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05/07/90 09/19/91 01/31/93 06/15/94 10/28/95 03/11/97 07/24/98 12/06/99 04/20/01 09/02/02 01/15/04
90
100
110
120
130
140
150
Date
Fore
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Exc
hang
e R
ate
(JP
Y/U
SD
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May90 Sep91 Jan93 Jun94 Oct95 Mar97 Jul98 Dec99 Apr01 Sep02 Jan0490
95
100
Month
CP
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5uhsqshk1nvj<hkpxqsp hkuv1p£hkj~n:wp%~tljyhkqau!k(rt, Zt)uhouQ<5Au È
drt = 1rt−>0[κ(θ − rt)dt + σdWt] + 1rt−=0dJt, r0 = r,
dZt = 1rt−=0[dt − ZtdJt
|dJt|], Z0 = Z,
O ¢
wpvp(Jt)
uKhSrtvp·!1qar*jt vp hkuv5jSrvp u1uw<u1pxwyhkvhknvp v5u1nv uK~tp r\pj~e<j£n:wtp0hn:pjl
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κθσrBhoj~
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hk~~t5n:5<jyhovp 5vpqspjn)nvwyhonahon)5p#hkun)<jpckr
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Zt
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Ùp #houvpaDpa<juv5~p n:wyhnxn:wp)nvqap)hk5vp hk~tluvr\p j.n5j jtíp jt p u nvwpa5jn:pjuvn?lºkn:wtp1qaru µp)hkvpSjkn5qaón:p~in:*un:1nv5lqapqs<lp u1u)QOpkxpÄr\<jpjn:hk ¢ ~5u1n:1n:kjuQ<n:wtpj.n:p:hk1v5zhknvqap u ¹wtue5uejtp p~p ~ä\p#hktuvpk hku*Dpm5u1p phon:pc5j n:wpmu1p nv<j ho\<tnjtqspv5#hkDhkrr5#hn:5kju LpÄr\<jp j.n:hkK~5u1n:1n:kju~Øjon)u1p p q nvØ\pchk5pPn:ºp hk~3n:
S·
uvwhkr\p ~l5p 5~1z<pu!w5pn:wp§<hkryhkjpuvp%\<j~qPhk1²kpnpÄwt5n:uuvwu1wyhkr\p uò'pnu~p³yjtpn:wtp=1tqsrc5jn:pjuvnFlº=<³, ´ k»µs» kâF¼ ¢
λt(Zt)u1wnvwyhon
Nt −∫ t
0
λs(Zs)ds
uh%qPho1n:5j.ho5pn:wPvpuvr\pnKn: n:wp³y5nv:hon:5kj(Ft)t≥0
wtp vpNt
u+n:wp!jqS\pDo/1qsrturn:Tn:5qsp
t
ñ kn:pTnvwyhonxjºn:wtu<p jtp :ho+u1pn1n:jtcn:wtpL1qarØuv¸p)u:hoj~<qe'tnxDpaqsh#luv5qsr5Àlen:w5ul£³Ät5jah) kju1n:hkjnG1qarcuv5¸ p
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.
p 1p+Dphku1uvqap+nvwyhonInvwprtv<yho5n?l~un:v5tnvkjQtjnv<jxkF1qar u1¸ puCuG(Zs, dx)
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
µ(dx, ds)Dp*qSunTwp ²¦nvwyhonSnvwp¶1qsr
r1 p u1u(Jt)
~tp u!jkn!whAz<phkjpÄtrt<u15z<p \p wyh#z<Kç@n!un:wp #hku1p wp jλ
~pu!jkn!~pr\p j~<j
shkj~n:wphovvzohknvqap uk1qaruQ<vq h1p jpKhktrvp u1uLwtwTwyhkuLho5Kh#lujkj·?pÄtrt<u15z<p
vp hk5¸#hn:5kju)=uvppkIQ<pÄthkqar5pk'òIhku1n hkj~ÙDvhkj~tn XA<<¶ 'rB .¢ ap vp hoÀn:p Ip)5hku1uvqapn:whonn5uj~tp p ~cnvwpx#hku1pk
S· ¸U¹»º½¼Ws¾p)hk1pSj.n:p 1p u1nvp ~Ø5jº<qartn:5jPnvwpTv1p j.nxrv5 p
P (r, Z, T )kh£¸p 1c kr\<jØ\<j~ºqsh·
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
ñ -DpDhkjnn: p unhk5u1wanvwp!Ãypljqshkjn-êxhkvprvpuvpjnhn:kjºQ<hk²Dhkv~£ê!<5qsk<kvAzp yhn:kj ¢ houvuv0honvp ~i5nvw ¡<¢ ¹C£~Pn:wyhn#|pnuvp#hk5Cnvwyhon!n:wpç@ne¿PQkvqS0hT5nvwV1qaruhkrrtp~*nv
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
nF5 px<jnvj<u15l£~¤µpvpjn:hk5pkñ äp hkji~pvz<p n:wtp n:pvq u1nvvn:vp%p yhn:kjhou!5jByQ<junhkjt pk|Ù¾À<v²m XA<kè.¢ ²p
hku1uvqap n:whonn:wtp vp%hkvp \<j~tuQ<u1pz<p:hkCqPhonvvn:p u5jenvwp%qshk1²kpn#D¹wp%rv5 p%honn:qaptk
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
Ãy1<q n:wpPç@ne¿iQ<1qS0h=uv5<w.n:lØqa<1pPkp jp:hkn:wyhoj3nvwyhonTho|Az<p ¢ Dp£<pnn:wpsQ<5A5jr1 p~tljyhkqauQkKn:wp
T·@\<j~BÈ
dfT = αT (t, rt, Zt)dt +
∫ ∞
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~
r~tp jon:pryhk1nv0hoW~pv5zhon:z<p u5nvwPvp u1r\p nnv%n:wp!³vu1nKhkj~£u1p <j~
zohov0hop%vpuvr\p n:z<p 5lkIhkj~Qjn:kjuhkvpSpzohk5yhonvp ~Øhonr\<5jn(t, r, Z)
j5p u1ukn:wp15uvpSuvr\p· ³yp ~µ
pjA horrlSn:wpç@n¿ QkvqS0hnve−
∫ t
0rsdsP (rt, Zt, T − t)
wtwsuvw<t~a\p!h qshk1nvj<hkpj~tp kw<5 pknvwpxvpQpvpj p qsp hkuvtvpkwpxu1pxn:wp=hknnvwyhon
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
·On:p 1q <tnhojp~£5jsn:wppÄr1p u1uv<jhk\Az<p.l-u1un:5nvtn:5<j-knvwpun:wyhkun:i~¤µpvpjnv0hk
dfTqSu1nP\pphk!n:¦¸ pv ¹wu
Dpºhkv15z<pnv3nvwp kj 5uv5<j-n:wyhn£nvwpiQjn:5<jfT (t, r, Z)
u:hon:5u1³ypuPn:wpiQ<555jryho1n:hkj.n:p<v·?~¤µpvpjnv0ho|p yhonv<jBÈ
−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. ¶<¢
jtQ<n:jyhn:p l<|hkjºhojyhk5lnv#hoIuvktnv<jenvsn:wpShk\Az<pp yhonv<jÇ î.¢ -µ ¶<¢ 5u!~!s5nn:P<·nhojB ¹wpi~!a nFlÂ5p u)jÅn:wpe=hknan:wyhonan:wp~<qshk5jïo~p³j5nv<jïon:wp\<j~-rv5 5jQjn:5<jcuh)jt<j·?unhkjt~yhkv~c<jpkÈLnvwpxj5<jckCn:wpn?Da<n:w<k<jyhkµuvpqsó·?5jt³yjnvpvpn:hkj<5p u
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
Bhkjt~in:wpjØhs³Ätp~Ør\kjnhkk<vnvwq n:u:hn:uÀlPn:wpx kjn:5jt5n?l£<j~n:5<j*<j
Σ
ñ kn:p!nvwyhonhonn:w5uDu1n:hk<p!Dpjpp ~£n:Su1r\p Àlλs(Zs)
hoj~*k5z<phoj*pÄr5 5npÄtrtvp u1uv<j*kIn#p u1wyhk5'hkuvu1qap 5j*n:wpxjpÄn!uvpnv<jcnvwyhon
λ5uknvwpQ<vq
λ(z) = α + βγzγ−1, Û.¢
wpvpα
hkj~β
hk1pr\<uvn:5z<p+<junhkj.n:u <hkjt~γuCh <jtu1nhojn&kvp#hn:p Invwyhkj X §ktqsr%5jnvp :hovvzohk
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
gihk1qa)pn!hk®'O <k¡.¢ hoj~cw5weuohSQvp pjn!u1p jcqshkj.l£vphk55ón?l£hkrtr5#honv5<ju
XA¡
 ÃÄ;If[%Å ÆÅf[*f]PÇ@ fÈrK^*[cdÇ K]Cc_a`É;
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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dZt = 1rt−=0
[
µZdt + σZdWt − Zt−dJt
|dJt|]
, Z0 = Z,
wpvpµZ
hkjt~σZ
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λ(Z)5uhk5Kh#lur\<u15n:z<pk
ñ n:wpx~ljhkqs5 ukCn:wpx\<j~er1px\p <qap
dfT = αT (t, r, Z)dt + σT (t, r, Z)dWt +
∫ ∞
0
γT (t, r, x, Z)µ(dx, dt),
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r +1
2σ2fT
rr] + 1r=0[µZfTZ +
1
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ZfTZZ ],
σT (t, r, Z) = 1r>0σfTr + 1r=0σZfT
Z ,
γT (t, r, Z, x) = 1r=0[fT (t, x, 0) − fT (t, 0, Z)].
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ZZ
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~¤µpvp jt pqapn:w~nvs³yjt~h\p<u1p z<p ~º<jz<pv<pj pQ<!hk5Irhk:hkqapnvp 1uzohk5p uDp n:v5p ~µ
ç?jØhk5In:wpSk:hkrwtuvpr\<1nvp ~ºwp 1pkµDpSuvp~mhsn:qap%wt<v5¸ <jo ¡k l<p#hk1u θ = 0.05
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
Time (years)
Yie
ld
¨ Ù©ªXd Ëc«æ Ð Àt9xqsrFqsx|cqt+a~jq¿hĤqcnt9kx~¬o+Ft9rk~jh | |c|cqcllkFqkmqsmhl-o+~(r, Z) = (0, 0)
t9rà±k~uqsro+~kmt9xl¬t+a~jqo+Ft9rk~jh jqsx λ(z) = 0.05 + z2 o+x (µZ , σZ) = (0, 0.002)
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0.01
0.02
0.03
0.04
0.05
0.06
time(years)
Yie
ld
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0
5
10
15
0
5
100
0.02
0.04
0.06
0.08
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(0, 2)
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100 200 300 400 500
2
4
6
8
x 10−3
100 200 300 400 500
2
4
6
8
10x 10−3
100 200 300 400 500
5
10
15
x 10−3
010
20
0
10
200
0.005
0.01
010
20
05
100
0.005
0.01
010
20
05
100
0.005
0.01
100 200 300 400 5000
0.01
0.02
0.03
0.04
0.05
0.06
100 200 300 400 5000
0.02
0.04
0.06
100 200 300 400 5000
0.02
0.04
0.06
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raijq¿rul~
kxqt+'vro+nhjl|ct9rruqclunt9xhl~t~jq|o9lqλ(z) = 0.05 + z2 o+x (µZ , σZ) = (0, 0.002)
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λ(z) = z2 o+x (µZ , σZ) = (0, 0.2)o+x6~jq°o9l~kJxpq¬~ut~jq|o9luq
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0
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Maturity (years)
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