implementing_interest_rate_models.pdf

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Transcript of implementing_interest_rate_models.pdf

Page 1: implementing_interest_rate_models.pdf

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Page 7: implementing_interest_rate_models.pdf

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|rwq uz r ~qwq|} uyuzwuv ��yvwuxy x� ��� � �qw � �� � � � q wpq s|uvq x� r �q|x�vx�sxy �xy~ x� �yuw �rvq �rt�q rw wu}q � � { uwp } rw�|uw� � � � �� � � vry�q ~qwq|} uyq~ �|x} wpq rzzx vurwq~ zs xw |rwq � �� � � ��

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{pq|q

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� e� fg nh n� � � n� �k �li�� � nhh ng��x| x�| s�|s xzqz {q rzz�}q wpq ~rut� � uqt~ ~rwr r|q �u�qy �x| �q|x�

vx�sxy �xy~z x� �r|ux�z }rw�|uwuqz � opq|q uz yx yqq~ � �x| q�r}stq �

Page 11: implementing_interest_rate_models.pdf

wx qzwu} rwq z�vp �uqt~ ~rwr �|x} x�zq|�q~ }r|�qw s|uvqz �x| zwry~r|~vx�sxy�vr||� uy� � xy~z � � uq t~ v�|�q �wwuy� { utt wp�z � q r zu}stq }rwwq|x� uywq|s xtrwuxy wp|x��p r �yuwq zqw x� s xuywz �

� uwp wpq ��tt�� puwq xyq��rvwx| }x~qt � wpq w|r~uwuxyrt }qwpx~ �x|�uqt~ v�|�q �wwuy� uz wx vpxxzq � ��� rz �u�qy �� ���rwuxyz ���� ry~ ���� ��uyvq wpq �x|{r|~ |rwq |q��u|qz ~u�q|qywurwuxy x� wpq vxywuy�x�zt� vx}�sx�y~q~ uywq|qzw |rwq � �� �� � r ~u�q|qywur� tq v�|�q � �� �� }�zw � q uywq|�sxtrwq~ �|x} wpq x�zq|�q~ }r|�qw |rwqz � �� �� � � � � � � � {pq|q ��~qyxwqz wpq }rw�|uw� ~rwq � �ywq|s xtrwuxy vry � q rvvx}stuzpq~ � �x| q�r}�stq � �� �zuy� ��zs tuyq |x�wuyqz z�sstuq~ �� zwry~r|~ y�}q|uvrt ryrt�zuzsrv�r�qz tu�q �� �� �

yq uzz�q x� |qvqyw uywq|qzw � sr|wuv�tr|t� uy wpq vxywq�w x� ��� }x~qtz �uz v�|�q vxyzuzwqyv�� � �� }x~qtz wr�q rz uys�w wpq uy uwurt � uqt~ v�|�q ��|x} {puvp wpq qywu|q q�xt�wuxy x� wpq � uqt~ v�|�q uz ~qwq|} uyq~ �xyvq wpq�xtrwutuw� ��yvwuxy prz � qqy zs qvu�q~ � � � qvqywt� uw prz � qqy qzwr� tuzpq~wprw � qtzxy��uq�qt v�|�qz r|q yxw vxyzuzwqyw { uwp ry� uywq|qzw |rwq }x~qt ������ u� wpq uyuwurt wq|} zw|�vw�|q uz zs qvu�q~ �zuy� r � qtzxy��uq�qt v�|�q �wpqy z��zq��qyw wq|} zw|�vw�|qz rz s|xsr�rwq~ �|x} wpq ��� q��rwuxyvryyxw � q q�s|qzzq~ rz �qtzxy��uq�qt v�|�qz � �q ~x yxw r~~|qzz ��|wpq|wpq uzz�q x� v�|�q vxyzuzwqyv� pq|q � zuyvq �x| x�| s�|s xzqz wpq u}srvw xyy�}q|uvrt rvv�|rv� uz yq�tu� u� tq �

� e� �ji��ik ��h� ��g�i�h njg� zz�} uy� wpq }qry |q�q|zuxy ry~ �xtrwutuw� vxq! vuqywz r|q vxyzwryw �

wpq zwry~r|~ ��tq| ~uzv|qwu�rwuxy x� wpq ��tt�� puwq xyq��rvwx| zpx|w |rwqq��rwuxyz tqr~z wx

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opq ~uzv|qwu�rwuxy wprw {q zprtt uyzwqr~ �zq uz wx �|zw �qyq|rwq �x|{r|~srwpz �x| � ��� rvvx|~ uy� wx

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�� ��� �� � � ��� � � �� �� � � ¬��¬ �� � ��� �¬ ����

Page 12: implementing_interest_rate_models.pdf

{pq|q � �� �� uz wpq uyzwrywryqx�z �x|{r|~ |rwq �u�qy �� ���rwuxy ���� ��vp} u~w ������ vtru} z wprw wpuz ~uzv|qwu�rwuxy ~x qz yxw uyw|x~�vq ~uz�v|qwu�rwuxy q||x|z rw wpq zr}stuy� wu}qz � ry~ uz s|q�q|r� tq wx wpq zwry�~r|~ ��tq| ~uzv|qwu�rwuxy � opuz }qwpx~xtx�� vry rtzx � q r~rswq~ wx~uzv|qwu�q wpq � trv��� r|rzuyz� u }x~qt �

� e� �� �n�i�h njg�y wpq ��tt�� puwq }x~qt � � ��� uz vx}stqwqt� ~qwq|} uyq~ �� wpq v�||qyw

�uqt~ v�|�q � �qwq|} uyuy� wpq |q}ruyuy� }x~qt sr|r}qwq|z � ry~ � uz|q�q||q~ wx rz wpq s|x vqzz x� vrtu�|rwuxy � �x| uywq|qzw |rwq ~q|u�rwu�qs|uvuy� rsstuvrwuxyz �{puvp uz wpq �x v�z x� }xzw x� wpq zwry~r|~ uy~�zw|�tuwq|rw�|q |qtq�ryw wx uywq|qzw |rwq }x~qtuy� � � wpq �qyq|rt s|uyvus tq uzwx vx}sr|q � �x| r zqtqvwq~ zqw x� �yryvurt uyzw|�} qywz x| } r|�qw ~rwr �wpq s|uvqz �x| |rwqz� x�zq|�q~ uy wpq }r|�qw { uwp wpxzq s|q~ uvwq~ ��wpq ��tt�� puwq }x~qt � �uyvq wpq pq~�uy� x� uywq|qzw |rwq ~q|u�rwu�qz uzr } r x| rss tuvrwuxy x� wq|} zw|�vw�|q }x~qtuy� � w|r~uwuxyrt vrtu�|rwuxy}qwpx~z wqy~ wx �zq � xy~ xswuxyz rz wpq vrtu�|rwuxy uyzw|�}qyw � �y wpqr�zqyvq x� r tu��u~ }r|�qw uy � xy~ xswuxyz �rz uy wpq � x|qry }r|�qw� �xyq vry txx� �x| z�||x�rwq uyzw|�}qywz � ��� �� � xy~ ��w�|qz � �x{q�q| ��xy~ ��w�|qz pr�q wpq ~uzr~�rywr�q x� � quy� uyzqyzuwu�q wx �xtrwutuw��

�y wpq r�zqyvq x� z�vp tu��u~ vrtu�|rwuy� uyzw|�}qywz � wpq �z�rt |q�vx�|zq uz wx |qt� xy puzwx|uvrt uywq|qzw |rwq ~rwr � quwpq| q�s tuvuwt� x| u} �s tuvuwt� � wpqzq �z�rtt� wr�q wpq �x|} x� zs xw |rwq wu}q zq|uqz x� ~u�q|qyw}rw�|uwuqz � �� xyq vxyzu~q|z rz uys�w wpq p uzwx|uvrt zs xw |rwq wu} q zq|uqz�x| xyt� r zuy�tq }rw�|uw�� wp uz }qwpx~ x� vrtu�|rwuxy vry � q |q�r|~q~ rzx vv�||uy� x�q| wpq �wu} q� ~u}qyzuxy � � twq|yrwu�qt�� u� xyq xyt� vxyzu~q|zwpq �r|ux�z zs xw |rwq }rw�|uwuqz rw wpq v�||qyw wu}q � wpuz vry � q |q�r|~q~rz vrtu�|rwuxy x�q| wpq �zsrvq� ~u}qyzuxy �x� zs xw |rwq }rw�|uwuqz� �

opq wp|qq �rzuv vrtu�|rwuxy }qwpx~z wprw {q vxyzu~q| r|q ��� �wwuy�wx wpq v�||qyw zsxw |rwq �xtrwutuw� v�|�q � ��� wpq }qwpx~ x� �qyq|rtu�q~}x}qywz �zuy� zpx|w |rwq wu}q zq|uqz � ��� � rt}ry � twq|uy� ry~ ��rzu�} r�u}�}�tu�qtupxx~ qzwu} rwuxy �zuy� puzwx|uvrt zsxw |rwq wu}q zq|uqz �x|�r|ux�z } rw�|uwuqz � � wpqzq � � qwpx~ � vx||qzs xy~z wx vrtu�|rwuxy x�q|wpq zsrvq ~ u}qyzuxy � � qwpx~ � wx vrtu�|rwuxy x�q| wpq wu}q ~u}qyzuxy �ry~ � qwpx~ � wx r vrtu�|rwuxy x�q| vx}�uyq~ zsrvq�wu}q �

��������� ���� �������� ��� � �q{|uwuy� ���rwuxy ���� uywpq �x|}

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Page 13: implementing_interest_rate_models.pdf

xyq x�wruyz wpq �x ttx{ uy� ~uzv|qwu�rwuxy �� �� � � ��

� � � �� � � �� ��� � ��� � ����{pq|q wpq ~uzv|qwq wu}qz r|q ~qyxwq~ � � �¬ � ry~ {q �zq wpq zu}stu�q~yxwrwuxy

� � � � ��� � � ����� � � � ��� � � ��� �� � ��� � ����

{ uwp wpq �� rzz�}q~ wx � q uy~qs qy~qyw zwry~r|~ �q|x�}qry � �yuw zwry�~r|~ ~q� urwuxy yx|}rt ~qyzuwuqz � � zu} utr| ~uzv|qwu�rwuxy x� wpq zsxw |rwqq��rwuxy ���� tqr~z wx

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� r|�qw �xtrwutuw� ~rwr �x| ~rut� zs xw |rwq vpry�qz x� ~u�q|qyw }rw��|uwuqz � � �� � vry � q x�wruyq~ �z�rtt� uy s��tuzpq~ �x|} � x| ~q|u�q~�|x} wu}q zq|uqz ~rwr �|x} r y�}�q| x� zwry~r|~ {r�z ���� �� wpq zr}�stq �r|uryvq �x|}�tr � q�sxyqywurtt��{qu�pwq~ }x�uy� r�q|r�q �x|}�tr � �vrtu�|rwuxy vry wpqy � q s q|�x|}q~ �� } uyu} u�uy� wpq x� qvwu�q ��yvwuxy

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{pq|q {q rzz�}q wpq �xtrwutuw� qzwu} rwqz x� wpq x�zq|�q~ ~rut� zs xw |rwqvpry�qz r|q x�wruyq~ �|x} wpq }xzw |qvqyw ��� ��� w|r~uy� ~r�z �rz uzvx}}xy s|rvwuvq uy uy~�zw|� � x� ~rut� zsxw |rwq wu}q zq|uqz ~rwr � �x|q�r}stq � ~qyxwuy� wpq wu}q zq|uqz x� ~rut� zs xw |rwq vpry�qz �x| r �u�qy

Page 14: implementing_interest_rate_models.pdf

}rw�|uw� �� �� �� � � � �� � � wpq rzzx vurwq~ zr}stq �r|uryvq uz ~q�wq|} uyq~ �|x}

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} uy � �� � � ����� �

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� uyrtt�� {q rtzx yxwq �x| |q�q|qyvq wprw wpq uyzwrywryqx�z �r|uryvq x�� �� � � vry � q vrtv�trwq~ � �|x} wpq vxywuy�x�z�wu}q ~�yr} uvz �x| � �� � �ry~ ��� � wx � q

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��������� ���� ��� ��� � ��� ����� � opq yq�w vrtu�|rwuxyrtwq|yrwu�q {q vxyzu~q| uz wx �zq p uzwx|uvrt zpx|w |rwq wu}q zq|uqz � opq}qwpx~ {q vxyzu~q| uz �rzq~ xy wpq �qyq|rt }qwpx~ x� }x}qywz ���� ��ryzqy ������ � ry~ prz � qqy �zq~ �� �pry � � r|xt� u � �xy�zwr� ry~�vp{r|w� ��x| ry q�rt�rwuxy x� wpuz }qwpx~ zqq wpq vprswq| xy vrtu�|rwuxyuy �r}qz ry~ �q�� q| ������� uy wpqu| { u~qt� vuwq~ {x|� xy xyq��rvwx|uywq|qzw |rwq }x~qtuy� �

� z ~uzv�zzq~ uy wpq qr|tuq| zqvwuxy xy �x|{r|~ srwp �qyq|rwuxy � wpqxyq��rvwx| ��tt�� puwq }x~qt vry � q ~uzv|qwu�q~ rz �x ttx{ z �

� �� � � �� ��� � � � � ��¬ ��� ��� � ��� �

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{pq|q �� � � ��� � � � � ��� � � ��� � uz r zq|uqz x� uy~qs qy~qyw zwry�~r|~ yx|}rt |ry~x} �r|ur� tqz x� }qry � ry~ �r|uryvq � � ry~ � �� ��~qyxwqz wpq uyzwrywryqx�z �x|{r|~ |rwq �u�qy �� ���rwuxy ���� � � u�qy

Page 15: implementing_interest_rate_models.pdf

r zpx|w�|rwq wu} q zq|uqz �­ � � � wpq vx||qzsxy~uy� wu} q zq|uqz �x|�� ­ �� � vry �q x�wruyq~ ~u|qvwt� �|x} ���rwuxy ���� ��q�yuy� ��� � � � �� � � �� ��� ����

u� wpq }x~qt uz vx||qvwt� zs qvu�q~ � wpqy �|x} ���rwuxy ���� {q vry q�sqvw��� � wx � q r �q|x�}qry �r�zzury { uwp �r|uryvq

�¬ � � ¬

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ry~

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Page 16: implementing_interest_rate_models.pdf

opq x� qvwu�q ��yvwuxy wx �q } uyu} u�q~ uz

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opq zpx|w |rwq �x| wpq � trv��� r|rzuyz� u xyq��rvwx| }x~qt uz �x�q|yq~�� wpq q��rwuxy

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opuz q��rwuxy vry �q �zq��t {pqy zqq� uy� ry ��tq| x| pu�pq|�x|~q| ~uz�v|qwu�rwuxy x� wpq � trv��� r|rzuyz�u }x~qt � rtwpx��p {q { utt yxw pr�qx vvrzuxy wx �zq wpuz ~uzv|qwu�rwuxy uy x�| vrzq �

� zz�} uy� vxyzwryw � ry~ � � uywq�|rwuy� ���rwuxy ���� tqr~z wx ��� � q�s �� ��� � � ���� ����

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����� � ��� uz wpq zxt�wuxy wx wpq |yzwquy��ptqy� qv� s|x vqzz �� � ��� �� �� �� { uwp � ��� � � �opq zpx|w |rwq uy wpq � trv��� r|rzuyz�u }x~qt �x ttx{ z r tx�yx|}rt s|x�

vqzz � uw vry �q |q�r|~q~ rz r vxywuy�x�z�wu} q �qyq|rtu�rwuxy x� wpq � trv� ��q|}ry � ry~ ox� }x~qt � opuz }x~qt rzz�|qz wprw wpq zpx|w |rwq { utt yxw� qvx} q yq�rwu�q � ry~ prz �xx~ �wwuy� s|xsq|wuqz wx }r|�qw ~rwr � } r��uy� uw r s xs�tr| }x~qt r}xy� s|rvwuwuxyq|z � �y�x|w�yrwqt�� �ytu�q wpq��tt�� puwq xyq��rvwx| }x~qt � wpq � trv��� r|rzuyz�u }x~qt uz yxw ryrt�w�uvrtt� w|rvwr� tq � ��� �� yx vtxzq~��x|} �x|}�tr �x| � ��� vry �q �x�y~ wprwq�rvwt� } rwvpqz wpq uyuwurt � uqt~ v�|�q ��ytu�q wpq xyq��rvwx| ��tt�� puwq}x~qt� � ry~ }x|qx�q| uw ~xqz yxw tqr~ wx vtxzq~��x|} ryrt�wuv �x|}�trz�x| � xy~z � �xwp vrtu�|rwuxy ry~ �uqt~ v�|�q �wwuy� r|q wpq|q�x|q w�s u�vrtt� s q|�x|}q~ { uwp y�}q|uvrt }qwpx~z �rzq~ xy r w|qq u}stq}qywrwuxyx� wpq zpx|w |rwq � �q�x|q ~uzv�zzuy� wpq vrtu�|rwuxy s|x vq~�|q � {q �|zwvxyzu~q| wpq w|qq vxyzw|�vwuxy s|x vq~�|q �x| wpq � trv��� r|rzuyz� u }x~qt �

� e� �i�� �jg�hilmhnjgopq w|qq vxyzw|�vwuxy }qwpx~ �x| wpq � trv��� r|rzuyz� u }x~qt � tu�q

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������ � � � �� � q�s �� � �� � � � � � {pq|q wpq � � �� r|q wpq yx~q �rt�qzx�wruyq~ �|x} �wr�q � �

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x� �x|{r|~ srwp �qyq|rwuxy �x| wpq � trv��� r|rzuyz�u }x~qt �

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� e� �� �n�i�h njgopq � trv��� r|rzuyz� u }x~qt uz vrtu�|rwq~ �zuy� zpx|w�|rwq puzwx|uvrt

~rwr � zu} utr| wx wpq rss|xrvp �x| wpq ��tt�� puwq }x~qt � opq ~uzv|qwu�q~�q|zuxy x� wpq zwrwq q��rwuxyz r|q �u�qy ��

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Page 26: implementing_interest_rate_models.pdf

vry �q x�wruyq~ �|x} wpq w|qq vxyzw|�vwuxy s|x vq~�|q �x| wpq � trv��� r|rzuyz�u }x~qt � � u�qy r zpx|w�|rwq wu}q zq|uqz �­ � � � wpq vx|�|qzs xy~uy� wu}q zq|uqz �x| �� ­ �� � vry � q x�wruyq~ ~u|qvwt� �|x}���rwuxy ���� � �����

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ry~

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� e �l ����� nh� ��j���mhji � jk� �opq }ruy r~�rywr�q x� wpq w{x��rvwx| ��tt�� puwq }x~qt uz wprw ry�

rt�wuv s|uvqz �x| �xy~z � ry~ wpq|q�x|q zs xw |rwqz � r|q r�rutr� tq � opqsr|wuv�tr| �q|zuxy wprw {q vxyzu~q| uz rtzx |q�q||q~ wx uy wpq tuwq|rw�|q rzwpq �� }x~qt � wpq q��u�rtqyvq � qw{qqy wpq �� ry~ x|u� uyrt ��tt�� puwq w{x��rvwx| }x~qtz uz qzwr� tuzpq~ uy �|u�x ry~ � q|v�|ux ���� ���{puvp {q z�}}r|u�q uy wpq rss qy~u� � � opq �� }x~qt qy x�z ry�}�q| x� r~�rywr�qz x�q| uwz vtrzzuvrt vx�ywq|sr|w � �qyq|rwuxy x� wpq�x|{r|~ srwpz |q��u|qz xy t� wpq �|zw�x|~q| ~q|u�rwu�qz x� wpq uywq|s xtrwq~uyuwurt zs xw |rwq v�|�q �wpq x|u� uyrt ��tt�� puwq w{x��rvwx| }x~qt |q��u|qzzqvxy~�x|~q| ~q|u�rwu�qz � uw uz {qtt��yx{y wprw pu�pq|�x|~q| ~q|u�rwu�qzuyw|x~�vq �|qrwq| uyrvv�|rv� ry~ y�}q|uvrt uyzwr� utuw� � � wpq }qry |q��q|zuxy sr|r}qwq|z vry � q zqtqvwq~ { uwp �|qrwq| q� u� utuw� �wpq x|u� uyrtw{x��rvwx| ��tt�� puwq }x~qt |q��u|qz wpq w{x }qry |q�q|zuxy vx q! vuqywzwx � q ~uzwuyvw� � ry~ �yrtt�� wpq ryrt�wuv �x|}�trz �x| � xy~ s|uvqz �ry~zs xw |rwqz� r|q }x|q w|rvwr� tq �

� ed ��g�� nm �l�h njg�opq ~�yr} uv q��rwuxyz �x| wpq w{x��rvwx| �� }x~qt r|q x� wpq

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Page 28: implementing_interest_rate_models.pdf

�� ��� � �� � ��� �� � � �� ¬ ��� ���� �� ��� � � �� �� � � ¬

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{pq|q � � r|q vxyzwrywz |q qvwuy� wpq |rwq x� }qry |q�q|zuxy � � � r|qwpq �xtrwutuw� vxyzwrywz � � �� �� ~qyxwqz wpq uyzwrywryqx�z �x|{r|~ |rwq�~q�yq~ uy wq|} z x� wpq zs xw |rwq v�|�q � �� �� rz � �� �� � � ��� � �� �� �� �� ��� � �� � ��� � �� ¬ ��� � � �� ry~ � ��� � � ��� � � � opq �� �x|�}�trwuxy prz r y�}�q| x� �qrw�|qz wprw }r�q uw rww|rvwu�q �x| ryrt�zuzry~ u}stq}qywrwuxy s�|s xzqz � �uyvq �� ��� � ���� uz r w{x�~u} qyzuxyrt |yzwquy��ptqy� qv� s|x vqzz � uwz zx t�wuxy uz yx|}rt { uwp }qry ry~ vx��r|uryvq �u�qy ��

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Page 29: implementing_interest_rate_models.pdf

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