Analytical Pricing of Defaultable Bond with Stochastic Default Intensity
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- Daniel Holland
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1 oki P o-ji Uiviy D o Ali Mhi My 5 Alyil Pii o Dll o wih Sohi Dl Iiy h wih Eoo Dl ovy O Hyo-hol Ni o i Si Ki Il S Uiviy Pyoy D. P.. o Ko D o Ali Mhi o-ji Uiviy Shhi hi Aho: O Hyohol 964- h; Ni 979- G S ooi Aho: l: ; F: E-il: li@.o Mili A: D o Ali Mhi o-ji Uiviy Shhi 9 hi. A: H w ovi lyil ii ol o oo ll o wih oh l i h wih ohi l iiy. I h wih o ho oo l ovy i PDE ho w v o ii ol o h ll o h oiio h l ovy i h wih l ovy; l iiy ollow o o 3 il o illo ol; 3 l iiy i ol wih i vl. h w iv i ol o i l w. A i h o ohi ho oo l ovy i PDE ho w v o ii ol o h ll o h oiio h l ovy i h wih l ovy; h ho ollow ik ol; 3 l iiy ollow o o 3 il o illo ol; 4 l iiy i ol wih i vl; 5 l iiy i ol wih ho. h w iv i ol o i l w. iv o i lyi oo. Kywo: Dll o; E l; U l; Sohi l iiy; Eoo l ovy; F vl ovy; Mk i ovy; i l w; i AMS Sj liiio: 355; 35K5; 94; 98; 93 JEL: G3; G33 $. Ioio h wo i oh o ii ll oo o; o i h l oh h oh o i h o oh. h l oh i iii y Mo lk o Gk qly vlo y Lo Shwz. I h l ho w hik h h l v o wh h i vl h i low hhol l i o h ov. Sh l. Aoi o h l oh i i h i vl i l h h l hhol vy h h w ol hik h i h o l ik. i liy h oo o i i lo h h h l zo oo o i i h i. hi i ol lio o h l ik o oo o o l oh o l ll o i ik i h h l oh i i ik. hi i h hooi o h l oh. I h -o oh h l i il v ov y l iiy o h l v h o y olio wih h i vl. So h l ih o o i vy low vl ho l v wol. Di - Silo Jow - ll Jow-Lo-ll o oh ol i h
2 O Hyo-hol Ni o ol. I h -o oh h l oiliy i iiiil i ivl i ll l iiy h i o oo o i o h l zo oo o i i oi o o o i ik l iiy i i i hi i. i h i o oo o i o h l zo oo o i ol o oly o i ik lo o oh o h h o ol ov i ik. hi i h hooi o h o oh. I o o ovo h hooi o vy oh iov h qliy o oli o o h i o oi l o h wo oh. l-shv-o h-el Jhl 3 lo 4 h-el Jhl 3 i PDE ho ovi i-lyil ii ol o ll o wih oh l wh h ho ollow I ol l iiy i lily o h ho. hy i h wih oo l ovy o olio o i vl ho hy h l ovy l ovy h. Ji-Lo 5 i h ol wih h-el Jhl 3 i hi ol ho ollow ik ol hy h olio o i vl ho i o ily zo ho hy ovi lyil ii ol i PDE ho. lo4 i PDE ho ovi lyil ii ol o ll o wih oh l i h wih o ho oo l ovy wh h l iiy i o. A hy lo ovi lyil ii ol i h h ho i ol wih i vl l ovy i oo wh h l iiy i o. hy i h l ovi h. I hi w y h ii ol o ll o wih oh l i h wih ohi l iiy. ilo 998 lih vy l ol o l iiy hz o i PDE ho ovi o ii ol o ll o wih l oly o l ovy. O h oh h o iv oo o olli-solik oi i liklihoo i o h wh l iiy ollow ik-lik ol ilo ol il i. hi o h l o ilo ol o l iiy i l. I hi w ovi lyil ii ol o oo ll o wih oh l i h wih ohi l iiy. I h wih o ho oo l ovy i PDE ho w v o ii ol o h ll o h oiio h l ovy i h wih l ovy; l iiy ollow o o 3 il o ilo ol; 3 l iiy i ol wih i vl. A h w iv ii ol o i l w. I h wih ohi ho h o ol oo l ovy i PDE ho w v o ii ol o h ll o h oiio h l ovy i h wih l ovy; h ho ollow ik ol; 3 l iiy ollow o o 3 il o illo ol; 4 l iiy i ol wih i vl; 5 l iiy i ol wih ho. A h w iv ii ol o i l w. v o i lyi.
3 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy h i o hi i oiz ollow: Sio 3 l wih o ho io 4 5 l wih ohi ho. I io w i h ii ol o oo zo oo o wih l oly. I i i o liy h o ohi l iiy. H w h l iiy o ollow ilo ol lih ii PDE ol o oo zo oo o wih l oly h i h l olio ho w ovi o l oiliy ol ii ol o ll o wih vl oo ovy k i oo ovy i h 3 il o ilo ol o l iiy. I io 3 w i h ii ol o h oo zo oo o wih oh o l l. H w h oo o oo o wih U Dl oly oi i vio io vil o o h h ik o l i lihi h ii PDE ol. A h i h i vil ho w ovi o l oiliy ol ii ol o ll o wih vl oo ovy i h 3 il o ilo ol o l iiy. I h o h io w iv i ol o i l w. I io 4 w i h ii ol o oo zo oo o wih l oly. I i i o liy h o ohi l iiy. H w h h ho ollow ik ol l iiy o ollow ilo ol lih ii PDE ol o oo zo oo o wih l oly. A h i h ho o i vil h l o io w ovi o l oiliy ol ii ol o ll o wih vl oo ovy i h 3 il o ilo ol o l iiy. N i h l olio ho w ovi o l oiliy ol ii ol o ll o wih k i oo ovy i oh 3 il o l h o o ilo ol o l iiy. I io 5 w i h ii ol o h oo zo oo o wih oh o l l. H i io 3 w h oo o oo o wih U Dl oly oi i vio io vil o o h h ik o l i lihi h ii PDE ol. A h i h h o i y l zo oo o h ho o i vil w ovi o l oiliy ol ii ol o ll o wih vl oo ovy i h 3 il o ilo ol o l iiy. I h o h io w iv i ol o i l w. I io 6 w iv o i lyi. $. h Pii oo o oo o wih U Dl oly. I o o liy h o ohi l iiy w oi h il h h i o oo zo oo o i o y h i vl oly y l. Mi io ollow: Aio : h ho- i i o. Aio : h l oiliy i i h l iiy ollow 3
4 O Hyo-hol Ni 4 o l. Aio 3: Dl ovy i iv h o o vl oo ovy : o o h o o k i oo ovy o i l i. Aio 4: O oo o i i iv y h io. A Fiil lio o io: o h h ol i o oly l wh w o o kow h i o h i vl lo vil o o h h ik o l v wh w kow vy wll o h i o h i vl how i io. I io 3 vl oo ovy h l ovy i h wih yi iy. h Divi o PDE ol. H w hv o y lyi wih whih o h h ik o. h oly i o o iviv o o i vil o h oly wy o o h oolio i y hi o o wih o o i iy ilo 98. L o h i o o wih iy i l ovy i y i ; i i. o oolio: Π h h i i ik- h i Π Π. i h o o l wih oiliy Π So l.3 h Π.4 ii h o l wih oiliy Π oii. li h hih o iiiil o h w hv
5 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 5. h l h o hi qio i io o o h ih h i io o o. h h oh i io i o iy. So w hv h qio o oo o wih l iiy :..6 Do y λ h i λ.7 λ i ll k i o ik i ll ik l i o h i i h ik l. i h o o ik l il.8 h i o h. h ov h oi yol ^ o iliiy oi h ovy io 3 h w hv h PDE ol o ii oo o wih l iiy oly:..9 I i il vl ol o o hooo oli qio wih vil oii. U i k v o h i. h w h o hooo qio wih ik :. U h o kow io û. h w h hooo qio wih ik ;.. U i h o kow io
6 O Hyo-hol Ni 6 û.3 h w..4 Fo...3 o ll o i i y.5 h i h i o o ll o i io o h vl o h o i h o o l h vl o h o i h o l i. So i.3 oiliy i o o l i h o l iiy h h qio.4 i PDE qio o o l oiliy wh l iiy i. I o o olv.4 w i olv io h ollowi i ilo:998: Aio 5: I.8 li o h i ;..6 h w hv h olio o 5 i h o o A..7 i h lii io o A. i.. / h /.8 A.9 h I h wh h l iiy ii o o 3 ol i.8 h i o oo o wih l iiy i iv y.. A A A wh A ivly iv y.8.9. k: h ol.. lo i h wh h l iiy ii h ol.6 o o iy o o h oiio o.8 lhoh i hi o hv lii ol.
7 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy h iil i o. i l io i h ov. h iil i o. i ollow: h i o. i h i o h o iv o o hol l o l o o o h o i h llow o l oiliy i. I o o h l i i h h i o h o i ly h i o l ovy. A how i ol. i h i i h i ohi o ov i o l h h o i i h o o zo oo o vivl oiliy; i h i i h i o y lo i o l h h o i i h wih zo oo o. No: h o k i ovy : h h ii qio.9 o. I i il vl ol o hooo oli qio wih vil oii. U i k v o h i h w h hooo qio wih ik :...3 hi qio h h y wih.4 o h io.6 i h h olio o ollowi y: H A iilly iv i h ov. A..4 $ 3. h Pii oo o oo o wih Dl i. I hi io w i h ii ol o h oo o oo o wih oh o l l. H w h oo o oo o wih U Dl oly oi i vio io vil o o h h ik o l i ivi h ii PDE. Mi Aio ollow: Aio : Sho- i o; Aio : h i vl oi o h o S h o zo oo o: S 3. ollow h oi owi oio i voliliy : o l 3. h i y o ivi i io o o i o i vl oioly. 7
8 O Hyo-hol Ni Aio 3: U l oiliy i i h l iiy ii.; Aio 4: E l o wh ; o : o. Aio 5: E l ovy i ; : o Aio 6: E h i l i o ol wih i vl. hi i vy likl io; Aio 7: O oo o i i iv y iii io ; A Fiil lio o io: Aio 4 h w oi h wo : h i i h h l oy i o o i h h l oy i h io vl o h iy l oy. Aio 5 l ovy i h wih yi iy. H io 3. i o h h ik o i Li- 4 wiho i o ill hol oo. h Divi Pii PDE Mol: H o ik o ll o o w h ho ik i h S h vil o h oo zo oo o wih l oly oi i h vio io i l ovy i o y. o oolio: o h Fo 3. w hv So Π S Π S Π. 3.3 Π 3.4 Π 3.6 i h o o l i i ivl : wih oiliy : y h wo iiol Iô ol w hv Π { { { } } }. 8
9 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 9 H w hoo h h h i 3.7 h w hv Π 3.8 ii h o l i i ivl wih oiliy ; I hi Si w hv h i h 3.7 io 3.6 h. Π 3.9 h oi h ov wo i ii h o 3.3 Π Π. oi h Π l hih o iiiil o lily o oh i h w hv. H i w oi h qio.6 o h w hv
10 O Hyo-hol Ni. 3. No : H i h ik l i o i.8 i lo h ik l i o. h i i o qio w oly ik l i o o h ik l il lhoh w wih h l hi o 3. i qio o i i. A hi h i h ik l wol h i vl ii ik l il 3. No : h h io 3. i o ii i io ho w lo 3. i wih h ohi iil qio.8 3. i ik l wol. So oii io 5 w hv iv h ii ol: Solv h ollowi ol i h io } : { Σ o : o : 3. hi i o-hooo qio wih vil oii ovi i. Glly ki h qio hv o olio ol. h ov io i il 4 5 oiio.6- o.6- o w olv i. A i h ov i w i k v o h i h h o kow io 3. h w h o hooo qio wih ik : o : o : 3.3 U i h h o kow io û 3.4 h w h hooo qio wih ik ;
11 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy o : o : 3.5 U i h h o kow io û 3.6 h o ll o i i y 3.7 h i h i o o ll o i io o h vl o h o i h o o l h vl o h o i h o l i. So i 3.7 oiliy i o o l i h o l iiy i l i i. h w h PDE qio o o l oiliy wh l iiy i l i i. o : o : 3.8 Si h l l o h l l o i o ol y io 5 w o l oiliy 3.9 Si 3.9 io 3.8 h w hv. So i w olv h ollowi wo ol h w h olio o 3.8. o : 3.
12 O Hyo-hol Ni 3. k: h w h i o l oiliy wih l iiy i o l oiliy wih l i. Si 3. i j h wih.4 w ily olv i h oiio -6-8 o h olio o 3. i: A. 3. wh A i iv y. i iv y. / h / hoh ii qio o o wih i wih ik ivi o. ily h olio i h ho i -ll iy o ow o i oio h. 8 ho o Ji LS 3 i h o o h ol 3 o OH 4 i h o.. : o N N N N 3.4 wh l l l l h w hv h o l oiliy ol h ii ol o ll zo oo oo o wih ohi l iiy l i:
13 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 3. o : N N N N A A k: h ol lo i h wh h l iiy ii h ol.6 o o iy o o h oiio o.8 lhoh i hi o hv lii ol. h iil i o 3.6 i l: h i o 3.6 i h i o h o iv o o hol l o l o o o h o i h llow o l oiliy i. I o o h l i i h h i o h o i ly h i o l ovy. A how i ol 3.7 o 3.6 i h i i h i ohi o ov i o l h h o i i h o o zo oo o vivl oiliy; i h i i h i o y lo i o l h h o i i h wih zo oo o. Ui h ho w iv i ol o i l w. oi i l w whih y i o l h l o i o h oo o hol i qi h oio y y h i o h i l w h i o h i o. L o o w y S h i ii h ollowi ol i h io } : { Σ. o : o : S S S S S S S S S S 3.7 Ui h ho h ov w S 3.8 H i h o l oiliy iv y 3.5. h iil i o 3.8 i l: i h w o iv ohi i o o l iv oio i o l h h i o h w
14 O Hyo-hol Ni o i h o o h oio l oiliy. k. h io 3 i io io 4 5 i io 3 o l i ovy io ol i ivi lyil ii ol. I h l ovy vl w o io vl o h i i h iil o h o hooo qio.9 o 3. io hooo qio. o 3.5. A i l ovy l ovy w o h o loi wol o wok i wol oh ho o l lo 4. k. I hi io h io 3. h h i vl i o o h o ok ll o io ol i hi h ik o i vl. H w h h ik o i vl i 3. h l ik i h vil o wih oly l o o qio 3.8 h o y i o l i iv i i o o o. k 3. I h wh h i q voliliy o l iiy li o o o iy o o h oiio.8 w hv i-lyil ol o o i i ii ol o hv lii ol. $ 4. h Pii oo o oo o wih U Dl oly -h wih Sohi Sho. I o o liy h o ohi l iiy w oi h il h h i o oo zo oo o i o y h i vl oly y l. Mi Aio ollow: Aio : ik ollow ik ol: θ µ ik l il 4. Aio : U l oiliy i i h l iiy ii ik l il 4. α β χ δ ε φ. 4.3 Aio 3: U l ovy i iv h ollowi wo o i F vl ovy: ; : :o h i h i h l i o l zo oo o. ii: Mk i ovy: o i h l i :o Aio 4: ovi: ov 4.4 Fho ivi h ii qio 4.3 w h 4.5 h i l i o ol wih h ho wh ivi h ii ol. hi i o o likl io w i o lyil ii ol; 4
15 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 5 Aio 5: h i o o oo ll zo oo o wih l oly i iv y iii io A Fiil lio o io: o h h ol i o oly l wh w o o kow h i o h i vl lo vil o o h h ik o l v wh w kow vy wll o h i o h i vl how i io. h Divi o PDE ol. U h oiio 4. h i ; o l zo oo o i iv ollow: µ θ 4.6 ;. A θ θ 4.7 H w hv o y lyi wih whih o h h ik o. i o l zo oo o h oly i o o iviv o o i vil o h oly wy o o h oolio i y hi o o wih o o i iy ilo 98. L o h i o o wih iy i l ovy i y i ; i i. o oolio: Π h h i i ik- h i Π Π 4.8 i h o o l wih oiliy Π
16 O Hyo-hol Ni 6. So l h i 4.9 h Π 4. ii h o l wih oiliy Π 4. I w oi i ii Π Π li h hih o iiiil o h w hv. H oi h. µ θ µ θ h l h o hi qio i io o o h ih h i io o o o h oh i io i o. h w hv. µ θ
17 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 7 L λ h i λ 4. h λ i ll k i o ik i ll ik l i o h i i h oii o i ohi iil qio o i h ik l il ilo 998 o o h io 4.3 h w hv h PDE ol o h i o o oo ll zo oo o wih l oly l iiy ovy : µ θ 4.3 Solvi: 4. h o F l ovy ; : :o; h 4.3 i h o µ θ 4.4 I o o olio ol w h h i voliliy o i o l o ho h i β ε i U i zo oo o i h i h h o kow io vil 4.5 oi 4.7 ivi oh i y h w h o hooo qio wih li o hooo ik :
18 O Hyo-hol Ni U i h h o kow io û 4.7 h w h hooo qio wih ik ; 4.8 U i h h o kow io û 4.9 h o ll o i i y } { 4. h i h i o o ll o i io o h vl o h o i h o o l h vl o h o i h o l i. So i 4. oiliy i o o l i h o l iiy i. h w h PDE qio o o l oiliy wh l iiy i 4. h h io w hv h olio o 4. i h o o A. 4. H i φ δ χ α 4.3 h A h olio o h ollowi oiy iil qio:. φ χ 4.4 A α δ 4.5 Fo w kow h A A how i io 4.4 h h ollowi olio h o iio o h oii o 4.3:
19 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy / h / H α α α δ δ : o δ : o. 4.6 h. h il ik ol α χ δ: o φ Ho-L ol χ δ: o φ Hll-hi ol χ δ: o φ. O i kow h A i iv y A α δ 4.7 h I h wh h l iiy ii h oiio o 4.6 h i o oo o wih l iiy i iv y A A A wh A ivly iv y k : h ol lo i h wh h l iiy ii h ol 4.3 o h oiio i 4.6 lhoh i hi o hv lii ol. k : h iil i o 4.8 i l io i h ov. h iil i o 4.9 i ollow: h i o 4.9 i h i o h o iv o o hol l o l o o o h o i h llow o l oiliy i. I o o h l i i h h i o h o i ly h i o l ovy. I l ovy i zo h i h h io o h ll o i l zo oo o i i h vy o l oiliy. I l ovy i ll h i h l v o o o h o i ll o i i h wih l zo oo o i. 4. h o Mk i ovy ; : :o; h 4.3 i h o θ µ H i o o olio ol w h
20 O Hyo-hol Ni h w i h olio o 4.3 i h o o A. 4.3 Glly i 4.3 i i ioil o i A liily i 3 io w i h A ivly iv ollow: i.. h wh wh wh ε δ β α ε δ β α ε δ β α 4.3 θ θ β ε 4.33 A δ α θµ 4.34 h o ll o i i 4.3. L o h o l oiliy y. I h o k i ovy l ovy i h 4.35 h o l oiliy i iv y 4.36 k 3: h ol lo i h wh h l iiy ii h ol 4.3 o o iy o o h oiio i 4.3 lhoh i hi o hv lii ol lik 4.3. k 4: h iil i o 4.35 i ollow: h o o 4.35 i h i o h o iv o o hol l o l o o o h o i h llow o l oiliy i. I o o h l i i h h i o h o i ly h i o l ovy. I l ovy i zo h i h h io o h ll o i l zo oo o i i h vy o l oiliy. I l ovy i ll h i h i 4.3 l v o o o h o i ll o i i h wih l zo oo o i.
21 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy No:ilo998 o h 4.4 h h olio o h o o wh l ovy. $ 5. h Pii oo o oo o wih oh E U Dl - h wih Sohi Sho. I hi io w i h ii ol o h oo o oo o wih oh o l l. H w h oo o oo o wih U Dl oly oi i vio io vil o o h h ik o l i ivi h ii PDE. Mi Aio ollow: Aio : ik ollow ik ol: θ µ. 5. Aio : h i vl oi o h o S h o zo oo o: S 5. ollow h oi owi oio voliliy o o: ik l 5.3 Aio 3: E l o wh ; : o wh i h i o l zo oo o. Aio 4: U l oiliy : i i : h l iiy ii ik l Aio 5: E l ovy i oo vl ovy: ; : :o Aio 6: i j ij i Fho w h 3 h i l i o ol wih i vlhi i vy likl io ivi h ii qio 5.8 o l oiliy qio 5.6 w h 3 h i l i o ol wih h ho wh ivi h ii ol. hi i o o likl io w i o lyil ii ol; Aio 7: oo ll zo oo o i i iv y iii io
22 O Hyo-hol Ni PDE Mol H o ik o ll o o w h ho ik i h S l zo oo o h vil o h oo zo oo o wih l oly oi i h vio io i l ovy i o y. o oolio: S 3 Π o h. 3 S Π Π 5.7 Fo 5. w hv 3 Π 5.8 So 3 Π 5.9 i h o o l i i ivl : wih oiliy : y h wo iiol Iô ol w hv { { Π } }. { } { } H w hoo 3 h h 3 h i
23 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 3 ; ; ; 3 5. h w hv. } { 3 3 Π 5. ii h o l i i ivl wih oiliy ; I hi Si w hv h i h 5. io 5.9 h 3 Π }. { 5. h oi h ov wo i ii h o 5.7 Π Π. oi h. Π l hih o iiiil o lily o oh i h w hv
24 O Hyo-hol Ni H i w oi h qio 4.3 o 4.6 o h w hv. 3 µ θ 5.3 No : h h io 5. i o ii i io ho w lo 5.3 i ik l wol. h w hv h PDE ol o ll o wih oh E U Dl i h wih ohi ho ohi l iiy : Solv h ollowi ol i h io { : } Σ 3 o : o o : o. θ µ 5.4 k: hi i o-hooo qio wih h oii o i vil ovi i. I hi ol { } oi o Σ w o o iv h oy vl h. I Fih ii i o 4 } { 4 i i j j ij i h i θµ / 4 3 h
25 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 5. Solvi: U i zo oo o i h i h h o kow io vil h 5.6 H i i i 4.7. Si io 5.4 oi h qio 4.6 o ivi oh i y h w h o hooo qio zo ik : o : o o : } { 3 3 X 5.7 U i h h o kow io û Σ 5.9 h w h hooo qio wih ik ; Σ o : o o : X 5. U i h h o kow io
26 O Hyo-hol Ni 6 û 5. h o ll o i i y 5. h i h i o o ll o i io o h vl o h o i h o o l h vl o h o i h o l i. So / i 5. oiliy i o o l i h o l iiy i l i i. h w h PDE qio o o l oiliy wh l iiy i l i i. Σ o : o o : 5.3 Fo h io 6 3 i 5.3 o 5.3 i h o Σ 5.4 w o l oiliy 5.5 Si 5.5 io 5.4 h w hv Σ. So i w olv h ollowi wo ol h w h olio o 5.7. Σ o : k: h w h i o l oiliy wih l iiy / i o l oiliy wih l i. U h io 5.5 h olio o 5.7 h h o o
27 Alyil Pii o Dll o wih Sohi Dl Iiy Eoo Dl ovy 7 A 5.8 h A i iv y A 5.9 i h ollowi il w i vio io;. / h / hoh ii qio o o wih i wih zo ik zo ivi i voliliy.! Σ Σ 5.3 o : * * 5.3 Ui h ho i -ll iy o ow o i oio h. 8 ho o Ji LS 3 w N N 5.33 l l 5.34 i o oiil vil h N N 5.35 l l Σ Σ Σ Σ
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