�
#/M���� AcQM�LK�
L;U �
#AxRQ�UK�
#/N���� A/Q�LK�
L;U �
#�AxRQ�UK�
#/N���#/M�������
DQG ; � P� BP� DQG h DV LQ ����������
$ IHZ UHPDUNV RQ WKLV HVWLPDWRU DUH LQ RUGHU�
p (TXDWLRQ ����� WKRXJK D ELW PRUH FRPSOH[ WKDQ WKRVH ZH HQFRXQWHUHG LQ 6HFWLRQ �� LV HDVLO\
HYDOXDWHG EHFDXVH �XQOLNH WKH H[DFW SDWKZLVH H[SUHVVLRQ ����� LW LV QRW UHFXUVLYH _ LW FDQ EH
HYDOXDWHG DW WLPH LK GLUHFWO\ IURP WKH VLPXODWHG IRUZDUG /LERU UDWHV DW WKDW WLPH DQG IURP
GHULYDWLYHV RI AxR WKDW FDQ EH SUHFRPSXWHG�
p 7KH HVWLPDWRU LQ ���� DSSOLHV WR WKH ?GLDJRQDO� JDPPD #� #/�N��� DQG XVHVAcQNM RQO\ ZLWK
M N� :H KDYH LQFOXGHG WKH PRUH JHQHUDO FDVH LQ ���� WR LQFOXGH WKH SRVVLELOLW\ RI HVWLPDWLQJ
DQ ?Rm�GLDJRQDO� JDPPD RI WKH IRUP #� �#/M���#/N����� )RU WKLV FDVH� UHSODFH AcQNN ZLWK
AcQNM LQ ����� DQG LQ WKH GHnQLWLRQ ���� RI BP UHSODFH N ZLWK M�
p 7KH UHFXUVLRQ LQ ���� GHWHUPLQHV WKH YDOXHV RI WKH AcQNM RQ WKH WLPH JULG I�� K� �K� � � �J ZKHUHDV
LQ ���� ZH KDYH LPSOLFLWO\ DOORZHG HYDOXDWLRQ RI WKHVH YDULDEOHV DW DUELWUDU\ WLPHV� ,Q SUDFWLFH�
RQH FDQ HLWKHU DUUDQJH WR KDYH DOO UHOHYDQW GDWHV OLH RQ WKH VLPXODWLRQ WLPH JULG RU HOVH
LQWHUSRODWH OLQHDUO\ EHWZHHQ JULG SRLQWV�
p 7KH HVWLPDWRU LQ ���� ZRXOG LQ IDFW EH XQELDVHG �IRU WKH GLVFUHWL]HG SURFHVV ZLWK WLPH VWHS K�
LI WKH IRUZDUG�GULIW DSSUR[LPDWLRQ KHOG H[DFWO\ �L�H�� LI WKH /LERU UDWHV ZHUH PXOWLYDULDWH JHR�
PHWULF %URZQLDQ PRWLRQ ZLWK WLPH�YDU\LQJ GULIW� SURYLGHG WKH IXOO�UDQN FRQGLWLRQ QHFHVVDU\
WR GHnQH hb� KROGV� +HQFH� ���� GRHV QRW HQWDLO DQ\ DSSUR[LPDWLRQV EH\RQG WKH IRUZDUG�GULIW
DSSUR[LPDWLRQ DQG WKH WLPH�GLVFUHWL]DWLRQ LQKHUHQW WR VLPXODWLRQ�
-XVW DV ZH GHULYHG WKH PL[HG JDPPD HVWLPDWRU E\ DSSO\LQJ /50 WR D SDWKZLVH GHOWD HVWLPDWH�
ZH FDQ GHULYH DQ DOWHUQDWLYH HVWLPDWRU E\ DSSO\LQJ /50 WR DQ /50 GHOWD HVWLPDWH� )RU H[DPSOH�
��
LQ WKH *DXVVLDQ VHWWLQJ VXUURXQGLQJ ���� ZH FRXOG GLmHUHQWLDWH WZLFH WR JHW
G�
Gt�(t>J�;�@
=J�[����[�P�t��h� G[ ����
=J�[�
���[�P�t��h�
��[�P�t��h���[�P�t��h� G[�
6LPSOH FDOFXODWLRQV VKRZ WKDW
���[�P�t��h�
��[�P�t��h� >�[bP�t���hb� BP@� b BP�
hb� BP� �[bP�t���hb� �P� ����
(YDOXDWLQJ WKLV H[SUHVVLRQ DW [ ; DQG PXOWLSO\LQJ LW E\ J�;� \LHOGV WKH /50 HVWLPDWRU RI WKH
VHFRQG GHULYDWLYH ZLWK UHVSHFW WR t� ,Q WKH /LERU PRGHO� ZH PDNH WKH FRUUHVSRQGHQFHV ���������
DQG
�PQ � b�IQ NJ
/�N���� K
Leb�;U �
#�AxRQ�UK�
#/�N���� Q �� � � � � 1� ����
ZLWK#�AxRQ�UK�
#/�N��� b�p�wQ�LK�w
�
N�LK�
�� � p/N������Is�LK� � N � QJ�
:H WKXV DUULYH DW
/50 *DPPD (VWLPDWRU
J�A/��W��� � � � � A/1 �WQ��Q>�; bP��hb� BP@� b BP�hb� BP� �; bP��hb� �P
R� ����
ZLWK ; � P� h� BP DV LQ ��������� DQG �P DV LQ �����
$Q /50 HVWLPDWRU IRU #� �#/M���/N���� FDQ EH GHULYHG VLPLODUO\� UHSODFH WKH VFDODU SDUDPHWHU
t RI � ZLWK D YHFWRU DQG UHSODFH ���� ZLWK WKH FDOFXODWLRQ RI #� �#tM#tN�� WKHQ PDNH WKH XVXDO
FRUUHVSRQGHQFHV WR FRQYHUW WR WKH /LERU VHWWLQJ�
��� 1XPHULFDO 5HVXOWV
$V PHQWLRQHG LQ 6HFWLRQ ���� D VWDQGDUG FDSOHW &Q FDQ EH KHGJHG XVLQJ MXVW WKH ERQGV %Q DQG %Q���
&RQYHUVHO\� %Q LV XVHIXO LQ KHGJLQJ &Qb� DQG &Q� 6HFRQG GHULYDWLYHV RI WKH IRUP #�&Qb� #%�Q DQG
#�&Q #%�Q DUH WKXV UHOHYDQW LQ FKHFNLQJ JDPPD HVWLPDWHV�
7DEOH � SUHVHQWV QXPHULFDO UHVXOWV IRU FDSOHW JDPPDV RI WKLV IRUP� 7KH SXUH /50 JDPPD
HVWLPDWRU SURGXFHV YHU\ ODUJH VWDQGDUG HUURUV LQ WKLV DSSOLFDWLRQ DQG LV WKHUHIRUH RPLWWHG IURP
IXUWKHU FRPSDULVRQ� ([DFW YDOXHV DUH FDOFXODWHG IURP %ODFNV IRUPXOD� 0XFK DV LQ 6HFWLRQ ����
WKH FRPSXWLQJ WLPH UHTXLUHG WR HVWLPDWH DOO JDPPDV XVLQJ nQLWH GLmHUHQFHV LV URXJKO\ �� WLPHV DV
JUHDW DV WKH WLPH UHTXLUHG WR HVWLPDWH DOO JDPPDV XVLQJ WKH PL[HG SDWKZLVH�/50 PHWKRG� 7KH
��
0HWKRG 'HOWD (VWLPDWRU 6WG� (UU� 506(
([DFW #�&� #%������ ������� b b
9DOXH #�&�� #%������ ������ b b
3DWKZLVH�/50 #�&� #%������ ������� ����� �����
0HWKRG #�&�� #%������ ������ ����� �����
cE #�&� #%������ ������� ����� �����
������ #�&�� #%������ ������ ����� �����
)LQLWHcE #�&� #%�����
� ������� ����� ���������� #�&�� #%�����
� ������ ����� �����'LmHUHQFH
cE #�&� #%������ ������� ����� �����
����� #�&�� #%������ ������ ����� �����
0HWKRGcE #�&� #%�����
� ������� ����� ���������� #�&�� #%�����
� ������ ����� �����
7DEOH �� *DPPD HVWLPDWHV IRU FDSOHWV� 7R EDODQFH WKH FRPSXWLQJ WLPH UHTXLUHG WR HVWLPDWH DOOJDPPDV� ZH XVH ������� UHSOLFDWLRQV IRU WKH PL[HG SDWKZLVH�/50 HVWLPDWRU DQG ������� IRU WKHnQLWH GLmHUHQFH HVWLPDWRUV� cE LV WKH LQFUHPHQW LQ %����� XVHG LQ WKH nQLWH GLmHUHQFH HVWLPDWLRQ�
UHVXOWV LQ WKH WDEOH DUH EDVHG RQ EDODQFLQJ WKH QXPEHU RI VLPXODWHG SDWKV IRU HDFK PHWKRG VR WKDW
WKH WRWDO FRPSXWLQJ WLPH WR HVWLPDWH DOO JDPPDV ZRXOG EH HTXDO DFURVV PHWKRGV� 7KH UHVXOWV IRU
WKH PL[HG PHWKRG DOVR XVH WKH FRQWURO YDULDWH = �eK�Le�b� BP� ZKLFK UHGXFHV WKH VWDQGDUG HUURU E\
DERXW ���� 7KH LQFUHPHQW LQ %Q XVHG IRU WKH nQLWH GLmHUHQFH HVWLPDWHV LV LQGLFDWHG LQ WKH WDEOH
E\ cE�
7KH UHVXOWV LQ WKH WDEOH DUH FRQVLVWHQW ZLWK WKH YLHZ WKDW HVWLPDWLQJ JDPPDV LV PRUH GLpFXOW
WKDQ HVWLPDWLQJ GHOWDV� WKH DFFXUDF\ ZH JHW IRU WKH JDPPD RI WKHVH /LSVFKLW] FRQWLQXRXV SD\RmV
LV VLPLODU WR ZKDW ZH JHW IRU WKH GHOWDV RI GLVFRQWLQXRXV SD\RmV� FRQVLVWHQW ZLWK WKH GLVFXVVLRQ
VXUURXQGLQJ ����� 7KH PL[HG SDWKZLVH�/50 PHWKRG LV VRPHWLPHV RXWSHUIRUPHG E\ WKH EHVW nQLWH
GLmHUHQFH HVWLPDWH� +RZHYHU� WKH nQLWH GLmHUHQFH PHWKRG LV YHU\ VHQVLWLYH WR WKH FKRLFH RI cE� DQG
D JRRG cE PD\ QRW EH NQRZQ LQ DGYDQFH� 8QOHVV cE FDQ EH FKRVHQ FDUHIXOO\� WKH PL[HG PHWKRG
DSSHDUV SUHIHUDEOH�
� 9HJD
:H QRZ WXUQ WR HVWLPDWLQJ VHQVLWLYLWLHV ZLWK UHVSHFW WR FKDQJHV LQ YRODWLOLW\� :H IUDPH WKH SUREOHP
E\ LQWURGXFLQJ D SDUDPHWHU t LQ WKH YRODWLOLWLHV wQ�W� DQG FRQVLGHULQJ GHULYDWLYHV ZLWK UHVSHFW WR
t� 6HWWLQJ #wQ�t� W� #t s � IRU VRPH Q �DQG DOO W� EXW #wL�t� W� #t s � IRU DOO L� Q FRUUHVSRQGV WR
D SDUDOOHO VKLIW LQ WKH YRODWLOLWLHV RI /Q� VHWWLQJ #wQ�t� W� #t s � IRU DOO Q FRUUHVSRQGV WR D SDUDOOHO
VKLIW LQ DOO YRODWLOLWLHV� 6HQVLWLYLWLHV WR YRODWLOLW\ EXFNHWV FDQ EH PRGHOHG E\ UHVWULFWLQJ QRQ]HUR
YDOXHV RI #wQ�t� W� #t WR W LQ VRPH LQWHUYDO�
:ULWH AcQ�W� s AcQ�t� W� IRU # A/Q�W� #t� 2QFH ZH KDYH FRPSXWHG WKH AcQ� WKH JHQHUDO SDWKZLVH
��
HVWLPDWRU WDNHV WKH IRUP LQ ��� EXW ZLWK AcQN UHSODFHG E\ AcQ� 'LmHUHQWLDWLQJ ���� \LHOGV
([DFW 3DWKZLVH $OJRULWKP� 9HJD
AcQ��L� ��K�
AcQ�LK�A/Q��L� ��K�
A/Q�LK�� A/Q��L� ��K�
t|#AxQ�LK�
#tb #wQ�LK�
#tw�
Q�LK�
}K�
#wQ#t
=L��
SK
u����
ZLWK LQLWLDO FRQGLWLRQ AcQ��� s ��
,Q ����� #wQ #t DQG wQ DUH URZ YHFWRUV DQG =L�� DQG w�
Q DUH FROXPQ YHFWRUV� 7KH GLmHUHQWLDWHG
GULIW DSSHDULQJ LQ ���� DEEUHYLDWHV WKH IXOO H[SUHVVLRQ
#AxQ#t
Q;
M s�LK�
�#AxQ
# A/M
AcM �G;
N �
#AxQ#wMN
#wMN#t
��
ZLWK wMN GHQRWLQJ WKH NWK FRPSRQHQW RI wM� 7KH SUHVHQFH RI WKH AcM LQ WKLV H[SUHVVLRQ PDNHV
VLPXODWLRQ RI ���� VRPHZKDW WLPH�FRQVXPLQJ� UHTXLULQJ HmRUW FRPSDUDEOH WR VLPXODWLQJ DQRWKHU
FRS\ RI WKH /LERU UDWHV ZLWK D SHUWXUEHG YDOXH RI t�
:H WKHUHIRUH FRQVLGHU WKH IRUZDUG�GULIW DSSUR[LPDWLRQ� 'LmHUHQWLDWLQJ AxRQ ZLWK UHVSHFW WR t
\LHOGV#AxRQ�LK�
#t
Q;M s�LK�
p/M���
� � p/M���
t#wQ�LK�
#tw�
M�LK� �#wM�LK�
#tw�
Q�LK�
u� ����
7KLV H[SUHVVLRQ LV LQGHSHQGHQW RI WKH VLPXODWHG SDWK DQG FDQ WKXV EH SUHFRPSXWHG� 5HSODFLQJ Ax
ZLWK AxR LQ ����� GLmHUHQWLDWLQJ DQG WKHQ VLPSOLI\LQJ WKH UHVXOWLQJ H[SUHVVLRQ \LHOGV
)RUZDUG 'ULIW $SSUR[LPDWLRQ� 9HJD
AcQ�LK� A/Q�LK�
�# L;
M �
|#AxRQ�MK�
#tb #wQ�MK�
#twQ�MK�
�
}K�
L;M �
#wQ�MK�
#t=M
SK
�$ � ����
%HVLGHV A/Q� WKH RQO\ WHUP RQ WKH ULJKW VLGH RI ���� WKDW FDQQRW EH SUHFRPSXWHG LV
L;M �
#wQ�MK�
#t=M
SK�
%XW HYDOXDWLQJ WKLV H[SUHVVLRQ DORQJ HDFK VLPXODWHG SDWK UHTXLUHV YHU\ OLWWOH HmRUW� PDNLQJ ����
PXFK IDVWHU WKDQ WKH H[DFW SDWKZLVH PHWKRG RU VLPXODWLRQ RI D VHFRQG FRS\ RI WKH /LERU UDWHV�
7DEOH � SUHVHQWV QXPHULFDO UHVXOWV IRU FDSOHW YHJDV #&Q #t ZLWK DOO #wL #t s �� FRUUHVSRQGLQJWR D SDUDOOHO VKLIW LQ WKH WHUP VWUXFWXUH RI YRODWLOLW\� ([DFW YDOXHV DUH FDOFXODWHG IURP %ODFNV
��
([DFW ([DFW 3DWKZLVH 3DWKZLVH $SSUR[LPDWLRQQ 9DOXH (VWLPDWRU 6WG� (UU� (VWLPDWRU 6WG� (UU�
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ����� ����� ���� ����� ����
�� ������ ������ ���� ����� ����
�� ������ ������ ���� ����� ����
�� ������ ������ ���� ������ ����
7DEOH �� (VWLPDWHG FDSOHW YHJDV #&Q #t DQG VWDQGDUG HUURUV �DOO LQ EDVLV SRLQWV�� (VWLPDWHV DUHEDVHG RQ ������� UHSOLFDWLRQV�
IRUPXOD� %RWK WKH H[DFW SDWKZLVH DQG WKH IRUZDUG GULIW DSSUR[LPDWLRQ PHWKRGV SHUIRUP ZHOO� 7KH
DSSUR[LPDWLRQ PHWKRG SURGXFHV ODUJHU ELDV DW ORQJ PDWXULWLHV� EXW VDYHV QHDUO\ RQH WKLUG RI WKH
FRPSXWLQJ WLPH FRPSDUHG WR WKH H[DFW PHWKRG�
� &RQYHUJHQFH RI 3DWKZLVH (VWLPDWRUV
,Q WKLV VHFWLRQ� ZH JLYH D WKHRUHWLFDO DQDO\VLV RI WKH �H[DFW� SDWKZLVH GHOWD HVWLPDWRUV RI 6HFWLRQ ��
:H VKRZ WKDW WKH GLVFUHWH�WLPH DOJRULWKP SURGXFHV XQELDVHG HVWLPDWRUV IRU GHOWDV RI WKH GLVFUHWH�
WLPH IRUZDUG /LERU SURFHVV� WKH FRQWLQXRXV�WLPH HVWLPDWRUV DUH XQELDVHG IRU WKH FRQWLQXRXV�WLPH
IRUZDUG /LERU SURFHVV� DQG WKH GLVFUHWH�WLPH HVWLPDWRUV FRQYHUJH WR WKH FRQWLQXRXV�WLPH GHOWDV
DV WKH VLPXODWLRQ WLPH VWHS GHFUHDVHV WR ]HUR� 7KHRUHWLFDO VXSSRUW IRU WKH �GLVFUHWH�WLPH� /50
HVWLPDWRUV LQ D *DXVVLDQ VHWWLQJ IROORZV IDLUO\ ZHOO�HVWDEOLVKHG OLQHV �H�J�� >�@� DQG LV WKHUHIRUH
RPLWWHG�
��� 8QELDVHGQHVV� 'LVFUHWH 7LPH
)L[ D WLPH LQFUHPHQW K DQG FRQVLGHU WKH SURFHVVHV GHnQHG E\ ���� DQG �����
��
7KHRUHP � 6XSSRVH J � 51 � 5 LV /LSVFKLW] FRQWLQXRXV� 7KHQ
(
�1;Q �
�#
A/Q�LQK�J�A/��L�K�� � � � � A/1 �L1K��
�AcQN�LQK�
�
#
#/N���(>J�A/��L�K�� � � � � A/1 �L1K��@�
IRU DQ\ L�� � � � � L1 � L�H�� WKH GLVFUHWH�WLPH SDWKZLVH HVWLPDWRU LV XQELDVHG�
3URRI� %HFDXVH J LV /LSVFKLW] LW LV GLmHUHQWLDEOH DOPRVW HYHU\ZKHUH VR WKH SDUWLDO GHULYDWLYHV HYDOX�
DWHG DW WKH A/Q H[LVW ZLWK SUREDELOLW\ RQH� 7R HPSKDVL]H WKH GHSHQGHQFH RQ LQLWLDO FRQGLWLRQV� ZULWH
A/Q�LK� A/Q�LK� /N����� DQG A/qQ�LK� A/Q�LK� /N��� � q�� 7KHQ
MJ�A/q��L�K�� � � � � A/q
1 �L1K��b J�A/��L�K�� � � � � A/1 �L1K��M � .J
1;
Q �
MA/qQ�LQK�b A/�LQK�M
IRU VRPH FRQVWDQW .J� 7KH WKHRUHP QRZ IROORZV IURP WKH GRPLQDWHG FRQYHUJHQFH WKHRUHP RQFH ZH
VKRZ WKDW
(> AcQN�LK�@ #
# A/N���(> A/Q�LK�@ ����
IRU DOO Q DQG L�
6LQFH Q LV DUELWUDU\� ZH OLJKWHQ WKH QRWDWLRQ E\ ZULWLQJ VLPSO\ A/�LK� A/Q�LK� /N���� DQG
A/q�LK� A/Q�LK� /N��� � q�� :H ZLOO XVH LQGXFWLRQ WR SURYH WKDW
�
q
nnnA/q�LK�b A/�LK�nnn � .L� ����
ZKHUH .L LV D UDQGRP YDULDEOH PHDVXUDEOH ZLWK UHVSHFW WR I=�� � � � � =LJ ZLWK (>.L@ ��� ,W LV HDV\WR VHH WKDW ���� KROGV IRU L �� $VVXPLQJ WKDW ���� KROGV IRU VRPH L� ZH KDYH
�
q
nnnA/q��L� ��K�b A/��L� ��K�nnn
�
qH[S
rb��wQ�LK�wQ�LK�
�K� wQ�LK�SK=L��
s nnnA/q�LK� H[S�Ax�A/q�LK���b A/�LK� H[S�Ax�A/�LK���nnn �
� z
q
KnnnH[S�Ax�A/q�LK����A/q�LK�b A/�LK��nnn�
nnnA/�LK�rH[S�Ax�A/q�LK���b H[S�Ax�A/�LK���
snnnL
� z&�.L �z
q
nnnA/�LK�&�rAx�A/q�LK��b Ax�A/�LK��
snnn
� z&�.L �z
q&�
KnnnrA/�LK�b A/q�LK�
sAx�A/q�LK��
nnn�nnnA/q�LK�Ax�A/q�LK��b A/�LK�Ax�A/�LK��
nnnL
� z&�.L � z&�.L � z&�.L�
:H KDYH GHnQHG z H[Srb��wQ�LK�wQ�LK�
�K� wQ�LK�SK=L��
s� ZKLFK LV D UDQGRP YDULDEOH LQGH�
SHQGHQW RI .L� :H KDYH DOVR XVHG WKH IDFW WKDW Ax�c� LV ERXQGHG� Ax�A/�c��A/Q�c� LV /LSVFKLW] �DVLQ ���� EHORZ� DQG WKH LQHTXDOLW\ MH[ b H\M � &M[ b \M IRU ERXQGHG [� \� 1RZ ZH FDQ GHnQH
.L�� z&�.L � z&�.L � z&�.L� DQG FOHDUO\� ZH KDYH (>.L��@ � �� (TXDWLRQ ���� QRZ IROORZVIURP WKH GRPLQDWHG FRQYHUJH WKHRUHP� (
��
��� 8QELDVHGQHVV� &RQWLQXRXV 7LPH
:H nUVW MXVWLI\ HTXDWLRQ ���� IRU WKH G\QDPLFV RI WKH cQN XVLQJ D UHVXOW IURP WKH WKHRU\ RI VWRFKDVWLF
oRZV �DV LQ &KDSWHU � RI 3URWWHU >��@�� 3LNRYVN\ >�@ KDV DOVR XVHG WKH VWRFKDVWLF�oRZ IRUPXODWLRQ
IRU WKH 0RQWH &DUOR HVWLPDWLRQ RI VHQVLWLYLWLHV� :ULWH / IRU WKH YHFWRU �/�� � � � � /1 � DQG c IRU
�c�N� � � � �c1N� ZLWK N n[HG EXW DUELWUDU\� ,QVHUW WKH LQLWLDO FRQGLWLRQV /��� DQG c��� DV DUJXPHQWV
RI WKHVH SURFHVVHV� :ULWH �N IRU WKH 1 �YHFWRU ZKRVH QWK FRRUGLQDWH LV �IQ NJ�
/HPPD � 7KHUH H[LVWV D XQLTXH SURFHVV �/�W� /�����c�W� �N�� VDWLVI\LQJ WKH VWRFKDVWLF GLmHUHQWLDO
HTXDWLRQ ���^����� )RU HDFK W� /�W� /���� LV FRQWLQXRXVO\ GLmHUHQWLDEOH ZLWK UHVSHFW WR /���� DQG
##/N���
/Q�W� /���� cQN�W��N��
3URRI� 6HW wPD[ VXSQ�W NwQ�W�N� ,W LV VWUDLJKWIRUZDUG WR VHH WKDW xQ�W�/Q�W� LV JOREDOO\ /LSVFKLW]�
EHFDXVH
NxQ�W� [�[Q b xQ�W� \�\QN � �1w�PD[N[b \N�
7KH nUVW GHULYDWLYHV
#
#[N�xQ�W� [�[Q�
�!!!!�!!!!�
� � N � s�W�pwQw
�
N[Q
���p[N��� s�W� � N � Q
pwQw�
Q[Q
���p[Q��� pwQw
�
Q[Q
��p[Q� N Q
� � N ! Q
����
DOVR VDWLVI\ D /LSVFKLW] FRQGLWLRQ� 7KXV� WKH OHPPD IROORZV IURP 7KHRUHP 9��� RI 3URWWHU >��@� ,Q
SDUWLFXODU� ZH FDQ ZULWH WKH VROXWLRQ RI ���� ���� DV
/Q�W� /Q��� H[S
�= W
�
xQ�V� GVb�
�
= W
�
wQ�V�w�
Q�V� GV�
= W
�
wQ�V� G:V
w����
cQN�W� �IQ NJ/Q�W�
/Q���� /Q�W�
= W
�
;M
#xQ�V�
#/M�V�cMN�V� GV� ����
(
:H QRZ VKRZ WKDW WKH FRQWLQXRXV�WLPH SDWKZLVH HVWLPDWRU LV XQELDVHG�
7KHRUHP � /HW �/�c��c�c�� EH WKH VROXWLRQ RI HTXDWLRQ ��������� 6XSSRVH J � 51 � 5 LV /LSVFKLW]
FRQWLQXRXV� 7KHQ
(
�1;Q �
|#
#/Q�WQ�J�/��W��� � � � � /1 �W1 ��
}cQN�WQ�
�
#
#/N���(>J�/��W��� � � � � /1 �W1 ��@�
IRU DQ\ W�� � � � � W1 � L�H�� WKH FRQWLQXRXV�WLPH SDWKZLVH HVWLPDWRU LV XQELDVHG�
��
3URRI� $V LQ 7KHRUHP �� LW VXpFHV WR SURYH
#
#/N���(>/Q�W�@ (>cQN�W�@� ����
'HnQH ;Q�W� ORJ/Q�W� VR WKDW
G;Q�W� �xQ�W�b�
�wQ�W�w
�
Q�W��GW� wQ�W�G: �W�� ����
'LmHUHQWLDWH WR JHW
G<QN�W� ;M
#xQ�W�
#;M�W�<MN�W�GW
;M
#xQ�W�
#/N�W�/N�W�<MN�W�GW� ����
)URP 7KHRUHP 9��� RI 3URWWHU >��@� ZH NQRZ WKDW WKHUH H[LVWV �;�W�� < �W�� VROYLQJ WKH HTXDWLRQV
��������� ZLWK <QN�W� #;Q�W� #;N��� /N���#;Q�W� #/N���� ,Q SDUWLFXODU� ZH KDYH
;Q�W� ORJ/Q�W� ORJ/Q��� �
= W
�
xQ�V� GVb�
�
= W
�
wQ�V�w�
Q�V� GV�
= W
�
wQ�V� G: �V�� ����
<QN�W� �IQ NJ�
= W
�
;M
#xQ�V�
#/M�V�
#/M�V�
#;M�V�<MN�V� GV �IQ NJ�
= W
�
;M
#xQ�V�
#/M�V�/M�V�<MN�V� GV�
����
DQG cQN�W�/N��� /Q�W�<QN�W�� 7RJHWKHU� ���� DQG ���� VROYH WKH 6'(V ��� DQG ����� 0RUH�
RYHU� xQ�c�/Q�c�� w�c�/Q�c� DQG3
M�#xQ�c� #/M�c��/M�c�<MN�c� VDWLVI\ D /LSVFKLW] FRQGLWLRQ� VR IURP
7KHRUHP 9�� RI 3URWWHU >��@ ZH KDYH
OLPq��
(N�/Q� <QN��W� �/N��� � q��N��b �/Q� <QN��W� �/N�����N��N� �� ����
7KLV JLYHV XV
OLPq��
(NcQN�W� �/N��� � q��N��bcQN�W� �/N�����N��N
OLPq��
(>�
/N���N�/N��� � q�cQN�W� �/N��� � q��N��b /N���cQN�W� �/N���� �N��@
bqcQN�W� �/N��� � q��N��N
� OLPq��
&�(>N/Q�W� /N��� � q�<QN�W� �/N��� � q��N��b /Q�W� /N����<QN�W� �/N���� �N�N@
� OLPq��
&�(>N/Q�W� /N��� � q�NN<QN�W� �/N��� � q��N��b <QN�W� �/N�����N���N@
� OLPq��
&�(>N<QN�W� �/N�����N��NN/Q�W� /N��� � q�b /Q�W� /N����N@
� OLPq��
&�
K(N/Q�W� /N��� � q�N�(N<QN�W� �/N��� � q� �N��b <QN�W� �/N���� �N��N
�L� �
� OLPq��
&�
K(N<QN�W� �/N�����N��N
�(N/Q�W� /N��� � q�b /Q�W� /N����N�L� �
��
��
EHFDXVH (N/Q�W�N� DQG (N<QN�W�N
� DUH nQLWH� 7KLV HVWDEOLVKHV WKH FRQWLQXLW\ RI (c�W� /N����� 2Q
WKH RWKHU KDQG� IURP /HPPD �� ZH KDYH
/Q�W� /N��� � q� /Q�W� /N���� �
= q
�
cQN�W� /��� � p� Gp�
7DNLQJ H[SHFWDWLRQ RQ ERWK VLGHV� ZH JHW
(>/Q�W� /N��� � q�@ (>/Q�W� /N����@ �(
�= q
�
cQN�W� /��� � p�Gp
w
(>/Q�W� /N����@ �
= q
�
(>cQN�W� /��� � p�@ Gp
ZLWK WKH LQWHUFKDQJH RI LQWHJUDO DQG H[SHFWDWLRQ MXVWLnHG E\ 7RQHOOLV 7KHRUHP �)XELQL IRU QRQ�
QHJDWLYH LQWHJUDQGV�� 7KLV UHSUHVHQWDWLRQ RI (>/Q�W� /N����@ LPSOLHV ���� WKURXJK WKH FRQWLQXLW\ RI
(>cQN�W� /����@� (
��� &RQYHUJHQFH IURP 'LVFUHWH WR &RQWLQXRXV 7LPH
:H QRZ VKRZ WKDW WKH GLVFUHWH�WLPH AcQN DUH DV\PSWRWLFDOO\ XQELDVHG HVWLPDWRUV RI WKH FRQWLQXRXV�
WLPH GHULYDWLYHV #(>/Q@ #/N��� DV WKH WLPH LQFUHPHQW K GHFUHDVHV WR ��
7KHRUHP � 6XSSRVH WKDW �/Q�W��cQN�W�� VROYH ���^���� DQG �A/Q�W�� AcQN�W�� VROYH ����^����� 7KHQ
ZH KDYH
OLPK��
(> AcQN�W�@ (>cQN�W�@� ����
3URRI� 'LVFUHWL]H <QN�W� XVLQJ
A<QN��L� ��K� A<QN�LK� �Q;
M s�LK�
#xQ�LK� A/�LK��
# A/M�LK�A/M�LK� A<MN�LK�K
A<QN�LK� �Q;
M s�LK�
p A/M�K�
�� � p A/M�LK���A<MN�LK�K� ����
&OHDUO\� ZH KDYH /N��� AcQN�LK� A<QN�LK�A/Q�LK�� )URP WKH IDFW WKDW x�c� LV SRVLWLYH DQG ERXQGHG�DQG WKH LQHTXDOLW\ H[ � � � [� [�H[��[ ! �� ZH JHW �ZULWLQJ $L IRU I=�� � � � � =LJ�
(
��nnnnn(
�A/Q��L� ��K�b A/Q�LK�
KM$L
�b xQ�A/�LK��A/Q�LK�
nnnnn���
(
��nnnnn(
�A/Q�LK�
K
rH[S�x�A/�LK��Kb �
�wQw�
Q � wQSK=L�b �
sM$L
�b xQ�A/�LK��A/Q�LK�
nnnnn���
(
�nnnA/Q�LK�nnn�nnnn�K
rH[S�x�A/�LK��K�b �
sb xQ�A/�LK��
nnnn��
� (
�nnnA/Q�LK�nnn�nnnn�K
rx�A/�LK��K� H[S�x�A/�LK��K��x�A/�LK��K��
sb x�A/�LK��
nnnn��
� &�K��
��
DQG
(
��
K
nnnA/Q��L� ��K�b A/Q�LK�b (
rA/Q��L� ��K�b A/Q�LK�M$L
sb wQ�LK�A/Q�LK�=L��
SKnnn�w
(
�MA/Q�LK�M�
K
nnnnH�xQ�A/Q�LK��Kb��wQw
�
QK�wQ=L��
SK� b H�xQ�A/Q�LK��K� b wQ�LK�=L��
SK
nnnn��
(
�MA/Q�LK�M�
K(
�tH�xQ�
A/Q�LK��Kb��wQw
�
QK�wQ=L��
SK� b H�xQ�A/Q�LK��K� b wQ�LK�=L��
SK
u�
M$L
��
(
�MA/Q�LK�M�
K
rH��xQ�
A/Q�LK��K�wQw�
QK� � H��xQ�A/Q�LK��K� � wQw
�QKb �H��xQ�
A/Q�LK��K� b �wQw�QKH�xQ�A/Q�LK��K�
s�
(
�MA/Q�LK�M�
K
rH��xQ�
A/Q�LK��K��HwQw�
QK b ��b �wQw�QKH�xQ�
A/Q�LK��K� � wQw�QK
s�
� (
�MA/Q�LK�M�
K
rH��xQ�
A/Q�LK��K��wQw�QK� H
wQw�
QK�wQw
�Q�
�K��b �wQw�QKH�xQ�A/Q�LK��K� � wQw
�QK
s�
� (
�wQwQMA/Q�LK�M�
rH�xQ�
A/Q�LK��K� b �s�w
�&�K
� &�K� �&�K�
$ VLPLODU DUJXPHQW VKRZV WKDW A<QN�c� DOVR VDWLVnHV WKHVH FRQGLWLRQV� )URP 7KHRUHP ����� RI .ORHGHQDQG 3ODWHQ >��@� ZH JHW
OLPK��
(nnnA/Q�W�b /Q�W�
nnn� �� OLPK��
(nnn A<QN�W�b <QN�W�
nnn� ��7KXV�
OLPK��
(nnn AcQN�W�bcQN�W�
nnn
�
/N���OLPK��
(nnn/N��� AcQN�W�b /N���cQN�W�
nnn
�
/N���OLPq��
(nnnA<QN�W�A/Q�W�b <QN�W�/Q�W�
nnn� �
/N���OLPK��
K(nnn A<QN�W� rA/Q�W�b /Q�W�
snnn�(nnn/Q�W�
rA<QN�W�b <QN�W�
snnnL
� �
/N���OLPK��
�t(nnnA<QN�W�nnn�( nnnA/Q�W�b /Q�W�
nnn�u� �
�
t( M/Q�W�M� (
nnn A<QN�W�b <QN�W�nnn�u� �
�
��
(
,Q ����� ZH FRXOG UHSODFH AcQN ZLWK DQ\ /LSVFKLW] FRQWLQXRXV IXQFWLRQ RI DOO �A/Q� AcQN� DQG REWDLQ
WKH FRUUHVSRQGLQJ UHVXOW� KRZHYHU� WKH /LSVFKLW] UHTXLUHPHQW UXOHV RXW HYHQ WKH LQGLFDWRU IXQFWLRQ
DSSHDULQJ LQ WKH SDWKZLVH HVWLPDWRU RI FDSOHW GHOWDV�
��
$FNQRZOHGJHPHQWV� :H WKDQN /HLI $QGHUVHQ RI *HQHUDO 5H )LQDQFLDO 3URGXFWV IRU VHYHUDO
YDOXDEOH GLVFXVVLRQV� 7KLV ZRUN LV VXSSRUWHG LQ SDUW E\ 16) JUDQW '0,�������� DQG DQ ,%0
8QLYHUVLW\ 3DUWQHUVKLS $ZDUG�
5HIHUHQFHV
>�@ %R\OH� 3�� %URDGLH� 0�� DQG *ODVVHUPDQ� 3�� 6LPXODWLRQ PHWKRGV IRU VHFXULW\ SULFLQJ� -� (FR�QRPLF '\QDPLFV DQG &RQWURO� ��� ��������� �������
>�@ %URDGLH� 0�� DQG *ODVVHUPDQ� 3�� (VWLPDWLQJ VHFXULW\ SULFH GHULYDWLYHV XVLQJ VLPXODWLRQ� 0DQ�DJHPHQW 6FLHQFH� ��� �������� �����
>�@ %UDFH� $�� *DWDUHN� '�� DQG 0XVLHOD� 0�� 7KH PDUNHW PRGHO RI LQWHUHVW UDWH G\QDPLFV� 0DWK�)LQDQFH� �� ������� �������
>�@ 3LNRYVN\� ,�� 0HWKRGV RI VWRFKDVWLF oRZV LQ FRQWLQJHQW FODLPV YDOXDWLRQ� ZRUNLQJ SDSHU� ������
>�@ )RXUQLuH� (�� /DVU\� -��0�� /HEXFKRX[� -�� /LRQV� 3��/�� DQG 7RX]L� 1�� $SSOLFDWLRQV RI 0DOOLDYLQFDOFXOXV WR 0RQWH &DUOR PHWKRGV LQ nQDQFH� )LQDQFH DQG 6WRFKDVWLFV� �� �������
>�@ )X� 0�� DQG +X� -�4�� 6HQVLWLYLW\ DQDO\VLV IRU 0RQWH &DUOR VLPXODWLRQ RI RSWLRQ SULFLQJ� 3URE�DELOLW\ LQ WKH (QJLQHHULQJ DQG ,QIRUPDWLRQ 6FLHQFHV� �� �������� �������
>�@ *ODVVHUPDQ� 3�� *UDGLHQW (VWLPDWLRQ YLD 3HUWXUEDWLRQ $QDO\VLV� .OXZHU 3XEOLVKHUV� 1RUZHOO�0DVV�� �������
>�@ *ODVVHUPDQ� 3�� DQG =KDR� ;�� $UELWUDJH�IUHH GLVFUHWL]DWLRQ RI ORJQRUPDO IRUZDUG /LERU DQGVZDS UDWH PRGHOV� )LQDQFH DQG 6WRFKDVWLFV� WR DSSHDU�
>�@ *O\QQ� 3�� DQG /(FX\HU� 3�� /LNHOLKRRG UDWLR JUDGLHQW HVWLPDWLRQ IRU VWRFKDVWLF UHFXUVLRQV�$GYDQFHV LQ $SSOLHG 3UREDELOLW\� ��� ����^����� �������
>��@ +HDWK� '�� -DUURZ� 5�� DQG 0RUWRQ� $�� %RQG SULFLQJ DQG WKH WHUP VWUXFWXUH RI LQWHUHVW UDWHV�D QHZ PHWKRGRORJ\ IRU FRQWLQJHQW FODLPV YDOXDWLRQ� (FRQRPHWULFD� ��� ������ �������
>��@ +R� <�&�� DQG &DR� ;�� 3HUWXUEDWLRQ $QDO\VLV RI 'LVFUHWH (YHQW '\QDPLF 6\VWHPV� .OXZHU3XEOLVKHUV� 1RUZHOO� 0DVV�� �������
>��@ -DPVKLGLDQ� )�� /,%25 DQG VZDS PDUNHW PRGHOV DQG PHDVXUHV� )LQDQFH DQG 6WRFKDVWLFV� �����^��� �������
>��@ .ORHGHQ� 3�� DQG 3ODWHQ� (�� 1XPHULFDO 6ROXWLRQ RI 6WRFKDVWLF 'LmHUHQWLDO (TXDWLRQV� 6SULQJHU�1HZ <RUN �������
>��@ 0LOWHUVHQ� .�5�� 6DQGPDQQ� .� DQG 6RQGHUPDQQ� '�� &ORVHG�IRUP VROXWLRQV IRU WHUP VWUXFWXUHGHULYDWLYHV ZLWK ORJQRUPDO LQWHUHVW UDWHV� -� )LQDQFH� ������� �������
>��@ 0XVLHOD� 0�� DQG 5XWNRZVNL� 0�� &RQWLQXRXV�WLPH WHUP VWUXFWXUH PRGHOV� )RUZDUG PHDVXUHDSSURDFK� )LQDQFH DQG 6WRFKDVWLFV� �� ������� �������
>��@ 3URWWHU� 3�� 6WRFKDVWLF ,QWHJUDWLRQ DQG 'LmHUHQWLDO (TXDWLRQV� 6SULQJHU�9HUODJ� 1HZ <RUN��������
>��@ 5HLPDQ� 0�,� DQG :HLVV� $�� 6HQVLWLYLW\ DQDO\VLV IRU VLPXODWLRQV YLD OLNHOLKRRG UDWLRV� 2SHU�5HV�� ��� �������� �������
>��@ 5XELQVWHLQ� 5�� DQG 6KDSLUR� $�� 'LVFUHWH (YHQW 6\VWHPV� :LOH\� 1HZ <RUN� �������
��