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drt = 1rt−>0[κ(θ − rt)dt + σdWt] + 1rt−=0dJt, r0 = r,

dZt = 1rt−=0[dt − ZtdJt

|dJt|], Z0 = Z,

O ¢

wpvp(Jt)

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λt(Zt)u1wnvwyhon

Nt −∫ t

0

λs(Zs)ds

uh%qPho1n:5j.ho5pn:wPvpuvr\pnKn: n:wp³y5nv:hon:5kj(Ft)t≥0

wtp vpNt

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t

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J0+ç@nvwp1qarcuv¸pxuvhkj~<qen:wpjeDpxwyhAz<p

Jt :=Nt∑

i=1

Ui =

∫ t

0

∫ ∞

0

xµ(dx, ds),

wpvp(Ui)i≥1

hkvpx:hoj~<q zohkvhk5p uvprvpuvpjn:5jan:wtp=1qareu1¸ pu µ(dx, ds)

5un:wtp qap#hku1vpko1qaruhkjt~e5n:u<qar\p juvhon:k

ν(dx, ds)5u~p³yjtp ~l

ν(dx, ds) := λs(Zs)G(Zs, dx)ds.

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ç?jn:wtpx#hku1pxwtp j

G(Zs, dx) = G(dx)n:wp1qarcrtv puvu kjuv5~p 1p ~5u*1un kqsr\<tj~e1p jpKhk

r1 p u1u ¹Cº<yhkvhkj.n:p pPnvwp*pÄu1nvp jpckn:wp£qsp hkuvtvp

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r1 p u1u(Jt)

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n:tvjtahonn:5qspT+¹w5urv5 pu<z<p je.l

P (r, Z, T ) := E[e−∫ T

0rsds|r0 = r, Z0 = Z].

¡.¢

ñ on:pxn:wyhn!\p hku1p(r, Z)

un:5qsp·?wkqa<kp jtp <uP (rt, Zt, T − t)

5unvwp rv5 p%honn:qaptkh

¸ pva kr\<je\<j~eqPhn:1jThonT

X<X

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rQuvppkyQ<pÄthkqsrtpkyé!,apk| <X rµ ¡kî<è.¢ u<z<p jcl

f(rt) = f(r0) +

∫ t

0

f ′(rs−)drs +1

2

∫ t

0

f ′′(rs)σ21rs−>0ds

+∑

0<s≤t

[f(rs) − f(rs−) − f ′(rs−)(rs − rs−)]

= f(r0) +

∫ t

0

f ′(rs−)drs +1

2

∫ t

0

f ′′(rs)σ21rs−>0ds

+

∫ t

0

∫ ∞

0

[f(rs− + x) − f(rs−) − f ′(rs−)x]1rs−=0µ(dx, ds)

Q<hkj.lf : R → R

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n:wtp ~tuv<j.n\<j~eqshon:tvjt)honn:5qspT

T·?\kj~BtQ<uvwt<1n ¢ #hkje\pxvpvnvn:pjhku

P (rt, Zt, T − t) =: fT (t, rt, Zt),

wpvp Dp%hku1uvqapxn:wyhnf

5uhau1qson:wcQjn:5<jek&5nvu ¡ zhkvhk5p u yhkj~T

u<ju1~pvp ~ihku!hryhkvhkqapn:p

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T·@\<j~BÈ

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∫ ∞

0

γT (t, rt, Zt, x)λ(Zt)G(Zt, dx)dt + dMt,

wpvp

Mt :=

∫ t

0

σT (s, rs, Zs)dWs +

∫ t

0

∫ ∞

0

γT (t, rs−, Zs−, x)(µ(dx, ds) − ν(dx, ds))

hkj~

αT (t, r, Z) = fTt + 1r>0[κ(θ − r)fT

r +1

2σ2fT

rr] + 1r=0fTZ ,

σT (t, r, Z) = 1r>0σfTr ,

γT (t, r, Z, x) = 1r=0[fT (t, x, 0) − fT (t, 0, Z)].

!pvp!n:wtpuvtuv 1rtnvuthoj~

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j5p u1ukn:wp15uvpSuvr\p· ³yp ~µ

pjA horrlSn:wpç@n¿ QkvqS0hnve−

∫ t

0rsdsP (rt, Zt, T − t)

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d(e−∫ t

0rsds) = −rte

−∫ t

0rsdsdt,

X

hkj~cp <pn.l*~1p n kqsrnhonv<j£nvwyhon

d(e−∫ t

0rsdsP (rt, Zt, T − t)) = d(e−

∫ t

0rsds)fT + e−

∫ t

0rsdsdfT

= −rte−

∫ t

0rsdsfT dt + e−

∫ t

0rsdsdfT

= e−∫ t

0rsds[−rtf

T dt + dfT ].

Ãyke−

∫ t

0rsdsP (rt, Zt, T − t)

n:S\phqshk1nvjt.hkpk.n:wpdt

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dfTqSu1nP\pphk!n:¦¸ pv ¹wu

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−rfT + fTt + 1r>0[κ(θ − r)fT

r +1

2σ2fT

rr] + 1r=0fTZ

+ 1r=0λ(Z)

∫ ∞

0

[fT (t, x, 0) − fT (t, 0, Z)]G(Z, dx) = 0, î¢

5nvwcnvwp hku1uv hon:p~e\<j~yho1l* kj~n:kjehnT

fT (T, r, Z) = 1. ¶<¢

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Θ1 = (t, r, Z) : 0 ≤ t ≤ T, r > 0, Z = 0hkj~

Θ2 = (t, r, Z) : 0 ≤ t ≤ T, r = 0, Z ≥ 0.¹wpm\<j~ r1pmu1w<5~ \pm<j.n:j<uehkj~ Dpmuvwyhk5nv1l n:³yj~ nvwpØQjn:5<j ww 5u kjn:5jt<u5j 5~5jSn:wtp1<j.n\kj~yhkl

Σ = (t, r, Z) : 0 ≤ t ≤ T, r = 0, Z = 0.pK hkjSuv<z<pn:wtpKpyhon:5<jS5j~pr\p j~pjnv5l%<jSp hkwTkn:wpnFvpnhoj<pu5nvw<tn&5qsr\<u1jth\<jt~yhk1lx<j~n:kj <j

ΣtnqPhn:w nvwp q wpvpk*puvwyho.p qarAlxh³yj5nvp~¤µpvpj pqspn:w~

<jΘ1

\n:wp*=pÄthkn ¢ qapnvw~kLwyhk:hon:pv5u1n:5 u!<jΘ2

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Σ

ñ kn:p!nvwyhonhonn:w5uDu1n:hk<p!Dpjpp ~£n:Su1r\p Àlλs(Zs)

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λ5uknvwpQ<vq

λ(z) = α + βγzγ−1, Û.¢

wpvpα

hkj~β

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n:5qspuKhkvppÄr\<jp j.n:hkls~5u1nvvttn:p~*5Choj~ckj5ls5λ

uD <junhkj.n%=®Rpoγ = 1

<β = 0

¢ ¹wp#houvp

α = 0 kvvpuvr\<j~tu£n:n:wtpÁp5~5u1nvvttn:5<jÅwtw DhkuPnvwpi<jpi<juv5~pvp ~ .l

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ç?j*n:w5uKuvpn:5<j*pxu1nv~tlPn:wp hkuvpwp jλ

5uKkIn:wpQ<1q <z<p j*jm Û.¢ hkj~*nvwp 1px5uK<jlP<jtpr\<u1uvtpG1qarsuv¸p

J0L!ADpz<p n:wtp!u:hkqap!qapnvw~s#hkjP\p!uvp~PQ<hkj.laj.n:p k:hk5pQjn:5<j

λk

Zhkj~hkj.l*~5uvvpnvpk³yjn:puvr\p n:1q k&r\kuvuv5p²1qarcuv5¸ pu

Ãy1<q6n:wtp!Qkvq kBp yhonv<j î¢ - ¶<¢ 5nu1p pqsuDjhon:vhkyn:Shouvuvtqspn:whonfT

ukµn:wpQ<vq

fT (t, r, Z) = 1r>0g(t, r) + 1r=0h(t, Z),

wpvpghkj~

hhkvpuv<5tn:5<juKknvwpQ<5A5jSp yhonv<jtu% <¢ hkjt~¦ è.¢ vpuvr\pn:z<p l\È

−rg + gt + [κ(θ − r)gr + 12σ2grr] = 0,

g(T, r) = 1,

k¢

hkj~

ht + hZ − λ(Z)h + λ(Z)g(t, J0) = 0,

h(T, Z) = 1,

è<¢

5nvwcnvwp hk~~t5n:5<jyhk|qshon:w5jT <jt~5nv<j

limr→0

g(t, r) = h(t, 0). .¢

õLyhonv<jÅ è<¢ uu1qarp n:*uvk5z<pkBrvAz~p~ºDp)hk1p#hk~li²jA6hkjØpÄrvpuvuv5<jØQ<g(., J0)

p%#hojiu1p n:wp qspn:w~owhk:hkn:pvun:5 u KÃ&1u1nDp%5jnvv~pxn:wp ~qaql£zohk10hop

shkjt~

Dpvn:pdh

ds=

dt

dsht +

dZ

dshZ .

ñ pÄn#IDpa5~p j.n:Àlinvwp) p<apjn:u kn:wpahk\Az<paphon:5<jØn:wØnvw<u1pakp yhonv<jï è.¢ p<pn

dt

ds= 1,

dZ

ds= 1,

dh

ds= λ(Z)h − λ(Z)g(t, J0).

p%hou1)vpv5nvpnvwpx\<j~yho1l* kj~5nv<jchou

t(0, u) = T,

Z(0, u) = u, u ∈ [0,∞),

h(0, u) = 1.

X#î

Ãy1<q n:wpuvp%p yhonv<juWDp% hkji5qsqap~honvp lP<pnhkjpÄr1p u1uv<jQ<thkj~

ZyQjn:kjk

(s, u)yhkuQ<5Au È

t(s, u) = T + s, XA.¢

Z(s, u) = u + s. X<X¢

¹wtp p yhn:kjcQ<h(s, u)

\p <qap u

dh

ds= λ(u + s)h − λ(u + s)g(T + s, J0),

w5wewyhounvwpQ<5A5jSuv<5tn:5<jBÈ

h(s, u) = e∫ s

0λ(u+x)dx

(

−∫ s

0

λ(u + v)g(T + v, J0)

e∫ v

0λ(u+y)dy

dv + K

)

,

wpvpK = K(T )

5u%he <junhkj.n%n:wyhn%qPh#lØkj5lº~p r\pj~3<jnvwpsryho:hkqapnvp T

hkj~#hkj¦\ppzhkhon:p~cQv<q nvwpx\<j~hk1l*<j~n:kjB+5j p

h(0, u) = 1p <j 5~pn:whon

K = 1

ñ Dp hkuvu1qspnvwyhonλ(Z)

ukCn:wpQkvq <z<p je5j Û.¢ +õLhon:5<j¦ î¢ \p <qspu

h(s, u) = eαs+β[(s+u)γ−uγ ]

(

−∫ s

0

[α + βγ(u + v)γ−1]g(T + v, J0)

eαv+β[(v+u)γ−uγ ]dv + 1

)

.

ñ kn:pnvwyhonDpT hkjØkpnxhkjpÄrvpuvuv5<jØkh

Qjn:kjºkLn:wpS<1<5jyhk'zohov0hopu(t, Z)

Ùlu1je XA.¢ hoj~ X<XA¢ n:wtpxwyhkjt<pxoCzohk10hktp

w = v + Thkj~ehoÀnvp Kuvqar5³y hon:5kjBpx<tn:hkj

h(t, Z) =eβZγ

eα(T−t)

(

1

eβ(Z+T−t)γ +

∫ T

t

eα(T−w)

eβ(Z+w−t)γ [α + βγ(Z + w − t)γ−1]g(w, J0)dw

)

. X ¢

ñ kn:pn:wyhon&phon:5<jc X ¢ 5qsrtp uIn:whon5j%n:wtpK hkuvpβ = 0

kOpkCQkCn:wpDqa~p n:w kju1nhojn

λn:wp\<j~£r1p!nvp j~uDnvS\p kju1n:hkjnx=pyhk\n: X¢ hku

λn:p jt~uDn: QOpk&wpj£n:wphAz<p:ho<p

n:5qspo&unhAlchn n:pj~unv ∞ ¢ p Khkj.nnvPuvpSha³j5nvp%~¤µpvpj pqapnvw~nvPuv<z<p£ <¢ p~p³yjpShsqspuvw

(gji )0≤i,j≤Nuvtwen:whonQ<hk

i, j

gji − gj

i−1 = δt

hkj~

gji − gj−1

i = δr,

wpvpNδt = T

hkj~δr

uKwkuvp jeu1wcn:wyhnJ0/(δr) = P

uhojc5jnvp <p n:wM

ÁÊË"¾O¼¬yâ=àáã0hov<pµn:whkj

PCp+ hkjxnvwp j 1p15n:pLphon:5<js è.¢ uv5juvkqsp³j5nvpL~5¤|p 1p jphkrrtvAÄ5qPhonv5<j

jtu1n:p hk~ktn:wp~pvzohonv5z<p u<hkjt~nv1ln:!uv<z<p5nCyhk².Khkv~S5j n:qapou1nhkn:5jQ1<qän:wp\<tj~yhkl kj~5nvkj

gjN = 1.

¹wtp%hk5<<vnvwq #hkje\pvnvn:pjhkuKQ<5Au È

X¶

X ¹hk²kp hkjhk15nv:hklP\<j~yho1l* kj~5nvkjehonr = 0 å

1uvpah£³yjn:pT~t5¤|p 1p jp)qspn:w~ºnv*³yj~g

<jºn:wpTj~p uxkkqap uvwØuv5j*nvwp)5Dp\kj~yhkl% <jt~5nv<j%<z<p jS5jTnvp r X hon

r = 0<hkuDphkun:wpDu1yhk\<jt~yhk1l%<j~n:5<j

hnt = T å

¡ 1uvpxnvwpzohk5p uQkj~hon!nvp r Q<g(., J0)

n:T³yj~h

tuvjt)nvwpQ<vqSh* X ¢ åî 1L5²

h(., 0)hou<jpäADp\kj~yhkl* <jt~5nv<jchon

r = 0 å¶ !É)yhk²Pn:sn:pr

penvvp~Ån:u1phkjïpÄr5 n)³yjn:pe~¤µpvpj pqapnvw~BDtnPnPhoqa<u1nahkKhAluap#ho~u)n:pÄr5<uv5<jB0!uv5j£hkjqar5 n<h*æDvhkj²- ñ w<5uv<j³yjn:p%~5¤|p 1p jpSu1wtp qapk|DpSkuvp1z<p ~ kjz<pv<pj p%kCn:wp hk5<<1n:wq n:wcvp#houv<jyho5l£uvqshk5

δthoj~

δr

Ì ÍÏÎ*]C8 f[%` Ðf> ` Ñ b*^CÒ;

ç?j n:w5uIuvp n:5<j Dp+n:v5p ~xn:uvk5z<p+Qk'n:wp\<j~%r1p<jxn:wp+wkp+~<qshkjxtuvjtnvwphkk<vnvwqktn:wpr1pz5<uuvpn:5<jBCpDw<u1pn:wtpDqa~pryhk:hoqspn:pvu<u1pnv!nvwp<jpuQ<jt~lÉ<vAz<<hkj~iò'jtpn:u1².l¦O kkî¢ wp j³nvnvj)nvwp 5qa~p 'k5jnvp vpu1n!:hn:puhku<rtnvkjB ñ hkqap l<Dp%u1pn

θ = 0.03σ = 0.02

hkj~κ = 0.2

¦p kqsrn:p ~Pnvwp\<j~£rv5 puKAz<p Kh%w<v5¸ <jPo ¡< l<p#hovu u1jmhn:qap·?u1nvp r

δto

1qa<jnvw ¡kÛ< nvqap·?un:pru ¢ Ópcw<u1phºjp1tqsrÂu1¸p

J0

k

0.005ww£uvpp qauKvp hk5u1n:5wp j£Dp<²)honn:wtp1p p j.n pjn:vhk|yhkj²WÜuKj.n:p 1p unK:honvpr\<5l<

!ADpz<p Pnvwpiu1¸pion:wp1qar-r1z<p~-n:¦wyh#z<ph<juv5~p vhk5pqaryhkna<jïn:wp<jto·@nvp vqhkulqartnvkn:5!zohopk'n:wplp~Bn:w£uvqshk5p¾1qar£uv5¸ pu1jkj%ADp D5<jo·On:p 1q lp~uhkuDp!qs5<w.npÄr\pn#áÎ^Ô[¦p!u1p ~*h :honvp·@u1n:pr

δro

0.001=wpjp

J0 = 5δr¢ ç@nuvp pqauD<rtnvqshktn:

u1p h1qsr£uv5¸ pn:wyhnuKhSqSn:5r5p!oIn:wp:honvp·@u1n:prBu1Tp~TjknwhAz<pnvTuvp5jnvp vr\k0honv5<jn:*<qartn:pn:wtpT\<j~Ør1p

g(., J0)honn:wtp01qarºuv5¸ pk%Ã&jyhoól<| <jt p 1jjt*n:wtpS³tÄp ~mr\<5jn

hk5<<vón:wqcµDp)uvp~¦5j5nvhk\<j~mr1p uh(., 0)

uvcn:wyhnxnvwpTl5p 5~Ø z<p)Khku hcun::ho<w.nx5jpQvkq

0%n:

1%jn:ho5ól<hoj~cDpx:hkj ¡k n:p:honvkjuKknvwp hk5<<vnvwq Onvp ru A· ¶<¢

Ã<tvp ¶ uvwAuIn:wp+ kjz<p 1<p jt pknvwphkk<vnvwq wp j Dp+ <ju1~pvp~h kju1n:hkjnλ = 0.5

®h#z<p vhk<pu1n:hAlPhon0k

2l<p hkvu ¢ wp#hkj£uvppn:wpuv puvuv5z<plp~P1z<puKn:wyhnDDp<tnhojsn:w

hkj jt5n:5hku1w<ncvhon:po0 ¹wtpØu1r\p p~ o <j.z<p 1<p jpmu1p pqsu*vp0honv5z<p5lÂ=hkun#pÄtp rneQk

kj<pqPhn:15n:p uç?jen:wyhn! hkuvpo\je\<j~r1 pn:pvqeyphk1px=hkhADhAl*Q1<q n:wp*¼Õ. n"âD³yjyhk\<jt~yhk1l3 <jt~5nv<jÇo

g(T, .) = 1hkjt~Å5nSn:hk²kp uTn:5qspsQ<Snvwpc5jtQ<vqshonv5kj¦n:Ø kqsp*yho²

Ï×Ö QòFç¬éíöðëçúèHê¨òFçIMÁüFöK"6éLä8çf÷êíòFç+°ðféÁê¨çèCaçç#"ÈçñêêíòFçéLòFõfèLêèÑðcêíç'ê¨õfõ;ðfñO(êíõ8çè¨õðêíçúèCfõ3äæãõ3üvê2õäëê1VéðèíçéLüëê÷êíòçDG+üðã+êíäëêþ J0

öðþð«¤çúñuêê¨òFç"Fè¨äJñuç-õ=äæãfêíçúèíçéÁêè¨ðêíç¬îvçúèíäæåcðêíäæå3çúé1&9ççûHäæãð82 ÿ3ù >µmõ3èðîvçê¨ðäæçúî"îväJé¨ñuüé¨éLäæõ3ã"õAê¨òFç'äëöK"ðfñêõçð3ñÑòöõvîvç1."ð3è¨ð3öçuêíçúèQõ3ã;Èõ3ãî!"Fè¨äæñçúé1

XAÛ

n:in:wtp£jtpt = 0

º¹wuqsp hkjunvwyhonn:wtp*5<jkp ·On:pvq lp~u%nho²kp£qs<1psn:5qapan: <j.z<p 1<pn:whkju1w<1nvp ·On:pvq l5p 5~u pShk5DhAlukuvp1z<p ~º <j.z<p 1<p jt pTu1n:hk1nvjaQvkqÆ~¤µpvp j.njn:ho kj~5nvkju tn)hV1~<tuwkp*k5jn:ho kj~5nv<jµjknSn:i=hk%Qv<q n:wp£pÄthkn)l5p ~ t1z<pkW5up u1uvp j.n:hkBn:akpnkuvp Kn:Tn:wpxuvktnv<j*5j*QpDpun:p rtu

¹wtp£zhkv5<uu1wyhkr\puTknvwpPlp~¦ t1z<p uSQ<%~¤µpvp j.nSjt5n:5hk+u1w<nTvhon:puThk1pPrvpuvp j.n:p~jmÃ&5<1p Û aÃ&<tvp 5unvwpau1<tnvkjºQ<n:wtp)lp~Ø z<pskjmnvwpawhk50·@rhkjpSkDr\kuv5nv5z<p

r

ñ on:pnvwyhon t5jPn:wppÄtr\kjp j.n:hkµ#houvpkjtTqshonvnvp w Dpw<uvpn:wpzhktpkλn:wtp<tn:hkjp~

lp~i z<p)hon¸ pv*5uhkKhAlu kj#h#z<pkµ5nvwj£rvk<jkp ~ir\p 1~k+¸p 1£lp~B¹w5u~p ujkn)ho<vpp£5nvw3n:wp£<uvp1zohn:kjÂknvwp* tvvpjns§<hkryhojp u1p*lp~Ç1z<powtwÂu

S·?uvwhkr\p ~

5nvw¦ho&l5p 5~uQ< hk\<tnxn:wpajpÄnT £l<p#hk1u%p yho+n:e¸ pvP¹wu 5jyhk~pyhklØkD< qa~pu~tp)nvcnvwp)uvwyhkr\p)knvwp)rvkyhk5n?leQjnv<jØkn:wtpapÄr\<jp j.n:hkL~5u1n:1tnvkjBµw5wm5uu1nvv5n:l¦r|kuvn:5z<p*hon

t = 0+uvºn:wyhonTn:wtpcrv<hk5n?lnvwyhonTDpc<pnahY1qarÂqaqap ~thon:plu

hkKhAlu!u1n:1n:5lcr\kuv5nv5z<po!¹C£1p <j 5p%k!qs~pI5nvwn:wp<tuvp z<p ~l5p 5~i t1z<p uµDpjp p~n:) kjuv5~p knvwp ~un:v5tnv<jtuDQ<Kn:wpx5jn:p:hovvzohk\n:qap uKko1qsrtu

50 100 150 200 250 300 3500

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Month

Yie

ld

Yield curve after 30 iterations

Initial assumption for the yield curve

¨ Ù©ªXd Ë½Ø Ð?| p|s|cqcllkFqkmqsm|shrFqcll~o+r~kxhrut9 o­|shrruqsx~lujt9r~ro+~uqt+®qsruto+xZ = 0

o+xt9±h~o+kxqc~jhrut9hvj"~jq¿hÄhqcnt9kx~Qo+Ft9rk~jh jqsx"~jqkx~uqsro+rrk9o+¤~kqslQt+9´ÑhnlQo+ruq2qÄÈnt9xqsx~k°o+¤kml~rk±hh~uqc

λ = 0.5~jqlut9h~kmt9x kml'~jq|shrqt9x ~jq~ut9nA

¹wtp£vpuvn:u5jn:wpP hkuvpPkh5jp hk wyho¸#hk1~3:honvpºα = 0

hkj~γ = 2

¢ hovpPrtvp u1p j.n:p ~Â5jÃ&5<vpu è hkj~ %pswt<uvp£hu1qPhk5zohk5pkjyhkqap 5l

0.025Q< pspjn

β&uvn:whonShoÀnvp

<jpl<p#houvr\p j.n 5jº¸p 1λ

5uu1nv .¶ Àn:pjºn:5qspuu1qPho5p nvwyhkjn:wpT<junhkj.nλ

k+n:wtpS³yvun#houvp ¢ ç?j3nvwu%#houvpknvwpPrv<hk5n?lØ~tp ju15n?lQtjnv<jQ< n:wtp£5jn:p:hovvzohkLnvqap u k1qaruu1n:hk1nvu+Qvkq6¸ pvhkjt~svpqPhojuLuvqshk5tQ<Luv<qapn:5qspo.uvxDp~%kuvp1z<phxrvk<jkp ~)¸ p 1o·@l5p ~

X

50 100 150 200 250 300 350

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Month

Yie

ldYield curve when the initial short rate is 0.04

Yield curve when the initial short rate is 0

¨ Ù©ªXd ËÉÙ Ð=Ú kmqsm)|shrqclt9±h~o+kxqcof~uqsrÝA±k~uqsro+~kmt9xlt+~jqwo+Ft9rk~jh t9r¤kJ·[qsruqsx~kxhk~k°o+Q9o+qclt+

r jqsx ~jqkx~uqsro+rrkvo+E~kqsl-t+´Ñhnl-o+ruqqÄÈnt9xqsx~k°o+ ¤kml~rk±hh~uqc λ = 0.5

0

0.010

0.020

0.030

0

150

300

4500

0.005

0.01

0.015

0.02

0.025

0.03

Initial short rateMonth

Yie

ld

¨ Ù©ªXd ËlÛ Ð'| hrÑÁo9|cq­t9rqc¾±y~jqÈkmqsmÃ|shrqclwt9rr±qs~ qcqsx¦±o+x¹ÝFÜ jqsxÃ~jqw´Ñhnl6o+ruq

qĤnt9xqsx~k°o+¤kml~rk±hh~uqc λ = 0.5

XAè

r\p 1~k+hk\<nhal<p hkTQuvp pSÃ&5<1p è.¢ hkj~n:wp% t1z<puS·?uvwhkr\p ~ÂQ <j.z<pÄ||n:wpj kj#h#z<p ¢

5nvwhkjcjtíypÄ<jcr\<5jnhov<jt~ X#< qskjn:wtu +¹wtp <j·@n:pvq hou1lqsrn:kn:5zohopknvwpxl5p ~u%u1n:5+5nvpPw<w X kqsryhovp ~¦n:wÇhkjÇhku1lqartn:on:azohopPo! wpj

λ = 0.5¢ º¹wtp

kjz<pv<pj pPuvpp qau =hku1nvp xn:wyhojnvwyhon jØn:wpapÄr\<jtp j.n:0ho hkuvpo&tn%qshAl\pa5nxux~panven:wtp~¤µpvp j.nu1wyhkr\p on:wtpl5p 5~c z<pk

50 100 150 200 250 300 3500

0.002

0.004

0.006

0.008

0.01

Month

Yie

ld

Yield curve after 30 iterations

Initial assumption for the yield curve

¨ Ù©ªXd ËÁÝ Ð?| p|s|cqcllkFqkmqsm|shrFqcll~o+r~kxhrut9 o­|shrruqsx~lujt9r~ro+~uqt+®qsruto+xZ = 0

o+xt9±h~o+kxqc ~jhrut9hvj ~jq¿hÄhqc)nt9kx~o+Ft9rk~jh jqsx λ

kmlkxqo+rλ(z) = 0.05z

~jqwlt9h~kmt9xkml~jq|shrqo+~~jq±t9~~ut96

wtp j£Dp<²)honDnvwpuvktnv<js<jPn:wpwyhk0·?rhkjtp!kIr\kuv5nv5z<pZ

=®Rpowp j*nvwp?&µçFÚ1Âwyhou\p pjØ5jrhkp%Q<u1<qap%n:5qsp ¢ WDpSuvppSnvwyhonnvwpl5p 5~1z<pn:p jt~unv£pz<k5z<pS5jnvsn:wpuvwyhkr\p<tn:hkjtp ~Øn:whc kju1n:hkjn

λhku

Zjt vp hkuvpu sç@n%vhku1p un:eqshk²kpT³y1u1n nvwpa¸p v·@lp~Ør\p v5~

~5u:hkrr\p hk nvwp jenvwp 5jtíypÄt5<jer\kjnkpn:u 5<uvpnva¸ pvPhoj~pz<p j.n:yho5l£~uvhkrr\p hkvun:iQuvp pn:wtp ~hkuvwp~ z<p u!5jeÃ&5<vp .¢

uv5jhkjmhÞajpλ

=jyhoqsp5lkλ(z) = 0.5 + 0.05z

¢ Dp)<pn%hojS·?u1wyhkr\p ~Øl5p 5~Ø z<pkn

j)jn:ho|r\p v5~ck¸ pvo·Ol5p 5~BL¹wtu5u~pn:Tn:wtpx <jtu1nhojnryhknkλ

wtweqPho²kp uu1nvv5n:lr\<u15n:z<pSnvwp)r1<yhkt5nFlenvwyhon

rt

5p#h#z<p ux¸p vcqaqap ~hn:plihoÀn:pvp#how5jcn#ç@DpTDhkjnn:\pehk5p£nvØ<uvp1z<pclp~Ç z<p u)5nvwïhoD5j5nvhk+n:p 1q lp~tuTp yhkKnvm¸p 1DpcqSun)jkn kjuv5~p wyhk¸ hkv~vhon:pn:we<ju1n:hkj.nn:p 1qsuÃy<

γ ≥ 2hkjt~

α = 0Dp <tn:hk5jp ~cl5p ~c t1z<p

uvwhkr\p u%uvqa5hkn:envwpajtp#hk hkuvpo&<j5ln:w¦hcqs<1p)rv<jt<jp ~S·?uvwhkr\pk ñ on:p)n:wyhn Dp

qshAletuvpThkj.l3=r\<u15n:z<p ¢ wyhk¸#hov~:honvp%Dp%Khkj.n#µr5P5n!jQ<1qS0h î¢ hkjt~ºktnhk5jn:wtpl5p ~ t1z<pxn:w1<<w*n:wpxuvhkqspxhk5<<vón:wqc

ß àâás]P[';'f_[ ]I_ 8 ã)]I_adwÉ87;B]Ifd6j8]P[cd²ä

5j pPpn:5<j ¡ IDpswyhAz<pPhouvuvtqsp~n:wyhonxn:wpa0honvp jlZ

jvp#houvp ~¦5jp hkvl5n:wØn:qapkChkjt~n:whonn:wpn:5qsphk5vp#ho~tlPuvr\p j.n!5jc¸p 1)Khkunvwpxkj5lP~pn:pvqajyhkj.nkCn:wtpx~5u1n:1n:kj£ko1qar

XA

50 100 150 200 250 300 3500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Month

Yie

ld

Z=0

Z=4

Z=2

Z=10

¨ Ù©ªXd ËÉå Ð=Ú kmqsm)|shrqclt9±h~o+kxqcof~uqsrÝA±k~uqsro+~kmt9xlt+~jqwo+Ft9rk~jh t9r¤kJ·[qsruqsx~kxhk~k°o+Q9o+qclt+

rlut9km |shrqclúo+x

Zo9ljqc |shrqclú jqsx λ(z) = 0.05z

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Zt

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dZt = 1rt−=0

[

µZdt + σZdWt − Zt−dJt

|dJt|]

, Z0 = Z,

wpvpµZ

hkjt~σZ

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dfT = αT (t, r, Z)dt + σT (t, r, Z)dWt +

∫ ∞

0

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wpvp

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r +1

2σ2fT

rr] + 1r=0[µZfTZ +

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ZfTZZ ],

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Z ,

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σ = 0.02

κ = 0.2J0 = 0.005

δr = 0.0005

δt = 30/500

hkj~δZ = δt

3 6 9 12 15 18 21 24 27 300

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

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Yie

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0.04

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0

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0

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100

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0.04

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2

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8

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100 200 300 400 500

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4

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0

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100

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20

05

100

0.005

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100 200 300 400 5000

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100 200 300 400 5000

0.02

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