"!
#$&%'$()%*&+,(-/.01234(5+21267-89%'$$:;+
$%<3$:=>39,:+,
?@BA"CD4EGFIHJA @LKMA NOEN?PD4EQCCR@BEKFGA @SD4A"F
TU$VXWYZM[WJ\]"^`_badc0^fehgc,a$ijkgflnmMo"p^fqLp4rXjXst^`gfiuBpbjvqnad_uB^`_Mtbadtbl"lwjxuBty^`_uBpbjg`^`c0^fuxmlwjkr{z|j}~adredyyrXa|^`c,eduB^fzdj
zdedg`tbeuS^fad_5ad}JadybuS^fad_blnmdjnlSyjxqX^fedg`gfo"pbjkr{j
}adrLc5tgfe|lqxed__bauJsJjijkrw^fzjxic,e|uBpbjkc,euB^`qnehg gf|\A"zjk_,o"^fuBp,uSpbjzdedg`tbed
uS^fad_ad}AQtrXadyjxed_qnedg`gadybuS^fad_bliM^flLuwahr{uS^`_b^`rLrXjndtgfedrL^fuS^ jxl$aMqnqktr\bahruSp^flqxelLjmXlwahtr{qxjnlad}qnad_zjkr{jk_bqnjMybeuLuLjXrL_bl
edrXj jxygfeh^`_bjniJ\$MtrXuSpbjXrLc,adr{j|mB^fu^flyr{adzjxiadrXijXrJad}qnad_zjXrXjk_bqXjah_bjJ}adrJuBpbj"adMFalLlBMFts^`_blLuwjL^`_ ddXc,aMijXg
elo"jXg`gel"}~adruSpbjuBrXjnj
ybedrXedc,jnuLjXr lLjXgfjxqnuS^fad_blad} dehrwrXadoed_bi0F
tbiiJ dBmded_bi0C"^fed_ dB\JC"pbjX_mMo jvijX_bj"_bjno
s^`_badc0^fedgJc,aMijXgflxmo"pbjXrXj0uBpbjqnedgfqktgfeuLjni,ahybuB^fad_2yrL^fqnjxlqnah_zjkr{|jlBc,aMauSpgf0uwa2uSpbj5]"gfe|qL$KMqLpbahgfjXllwadg`tbuS^fad_
ed_bi#r{jkc,edrLhehsgfmo j2jxzjX_>e|qLp^fjnzjadrXijXr5ad}qxad_zjXrXjX_bqxjuBo"a2o"^fuBpctbqLp&lSc,edg`gfjkr
^`_^fuS^fedg jkrwrXadrd\4NGauLedsgfm
lLadgfjXgfuSpbj"}ahrwctgfelJuwavijnuLjXrLc0^`_bjuBpbjqxad_blLuwed_utyed_bi0iado"_}eqxuwahr{lqLpbed_bj\J^`_bedg`gfmdedg`guBrXjnjehyyr{ae|qLpbjnl
edrXj5qxadc0ybedrXjni0o"^fuSp>rXjnlSyjxqnuuLa,lSyjxjni2ed_bieqxqXtrXeqn0qxedgfqXtgfe|uB^`_brXjXgfe|uB^fzjGrXa$a|uBc,jxed_$lB|tbexr{jxi0jXrLr{ahr
ad}edy
yrXa|^`c,euS^fzjadybuS^fad_zdedg`tbjnl}~adr0elwehc0ygfj0ad}rXed_biadc0gflwjkgfjnqnuLjni#ybedrXedc,jnuLjXrXle|qXrXalwlve>lLjnuad}rXjX_bjkc,jX_uBlx\
EyyrXad^`c,e|uB^fad_2ad}Ec,jXrL^fqneh_uSyJj5adybuS^fad_blGo"^fuBpuBpbjv_bjnoc,aMijXgfljxp^`s^fuLlahr{ijkrah}qnad_zjXrXjk_bqXj
ah_bjstbu
lSc,edg`gfjXr"^`_^fuB^fehgbjXrLr{adruSpbed_,yrXjnz|^fadtblSgf
jx^flLuS^`_bs^`_badc0^fedgMc,a$ijkgfln\
b #X ¡
¢£¤"¥w¡
¦X§©¨nªQ«5¬J~®n¨x¯«0°±²³®n´~¨x®²µ«5¶Q®nB·d~§µ2¨nª«|°b®n¸J¹¨xª«0¶®n«|ºn«§J¨¬b²³»¯Q«5°³±²2¼J«®n~¬b²³¨n~¬b«ºx«$·d¯®x~¨X¸>~º¼J«®x¬«$¼©´J¸>·²³»½·h¯»B²¨n~§µ
¨xª«0~§~¨nB²»J·d°bºx¨°±"ºn°b¾«5¼J¸J§ ²³¾B·¶"«®X±w«M·d¨n»~¸©®n«¶»~B·²³¨n~§µ2¶ °®n¨k±w°b»~~° ¹M·d°b§ºx~ºn¨n~§µ#°± ¶®x°b¶"°b®x¨n~°b§º~§©¨nª«0¯§"¼J«®n»~¸J~§µ
ºx«$·d¯®x~¨X¸¿²§ ¼+²³¾°b¯Q§¨xº5°³±5·²ºxª¿À0ªB·xª·xª ²§µ«>À0~¨xª9¨x~¾«bÁ0«§ ·h«>¨xª~º,¶"°b®n¨k±w°b»~~°~º0®n~ºnÂJ»~«ºxºn»~¸+§J¨x«®·xª ²³§µb«$²³´»~«
À0~¨xª>¨xª«5°¶¨n~°§>~¨nºx«»Ã±®x«µ²³®¼J»~«ºxº°±¨nªQ«,²b·d¨x¯ ²»~»~¸2°Ä·|·d¯®n~§µ#ºn¨²³¨n«º¼J¯®n~§µ2¨xª«0»~f±L«¨n~¾«°³± ¨nª«0°¶¨n~°§
Á-°À5«|¬b«®M¹
²³§~§JÅ §Q¨x«,§J¯¾´"«®°±
¶"°bºxºn~´»~«±w¯¨x¯®n«2ºn¨d²¨x«º0¨n®d²§ºx»½²³¨n~§µ©~§¨x°>ºx¯¶¶"°bºx«$¼+¶®x½·h«4¾°¬b«¾«§J¨nº°±¨nª«2¯§"¼J«®n»~¸J~§µ
ºx«$·d¯®x~¨X¸¿¼J«dÅ §Q«>¨xª«>¶"°b®x¨X±w°»~°"Æ º,·d°b¾¶ °ºn~¨n~°b§²³§ ¼+¬b²®xB²¨n~°§º$,Ç-°À/²ºxºn¯¾«¨nª ²³¨,¨xª«ºn«¶Q®nB·d«©¾°¬b«|¾«§J¨nº0²®x«
¼Ä«º·d®x~´ «$¼È¾9²³¨nª«¾+²¨x½·|²»~»f¸´¸É²ºx¨n°·xª ²ºx¨nB·¼Äfʯºx°§¿¶Q®n°·d«ºxº$Ë0ª«§Ì¨n«M·nªQ§B·²»~»~¸¹G²§J¸¶®x½·h«·nª"²§µ«·²§Ì´"«
¼Ä«$·d°b¾¶ °ºn«$¼©~§J¨n°©²§~§JÅ §~¨x«0ºn«$ͯ«§"·d«b0ΰb§ºx«$ͯ«§J¨n»~¸J¹²³§²®x¨nÃÅ·dB²»"¼J¸J§ ²¾½·v¶"°b®n¨k±w°b»~~°2À0~¨nª²·d°®n®x«ºn¶"°b§ ¼Ä§Qµ
ºx«$ͯ«§ ·h«>°³±¶Q®n°b¶"°b®x¨n~°b§É²b¼Ïd¯Qºn¨n¾«§J¨nº,²·nªQ«|¬b«º¨nª«>¼Ä¯¶»~B·²¨x°§¨d²ºnÂ0ΰb§Qºn~ºn¨x«§J¨n»~¸¹QB¼J«$²»~~Ы$¼ÑÅ §"²§ ·dB²³»¾9²³®XÒ
«¨nº2¾¯ºx¨0´ «²ºxºn¯¾«$¼¹À0ªQ«®n«²·d°§J¨n~§J¯°b¯ºÓ °À<°³±5ºn¯Q¶¶»~¸²§ ¼¼Ä«¾9²³§ ¼+¨n°+²ºxºn«¨xº>²®x®n~¬b«º²³§ ¼·d»~«$²³®nº²¨²
ºx«$ͯ«§ ·h«°±«$ͯ~»~´Q®n~¯¾Ô¶®xB·d«º,~§ºx¨²³§¨x»~¸9ͯ°b¨x«$¼,Õ¬B¼J«|§¨x»~¸¹J~¨,~º0¨xª~º,¬b«|®n¸²ºxºn¯¾¶¨n~°b§°±5·d°§¨x~§¯Q°b¯º2²§ ¼
±L®nB·d¨n~°§»~«ºnº¶®x½·h«,¾°¬«¾«|§¨xº~§&·d°§J¨n~§J¯°b¯ºv¨n~¾«¾+²®xÂb«¨xº
À0ªQ½·xª²»~»~°À0ºG±w°b®¨nªQ«,¼J¯¶»~B·²¨x~°b§©°±²§J¸¯§ ·h«®n¨d²~§
~§ ·h°b¾«2ºx¨n®n«M²¾¿2Ö-±×·d°¯®nºx«b¹¨x®²¼J~§µ9²³¨0Å § ²³§ ·dB²»¾+²®n«¨nº,¼Ä°«|º§°b¨°··d¯®·d°§¨x~§¯Q°b¯ºx»¸+~§¨xª«>ºx¨n®x½·h¨4ºx«§ºx«b
ز¨xª«®$¹J¨xª«2Ó °ÀP°±¾¯»~¨n~¶»~«ºn~§µb»~«¼J«$²³»~º0¨n®²³§ºn»B²³¨n~§µ©§J¨x°>¶®xB·d«º,~º0¾°Ä¼Ä«»~»«M¼©§+¨xª~º0ÀײM¸¾+²¨nªQ«¾9²³¨nB·²»~»Ã¸
Ù »~¨n«®x§ ²¨x¬«»~¸¹¼Ä«º·d®x~´~§µÌ¨nª«±w°®n¾+²¨n~°§°±2¯§ ·d«®x¨²³§O±w¯Q¨n¯®x«9¶Q®nB·d«º·²³§Ú´ «·²³®n®n~«$¼O°¯¨>¾¯"·nªÌ¾°b®x«ºn~¾Ò
¶Q»ÃÅ «M¼Ì¦X¾9²³µb~§«b¹¨xª ²¨²ºn«¨°±,±L¯¨n¯®x«+¶®nB·d«º~º°´¨²³§Q«$¼Ì´J¸Û²É¼J«$·d°¾¶"°bºn~¨x°§Û·d°b§Qºn~ºn¨x§QµÌ°±²ÉÜv«®n§°¯»~»~
ºx«$ͯ«§ ·h«b0Ç,²¨x¯®²³»~»¸J¹°§»~¸²»~~¾~¨n~§µ»B²¨n¨xB·d«2ºn¨n®x¯ ·d¨x¯®n«À0~¨xª9²§~§JÅ §~¨n«2§J¯¾´"«®°³±Üv«®n§°¯»~»~ºn¨n«|¶º,ºx¯ ··d««M¼Jº
¨x°9·²³¶¨n¯Q®n«©§ÄÅ §~¨n«©ºn¨d²¨n«|º4°³±5§ ²³¨n¯®x«bܯQ¨4ñ¨nªQº2»B²¨x¨nB·d«~º2·d°b§ºx¨n®x¯ ·d¨n«M¼Û·d°®n®n«|ºn¶"°b§ ¼J~§µ»~¸9¨x°+¨nª«·d°b§J¨x§J¯°¯º
±L®²¾«À°b®n¹´"°b¨xªÌ¾°¼J«»~º2¾9²M¸·d°~§ ·dB¼J«~§¿¨xª«»~~¾~¨M&ݲ®nµ~§ ²»~»~¸Íb¯Q°b¨n«M¼¹¨n«$·xª§Q½·|²»~»~¸°b§»~¸¾°¼J«»~º2À0ª«®n«
¬«®n¨xB·d«º,®x«$·d°¾´~§«,¶"«®n¾+²§«|§¨x»~¸²®n«#¨n®²·x¨d²´»~«bË0ªQº0¶Q®n°b´Q»«|¾ÔÀ5«²³ºnºn¯Q¾«²MÀײM¸ªQ«®n«b
Þßdàwá{âbD2edr{qLp4ãh$äM\JC"p^flzjkr{lS^fad_åJD2edr{qLp4æhçmãd|$äM\
èé"êÉë ì ßdíXíXî ïðLßdà~î½ñxòâóãd\
ô0á{õ0ö
ñx÷Lødí5ßdòbøùMú÷wßdíá{íXâs^`_bahc0^fedg$c,aMijXgBmahybuB^fad_,zdedg`tbe|uB^fad_mxadr{ijkrad}qnad_zjkr{jk_bqXjmdqnad_zjkr{jk_bqXj"ybeuLuwjkrw_\
û#jedrXjdrXeuwjk}ftg$uLaRed_bl"adü g`g`c,jkrmdKMuwjk}ed_5HJaMadJmed_bi?G^fjnuLjXrKMah_bijXrLc,ed_$_e\ a³\b}~adr qnahc0c,jX_uwlJed_bielLlB^flLuweh_bqnj\J^`_beh_bqX^fedg$lSty
yJadrXub}Ãr{ahc¿uBpbj"?jXtbuLlwqLpbj"adrXlwqLpMt_blLjLc,jX^ _blLqLped}~uXmnKMad_bijXrL}ahr{lLqLpt_bdlBsJjLr{jk^fqBpçeuF
pbjX^`_^flLqLpbjJMrw^fjxiMrw^fqLpBû^`g`pbjkg`c×lSBý_^fzdjXrXlS^ uxeü u
]ah__0^fldrXeuLjX}ftg`gfveqL$_bado"gfjxijni\b?a$qktc,jX_uuB|yjxlwjxu^`_2H þdC Aÿ \ Qgfjne|lwjeiiMrXjnlLl qxadrLr{jxlByJad_bi$jk_bqXjuwauSpbj
lLjnqxad_bi0edtbuSpbadrd\
Ù d· ¨n¯ ²³»~»¸J¹5²È´~§°b¾B²»¨x®n««É°±ÅÄ«M¼Ú»~«§µb¨xª ²¶Q¶®n°~¾9²³¨n«º©¨nª«É·d°b§J¨n~§J¯°¯ººx«¨°³±>¨n®d²b¼J~§µÌ°··d¯Q®n«§ ·h«º ²³§ ¼
ºx«$·d¯®x¨X¸+¶®xB·d«º²»~À0²M¸º0´J¸+·d°¬b«®x~§µ´¯Q¨,ºn°¾«-Å §~¨n«®d²§µ«°±ºx«$·d¯®x¨X¸¶®nB·d«º2À0~¨nª²¼J~º·d®x«¨n«©µb®x½¼ºx¨n®n¯"·d¨n¯®x«
°³±
·d°b§ºx¨²³§¨0»~°µ ºx¨n«¶Qº,²¨0¼J~º·d®x«¨n««MÍb¯Q½¼Äºx¨²³§¨¨n®d²b¼J~§µ©~§ºn¨d²§ ·h«º$ I~¨nª«¬«®n¸²¶Q¶»~½·|²¨n~°§>°³±
´~§°¾B²»J¨n®x««ºM¹
°§«~§«¬J~¨²´Q»¸¾¯ºn¨2« ²¾~§«>¨xª«²¶Q¶®n° ~¾9²³¨n~°b§Í¯ ²»~~¨X¸´J¸Ì·²®x«d±w¯»0·d°b§QºnB¼J«®d²¨n~°§Ì°±,¨nªQ«²¶¶®x° ¾+²¨x°§
®x«ºn¯»~¨À0~¨nª·xª ²§µ~§µ,¨x®n««0®x«dÅ §«¾«§J¨nºM¦X§ ·dB¼J«§J¨²³»~»¸2ºx¨²¨x«$¼¹¨x°2°b¯®°b¶Q§Q°§2ºn¯ ·xªºn~¾¯»B²³¨n~°b§º¼J°2§°¨¼Ä«º·d®x~´ «
ª°À ¼J~º·d°§J¨n~§J¯~¨n~«º2®n«ºx°b»~¬b«²³¨Å § ²§"·dB²»¾+²®n«¨nºMØ,²¨xª«®M¹ºx°b»~«»~¸¶®x°b¶"«®n¨x«|º>°³±×¨xª«²¶¶Q®n° ~¾9²¨x~°b§+¨nª«|¾«
²³®n«2ºnª°À0§G
Ë0ª«+²¶¶Q®n° ~¾9²¨x~°b§+®n«ºx¯»~¨~º« ¶®n«|ºnºn«M¼Ú´J¸¿¨xª«+¶ ²®d²¾«|¨n«®xº>¼J«ºd·d®n~´~§µÉ¨nª«¼J¸J§ ²³¾B·2®x«¶»~B·²¨x°§Ûºx¨n®²³¨n«µ¸
Ë0ª~ººn¨x¯ ¼J¸+~º0®n«ºx¨n®nB·d¨x«$¼¨n°©¨nª«!« Q²³¾~§ ²³¨n~°b§©°±¨nª«°¶¨n~°§¶®x½·h«°b§»~¸J
¦X§J¬b«|ºn¨n~µJ²¨n~°§º®x«¬b«$²³»X¹,¨xª ²¨+¨nªQ« ²·Í¯~®n«$¼'¼J«µ®n««Ì°³±>¶®n«M·d~ºn~°b§<~§I´~§°¾B²»~»Ã¸Û·d°b¾¶¯¨n«M¼Ú°b¶¨x~°b§I¶®xB·d«º~§
·d°¾¶ ²³®n~ºn°§¿¨x° ²·d°b§J¨n~§J¯°¯ºn»~¸Ì·²»B·d¯»B²³¨n«$¼È°b¶¨x°§Ì¶®nB·d«+¬b²³®n~«º©À0¨xªÌ¨nª«®n«dÅ"§«¾«|§¨°³±,¨nª«´~§°b¾B²»¨n®x««º
~§¿²´¯¾¶J¸9¾+²§§Q«®$#Ë0ª«>°¶¨n~°§¶®nB·d«|º>¯§Qºn¸J¾¾«¨n®x½·|²»~»~¸°bº·h»~»B²¨x«À0~¨nªÌ·xª ²§µ~§µ²¾¶Q»~¨x¯ ¼J«²®x°b¯§ ¼¨nªQ«
Üv»B²b·xÂ#Ò "·xª°»~«º0ºn°»¯Q¨n~°b§Ñ±w°b®0²Õ¯®x°b¶"«$²§¿·|²»~»°b¶¨x~°b§
Ö¯®¶ ²¶"«®2µb°J«º´ «¸°b§ ¼¨nªQ«2Å § ¼J~§µº,°³±¨xª«©« ºx¨n~§µ»~¨x«®²³¨n¯®x«~§ºn«¬«®²³»ÀײM¸Jº$$ Iª«§¿·h°b§ºx½¼Ä«®n~§µ»½²³¨n¨x½·h«
²³¶¶®n°J²b·xª«º0¶Q®n~¾9²³®n~»~¸>²º0²¾«$²§Qº5¨x°&¼J«|ºn~µb§+²»~~¾~¨¼J~ºn¨x®n~´¯¨x°§¹À5«¼J«ºx~®n«2¨nª ²³¨,²§+²¶¶®x° ¾+²¨x°§¾«|¨nª°¼
ºxª°b¯»B¼ª ²M¬«²·d°b§J¬«®nµ«§ ·d«ºn¶"««$¼©²º
±{²ºx¨²º¶"°bºxºn~´»~«À0ªB·xª2~º¾«M²ºn¯Q®n«$¼´J¸,¨xª«×¼Ä«µb®x««5°³± ·nª"²§µ«
À0~¨xª~¨n«®d²¨n«M¼
®x«dÅ §«¾«§J¨~§²´ºx°b»~¯¨x«¼ÄfÊ«®x«§ ·d«0°³±Ä´Q§Q°b¾B²³»¶®nB·d«0²§"¼4·h°b§J¨n~§J¯°b¯Qººn°b»~¯¨x~°b§ %Q¯®x¨nª«®x¾°®n«b¹MÀ0~¨nª2ºn¾°J°b¨nªQ§«ºnº
°³±v·h°b§J¬b«®xµb«§"·d«>¨xª«>²³¶¶®n° ~¾9²³¨n~°b§®x«ºn¯Q»¨xº,~¾¶Q®n°¬b«À0~¨nª+«$²b·xª~§ ·h®n«$²³ºn«©°±¨nª«#®n«dÅ §Q«¾«§J¨®n«µJ²®¼Ä»«|ºnº2°±¨nªQ«
µ¬«§¶"²®²³¾«¨x«®,·d°§ºn¨x«»~»½²³¨n~°b§G¹J«ºn¶"«$·dB²»~»~¸~§ ¼Ä«¶ «|§ ¼J«§J¨n»~¸°±5²ºx¶ «M·dB²»
·xª°B·d«¨n°©¨nªQ«ºn¨x®n~Âb«2¶®x½·h«b
%Q°b®ºx«¬b«|®²»!« Ä~ºx¨n~§µ»B²¨x¨nB·d«2²¶¶®x°²·nªQ«º5¨xª«,°®¼J«|®0°±
·d°§¬«®nµ«§ ·d«2~ººxª°À0§9²³§ ¼¶®x°¬b«$¼
Ù ºn¨n°§~ºnª~§µ»~¸¹b·h°b§JÒ
¬«®nµ«§ ·d«ºx¶ «|«$¼°±
´Q§Q°b¾B²³»»~¸·d°¾¶¯¨x«$¼°¶¨n~°§¶®nB·d«ºª ²º§°¨0´ ««|§9!« Q²³¾~§«$¼©¨x«$·xª§B·²³»»~¸ºx°©±{²®M¹²³»¨xª°b¯Qµbª
¨xª«®n«!« Ä~ºx¨nº²¬b²³ºn¨2¾9²³¨nª«¾+²¨x½·|²»"»~¨x«®²³¨n¯®x«>°§¿·d°b§J¬«®nµ«§ ·d«ºn¶"««$¼É~§¿·d°§§«$·d¨x~°b§ÉÀ0¨xª¿·d«§J¨x®²»v»~¾¨¨nªQ«°Ò
®x«¾ºM %¯Q®n¨nªQ«®n¾°b®n«¹®n«$²³ºn°§º±w°®¯§ºd²¨xºk±{²b·h¨n°b®x¸>·d°§¬«®nµ«§ ·d«0¶"²¨n¨x«®n§Qº5²®x«×¼J~ºd·d¯ºnºx«$¼vË0ª«§¹¨xª«0¶®x«ºn«§J¨d²¨n~°§
°³±5¾«¨xª°¼Jº2¨n°²b·xª~«¬«>¾°¼J«»~º2À0¨xªºn¾°J°b¨nªÉ·d°b§J¬b«|®nµb«|§ ·d«¶"²¨n¨x«®n§Qº±w°»~»°À0ºMÇ0« ¨M¹
²¾°¼J«|»
À0~¨xª¿²ª~µbª«®
°®¼J«®2°³±v·h°b§J¬b«®xµb«§"·d«²§ ¼ ºn¾°J°¨nª·d°b§J¬«®nµ«§ ·d«¶ ²³¨n¨x«®n§É~º,¶®x«ºn«§J¨x«$¼Ý+°b®x«°¬b«|®$¹²»~»¨xª«©¶®n«ºx«§J¨n«$¼È¾°¼J«»~º
ª ²M¬«0¨nª«2ºd²¾«,·h°b¾¶Q¯¨²³¨n~°b§©ºn¶"««$¼µb~¬«§¨xª«,ºd²¾«0¨x®n««2®x«dÅ §«¾«§J¨$¹J´"«$·²³¯ºn«2°b§Q»¸¨xª«0±L°b®n¾¯Q»½²³º¨n°>·²³»B·d¯»B²¨x«
¨xª«2¨n®n«|«¶ ²®d²¾«¨n«®xº,·xª ²§µ«b
Üv¯¨0~¾¶"°b®x¨²§J¨x»¸J¹»B²¨x¨nB·d«²¶Q¶®n°J²b·xª«º0«ºx¨²³´»~ºxª+¾¯ ·xª¾°®n«¨nª ²³§ ²#¬b«|ªB·d»~«¨n°²b·xª~«¬«²·h«®n¨d²~§»~~¾~¨¼J~ºn¨n®xÃÒ
´¯Q¨n~°b§+°±G±w¯¨x¯®n«>²³ºnºx«¨2¶®nB·d«ºM0Á0«®x«b¹"¨nª«>²³®n´~¨x®²µ«2®n«»B²¨x~°b§ºxª~¶º0À0ªB·xª~¾¶»~¸©¨nª«2®n«|¶»~½·|²¨n~§µ¶"°b®x¨X±w°»~°·²³§
´"«&·xª ²³®²·d¨n«®x~ºn«$¼O·d»~«$²®x»¸J¹"À0ª«®x«$²º¨xª~º2¨nª«°®n«¨xB·²»5·h°b§ºx¨n®n¯"·d¨n~°b§Ì~º2ºn°¾«À0ª"²¨4·h°b§ ·d«M²»~«$¼É~§¨nªQ«&·d°§J¨n~§J¯°b¯º
ºx«¨n¯¶GÛÇ0°¨²´Q»¸J¹G¨nª«ºx«+¶®n°¶ «®x¨n~«º²³®n«®x«¨²³~§«$¼O«§J¨n~®n«|»¸Ì~§Ì¨xª«+§«À0»~¸¿«|ºn¨²³´»~~ºnª«$¼O¾°Ä¼Ä«»~ºÀ0~¨nªÌ~¾¶®x°¬b«$¼
²³¶¶®n° ~¾9²³¨n~°b§©¶®x°b¶"«®n¨x«|º$& %Q®n°b¾ ¨nªQ«¨nª«|°b®n«|¨nB·²»v¶ °§J¨0°³±¬J~«À/²³»»¨xª«·d°b§ºxB¼J«®n«M¼¾°Ä¼Ä«»~º,·²§´"«¯ºx«$¼§ÄÒ
¨x«®·xª ²³§µb«$²³´»~¸Î°b§Qºn«$ͯ«§J¨x»¸J¹À5«ª ²M¬b«0ºxª°À0§ª°ÀÚ¨nª«2²¶¶Q»B·²³´~»~¨X¸2°³±»B²¨x¨nB·d«2²¶¶®x°²·nªQ«º×·|²§´"«,~¾¶®x°¬b«$¼
¨x®n«¾«§ ¼J°¯ºn»~¸J
%G~§ ²»~»~¸J¹À«,µb~¬«5ºx°b¾«§J¯¾«|®nB·²»Q« ²¾¶Q»«|º
¨x°2¯§ ¼J«®x»~§Q«,¨nªQ«0ºn¨n®x«§µ¨nª+°±¨nª«0§Q«À ²¶Q¶®n° ~¾9²³¨n~°b§ Ù ¾«|¨nª°¼
®x«$·d«§J¨n»~¸Ì¶®x«ºn«|§¨x«$¼O´J¸¿Ü®x°²¼J~«²§ ¼ '«¨x«¾¶»~« (*),+-±w°b»~»~°À0º$¹"À0ª«|®n«°¶¨n~°§¿¶®xB·d«º²®n«·h°b¾¶Q¯¨n«$¼±L°b®²+»B²®xµb«
ºd²¾¶Q»«0°³±
®d²§ ¼J°¾Ô¶ ²³®²¾«¨n«|®nº,²§"¼¨nªQ«§¨nªQ«4®x«»B²¨x¬«2ºn¨²³§ ¼²³®¼+¼J«¬JB²¨x°§¨x°>¨xª«2¨n®n¯Q«ºn°»¯Q¨n~°b§º0·²³»½·h¯»B²¨n«M¼
²³§ ¼Ì·d°b¾¶ ²®x«$¼¨n°¿·d°¾¶¯Q¨²¨x~°b§¨x¾«>À0~¨xªÌ§"·d®n«$²³ºn~§µ®x«dÅ §«|¾«§J¨$
.®²¶QªB·²»~»~¸+¨#º#ºnª°À0§Ì¨xª ²¨¶Q®n«¬J~°b¯º
¾°Ä¼Ä«»~º0ºn¨²M¸+´"«ª~§ ¼É¼J®²³ºn¨x½·|²»~»~¸À0~¨nª®x«ºn¶"«$·d¨©¨n°+²b··h¯®²·d¸Ö§É¨n°¶¹"¯ºn~§µ¨nª«©º²³¾«ºd²¾¶Q»«2°³±¶"²®²³¾«¨x«®nº
~¨~ººxª°À0§Ì¨nª ²³¨>°¯®¨x®n««+¾°¼J«|»º2¶"«®X±L°b®n¾ ´ «¨x¨n«®¨xª ²§Û²³»~»5¶®x«¬J~°b¯º»B²³¨n¨nB·d«+²¶¶Q®n°²·xª«º©À0ª«§ ·h°b¾¶Q¯¨n~§µ
Ù ¾«®x½·|²§¨X¸J¶ «2°¶¨n~°b§¶®nB·d«|º$5Á-«®n«#º¨nªQ«4ºx¶ «M·dÃÅ·²¨x¨n®d²b·d¨x°§°³±±{²³ºn¨0¶"«®X±w°®n¾§Qµ¾°¼J«»~ºM¹´ «M·²¯ºx«>´Q§Q°b¾B²³»
¾°Ä¼Ä«»~º5²¶¶Q®n° ~¾9²¨x«0¶®nB·d«|º0±w°®0À0ªB·nª+« ¶»~B·d~¨v±w°b®x¾¯»B²º5·²³§§°¨0´ «¼J«®x~¬b«$¼+~§¨nª«>·h°b§J¨n~§J¯°b¯Qº5ºx«¨n¯¶G
Ù ´~§°¾B²»G°b¶¨x°§¿¶Q®nB·d~§µ+¾°¼J«»À0²º2Å"®nºn¨©¼J«¬b«|»°¶ «M¼Éºn~¾¯»~¨²³§«°b¯Qºn»~¸+´¸Éΰ
¹Ø0°ºnºM¹
²³§ ¼¿Ø-¯´~§ºx¨n«~§ (*/)01 Î0Ø,3
Ø 20²³§ ¼Ø-«§ ¼J»~«¾+²§9²³§ ¼Ü²®x¨n¨n«|4® (*/)0-kÎ0Ø,Ø'¶®x«ºn«§J¨x«$¼É¨nª«±w¯§ ¼²³¾«§J¨d²»«$·d°§°b¾B·,¶Q®n~§ ·d~¶»~«º°±°b¶¨x~°b§
¶®xB·d~§µ,´J¸²®x´~¨n®²³µb«5·h°b§ºx½¼Ä«®²³¨n~°b§º~§¨xª«5¾°bºx¨ºn~¾¶»~«ºx¨¾+²§§«|®$Üv¸²¶¶»~B·²³¨n~°b§2°±²,·d«§J¨x®²»Q»~¾¨"¨nªQ«°b®x«¾
¨xª«¸Ì¶®x°¬b«$¼È¨nª ²³¨>¨xª«~®¾°¼J«»v¾«®xµb«º©~§¨x°¨nªQ« Üv»B²b·xÂÛ²§"5
¼ "·xª°»«|º>¾°Ä¼Ä«»5À0ªQ«§Û¨xª«+¨n~¾«ºx¨n«¶º´"«¨XÀ««§
ºx¯ ··d«ºxºn~¬b«#¨n®²¼J~§µ©§Qºn¨²³§ ·d«º2²¶¶Q®n°²·xªÐ«®x° Ù ¼¼J~¨x°§ ²»~»~¸J¹$¨xª«2¾°¼J«»"À0²º¯ºn«$¼¨x°>«¬b²³»~¯ ²¨x« Ù ¾«®nB·²³§¨X¸J¶"«
°¶¨n~°b§Qº×²³§ ¼°b¶¨x°§º°b§²ºxºn«¨xº,À0~¨nª·d°§¨x~§¯Q°b¯º,¼J~¬JB¼J«§ ¼Ñ¶ ²M¸J¾«§J¨xº$
¦X§É¨nªQ«¾«M²§J¨n~¾«¹~§§J¯¾«|®²´Q»«©·d°b§J¨n®x~´¯¨n~°§º¨n°+»B²¨x¨nB·d«²³¶¶®n°J²b·xª«º#ª ²M¬b«´"««§O¶¯´»~~ºnª«M¼Ë0ª«®x«d±w°®n«b¹À«
¾¯Qºn¨« ·d¯ºn«#¨nª ²³¨0§°b¨,²³»»°³±
¨xª«¾)·²§+´"«¾«§J¨x°§«$¼©ª«|®n«b
J²®x®n°À ²³§ ¼ÚØ0¯ ¼¼ ( -,·d°b§Qºn¨n®x¯ ·d¨x«$¼I²É´~§°¾B²»¾°¼J«|»5À0ª«|®n«¨xª«Å ®nºx¨¨XÀ5°É¾°¾«§J¨nº2°³±4¨nªQ« ¼Äºd·d®n«|¨n«Ì²§ ¼
·h°b§J¨n~§J¯°b¯Qº4¾°¼J«»
·h°b~§ ·dB¼J«©´J¸¿·d°b§Qºn¨n®x¯ ·d¨x°§ %Q¯®n¨xª«®x¾°b®x«b¹¨xª«>¶Q®n°b´"²´~»~~¨X¸9¾«$²ºx¯®n«#º2«$ͯ ²³»¨n°+°§«>ª"²»Ã±
"ħ"·d«>~§¨xª«Î0Ø,Ø×Ò{¾°¼J«»X¹J¨nª«©¬b²®xB²§ ·d«©°±¨nª«²ºxºn«¨®x«¨n¯Q®n§Û·d°§¬«®nµ«º¨x°Àײ³®¼Jº¨nª«¬b²³®nB²§ ·h«~§¨nª«Üv»B²b·xÂÒ
"Q·nªQ°b»~«º¾°Ä¼Ä«»b°§»~¸~§>¨xª«0»~¾~¨$¹d¨nªQ«¸&·h»½²³~¾<¨nª ²³¨¨nª«|®v¾°¼J«»ºnª°¯»B¼©ª ²M¬b«0
² b´"«¨n¨x«® >·h°b§J¬b«®xµb«§"·d«,´"«ª ²M¬J~°b®M
Ë0ªQ«4«|ºnºn«|§ ·d«°±´ «|¨n¨n«|®4·d°§¬«®nµ«§ ·d«©´"«ª ²M¬J~°b®0~º5§Q°b¨0¨²·x»~«$¼
Üv°¸J»~« (-
·h°b§ºx¨n®n¯"·d¨n«$¼O²¨x®n~§°b¾B²»"»½²³¨n¨x½·h«b¹"À0ªB·nªº-Å «$¼+¯¶É¨n°ºn°¾«²®x´~¨n®²³®n¸+¶ ²³®²¾«¨n«|®,»B²¾´¼²¹JÀ0ªQ½·xª
~º,¼J«|¨n«®x¾~§«$¼ª«¯Q®n~ºn¨x½·|²»~»~¸ Ù »~¨nªQ°b¯µª¨nªQº0¾°¼J«»»B²b·xÂJº,²¯Q§~¬b«®xº²»Gºn°»¯Q¨n~°b§¹JªQ«4®x«$²»~~Ыº~§ ¼J««$¼¨xª ²¨0¨xª«®n«
²³®n«2¶"°b¨n«|§¨xB²»~~¨n~«º¨n°~¾¶®x°¬b«0»B²¨x¨nB·d«²¶Q¶®n°J²b·xª«º´¸²³§+§Qµb«§~°¯º×·xª°B·d«°±¶ ²®d²¾«¨n«®xº$
Ö¾´"«®nµ (-¼J«$¼J¯ ·h«$¼¿²#À0ª°»«-±{²¾~»¸#°±
»B²³¨n¨nB·d«2¨x®n««º¯ºn~§µ©¨nªQ«4¨x«$·xª§Bͯ«°³± .²³¯ºn#º ÄÁ0«|®n¾~¨x«4ͯ ²b¼Ä®²¨x¯®n«²³º
ºx°b»~¯¨x°§+¨n°¨nªQ«>´ ²·xÂÀ0²®d¼®n«$·d¯Q®nºn~¬«§J¨x«µb®d²¨n~°§9¶Q®n°b´Q»«|¾¿ -§J±w°®n¨n¯Q§ ²¨x«»~¸¹Q¨n®n«|«ºÀ0~¨nª±w°¯®,°®0¾°b®x«2¬b«®x¨nB·d«º
¼Ä°>§°¨0®n«$·h°b¾´~§«¶®n°¶ «®x»~¸9²§"¼~§J¨n«|®n¶"°b»B²¨x°§¾«¨nª°¼Jº¾¯ºx¨5´"«²¶¶»~~«$¼Ñ¨n°Âb«|«¶²©¨n®²·x¨d²´»~«2µb®xB¼°±²ºxºn«¨
¶Q®nB·d«ºM,Ç0°¨²´Q»¸J¹ªQ«®n«$·h°bµb§QÐ|«º2¨nª ²³¨,«¬«§À0~¨nª¿² ¨xª°b®d¼J«® .#²¯ºxº ÄÁ-«®n¾~¨n«Ïd¯¾¶¶®x°Ä·h«ºnº¨xª«»~°·²¨x~°b§+°±
¨xª«2« «®·hºx«>¶Q®nB·d«À0~¨nªQ§¨nª«2¨x®n««ºx¨n®x¯ ·d¨n¯Q®n«©¾9²M¸·²¯ºx«¨n®x°b¯´Q»«
Ë0B²³§ (*)-¶®n°¶ °ºn«$¼Ú´Q§Q°b¾B²³»²³§ ¼O¨n®x§Q°b¾B²³»
¾°¼J«»~º>À0ªQ«®n«¨xª«+¾°¼J«»¶ ²³®²¾«¨n«|®nº²®x«¼J«®n~¬«$¼Ú²º©¯§Bͯ«
ºx°b»~¯¨x°§º¨n°«$ͯ ²³¨n~°b§+ºn¸Jºx¨n«¾º$¹«|ºn¨²³´»~~ºnª«$¼9±w®n°¾Ôºn¯ 9·h«|§¨2·d°b§"¼J~¨n~°b§º0¨x°²b·Í¯~®x«4À«$²³Â9·d°§¬«®nµ«§ ·d«¼J¯«#¨n°
¨xª«
§"¼J«´"«®nµ¨nª«°®n«¾É¹ºx¯¶¶»~«¾«§J¨n«$¼+¨x°¯ºn«©®n«|¾9²~§~§µ©¼J«µb®x««º2°±±w®x««$¼J°¾)¨n°«$ͯ ²»~~Ы±w¯®x¨nª«®2¾°b¾«|§¨xº
°³±5¨nªQ«&·d°§J¨n~§J¯°b¯º²³§ ¼¿¼Äºd·d®n«|¨n«²ºxºn«#¨ ļJ~ºx¨n®n~´¯Q¨n~°b§ºM -§J±w°b®x¨n¯§"²¨n«|»¸J¹¨xª~º,~§J¨n«®x«ºn¨x§Qµ¿·d°b§J¨x®n~´¯¨x°§»B²b·xÂJº2¨n°
ºx¯¶¶"°b®x¨0¨nª«B¼J«M²º0´J¸¾+²¨nªQ«¾9²³¨nB·²»²³®nµb¯Q¾«§J¨nºM
Ù ¨x°b¨²³»~»¸¼JÃÊ«®n«|§¨©²¶¶®x°²·nªÉ¨n°+~¾¶Q®n°¬b«2¨xª«²b··h¯®²·d¸¿°³±×´Q§Q°b¾B²³»¾°¼J«»~º0À0²º~§ ²³¯µb¯®d²¨x«$¼+´J¸¿Á0¯Q»»²§ ¼
Pª~¨n« (0-n¹À0ª°¨n®²³§ºX±L«®n«$¼É¨xª«·d°§¨x®n°» J¬b²®x½²³¨n«2¨n«M·nªQ§BÍb¯Q«©±w®n°¾)¨nª«Ý+°b§J¨x«&Î0²³®n»~°¾«¨nª°¼ Ù §É~§J¨n«®x«ºn¨x§Qµ
²³¶¶®x°²b·xª¹J´Q¯¨0~¨,¼JÃÊ«®xº,«§J¨x®x«»~¸±w®x°b¾Ô¨xª«2»~§Q«4°³±
¨xª°b¯µªJ¨5¶"«®nºx¯«$¼ª«®x«b
Ç-¯¾«|®n°b¯Qº&²¼Ïd¯ºx¨n¾«|§¨xº>ª"²M¬b«´"««§P§J¨x®n°¼J¯ ·d«M¼Û¨x° ²³¶¶»~¸Ì»B²¨x¨nB·d«9²³¶¶®x°²b·xª«|º¨n°Ì¬b²®x~°b¯º¨X¸J¶"«º°³±,°b¶¨x~°b§ºM
Ë0ªQ«®n«2~º¨nªQ«4´Q®n°²¼#Å «»B¼°³±« °b¨x½·-°b¶¨x~°b§ºM
ΰ ¹JØ0¯´~§ºx¨n«~§ (- ¶®x«ºn«|§¨x«$¼²³§9²b¼|Ïd¯ºn¨x¾«§J¨±L°b®¨xª«,¬b²³»¯"²¨n~°§
°³$± '°À0§ ²§ ¼ °b¯Q¨·|²»~»~º$+Á0¯»~»5²³§ ¼ Iª~¨n« ( )-5¾°¼JÃÅ «$¼+¨xª«°®n~µb~§ ²³»Î0ØØÒX¾°¼J«»~»
±w°b®¨xª«¶®x½·h§Qµ°±,¶ ²³¨nª
¼Ä«¶ «|§ ¼J«§J¨2« °b¨x½·°b¶¨x°§º0´J¸+»~§«M²®0°b®2ͯ ²b¼J®d²¨x½·~§J¨n«®x¶ °»B²¨n~°§0Ø0«$·d«|§¨x»~¸¹"Ϊ«¯Â²§""
¼ !°b®xºn¨ ( ),+-¶®x«ºn«|§¨x«$¼
²¾°¼J«|»5À0ª«|®n«+¨nª«+¶ ²M¸°ÊI°#± °J°bÂJ´ ²·xÂ̰b¶¨x°§º©¨xºn«»Ã±2º©¾°¼J«|»»~«$¼É~§Ú²»B²¨x¨nB·d«b¹v¨nªJ¯º®x«ºn°»¬J~§µÉ¨nª«+¶ ²³¨nª
¼Ä«¶ «|§ ¼J«§ ·h¸ Iª«®x«$²º¨xª~º¶ ²³¶ «®¼J°J«º§°¨±L°Ä·h¯º5¼J~®n«M·d¨n»~¸©°b§©¨nª«0¶®xB·d~§µ°±·d°b¾¶»~« >¶ ²M¸°Ê¨nª«¾«º$¹À«0¬~«À
°¯®,·d°§¨x®n~´¯¨x~°b§9²³º,²©ºn¨d²®n¨x~§µ>¶"°b~§J¨±w°®5¨xª«>¼J«®x~¬b²¨n~°§°±¾«¨xª°¼JºÀ0¨xª+ºn¯¶"«®n~°®,²b··d¯Q®²b·h¸9¨xª«®x«b %¯®x¨nª«|®
«!Ĩx«§ºx°§º¨n°2¨nª«-Å «»B¼©°±»B²¨x¨nB·d«,²¶Q¶®n°J²b·xª«º~§J¬b°»~¬b«¨nª«0¨x®²§QºX±w«®¨n°2¨xª«,¶®xB·d~§µ2°±
¼J«®x~¬b²¨n~¬«×·d°§J¨n®²·d¨nºÀ0~¨nª
¾¯Q»¨x~¶»~«
¯Q§ ¼J«®x»¸J~§µ2ºn«M·d¯®n~¨x«|º 1 ºn««2Á0« (*)0- 2Ö¨nªQ«®×²³¯¨nªQ°b®nº¼J«¬b°¨n«0®x«ºn«$²³®·xª¨x°2¨nª«,·d°§ºn¨x®n¯ ·h¨n~°b§°± ºn~¾¶»~$« ´Q§Q°b¾B²³»»½²³¨n¨x½·h«º$¹G¨nª ²³¨2º·d°§ºn¨x®n¯ ·h¨n~°b§O¶®n~§ ·d~¶»~«º2À0ª«|®n«¶Q®nB·d«¶ ²³¨nªº#®n«$·d°¾´~§«©¶®n°¶ «®x»~¸¿«¬«§ÌÀ0ª«§É¾°®n«
·h°b¾¶Q»!« ¾°¼J«»~º,ºx¯ ·xª ²³º¾°¼J«»~ºÀ0¨xªºn¨d²¨x«¬b²®n¸J~§µ¬b°b»B²³¨n~»~¨X¸Ñ±w¯§ ·h¨n~°b§º²³®n«·d°b§ºxB¼J«®n«M¼ 1 « µ"ºx««9Ç-«»~ºn°b§G¹
ز¾+²ºnÀ0²¾©¸ ( )-{¹ (*)- 2|
&
%('¡ ))¥B *P£ )¿ '+̤¡¥B£¡
£ ,-J ./(+0&¡
132¿¤¡ ,| "£¤"¥w¡4
) ¥B¤"5¥ )76+,©¡ -8,$9¡ /(+:) +7;¥w¡
£ ,
6.J¥w¤ +<.)7) ¡ .¤ '+,
Ö§ ·h«&²³µ²~§¹"Ü»B²·n²³§ ¼ "·xª°»«|º4²ºxºn¯¾«¨nª"²¨,¨x®²¼J~§µ9²³¨0Å § ²³§ ·dB²»G¾9²³®nÂb«|¨nº0¶®x°Ä·h««$¼Jº>·h°b§J¨n~§J¯°b¯Qºn»~¸+§+¨x¾«b¹
¨xª«¾+²®n«¨0®²³¨n«°³±§J¨x«®n«ºx¨2®,~º,·d°b¾¾°§»~¸ÂJ§°À0§¿²§"¼Å Ä«M¼+°¬b«®¨n~¾«¹J¶ ²M¸J¾«§J¨nº°b¯Q¨,°±¯§"¼J«®n»~¸J~§µºn«M·d¯JÒ
®x~¨n~«º$¹J« µ"¼J~¬J½¼Ä«§ ¼JºM¹ ¼J°©§°b¨0!« Ä~ºx¨$ "J¨x°Ä·xÂÒX¶®nB·d«>¼Ä¸§"²¾B·dº0²®x«¼J«º·h®n~´ «M¼´J¸
1 4 2
=> 15? 2A@3B > 15? 2 = ?DCFE > 1G? 2 =H 15? 2
À0ªQ«®n«2®0~º5¨xª«2~§ºn¨d²§J¨²³§«°b¯Qº0«¶ «$·h¨n«$¼+®x«¨n¯Q®n§°³±¨nª«2¯§ ¼Ä«®n»~¸J~§µ&²³ºnºx«¨ > f±~¾¾«$¼JB²³¨n«»~¸2¨nª«2®xºxÂÒ{§«|¯¨n®d²»~~¨X¸
²³®nµ¯¾«§J¨°³±5Á,²®x®n~ºn°§9²³§ "
¼ I»~ºxÂb² (!- ~º0¯ºn«M¼¹ E ~º¨nª«#§Qºn¨²³§J¨n«§«|°b¯º¬b²®nB²³§ ·d«,°³±
¨xª«®n«|¨n¯®x§¹ ²³§ ¼+¼~º,²
ºx¨²³§ ¼²®d¼ .²¯ºxº I~«§«®0¶®x°·d«ºnºM
P¨xª~§¨nªQ«~®4¾°¼J«»X¹²ª«$¼Äµb«¶"°b®n¨k±w°b»~~°·²³§Ì´ «·d°b§Qºn¨n®x¯ ·d¨x«$¼Û·d°§J¨²~§~§µºn°»«|»¸¨nª«©¯§ ¼J«|®n»~¸J§Qµ ²³ºnºx«¨ > ²§ ¼
²º²M¬J~§µbº²··d°¯§J¨À0¨xªÛ®xºxÂJ»«|ºnº´"°b®x®n°À0~§µÌ²§ ¼O»~«§ ¼Ä§Qµ ²³¨ BJ¹À0ªB·xªÚ¶ «®k±w«$·d¨x»~¸ ®x«¶»~B·²¨x«º¨xª«+¬b²»~¯«°±>²
Õ¯Q®n°b¶"«$²³§ ·|²»~»°b¶¨x~°b§²¨«$²b·xª~§ºx¨²³§¨°±¨x¾«b¹ª ²M¬J§Qµ>ºx¨n®n~«4¶Q®nB·d« ²§"¼¨n~¾«,¨n°¾+²¨x¯®n~¨X¸ ?
¦X¨~º¨nªQºG«$ͯ~»~´Q®n~¯¾ ·d°§§«$·h¨n~°b§À0ªQ½·xª2®n«|ºn¯»~¨nºG~§4¨xª«5Üv»B²b·xÂ#Ò "·xª°»~«º
¼JÃÊ«®n«|§¨xB²»b«MÍb¯"²¨n~°§4À0ªQ«®n«¨xª««¶"«$·d¨x«$¼
®x«¨n¯®x§°b§¨nª«#°b¶¨x°§~º0« ¶®x«ºnºx«$¼~§¨n«®x¾º0°±¨nª«#°b¶¨x°§¶®xB·d«2±w¯§"·d¨n~°b§+²§ ¼~¨nº0¶"²®n¨xB²»¼J«®x~¬b²¨n~¬«º$
C B > C E > @ B 1 2
?
>
>
Ë0ª«2ºx°b»~¯¨n~°§¨n°¨xª«>¼JÃÊ«®n«|§¨xB²»«$ͯ ²³¨n~°b§+À0~¨nª+´"°b¯§ ¼Q²®n¸+·d°b§"¼J~¨n~°b§
~ºµb~¬b«§+´J¸¨nªQ«>Ü»B²·nÂÒ "·xª°b»~«º0°¶¨n~°§¶®x½·h§Qµ©±w°b®x¾¯»B²
15? >
1 2
@
2
= @
> 1 = 2 !#"%$'&()*%+', 1 = (0/21 143 $58 )6*7+ 2 C 1 B C E 2 1 . ? 2 E:9 ; ? @ ( > 1 32 -̲º0¶"²M¸b°Ê±w¯§
·d¨x~°b§
E - . ? 2
À0ª«|®n« ~º5¨xª«>·d¯¾¯Q»½²³¨n~¬b«0ºx¨²³§ ¼²®d¼§°b®x¾9²»¼J~ºn¨x®n~´¯¨x~°b§±L¯§ ·d¨x°§
Ë®²³§ºX±L«®n®x§Qµ¨nªQº±w®d²¾«À5°®n§J¨x°¨nªQ«>ºx¾¶»~ñw¸J§Qµ´~§°b¾B²»ºn¨x®n¯ ·h¨n¯®x«§"¼J¯ ·d«º#ºn«¬«®²³»5²b¼Ïh¯ºn¨x¾«§J¨nºMÁ0«®x«b¹
¨xª«¿¾°¼J«»,«|¬b°b»~¬«ººx¨n«¶'´J¸ ºx¨n«¶/²³¨9·d«|®n¨²³~§ ºx¶ °¨nº+°±>¨x~¾« ?<=?> ? ?@A CB ? ·nª"²®²·d¨n«|®n~ºn«ºÌ²ÉÅÄ«M¼
»~«§µ¨nª+°±¨n~¾«0¶"²ºnºx~§µ´J¸´"«¨XÀ5«|«§ºn«MÍb¯Q«§J¨nB²»G¾°¼J«»"«¬b«|§¨xº$0Ý+°¼J«»«|¬b«§J¨nº2·d°b§Qºn~ºn¨°±
¶Q®nB·d«>·xª ²³§µb«º0~§+¨nªQ«
¯§"¼J«®n»~¸J~§µºx«$·d¯®x~¨X¸ ²³§ ¼°¶¨n~°§¶®nB·d«&ΰ§ºn~ºx¨n«§J¨n»~¸J¹¨nªQ«>®x«¶»~B·²¨x§Qµ9¶"°b®x¨X±w°»~°®x«$ͯ~®n«º²b¼|Ïd¯ºn¨x¾«§J¨xº,¨nªQ«®n«
°§»~¸,ΰ§J¨n~§J¯°b¯º0¶"²®²³¾«¨x«®nºM¹X «b( B²§"¼ E ? ²®x«¨n®d²§ºx»½²³¨n«$¼¨n°¶ «|®4¶"«®x°¼¬b²®x½²³´»~«Eº BD ¹ E 1 GFH122Ϋ§J¨n®d²»
ºxµ§ÃÅ·²³§ ·d«¼J«¬b°»~¬b«º¯Q¶ °§¿¨xª«·d°b§Qºn¨n®x¯ ·d¨x°§Ì°±5¶Q®nB·d«¾°¬«¾«|§¨xº$& .2«|§«®²³»~»¸J¹"À0~¨nª~§Ì²»~»´~§°b¾½²³»¾°¼J«»~º
¨xª«2ºn¨n°·xª ²³ºn¨nB·2¶Q®nB·d«´"«ª ²M¬J~°b®~º0¾°¼J«»~»~«$¼©´J¸
> 1G?
1 +2
À0ª«|®n«
< 7IJ%12@
> 1G? 2 K 1 122L = 1 12 <
L
MIJ120¼Ä«§°b¨x«º2¨nª«©²ºnºx«¨2¶®nB·d«©²¨,¨x~¾« ?< M² ±w¨x«®NI¯¶JÒX¾°¬«¾«§J¨xº5~§²O1QÒ{ºx¨n«¶É´~§°¾B²»¨x®n««¹À0~¨nª
? ²º0»~«§µ¨nª+°±¨nª«#¨n~¾«2²~º$ Iª«|§À5«¼J«hÅ §« P D < ²³º0¨nª«®d²§ ¼Ä°b¾»~¸¼J~ºn¨x®n~´¯¨x«$¼+°b§Q«dÒ{¶"«®x°¼®n«|¨n¯®x§±w°®
;
> 15?<
¨xª«2¯§ ¼J«®x»~¸~§µ²³ºnºx«¨,À«2ª M² ¬b«
2 À0~¨nª+¶®n°´ ²´Q»~~¨X¸
S 1 12
1 2
P D < @RQ K= 11 112
À0~¨nª·d°b¾¶»~«¾«|§¨d²®x¸>¶®x°b´ ²³´~»~¨X¸ TUS 1 12VXW 1 12ZY @ [ 1
À0ª«|®n« 1ɼJ«§°¨n«º¨nª ²³¨0¨nª«|ºn«¶ ²³®²³¾«¨x«®nº0´"«»~°b§µ¨x°&²©ºx¶ «M·dÃÅ·2´~§°¾B²»¨x®n««©²§ ¼+·xª ²§Qµb«2À0~¨nª+®n«dÅ"§«¾«|§¨\1
Ù ¶ ²®x¨,±L®n°b¾ ¨nª"²¨$¹
¨nª«ºx«¶ ²®d²¾«¨n«®xº,®x«¾9²³§É·d°b§ºx¨²³§¨2¨xª®n°¯µbª°¯¨4²Ñ¨n®x««bÕ¬JB¼J«§J¨n»~¸J¹ ²ºn¶"«$·dB²»°®¼J«®x~§µ+°±
¯¶ IJ§"¼¼J°À0§ J¾°¬«¾«§J¨xº¼Ä°«|º,§°¨,²MÊ«$·d¨,¨x«®n¾~§ ²»²ºxºn«¨¶®nB·d«ºM
I~¨xª~§Û¨nªQ«ºn«Ì¾°Ä¼Ä«»~º$¹5²Èª«$¼Jµ«¿¶"°b®x¨X±w°»~°Ì·²³§I´ «É·d°b§ºx¨n®x¯ ·d¨n«M¼'À0ªB·nªP¶ «®k±w«$·d¨x»~¸I®n«¶Q»B·²³¨n«º+¨xª«¬b²»~¯«+°³±&²
Õ¯Q®n°b¶"«$²³§ ·|²»~»5°¶¨n~°§ ²¨©«$²·nªÚ¼J~º·d®x«¨n«¶ °§J¨©°±2¨n~¾« ? < ®n«|µ²®d¼J»~«ºnºÀ0ªQ«¨nª«|®¨nªQ«9²ºxºn«¨¶®nB·d«+~§ ·d®x«$²ºx«º¨x°
> 15?< 2 @]K 1 12 > 1G?< 2°b®+¼J«$·d®x«$²ºx«ºÉ¨n° > 1G?< 2<@ = 1 12 > 15?< 2 I²M¸b°³Ê ®n«¶Q»B·²³¨n~°b§PÀ0¨xªI®n«|ºn¶"«$·d¨¨x°Û¯¶JÒ°b®
¼J°À0§ÄÒ{¾°¬b«¾«§J¨xº®n«$ͯ~®x«º
B K 1 12 > 15?< MI,2 C B_^ @ ` 1 > 15?< 7I C 2 2
B = 1 1 2 > 15? < MI,2 C B_^
@
`
1 > 15?
< 7I2 2
¨x°>ª°»B¼
%Q®n°¾<¨nª~º$¹¨nªQ«0¶®n°¶ °®n¨n~°§EB<°±¨nªQ«0¯§ ¼J«®x»~¸~§µ²³ºnºn«|¨×²³§ ¼>²³¾°¯§J¨(^'°±·²³ºnª~§©¨nª«0®x«¶»~B·²¨x~§µ2¶ °®n¨X±L°b»~~°4·²³§
´"«¼J«®n~¬«$¼¹J¨nªQ«4¬b²³»~¯«4²³¨ ? < ~º5~§J¨n«|®n¶®x«¨²³´»~«²³º×¼Äºd·d°b¯Q§¨x«$¼«¶ «M·d¨n«$¼¬b²»~¯«0°³±
¶®xB·d«º,²¨ ? < À0~¨nª¾9²®x¨n~§µJ²»~«
¾«$²ºx¯®n««$ͯ ²»G¨na° S 1 12 @ 1 BND = 1 12 2F 1 K 1 12 = 1 12 2
Á-«®n«b¹
¨nª~º,«MÍb¯Q»~~´®x¯Q¾ ·d°b§Q§«$·d¨x°§¿~ººn¯´ºx¯¾«M¼±w°b®,²³»~»
¼J~º·h®n«¨x«ºn¨n«|¶º2§É²©´~§°b¾½²³»±L°b®n¾¯Q»½²Q¹J°±À0ªB·xª¨nªQ«
Å ®xºn¨0À0²º0¶®x«ºn«|§¨x«$¼´¸+Î0Ø,bØ 17j 2
@ 1
@
e
5
1
?
` > 2c%12 d
@ BD
S 1 1 2 1 TS 1 1 2 2 @ L( > 1 N 7IJ%12 - #
I
i
Lf h g
15?
²³§ ¼«$ͯ~¬b²»~«§J¨n»~¸:
1 /2
` 1G? > 1G? 2 12 @ > 15? 2 (%1S 1 12 - B @ (16S 1 12 @
À0ª«|®n«ZS @ K 1 BND 1 2 = 1 = 11 2 12 S @ K 1 B D 12 S 1 1 2 #@P¦X§J¨
/2/21 1 1 1 K 1 1F > 2 F = = 1 1112 2 2
³² § ¼ ( -¼J«§Q°b¨n«|º,¨nªQ«>·d°b¾¶»~«¾«|§¨d²®x¸´~§°b¾B²» ¼J~ºx¨n®n~´¯Q¨n~°b§
Ë®d²§ºx¨x~°b§¨x°9¨xª«·d°§¨x~§¯Q°b¯º2¾°¼J«|»
«|¬b«§J¨n¯"²»~»~¸9~¾¶»~«|º,¨n®d²§ºx~¨n~°b§±w®n°¾)¨nª«´Q§Q°b¾B²³»·d°b¾¶ °§«§J¨nº2¨n°+¨xª«
ºx¨²³§ ¼²®d¼§Q°b®n¾+²»·d°b¾¶ °§«§J¨nºÀ0ªB·nªÉ®n«|ºn¯»~¨2±w®x°b¾)¨xª«>¨x«®n¾~§ ²»¼J~ºn¨n®x~´¯¨n~°§¿°³±×²ºxºn«¨¶Q®nB·d«ºM>Ë0ªQ«>¨x«®n¾§"²»
²³ºnºx«¨,¶®xB·d«¼J~ºn¨n®x~´¯¨n~°§9®x«ºn¯Q»¨xº0±w®x°b¾Ô®x«¨n¯Q®n§ºx«$ͯ«§ ·d«|º4®d²§ ¼Ä°b¾»~¸·xª ²§µ~§µ>´"«¨XÀ««§ K 1 120²³§ ¼ = 1 12c
15? 2 P D <
> 1 32 @ >
> 1 32
@
> 15? 2
<f
@
PD <
ΰ§ºnB¼J«|®,²©®²§"¼J°b¾ Üv«®x§°b¯»~»~¬b²®xB²´»~«,À0~¨xª
À0ª«§ P D < @ K 1 12
1 Y 12A@ Q
À0ª«§ P D < @ = 1 12
À0ªQ½·xª·d°¯§J¨nº0¨nªQ«§J¯¾´ «|®5°³±
¯¶JÒX¾°¬«¾«|§¨xº$ '«dÅ"§~§µ&²©ºx¯¾ ¬b²®nB²³´»~«,À«4ª"²M¬b«
1 2
I 1 Y %12 @ 1 H 12 C 1 [12 C$C 1 Y%12 C$C
Ç-°À ·d°§ºnB¼J«®
> 15?@ MI %12
@
> 15? 2
Üv¸¨x®²³§ºX±w°®n¾+²¨n~°§¨n°
1 1%12
K 1 1 2 L= 1 1 2 @ L
» § > 15?> @ 15?MI 2%12 @;I»~§ K= 11 1122 C 1»~§ = 1 12
À«ª ²M¬« I @ ^ 1 16S2|¹À0ªB·xª~º4²ºx¯¾)°±~§ ¼J«|¶ «§"¼J«§J¨B¼J«§J¨nB·²³»~»¸+¼J~ºn¨x®n~´¯¨x«$¼¬b²³®nB²´Q»«|º0¨n°À0ªQ½·xª¿²·d«§J¨n®d²»
»~~¾~¨¨xª«°®n«¾ ·²§©´ «0²¶Q¶»~«M¼Üv¸ºn~¾¶Q»«G»~§Q«$²®G¨n®²³§ºX±L°b®n¾+²¨x°§2¨nª«»~~¾~¨¼J~ºn¨x®n~´¯¨x°§2±w°®»~§ > 1G? @ MIJ12 F > 15? 2
·|²§+´ «¼J«|®n~¬b«$¼
ΰ§ºn~ºx¨n«§J¨n»~¸J¹5¼J¯Q«¨n°É¨nª«É²ºnºx¯¾«M¼Û²ºxºn«¨¶®nB·d«É¼J¸J§ ²¾½·hº$¹À«¾¯ºn¨©ª ²M¬«¨nª"²¨»~§ > 1 32 F > 15? 2©~º§°®n¾9²³»~»¸
¼Äºx¨n®x´Q¯¨n«$¼À0~¨nª
»~§ >> 151 ? 3 2 2_i @ B E i 1 . ? 2c E - ; ? g
g
Õºnºn«|§¨xB²»~»~¸¹« ~ºn¨x§QµÌ»½²³¨n¨x½·h« ²³¶¶®x°²b·xª«|º9°§»~¸Û¼JÃÊ«|®~§Ú¨nª«ÉÀײM¸ÌªQ°À)¨nª~º»~~¾~¨®n«|ºn¯»~¨~º+²b·Í¯~®n«M¼'Ë0ª«
¶Q®n°·d««$¼Ä§Qµ~§J¬b°b»~¬«º×¼ÄfÊ«®x§Qµ&¼J«hÅ §~¨n~°b§+°³±
¨nªQ«4¨x®n««©¶ ²³®²¾«¨n«|®n\º K 1 12²§ ¼ = 1 12|
Î0Ø,Ø (*/)0Ø (Ë0¦ Ù Ç ( )1 )2
K
=
@
@
«¶ E 9 B
«¶ E 9 B
@
«¶ B
=
@
«¶
@
BN E
K
?
?
?C E
9 B
B ? E9 B
K
?
?
@
=
@
@
@
C
C C
T !"
! C C %$
8"#
#
«¶ > B_B c? A
«¶'& E B ?)(
Ë0ªQ«0ÀײM¸°³±
¼Ä«$¼J¯ ·d¨x~°b§©±w°®
¨xª«,¶"²®²³¾«¨x«®nº~§©¨nª«2Î0Ø,ØÒX¾°¼J«|»b~º¯§ÂJ§°À0§¹¨nª°¯µbª¼J«¶"«§ ¼Ä«§ ·d¸+²§©¬b°b»B²³¨n~»~¨X¸
²³§ ¼ºn¨x«¶ºn~Ы#ºn««¾º0ºn¨x®²~µª¨k±w°b®xÀײ³®¼
0.02
1
0.018
0.9
0.016
0.8
0.014
0.7
0.012
0.6
F(S)
f(S)
Ë0ª«¶"²®²³¾«¨x«®nº#°±²®x®n°ÀÔ²§"¼ÌØ0¯ ¼¼É·²³§Ì´ «+¼J«®n~¬«$¼É´J¸¿ºx°b»~¬J§Qµ ²³§¿«MÍb¯"²¨n~°§Ìºn¸Jºn¨x«¾ «$ͯ ²¨x§Qµ+Å ®nºx¨&²³§ ¼
ºx«$·d°b§"¼+¾°¾«§J¨nº°³±v·h°b§J¨n~§J¯°b¯Qº×²³§ ¼+¼J~º·d®x«¨n«©¾°¼J«»"ª ²M¬J~§µ#Å «$¼S 1 12 @S @*FQ
Ë0B²§+« ¶»~B·d~¨n»~¸¼J«®x~¬b«º2¨nªQ«>¶ ²³®²³¾«¨x«®nº´¸+«$ͯ ²³¨n~§µ¨nª«#Å ®nºx¨2¨nª®x««¾°¾«§J¨nº0²§"¼+¯ºn~§µS C W @Ô,¦X§J¨x«®XÒ
«ºx¨n~§µ»¸J¹®n«$²³®n®²³§µb~§µ¨xª««$ͯ ²³¨n~°b§¨n°¼J«|®n~¬b«2¨nªQ«¾9²³®n¨n~§µJ²»~«¾«$²³ºn¯®x«4¸J~«»B¼Jº
1 2
>> 151 ? 3 2 2) @ ( S 1 12 K 1 12 C W 1 12 = 1 12 - @ @dB D @
À0ªB·xª9·h°b®n®x«ºn¶"°b§"¼Jº2¨n°
1 b4 2
»~°bµ g >> 1G1 ?3 2 2 i @X1:S 1 12J»~°bµ K= 11 1122 C »~°bµ = 1 12 g »°µ B E i ?
À0ª«|§9·d°§ºnB¼J«®x~§µ>¨xª«,»~°µ²®x~¨nª¾*°±
¨xª«,®x«¨n¯Q®n§Î°b§ºx«$ͯ«§J¨n»~¸J¹b¨xª«Å ®nºx¨5¾°b¾«|§¨G~ºÅ «$¼Ñ~§ºn¨d²§J¨n»~¸#±w°b®«¬b«|®n¸
¾°Ä¼Ä«» À0ªQ½·xª±w¯»ÃÅ »~»~º¨xª«§°J²®n´Q¨x®²³µb«·d°b§"¼J~¨n~°b§
Õͯ~¬b²³»«|§ ·d«À0~¨nª+¨xª«>ºx«$·d°§ ¼¾°b¾«§J¨ª°b»B¼Jº0«|¨xª«®2´J¸·d°b§ºx¨n®x¯ ·d¨n~°§ ²³º,À«»~» 1 ºx«« ²®x®n°À¹Ø0¯"¼¼+²§ ¼+Ë0B²³§ 2
°®5~º,²b·xª~«¬«$¼~§9¨xª«2»~¾~¨ 1 ºn««Î0Ø,Ø 2 I~¨nª¨nª ²³¨$¹ ²³»~» ®x«$ͯ~®n«¾«§J¨nº0¨n°²·Í¯~®n«2¨nªQ«º²³¾«,»~~¾~¨¼J~ºn¨x®n~´¯¨x~°b§
²³®n«2µb~¬b«|§
Üv¯¨´ «¸°b§ ¼¨xª ²¨M¹J¨nª«¼J~ºn¨x~§ ·d¨0²¶¶®x°²·nªQ«º,¼J°©§°b¨®n«¬«$²»G¶®n°¶ «|®n¨n~«ººn¯µµb«ºx¨n~§µ©ºn¯¶"«®x°®n~¨X¸°b®0~§J±L«®n~°b®x~¨X¸>~§
¨x«®n¾º5°³±
·d°b§J¬b«|®nµb«|§ ·d«>ͯ ²³»~¨X¸©« ¶ «M·d¨,~§ºx°b¾«,·²ºx«º Ö§¨xª«·d°b§J¨x®²®x¸¹Jºx~¾¯»B²¨n~°§º§"¼JB·²¨x«,¨nª"²¨¨nª«2¨nªQ®n««
¾°Ä¼Ä«»~º4´"«ª ²M¬«+ºn~¾~»B²®x»Ã¸+À0¨xªÌ®n«ºx¶ «M·d¨¨x° ·d°§J¬b«®d·d«§ ·d«ºn¶"««$¼¹0²+®x«ºn¯»~¨À0ªB·xªÌÀ0~»~»5´"«ºx¨²¨x«$¼Ú²§ ¼É¶Q®n°¬b«M¼
~§Ì¨nªQ«9§Q« ¨>ºx«$·d¨n~°§ÛÕºxºn«§J¨nB²³»»~¸J¹«MÍb¯"²¨n~§µÉ¾°¾«§J¨xº,¾«|®n«»~¸Ì²ºnºx¯®n«|º9·d°§¬«®nµ«§ ·d«+¨x° ²¼Äºx¨n®x´Q¯¨n~°b§OÀ0~¨nª
¾+²¨·xª~§µ#¶ ²®d²¾«|¨n«®xº$ 5«¨M¹ ·d°¾¶¯¨x~§µ>°¶¨n~°§9¶Q®nB·d«ºÀ0¨xª~§²©¨n®n«|«&·d°§ºn¨x®n¯ ·h¨n«$¼É¨xª~º0ÀײM¸+¼J°J«º,§°¨0»~«$²b¼+¨x°
¨xª«2´ «ºx¨4²·nªQ«|¬b²´»~«2«ºn¨x~¾9²¨x~°b§®n«|ºn¯»~¨nºM
ήx¯ ·dB²»~»~¸¹J²»~ºx°¨nª«²b·|·d¯®²·d¸°³±²¶Q¶®n° ~¾9²³¨n~°b§©~º0§ÄÓ ¯«§ ·h«$¼´J¸+²©ºn«$·d°§ ¼±{²·d¨$¹Q~§ª«®x«§J¨,~§²»~»¨n®x««¾°¼J«|»ºM
Ë0ª«2«§J¨x®x«2¶®n°´ ²´~»~~¨X¸>¾+²ºxº~º,·d°b§ ·h«§J¨n®²³¨n«$¼ « ·d»~¯ºn~¬b«|»¸²³¨0¨nª«2¨x«®n¾§"²»"§°¼J«º0°±¨nª«2¨x®n««¶Q®n«ºx¯¾~§µ¨xª ²¨
°¯¨·d°¾«º0¼J«¬JB²¨x~§µ2±w®x°b¾Ô¨xª«ºn«#§°¼J«º,·²³§§°b¨0°··h¯®$vË
®d²§ºx»B²¨n~§µ#§J¨x°&²#´~§°b¾B²»¼J~ºn¨x®n~´¯¨x°§±w¯Q§ ·d¨n~°§À«
ª ²M¬«4²©¶~«$·h«À0~ºn«>·d°§ºn¨d²§J¨0±L¯§ ·d¨x°§À0~¨nªÏd¯Q¾¶º5²³¨0«$²b·xª¨x«®n¾~§ ²»"§°¼J«b¦X»~»~¯ºn¨x®²³¨n~§µ©¨nª«>¼Äºd·d®n«|¨n«ºx¨n®n¯"·d¨n¯®x«
¨d²Âb«©¨nªQ«ºn«¨°³±5¶®x°b´ ²³´~»~¨x«|º4·d°§§«$·d¨x«$¼ÌÀ0~¨xª ²+·d°®n®n«|ºn¶"°b§ ¼J~§µºn«¨°³±×¨x«®n¾~§ ²»²ºnºx«¨©¶®nB·d«º#¨n°·d°b§Qºn¨n®x¯ ·d¨²
ª~ºx¨n°bµ®²³¾Ô°b¯¨x»~§Q§Qµ¨nª«©²¶¶®x° ¾+²¨x°§©°±¨xª«·d°b§J¨n~§J¯°¯º,¼J«§ºx~¨X¸
Ç-°b¨x½·h«4¨xª ²¨0¨xª«¶®x°b´ ²³´~»~¨X¸©¾9²³ºnº°±«$²b·xª®x«$·d¨²³§µb¯Q»½²³®0~º,·d°b§JÅ"§«$¼+¨x°&²©¨x«®n¾§"²»"§°¼J«b
0.01
0.5
0.008
0.4
0.006
0.3
0.004
0.2
0.002
0.1
0
¥ *G£ +-ª~ºn¨x°bµb®d²¾Ô¬JºM¼J«§ºx¨X¸Ñ±w¯§ ·h¨n~°b§
60
80
100
120
140
asset price S
160
180
0
¥ *G£ +-ºn¨x«¶+¬ºM¼J~ºn¨x®n~´¯¨x°§±w¯Q§ ·d¨n~°§
60
80
100
120
140
asset price S
160
180
Üv¯¨0¨nªQ«ºn«¨°±¶ °ºnºn~´»~«>²³ºnºx«¨,¶®xB·d«º,ºxªÃ±w¨nº0À0~¨xª9«M²b·xª+¨x«®²³¨n~°b§+°³±
¨nªQ«¨n®x««®n«hÅ §«¾«§J¨$5Á²M¬J§QµÅ «$¼«°µb«dÒ
§°¯ººx¨n®n~«¶®nB·d«²³§ ¼Ó ¯ ·d¨x¯ ²¨x~§µ»~°·²¨x°§º2°±Ïd¯¾¶º2§É¨nªQ«¼J~ºn¨n®x~´¯¨n~°§É±w¯§ ·d¨x~°b§¹¨xª«ºx«¶ ²³®²¨x~°b§Ì°±0¨nªQ«
¶®x°b´ ²³´~»~¨X¸+¾+²ºnº2´"°b¯§"·d«º©´ ²b·x¿²³§ ¼±w°b®x¨nªÌÀ0~¨nªÌ·xª ²§Qµb~§µ+ºn¨x«¶Û¶ ²³¨n¨x«®n§ºM¹´ «M·²¯ºx«9²³®n´~¨n®d²®x¸¿»~°·²¨x~°b§º·²³§
°··d¯Q®$
Ë0ªJ¯ºM¹ ¨xª«©´~§°b¾½²³» ºx¨n®x¯ ·d¨n¯Q®n«~¨xºn«»Ã±5~§¿·d°¾´~§ ²¨x~°b§À0~¨xª¿²ºx¨n®n~«>¶®xB·d«>»~°·²³¨n«$¼+~§ ¼Ä«¶ «|§ ¼J«§J¨n»~¸°³±5¨nªQ«>¨x®n««
Crw^f|jnadrXd^fl0 $ãlBpbadol0uSpbeu^`_+uBpbj2"FF
c,aMijXg"_bjn|euB^fz|j0adybuB^fah_>yrL^fqnjxlc,ed©a$qxqXtr5o"pbjk_9uSpbjXrXj0^fl0ezjkr{#gfad_buS^`c,j,uLa
c,euStrL^fuBqnahc5s^`_bjxi5o"^fuSp,gfado+uBrXjnj
r{jk_bjXc,jk_ux\
µ®nB¼~§ ¼Ä¯ ·d«º0~®n®x«µb¯Q»½²³®n~¨n~«ºÀ0ªB·xª9¼J~ºx¨n°b®x¨0¨nª«²³¶¶®n°~¾9²³¨n~°b§#®n«ºx¯»~¨$5Õ Q·h«¶¨0°±ºn¶"«$·dB²»·²ºx«º$¹Q¨nª«·d°¾¶¯Q¨n«$¼
°¶¨n~°§¶®nB·d«º0°º·d~»~»B²¨x« ¹²§"¼·d°§¬«®nµ«§ ·d«#ÀײM¬J¸©¨n°©¨nª«Üv»½²·xÂbÒ "·xª°»«|º5ºx°b»~¯¨n~°§ Ë0ª«®n«¾+²~§~§µ2¶ ²®x¨5°³±
¨xª«
¶"²¶"«®0~º,¼J«¬b°¨n«$¼ « ²b·d¨x»¸¨n°©¨nª«|ºn«©¨XÀ5°¼J«dÅ·h«|§ ·d~«º$
Ù ¨2¨xª«º²³¾«¨x¾«b¹²b·d¨x¯ ²»~»~¸¨xª«ºn«©Å"§ ¼J~§µbº2~§ºx¶~®n«$¼É¨x°9ºx«$²®d·xªÉ±w°b®·d°§ ¼J~¨n~°b§Qº4¯Q§ ¼J«®À0ªQ½·xªÓ ¯ ·d¨x¯ ²¨x~°b§º~§
°¶¨n~°§¶®nB·d«2²¶¶®x° ¾+²¨x°§·²§´ «2²M¬b°bB¼J«M¼²§ ¼Ñ´ «¸°b§ ¼©¨xª ²¨ª~µbª²··d¯®d²§ ·h¸9²b·xª~«¬«$¼¦X§§J¯¾«®n°¯º5·²³ºn«º
¨xª«0²¶¶®x° ¾+²¨x°§º"°bº·h»~»B²¨x«5²»~¾°bºn¨"« ²b·h¨n»~¸>²®x°b¯§"¼2¨nª«¨n®x¯«0¬b²»~¯« %¯Q®n¨nªQ«®n¾°b®n«¹$À0~¨xª&·xª ²³§µb~§µ,²³¾¶»~~¨n¯ ¼Ä«
°³±
¨xª«À0²M¬b«ºM¹J¨nª«²¶Q¶®n° ~¾9²³¨n~°b§º²b··dB¼J«|§¨xB·²»~»~¸²¶¶Q®n°²·xª9¨xª«2¨n®x¯«¬b²»~¯«®n«¶"«$²³¨n«$¼J»~¸J
./¥B¥B*Ú'+É¡ ¢+Q 9¡ -¿¤¡;+Q * +¤+
¦X§+¨xª~º4·xª ²¶Q¨n«®M¹ À«2À0~»»"Å"®nºn¨2¼J«dÅ §Q«¨nª«#°b®¼Ä«®2°±·h°b§J¬b«®xµb«§"·d«b0Ë0ª«|§À5«#À0»~»¶®x°¬b«²©µb«|§«®²³»®n«ºx¯»~¨,°b§+¨xª«
°®¼J«|®°±·d°b§J¬«®nµ«§ ·d«bvË0ª«0ÂJ§°À0§»B²¨x¨nB·d«0²¶¶®x°²·nªQ«ºÅ"¨~§¨x°2¨nª«±w®d²¾«À5°®nÂ2°±¨nªQ«,¾9²³~§4¨xª«°®n«¾bºx°~¨
À0~»~»
´"«²¶¶Q»~«$¼¨x°¨nª«¾É Ù ¨0¨nª«#«§ ¼¹QÀ5«2À0~»~»"±w¯®n¨xª«®x¾°b®x«2µb~¬b«ºn°b¾«2ºn~¾¯»B²¨x°§º²§ ¼«¶»B²³§G¹Jª°À ¨xª«2®n«ºx¯»~¨nº
·h°b¯»B¼´"«¯ºx«$¼
Ì«À0~»»²¼J°b¶Q¨5¨xª«2±w°»»~°À0~§µ2§°¨²¨x~°b§ Ì«5®x«¾«¾´"«®¨nª"²¨¨xª«0ºn«$·h¯®n~¨X¸©±w°»»~°À0ºG¨nª«¶®nB·d«0¶®x°·d«ºnº ) À0ªB·xªº¨xª«,ºx°b»~¯¨n~°§>°³±¨xª«,ºx¨n°·xª ²ºx¨nB·
¼JÃÊ«®x«§J¨nB²»«MÍb¯"²¨n~°§
) B = ? C ) E =7H
)
) )
Ë0ª«2¶®n°´ ²´~»~~¨X¸>¾«$²ºx¯®n«2~º,¼J«§Q°b¨n«M¼+´J¸È
1
QN 2 ºxª ²»~»´"«2¨nª«¶"²M¸b°Êb±L¯§ ·d¨x°§¹ ´"«¨xª«2ºn¨n®x~Âb«b
QË0ª«¼J~º·d®x«¨n«©¶®n°·d«|ºnº2~º,¼J«º·h®n~´ «M¼É´¸ @ À0ª«®n«©º0¨xª«¬b²³»¯Q«,°±¨xª«>¼J~º·h®n«¨x«>ºx«$·d¯®x¨X¸+¶®xB·d«
¶®n°·d«ºxº4²³¨0¨n~¾« ? Ë0ª«¸9²³®n«®d²§ ¼Ä°b¾Ô¬b²³®nB²´»~«ºv±w°®3
Ë0ª«2¶®x°b´ ²³´~»~¨X¸©¾«M²ºn¯Q®n«2~º
1 2
) @ C
L¦ ±,«!Q²³¾~§~§µ²É·d«®n¨d²~§Û»B²³¨n¨nB·d«+²¶¶Q®n°²·xªÌ±w°®&²ºn¶"«$·dÃÅ·ºn«M·d¯®n~¨X¸J¹¨nª«°b§»~¸Ì·nª"²§µ§Qµ¿¶ ²³®²³¾«¨x«®~º>¨xª«+®n«dÒ
Å"§«¾«|§¨ 1ɰb®«$ͯ~¬b²»~«§J¨n»~¸¨xª«©ºn¨n«|¶Ìºn~ЫOB ? ,¦X§+±X²b·d¨M¹ ¨xª«©°b¶¨x°§¶®x½·h«>~§¿²¼J~º·d®x«¨n«»B²¨x¨nB·d«>²³¶¶®x°²b·xªÉ~º4²
±L¯§ ·d¨x°§°³± 1GË0ª«Ü»B²b·xÂÒ "·xª°b»~«ºkÒ{¬b²³»¯Q« 1G? > 2~º5¨xª«2« ¶ «M·d¨n«$¼¶ ²M¸°Ê¿²¨¨n~¾«0Ë'¼J~º·d°¯§J¨n«$¼+¨x°>¨x~¾«0¨¨nª ²³¨
~º
1 2
15? > 2N@ "$'&#)6+ "!5 1 & ) 3 2$#
Ë0ªQ«4»B²³¨n¨nB·d«2¬b²³»¯Q«,~º0¨nª«2« ¶"«$·d¨x«$¼¶"²M¸b°Ê̲¨¨n~¾«Ë ¼J~ºd·d°b¯§J¨x«$¼¨n°¨n~¾«,¨0¨nª"²¨0~º
1 +2
#"$ &()+ ( 1 @ 2 "ħ"·d«À«>²®x«°b§»~¸§J¨x«®n«ºx¨n«$¼~§¨nªQ«°b¶¨x~°b§+¶®nB·d«¹ À«À0~»~»« ²¾§Q« @ ²º«®n®x°b®²¨0®x«dÅ §«¾«§J¨ 1̼J«hÅ §«$¼´J¸
¨xª«²´ºx°b»~¯¨x«4¬b²³»~¯«,°³±¨xª«¼JÃÊ«|®n«§ ·h«>´"«¨XÀ««§¿¼J~ºd·d®n«¨x«&²³§ ¼+·d°b§J¨x§J¯°¯º¶®nB·d«b
% uSp^fl"yr{ahyjXrXuB^fllLadc,jxuB^`c,jxlqnedg`gfjxi'&jnz|jX_Ba$iiM~yMr{ahs$gfjkc(&
) @~_2adrXijXruwaz^flStbedg`^+x* juBpbeulLadgfjkgf5uSpbj, tbqxuBtbeuS^`_blwjkybedrXeuB^fah_5qnedtblLjnl"jn|^flLuB^`_bqxad_zjXrXjX_bqXj"ybe|uwuLjXrL_bluLed$juSpbj]"gfeqLMSKMqLpbahgfjnl
}~adrLc5tgfeeh_bi,eiedybuuSpbjlLuBrL^`$jGyrL^fqnjG^`_uBpbjG_badrLc,edgqxadc0yah_bjX_$uLl uLauBpbj^`c0yg`^fqk^fuwjGlwjkybedrXeuB^fah_5rLtgfjad} uSpbjGs^`_badc0^fedgc,aMijXgflxmytbu
ah_>uwahy>e,qxad_uS^`_Mt^fuB0qnadrLrXjnqnuS^fad_2ad}ah_bjpbedg`}L\@~_uwjkr{jxlwuS^`_bdgfdmed_,jn|^flLuB^`_b0qxad_z|jXrXjX_bqxjybeuLuwjkrw_blGqned_2sJjrXjXyrXa$iMtbqxjniJmMiM^flBygfeqxjni
o"^fuSp2r{jxlByJjnqxuuwa0uSpbjiM^flLuBrL^`stbuB^fah_×jkrwrXadr
ah_gf\"ad_blLjn|tbjX_uBgf|mdc,jXrXjXgf0uSpbj
lLjXybehr{euS^fad_0yrXadsgfjXcÛlS^fd_bl
r{jxlByJad_blB^`sgfj
}adredg`gjn|^flwuS^`_b
qxad_zjXrXjX_bqXjGybeuLuLjXrL_bln\C"pbjiM^flLuBrL^`stbuB^fah_,jXrLr{adrqned_2sJj5|tbed_uB^`bjxi^`_bijnjnis,eiehybuB^`_buBpbjv]"gfeqL$wKMqLpbadgfjxl}~adrLc5tgfead_bqnjc,adr{j|m
_badoÉo"^fuBpelLjXrL^fjnljn|ybed_blS^fad_\5C"pMtblxmJs^`_badc0^fedgadybuS^fad_2yrw^fqxjnlqned_©sj0iMtyg`^fqxeuLjni0suSp^fluBo ah}adgfi©eiedybuLeuS^fad_'\ -Glwqk^`g`gfeuB^fah_^fl
yrXaMiMtbqnjnisjnqxedtblLjuBpbjGlwuSrw^`M/j .Btc0ybl"adzjXr"rXjnqnuLed_bdtgfexrXlJo"^fuBp2jnzjk_,ed_bi0a$iirXjX_bjkc,jX_uwlxmno"edzjnlijxlwqkrw^`sJjuBpbj
r{jkgfeuB^fz|j"c,az|jXc,jX_u
sJjnuBo"jnjk_,lBtrLrXadt_biM^`_buLjXrLc0^`_bedgM_ba$ijxlnmo"p^fqLp2qLpbed_bj^`_2zdedg`tbjqxad_uS^`_MtbadtblBgfuSpbadtbdp\
Ǭ
1 2
@ @ "7& !5 1 & 3 2$#h ( 1 @ 2 - ´"«2¨nª«
Ù »B²¨x¨nB·d«2²¶¶®x°²·nª"!-ñ²§"¼°§»~¸Ã±±w°®,²»~»¶ ²®d²¾«¨n«®xºG B E N À«2ª ²M¬b«
@$»~#~¾ % @ @ Ë0ª~º·d°§ ·d«¶¨#°±,·d°b§J¬b«|®nµb«|§ ·d«~º2« ²b·h¨n»~¸¨nª"²¨¯ºx«$¼É~§¿¾+²¨xª«¾9²³¨nB·²³»»~¨x«®²³¨n¯®x«b©¦X§¿°´ºn«®x¬J§Qµ¿·d°b§J¬«®nµ«§ ·d«
~§É¨nª«®x«dÅ §«¾«§J¨ 1¹°§«¨X¸J¶B·²»~»~¸+°b´ºx«®n¬«º>À0²M¬J¸¶ ²³¨n¨n«|®n§ºMË0ª~ºÀ0²º²³»®x«$²b¼Ä¸ ¼J~ºd·d¯ºnºx«$¼Ì~§É¨nª«¶Q®n«¬J~°b¯º
ºx«$·d¨n~°§ Ù §É²¶¶®x° ¾+²¨x°§À0ªB·xª~º,®²³¨nª«|®4·d»~°bºx«>¨x°¨nª«Üv»½²·xÂbÒ "·xª°»«|º,¬b²»~¯«2¾+²M¸©±w°b»~»~°À ²§Q°b¨nªQ«®,µ°°¼°b®
«¬«§²©À5°®nºx«²³¶¶®x° Ä~¾+²¨n~°§
Á-°À5«¬«®0À«4®x«¾«|¾´ «®¨nª"²¨,·d°§¬«®nµ«§ ·d«#« ²b·d¨x»¸º²M¸Jº0¨nª"²¨0~§+µb~¬J§Qµ>²·d«®x¨²~§+«®x®n°b®´ °¯§ ¼¹QÀ5«·²³§Å § ¼+²
®x«dÅ §«¾«§JG¨ 1 ºx¯ ·xª¨nª ²³¨5«M²b·xªÅ §«®0°§«2ª ²º,²³§«®x®n°b®0À0ªQ½·xª¼J°J«º0§°¨5!« Q·h««$¼+¨xª«2´ °¯§ ¼Üv¯¨,ªQ°À ¼J°J«º0¨nªQ«
®x«dÅ §«¾«§J¨,¼J«¶"«§ ¼+!« Q²·d¨n»~¸'&+˰>¾+²Âb«¨nª~§µº5¶Q®n«$·d~ºx«b¹"À5«2ºnª"²»~»º²M:¸ 0Õ¯®n°¶ «M²§Ì·²»~»°¶¨n~°b§Qº$¹·d°b¾¶¯¨x«$¼À0~¨xª¿²»½²³¨n¨x½·h«&²³¶¶®n°J²b·xª()$*+,-.0/)"B ñ
¨xª«®n«#«ºx¨nº,²·d°§ºn¨d²§J¨ ` F©ºx¯ ·xª¨nª ²³¨
1j 2
1 1 @2 1 ` "
3$4(567
b0¦X§#°b¯®Ë0ª«°®n«¾*´ «|»°ÀÌÀ«À0»~»ºn««¨nª ²³¨¨nª««ºn¨x~¾9²¨x~°b§2°± ¨xª««®n®x°b®·²§©´"«×¼J«M·d°b¾¶ °ºn«$¼#~§¨x°4²·d°b§ºx¨²³§¨
` 3>¼J«|¶ «§"¼²§J¨,°³±
¨xª«ºn¶"«$·dÃÅ·°b¶¨x°§9²³§ ¼¨nª«°®¼J«|(® B>¼J«|¶ «§"¼²§J¨,°³±
¨xª«>·xª°bºx«§+»B²¨n¨xB·d«²¶¶®x°²·nªG
(
I»«M²ºn«©§°¨n«©¨nª ²³¨4·d°§¬«®nµ«§ ·d«~º2~¾¶»~~«$¼Ñ´¸²³§¸+°®¼J«®2µ®n«$²³¨n«®¨xª ²:
§ Ý+°®n«°¬«®2À5«®x«¾9²³®nÂ+¨nª"²¨²
2
»B²³¨n¨nB·d«©²¶¶®x°²·nªÉÀ0~¨nª°®¼J«|® BѪ ²º²³»ºx°°®¼J«®B 8 BJ Ù ªQµª«®2°b®d¼J«®2¾«M²§º ͯB·xÂb«® 9·d°§¬«®nµ«§ ·d«
˰²b·xª~«¬«²©·d«®x¨²³§¶®n«$·hºx~°b§+»~«¬b«»X¹J¨xª«>·d°§ºn¨d²§J\¨ ` ²§ ¼Ñ¨nª«°®¼J«|®0º0°³±
~¾¶"°b®x¨²³§ ·d«b
0Ë0ªQ«¾°ºn¨2~¾¶"°b®x¨²§J¨2±X²b·d¨~ºM¹¨nª"²¨©§Ì¶Q»°¨n¨x§Qµ @ ²³µ²³§Qºn¨©¨nª«®n«dÅ §Q«¾«§JO
¨ 1Ú°b§Ú²+»~°bµ³Ò{»~°bµ³Ò{ºd·²»~«b¹G¨nªQ«
´"°b¯Q§ ¼J~§µ#±w¯§ ·d¨x~°b§:@69 ; ´ «M·d°b¾«º,²©ºn¨x®²~µª¨»~~§«,À0~¨nª+ºx»°¶ «2«$ͯ ²³» 1 (B20²§"¼ºxªÃ±wN¨ `¦X§+§°b¨xf±L¸~§µ#¨nª~º$¹
~¨´ «M·d°b¾«º0«$²ºx¸¨x°°b´ºx«®n¬«¨nª«#°b®¼Ä«®XÒX°±BÒn·d°§¬«®nµ«§ ·d«#§+ºx¾¯Q»½²³¨n~°b§Qº$
=<>5?4(@,"
10.3
1
CRR
4/t
10.2
0.1
10.1
0.01
error
price
BS
CRR
10
0.001
9.9
0.0001
9.8
1e-05
9.7
20
¥ *G£ +(AÎ0Ø,Ø
40
60
80
100
refinement n
J¶®nB·d«
120
140
1e-06
10
¥ *£ +CB ,Î0ØØ
100
refinement n
J«®n®x°b®
> @* [ @/ @*J B(@ [ E @3 ¦X§+¨nªQº,·|²ºn«¹¨nªQ«4°®¼J«|® °³±Ä·h°b§J¬b«®xµb«§"·d«>~º«$ͯ ²»bË0ª«2¶®x°J°±~º5µ¬«§+»B²¨n«|®0§¨nª~º,·xª ²³¶¨n«®M
Á-°À ·d°b¯Q»½¼°§«¼J«¨n«|®n¾~§«2¨xª«2°b®¼Ä«®,°³±²©»B²¨x¨nB·d«dÒn²¶¶Q®n°²·xª9¾+²¨xª«¾9²³¨nB·²³»f»~¸?&
1000
-ºn~§µ¨xª«0®n«|¶®n«ºx«§J¨²³¨n~°b§+°±¨xª«,¼J~º·h®n«¨x«4¶Q®nB·d«,À0~¨xª¨nª«,·d¯Q¾¯»B²¨n~¬«5´~§°¾B²»J¼J~ºn¨x®n~´¯¨x°§>¾«§J¨n~°b§Q«$¼~§¨xª«
¶Q®n«¬J~°b¯º2·xª ²¶¨x«®$¹"°§«&·h°b¯»B¼« ²¾§Q«4¨xª«°®¼J«®2°³±v²³¶¶®x°Ä~¾+²¨n~°§°±¨nª«#®n«ºx¶ «$·h¨n~¬b«¼J~ºx¨n®n~´¯Q¨n~°b§±w¯§"·d¨n~°b§G
Üv«®x®n
¸ ( + -/Õºnº«| «§ ( +-« ²¾~§«$¼©¨nªQº¨nªJ¯ºµb«¨x¨n~§µ©°b®¼Ä«®
Á-°À5«|¬b«®0À«2ª ²M¬b«²»~®x«$²b¼J¸ºn¯µµb«ºx¨n«$¼ °b®¼Ä«®>2~§¨xª«>²³´ °¬«4!« Q²³¾¶»~«0°³±¨xª«Î0Ø,Ø'²¶¶Q®n°²·xª "J~§ ·h«4À«2¶®n°¬«
¨xª«2´ «|¨n¨n«|®4®x«ºn¯Q»¨0´J¸°b¯®0¨xª«°b®x«¾)À«²³®n«2§°¨0µb°b~§µ#¨n°¶®x«ºn«§J¨2¨nªQº0B¼J«M²~§+¾°®n«4¼Ä«¨²³»X
Ö¨xª«®>²³¶¶®n°J²b·xª«º#ºn¯ ·xªÛ²º#¦X´®²³µb~¾°¬ ( j_j -
²³®n«« ²¾~§~§µ>¨xª«·xª ²®d²b·d¨x«®n~ºx¨nB·©±w¯Q§ ·d¨n~°§¹¨nªJ¯º#¯ºn~§
µ %Q°b¯®x«|® Ù § ²»~¸Jºn~º°±¨nªQ«¼Äºx¨n®x´Q¯¨n~°b§Ñ±w¯§ ·h¨n~°b§Ë0ªQº0¸J~«»B¼Jº,·d°§ ¼J~¨n~°§º5À0ªQ½·xª²®x«¼J 9·d¯»~¨5¨x°¬b«®xñw¸
"ħ"·d«,¨xª«0°b¶¨x~°b§¶®x½·h«,~º¨nªQ«4¼J~º·h°b¯§J¨n«M¼« ¶"«$·d¨²³¨n~°b§°±¨xª«0Å §"²»¶ ²M¸°Ê¹²³§ ¼©¨nªQ«,»~°bµJ²®n~¨xª¾ºG°±¨nªQ«,ºn«M·d¯®n~¨X¸
²³®n«§°b®x¾9²»~»~¸ 1 Ç 25²³§ ¼©´~§°b¾B²»~»~¸ 1 Ü 2¼J~ºn¨n®x~´¯¨n«M¼®d²§ ¼Ä°b¾ ¬b²®x½²³´»~«º®x«ºn¶"«$·d¨x¬«»~¸¹J°§«,~º»~«$¼Ñ¨n°©« ²¾§Q«5¨xª«
°®¼J«|®0°±¨nª« À5«$²³Â9·d°§J¬b«®xµb«§ ·h$« °³±
¨nªQ« '«$Ý+°b~¬J®n«hÒ G²¶»B²b·h«4Ë0ªQ«°b®x«¾¿¹¨nª ²³¨0~º±w°b®x¾¯»B²º°³±
¨nªQ«ÂJ§"¼
( 1 2 - ( 1 ^ 2 - À0~¨nª E³¦ Ø ¦ Ø Üv¸Û²+¾°b®n«¼Ä«¨²³»~«$¼È« ²¾§"²¨n~°§¹²º©Àײ³º&¼J°§«´J¸ÌܯQ¨nЫ
® )Á,²³ª§ (*/-n¹G¯ºn~§µ¨nª«Ö¶ «|®²¨x°b®kÒkÝ+«|¨nª°¼O°±
Ë®x°b¨n¨x«® ( )-5¨x°¶®x°¬b«¨xª«9·h«§J¨n®²³»0»~¾~¨,¨nªQ«°b®x«¾¿¹G°b§«µb«¨xº>¨xª«°®¼J«|®°±,·d°b§J¬b«|®nµb«|§ ·d«°³±4²´"°¬b«¨n«|®n¾ºM+¦X§
«|ºnºn«|§ ·d«¨xª~º5®x«$ͯ~®n«º0¨xª«0±w¯Q§ ·d¨n~°§ >¨x°>´"«,ºx¯ 9·d~«§J¨n»~¸ºn¾°J°b¨nªG¹~§°b®¼Ä«®0¨n°¾+²Â«,²2˲M¸»~°®XÒX« ¶ ²§ºx~°b§ "ħ"·d«
°¯®0¶ ²M¸°Ê+±w¯§ ·h¨n~°b§+~º5§Q°b¨,¼JÃÊ«®n«§J¨x½²³´»~«²¨ > @ ²¨,²³»~»{¹¨nªQº0B¼J«M²©º§°¨,²¶¶»~B·²³´»~«4¼J~®n«M·d¨n»~¸J
Á-°À5«|¬b«®¨nª«|ºn«4²³¶¶®x°²b·xª«|º×¼J°#§°b¨¾9²³Âb«¯ºx«2°±¨nª«-±{²b·d¨M¹b¨xª ²¨À«,²®n«~§¨nª«ºn¶"«$·dB²»"ºn~¨n¯"²¨n~°§°±
²2ºx¨n°·xª ²ºx¨nB·
¶Q®n°·d«ºxº$+¦X§Ì¨nªQ«·²ºx«°±,²+ºx¨n°·xª ²ºx¨nB·©¶®n°·d«|ºnº²§"¼¿²±L¯§ ·d¨x°
§ À0¨xªÌ¶ °»~¸§Q°b¾B²³»´ °¯§ ¼J«$¼Ì¼Ä«®n~¬b²¨x~¬b«º 1 °±
ºx°b¾«,°b®d¼J«® 2¹ »~°J«$¼J«
§ I»B²¨n«|§ (*)-¶®n°·d««M¼¼JÃÊ«|®n«§J¨n»~
¸ Ë0ª«¸9¼Äºd·d®n«|¨n~Ы©¨nª«2¨n~¾«,² ~º$¦X§¨xª~º0ÀײM¸¨xª«¸
²³®n«²´Q»«2¨x°®n«¶®x«ºn«|§¨2¨xª«>²´"°¬b«²³º,²©ºn¯¾ °±¨nª«>¼ÄfÊ«®x«§ ·d«º2~§+«$²b·xª+ºn¨x«¶¦X§+«$²b·xª+¨n~¾«,ºn¨x«¶¨xª«¸+«¬b²»~¯ ²³¨n«
~¨´J¸¾9²ÂJ~§µ²#˲M¸J»~°b®kÒkÕ Ä¶"²§ºx°§²º0²´"°¬b«bË0ªQº~º0¶ °ºnºx´Q»«´ «M·²¯ºx«>~§¨nª«~®,·²³ºn«2¨nªQ«¸9·²³§+¾9²Â«0¯ºn«2°³±²
¨xª«°®n«¾ °±5Ý+B·d¯»~«¬JB·d~¯º«§ºx¯®n~§µºn¯ 9·d~«§J¨,¼JÃÊ«®n«|§¨xB²´~»~~¨X¸©±w°b®¨nª«|®¶¯®x¶ °ºn«ºM
-§J±w°®n¨n¯Q§ ²¨x«»~¸¨nªQº0~º0§°¨,¨xª«·²ºx«>ª«|®nJ« ª°À«¬b«|®$¹°´ºn«|®n¬J~§µ¨nª ²³¨0¨nª«©Ü»B²b·x "Q·nªQ°b»~«º,¶Q®nB·d«~º0ºn¾°J°b¨nªG¹b¨xª~º
~º0¨xª«²¶¶®x°²·nª+¨xª ²¨,²³»»~°À0º¨x°·d~®·d¯Q¾¬b«§J¨¨nª«¶Q®n°b´Q»«|¾Ô°±§°§ ¼JÃÊ«|®n«§J¨nB²³´~»~¨X¸©°±°b¯Q®0¶ ²M¸b°³Ê9±L¯§ ·d¨x°§
Ø-«¾9²³®nÂb²³´»~¸¹²»~¨xª°b¯µªÀ«²®n«2¬b²³»¯Q§Qµ>Õ¯®x°b¶"«$²§·²³»»ÃÒX°b¶¨x~°b§º~§¨nªQº0¶"²¶"«®0°b§»~¸J¹b¨xª«²¶¶Q®n°²·xª9ªQ«®n«>²³»~»°À0º
¨x°« ¨n«§"¼~¨0¨x°¶ ²¨xª ļJ«|¶ «§"¼J«§J¨,°¶¨n~°b§Qº0«$²ºx»~¸J¹b´"«$·²³¯ºn«©À«4¾+²Â«0¯ºn«°³±
¨xª«À0ª°»«¶®nB·d!« ¶®x°Ä·h«ºnºM
'~ºn¨x®n~´¯¨x~°b§º5²³®n«,·d°¾¶»~«¨x«»~¸>·xª ²®d²b·d¨x«®n~ЫM¼+´J¸>¨xª«~®¾°¾«§J¨xº$
%Q°b®!« Q²³¾¶»~«¨nª«0§Q°b®n¾+²» ¼Äºx¨n®x´Q¯¨n~°b§©´J¸~¨nº
Å"®nºn¨0¾°b¾«§J¨ 1 ¨xª«2¾«$²³§ 2²³§ ¼ºx«$·d°b§"¼+¾°¾«§J¨ 1 ¨xª«2¬b²®nB²³§ ·d4« 2|Ë0ª«®n«h±w°b®x«~§¨nªQ«·h«§J¨n®²³»»~~¾~¨¨nª«|°b®n«|¾ ¨xª«
¾°b¾«§J¨nº°³±¨xª«¼J~º·d®x«¨n«©®²³§ ¼J°¾ ¬b²³®nB²´»~«2²¨0»~«$²ºx¨,§«|«$¼+¨n°²³¶¶®x° Ä~¾+²¨n«¨nª°ºn«°³±
¨xª«§°®n¾9²³»¼J~ºn¨x®n~´¯¨x°§
ΰ§J¬b«®xµb«§ ·h«~º0«§ºn¯Q®n«$¼+´J¸¨nª#« bÏ|²¶¯§Q°Ê ·d°§ ¼J~¨n~°§¹JÀ0ªB·xª~º,²©ºn¯ 9·h«|§¨,·h°b§ ¼J~¨x°§°§9¨xª«>·d°§J¬b«®xµb«§ ·h«°±
°§«2ª~µbª«|®0¾°b¾«§J¨$
¦X§²#»B²¨n¨xB·d«²¶¶®x°²·nªG¹J¨nª«°®¼J«|®,°±·d°b§J¬«®nµ«§ ·d«~º,·d°¾¶»~«¨x«»~¸¼J«¨x«®n¾§Q«$¼´¸¨xª«2±w°»~»°À0~§µ±{²·d¨n°®nº
,5 Ì
«4·²³»~»
1 4 / 2
@
@
1 2
@
@
1 4 ) 2
@
@
i E
g
g
i g
g
1 2
S @ @
1 2
S @ @
°b¯® 4(4()!,²§"¼
°b¯® ! /Q 4(4(> !
»§ i i
g
g
»~§ i i
g
g
i
i
3 $4(5674Ç0°¨nB·d«,¨xª ²¨¾°¾«§J¨xº²³§ ¼¶Qºn«¯ ¼Ä°b¾°¾«§J¨xº¼Ä°b§°¨×¼J«|¶ «§"¼°§ºn¶"«$·dÃÅ·
@ @3´"«$·²³¯ºn«#°±¨nªQ«
®xºx§«|¯¨n®d²»~~¨X¸²®nµ¯¾«|§¨°±Á,²³®n®n~ºx°b§ I»~~ºnÂb²(1 -$4Ú(
«¨ > @ @ @ A À0~¨nª @ @ @ > ¼J«|§°b¨x«¨nª«¼J~ºd·d®n«¨x«>¶Q®nB·d«2¶®n°·d«|ºnº2°±²©»½²³¨n¨x½·h«²³¶¶®x°²b·xª
Ë0ª«2°®¼J«®°±5·d°§¬«®nµ«§ ·d«©~º0¨nª«2ºx¾9²»~»~«ºx¨5°®¼J«®2·d°b§J¨d²~§«$¼~§ @ @ °b® S @ ®x«$¼J¯ ·d«M¼´J¸¿¹´¯Q¨0§°b¨0ºx¾9²»~»~«®
¨xª ²§Ìb¹J¨xª ²¨0~º Ë0ª«|®n«2« ~ºn¨nº,²©·d°§ºn¨d²§JG¨ `ѹJ°b§»~¸¼J«¶"«§ ¼J~§µ°§ > % B_ E Oºn¯ ·xª¨xª ²a¨ @'2 ` @ C @ C B ?S @ C 1 B ? 2 i
g
> , ~ºµb~¬b«§+~§ Ù ¶¶"«§ ¼J Ü Ì«
Ïh¯ºn¨0§°¨n«¨xª ²¨¨nªQ«>²´"°¬b«2¾«|§¨x~°b§«$¼©Ë²M¸J»~°b®#ÄÕ ¶ ²³§ºn~°b§¸J~«»B¼Jºa
@
@
e
@ 2 #" ) ! 8 1G? 2 # @ C e #" ) ! 8 1G? 2 # @ C
f f
@
e _") ! 8 15? 2 # @ C f C ! P 1G? 2 # C
@
e
f #" ) !'P
15?
2
#
g 1Ti
À0ª«|®n«P ²®x«4®x«¾9²³~§ ¼J«®#¨n«|®n¾ºM
3$4(567
b0Ë0ªQ«4Ë0ªQ«°b®x«¾ ºx«¶ ²³®²¨x«º·d°b§J¬«®nµ«§ ·d«~§J¨n°©¨XÀ°¶ ²®x¨nº
Ò¨xª«>·d°§ºn¨d²§J¨0~º,¼J«¶"«§ ¼J«§J¨2°b§¨nª«#¨X¸¶"«2°±°b¶¨x°§ ª«®x«>²©Î0²»~»ÃÒkÖ¶¨x°§
Ò¨xª«¾°¾«§J¨xº²³§ ¼¶ºn«|¯ ¼J°b¾°b¾«§J¨nº5·d°§J¨²~§+¨nªQ«¼ÄfÊ«®x«§J¨,»B²¨x¨nB·d«²¶¶Q®n°²·xª
,ݲ~§»~¸+¨nª«Ë0ª«°b®x«¾ ºn¨d²¨n«|º>¨xª ²¨°®¼J«|®°±4·d°§J¬b«®xµb«§ ·h«°b§Q«º#§Qª«®n«|§¨x»~¸ ·d°§J¨²~§«M¼É§Ì²»~»´Q§Q°b¾B²³»
»B²³¨n¨nB·d«,²³¶¶®x°²b·xª«|º$¦X§J¨x«®n«ºx¨n~§µ»¸¨xª«2§« ¨ºn«$·d¨x~°b§+ºnª°À0ºª°ÀÚ¨nªQ«4°®¼J«|®5°³±·d°§¬«®nµ«§ ·d«·²³§´"«2¨n¯§«M¼
´J¸²©¬«®n¸ºn¶"«$·dÃÅ··d°§ºn¨x®n¯ ·h¨n~°b§¶Q®n~§ ·d~¶»~«b
0¦X§¨nª«»B²¨x«®2ºn~¾¯»B²¨x~°b§º0À«>À0~»~»ºx««¨nª"²¨2°b®d¼J«®·²³§§°b¨2´"«>~¾¶®n°¬«$¼+~§¨nªQ«²¶¶®x°²·nªQ«º2§Q°À0§¿~§
»~~¨n«|®²¨x¯®n«Ý+°b®x«°¬b«®À«4À0~»~»« ¶»B²³§»B²¨n«|®5~§¨xª«,ºx~¾¯»B²¨n~°§ºª°ÀI¨n°·d°§ ¼J¯ ·d¨0« 9·d~«§J¨n»~¸·d°b§J¬«®nµ«§ ·d«
ºx¶ «|«$¼Ú¾«M²ºn¯Q®n«¾«§J¨nº~§ ²³¶¶»~B·²¨x°§º$IÇ-°b¨d²´»~¸ÛÀ«+¶®n°¬«$¼O°b§»~¸O¨nª ²³¨°b®¼Ä«®°³±·d°§¬«®nµ«§ ·d«É«$ͯ ²³»º
²³¨2»«M²ºn¨#°b§«b¹G¶"°bºnºx~´»~¸¿ª~µª«®°®¼J«®©·d°b¯»B¼É´"«·d°b§J¨d²~§«$¼¹¨xª°b¯QµbªÉºn~¾¯»B²¨x°§º0~§ ¼JB·²³¨n«¨nªQ«°b¶Q¶ °ºn~¨n«b¹
«|§¨x~®n«»~¸J
+ 0˰+²b·xª~«¬b«°®¼J«|®°±,·d°b§J¬«®nµ«§ ·d«°b§«b¹G¨nªQ«¨nª«°®n«¾ ºn¨²³¨n«º#¨nª ²³¨¨nª«²³¶¶®x° Ä~¾+²¨n~§µ¾°b¾«§J¨nº0°±0¨nªQ«
¼Äºd·d®n«|¨n«¶®x°·d«ºnº¾¯ºn¨·d°b§J¬b«|®nµb«2À0~¨nª+°®¼J«®0¨XÀ°©¨n°À0²®¼Ñ¨nª«2¾°b¾«|§¨xº°±
¨xª«·d°b§J¨x§J¯°¯ººn«$·d¯Q®n~¨X¸+¶®n°³Ò
·h«ºnºM¹´"«$·²¯Qºn«°§«¼J«µb®x««2~º5»~°bºx¨5À0~¨xªºn¯Q¾¾9²³¨n~°b§©°¬b«|®5¨x¾«b %Q¯®n¨xª«®x¾°b®x«b¹J°b§Q«4§Q««$¼Jº0¨xª«2º²¾«,°®¼J«®
°³±×·d°§¬«®nµ«§ ·d«+~§É¨nª«¶ºx«¯ ¼J°¾°¾«§J¨nºMË0ª~º~º2§°¨4°§»~¸¿²¨n«$·xª§Q½·|²»¾9²¨x¨n«®2´¯Q¨4!« ĶQ»½²³~§ºÀ0ªJ¸¨nªQ«
¾°¼J«»"¶®n°¶ °ºn«$¼+´J¸Ë0B²§+¼J°J«º,§Q°b¨0¶"«®X±w°®n¾ ´ «|¨n¨n«|®$
>! Ë0ª«Î0Ø,Ø&(*/ )-¾°¼J«»·d°b§J¬«®nµ«º,À0~¨nª+°®¼J«®
>! Ë0ª« &Ø (-¾°¼J«»·d°§J¬b«®xµb«ºÀ0¨xª°®¼J«®©b
>! Ë0ª«2Ë0½²³§ (*)0-¾°Ä¼Ä«»·h°b§J¬b«®xµb«ºÀ0~¨nª+°b®d¼J«®b
Ë0ª«2¶®x°J°±wº,²³»»¯Qºn«2¨nª«©²´"°¬b«,Ë0ªQ«°b®x«¾ ²§ ¼ÑÀ0~»»´"«2µb~¬b«|§9~§ Ù ¶¶"«§ ¼J Î2
>
4 @
5? v×Ç0°¨nñw¸J~§µÌ¨nª"²¨À«¿²³®n«µ«¨n¨x§Qµ ²»~À0²M¸Jººn~¾~»B²®©¶B·d¨n¯Q®n«º±w°®9·d°§¬«®nµ«§ ·d«É~§I¶®x½·h«b¹,À«À0~»~»
¶®x«ºn«|§¨°b§»~¸¨XÀ°¶B·d¨n¯Q®n«ºÀ0¨xª¨xª~º0¨n°b¶Q½·³
10.16
10.3
BS
Tian
10.14
BS
JR
10.2
10.12
10.1
10.1
10.06
price
price
10.08
10.04
10.02
10
9.9
10
9.98
9.8
9.96
9.94
¥ *G£ +-Ë0B²§J¶®nB·d«
50
100
150
refinement n
> @/ [ @*b @/H
200
9.7
250
300
20
¥ *G£ +
40
60
> @/ % @*b
B @ E 3
@ 80
100
refinement n
120
140
Ø J¶®nB·d«
F@<H B@ E @ ¦X§ÉÀ0ª ²³¨,±L°b»~»~°À0º,À«À0»~»¼J«¾°b§ºx¨n®d²¨n«©¨nªQ«¨nª®x««»B²¨x¨nB·d«²¶¶Q®n°²·xª«ºÀ0~¨nªÉ¨nªQ®n««+¼JÃÊ«®x«§J¨ºn¨x®n~Âb«|º$Ö§«·²³§
®x«$·d°µb§~Ы~§¨nª«ºx~¾¯»B²¨n~°§º¨nª ²³¨0¨nª««|®n®n°®,·²§²³»À0²M¸Jº5´"«>¼J°¾~§ ²³¨n«$¼©´J¸9²³§¯¶Q¶ «®´ °¯§ ¼°±°b®d¼J«® @
1
0.01
CRR
4/t
2nd moment
3rd moment
4th pseudo-moment
0.1/t**2
0.001
0.1
0.0001
error
error
0.01
1e-05
1e-06
0.001
1e-07
0.0001
1e-08
1e-05
10
¥ *G£ +Î0Ø,Ø
1e-09
100
refinement n
1000
10
1
Î0Ø,Ø ¾°¾«§J¨nº
100
refinement n
1000
0.01
Tian
4/t
0.1
0.01
0.0001
0.001
1e-05
0.0001
1e-06
1e-05
1e-07
1e-06
1e-08
1e-07
10
¥ *G£ +-Ë0B²§J«®n®x°b®
100
refinement n
2nd moment
3rd moment
4th pseudo-moment
0.1/t**2
0.001
error
error
«®x®n°b®
¥ *G£ +
> @* % @ )[
@*J B(@3 E @ 1e-09
1000
10
¥ *G£ +0Ë0B²³§¾°b¾«§J¨nº
> @* % @/ [
@*J B(@ [ E @ 100
refinement n
1000
1
0.01
JR
4/t
2nd moment
3rd moment
4th pseudo-moment
0.1/t**2
0.001
0.1
0.0001
error
error
0.01
0.001
1e-05
1e-06
0.0001
1e-07
1e-05
1e-08
1e-06
10
¥ *G£ + 1e-09
100
refinement n
1000
10
Ø «®x®n°®
¥ *G£ +
> @* [ @/ @*J B(@ [ E @3 100
refinement n
Ø ¾°b¾«§J¨nº
1000
I»~«$²³ºn«§°¨n«b¹¨xª ²¨#¨nª«»B²¨x¨nB·d«²³¶¶®n°J²b·xªÌ´J¸Ë0B²§Éª ²º#«¶»~½·h¨x«»~¸ÅÄ«M¼É¨nª«©Å ®xºn¨©¨XÀ5°+¾°b¾«|§¨xº$¹"¨nªJ¯º#´ «~§µ
« ²·d¨n»~¸ Ý+°®n«°¬«®2À5«©®n«|¾9²®xÂ+¨nª ²³¨4¨xª«©¾«¨xª°¼+°±×¼Ä«¨n«®x¾~§~§µ¨xª«>°®¼J«|® °³±Ä·h°b§J¬b«®xµb«§"·d«©±w®n°¾)¨nª ²³¨2°±
¨xª«2¾°b¾«§J¨nº5²³§ ¼¶ºn«|¯ ¼J°b¾°b¾«§J¨nºÀ°b®xº0¬«®n¸À5«|»»X
+ &
%4¡ ,| "£¤"¥w¡Ú¡ - ¥B¡ /5¥ .6"/4¡
¢ +76,0¥B'Ì¥5/) ¡;+¢I¤¡;+ * +¤+8) ¡)7+ J¥5+,
"J° ±{²®M¹À«9¼J«|º·d®x´"«$¼O¨nªQ«ºn°¯®·h«º°±,·d°b§J¬«®nµ«§ ·d«+¶ ²³¨n¨n«|®n§º~§Ì« ~ºn¨x§Qµ»B²¨n¨xB·d«²¶¶Q®n°²·xª«º$ÉÝ+°b®x«°¬b«|®$¹À«
¼J«|®n~¬b«$¼Ñ¨nª«°b®d¼J«®0°±
·h°b§J¬b«®xµb«§"·d«2§¨xª«2¶®n«|¬~°¯º5ºx«$·d¨x°§Á0«|®n«²»~¨n«|®n§ ²³¨n~¬b«0¾«¨nª°¼Jº¨n°©·d°b§ºx¨n®x¯ ·d¨,´Q§Q°b¾B²³»
¨x®n««ºÀ0~»»´"« ¼Äºd·d¯ºxºn«$¼IÖ¯®²³~¾ >¼Ä«dÅ §~¨n~°§Û°±2´~§°b¾B²»¨n®x««º$¹-À0ªB·nªO´ «h±w°b®x«ª ²§"¼Û°´ «|¸Û¨nªQ«9ºx°b¯®d·d«º°±
~®n®x«µb¯Q»½²³®n~¨n~«º0ºx~¾¶»~¸´J¸¼JÃÊ«®x«§J¨2´¯¨,°§»~¸ºx»~µª¨x»~¸¾°¼JÃÅ «M¼¼J«dÅ"§~¨n~°b§°³±¨n®n«|«>¶ ²³®²³¾«¨x«®nº K 1 12,²§"¼ = 1 12
¦X§Éºn«$·h¨n~°b8
§ À5«©º²MÀ*¨nª ²³¨,~®n®x«µb¯»B²³®n~¨n~«º2«¬°b»~¬b«©´ «M·²¯ºx«¨nª«®x«»B²¨x¬«¶"°bºn~¨x°§¿°³±5¨nªQ«>ºx¨n®n~«À0¨xª~§¨nªQ«>¨x®n««
¬b²³®n~«º$Ç-°À¹JÀ5«2ºx¯µbµ«ºn¨¨nª ²³¨0¨nª~º®n«»B²¨x~¬b«2¶ °ºn~¨n~°§ºnªQ°b¯»B¼´ «Å «$¼§ºn°¾«,ºx«§ºx«b
Ë0ª«,·h°b§ºx¨n®n¯"·d¨n~°b§+°³±¨x®n««º0¨x°>²b·xª~«¬«,·d°b§J¬«®nµ«§ ·d«2¨n°©Ü»B²b·xÂÒ "·xª°b»~«º0¼J°J«º§°b¨5¼Ä«¶ «|§ ¼°§²§J¸>¶"²®n¨xB·d¯»B²®µ®nB¼
°®4µ®nB¼+ºx«$ͯ«§ ·d«À0ª ²¨xºn°J«¬«®$ !,B·d«¬«®nºd²À5«©¶ °ºn¨x¯»B²¨n«¨xª ²¨2¨xª«&·h°b§ºx¨n®n¯"·d¨n~°b§Ì°³±5ºn«MÍb¯Q«§J¨nB²»¨x®n««|º>¾+²M¸´"«
»~~§Âb«M¼¨n°²·xª~«¬b«2ª°¾°bµ«§«~¨X¸©À0~¨nª¨xª«»~°·²¨x~°b§°³±
¨xª«ºn¨x®n~Âb«
Ë0ª~º5·h°b§ ·d«|¶¨0´®n~§µº5¯Q¶¨nªQ«4ͯ«ºx¨n~°b§¨n°©¨nª«2·d°b®x®n«$·d¨0®x«»B²¨x¬«,¶"°bºx¨x~°b§°±¨xª«,ºx¨n®x«b I~¨nªQ°b¯¨!« Ĩx«§ ¼J~§µ©¨nªQº
ͯ«ºx¨n~°b§G¹ºn°¾«®n«dÓ «M·d¨n~°b§Qº4ºx¶ «M²Â+~§±X²M¬b°b®0°³±²Ñ»°·²³¨n~°b§¶®n«M·d~ºn«»~¸+~§9¨xª«¾½¼Q¼J»~«,°±¨XÀ°>ºx¯®n®x°b¯§"¼J~§µ§°¼J«|º$
Ë0ª~º2·d°b¾¾~¨n¾«§J¨®x«ºn¯Q»¨xº~§+Å ~§µ¨nªQ«>®d²§ ¼J°¾)¬b²®x½²³´»~h« I¨n°~§J¨n«|µb«®2§J¯¾´"«®nºÀ0¨xª¿·d°§¨x~§¯Q¨X¸+·d°b®x®n«$·h¨n~°b§
°³±
°b§Q«4ª"²»Ã±Ý+°b®x«°¬b«®M¹~§»~~§«2À0~¨nª¨nª«ªQºx¨n°µb®²³¾ ·d°§ ·d«¶¨M¹"®n«$·d¨d²§µ¯»B²®xº0®n«¾+²~§¯§"¼J~¬B¼J«M¼
Üv¯¨,À0ªQ«®n«©ºnª°¯»B¼+À«2Å 9¨xª«©ºn¨n®x~Âb«°¬«®²³»»& IªJ¸+§°b¨0~§¨xª«>·d«§J¨x«®2°±5¨xª«¨n®x«« &OË0ª~º,¶Q®n°·d««$¼Ä§Qµ~¾¶»~~«º
¨xª ²¨G¨nª«ºx¨n®x«5~º²»~ÀײM¸Jº·d°§J¨²~§«M¼2~§¨nª«v´~§°b¾½²³»$¨x®n««µb®xB¼
ΰ§ºn«MÍb¯Q«§J¨n»~¸¹M¨xª«5¶"²¨nªQººn¯Q®n®n°¯§ ¼J~§µÍ¯ ²³§¨xñw¸
«¶»~B·d~¨n»~¸¨nª«¶®n°´ ²´Q»~~¨X¸0±L°b®¨xª°bºx«×²³ºnºx«¨¶®nB·d«ºÀ0ªB·xª#Å §~ºnª~§¨xª«§«$²®°³±Ä¨xª«ºn¨n®x~Âb«bÇ0°¨²´Q»¸J¹Mª«®x«×²·d¨n¯ ²³»~»¸
¨xª«¾°ºn¨2ºn«§Qºn~´»~«ºx¨x¯ ²¨x~°b§º>²³®n~ºn«Î°b§ºx«$ͯ«§J¨n»~¸J¹¨xª«¾°Ä¼Ä«»~¨nºx«»Ã±×ºx¶ «$²³ÂJº>~§É±{²M¬°b®2°±,¨xª~º>·d°b§Qºn¨n®x¯ ·d¨x°§
Üv«ºnB¼J«|º$¹ºn~§ ·h«¾°bºx¨,°³±×¨xª«>¨x~¾«¨x®²b¼Ä§Qµ9~§É°b¶¨x~°b§º2°··d¯®xº>~§Ì²¨kÒ{¨xª«dÒX¾°b§Q«¸²§ ¼§«$²³®XÒX¨nª«dÒX¾°§«¸°¶¨n~°§º
°§»~¸¹²b·h¨n¯ ²³»»~¸2¨nªQº®x«$ͯ~®n«¾«§J¨·xª ²³§µb«|º
°§»~¸¾~§°®n»~¸0¨nªQ«0ºn¨n®x¯ ·d¨x¯®n«bvË0ª«ºx«0®n«dÓ «M·d¨n~°b§Qº5»~««$¼#¨n°2¨xª«,¼J«dÅ §Q¨x~°b§
°³±
¨nªQ«,±L°b»~»~°À0§Qµ¾°¼J«»X
1 2
K 1 12 S 1 12 C = 1 12 W 1 12 @P« ¶ > BNFJ1 A V 1 2
K 1 12 S 1 12 C = 1 12 W 1 12 @P« ¶ > E GFH1 A V 1 2
S 1 12 C W 1 12 @*
1 +2
K 1 12 = 1 12 @P« ¶ > HFJ1"»~§ 1 EF > 2 A
¦X§+²b··d°®¼²³§ ·d«¨x°>¨xª«²¶¶Q®n°²·xª°±Ë0B²§À«¼Ä«dÅ §«²§«$ͯ ²¨x~°b§ºx¸Jºn¨n«|¾ À0ª«|®n« 1 2²§ ¼ 1 2GÅ 9¨nªQ«,Å"®nºn¨
¨XÀ°¾°b¾«|§¨xº,²º2ºn¯ 9·d~«§J¨4´Q¯¨2§°b¨2§«$·h«ºnºd²®n¸Û·h°b§ ¼J~¨x°§º,¨x°²b·xª~«¬b«·d°§¬«®nµ«§ ·d«¨n°+¨nªQ«>µ¬«§¿·d°§J¨n~§J¯°b¯º
¼J~ºx¨n®n~´¯Q¨n~°b§'Õͯ ²¨x~°b§ 1 2©« ¶®n«|ºnºn«|º¨nª ²³¨¨nª«¶ °§J¨¶®x°b´ ²³´~»~¨x~«º>ºx¯¾6¯¶Ú¨n°Ì°§«b'¦X¾¶"°b®x¨²³§¨x»~¸¹~§
¼ÄfÊ«®x«§ ·d«©¨n°Ë0B²§G¹À0ª°À0²ºx¨n«$¼+¨xª«®x«¾9²³§Q§Qµ&¼J«|µb®n«|«°±±w®x««$¼J°¾)¨n°©Å ¨nªQ«¨nª~®d¼¾°b¾«§J¨$¹À5«2~¾¶»~«¾«§J¨
²·h°b§ ¼J~¨x°§µb¯"²®²³§J¨n««~§µ¨xª ²¨¨nª«©ºn¨x®n~Âb«#º¶ °ºn~¨n~°§«$¼+²¨¨nª«·d«§J¨n«|®4°³±¨nª«©¨n®x««±L°b®0«¬«®n¸¨x®n««©®n«hÅ §«¾«§J¨
²³¨2¾9²¨x¯®n~¨X¸J I~¨nªÉ«¬b«§Ì§J¯¾´"«®2°±0ºn¨x«¶º©¨nª~º¶"°bºx~¨n~°b§Ì·d°b§J¨d²~§º>²¨n«|®n¾~§ ²³»§°¼J«9²³§ ¼À0~¨xª¿°¼¼É§J¯¾´"«®
°³±ºn¨n«|¶º,¨xª~º,¶"°bºn~¨x°§+¶®n«$·hºx«»~¸+ºn«¶ ²³®²³¨n«º2¨XÀ°¨n«®x¾~§ ²»§Q°Ä¼Ä«º$Ë0ªJ¯º$¹"À«2¾9²M¸®n«|ºn¨n®xB·d¨2°b¯®xºn«»Ã±w«º¨n°«¬«§É°b®
°¼¼®x«dÅ §«¾«§J¨nº°b§»~¸9¼Ä«¶ «|§ ¼J~§µ9²³§+¨nª«¼J«ºx®x«$¼+ºx«¶ ²®d²¨x°§®n¯Q»«5Ë0ªQº0¶Q®n°·d««$¼Ä§Qµ ¼Ä°«|º,§°¨,¼J«¬b²»~¯ ²³¨n«2°b¯Q®
²³¶¶®x°²b·xªÉºn~§ ·d«©¨n®x««·²³»½·h¯»B²¨n~°§º4¼J°+§°¨4¼J«|¶ «§"¼É°b§Ì²§J¸+ºn¶"«$·dÃÅ··xª°bB·d«°³±5®n«hÅ §«¾«§J¨$>Üv«ºx½¼Ä«º$¹G«¬b«|§Û²§ ¼
°¼¼®x«dÅ §«¾«§J¨nº2·d°b§J¬b«|®nµb«#¾°b§Q°b¨n°§B·²»~»~¸J¹$®x«ºn¶"«$·d¨x~¬b«»~¸J,Ü«|»°À¹QÀ5«2¶®x«ºn«|§¨2¨xª«« ¶»~B·d~¨0« ¶®n«|ºnºn~°§º±w°b®0¨xª«
¨x®n««#¶ ²®d²¾«|¨n«®xº,²º0¯§Bͯ«2ºn°»~¯¨n~°b§°±¨nª««MÍb¯"²¨n~°§ºn¸Jºx¨n«¾ ²³´ °¬«b
1 2
1 2
1 2
1 + 2
K 1 1 2 @ ( C 1 C = 1 1 2 @ FJK
S 1 1 2 @ K 1 B D 1 2 = 1 = 11 2 12
W 1 1 2 @/TS 1 12
À0ªQ«®n« @ + 2
C 9 ,-F
(
9
C
-
Î °§ºnB¼J«|®$¹M¨nª«0·d°§¬«®nµ«§ ·d«¶ ²³¨n¨n«|®n§©°±¨nª«Î0Ø,ØÒX¾°¼J«|»±w°®
²³¨XÒX¨nª«dÒX¾°§«¸0°¶¨n~°b§Qº$GË0ª«®n«~º¾«®x«»~¸,°º·d~»~»B²¨n~°§
°³±5¨nªQ«°b¶¨x°§¿¶Q®nB·d«©À0¨xª°b¯Q¨²³§J¸ÀײM¬«º$ Ù »°§µ«|¬b«§Û²³§ ¼°¼¼®n«hÅ §«¾«§J¨nº>²³»°§«b¹À«©ª ²M¬b«²¾°b§Q°b¨n°§B·²»
·h°b§J¬b«®xµb«§"·d«+¶ ²¨x¨n«®x§¿Ç-°b¨²³´»~¸¹
¨nª~º>·d°§J¬b«®xµb«§ ·h«9¶"²¨n¨x«®n§O~º>·d°b§Qºn«®x¬b«$¼È±w°®>²§J¸Ì·nªQ°bB·d«°³±×¨xª«ºx¨n®x«ª«|®n«b
Ø-«¾9²³®nÂb²³´»~¸¹Mºn¯"·nªºn¾°J°¨nª·d°§¬«®nµ«§ ·d«©¶ ²³¨n¨n«|®n§º2·²§+ºn«|®n¬b«±w°b®0¨xª«²¶¶»~B·²³¨n~°b§°±
!« Ĩx®²³¶ °»½²³¨n~°b§¾«¨xª°¼Jº$
Ø-«¾«|¾´ «®©°b¯Q®®n«hÓ «$·d¨x°§º«$²³®n»~~«®$ Ù ´"°¬b«É²»~»X¹5À«¿¼J«|ºn~®n«$¼P¨n°O~¾¶®x°¬b«+¨nª«É²b··d¯Q®²b·h¸ °³±#²³¶¶®x°Ä~¾+²¨n~°§
-§J±w°®n¨n¯Q§ ²¨x«»~¸¹"¨nªQ«¾°¼J«»²´"°¬b«¼J°J«º§Q°b¨2ºn¯ ·|·d««$¼È§Ì²b·|·d«»~«®²³¨n~§µ¿·h°b§J¬b«®xµb«§"·d«ºn¶"««$¼+Á0«|®n«b¹GÀ5«¶Q®n°b¶"°bºx«
²³§«§J¨x®x«»~¸§«|À/²¶¶®x°²·nª¯ºn~§µ©±X²~®n»~¸©°b»B¼©Å §"¼J~§µbº0°³±
¾9²³¨nª«|¾9²¨xB·²» ²³¶¶®x°Ä~¾+²¨n~°§>¨xª«°b®x¸Üv«$·²³¯ºn«#°±~¨nº
ºx~¾¶»~B·d~¨X¸¹¨nªQ«>´~§°¾B²»¼J~¨n®x~´¯¨n~°§ ²³»~ÀײM¸Jº,ª"²º,ºx«®n¬«$¼Ì²º2²¬«®n¸¶"°b¶Q¯»B²®4¼Äºx¨n®x´Q¯¨n~°b§G,Ç0°¨²´Q»¸J¹ ²·d¨n¯"²»~»~¸
¨xª«Å ®nºx¨ ·h«§J¨n®²³»»~~¾~¨>¨xª«°®n«¾À0²º¶Q®n°¬b«M¼Ú±w°®¨xª~º ¼Äºx¨n®x´Q¯¨n~°b§P¨X¸¶"«b ',«|ºn¶~¨n«É°±©¨nª«Éºn~¾¶Q»B·d~¨X¸J¹¨xª«
²³¶¶»~B·²¨x~°b§Û°³±¨nª«+±L°b®n¾¯Q»½² º·h¯¾´ «|®nºn°¾«b¹v´ «M·²¯ºx«¿¨xª«¿·d°¾¶¯Q¨²¨x~°b§Û¾~µbªJ¨~§J¬b°b»~¬«±{²·d¨n°®nB²»~º°±»B²®xµb«
~§J¨n«|µb«®xº4°®,¨xª«>ºx¯¾9²³¨n~°b§+°³±v²Ñ»B²®nµ«§J¯¾«|®,°±§"¼J~¬B¼J¯"²»¨n«|®n¾ºM,Ë0ªQ«®n«d±L°b®$¹
§°b®x¾9²»²³¶¶®x° Ä~¾+²¨n~°§º5¨x°¨xª«
´Q§Q°b¾B²³»¼J~ºn¨n®x~´¯¨n~°§¿À«®n«+¼J«®x¬«$¼Õºn¶"«$·dB²»~»~¸¹G¨nªQ«Î0²¾¶ÄÒ I²¯Q»ºx°b§ (Q-¾«|¨nª°¼É²§ ¼¨nª«²³¶¶®x° Ä~¾+²¨n~°§º
°³± I«|Ð|«®4²³§ "
¼ I®²³¨n4¨ ( j -
®x«¬b«M²»
²©®x«¾9²³®nÂb²³´»~«4ͯ ²»~~¨X¸°±5²··d¯®d²b·d¸ "į¾¾+²®n~ºx§Qµ ¹«¬«§J¨n¯ ²³»~»¸¨nª«|ºn«©§°b®x¾9²»
²³¶¶®x° Ä~¾+²¨n~°§º¼J«¨x«®n¾§Q«0¨nª«§Q¶¯¨°³±¨nª«ºn¨²³§ ¼²³®¼©§°®n¾9²³»±w¯§ ·d¨x~°b§À0ªB·xªºn¯¶Q¶ °ºn«$¼J»~¸²³¶¶®n° ~¾9²³¨n«º¨xª«
´Q§Q°b¾B²³»d±w°b®x¾¯»B²À0~¨nª2ºn¾+²»~»²§ ¼2¼J«$·h®n«$²³ºn~§µ2«®n®x°b®M
Üv¯¨ªQ«®n«b¹³°b¯®G¶®n°´»~«¾ ®x«¶®x«ºn«§J¨xº¨nª«v°b¶¶"°bºx¨x«5¼J~®n«M·d¨n~°b§G
ΰ¾¶¯Q¨²¨x~°b§°³±´~§°b¾½²³»b°¶¨n~°b§©¶®x½·h«º0«¬b«|§¨x¯ ²»~»~¸©§J¬°b»~¬b«º¨xª ²¨§Q°b®n¾+²» ·h°b¾¶"°b§Q«§J¨nº5²³®n«,²¶Q¶®n° ~¾9²³¨n«$¼#´J¸
´Q§Q°b¾B²³»Ä·h°b¾¶"°b§Q«§J¨nº$ IG«~Ы|®,²§ "
¼ I®d²¨n¨,¼Ä«®n~¬b«M¼¨xª«2~§¬«®nºx~°b§±L°b®n¾¯Q»½²¨x°>¨xª«>Î0²¾¶JÒ I
²³¯»~ºn°§>¾«¨nª°¼²§ ¼
ºx¶ «M·dÃÅ «$¼+¨xª«~§J¬b«®xºn~°b§Ñ±w°b®x¾¯»B²,°³±
¨xª«~®0¾«¨xª°¼~§+¨nª«·²³ºn«À0~¨xªB¼J«§J¨nB·²³»§J¯¾´"«®°±
ºx¯ ··d«|ºnºn«|º>²§ ¼Ñ±{²³~»º
Ç-°À¹JÀ5«2À0~»~»¼Ä«¾°§ºn¨x®²¨x«2ª°À ¨xª«ºx«2Å § ¼J~§µº,·²§+´"«2¯ºn«$¼ ¨n°·d°§ºn¨x®n¯ ·d¨2Î0Ø,Ø×Ò{»~~Âb«´~§°b¾½²³»J¾°¼J«»~º$
%Q°b®0²2µ¬«§®x«dÅ §«|¾«§J¨0¨nªQ«4~§J¬b«|®nºn~°§©±w°b®x¾¯»B²º²´"°¬b«ºn¶"«$·dñw¸¨xª«¼J~ºn¨x®n~´¯¨x°§¶ ²®d²¾«¨n«
® S¨n°²¶¶®x° ¾+²¨x«
1 2vÀ0~¨n
ª ^ 1 1S2À0ª«|§9¨xª«2ºn«¶"²®²³¨n~§µ¬b²³®nB²´»~T« I>~ºvÅ «$¼ ΰb§ºx~ºn¨n«|§¨x»~¸¹Å ~§µ I©¾¶»~~«º¶"°bºn~¨x°§~§µ¨xª«
ºx¨n®x«2ºn°b¾«À0ª«®x«,À0~¨nªQ§+²´Q§Q°b¾B²³»¨x®n««5Ö§ ·d«>²³µ²³§G¹bÀ«2»°·²³¨n«¨nª«2ºn¨x®n~Âb«²³¨5¨xª«·d«§J¨n«®°±¨nª«2¨n®x««>²³º5À«
Ïh¯ºn¨xfÅ"«$¼~§¨xª«©®n«dÓ «M·d¨n~°b§Qº«$²®x»~«|®$Ý+°®n«°¬«®$¹¨xª~º2¶®n~§ ·d~¶»~«>²³»~»°À0º0¨xª«©¯º²³µb«©°±5¨xª« IG«~Ы|®XÒ I®²³¨n¨¾«¨nª°¼
À0~¨xª« ¶»~B·d~¨0~§¬«®nºx~°b§®x¯»~«b¹À0ªQ«§+À5«2®n«|ºn¨n®xB·d¨2¨nª«2ºn«|¨,°±®n«dÅ"§«¾«|§¨xº,¨x°°Ä¼Q¼~§J¨n«|µb«®xº
C"pbjrXjneijkr×o"^`g`g_bilLadc,jr{jkc,edrwluLa>uSpbj2ijXrL^fzdeuB^fah_>ad}5uSpbjnlLj,edyyrXad^`c,e|u~^fad_led_biuSpbj4qk^fuwe|uB^fad_ad}g`^fuLjXrXeuStr{jv^`_uBpbj
ehyyjX_biM^f
^`_2uBpbjGtblStbedgt_bijXrXlwuLed_biM^`_ad}uBpbjGs^`_badc0^fedgiM^flLuBrL^`stbuB^fah_
@~_2uBpbjGtblStbedgblLjnuSty5d^fzjxluBpbjG_Mtc5sJjXr ad}lStbqnqxjnlLlwjxl^`_uBrL^fedgfldpbjXrXjm
^fl"^fijX_uB^`bjxi5o"^fuSp4uSpbjG_$tcsjkrah}ty~c,adzjkc,jL_uLlX\
-GuSpbjXrXo"^flwj"uBpbj"^`_zjkr{lS^fad_
qxadtgfisJj eqLp^fjxzjniG_Mtc,jXrL^fqnedg`gf|\K|^`_bqxj uSp^fl^`_zjkr{lS^fad_^fljk_bjXrXedg`gfzdedg`^fiuLaed_GybedrXedc,jxuwjkrblwjkgfjnqnuS^fad_m
^fuQqnadtgfisJjuwehstgfeuwjxiadr edyyrXad^`c,eduLjniyJadgf|_badc0^fedg`g`}ahrb$jxi ,lS^`c0^`gfedruLa
uSpbjQyr{aMqnjxjniM^`_bo"^fuSp×uSpbjlwuLed_biedrXiG_badrLc,edgd}ft_bqnuS^fad_\
7
© 564( 5@! >0! 1 ¯Q§~¬b«®xº²»~»~¸¬b²»~B¼ 2
) I42 C 1 )% I2 5)% S @ i (*)%hI- (*) - C (*) ( 1 Ú
- 5) g
À0~¨n
ª @X1 I¹ @.I C ¹ ²º0~§¶¯¨0°³±
¨xª«ºn¨d²§ ¼Q²®¼§°®n¾+²»Q±w¯§ ·d¨x~°b§
$M 5?d% ->/ >0! f·²ºx«JI
N1 @ 4I C
S @ «!ĶQ 1 C i 1 C j i g
g
$M 5?|
% -))/
0! ·²³ºn«J I C @X1O2 I C N1"@ I C S @ ! «¶#$ %" 1 C C 1 C j i'& (
g
)+,*
$ @ +
Ç-°b¨²³´»~¸¹¨nªQ«Î0²³¾¶JÒI²¯»~ºx°b§©±w°®n¾¯»B²2·²§+´"«>²¶Q¶»~«M¼©±w°®,²®n´Q¨x®²³®n¸9·xª°B·d«2°±
®x«dÅ §«|¾«§J¨$
-ºn~§µ²¶Q¶®n°~¾9²³¨n~°b§©®n¯»~« Ù ¹ ܹ°®,Î'À5«2°b´Q¨²~O
§ S²§"¼ S ²³º4¼J~ºn¨x®n~´¯¨x~°b§+¶ ²®d²¾«¨n«®xº0°±¨nªQ«¨XÀ5°´Q§Q°b¾B²³»
·d°¾¶"°b§«|§¨xº~§¨xª«,´~§°¾B²»J°¶¨n~°b§¶Q®nB·d~§µ#±w°b®x¾¯»B²"Ë0ª«§¹JÀ«¼J«®x¬«,¨x®n««2¶ ²³®²¾«¨n«|®nNº K 1 125²³§ ¼ = 1 12v´J¸²
ºx¾¶»~«¨n®xB·nÂË0ª«0§Q°²®x´~¨n®d²µb«,·h°b§ ¼J~¨x°§>~¾¶»~«|º
¨xª ²¨ S 1 12 @ 1 B D = 1 12 2 F 1 K 1 12 = 1 12 2Gª°»½¼Äº$ %¯®x¨nª«|®n¾°®n«b¹
S º0¼J«dÅ"§«$¼+¨x° S @ KF B D S˲ÂJ~§µ¨xª«ºn«#¨XÀ5°©®n«»B²³¨n~°b§º,²³º5«MÍb¯"²¨n~°§9ºx¸Jºn¨n«|¾ À0ªB·xª9·|²§+´ «2ºx°b»~¬b«$¼¯§Q½Í¯«|»¸
À0~¨nª®x«ºn¶"«$·d¨¨nO° K 1 12²§"¼ = 1 12|¹bÀ«,ºx¯ ··d««M¼+¨n°²b·Í¯~®x«4²2§«|À ´~§°¾B²»¾°¼J«»XË0ªQ«,±L°b®n¾¯Q»½²³º´"«»~°ÀIºn¯Q¾7¯Q¶
¨xª«2¾°¼J«»¶ ²³®²³¾«¨x«®nºMÇ0°b¨xB·d«b¹J¨nª"²\¨ 1 7I 1 12 20¼J«§°¨n«º2¨nªQ«·xª°ºn«§§J¬«®nºx°§±w¯Q§ ·d¨n~°§
1 + 2
S @X 1 = MI 1 12 2
C @
1 2I
-
C 1 + 2
1 + 2
1 + + 2
S @X 1 = MI 1 12 2
K @ B S S 1 1 112 2
BND S 1 12 K 1 12
= @
T S 1 12
"J«|«¾~§µ»¸J¹M¨nª«2®x«ºn¯»~¨x§Qµ´~§°b¾½²³»J¨n®n«|«4¶"²®²³¾«¨x«®nº,¼Ä¬«®nµ«,°§»~¸¬b«®x¸»~¨x¨n»~«0±w®x°b¾Ô¨xª°bºx«2°±¶®x«¬J°¯º0¾°¼J«|»ºM¹
´¯Q¨²³ºn¨x°b§~ºnªQ§Qµb»~¸¹|¨nª«,·h°b§J¬b«®xµb«§"·d«,¶®x°b¶"«®n¨x~«º5À0~¨xª¨nª«,·h°b¾¶Q¯¨²³¨n~°b§2°±°b¶Q¨n~°b§¶Q®nB·d«º5·xª ²³§µb«|º5¼J®²³¾9²³¨nB·²»~»~¸J
Ç-«¬b«®x¨nª«|»«|ºnº$¹À0~¨xª~§¨nª~º·d»B²ºxº°±¾°¼J«»~º,¨xª«¶ ²³®n¨nB·d¯Q»½²³®,¨xª«°b®x«¨nB·²³»5´Q¯~»½¼Ä§Qµ´»~°Ä·xÂJº±w°b®2À0ªB·xªÉ¨nª«Î0ØØÒ
¾°Ä¼Ä«»´ «$·|²¾«Ñ±{²¾°b¯º2²®x««§J¨n~®x«»~¸¿¨x®²§QºX±w«®x«$¼O´J¸ ·d°§ºn¨x®n¯ ·h¨n~°b§ÉܯQ¨>¾°b®n«|°¬b«®M¹¨nª~º©¾°¼J«»·d°§ºn¨x®n¯ ·h¨n~°b§
¶®x°Å ¨xº©±w®n°¾ ¨xª«¿²³¨n¨n®x~´¯¨n«|º°±2¨nª«É·nªQ°bºn«|§I§°b®x¾9²³»×²³¶¶®n° ~¾9²³¨n~°b§GÛÜ«|»°À¹¨xª«+Å µ¯®n«º+¼J«¾°b§ºx¨n®²³¨n«+¨nªQ«
ºx¨n®n«|§µb¨xª°±¨nª«¾«¨nªQ°Ä¼Ñ~§9²¶Q¶®n° ~¾9²³¨n~§µ2°b¶¨x~°b§¶Q®nB·d«º§+·d°b¾¶ ²®x~ºn°b§¨n°¶®x«¬J~°b¯ºx»¸!« Ä~ºx¨n~§µ¾°¼J«|»ºM
19.699
1
BS
PP1
PP2
CP
19.698
19.697
PP1
PP2
CP
1/t
1/t**2
0.1
0.01
19.696
19.695
error
price
19.694
0.001
0.0001
19.693
1e-05
19.692
1e-06
19.691
19.69
40
¥ *G£ +A-¶®x½·h«
60
80
100
refinement n
120
1e-07
140
10
> @* % @ )[
@*J B(@3 E @ 14.234
¥ *G£ +0B -«®n®x°b®
100
refinement n
1000
1
BS
PP1
PP2
CP
14.232
PP1
PP2
CP
1/t
1/t**2
0.1
14.23
0.01
error
price
14.228
14.226
0.001
0.0001
14.224
1e-05
14.222
1e-06
14.22
14.218
40
¥ *G£ + -¶®x½·h«
60
80
100
refinement n
120
1e-07
140
10
> @* % @/ [
@*J B(@ [ E @ 10.022
¥ *G£ + -«®n®x°b®
100
refinement n
1000
1
BS
PP1
PP2
CP
10.02
PP1
PP2
CP
1/t
1/t**2
0.1
10.018
0.01
error
price
10.016
10.014
0.001
0.0001
10.012
1e-05
10.01
1e-06
10.008
10.006
40
¥ *G£ + -¶®x½·h«
60
80
100
refinement n
120
140
> @* % @/ [
@*J
1e-07
10
B(@ [ E @ ¥ *G£ + -«®n®x°b®
100
refinement n
1000
Ù ¨5¨xª«2¾°¾«§J¨À«4²³®n«§°b¨0²´»~«¨n°µ¬«4²2ºx¨n®x½·h¨0¶®n°J°±°³±
¨xª«,µ®n«$²³¨n«®0°®¼J«®0°³±
·d°b§J¬«®nµ«§ ·d«bÁ0°À«¬b«|®$¹JÀ«
´"«»~~«¬b«2¨nª"²¨¨0ª"²º´ «$·h°b¾«·d»~«$²³®±w®n°¾Ô¨nªQ«²³´ °¬«2ºn~¾¯»B²¨x°§º «ºn¶"«$·dB²³»»~¸f±°§«·d°b¾¶ ²®x«º0¨nªQ«¾ÔÀ0~¨nª+¨nªQ«
¶®x«¬J~°b¯º0ºx¾¯Q»½²³¨n~°b§Qº
°³±¨nª«2¾°¼J«|»º~§»~~¨n«®d²¨x¯®n« ¹J¨xª ²¨0°¯®0¾°¼J«|»º´"«ª ²M¬b«2¾¯ ·xª´ «¨x¨n«®O
¨xª«2°b®d¼J«®,°³±·d°§J¬b«®xµb«§ ·h«~º0§"·d®n«$²³ºn«$¼ ´¸°b§«
¨xª«·d°b§Qºn¨²³§J¨,ÎI~º,²´"°b¯Q¨ °³±
¨xª«2¯ºn¯ ²³»
·d°§ºn¨d²§J¨
¨xª«4·h°b§J¬b«®xµb«§"·d«2ºnª°À0º¬«®n¸»~~¨n¨x»~«0°bº·h»~»B²¨x~§µ2¶ ²¨x¨n«®x§º×²³§ ¼~º~§©±{²·d¨¾°b§°¨n°b§Q½·|²»~»~¸4·d°§¬«®nµ~§µ>¨x°¨nªQ«
Üv»B²b·xÂ#Ò "Q·nªQ°b»~«º0¶®x½·h«
Ö¯®¨nª«°®n«¾ ~§ ¨xª«É»½²³ºn¨ºn«$·d¨x~°b§ ¼J°J«º+§°¨¸J~«»B¼Ú²Ì¶®x°°³±2±w°b®´ «¨x¨n«®·d°§J¬b«®xµb«§ ·h«É±w°b®¨n«$·xª§B·²³»2®n«$²³ºn°b§Qº$
Á-°À5«¬«®$¹À«·²§Ú¯ºx«~¨>¨x°Ì« ¶»B²~§Û¨xª«´"«¨n¨x«®9·d°§J¬b«®xµb«§ ·h«b'Ë0ª«+¶®n°J°±~º©¾9²~§»~¸Ïd¯ºx¨¯ºn~§µÌ²É˲M¸»~°®XÒ
Õ ¶ ²³§ºn~°§ "ħ"·d«0¨nª~º°§«,~º!« Q²·d¨$¹¯¶¨x°¨nª«¯§ÂJ§°À0§©®n«¾+²~§ ¼J«|®¨n«®x¾ºM¹b°§«0« ¶ «M·d¨nº0¨xª ²¨5²³»~» ·d°b§J¬«®nµ«§ ·d«
¶ ²³¨n¨x«®n§ºM¹ºx¯ ·xª9²º°bºd·d~»~»½²³¨n~°b§²³§ ¼©°b®d¼J«®,²³®n«,®x«dÓ «$·h¨n«$¼~§¨nª«¼J«®x~¬b²¨n~¬«º$ Ì«,®x«¾«¾´"«®±L®n°b¾ ¨xª«2¶®n°J°±¨xª ²¨
¨xª«2«®n®x°b® @ ~º×¼J°¾~§ ²³¨n«$¼Ñ´¸
@
@
e
2
C
#" ) !
8 1G? 2 #
f @
e _") ! 8 15? 2 f C ! P 1G? 2 # @
@ C e
# @ C C
#" ) !
8 1G? 2 # @ f
@
e #" ) !'P 15? 2
f
#
g 1Ti
À0ª«|®n«
"$'&#)6+ «¶ > 1 = E 9 . ? 2 A
9 E 9 . ?
8 15? > 2 @ 9 E"1$'&#)6+ ? ! 1 = E 9 . ? 2 C E 9 . ? # «¶ > 1 = E 9 . ? 2 A
2
¦L±×À«ºn«¨ 8 @C $ " +2$'&#)6+ ±w°® ?= ([ - 1 ºx««
«¾¾+²§ Ù ¶Q¶ «§"¼J ÈÜ 22À«§°b¨x«>¨xª ²¨ ¨xª«¸Ì²®x«
8
15? > 2
@
20000
c2
c3
15000
10000
5000
0
-5000
-10000
-15000
-20000
90
¥ *£ + 95
8
100
S
15? > 2,²³§
105
110
¼ 8 15? > v2 ±w°b®F ºx¸¾¾«¨x®nB·²³» ²®x°b¯§ ¼ Ì8
Ý+°®n«°¬«®À«9®x«¾9²³®n¹¨xª ²¨ 8 15? > 2²§"¼ 8 15? > 2²®n«É·d®n~¨nB·²³»0~§ ? ²º ? ¹¨xª ²¨~º²º¨xª«+®n«¾+²~§~§µ
¨x¾« ¨x° J¾²¨n¯Q®n~¨X¸´"«$·d°b¾«º(
Ë0ª«2¨x¾« ¶"°b~§J¨¨0®n¯»~«º¨nª«¾+²~¾>¯Q¾/°±¨xª«¨XÀ°©±w¯§ ·d¨x~°b§º¾°®n«°¬«®$¹Jºx~§ ·d«¨xª««¶"°b§«§J¨x½²³»±w¯Q§ ·d¨n~°§9´"«dÒ
·d°¾«º0¼J°¾~§ ²¨x~§µ2¬b«®x¸9ͯB·x»~¸J¹b~¨0µ¬«º0¨nªQ« À0B¼J¨n7ª ©°±¨xª«±w¯§ ·d¨x~°b§ºM
Ì«²»~ºx°®n«$·d°µb§~Ы¹¨nª ²³¨+
8 15? > 2~º0¶"°bºn~¨x¬«2±w°® > 8 ²§ ¼§«µJ²¨x¬«°b¨xª«®xÀ0ºx«bË0ª«¾9²0¾+²2²®n«#»¸J~§µ~§
#² ®²³§µb«2»~«ºxº,¨nª"²§ 8 )
Ì«©´"«»~«|¬b«¨xª ²¨#¨nª~º>·xª ²³§µb~§µ+ºn~µ§Ìº2®x«ºn¶"°b§ºx~´»~«±w°®°bºd·d~»»~B²¨x~§µ9·d°§J¬b«®xµb«§ ·h«9¶"²¨n¨x«®n§Qº$+¦X¨~º«¨x®n«¾«»~¸
·h®n~¨nB·²³»"±w°b®0»~~¨n¨x»«¬b²»~¯«º0°³± ? ¹Jºx§"·d«¨nªQ«§+¨nª«¸²³®n«2«¬b«|§¿²¾¶»~fÅ"«$¼©´J¸ ) ®x«ºn¶"« ·d¨n~¬«»~¸ )
˰©ºn¯¾¾9²³®n~Ыa¨nªQ«4´"«ª ²M¬J~°b¯Q®0º0«!Ĩx®n«¾«»~¸·d®n~¨x½·|²»~§+²©®²§Qµb«2°±°b®¼Ä«® 8 )
Ù ¨0°¯®0»B²ºx¨5´¯Q¨0°b§«¨x~¾«0¶"°b~§J¨ ? @ @ 1 1 2 & @ À«ª ²M¬b«¨nª ²³¨0¨nª~º,·d®x~¨nB·²»G®²³§µb«2~º5°³±°b®¼Ä«®0°±¯²§ ¼¼
Á-°À5«|¬b«®M¹5¨xª~º¼J°J«º§°¨§««$¼Ú¨x°¿¶Q®n«ºx«§J¨ ²¶®x°b´»~«¾É¹¾°b®x«°¬b«|®&·d°®n®x«$·d¨n»~¸I²b¼|Ïd¯ºn¨x«$¼O~¨º²Ì·xª ²§"·d«¨n°Éµb«|¨
¾°b§°¨n°§B·²»~»~¸&²³§ ¼ÉÍb¯Q½·x¿·h°b§J¬b«®xµb«§"·d«bÜ«M·²¯ºx«°³±5¨nª«©ºn¸J¾¾«¨x®n¸°³±×¨xª«#±w¯§ ·d¨x~°b§º2À«>¾+²M¸«¬«§¿ªQ°b¶"«¨n°
µ«¨ ( 8 15? 2 - @ Ù ·d¨x¯ ²»~»~¸¨xª~º~º§°b¨2¶"°bºxºn~´»~«b¹´¯¨#¨nª«®x«~º2§««$¼È°±0¨nª«ÂJ§°À0»~«$¼Jµb«©ª°À ¨n°²¼Ïd¯ºx¨
¨xª«¶ ²³®²³¾«¨x«®nº0¯¹¼¶®x°b¶"«®n»~¸J¹ºn¯"·nªÉ¨nª"²¨0¨nªQ«¸Å ¨,´"«ºn¨M0Ë0ª~º0~º4¼J°§«´J¸¨xª«>²¼Ïd¯ºx¨n~§µ©±w¯§"·d¨n~°b§+À0~¨nªQ§+¨xª«
§Q°b®n¾+²»²¶¶Q®n° ~¾9²¨x~°b§º 1 ºx«« Ù ¶¶ «|§ ¼J Ù 2
Ì«²³®n«§°¨,µ°b~§µ©¨n°«!Ĩx«§
¼ ¨nª~º0« ²³¾~§ ²¨x~°b§Üv¯¨2À5«#Àײ³§¨0¨x°¶®n«|ºn«§J¨¨xª««dÊ«$·d¨2~§ºn°¾«ºn~¾¯»B²¨x°§º0²¨
¨xª«2¾°ºn¨0·d®n~¨nB·²³»¨n~¾«¶ °~§¨ ? @ 13500
80000
CRR
PP2
PP1
CP
13000
12500
CRR
PP2
PP1
CP
60000
40000
12000
20000
11500
11000
0
10500
-20000
10000
-40000
9500
-60000
9000
8500
10
¥ *G£ + 20
30
-80000
40
S
50
( 8 15? @ @
60
2-
70
10
¥ *£ + 20
30
40
S
50
( 8 15? @ @
60
2-
Ö§«2ºx««º2¬b«®x¸À«»~»¨nª ²³¨0°b¯®,²³¶¶®x°²b·xª«|º,«ª~´~¨°b§»~¸§«µ»~«$·d¨n~§µ°º·d~»~»B²¨x°§º§·d°¾¶ ²³®n~ºn~°§>¨x°Î0Ø,Ø
70
4£/+ ¥w¤.6+ +,|£6,
¦X§¨nª~º0ºn«M·d¨n~°b§À5«2¶®x«ºn«§J¨2·d°b¾¶¯¨²³¨n~°b§"²»"®n«ºx¯»~¨nº$ Ì«4·d°¾¶ ²³®n«¨nª«¨xª®n«|«´~§°b¾B²»J¾«¨xª°¼Jº,·d°§ºnB¼J«®x«$¼+~§
ºx«$·d¨n~°§Ì¨XÀ5°À0~¨xª¿¨xª°bºx«§«À0»~¸Ì¼J«¬«»~°b¶"«$¼+Ü«»~°À¹G¨nª«|®n«+~º² ¨²´Q»«·h°b§J¨²³§Q§Qµ9!« Q²³¾¶»~«>·h°b¾¶Q¯¨²³¨n~°b§º±w°®
Õ¯Q®n°b¶"«$²³§¿·²»~»²§ ¼¶¯¨°b¶¨x~°b§º0À0~¨nªÉ²©ºn¶"«$·dÃÅ·#ºn«»~«$·d¨x~°b§°³±5¶ ²®d²¾«¨n«®xº$0ΰb¾¶¯¨x§Qµ´~§°b¾B²»"¶®nB·d«º-±w°b®2²
Å «$¼ ¨n®n«|«®n«dÅ"§«¾«|§¨®n«¶Q®n«ºx«§J¨nº°§»~¸9²©ºn¾+²»~»"À0§"¼J°À °³±¨nª«À0ªQ°b»~«²¶¶®x° ¾+²¨x°§>¨xª«¾«4À0~¨xª ²··dB¼J«§J¨d²»
¼J«|µb®n«|«º0°±²b··h¯®²·d¸Ç-«¬b«|®n¨nªQ«»~«ºnºM¹ À«0µb~¬«,²2¨²³´»~«0¨n°¨xª«4·d°§J¬b«§~«§"·d«,°³±¨nª°ºn«,®x«$²b¼Ä«®nº0¶®x°bºx«$·d¯¨x§Qµ>¨xª«0~¾Ò
¶»~«¾«§J¨²³¨n~°b§2°±¾«|¨nª°¼JºM
Õ¬«§À0~¨nª©¨nª«0¬«®n¸#»°ÀO¨n®x««0®n«dÅ"§«¾«|§¨v°± 1"@ ¹³¨nª«0°¯¨nºx¨²§"¼J~§µ2¶ «®k±w°b®x¾9²³§ ·d«
°³±¾°¼J«»~º5¯Qºn~§µ§°®n¾9²³»²¶¶Q®n° ~¾9²¨x~°b§º5·|²§´"«®n«M·d°bµ§~Ы$¼Oר-«¾9²³®nÂb²³´»~¸¹¾°b®x«4¼J~µb~¨xº0¾¯ºn¨0´"«>¼J~ºn¶Q»½²M¸«$¼
¨x° ·|²¨·xªÌ¨xª«&¼Ä«µb®x««°³±-²··d¯®d²b·d¸JÇ0°b¨d²´»~¸J¹·²®x«¾¯ºx¨,´"«¨²Â«§É°±0¨nª«©¾«¨xª°¼¨x°·²»B·d¯»B²³¨n«©¨nª«ºx¨²§"¼²®d¼
§°®n¾+²»"±w¯§"·d¨n~°b§+~§°®¼J«®¨n°+²M¬b°bB¼+¼J~ºn¨x°b®x¨n~°b§+´J¸+¨nª«©ºn¯¶Q¶ °ºn«$¼J»~¸+¨n®x¯«©ºn°b»~¯¨x~°b§0Ë0ªJ¯ºM¹"¨nª«>·xª°ºn«§É¾«|¨nª°¼
µ¯ ²®d²§J¨n««|º¾90² Ä~¾²³» «®x®n°®°3± /¼J~µ¨xº$ Ù »~¨nª°¯µbªÉ¨nªQ«>¨x®n««+²b¼Ïd¯Qºn¨n¾«§J¨,¶®x~¾9²®x~»¸ºx«®n¬«$¼É±L°b®,¨xª«~¾¶®x°¬b«$¼
²³¶¶®n° ~¾9²³¨n~°b§°±Õ¯®n°¶ «$²³§Úºn¨²³§ ¼²³®¼Ì°¶¨n~°§º$¹vÀ5«ºnª°ÀÔ¨xª ²¨¬b²³»~¯ ²´»~«~¾¶Q®n°¬b«|¾«§J¨nº±w°b®©¨nªQ«9¶Q®nB·d~§µÉ°±
Ù ¾«®x½·|²§©¨X¸¶"«,°¶¨n~°§º5²®x«,·d°b§J¨d²~§«$¼Ë®x¯« Ù ¾«®x½·|²§©°b¶¨x°§¬b²»~¯«ºvÀ5«®x«4¼J«|®n~¬b«$¼Ñ¯ºn~§µ#¨nª«,Î0ØØ ¾«¨nªQ°Ä¼
¯ºx~§µ9 #¨n®n«|« ºx¨n«¶ºM
"Ĩn®n~«
) ) ) Î0Ø,Ø
Ø
Ë0B²§
ºx¾°
ÎI
5?P 5?@
@ !
*/,+ Q / j */ j / jJj + Q / j j ~ / j j + j j ~ ,+),+ j Q / j j ) j 4)
Q / Q + ,+ j Q Q ,+
j Q j + j + j j Q ),+ j Q )
Q +)
Q Q /
Q /) )
5?I + !
*)) j
Qb + Q* ) b~ + )
Q ))+
Q ),+ ~ ,+
Q j j */ //Q j / j ) j j * / j +/ j * // j j ))
/ Q ,+ + ) j ,+) Q
4)~) 4)Q j )+ ) 4 ) ,+ j 4 )Q 4( 5?I ,!
bQ + j + Q* ) ) j b~ j ,+
~ j Q + +/
Q~ + j + ,+ /
Q4 / j
/~ /Q +"
/Q /
/~ /Q )
Q Q Qb * ) )/ ) / Q )+
*/+ + Q /,++/) * / jHj * / Q j +
D. 6+ -¶ ²³®²¾«¨n«|®nº > @/ B@ /[ E @ @
I I0
I I Ë ®n¯«
¬b²³»~¯«
*/
j ) )"+
~ j
Q*)+ )
Q
*//
j ~ /
~ ++
*)
jHj
Q // ))
j )) j Q //
Q ),+),+ j
Q jHj
Q )
j j ) /
~ j +
) ~ b/)
+ j
j j )+),+
~ jJj /
4)~ j Q
j + Q +" j j )+
Q jJj j
4)Q ,+
+ j + b,+ /
Q~ ~ )
/Q /)Q
* )Q j *)
j // j j / j +)
BH1 @ 2±w°®,²»~»
b /
~ / * )
* /Q4 /
Õ ²·xªÛ·d°b§ºxB¼J«®n«M¼Ìºn~¾¯»B²³¨n~°b§®n«ºx¯»~¨>¾+²M¸ ¼Ä«¶ «|§ ¼Ìºxµ§ÃÅ·²³§¨x»~¸9°§Û²§Ì²b··h½¼Ä«§J¨nB·²»~»~¸¿·xª°ºn«§O¶ ²®d²¾«|¨n«®2ºn«|¨$
Ë0ªJ¯º0À«»~°J°bÂb«M¼©±w°®,²©¶®n°·d«$¼Ä¯®n«©¨n°¨n«ºx¨,ºn~¾¯»~¨d²§«°¯ºn»~¸²b·d®x°bºxº4²©À0ª°»«2ºx«¨,°³±¶"²®²³¾«¨x«®nºM Ì«2ºn¨xB·nÂ+¨x°²§
²³§ ²»~¸Jºn~º®x«$·d«§J¨x»¸·d°§ ¼J¯ ·d¨x«$¼´J¸>Üv®n°J²b¼J~«0²§ ¼ '«¨n«|¾¶»~« ( )),+-À0ªQ°4¨x«ºn¨x«$¼©ºn«|¬b«®d²»"¾«¨xª°¼JºG±w°b®¨nªQ«5¶®xB·d~§µ2°±
Ù ¾«®x½·|²§°b¶Q¨n~°b§ºMË0ªQ«®n«b¹QÀ0~¨nª~§°b§Q«4²§"²»~¸Jºn~ººn«|¬b«®d²»¾«|¨nª°¼Jº¯ºn~§µ©²»B²®xµb«0ºd²¾¶Q»«0°³±®²³§ ¼J°b¾»~¸ºn«»~«$·h¨n«$¼
¶ ²³®²³¾«¨x«®nº,²³®n«>·d°¾¶ ²³®n«$¼ºx¾¯Q»¨d²§«|°b¯ºx»¸©°¬b«|®,®n«hÅ §«¾«§J¨nº,À0~¨xª9¾«$²ºx¯®n«|¾«§J¨0°±5·d°¾¶¯Q¨²¨x~°b§ºx¶ «|«$¼¿²³§ ¼
²³¶¶®n° ~¾9²³¨n~°b§«®n®x°b®$Iΰ¾¶¯Q¨²¨x~°b§Éºn¶"««$¼Ú~º« ¶®x«ºnºx«$¼Ú´J¸Û¨xª«+§¯Q¾´ «®©°±2°b¶¨x~°b§Ì¶®x½·h«º9·²³»B·d¯»B²¨x«$¼O¶ «®
ºx«$·d°§ ¼ "J~§ ·d«À«+ºn¨x½·xĄ̂x°¿¨x®n««+¾°¼J«|»º#À0¨xªÛB¼J«§J¨x½·|²»0ºn¨x®n¯ ·h¨n¯®x«« ·d«¶¨©±L°b®©¨nª«+¨n®x««+¶ ²®d²¾«¨n«®xº$¹±w°b®²»~»
¾°¼J«»~º5ªQ«®n«b¹"À«2¯ºn«©¨nª«2ºx¶ ««M¼®x«ºn¯»~¨xº4°³±Üv®n°J²b¼J~«b¹',«|¨n«¾¶»~«,±L°b®,Î0Ø,Ø#JË0ªJ¯º$¹QÀ5«2§««M¼§°¨,·²®x«°b§+¨x¯§~§µ
°¯®·d°b¾¶¯¨x«®~¾¶»~«¾«§J¨²³¨nf°§+°±0¾«¨xª°¼Jº$Ë0ªQ«²¶¶®x°¾+²¨x°§«®x®n°®>~º2¾«$²³ºn¯®x«$¼É´J¸¨nªQ«®n«»B²¨x~¬b«®n°J°¨ ¾«$²§ ºÍ¯ ²³®n«$¼ 1 Ø
Ý " 2«®x®n°b®M5Ø
Ý " «|®n®n°®0~º,¼J«dÅ §«M¼+´J¸
ØÝ" @ e <
< f
À0ªQ«®n« < @ 1 D
< < 2F < ~º¨nª«2®x«»B²¨n~¬«0«®n®x°b®$¹ < ~ºn¨¨nª«¨n®n¯Q«4°¶¨n~°§¬b²»~¯«b\
D < ~ºn¨¨nª««ºn¨x¾+²¨x«$¼©°b¶¨x~°b§¬b²³»~¯«b
˰¾+²Â«®n«»B²¨x~¬b««®x®n°b®¾«$²§Q§Qµ±w¯»X¹¨xª ²¨©~º>¨x° ²M¬°bB¼Éºn«|§ºn«»~«ºxº9¼J~ºx¨n°b®x¨n~°b§Qº>´"«$·²³¯ºn«°³±,¬b«®x¸Ûºx¾9²»~»°b¶Q¨n~°b§
¶Q®nB·d«ºM¹¨nªQ«ºn¯¾¾9²³¨n~°b§~º0¨d²Âb«|§9°¬«®0°b¶Q¨n~°b§º0~§¨nª«>¼Q²¨²³ºn«¨0ºd²¨xºk±w¸J~§µ ` <
Ì«·xª°bºx«¨nª«©±L°b»~»~°À0§Qµ¼J~ºn¨n®x~´¯¨n~°§¿°³±5¶ ²®d²¾«¨n«®xº$
!v°b»B²¨x»~~¨X¸~º¼J~ºn¨x®n~´¯¨x«$¼É¯§Ã±w°®n¾»¸´ «|¨XÀ5««|§
²§ ¼
0j GË0¾«0¨n°©¾9²³¨n¯®x¨X¸#ºM¹JÀ0~¨nª¶Q®n°b´"²´~»~~¨X¸ Q /¹¯§Ã±w°®n¾ ´ «|¨XÀ5««|§ 2²§ ¼ 2¸b«M²®nº,²³§ ¼¹JÀ0~¨nª¶®n°´ ²´Q»~~¨X¸
¹"¯§Qf±L°b®n¾)´"«¨XÀ««§I ²§ :
¼ Ѹb«$²³®nº$ Ì«2Å ¿¨nª«ºx¨n®x«>¶Q®nB·d«²a¨ @ ²§ ¼¨²³Âb«©¨nª«©~§~¨nB²»²ºxºn«¨
¶Q®nB·d« > V > ¨x°´ «2¯§Ã±w°®n¾Ô´"«¨XÀ««§ /²§ ¼ Ø0«»B²¨x~¬b«,«|®n®n°®nº2¼J°§°b¨,·xª ²³§µb«2ñ > ²³§ E¼ ²®x«2º·²³»«M¼+´J¸
¨xª«0º²³¾«v±{²b·h¨n°b®M¹bX «b~¹°§»~¸¨nª«0®d²¨x° > FH ºv°±§J¨x«®n«ºx¨$vË0ª«0®n~ºnÂJ»~«ºxº5®d²¨n(« B~º$¹³À0¨xª¶®n°´ ²´Q»~~¨X#¸ ¹³¯§Ã±w°b®x¾
´"«¨XÀ««8
§ ²§"¼ ²³§ ¼¹"À0~¨nª+¶®x°b´ ²³´~»~¨X"
¸ ¹Q«$ͯ ²»G¨n° ,Õ²·xª¶ ²®d²¾«¨n«®~º,ºn«|»«M·d¨n«$¼É~§ ¼Ä«¶ «|§ ¼J«§J¨n»~¸
°³±
¨xª«°b¨xª«®xº$Ë0ªQººn«|»«M·d¨n~°b§°±
¶"²®²³¾«¨x«®nº0!« Q²·d¨n»~¸¾9²¨d·xª«º0¨nªQ«·xª°B·d«°±Ü®x°²¼J~«b¹ '«¨n«¾¶»~«b
%G~µb¯Q®n
« ®n«¶"°b®x¨nº¨xª«0®n«|ºn¯»~¨nº±w°b®Õ¯®x°b¶"«$²§·|²»~»J°b¶¨x°§º¨n°2À0ªB·xª>¨xª«,²§"²»~¸Jºn~ºÀ0²º¼J«¬°b¨x«$¼©«ºn¶"«$·dB²³»»~¸ºx°±{²®M
Ö-±·d°b¯®xºn«b¹"ºx¾~»½²³®®n«ºx¯»~¨nº4·h°b¯»B¼´"«2¶®n«|ºn«§J¨n«M¼+±w°®×Õ¯®n°¶ «$²³§+¶¯¨0°b¶Q¨n~°b§ºM Ù ¾+²Ð~§µ»¸J¹M¨nª«2§«|À0»¸+¼J«¬«»~°b¶"«$¼
1000
CRR
JR
Tian
smo
CP
PP1
PP2
Speed
100
10
1
0.1
1e-08
1e-07
1e-06
1e-05 0.0001
RMS
0.001
0.01
0.1
¥ *G£ + !"#%$&')(*$+
,)%-.0/-1324#!51#%$$,326!4
7 891 < @
> M [ + j [ / ) [ A
¾«¨nªQ°Ä¼Äº°b¯¨x¶ «|®X±w°®n¾6²»~»¨n«|ºn¨n«M¼I²¶¶Q®n°~¾9²¨x~°b§º§O¨n«®x¾º©°±4²··d¯®d²b·d¸J5Ì«®x«¶®x°Ä¼Ä¯ ·d«$¼Ú¨xª«Å § ¼J~§µ¨nª ²³¨
¨xª«+ºn¶"««$¼ ²b··h¯®²·d¸Ú»~§«Ñ±w°b®©¨nªQ« Î0ØØ J¾°¼J«»~º»~§Q«$²®~§Ú²¶¶"«$²®d²§ ·h«bÚË0ª«ºx«Å"§ ¼J~§µbº©¨²³Âb«°¬b«®¨x°¿¨xª«
JØ J¾°¼J«»X¹Ë0B²§ ¾°¼J«|»{¹²§ ¼°b¯Q®,²¶¶®x°²·nªÑ±w°b®0ºx¾°J°b¨xª·d°b§J¬«®nµ«§ ·d«b0Ö´bÏd«$·h¨n~§µ ¹Q¨nª«2ºx¾°J°b¨xª»~~§«4¼J«|¬b«»~°b¶Qº
±L®n°b¾ ¨nª«²M¬«®²³µb~§µ°¬b«®2¨nªQ«®n«|ºn¯»~¨nº#°±0¨nª«À0ª°»~«>ºd²¾¶Q»«>˲³Â~§µ°b§»~¸¿²ºx§Qµb»~«>¶"²®²³¾«¨x«®·d°b§Qºn¨n«|»»B²³¨n~°b§
¸J~«»B¼Jº5²#¶B·d¨x¯®n«b¹JÀ0ªQ«®n«2¨nªQ«4·d°§¬«®nµ«§ ·d«#¶ ²¨x¨n«®x§º×¼Ä«º·d®x~´ «$¼ «$²®x»~«®«¾«®nµ«,²µ²³~§¹bÀ0ªQ«®n«$²³º0¨nª«»~§Q«º±L°b®¨xª«
§Q«À ¾«¨nªQ°Ä¼Äº0®n«¾+²~§ºn¨d²´»~«b
1000
CRR
CP
PP1
PP2
Speed
100
10
1
1e-08
1e-07
1e-06
¥ *G£ +6A
4 $ )$ %
1e-05
RMS
0.0001
26#!-1#% 3-
7 9
0.001
1<
0.01
@ ( -
1000
CRR
JR
Tian
smo
CP
PP1
PP2
Speed
100
10
1
0.1
1e-05
¥ *£ + $B 0.0001
0.001
RMS
0.01
0.1
3
4 !"#%$&')(*$+
!-#%'2/* 26%
<
>
j
9
1
@
[
M
[
+
[
/
) | A
7
G% ~§ ²³»»~¸J¹vÅ µb¯®x«
Ú
+ ®n«|¶ °®n¨nºÉºn¶"««$¼Ä ²b·|·d¯®²·d¸<¶®x°b¶"«®n¨x«|º¿À0~¨nª'¨nªQ«Û·²»B·d¯»B²³¨n~°b§I°³± Ù ¾«®xB·²§'¨X¸J¶ «É°b¶¨x~°b§ºM
Pª«®n«M²º$¹ºx¾~»½²³®2®n«ºx¯»~¨nº9²³®n«+°b´Q¨²~§«M¼É±w°®¨nª«+¶®x«¬J°¯º¾°¼J«|»ºM¹¨nª«+§«|À ¾°Ä¼Ä«»~º&·d°§J¨²~§O°b®¼Ä«®°±·d°b§JÒ
¬«®nµ«§ ·d«°b§«©ª«®x«°b§»~¸´Q¯¨2À0¨xª¿ºx¾9²³»»G~§~¨nB²»G«®n®x°b®$Ç,²¨x¯®²³»~»¸J¹"¨nª«¼Ä«ºn~µb§Ìºx°b»~«»~¸²ºxºn¯®x«$¼Ìª~µª¿²b··h¯®²·d¸
À0~¨xª®n«ºx¶ «$·h¨,¨n°#¨nª«¨n«®x¾~§ ²»Q¶ ²M¸b°³Ê ¼J~ºn¨x®n~´¯¨x~°b§ Ù ¶¶Q®n°~¾9²¨x~°b§°³±«$²®x»¸©««®d·d~ºn«2¶®n«|¾~¯¾º~§J¬b°»¬«5°®n~µbÃÒ
§"²»"ºn°b¯Q®·d«º°±~®n®x«µb¯Q»½²³®n~¨n~«ºMÇ0«¬«®n¨xª«»~«ºxº$¹"¨nª«¯§« ¶"«$·d¨d²´»~«ºx¨²´Q»~~¨X¸©ºnª°À0º¨nª ²³¨5¨xª«²¶¶Q®n° ~¾9²¨x~°b§«®x®n°®
·xª~«dÓ"¸9²®x~ºn«º-±w®n°¾ ¼J«dÅ·h«|§ ·d~«º,~§·d°§§«$·d¨x~°b§+À0~¨nª+¨nª«2¨x«®n¾§"²»¶ ²M¸°Ê¿¼J~ºn¨x®n~´¯¨x°§
j %4¡¤ 6£ ,¥w¡
ΰ§J¬b«®xµb«§ ·h«ºn¶"««$¼'²§ ¼Ú·d°§J¬b«®xµb«§ ·h«¿¶ ²³¨n¨x«®n§º°±¨nªQ®n««¶Q®n«¬J~°b¯ºx»~¸Û« ~ºn¨x§QµÌ»½²³¨n¨x½·h« ²³¶¶®x°²b·xª«|ºÀ«®n«!« Ò
²³¾~§«$¼
.«§«®d²»~»~¸¹GÀ5«©Å"§ ¼É°b®d¼J«®©°±,·d°b§J¬b«|®nµb«|§ ·d«+°b§« -§J±w°b®x¨n¯§"²¨n«|»¸J¹·d°b§J¬b«|®nµb«|§ ·d«~º>¼Äºx¨n°®n¨n«M¼Ì´J¸°¬b«|®
¨x®n««©®n«hÅ §«¾«§J¨nºÓ ¯ ·d¨x¯ ²¨x~§µ®n«»B²³¨n~¬b«¶"°bºx~¨n~°b§º°±¨nªQ«>ºx¨n®n~«¶®nB·d«3 Ì«2ºn¯"··d««$¼Ä«$¼Ì¨x°9·d°§ºn¨x®n¯ ·d¨²Ñ´~§°b¾½²³»
¾°¼J«»À0ªQ½·xªÌ« ª~´~¨xº4ºx¾°J°b¨xª¿·d°§¬«®nµ«§ ·d«Ý+°b®x«°¬b«|®$¹GÀ5«¶®x«ºn«|§¨x«$¼Û²ºx¾°J°b¨xª»~¸9·d°§¬«®nµ~§µ+¾°¼J«»À0~¨nª
·h°b§J¬b«®xµb«§"·d«2°b®¼Ä«®0°b§«´¯¨~¾¶®x°¬b«$¼©·d°J« 9·d~«§J¨$ΪQ«hÓ ¸¹À«,¶®x«ºn«§J¨x«$¼²2ºx¾°J°b¨xª»~¸>·d°b§J¬«®nµ§Qµ>¾°Ä¼Ä«»À0~¨nª
°®¼J«|®,°±·d°b§J¬«®nµ«§ ·d«©¨XÀ5°" %G~§ ²³»»~¸J¹bÀ«2»~ºx¨n«$¼ºx¾¯Q»½²³¨n~°b§©®n«|ºn¯»~¨nºM0Õºn¶"«$·dB²»~»~¸À«·d°b§"¼J¯ ·d¨x«$¼²§+« ²³¾~§ ²¨x~°b§
°³±4·d°b¾¶¯¨d²¨n~°§¿ºx¶ ««M¼I²§ ¼Ì²··d¯®d²b·d¸È±w°b®²+»B²®xµb«º²¾¶»~«°±,®d²§ ¼J°¾»~¸ºx«»~«$·d¨n«M¼Ú¶ ²®d²¾«¨n«®©·d°b§ºx¨n«»~»B²¨x~°b§ºM
Ø-«¾9²³®nÂb²³´»~¸¹¶®nB·d«·d°¾¶¯¨d²¨x°§°³± Ù ¾«®nB·²³§9¨X¸J¶"«>°¶¨n~°§~º,~¾¶®x°¬b«$¼
5Ë®d²§ºk±w«®n~§µ¨nª«|ºn«©Å § ¼Ä§Qµbº,¨x°¨xª«
¬b²³»~¯ ²¨x°§°±·d°b¾¶»~« °b¶¨x~°b§¨X¸J¶"«º,®x«¾9²³~§º±w°®±w¯¨n¯Q®n«2®n«ºx«$²®d·xª
¡
/(.6 )7) ¡ ¥5/(.J¥w¡
,
#% 2 #!/$ )% 2 2 -1 '# % Ë0ª~º²³¶¶®x° Ä~¾+²¨n~°§¶®x°Ä·h««$¼Jº2±L®n°b¾)¨xª««$ͯ~¬b²»~«§"·d«>°³±×²·h¯¾¯»B²¨x¬«2´ÃÒ
§°¾B²»G¶®n°´ ²´Q»~~¨X¸¨x°¿²§É~§ ·d°b¾¶»~«¨n«´"«¨d²MÒL±w¯§ ·d¨x~°b§Ì®²³¨n~° 1 «§ ¼Ä«»~»{¹ "Ĩn¯ ²³®n4¨ (*/ /-n¹¶GÌ 22²§ ¼É¨xª«§ ·d«+¨x°¿²
¶®x°b´ ²³´~»~¨X¸~§¨x«µb®d²»°±¨nª«©¬b²®x½²³§ ·d«©®²³¨n~°¿¹ 1 «§ ¼²³»~»{¹ "J¨x¯ ²®x4¨ (*//0-n¹ ¶G& + /2|# 0ºn~§µ+²§Ì²¶¶Q®n° ~¾9²¨x~°b§¨x°
¨xª«>~§J¨n«|µb®²³»°±Ô¼J«¬«»~°b¶"«$¼É´J¸ I²¯»~ºn°§ ( + 0- 1 «§"¼²»~»X¹ "J¨x¯ ²®x4¨ (*//0-k¹"¶ +" 2-À0ª°+~§¨n¯®x§Ì¯ºn«M¼ I~»ºx°b§¿²³§ ¼
Á-»Ã±w«|®n¨X¸Æ º4(!-²¶¶Q®n° ~¾9²¨x~°b§2±w°®¨nªQ«¼Äºx¨n®x´Q¯¨n~°b§°³±·xªÃÒXºÍ¯ ²®x« 1 «§"¼²»~»X¹ "J¨n¯"²®n¨ (*//-n¹¶G )) 2²§"¼¨nªQ«4®x«ºn¯Q»¨
°´¨²³§Q«$¼©´J¸ %~«»~»~«® ( -²³§ ¼ .«$²®x¸ (-·d°§ ·d«®x§~§µ¨nªQ«,®²³¨n~°°³±¨XÀ5°©§°b®x¾9²³»»~¸¼J~ºn¨x®n~´¯¨x«$¼¬b²³®nB²¨x«º 1 «§ ¼²³»~»{¹
"J¨x¯ ²®x4¨ (*//0-n¹b¶ 2¹Î0²¾
¶ (!-¼J«¬«»~°b¶"«$¼+²§+« ¶»~B·d~¨5« ¶®x«ºnºx~°b§À0ªB·xª+¾9²M¸©´ «#À0®n~¨n¨x«§²º±w°»»~°À0º )7)7+¢G¥ g 9 i
@ 1 1 I 2 1 I C S 2 1 TS2 - (*)h 1 1 I,2 C 1 I C 42
S 1 TS2 -
C
@ ( 1 1 I 2
1 I C 2
1 1OI2
1 I C 42
ºx««4²³»ºx° 2. «´Qª ²®d¼J¨4( j )0-
²³§ ¼IG«~Ы®M¹7I®²³¨n¨4( j -nDIG«~Ы|®,²§ ¼I®d²¨x¨4( j 0-¼J«®x¬«$¼¨xª«,~§J¬b«|®nºn~°§©±w°b®x¾¯»B²0¶®x«ºn«§J¨x«$¼
^ 1 JI 16S2
@
~§ºx«$·d¨x°§+
) -0 -# 2 2 -1!'# !4
«¨ ´"«É¨nª«É¨n®x¯«Ì´¯¨Ñ±w¯§ ·d¨x~°b§ ²³»»~¸O¯§ÂJ§°À0§'§Q¶¯¨°³±>¨nªQ«¿ºx¨²§"¼²®d¼
§°®n¾+²»Q±w¯§ ·d¨x~°b§+¨n°²³¶¶®n°~¾9²³¨n«,²©¬b²³»¯Q«4°³±¨nª«>·h¯¾¯»B²¨x¬«0´~§°b¾B²» ¼J~ºx¨n®n~´¯Q¨n~°b§±w¯§"·d¨n~°b§G "J¨d²®n¨x~§µÀ0~¨nª
²³¶¶®n° ~¾9²³¨n~°b§ @ ( 1 I C 2 .1 S-F 9 B%S#WJ¹À0ªQ«®n« I C ¼J«§Q°b¨n«|º>¨xª«§J¯¾´ «|®4°³±×ºx¯ ··d«|ºnºn«|º~§ 1OÜ«|®n§°¯»~»
¨x®nB²»~ºÀ0¨xª·d°b§J¨n~§J¯~¨X¸·d°®n®n«M·d¨n~°b§G¹¨nªQ«¸·d°b®x®n«$·d¨-±w°b®0¾~ºn¶»B²b·h«¾«§J¨°±¨xª«2¾«$¼JB²³§´¸
I C 1 S C @
9 1 S#W
²³§ ¼±w¯®x¨nª«|®4~§J¬b«|ºn¨n~µJ²¨n~°§º,ºx¯µbµ«ºn¨2®n«|¶»B²b·d~§
µ 1É´J
¸ 1 C §¨nª«¼J«§°¾~§ ²³¨n°b®M,Ë0ªQ°b®n°¯µbª§J¬«ºn¨xµJ²¨x°§+°±
²³¶¶®n° ~¾9²³¨n~°b§#¶ ²¨x¨n«®x§º,À0~¨nª F ®n«¬«$²»
±w¯§ ·h¨n~°b§ ²³»~»¸©« ¶®x«ºnºd²´»~«ºx¾¶»~«0¶ ²¨x¨n«®x§º$¹QÀ0ªB·xª9«|¬b«§J¨n¯"²»~»~¸»«M²b¼
¨x°>¨xª«2±w°»»~°À0~§µ²b¼|Ïd¯ºn¨x¾«§J¨ ( 1 I C 2 1 S C I C C S 1 1I C 2 C W @
1 1 C 2 S#W
g 1 S i g 1 W i À0ª«®x« 1 2 @ 1 T 2 1 T C »~§ 2
%Q¯®n¨xª«®0¾°Ä¼ÄfÅ·²¨x°§¨n°¨xª«Å ®nºx¨,¶ ²³®n¨,¼J«»~~¬b«|®nº,²©ºn«M·d°b§ ¼²¶¶®x° ¾+²¨x°§
@ L @ "! # ( 1 I C 21 S C - C ! L #
@
1 1 C 22SW
¦X§+¨nªQ«·|²ºn«#À0ª«®n« I C @ 1 1 I C 2¨nªQ«ºn«2±L°b®n¾¯Q»½²³º®n«M¼J¯ ·d«¨x°
0»§ 1 +SW2 @ 1 1 C 2 C 0»§ 1 +SW2 1 C À0ª«|®n«¨xª«2ºn~µb§+~º0¨n°©´ «©·nªQ°bºn«|§¨n°²µ®n««À0~¨xª¨nªQ«ºn~µb§°± W Q
@
1 1
C
1
2
Ö§»~¸¨nªQ«§¹~§J¬b«|®nºn~°§±L°b®n¾¯Q»½²³º5²º0¶Q®n«ºx«§J¨n«$¼~§ºn«M·d¨n~°b§ &+ ·²³§+´ «>¼Ä«®n~¬b«M¼
$ uSp^flijnlLqXrL^`ybuB^fad_v^fladsbuLed^`_bjnivc,ed^`_gf}Ãr{ahc Fe&% ä('dBmdjnlSyjxqX^fedg`gfr{jk}jXrXjX_bqxjnluLa*)
jk_biedg`gBm$KMuStbedr{ux xo"jXrXjlBtyygfjkc,jX_$uLjni
+ lLjnj jX+^ *xjXrdm rXeuLun '|deh_bi Qr{euLux 'd}ahruBpbjxlwjGedyyr{ad|^`c,eu~^fad_l
)7)+¢¥ ¦X§¨nª~º Ù ¶¶ «|§ ¼J À5«2À0~»~» ¶Q®n°¬b«¨nª«2¨xª«°b®x«¾ ±w®n°¾Ôºn«$·h¨n~°b§ ©²´"°b¯¨°b®d¼J«® J°±·d°b§J¬«®nµ«§ ·d«bvË0ª«·d°b§"¼J¯ ·d¨x°§
°³±5¨nªQ«¶®n°J°±0~º2®²¨xª«®>·h¯¾´ «|®nºn°¾«>²³§ ¼ºn°b¶Qª~ºn¨nB·²³¨n«$¼É¾+²¨xª«¾9²³¨nB·dº,²³®n«©§J¬°b»~¬b«$¼
0§Ä±w°b®x¨n¯§ ²³¨n«»~¸J¹"±w¯®x¨nª«|®
~§J¨n®x°¼J¯ ·d¨x°b®n¸§°b¨d²¨n~°§~º4§Q««$¼J«$¼
Ë0ª«©¶®n°J°±0~º2ºn«¶ ²³®²³¨n«$¼É~§J¨n°+ºn«|¬b«®d²»¶ ²³®n¨xº$©Ë0ª«²b·d¨x¯ ²»¶Q®n°J°±0~º4«|§ ¼J~§µ
²±w¨n«|®¾9²³Â~§µ¨nª«Ë²M¸»~°® ÄÕ Ä¶"²§ºx°§Ë0ª«|§ ·d°§¬«®nµ«§ ·d«~º>²»~®n«M²b¼J¸Éºn«¶"²®²³¨n«$¼O~§J¨n°²§Ì°¶¨n~°§Û¼J«¶"«§ ¼J«§J¨
²³§ ¼+²»B²³¨n¨nB·d«¼J«|¶ «§"¼J«§J¨2¶ ²®x¨$
Ë0ªQ«2±w°b»~»~°À0~§µ»~«¾¾9²¨d²²®x«®²³¨nª«|®4¨x«$·xª§B·²³»{-Ë0ª«¸µ¬«2«ºn¨x¾+²¨x«º0±w°®0¨nª« ·h°b§ºx¨²§J¨ ~§+¨nª«¼J«dÅ §Q¨x~°b§+°±
°®¼J«|® °³± Ä·h°b§J¬b«®xµb«§"·d« 1 ºx«««$ͯ ²³¨n~°b§Ì j 2 Ì«2À0~»»²b¼Ä°b¶¨0¨xª«±w°b»~»~°À0~§µ
56 ) ºnª ²³»~»´ «2¨nªQ«ºn°»¯Q¨n~°b§°±
¨xª«2ºn¨x°Ä·xª ²³ºn¨x½·#¼JÃÊ«®x«§J¨nB²»G«$ͯ ²¨x~°b§
) B = ? C ) E =H )
) )
Ë0ª«2¶®n°´ ²´~»~~¨X¸>¾«$²ºx¯®n«2~º,¼J«§Q°b¨n«M¼+´J¸È
Q 15? > 2\@ #"$ &()+ "!5 1 ) 3 2$©
# ~º,¨nªQ«>Ü»B²·n "·xª°b»~«º# I®x½·h«>°³±v²·|²»~» °b¶Q¨n~°b§À0¨xªºn¨x°·nÂ+¬b²³»¯Q« > ²¨
n¨ ~¾« ? ²§ ¼¾+²¨n¯Q®n~¨X¸&²³¨0¨n~¾«N&
QË0ª«5¨x~¾«5²0Ä~ºGÀ0»~»´ «,¼J~ºd·d®n«¨x~Ы$¼~§ºx¨n«¶Qº°±»~«§µb¨xªB ? @ & @ Ë0ª«,¼Äºd·d®n«|¨n«0¨n~¾«¶ °~§¨xºÀ0»~»´ «,¼J«|§°b¨x«$¼
´¸ ?< @ Y B ?
+" Ì«,¯ºx«¨nª«²³´´®n«|¬b²¨n~°§ @ ) ±w°®5§°¨²³¨n~°b§ ²³» ºx~¾¶»~B·d~¨X¸
QË0ª«2§Ä±w°b®x¾9²¨x~°b§©ºn¨n®x¯ ·d¨x¯®n«©~º0¾°¼J«)»~»~«$¼© ´J¸ @ E 1 < I 2 2|
j Ǭ
15? > 2 @ #> 15? > 2 8 15? > 2 @ > 15? > 2
15? > 2 @ > 15? > 2 8 15? > 2 @ > 15? > 2
15? > 2 @ > 15? > 2 8 15? > 2 @ > 15? > 2
) @
I»~«$²ºx«4§Q°b¨n«
»§a8 3
= 1> 2 @
C
B C
E 9 ?
!
C
#
?
"J¯´ºx«$ͯ«§J¨n»~¸2À5«À0~»~»¾9²³Âb«§Q°,¼JÃÊ«®x«§ ·d«´"«¨XÀ5«|«§ < 1G? > 2c 8 < 1G? > 2G²§ ¼ < 15? = 1 = 2 2®x«ºn¶G 8 < 15? = 1 = 2 2
²ºG±w¯§ ·h¨n~°b§º~§ 1G? = |2
Á-°À5«¬«®$¹À0ª«§¨d²ÂJ~§µ,¶"²®n¨xB²»¼J«®x¬b²³¨n~¬b«|º
°§«§««$¼Jºv¨n°2¶ ²M¸ºx°b¾«²¨n¨x«§J¨n~°b§¦X§
°b¯®0§°¨²³¨n~°b§À5«2ª ²M¬«4´J¸¨nª«>·xª ²³~§9®x¯»~«O > < @ = < = > J" ~§ ·d« 15? > N2 @ =
? = 1 > 2 ´ «$·h°b¾«|º
¾°b®x«,ºxµ§ÃÅ·²³§¨»B²¨x«®$¹JÀ«À0~»~»¼J«|§°b¨x«~¨5´J¸9²#ºn¶"«$·dB²³»
»~«¨x¨n«®M
/QǬ\P 1G?
QǬ
2N@ 1 *
> 2
#
15? >
>
1G? 2
@
1G? 2
@
2 =>
1 = ¦Ø #" ) 9 E $ " +)
+
9 E
1G? 2 @
1 N? = ( k-
") 1
9 E
C
E 2
)(
Ǭ
I
E @ @ i
i g
g
@ @ i i g g S @ @ »§ i I i g
g
S @ @ »~§ i I i g
g
Ç-°b¨x½·h«4¨xª ²¨0¨xª«¸+¼J°b§°¨,¼J«¶"«§ ¼+°§+ºn¶"«$·dÃÅ·
Q Ì«°±w¨x«§I§««M¼'¨n°Û«|¬b²»~¯ ²¨x«¿~§J¨n«|®n¬b²»~º ( -À0~¨nª ²¨+®d²§ ¼J°¾¿ "J°Û¾+²M¸J´ «+¨nªQ«®n«Ì~º+«¬b«|§
J¨nªJ¯ºÀ5«2¯ºx«,±L°b®0ºn~¾¶»~½·h¨X¸#¨nª«>¼Ä«dÅ §~¨n~°§
( - @ ( -
ñ @
@
Ì«À0~»~»§°À<ºn¨d²¨x«¨nª«¨xª«°b®x«¾
@ @ ~§ºx¨n«$²¼°³± @ @
¶»~«$²ºx«>§°¨n«¨xª ²¨2À«>ºx¨²¨x«~¨,ª«®x«~§ ²ºn»~~µb¨x»~¸¾°®n«$56@±w°b®x¾ ¯ºx~§µ
1 - $4Ú(
«¨ > @ @ @ A ´ «>²#»B²¨x¨nB·d«²¶¶Q®n°²·xª9À0~¨xª @ @
«¨ @ ´"«2¨nª««|®n®n°®,~§¨xª«¶®xB·d«2°±²Õ¯®n°¶ «$²³§¿·²³»»°¶¨n~°§¨nª"²¨0~º
@ @ #"M& '( 1 @ 2 -# ! 1 1 1
* 2 #
&
Ë0ª«|§9¨xª«®x««~ºn¨nº2²>·h°b§ºx¨²§J¨ `¹J°b§»~¸¼J«|¶ «§"¼J~§µ°b§ > % B_ E Úºn¯"·nª+¨xª ²¨a
@'2 `
0-) J" ~§ ·h« 1 @
@
C @ C
1 G @ 2 À5«2µ«¨
( 1 @ 2 #"7& ! 1 & * 2 #
2D@
Ë0ª«|®n«d±w°®n«E
@
'
@
@
Ì«§«¨,°´ºn«®x¬b«¨xª
g
B
?
S@
C 1B ?2
i
( 1 G @ 2 - ²§ ¼
1 2
@ ( 1 2 @
@
"7& 1 N @ 2 1 2 @ e
#" ) & #") 1G? 2 1G? 2 (
f ²¨¨nªQ«>Ü»B²·n " ·xª°b»~«º0¶®xB·d«~º5®x~ºnÂJ»~«ºnºM¹À0ªB·xª+¾«$²³§º
! #") 1G? 2
1G? ) ) 2
#
@ ¦X¨±L°b»~»~°À0º\
@
e #" ) ' ! #" ) 15? 2 1G? 2 #
f ! _" ) 15? 2 15? [ ) ) 2 # 5
$
)
c+
f
@
@
e #" ) ! 1G? 2 #h ! 15? 2 # f
@ @
e #" ) ! 1G? 2 15? 2 #
f ! 15? 2 15? 2 # Ë0ªQ«»½²³ºn¨¨x~¾«©¶ °§J¨ 8
@ 1<
~º«¬b²»~¯ ²³¨n«$¼ºn«|¶ ²®d²¨n«M¼J»~¸Ì§
«¾¾+²²§"¼)9²³º @ +Ë0ª«°¨nª«®©¨n~¾«
¶"°b~§J¨nº,²³®n«2«¬b²³»¯"²¨n«M¼´J¸9²©Ë²M¸J»~°b®ºx«®n~«º«¶ ²§ºx~°b§²®x°b¯§ ¼ Ë0ª~º¸J«|»½¼Äº
@
@ 2 1T i C e #") ! 1G? 2 1 2 C 15? 2 1 2
g
f C 1G? 2 1 2 C P 15? 2 #
! 1G? 2 1 2 C 15? 2 1 2
C 1G? 2 1 2 C P 15? 2 # @
Ö§«2ª ²³º
@
8 15? 2 ' # @ 8 15? 2 Ö§«2µ«¨nº2²§ ²³»°µb°¯º5®x«ºn¯Q»¨xº0±w°®5¨x«®n¾º0À0~¨nªC
8 ² § ¼ 8 Ë0ª«®n«h±w°b®x«E
@
@ 2 1 i C ' e #") " 8 15? 2 @ C 8 1G? 2 @ C 8 15? 2 @
g
f C ! P 15? 2 # ! P 1G? 2 # @
2 1T i C e #" ) ! 8 15? 2 # @ g
f @
C
e #" ) ! 8 15? 2 # @ f @
C
e #" ) ! 8 15? 2 # @ f @
C
e #") " ! P 1G? 2 #
f C ! P 1G? 2 # ! 15? 2 1 2
"! 15? 2 1 2
#
@
Ë0ª«2¶®x°J°±§°ÀÚ±w°»~»°À0ºv¾¾«$¼Ä½²³¨n«»~¸2±L®n°b¾ ¨nª«±w°»»~°À0~§µ»~«¾¾9²M¨d²
( 4(4(5 2Ë0ª«§
¾90² ¦ Ø @
¾90² c @ _ ¾9 0² 1 2 @
%Q¯®n¨xª«®±w°® ? O
1 = F »~§ ? »~§ ? 9 ?
1 = F ( ~» § ? ~» § ?
1 = F ( »~§ ? ~» § ?
²³§ ¼ ?
? 9 ?
> V b¹ 1 2A@ ¹ 1 2A@ ²³®n«¨xª«2Å ®xºn¨2¨nªQ®n««Á0«®x¾~¨n«! IG°b»~¸J§°b¾²³»ºMË0ªQ«2Å ®nºx¨2¶ ²®x¨,§°À
±w°»~»°À0º-±w®n°¾ ¨xª«¶®n°¶ «|®n¨n~«º©µb~¬«§¿~§ Ù ´®d²¾°À0¨xÐ>²§"¼ "J¨x«µb¯Q§ ( j 0-k©Ý+°b®x«°¬b«|®°b§«µ«¨nº±w®n°¾ ¨nªQ«®n«b¹¨xª ²¨
¨xª«2±w¯§"·d¨n~°b§Qº,~§+¨nª«ºx«$·d°§ ¼¶"²®n¨,²³®n«ºx¨n®x½·h¨n»~¸¼J«$·d®x«$²ºx§Qµ9~§+¨nªQ«~®,®n«|ºn¶"«$·d¨n~¬«®²§Qµb«bË0ª«®x«d±w°®n«©¨nª«¸·²³§´"«
»~~¾~¨n«M¼´J¸¨nªQ«~®4¬b²³»~¯«>²¨¨nª«´"°b¯Q§ ¼²®x¸+°±0¨nª«~®2®n«|ºn¶"«$·d¨n~¬«®²§Qµb«b "J~§ ·d« 1 ? »~§ ) 9 ?
¨xª~º0¶®n°¬«º0¨nª«2»~«¾¾9²
( 4(4(5 8 15? > 2 @ 9 "E 9 $'&#)+ ? _
.
8 15? > 2 @ "$'&#)+ 1 = E 9 . ? 2 C E 9 ; ? # _ 15? > 2
@
Ý+°®n«°¬«®a
9 E,1 ; ? 2 !
$ " +2$'&#)6+
9 E1 . ? 2
> 3Ì«®n«¾+²®n¨xª ²¨
À0ªB·xª~¾¶»~«|º
Ì«ºn«$·d°§
= 1 > 2A@
!
1 = C E-
8 15? > 2 2
15? > 2 2
»§ 8 3
.
1 ; ? 2
?
1 ; ? 2
; ?
9
C 1 B C 2 1 . ? 2
E 9 ?
¼J»~¸+®n«¾+²®x¹¨nª"²¨
E9 ;
?22
8 1G? > 2 2
= @
>
> E
<
>
@
?1= C E
T
?2 C
E 1 ? 2 #
1 ; ? 2
?
(& ) $5" 2+ $'&#)+
> @X
9 .
< = = >
9 ;
?
J´ ¸©¨nª«·xª ²~§J®n¯»~«bÇ-°ÀÚ°b§«µb«¨xº0¨nª«¼J«®x~¬b²¨n~¬«º5´J¸©ºn~¾¶Q»«,·|²»B·d¯»B²¨x~°b§ºMË0ª«2«ºx¨n~¾9²³¨n«º±w°b»~»~°ÀÛ~¾¾«$¼JB²¨x»~¸
±w®x°b¾Ô¨xª°bºx«b
( 4(4(5¿ P 15? 2 2 »~§ i 1 2 ¾9² 1G? > 2 g
3 * ,
> 3Ì®x~¨n~§µ 1G? = µ«¨0´J¸¨xª«·nª"²~§+®n¯»~«O
Ë0ªJ¯º0À«2µb«¨a
15? >
>
15? > 02 À«&·|²§¯§"¼J«®nºx¨²³§ ¼ ²³º4² Q% ¯§ ·h¨n~°b§+~§
1 > 2 2 @
2 @
P
1 15?
>
1 = 2
G1 ? > 2 @
2
= = 2
51 ? > 2
=> 1 > 2A@
15? = 2-¦X§¨nª~º-±w°b®x¾
=> 15? = 2
1 > 2 15? > 2 => >
*
1 > 2 15? = 1 > 2 2 = 1 > 2 =>
>
*
$ +
= 1 > 2 1G? > 2 2 => $ *+
À«
1> 2
@ @
@
´J¸¨xª«¨n®d²§ºk±w°b®x¾9²³¨n~°b§ > = 1 > 2
"ħ"·d« = ~ºGºn¨n®xB·d¨n»~¸>¼J«M·d®n«$²³ºn~§µ"¹¯¶¨x° ¨xª«§J¨x«µb®d²§ ¼2~ºG§«µ²³¨n~¬b«« ²·d¨n»~¸À0ª«|§ = 1 2( = 1 2
Á-°À5«|¬b«®2~§¨nªQº·²³ºn«¨xª«>~§J¨n«|µb®²³¨n~°b§´ °¯§ ¼²³®n~«º°³±5¨nª«Ø0~«¾+²§§Q½²³§~§J¨n«µ®²³»§««$¼¨n°+´"«&·xª ³² §µb«M¼¹À0ªQ½·xª
·h°b¾¶"«§ºd²¨x«º0¨nª«#¾~§J¯ººn~µ§ "J°©°b§«2ª ²³º
P
15?
$ + $ *+
2 2
2
1 = 1 > 2 2 +
¾ ² 3 * ,
? =
1G? > 2
1 > 2 =>
$ +
=
$ *+
Ë0ªQº0~¾¾«$¼JB²¨x«»~¸~¾¶»~~«º¨xª«2»«|¾¾9²Q
( 4(4(
5 G
«¨ @ ! Bh # & @ ¹"Î0ØX @ = ! ( E 1 »~§ ) 2 B ? E
E & @ ²§"¼ 1 »~§ 2 B ? + E
"į¶¶"°bºn«#¨nª ²³¨
«
¨ '´"«>²®d²§ ¼Ä°b¾ ¬b²®nB²³´»~«,À0ªB·xª~º0) »~°bµ§°b®x¾9²»~»~¸¼J~ºn¨x®n~´¯¨x«$¼¹ºx¶ «M·dÃÅ·²»~»~¸
»~§ ! E & @ #
Ë0ªQ«
§ Î0Ø 1 »~§ 32 1 2 2 B ? »~§ )
>
«¨ P8 E@ F E 1 »~§ ) 2B ? P 8 @ ( E 1 »~§ ) 2 B
«\
¨ P @ + E 1 »~§ ) 2B ? = P 8 NP @ + E 1 »§ ) 2B ?
Ë0ªQ«
§ Î0Ø 1 »§ 32 1 .Ú4 2 @ 9 E B ? 1 »~§ & 2 1 ' & 2 9
.- E B ?
"$! #% " !
"$#
1 ~» § 2B ? #
)
?
= P8
, + ) = ) ( )* ( ) .- & ) ) = C
9 E B ?
"
&
) ) = ´J¸²©»~~§«$²³®5¨x®²§QºX±w°®n¾9²³¨n~°b§G
Üv«$·²³¯ºn«°± ¨xª«0ºn¸J¾¾«¨x®n¸J¹¨nªQ«0ºn«$·d°§ ¼©¶ ²³®n¨~º« ²b·d¨x»~¸B¼J«§J¨nB·²³»¨x°2¨nª«Å ®xºn¨¶"²®n¨bºx°2À5«0°§»~¸4§Q««$¼©¨n°2«ºx¨n~¾9²³¨n«
¨xª«Å ®nºx¨,¶ ²³®n¨M
&
&
"ħ"·d« 1 »~§ 2 B ? + E °b§Q«ª ²º + E 1 »~§ 2 B ? Ô jE B ? %Q°b® jE B ?
)0/ )
)
)
&
Ë0ª«|®n«d±w°®n«J )
/
9
E
B
?
&
)
&_ ) " #
&
) #"@ & )
) = @ 9 ) =
1 E 9 JB ? 2
9 E B ?
" #
" #
9
E
?
H
B
2B ?
9 0
À
x
¨
ª
@ + E 1 »§
9 B ?
f f $ )+ B ? »~§
h
+
/
@
)
@ +
9 !
&
B
+
)#
!
&
?
Ë0ª«©»B²ºx¨4~§«MÍb¯"²»~~¨X¸9~º²§É«ºx¨n~¾9²³¨n«2±w°®2¨nª«¨²~» ¶Q®n°b´"²´~»~~¨X¸©°±0¨nª«©§°b®x¾9²³»
¼J~ºn¨x®n~´¯¨x~°b§©¦X¨2~º4¶Q®n°¬b«|§¿~§
%Q«»~»~«® ( /-±w°®5¬b²³®nB²§ ·h« @ bË0ª«·²³ºn«§Q««$¼J«$¼ ª«®n«±w°b»~»~°À0º5«M²ºn~»~¸´J¸9²#»~§Q«$²®0¨x®²§QºX±w°®n¾9²³¨n~°b§G
( 4(4(
5 @
e ") ! 8 15? 2 # 2 E C # 1 À0ªQ«®n«
1 2
@
! 9 S#W
f > %Q°b®¨nª«#¾°b¾«§J¨$¹Mºn¯Q¶¶ °ºn« ? ¾§ > NM A
Ù º0ºnª°À0§+~§
«¾¾+²#E 8 15? > 2 @ 8 ; $ &&#)#)2+ &#)
Ǭ 8 @ $5" +2$'&#)+
Ö§«·²³§9«M²ºn~»~¸¬«®nñw:¸ = 1 > 2 E 9 . ? @ > @ 8 &#)
«¨,Î0X
Ø @ 8 &() = ! ( »~§ &#) »~§ &#) #
@
@
«¨ @]( E 1 »~§ 2 1 . ? 2 ?@ E 1 »~§ 2 1 . ? 2 @ @ @ @ Ô´"«¨xª«§°®n¾9²³»
¼J~ºx¨n®n~´¯Q¨n~°b§2±L¯§ ·d¨x°§& #)"J~§ ·d«0»~/§ ~º´~§°b¾½²³»~»f¸0´&~ºn#)¨x®n~´¯¨x«$¼¹À5«ÂJ§°À̱L ®n°b ¾< ¨xª«3'«$Ý+ °¬J ®x «dÒ G²¶»B²·d«5Ë0ªQ«°b®x«¾
1 %«»~»~«&
® (/0-n¹¶04 /2|¹J¨nª ²³¨ ( 2 /21 2 -·²§+´"«>²¶¶Q®n° ~¾9²¨x«$¼©´J¸ 1 2 1 2
"J¯Q®n«»~¸À«2ª ²M¬b«
E 1 ; ? 2X»~§ &()
( 2 »~/§ 2 - 2 1 1 2 1 2 2 @3 = 2 1 2 @ 9 1 S#W @ 1 SW
9
E 9
2 9 S#W ?
%Q°b® >
= F Î0ØP°b§Q«4µ«¨nº-±w®n°¾ «|¾¾9²©²³§ ¼ [ Ë0ª«|§ ! 8 1G? 2 #
@
8 15? > 2 8 1G? 2 f + 2 & #)
( = 0Î Ø - C 2 9 C $'&#)6+
@
J" ~§ ·h« 9 +
1 À«ª ²M¬«,¶®x°¬b«§+¨nªQ«»~«¾¾+²
f
( 4(4(
5 @
e #" ) ! ! P 1G? 2 # # 2
!
E
9 #S W
C
C
8 1G? 2 ( - # 1 À0ª«®x«
@
1 2
> Ù º0ºnª°À0§+~§
«¾¾+²# 15? > 2 @ 9 $ " E+ ) 1 = C E 9 E 9 ;
?2
Ǭ 8 @ $5" +2$'&#)+
Ö§«2«$²³ºn~»~¸¬«®nÃÅ «º = 1 > 2 C E 9 . ? @
«¨,Î0X
Ø @ 8 &#) = ! » § & #) %Q°b® = F Î0ØP°b§Q«4ª"²º = F 8 &#) ºx°©±w°b® = ( -M ¾9²0 15? 2 2 »~§
! ! P 15? 2 #
@ ! P "ħ"·d«
¾9 ²0 @
e
f #") ! ! P
2 9 E S W 1 2 S @
À0ªQ«®n«
@
1 20²³º0~§ «|¾¾9²
>
«¨Î0Ø 1 2N@ #
#
»§
j¹
C
@
!
C
#
#
+ #
-
?2
1B ?2
»§ B ?
C
S @ ~» §
&() = ( ~» § ;
?
;
À0ªQ«®n« 8 ³² º0§"«¾¾9² j
Ù º0~§
«¾+
¾ ² j °§«2«$²ºx»~¸¶®n°J°±Lº Q% °b® = F Î0Ø'²§ ¼ = 0Î Ø 1 2N
¾+² 1G? 2 ,
Ë0ªQ«®n«d±L°b®n«©²§ ¼À0~¨nª"«¾¾9² ²§ ¼ +#±w°b®' = F Î0Øb
# &@
BN
&
_
µb«|¨nº ¨xª«
«¾¾+² ²³º ~§ ¨xª« ¶®x°J°± °±
. 9 + 1 .
Î0ØX @ 0Î Ø 1 2c À0~¨nª = Q 8 ! P = ! ( E 1 ~» § ) 2B ? E 1 ~» § ) 2B ? #
&
E 1 ? 2 2
C &#)
2 15?
T
&#)
2 °§« ~¾¾«$¼JB²M¨n«|»¸
1
«¾¾+²
( 4(4(5"
?2 C 1
~» § # & #)
»~§ »~§ # &()
&()
´J¸ «|¾¾9²©²³§ ¼ Q
&()
= ! ( 2,
Ë0ªQ«®n«d±L°b®n«¹ À0~¨xª «|¾¾9² [
&
8 > @
9 ?1= C E
?
?
! P
15?
2 2 #
1 ´J¸ «|¾¾9²&2
1G? 2 ~» § i I i g
g
& ,
@ 9
¾
²
0
~
»
§
$ %+ i g i g
& ,
+ ) »~§ I C +
¾
² ~
»
§
i g i $ + & 2,
( g
( + .- - B ? »~§ C S @
9 + 1 .
B ? ? 2
%Q°b® = Î0Ø'°b§«2µ«¨nº±w®x°b¾ «¾¾9² ©²§ ¼ +²º,²³´ °¬«
¾9²0
! P
15?
2 #
.
!
0( 1
@ 2 1 @
@
2
S @ ¾+² ? S @1 2
.
Ù º0§"«¾¾9² j °b§Q«4§Q°À ~¾¾«$¼JB²M¨n«|»¸#µb«¨xº,¨nªQ«
«¾¾+²
( 4(4(5
?
- #
@
>
«¨ @ 1 Bh 2 & @ «¨ 8 @ $ »~§ + ) ¹ 8 @
Ǭ
'´ «©²»~°bµ,§°®n¾+²»~»~¸¼Äºx¨n®x´Q¯¨n«$¼ ®²§"¼J°b¾Ô¬b²³®nB²´Q»«¹b»~§ E
Ì«ª ²M¬b«
¾+² > A 2 '- - ¾9² > 1 2
2 - - 1 _ 2 2 - - ) 1
$ »§
&@
1
2
+ )
1 2
A
À0ª«|®n«¨xª«ºn«M·d°b§ ¼~§«$ͯ ²³»~¨X¸©±w°»~»°À0º´"«$·²¯Qºn«©»§ 1 -2 ª ²º0¾«$²§ ²§"¼~º0¨xª«®n«h±w°b®x«&¼J~ºx¨n®n~´¯Q¨n«$¼ºn¸J¾¾«¨XÒ
®x½·|²»~»~¸
"J~§ ·h« 1 = ¦ Ø 2 - - °b§«2ª ²³º
! @ @ @ # @
@ ( ; - 2 @ »~§ ¾9²0 > h A
!
"$# %&' ( )!*
2
2
@
1 »~§
@
2
f $ + E
E C 1 B ¾9²0 > A '- - )
2
1
Õ Q ²·d¨n»~¸+²º0~§
«¾¾+² j °§«2¶®n°J°³±wºa
²³§ ¼
Ë0ªQ«®n«d±L°b®n«¹ ºx~§ ·d«
!
'
!
2
! @ #
2 @
@ #
8! # @ #
@
(
- - 9 1 @
- - 9 1
-
8 @
#
" I
# @
!
! 1 @ 2 1 @
_ - , 2 +
@ #
(
8 ! # ! @ "c
C 8!
8 ,
@ 2 1 @ 2 @ f
+ - 6,
( @ - " C 9 - - 1 C ' ! 8 ! 8 8 ! # @ #
! ! ;
8 # + 8 8! # ,
f ; " E C BN EUi '- - ) 1 C 9 - - 1 g
C 8!
2
! @ @
8! # ( 8!
@ 2 1 @
1
1
2 ·h°b§1 ºx¨$
"ħ"·d«2À5«·²³§9«|ºn¨n~¾+²¨n«²³º0ºnª°À0§+~§¨xª«#I®n°J°±°³± «|¾¾9²&E
E 1 » § ) 2 B
8
8
( 2 @ 2 - 2
9 1 SW
²³§ ¼ ~» § ) 2 9 B ? x¨ ª«#«¾¾9²~º0¶®n°¬«§
( 4(4(5¿!
> !
! !5 1 @ 2 1 @
2 ' 8 C 2
2
8 ( @
f
"
-
; g
2 " Ú
+ & B 1 "
C
@ #
#
8 = @
C
9 1 S#W
9 1 S#W
'
@
#
?
1
1
@ I
@
@ I
g @ i
2
8 (
@ i
@
8 K/ @
@
@ #
9
@ #
C @
@ #
8 ! @ @
(
2 ! 1 @ 2 1 @
@
Ë0ª«2ºx«$·d°b§"¼+~§«$ͯ ²»~~¨X¸#±w°b»~»~°À0º5ºx~§ ·d«2¨nª«#¶®n°´ ²´~»~~¨X¸>°³±²©ºx§Qµb»~«,°¯¨·d°¾«2·²§+´"«4«|ºn¨n~¾+²¨n«M¼9²³º @ 1 %«|»»~«®
(/0- 2Ë0ª«#»½²³ºn¨~§«$ͯ ²»~~¨X¸~º,Á °»½¼Ä«®$Æ º$
@
1
!
"J~§ ·h« ! 1 2 @ # 2 @ C
¨xª«
«¾¾+²,~º0¶®x°¬b«§G
@ I42 @ #
f $ + ±w°b»~»~À0º×²³º0ºn¨d²¨n«|¾«§J¨§
«¾¾+²>
( 4(4(5 Ä
' ! # 2 1 ?
> ! # @ L i L Lf 2
ºx§"·d« 3
!
*
#
@
2
2 1 C 1
2 @ C
( 4(4(5 "
S @
@
g L Ú
g i
@
2
I
g i
f $ +
@
2 @
B ? 2B ?
«©²»~°bµ§°b®x¾9²³»»~¸>¼Äºx¨n®x´Q¯¨n«$¼+Ø9!>¹»~§ @ E B ?
¦ ØZ b 2 »~§ ¾+² & ( / 1 »~§/
2 1 I2 2 1 »~§ 2 ¾9² & (
1 2 1 E B ? 2À«2ª ²M¬b«
S @ @ ¾90² 2 .- - ¾9²0 & 1 2 1 2 ( 2 .- - 1 _ 2 @ .- - )
À0ª«|®n«¨xª«ºn«M·d°b§ ¼~§«$ͯ ²³»~¨X¸©±w°»~»°À0º´"«$·²¯Qºn«©»§ 1 2-ª ²º0¾«$²§ ²§"¼~º0¨xª«®n«h ±w°b®x«&¼J~ºx¨n®n~´¯Q¨n«$¼ºn¸J¾¾«¨XÒ
®x½·|²»~»~¸
Ë0ª~º~¾¶»~~«º 1 »~§ 32 1 .I4 2 2 1 »~§ 2 ¾90² & ( 2 »~§ 32 ¾90² & ( !#"$&%' 1 E 9 B ? 2 / # .- - ) # @
!
!
>
Ǭ
'´
%Q°b'
® =
"J~§ ·h«4»~§
Ǭ @ 1 Bh
@
1B ? 2
)7)+¢¥ %
Î0Ø,Ø'·d°§¬«®nµ«º,À0~¨xª9°®¼J«|®b
-1 Ì«°b§»~¸ª ²M¬«2¨n°¬b«|®nñw¸:
1 2 @ @
g 1 i
1 2 @ @
g 1 i
1 2S @ @
1g i
%Q°b®ºn~¾¶»~½·h¨X¸J¹ºx«¨ E@ " ) ¹ @ )
1 2 "J~§ ·dN
« S#K C W = @°b§Q«4µ«¨nº -126 %
S#K
/
S#K
C
C
W 1K
W=
/
K
²§"¼
= 2 C W= @ =
1 K C = 2 1 K = 2 1 S C W2
@
1 K C = 2I
SK C W = @ K C = = 2
@
@
@
@
@
@
S 1K
g 1 i
g 1 i
±w®n°¾Ô¨nª«2ºx«®n~«º2«¶ ²³§ºn~°b§°±¨nªQ«Õ Ķ"°b§Q«§J¨nB²» %Q¯§ ·h¨n~°b§
1 2 @ @
1g i ~º¶®n°¬«§+§+¨xª«2º²¾«,À0²M¸
1 2 %®x°b¾ ²#ºn«®x~«º2« ¶ ²§ºx~°b§°³±¨nª«Õ ¶"°b§«§J¨x½²³» %Q¯§ ·d¨x°§°§«2¾¾«$¼Ä½²³¨n«»~¸2µb«|¨nº
= @
K @
g 9 1 i
g 9 1Ti
@ "J~§ ·d«2»~#§ K @ 0»~§ = @ E 9 B ? °§«2µb«¨xº S @ @
-126 % Ø<·h°b§J¬b«®xµb«ºÀ0~¨nª°®¼J«|®>b
-1 Ì«°b§»~¸ª ²M¬«2¨n°¬b«|®nñw:¸
1 2 @ @
1g i
1 2 @ @
g 1 i
1 2S @ @
g 1 i
%Q°b®ºn~¾¶»~½·h¨X¸J¹ºx«¨ E@ " ) ¹ @ )
1 2 Ù º0~§¨xª«¶®x¬J~°b¯Qº5¨xª«°b®x«¾)°b§Q«4µ«¨nº
S#K
/
C
1 2
W=
@
@
1 K C = 2 1 K = 2
@ 1 K C = 2
S#K C W = ) C = ) ) K @
@
@
) @
g 1 i
g 1 i
´¸²©ºx«®n~«º«¶ ² §Qºn~°b§+°±¨nª«©Õ ¶ °§«§J¨nB²» %¯§"·d¨n~°b§G
@ @
1 i ~ º¶®n°¬«§+§+¨xª«2º²¾«,À0²M¸
g
)
)_" ) $ " + ) %
1 2 %Q®n°¾
²#ºn«®x«|º,«¶ ²³§ºn~°b§+°³±¨xª«Õ ¶ °§«§J¨nB²³» %¯Q§ÂJ¨n~°b§°§«2¾¾«$¼Ä½²³¨n«»~¸µ«¨nº
9 B ?# = @
9 B ?#
K @
!
!
9 B ? #¹»~§ = @
9 B ? #¨xª~º0¶®n°¬«ºS @ @
"ħ"·d«2»§ K @
B ?
!
!
0-26 ! Ë0B²§9·h°b§J¬b«®xµb«ºÀ0~¨nª+°b®d¼J«®b
0-) "J~§ ·h«>Ë0B²§ª ²º2« ¶»~B·d~¨n«»~¸+Å Ä«M¼¨xª«Å"®nºn¨#¨nª®x««¾°¾«§J¨nº @ @ @ @ @ @ ¹À«©°b§»~¸+§««$¼É¨x°
¬«®nñwa¸ S @ @ @
%Q°b®ºn~¾¶Q»B·d~¨X¸J¹ºn«|¨ E@ " ) ¹ @ )
2 #°b§«ª"²º »~§ 1 2 1 Ù ´®n°¾°À0~¨nв³§ %Q°b(
® ¼ "J¨x«µb¯§ ( j - 2Ë0ª ²³¨,¾«$²§º-±w°b®( + 4/ 2
§ ( j 0- 2
»~§ 2 I
Ý °b®n«|°¬b«®±w°® ©°§«2ª ²º0»~§ 1 C 2 1 Ù ´®d²¾°À0~¨xÐ,²§ ¼ "J¨x«µb¯
»~§ Ú 1 +"4 /
1 >»~§
I
Ë0/ ª~º~º¨nª«·²³ºn«±w°® 1 Ì/ «0ª"²M¬b« = K ²§ ¼©¨xª«®x«d±w°b®x« 1 »~#§ K2 2 1 K Ì4 2 ¹ 1 »~§ = 2 2 1 = Ì4 2
"J~§ ·h«
S @ @ »§ i i g
g
2 » § i ' i g
g
~¨º0ºx¯9·d~«§J¨0¨n°¶Q®n°¬b«H KI @ ! 9 B ? # ¹ = Ú(@ ! 9 B ? #
%Q°b®x¾ ²ºx«®n~«º2«¶ ²³§ºn~°§°b§Q«µb«¨xº
C @ C 1 B ? 2 C C 1 B ? 2
²³§ ¼
KI
@
@
@
@
C
C
!"
1
1B ?2
¦X§+¨nªQ«4ºd²¾«,À0²M¸°b§«2¶®x°¬b«º = Ú
@
C -
C 42 C
1B ?2
C
C
1 B ? 2|
C 1 B ? 2
@
1B ?2
Ú
C
@
1B ?2
I
#
!#"$&%' C
1B ?2
Ú
C
1B ?2
(+7-+ +Q¤+,
5?4(?*+, %*0> "/!³ Ä!
¹ #%( # 94 # 1#%$ /3 !
¹ ¨xª '°¬«®I®n~§J¨XÒ
§Qµ ¹"4 ) j
+B© ! Ë0ª« Ù ··d¯®d²b·d¸P°±¨nª«.²³¯ºnºx½²³§ Ù ¶¶®x°¾+²¨x°§Ì¨n°O¨nª«"J¯¾ °±>¦X§"¼J«¶"«§ ¼J«§J¨
!5²®x½²³´»~«º "¹ -#!
'- # 9 ) +)Q¹ )+ ¹J¶¶~ j
>@
-0 ! JÖ¶¨n~°b§ !5²»~¯ ²³¨n~°b"
§ 0ºx§Qµ²©Ë0ª®n«!« b¯Q¾¶ I®n°·d«|ºnº "¹ -# ! #!$26!
!!/-(
#!$7Q¹4 )¹¶¶ /0 Q
5/,%*5 4(@
"# ! ! "
Ù ¾«|®nB·²§Ö¶¨x°§&Õ¬b²»~¯ ²¨x~°b§ Ç0«|À Üv°b¯Q§ ¼Jº$¹ Ù ¶¶®x° ¾+²¨x°§º$¹
²§ ¼O²¿Î°¾¶ ²³®n~ºn°§Ì°±Õ Ä~ºx¨n~§µÌÝ+«¨nªQ°Ä¼Äº ¹%$ !-&! #26- %$ /'#(' 6!) -3
38 *,+ 7.- %-&*/+ - *
012143"5|
06
-(07(-985?-:."! -5?@;< ! Ö§<¨nª«Ûز¨x«Ì°± Ù ¶¶®x° ¾+²¨x°§I~§ ¨xª« Ϋ|§¨x®²³»
~¾~¨Ë0ª«°®n«¾ ¹=!/-#%$, 2 2 - 94!-© Q¹4 ) /¹¶¶ /7+
5?4( >8 Ä! Ù ¶¶®x° Ä~¾+²¨n~°§4¨x°>¨xª«,¶"°b~§J¨´~§°¾B²» "¹ 6#%$ # 94 # #%$ # 8
3
7¹ Q¹ )¹J¶¶0 7
- 71©?8~.3 0! 21©B©./ !"! Üv§Q°b¾B²³»$Ý+°¼J«»~º±L°b® "J°b¾« I²³¨nª ',«|¶ «§"¼J«§J¨
Ö¶¨x°§º "¹$ !-@ #26-A -% / @* .0-1# /*
B' 6!) -3
38 DC -( #!D*EC 26!-FHG323HI
?<# >! !4
~+ Ä! bË0ªQ« !5²³»~¯ ²¨x°§2°±Ö¶¨x°§ºG±w°b® Ù »¨x«®n§"²¨n~¬« "J¨x°·nª"²ºn¨xB· I®x°·d«ºnºx«º "¹!/-(
#!$ #! #%$ . !%3
+ )/ j ¹¶¶0 + jJj
?<# >! !4
~+ ,! Û%* ! !
JÖ¶¨n~°b
§ I®nB·d~§#µ Ù "J~¾¶Q»ÃÅ «M¼ Ù ¶¶Q®n°²·x7ª ¹!/- #!$
#! #%$ . !%3
/Q¹4 ) /)¹J¶Q¶ ) j Q
?<# ,! Û%* ! JÖ¶¨n~°b§Qº,ݲ®xÂb«¨xº ¹ I®n«|§¨xB·d« ÄÁ²»~»X¹ Ç-«À «®nºx«¸J
!!=J >S: ! %°¯®n~«® Ù § ²³»¸JºxºG°± '~ºn¨x®n~´¯¨x~°b§ %Q¯§ ·d¨x~°b§º ¹ 3# # 9 //Q¹4 ),+ Q¹¶¶
. ,@
@,60> ! Ë0ª« '~ºn¨x®n~´¯¨x°§,°³±Ä¨xª«¦X§ ¼Ä« §2²0Ç0°®n¾9²³»bÜ~¬b²³®nB²¨xD« IG°b¶¯Q»½²³¨n~°b§ ¹K !'-? #¹
+ ¹ )¹¶¶ + +
.>@,@
6: ! 6
Ù §¦X§¨x®n°¼J¯ ·h¨n~°b§Ì¨x° I®x°b´ ²³´~»~¨X¸+Ë0ª«°®n¸¿²³§ ¼¦X¨nº Ù ¶¶Q»B·²³¨n~°b§Qº ¹L$ $ -
"
-1*1#* $ #%(L # 96'# #!$ # 3
h¹ !°»J¦ C ¦x¦¹ I~»~«¸ "J°§¹¦X§ ·Ç-«
0
À °b®n¹
°b§ ¼Ä°b§²§ &¼ "Ľ¼Ä§«¸J
: 5?
5©! " bË0ªQ« %Q®n«$ͯ«|§ ·d¸ ',~ºx¨n®n~´¯Q¨n~°b§°³± ¨nªQ«M,¯Q°b¨n~«§J¨°±"Ë0À5°Ç0°®n¾+²» !²®nB²³¨n«º ¹"!/- #!$
96LC ! #!$ # #!$ ¹ )Q¹4 )¹J¶¶G ++ 0 j
: 6-5?0/. ! !
"J°¾«Ç0¯¾«®nB·²³»×ΰ¾¶ ²³®n~ºn°§º°± "Ä«¬b«®d²» Ù ¶¶®x° ¾+²¨x°§º2¨n°É¨nª«+Ü~§°¾B²»
',~ºx¨n®n~´¯Q¨n~°b7§ "¹N%/ -#!$ 94 '-#! # #!$ 1
) # !¹Ü¼ j + ¹ ) j )¹¶¶0 j 0 j + j
8 56! # @
! 75Ì< ! 6
ݲ³®n¨n~§µJ²»~«º5²³§ ¼ "Ĩn°·xª ²ºx¨nB·0¦X§J¨n«|µb®²³»~º5~§¨nªQ«4Ë0ªQ«°b®x¸°±
ΰb§J¨x§J¯°¯º
Ë
®d²b¼J~§µ ¹ )394# 1 0 -1 )3
#! (' 94- 2*2$ # !
¹G4 )Qb
8 E8 5> ! !"
Jΰb§J¬b«|®nµb«|§ ·d«#±w®n°¾ '~º·d®x«¨n«hÒרx°9ΰb§J¨x§J¯°¯ºXÒXË0~¾«,২x~§µb«§J¨,λB²~¾º I®nB·d«|º ¹OC &
)) 7 L #!#!$ /4(%3
¹ ))¹Ç0°" + ¹J¶¶G 0 ,+ j
8 @
@P# ! !"
JÖ,¶Q¨n~°b§ºM¹ %Q¯¨n¯Q®n«ºM¹ ²§"¼9Ö¨xª«® '«®x~¬b²¨n~¬« "J«$·h¯®n~¨n~«º ¹ I®n«§J¨x½·h« ÄÁ²»~»X¹ Ç0«|À «®nºx«¸
8 @
@P#E: -,+ ! ! JÕ 9·d~«§J¨ I®n°·d«$¼Ä¯®n«º-±w°bA® !5²³»¯Q§Qµ I²³¨nª ',«|¶ «§"¼J«§J¨,Ö¶¨n~°b§Qº ¹$ !-&! #26-@*& # /$ L #%# 'Q&' 67) -
Q !-1%
8 @
@P#E: -,+ # Ë0ª« Ù ··d«|»«|®²¨x«$¼ÉÜ~§°¾B²»Ö¶¨n~°b§ I®xB·d~§µÝ+°¼J«» ¹=%/ -#!$L #%#!$ #! (
R /6#% 8# !) #%$ j ¹ ))Qb¹¶¶0 j +"
H5?
4(?> ! JÖ,§¨nª« Ù ··h¯®²·d¸¿°³± .²³¯ºnºx½²³§ Ù ¶Q¶®n° ~¾9²³¨n~°b§¨n°¨xª« ',~ºx¨n®n~´¯Q¨n~°b§ %Q¯§ ·h¨n~°b§º°±
"J¯¾º°±¦X§ ¼J«¶"«§ ¼Ä«§J¨ !²®nB²³´»~«º "¹ !- 0
-1* 2 2$ b¹4 ) jJj ¹b¶¶G )0 )/
0! @
@P#< Ä! 6
S 94!- T #!#%$U %> #! ¹4Ø-°À0¾9²³§ ~¨n¨x»~«dÅ «»B¼¹Ë°b¨x°ÀײQ¹×Ç-«À
b«®xºn«¸J
#>5??*A
+
//Ú+ ! "
B 26%0 -¹Á0°¾«À°J°¼¹b¦X»~»~~§°b~ºM¹ )Q¹b¶Q¶~ 0
#>-! ~(0WV ÉW ! !
'~º·d®x«¨n« '~ºn¨x®n~´¯¨x°§º "¹L
$ $ -
' 0 -1*# $ #%(A # 94&
# #%$ # 8
13
¹ I~»«|¸ "J°§¹J¦X§ ·Ç0«À 5°b®xÂ7¹ °b§"¼J°b§+²§ ¼ "JB¼J§«|¸
V 56@,
! "!
Ù %G~®nºn¨Î°b¯®xºn«~§ "J¨n°·xª ²³ºn¨nBA· I®n°·d«|ºnºn«|º ¹ # (*" 0 -
¹Ç0«À 5°b®xÂ4²§"#¼ °§ ¼J°b§G
V/5?@,@;>,%*5+> 5?A Ä! S 94 (4) #%1(> 96!- # ¹ !°»X¹Îª ²³®n»~«º .2®n § ΰ¾¶ ²³§J¸¹ °§ ¼J°§
V@
>?/)
-27 @
5?W2 Ä!"! + /'-#!$ !$ /*!' )394# U -1 #!$!./6# %
¹0"Ķ®n~§µ«®
!v«®n»B²µ"¹ ))
( ;
+ ! ! JÜ~§°¾B²» Ù ¶¶Q®n° ~¾9²¨x~°b§ ΰ¾¶¯¨d²¨x°§ ²»"J~¾¶Q»B·d~¨X¸5²³§ ¼Î°§¬«®nµ«§ ·d« ¹E$ !-&! #
26-F36104*E (*-#!$C )-H) K #!6
$ H)!$ #! (¹4) )Q
@ ! 75 5?4(5! *5?4 EV< Ä!"! "J~¾¶»~«
Üv§Q°b¾B²³» I®x°·d«ºnºx«º0²º 'Ãʯºn~°b§ Ù ¶¶®n°~¾9²³¨n~°b§Qº
~§ %~§ ²³§ ·dB²»Ý °Ä¼Ä«»~º ¹C H) 7 #% #!$ /4(%3
¹~ ))Q¹b¶Q¶ )0+
+4 $O2 ! Õ 9·d~«§J¨ ',~ºd·d®n«¨x«>Ë0~¾(« b¯Q¾¶ I®n°·d«|ºnº>Ý+°¼J«|»º~§ Ö¶¨x~°b§ I®xB·d~§µ "¹!/-#%$,
#!#!$#!( R /4#% # 7) #!$ ¹ !°b»X ¹Ç-° ¹ ¯§#« Q¹J¶¶0 j 4 /,+
5 @ ! /)* 5?0/ ! Ù § Ù ¶¶®x° ¾+²¨x«Ç0°b®x¾9²³»~Ð$²³¨n~°b§°±4¨xª« Ù § ²»~¸Jºn~º©°± !5²®x½²³§ ·d« ',~ºx¨n®n~´¯ÄÒ
¨x~°b7§ "¹ 6 #!$ L # 94 # #%$ # ¹ Q¹4 ),+ ¹¶¶ 0
06~ 5?=# : Ä! Ù Ç0°b®x¾9²³» Ù ¶¶Q®n° ~¾9²¨x~°b§±w°b®>Üv~§°b¾B²»X¹ %¹
Üv«¨d²¹5²³§ ¼ÛÖ¨xª«®
ΰ¾¾°§9Ø0«|»½²³¨n«$¼Ë²~D» I®n°b´"²´~»~~¨n~«º$¹"¦ "¹< 96!/-#%$0 96 '-#! # 8
1#!$ 1
) # !¹Ü¼
j ¹"4 ) j ¹J¶Q¶0 +" j + j
5? =# : Ä! Ù Ç-°b®n¾+²» Ù ¶Q¶®n° ~¾9²³¨n~°b§±w°®Üv~§°b¾B²»X¹ %¹bÜ«|¨²¹J²§"¼Ö,¨xª«®5ΰ¾¾°b§¹Ø-«»B²¨n«M¼
˲³» I®n°b´"²´~»~~¨n~«º$¹M¦x¦ ¹ 94 %/-#!$ 94 '-#! # #%$ # !Q¹bÜ5¼
j ¹J ) j Q¹M¶¶ + /
+
35 %*B< Ä! Ö§ Ù ¶¶®x° ¾+²¨x§Qµ0¨nª« IG°b~§J¨Üv~§°b¾B²» ¹ '-#! # 1#%$ ))# %B!/-(
#!$f¹ ¯§« ) j
35 %*B< Ä! bË0ª«Î°¾¶ ²³®²³¨n~¬b« Ù ·|·d¯®²·d¸É°3± "J«¬b«|®²» Ù ¶Q¶®n° ~¾9²³¨n~°b§º0¨x°+¨nª«Î¯¾¯Q»½²³¨n~¬b«ÜvÃÒ
§Q°b¾B²³» 'ºx¨n®x´Q¯¨n~°b§ ¹ 0§¶¯Q´»~ºxª«$¼+ݲ³ºn¨n«|®$Æ º,¨xª«ºn~ºM Û²ºxª~§µb¨x°b§ 94 -#! &' 6!) -
¹"4 )Q
3,4(6%*0>5?/4(5?W=V Ä!"!"
'°À0§ Ù § ¼ ÄÖ¯¨4Î0²³»~» "¹ " %Ü ¹ '~º·h¯ºnºx°§ I²¶"«®Ç0°",Ü )Q¹ -§~¬b«®xºn~¨X¸°³±Üv°b§§
3,4(6%*0>5?/4(5?W=V Ä!"! "
Õ¯®n°¶ «$²³§I²§ ¼ Ù ¾«|®nB·²§ÚÜ5²³®n®x«|®Ö,¶Q¨n~°b§º ¹ " %Ü Q¹ ',~ºkÒ
·h¯ºnºx°§ I
²³¶ «|®4Ç0°"
Ü /Q7¹ 0§~¬b«|®nºn~¨X¸°±Üv°b§Q§
3 /@
4(5?+7Ñ556 B ! Ä!
bË0À° "Ĩ²¨x«>Ö¶¨n~°§ I®nB·d~§µ "¹!/- #!$ D #! 1 ,+ '«$·d«¾ÑÒ
´"«® )/)Q¹J¶¶0 )0b Q
! E%* ! !"
JÕ °b¨nB·2Ö¶¨n~°§º "¹$ !-@ 4 #324-S' 6!) -
' #%$ )%-6#'# WK -& $
1 ,5? < Ä!"! Ù Ý+°¼JÃÅ «M
¼ ²³¨n¨x½·h« Ù ¶¶®x°²b·xªÚ¨n°ÛÖ¶¨x~°bF
§ I®nB·d~§µ "¹/!/- #!$ A /* /-13
Q #%-&% 3 h¹
!v°b»X Q¹Ç0°" Q¹J¶¶ j 7/ /
1 !(>!Q !"! 6
Ù °bµ³Ò{¨x®²³§ºX±w°®n¾«$¼ÌÜ~§°¾B²»Ç-¯¾«®xB·²» Ù § ²³»~¸ºx~º&Ý+«|¨nª°¼É±L°b® !5²»~¯~§µ
ΰ¾¶»~« ¿Ý+¯Q»¨xÃÒkÖ¶¨x°§¿¦X§J¬«ºn¨x¾«§J¨nº ¹!/-#%$ #!#%$ #%( R /4#% # 7) #!$ 38
j ¹Ç0° "
¹
"Ä«¶¨n«|¾´ «® ))Qb¹¶¶ )7 j
1 6 8. Ä! !
Ù §Õ»~«¾«§J¨d²®n¸ I®x°°³±°±¨xª«4Ϋ§J¨x®²» ~¾¨GË0ª«°®n«¾ -9 # 9 Q¹ ))Q¹
¶Q¶ j +"
: @! 25ÑE8,@ H%*%* #bË0ª
« '~ºn¨x®n~´¯¨x°§Ú°±>ΪÃ#Ò "QÍb¯"²®n« ¹ 0-1 1(% 94 + # %#!$
# (*" 3
h¹G4 /Q¹ )¹J¶¶ j + j
ZY|W
W
T
#ZW
Z|W`[V!
`VX[
""Y
bY`[#%$&
V'#%()
MYV*`W,Z+ W.-/,$&
01
MYVXWYZVXV*
2
#*34565637-198
MYZ:
é;=×< ßdî~ìßøøh÷{áLíXíå?>A@BDC6@EAFBGE6HB*I?J=BDKDLMJON6EBDPQ6RE6E,JSH6@
ZY|W
WTMU W|ZW` VXW` [V!
` VX[
T" Y
b Y` [#'$V
V'#%()$
YV*` W Z+ WT-JT
ZWM
YZ
XYZYX?[#\34565]4
-/^8M
Y Z:
é;=×< ßdî~ìßøøh÷{áLíXíå?_A@B*L`@6_]FKH6HBJaR_,JONDEbBDPcQREE?JSH6@
© Copyright 2026 Paperzz