LIBORマーケット・モデルのインプリメンテーションについて―本邦の金利派生商品データを用いた具体例を基に―

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1 LIBOR LIBOR LIBOR E-mal:

2 LIBORLIBORLondon InerBank Offered Rae BGMBrace, Gaarek and Musela LIBOR LIBOR BlackBlack and Scholes LIBOR LIBORLIBOR LIBOR LIBOR LIBOR LIBOR Brace, Gaarek and MuselaHJMHeah, Jarrow and Moron BGMJamshdan BGM

3 LIBORLIBOR LIBOR LIBOR LIBOR LIBOR LIBOR

4 LIBOR LIBOR LIBOR LIBOR {T =,, M} δ = T + T, =,, M LIBORLIBOR L δ =L T δ L T T + δ L T T D, 0 T LIBORD =+δ L D + D D + L =, T, δ D + T L T D + L δ L T T+ D D + D + = D, +δ L

5 LIBOR D + = D +δ L = D +δ L +δ L = = D m Π, = m +δ L m D m LIBOR L LIBORT + D + LIBOR L dl = σ dw +, L W + T + D + Mσ M σ σ LIBOR

6 LIBOR L KLIBORL 0 C L 0, K, γ N. LIBOR C L 0, K, γ = δ D 0[L 0Nd KNd ] d L0 log + γ = K γ, d L 0 log = K γ γ γ = T 0 σ d LIBOR C L 0, K, σ = δ D 0[L 0N d KN d ] d L0 log + K = σ T σ T, d L0 log = K T σ σ T. γ σ T LIBOR γ γ = δ σ T + +δ σ T,

7 γ = σ T, γ LIBOR LIBOR LIBOR LIBOR LIBOR LIBOR W + W W + W δ L dw = dw + ρσ d, +δ L ρmρ, k k T dw T + dw + d Pelsser ρ

8 dl L δ L = σ d dw ρ σ + σ, L, + δ + T L T + d L L LIBOR LIBOR LIBOR LIBOR LIBORL,, L M T M+ L dl L M M+ = dw ρ d+, =,,, σ σ = + + L δ = M δ L σ M T, LIBORL M T M+ D M+

9 dl M = σ M dw M+, L M < Md =M σ M L M + L M = σ M W M+ + W M+, L M L M + =L M +L M σ M W M+ + W M+. W M+ M W M+ + W M+ N M 0,ρ, ρn M L + < M M δ L L + = L σ L ρ + δ L = +, σ + σ L W M + + W M+, δ σ L L + + T L log L M δ L, ρ σ M+ dlog L = σ σ d + dw = + + δ L σ, εw M+ + W M+ ε Kloeden and Plaen

10 M δ L L + = L exp{ σ ρ, + δ L + σ W M+ = + + W M+ }. σ σ M= 4 LIBOR =0.5δ =0.5σ =0.5LIBOR + = W M W M+ 0L 0=5%L 0+0.5, =,, 4 L 0.5L 0.5L 3 0.5L 4 0.5L.0 L 3.0 L 4.0 LIBOR LIBORLIBOR T 0 L 0 T 0,, L 4 T 0 L T L T L 3 T 3 L 4 T 4 T,, T 4 LIBOR T = 0.0 T = 0.5 T =.0 T =.5 T =.0 L L L L L W

11 T = 0.0 T = 0.5 T =.0 T =.5 T =.0 D D D D D LIBOR LIBOR LIBOR LIBOR M M M= 60 T = 0.0 T = 0.5 T =.0 L L L L M L M L M L T L T L T L M T L T L T L M T L T L M T LIBORHJM L M T 0

12 LIBOR K=5.0% L 4 T 4 =5.7% T 5 C T 5 =max{l 4 T 4 K, 0} T 5 T 5 N T 0 C CAPLET 4 T 0 C T 5 N C 4 CAPLET T 0 =δd 5 T 0 C T 5, N = δ D 5 T 0 K=5.0% LIBOR C T =max{l T K, 0}, =,, 4 D 5 T 5 C T C T =C T /D 5 T C T 3 =C T 3 /D 5 T 3 C T 4 =C T 4 /D 5 T 4 C T 5 =C T 5 /D 5 T 5 =C T 5 D T = T 5 D T D T D T CT CT CT CT T T T T T

13 N T 5 T 0 D 5 T 0 T 0 N C T T 0 C CAP T Y 0 N Y T 0 =δd 5 T 0 {C T +C T 3 +C T 4 +C T 5 }, C CAP N = LIBOR ρm M MM LIBORd d M Z Rebonao d dmm db T M+ D M+ Mρ T 5 T 5 LIBOR

14 dw M+ dw M+ =ρd, BB BdZ BdZ =BdZ dz B =BB d, ρ B =BB ρ, BdZ ρm dmθ q, q =,, db qb q b q q cosθ = q snθ = q = snθ, q = d., q =,, d d = b b b b B = b M b M = cosθ cosθ cosθ M snθ snθ snθ M, d =3 b b B = b M b b b M b b b 3 3 M3 cosθ cosθ = cosθ M cosθ cosθ cosθ M snθ snθ snθ M snθ snθ snθ M snθ snθ snθ M, BB Z W M+ T M+ D M+

15 B d σ q = σ b q, =,, M, q =,, dz q Z q T M+ D M+ d σ q d σ B σ q = σ b q BB ρ LIBOR ρ B LIBORW M+ ρ } exp{ = = + = = = + d q q q q d q q M d q q q Z Z L L L L σ σ δ σ σ δ, dz d L L L dl B M B B σ σ δ δ σ + + = + =. dw d L L BdZ d BB L L dz B d B L L B L dl M = + + = M + = σ ρσ δ δ σ σ σ δ δ σ σ σ δ δ σ M + = M + =, ρ, =α + α exp{ β β max T,T. T T }, α=0.3β = 0.β = 0.005,

16 ρρ B ρ B Bd =3 ρ B ρ ρ B θ θ B B, =,, 3 B B B d =3

17 B B B LIBORLIBOR Jamshdan LIBOR Rebonao LIBOR T T n S,n LIBORS,n n n S, n δ D + = δ L D +. = = D + D + L δ S,n L n T T + T n T n+ δ n S,n L n δ n S,n

18 S, n n = δ D + = n = = D D D D+ δ D δ D n S,n D D n+ S, n = n. δ D + = n P,n δ D +, = P,n LIBOR P,n S,n ds,n = σ,n dw,n, S,n σ,n T T n T S,n W,n P,n S,n K PS, n 0 PS, n Γ S, n 0, K, Γ = P [ S 0 N d KN d ],, n d T =, n, n 0, n S log =, n, n 0 + K Γ, n Γ, n σ d log, S n, d S log = 0 K Γ, n, n Γ, n σ,n

19 LIBOR T T n T σ,n Γ,n Γ,n = T σ,n, LIBOR LIBOR LIBOR LIBORLIBOR LIBOR Rebonao Brgo and Mercuro Brgo and MercuroRebonaoHull and Whe Rebonao Rebonao

20 T T n T σ,n LIBOR σ,n n n k σ, n w, n 0 w, n 0 L 0 Lk0 ρ k l l k T, l S δ σ, n0 = + k + l σ = =, w, n 0 = n δ D k= k + δ D 0 k+ 0, ρ,k k D LIBOR σ ρ,k LIBOR LIBOR LIBOR LIBOR

21 LIBOR LIBOR YYY M YYYYYM nymn 0.5 Y LIBOR LIBOR LIBOR

22 M M LIBOR M LIBOR YKM LIBOR log [L + /L ] L + L + L + L log = L L L L + /L = L + L /L x,, x N x SK 3 4 N x N x, x x S = K = N = / = / N N N N N

23 YYYYM

24 - Normal Dsrbuon Normal Dsrbuon Normal Dsrbuon Normal Dsrbuon QQQuanleQuanlenkx k,n x k,n, N n k+/n+n

25 LIBOR LIBOR LIBOR LIBOR LIBOR LIBOR LIBOR LIBOR

26 LIBOR LIBOR LIBOR MID MMID MIDTelerae BIDASK YM M

27 σ CAP σ CAPLET C δ D 0C L 0,K, σ CAP = δ D 0C L 0, K, σ CAPLET. = = = δ D 0C L 0,K, σ CAP =δ D 0C L 0, K, σ CAPLET, σ CAP = σ CAPLET = δ D 0C L 0,K, σ CAP +δ D 0C L 0, K, σ CAP =δ D 0C L 0,K, σ CAPLET +δ D 0C L 0, K, σ CAPLET,

28 σ CAPLET σ CAPLET γ γ = T σ CAPLET, γ LIBOR LIBORσ γ = δ σ T ++δ σ T σ γ LIBORσ ρ,k LIBORσ ρ,k LIBOR LIBOR L 0,, L M 0D 0S,n 0 γ = δ σ T ++δ σ T LIBOR ρ,k :, k =,, M σ T,, σ T, =,, M σ : MM+/ Brgo and MercuroLIBOR LIBOR

29 0 T T < T T T < T M < M T M L σ L σ σ L σ σ σ L M σ M σ M σ M σ M M σ LIBORv 0 T < T < L T v σ T T L L v σ v σ v σ v σ v σ L M v M σ M v M σ M v M σ M T M < T M v M σ LIBORσ v LIBOR σ σ M v v T σ CAPLET LIBOR σ

30 T σ CAPLET = δ σ T σ CAPLET = δ σ + δ σ T σ CAPLET = δ σ + δ σ + + δ σ, σ σ v T σ CAPLET = v δ σ T σ CAPLET = v δ σ + δ σ T σ CAPLET = v δ σ + δ σ + + δ σ, YYv, v LIBORT σ σ T =at +de bt +cabcd. LIBORv

31 0 T < T < T T T T M < T M L v σt ~ L v σt ~ v σt ~ L v σt ~ v σt ~ v σt ~ L M v M σt ~ M v M σt ~ M v M σt ~ M v M σt ~ M = 0 a bcd σ T σ, n = { w 0 wk 0 L 0 Lk0 ρ, k σ σ d k }. S 0 0 T, n = + k= + n n N N T T T T σ σ d k σ l σ k l. 0 l = N N N

32 LIBOR δ = 0.5LIBOR Md = 3 θ,, θ 0 θ,, θ 0 =,, 0 γ = δ σ T = v = T δ σ CAPLET σ + + δ σ T + + δ σ, v Bloomberg

33 v = T δ σ + + δ σ CAPLET σ. σ,,σ 0 θ,,θ 0 θ,,θ 0 σ,, σ 0 θ,,θ 0 θ,, θ 0 π/v σ, v σ σ v v v σ LIBOR B v

34 B B B γ = = = T T σ 0 d T v 0 σ T CAPLET σ. d v

35 v = T CAPLET σ, T σ T 0 d N = 00 a, b, c, dθ,,θ 0 θ,,θ 0 σ a, b, c, dθ,, θ 0 θ,,θ 0 π/ v σ, v σ σt - v v LIBOR σ T σ T σ σ T a, b

36 LIBOR B B B B

37 LIBOR LIBOR LIBOR LIBOR Brgo and Mercuroθ v =0. Rebonaob

38 LIBOR LIBOR Glasserman and KouLIBOR Andersen and AndreasenLIBORCEV LIBOR LIBOR Glasserman and KouAndersen and Andreasen Glasserman and KouLIBOR LIBOR/λ ms CEVconsan elascy of varance Glasserman and Kou Glasserman and Merener Glasserman and Kou σ σ,, σ M σ σ =σ + +σ M

39 C JUMP 0 = δ L = 0 e λ T λ T! Cˆ L 0, K, γ T λmt 0 = + m L 0 e γ = σ + 0 d T s C γ s m λ zm s s log s + z s s + m z + s m s + + m s m s + + λ > log λ + + max0, z. m T δ γ λ m s

40 LIBOR m m m m m m =0 m<0m>0 Glasserman and Kou Andersen and AndreasenLIBOR

41 LIBOR α dl = L α σ dw +, αlibor d α a = K α γ L0 log + γ K = γ, b =, α, d α L c = α γ L0 log K = γ, γ, γ = 0 T σ d LIBOR 0<α<L =0 C CEV L 0, K, γ = δ D 0[L 0 χ a,b+,c Kχ c,b,a α = C CEV L 0, K, γ = δ D 0[L 0 Nd KNd ] α > C CEV L 0, K, γ = δ D 0[L 0 χ c, b,a Kχ a, b,c] N. χ.,d,λ λd L = 0L δ/ v δ k = δ / χ x, d, δ e v x, d + k k = 0 k! Dng

42 α < Andersen and Andreasen dl = ϕ L σ dw +, ϕ x= x.mnε α, x α, ε > 0 ε α <α > ε Lmed CEV Lmed CEVAndersen and Andreasen Lmed CEV LIBOR LIBOR LIBORHJM Andersen LIBOR Hun, Kennedy and Pelsser SV Josh and RebonaoRebonaoLIBOR

43 SV LIBOR Rebonaoa,b LIBOR LIBOR Glasserman and Kou Andersen and AndreasenEV LIBOR CEVSVLIBOR

44 LIBOR LIBOR L C C T + = maxl T K,0 C D + C /D + C T + T + C = E D + T + C T + D + T +. E T+ T + T +

45 D + T + = C =D + E T+ [C T + ]. C T + LIBOR dl L L T L T dl 0 L T T T L T T + LIBORLIBOR L T + D +

46 0 dl L dl L T T T dw dw D D + LIBOR σ W+ σ L dl = L σ dw+. σ

47 D + L L D + L L L T L T L 0 T T T L dw δ L dw = dw + ρσ d +δ L D + L LIBOR LIBOR T 5 T 5

48 S. N. J. BGM Andersen, L., A Smple Approach o he Prcng of Bermudan Swapons n he Mul-Facor LIBOR Marke Model, Journal of Compuaonal Fnance 3, pp.5-3, 000.,and J. Andreasen, Volaly Skews and Exensons of he LIBOR Marke Model, Appled Mahemacal Fnance 7, pp.-3, 000. Black, F., The Prcng of Commody Conracs, Journal of Fnancal Economcs, 3, pp.67-79, 976., and M. Scholes, The Prcng of Opons and Corporae Lables, Journal of Polcal Economy, Vol. 8, pp , 973. Brace, A., D. G aarek, and M. Musela, The Marke Model of Ineres Rae Dynamcs, Mahemacal Fnance, Vol. 7, pp. 7-55, 997. Brgo, D., and Mercuro, F., Ineres Rae Models Theory and Pracce, Sprnger Fnance, Sprnger- Verlag, 00. Dng, C. G., Algorhm AS75: Compung he Non-Cenral χ Dsrbuon funcon, Appled Sascs, 4, pp , 99. Glasserman, P., and S. G. Kou, The Term Srucure of Smple Forward Raes wh Jump Rsk, workng paper, Columba Unversy, 000., and N. Merener, Numercal Soluon of Jump-Dffuson LIBOR Marke Models, workng paper, Columba Unversy, 00. Heah, D., R. Jarrow, and A. Moron, Bond Prcng and he Term Srucure of Ineres Raes: A New Mehodology for Conngen Clams Valuaon, Economerca, Vol. 60, pp , 99. Hull, J., and A. Whe, Forward Rae Volales, Swap Rae Volales and he Implemenaon of he LIBOR Marke Model, workng paper, Joseph L. Roman School of Managemen Unversy of Torono, 999. Hun, P., J. Kennedy, and A. Pelsser, Markov-funconal neres rae models, Fnance and Sochascs 4, pp , 000. Jamshdan, F., LIBOR and Swap Marke Models and Measures, Fnance and Sochascs, Vol., pp , 997. Josh, M., and R. Rebonao, A sochasc-volaly, dsplaced-dffuson exenson of he LIBOR marke model, workng paper, Quanave Research Cenre, 00.

49 Kloeden, P.E., and E. Plaen, Numercal Soluon of Sochasc Dfferenal Equaons, Sprnger, 995. Pelsser, A., Effcen Mehods for Valung Ineres Rae Dervaves, Sprnger Fnance, Sprnger- Verlag, 000. Rebonao, R., Calbrang he BGM Model, RISK March, 999b., Volaly and Correlaon -In he Prcng of Equy, FX and Ineres-Rae, John Wley & Sons, Ld, 999b., The Sochasc Volaly Lbor Marke Model, RISK Ocober, 00.

IMES DISCUSSION PAPER SERIES LIBOR Discussion Paper No J- -J-2 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN

IMES DISCUSSION PAPER SERIES LIBOR Discussion Paper No J- -J-2 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN IES DISCUSSION PAPER SERIES IBOR Dscusso Paper No. 00-J- -J- INSTITUTE FOR ONETARY AND ECONOIC STUDIES BANK OF JAPAN 03-8660 30 IES Dscusso Paper Seres 00-J- 00 IBOR * IBOR IBOR IBOR JE classfcao: C5