IMES DISCUSSION PAPER SERIES LIBOR Discussion Paper No J- -J-2 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN
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1 IES DISCUSSION PAPER SERIES IBOR Dscusso Paper No. 00-J- -J- INSTITUTE FOR ONETARY AND ECONOIC STUDIES BANK OF JAPAN
2
3 IES Dscusso Paper Seres 00-J- 00 IBOR * IBOR IBOR IBOR JE classfcao: C5 E43 G3 *E-mal: kouarou.shyama@bo.or.p
4 ..... IBOR IBOR IBOR IBOR... 8 IBOR... 8 IBOR IBOR IBOR IBOR GASSERAN AND KOU[000] CEV ANDERSEN AND ANDREASEN[000]
5 . IBOR IBORodo IerBak Offered Rae BG Brace Ga arek ad usela[997] IBOR IBOR Black[976]Black ad Scholes[973] IBOR IBOR IBOR IBOR IBOR Brace Ga arek ad usela [997] HJ Heah Jarrow ad oro[99] BG Jamshda[997] BG
6 IBOR 3 4IBOR 5 IBOR IBOR 64 7IBOR 89. IBOR
7 VaR IBOR 990 IBOR IBOR IBOR IBOR IBOR IBOR IBOR { } T T T 3
8 IBOR IBOR 0. 5 T T T T T D 0 T IBOR D D D D T D - T T IBOR D D T T - D D D - IBOR D m D D m m D -3 D m 4
9 IBOR IBOR 3 IBORT D IBOR d dw -4 W T D 4 0 IBOR IBOR -4 5 K 0 IBOR 0 C 0 K γ N [999] [999] 5
10 6 IBOR T d K d K d d KN d N D K C 0 0 log 0 log ] 0 0[ 0 γ γ γ γ γ γ -5 IBOR T T K d T T K d d KN d N D K C ˆ ˆ 0 log ˆ ˆ ˆ 0 log ˆ ] ˆ ˆ 0 0[ ˆ 0 ˆ γ T ˆ IBOR -5 γ T T γ -7 T γ ˆ -8 γ IBOR 6
11 3 IBOR IBOR IBOR IBOR IBOR IBOR -4 W W - 6 W W dw dw ρ d ρ k 7 ρ k -9 T dw -9T dw d -9-4 d ρ d dw Pelsser[000] 7-9 ρ 34 7
12 -0T T d -0-5 IBOR 3 IBOR 3. IBOR IBOR IBOR IBOR IBOR T 8-9 d ρ d dw 3- d T 3- dw 3- < d 8 IBOR T D 8
13 9 3- W W W W 3-3 W 0 ~ ρ N W W 3-4 ρ N 9 < 3- W W ρ T log 0 3- log dw d d ρ ε 0 W W ~ ε 0 Kloede ad Plae[995]
14 exp{ ρ W W 3-7 } 3-7 IBOR IBOR IBOR W W 0 0 5% IBOR IBOR IBOR T 0 0 T0 4 T0 T T 3 T 3 4 T 4 T T4 IBOR IBOR W 5 0 T 0 T T T 3 T
15 D D D D D 5 T 0 T T T 3 T 4 3 IBOR IBOR IBOR IBOR T T 0 0 T 0 T 0 T T 0 T T 4 0 T T T T IBOR HJ T 0
16 3 IBOR K 5.0% 4 T 4 5.7% T5 C T5 max{ 4 T4 K 0} T 5 T 5 3 N CAPET T 0 C4 T 0 T C 5 C 3-8 N CAPET 4 T0 D5 T0 C T5 N D 5 T 0 K 5.0% IBOR C T max{ T K 0} 4 D5 T 5 C T C T C T 3 C T 4 C T 5 C T C T 3 C T C T 4 5 / D / D 5 / D / D T T 3 T T 4 5 C T D T T
17 D 5 T D 5 T 3 D 5 T 4 C T C T C T C T T T T 3 T 4 T 5 N T 5 T 0 D 5 T 0 T 0 N C T CAP T 0 C Y T 0 N CAP C Y T0 D5 T0 { C T C T3 C T4 C T5 } 3-0 N IBOR ρ IBOR d d Z Reboao [999a b] 5 T5 T 5 IBOR
18 6 d d d B T D ρ dw dw ρd B B 3- BdZ BdZ BdZ dz B 3- BB d B ρ BB ρ 3-3 B d Z ρ 7 d θ q q d B q b q b q q q cosθ q sθ sθ q d q d 3-4 d [00] 7 Z W T D 4
19 5 s cos s cos s cos b b b b b b B θ θ θ θ θ θ d s s s cos cos s s s cos cos s s s cos cos b b b b b b b b b B θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ 3-6 B B B 3-7 d } exp{ d q q q q d q q d q q q Z Z d q b q q Z q Z q 3-7 T D d q d B 3-7 dz d d B B B b q q ρ B B 3-3-
20 d B B d B dz BB d BdZ ρ d dw IBOR ρ B IBOR W ρ ρ α αexp{ β β max T T T T } α.3 β 0. β ρ B 3-0 B d ρ ρ B 8 6 d 3 6
21 ρ B 3 ρ 3 ρ B θ θ 4 B B 3 B 3 B B B3 4. IBOR IBOR Jamshda[997] IBOR Reboao[999b] IBOR 7
22 T T S IBOR S 5 D D S 4-5 IBOR D D T T S T T S S - S D D D S S D D D D D D D D P 4-4 P IBOR S P 8
23 ds dw S 4-5 T T T S W 9 P S K 0 PS 0 PS S 0 K Γ d P [ S S log T 0 N d KN d ] 0 K Γ Γ d S log Γ d log 0 S 0 K Γ Γ 4-6 IBOR T T T ˆ Γ Γ 4-7 T ˆ IBOR IBOR 4-0 IBOR 9 0 Brgo ad ercuro [00] 9
24 IBOR IBOR IBOR IBOR Reboao[999b] T T T ˆ IBOR ˆ Reboao[999b] k ˆ w 0 w 0 0 k 0 ρ k l l k l T 4-8 S 0 k l ρ k w 0 D k k D 0 k k D IBOR ρ k IBOR Brgo ad ercuro[00]reboao[999b]hull ad Whe [999]Reboao[999b] Reboao[999b] 0
25 6 5. IBOR IBOR IBOR IBOR IBOR IBOR 3 0Y 0.5Y.0Y9.5Y Y3Y5Y7Y0Y 6 6 Y Y 3 IBOR TIBOR IBOR
26 Y 6 6 IBOR 5 log log -4 6 x x x S K N S N 3 N N x x x x K / N N N N / N 4
27 IBOR IBOR 999 YK IBOR 7 IBOR 3
28 6 Y3Y5Y7Y QQ 7 7 QQQualeQuale k x k 4
29 9 QQ Y 3Y Y 7Y QQ IBOR IBOR IBOR 7 N k / x k N 5
30 3 IBOR IBOR IBOR IBOR VaR QQ IBOR 6
31 6. 4IBOR IBOR IBOR ID ID 6 : IBOR[00/0/3] 8 ID Telerae58376 BID ASK Y 3 7
32 8 CAP ˆ CAPET ˆ Ĉ -6 K C D K C D CAPET CAP ˆ 0 ˆ 0 ˆ 0 ˆ ˆ 0 ˆ 0 ˆ 0 ˆ 0 CAPET CAP K C D K C D 6- CAPET CAP ˆ ˆ ˆ 0 ˆ 0 ˆ 0 ˆ 0 ˆ 0 ˆ 0 ˆ 0 ˆ 0 CAPET CAPET CAP CAP K C D K C D K C D K C D 6-3 CAPET ˆ CAPET ˆ γ CAPET ˆ T γ 6-4
33 γ -5 IBOR IBOR - γ T T γ IBOR ρ k ρ k IBOR 3IBOR IBOR 0 0 D 0-3 S γ T T IBOR ρ : k T T k 7 / 9
34 T 7 IBOR T 3 T3 0 T T T < < 3 < T Brgo ad ercuro [00] 7 IBOR IBOR IBOR 8 IBOR v 8 T 0 T T T 3 v v 3 3 < T v 3 < T 3 v v v v v3 v < v T IBOR v 3 IBOR 3 8 v 30
35 9 v T 0ˆ CAPET IBOR -7-8 T ˆ T T ˆ CAPET CAPET ˆ CAPET YY 3Y 8 v -7-8 T ˆ T ˆ CAPET CAPET CAPET T ˆ v v v 6-6 YY v v 8 IBOR
36 IBOR T ~ b T T a T d e c a b c d IBOR v T v ~ T v ~ T v ~ 3 T3 v ~ T T < T v ~ T v ~ 3 T3 v ~ T T < v ~ T v ~ T T T < v ~ T T a b c d.48 ボラティリティ.8.6 キャプレット ボラティリティ.4 推定値 オプション期間 4-7 T ˆ { w 0 wk 0 0 k 0 ρ k k d} T S 0 k N 3
37 T 0 N T T T k d l k l 6-9 N N N l IBOR IBOR d 3 θ θ θ θ 0 0 IBOR ID Bloomberg 33
38 γ T v T ˆ CAPET 6-0 T v v T CAPET ˆ θ θ θ θ θ θ 0 θ θ 0 π / v 3 v v 0.5Y 8 v / v 34
39 3 相対誤差 0% 0%.00 0% % -0% -30% -40% -50% 原資産となるスワップ期間 IBOR B B B B IBOR
40 γ T 0 d 6- T 0 T ˆ v ~ T d CAPET v v T ˆ CAPET T ~ T 0 d N a b c d θ θ 0 θ θ 0 33 ~ a b c d θ θ 0 θ θ 0 π / v 5 v v IBOR T 5 T T a b 0.5Y 36
41 6 相対誤差 30.0% 0.0% 0.0% 0.0% -0.0% -0.0% -30.0% -40.0% -50.0% -60.0% 原資産となるスワップ期間 IBOR B B B B IBOR
42 IBOR IBOR IBOR IBOR 5 IBOR 34 Brgo ad ercuro [00]θ v ± 0. Reboao [999b] 38
43 IBOR 5 3 Glasserma ad Kou[000]IBOR Aderse ad Adrease[000]IBOR CEV 35 IBOR IBOR Glasserma ad Kou[000] Aderse ad Adrease[000] Glasserma ad Kou[000] Glasserma ad Kou[000]IBOR CEVCosa Elascy of Varace 36 Glasserma ad Kou[000] 4 37 Glasserma ad ereer[00] Glasserma ad Kou[000] 39
44 IBOR / λ 38 m s 39 C JUP e λ T λ T! λ m 0 e mt γ Cˆ 0 K γ T d s T 0 7- Ĉ -6 γ s m λ z m s s m m m m log z z s s s s s s s 7- λ > log max0 z λ 7- m [987] 39 40
45 T % γ 0.05 λˆ.0 m s IBOR m 0 0 m < 0 m > 0 Glasserma ad Kou[000] 4
46 CEV Aderse ad Adrease[000] Aderse ad Adrease[000]IBOR CEV 40 4 IBOR CEV -4α d dw α 7-3 α IBOR CEV α K a α γ d b α 0 log γ K γ d c α γ α 0 log γ K γ γ T 0 IBOR 7-3 a 0 < α < 0 4 CEV C 0 K γ D 0[ 0 χ a b c Kχ c b a] b α [ 0 N d KN d ] CEV C 0 K γ D 0 c α > CEV C 0 K γ D 0[ 0 χ c b a Kχ a b c] N χ D λ D 43 d λ / v 4
47 α < 44 Aderse ad Adrease[000]med CEV d ϕ dw ϕ x x m ε α x α ε > 0 ε α < α > 7-5 ε med CEV med CEV Aderse ad Adrease [000]7-4 med CEV 8. IBOR IBOR IBOR HJ Aderse[000] IBOR Hu Keedy ad Pelsser[000] χ x d e k 0 / k! k v x d k Dg[99] 44 43
48 SV Josh ad Reboao[00]Reboao[00]IBOR SV 9. IBOR Reboao[999ab] IBOR IBOR Glasserma ad Kou[000] Aderse ad Adrease[000] CEV IBOR 44
49 CEV SV IBOR 45
50 IBOR [00] /494/4965/49 IBOR C C T max T K0 C D C / D C T T T E C E D T C T D T A- T 46
51 45 D T A- T C D E [ C T ] A- C T 3 IBOR 9 IBOR d d T T T T T T T 45 T [99] 47
52 IBOR IBOR T D IBOR d T d T T dw dw D D 0 IBORA-3 W 0 0 d dw A-3 A-A [993] 47 48
53 D D T T T dw dw T T dw ρ d D IBOR IBOR T 5 T
54 S. N.00. J BG IES Dscusso Paper SeresNo.99-J Aderse. A Smple Approach o he Prcg of Bermuda Swapos he ul-facor IBOR arke odel Joural of Compuaoal Face Aderse. ad J. Adrease Volaly Skews ad Exesos of he IBOR arke odel Appled ahemacal Face Black F. The Prcg of Commody Coracs Joural of Facal Ecoomcs 3 pp Black F. ad. Scholes The Prcg of Opos ad Corporae ables Joural of Polcal Ecoomy Vol. 8 pp Brace A. D. Ga arek ad. usela The arke odel of Ieres Rae Dyamcs ahemacal Face Vol. 7 pp Brgo D. ad ercuro F. Ieres Rae odels Theory ad Pracce Sprger Face Sprger-Verlag 00. Dg C. G. Algorhm AS75: Compug he No-Ceral χ Dsrbuo 50
55 fuco Appled Sascs Glasserma P. ad S. G. Kou The Term Srucure of Smple Forward Raes wh Jump Rsk workg paper Columba Uversy 000. Glasserma P. ad N. ereer Numercal Soluo of Jump-Dffuso IBOR arke odels workg paper Columba Uversy 00. Heah D. R. Jarrow ad A. oro Bod Prcg ad he Term Srucure of Ieres Raes: A New ehodology for Coge Clams Valuao Ecoomerca Vol. 60 pp Hull J. ad Whe A. Forward Rae Volales Swap Rae Volales ad he Implemeao of he IBOR arke odel workg paper Joseph. Roma School of aageme Uversy of Toroo 999. Hu P. J. Keedy ad A. Pelsser arkov-fucoal eres rae models Face ad Sochascs 4 pp Jamshda F. IBOR ad Swap arke odels ad easures Face ad Sochascs Vol. pp Josh. ad R. Reboao A sochasc-volaly dsplaced-dffuso exeso of he IBOR marke model workg paper Quaave Research Cere 00. Kloede P.E. ad E. Plae Numercal Soluo of Sochasc Dffereal Equaos Sprger 995. Pelsser A. Effce ehods for Valug Ieres Rae Dervaves Sprger Face Sprger-Verlag 000. Reboao R. Calbrag he BG odel RISK arch 999a. Reboao R. Volaly ad Correlao I he Prcg of Equy FX ad Ieres-Rae Joh Wley & Sos TD 999b. Reboao R. The Sochasc Volaly bor arke odel RISK Ocober 00. 5
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COE-RES Discussion Paper Series Center of Excellence Project The Normative Evaluation and Social Choice of Contemporary Economic Systems Graduate School of Economics and Institute of Economic Research
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[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
More information(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0
(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP
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211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
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