IMES DISCUSSION PAPER SERIES 相互取引に伴う債権債務の依存構造を考慮した金融 機関の与信評価について にしでかつまさ 西出勝正 Discussion Paper No. 2015-J-6 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN 日本銀行金融研究所 103-8660 東京都中央区日本橋本石町 2-1-1 日本銀行金融研究所が刊行している論文等はホームページからダウンロードできます http://www.imes.boj.or.jp 無断での転載 複製はご遠慮下さい
備考 : 日本銀行金融研究所ディスカッション ペーパー シリーズは 金融研究所スタッフおよび外部研究者による研究成果をとりまとめたもので 学界 研究機関等 関連する方々から幅広くコメントを頂戴することを意図している ただし ディスカッション ペーパーの内容や意見は 執筆者個人に属し 日本銀行あるいは金融研究所の公式見解を示すものではない
AE 勝 AE 正 IMES Discussion Paper Series 2015-J-6 2015 年 6 月 相互取引に伴う債権債務の依存構造を考慮した金融機関の与信評価について にしでかつまさ 西出 E E E A* 要 旨 本論文では いわゆる Merton (1974 や Black and Cox (1976 型の構造モデルを用いて 金融機関の融資先が相互に商取引をしている場合の与信 ( 融資金 の評価モデルを提示し 融資先企業の債権債務の依存構造が融資継続 回収方針や実行時点でのスプレッドに与える影響を考察する 数値分析によって 企業が持つ商取引債権は (1 資産の分散効果 ( 正の影響 (2 同時倒産効果 ( 負の影響 の 2 つの効果があり それぞれの影響度を考慮して与信判断を行う必要があるという示唆が得られる キーワード : 融資評価 構造モデル 相互取引 債権債務 JEL classification: G13 G32 G33 L14 * 横浜国立大学大学院国際社会科学研究院教授 (E-mail: knishide@ynu.ac.jp 本稿は 日本銀行金融研究所からの委託研究論文であり 作成に当たっては八木恭子氏 ( 秋田県立大学 吉羽要直氏 ( 日本銀行 内田善彦氏 ( 金融庁 から貴重なコメントを頂いた 特に, 竹山梓氏 ( 日本銀行 には 本稿作成までに至る議論を始め先行研究の紹介や細部に亘る論文のチェックなど大変有益なコメントを頂いた ここに記して感謝したい ただし 本稿に示されている意見は 筆者個人に属し 日本銀行の公式見解を示すものではない また ありうべき誤りはすべて筆者個人に属する
1 2000 (2013 1989 2013 7 1 1 1989 2013 25 9 64 64 68 41 (2013 ( *1 2 *1 (2000 1
Gordy (2000 Rösch and Winterfeldt (2008 Bade et al. (2011 *2 Eisenberg and Noe (2001 Eisenberg and Noe (2001 (clearing payment vector Shin (2008, 2009 Elsinger (2009 Rogers and Veraart (2013 Suzuki (2002 Fischer (2014 Karl and Fischer (2014 Nishide et al. (2014 Nishide et al. (2014 Black and Cox (1976 ( ( *2 (2008 2
Diamond (1991 Mella-Barral and Perraudin (1997 Mella-Barral (1999 (2007 *3 (1 *4 (2Merton *5 (1 (2 2 2 3 4 5 A *3 Calomiris and Kahn (1991 Diamond and Rajan (2001 *4 Mella-Barral and Perraudin (1997 *5 Diamond (1991 3
2 2.1 i Q i = {Q it } ˆP Q i *6 dq it = ρq it dt + σ i Q it dŵ it (2.1 ρ σ i Ŵ i ˆP ρ *7 i ( D i ( T ˆt ( *8 i (Q iˆt t = 0 T Q it B it i T 2 δ T K T T *6 ˆP hat *7 Q i ( ρ *8 Merton (1974 Black and Cox (1976 2 1 4
R it R it = 1 {QiT B it }D i + 1 {QiT <B it }L T (Q it L T (Q it = (1 δ T Q it K T Merton B it = D i l T > 0 B it = D i /(1 l T (1 l T Q it < D i (2.2 l T > 0 (fire sale l T (2.2 ˆt ˆt (0,T i Q iˆt *9 R it ˆt 1 Φ 1 A iˆt (Q iˆt =ʈt [ e ρ(t ˆt R it ] =e ρ(t ˆt D i Φ 1 (d i,t ˆt (Q iˆt;b it + (1 δ T Q iˆt Φ 1 ( d + i,t ˆt (Q iˆt;b it e ρ(t ˆt K T Φ 1 ( d i,t ˆt (Q iˆt;b it ʈt ˆt ˆP ( ( ( log Q d i,t ± (Q;B = B + ρ ± σ i 2 2 t σ i t *9 ( (2008 5
i T δˆt Kˆt Lˆt Q iˆt = (1 δˆt Q iˆt Kˆt A iˆt (Q iˆt Lˆt (Q iˆt A iˆt (Q iˆt < Lˆt (Q iˆt ˆb iˆt (B it * 10 Q iˆt ˆb iˆt (B it Q iˆt < ˆb iˆt (B it ˆb iˆt (B it b ( ˆψ i (b;b it =e ρ(t ˆt D i Φ 1 (d i,t ˆt (b;b it + (1 δ T bφ 1 d + i,t ˆt (b;b it e ρ(t ˆt K T Φ 1 ( d i,t ˆt (b;b it (1 δˆt b + Kˆt = 0 (2.3 0 P i ] [ ] P i = Ê 0 [e ρ ˆt 1 {Qiˆt ˆb iˆt (B it } A iˆt + Ê 0 e ρ ˆt 1 {Qiˆt <ˆb iˆt (B it } Lˆt(Q iˆt 2 1 κ = ˆt/T 0 i Q i0 = q i P i *10 3 B it 6
P i (q i ;B it =e ρt D i Φ 2 (d i,ˆt (q i; ˆb iˆt (B it,di,t (q i;b it ;κ + (1 δ T q i Φ 2 (d + i,ˆt (q i; ˆb iˆt (B it, d i,t + (q i;b it ; κ e ρt K T Φ 2 (d i,ˆt (q i; ˆb iˆt (B it, di,t (q i;b it ; κ ( + (1 δˆt q i Φ 1 ( d + i,ˆt (q i; ˆb iˆt (B it Kˆt e ρ ˆt Φ 1 d i,ˆt (q i; ˆb iˆt (B it Φ 2 (, ;κ κ 2 A 0 i P i ŝ i = 1 T log ( Di P i ρ ρ + ŝ i i 2 2 0 D i ρ + ŝ i ˆt Q iˆt < ˆb iˆt (B it T D i L T (Q it 2.2 3 3 * 11 *11 3 7
3 Q i0 D i T ˆt B it ρ σ i δ T δˆt K T Kˆt 180 100 1 0.5 100 0.03 0.5 0.7 0.5 0 30 2.2.1 ˆt ˆb iˆt ˆb iˆt δ T ( 1 1 ˆb iˆt δ T 140 130 120 110 b it 100 90 80 70 60 0.2 0.3 0.4 0.5 0.6 0.7 δ T ˆt ˆb it δ T δ T ( 8
b iˆt δˆt ( 2 2 ˆb iˆt δˆt 140 130 120 110 b it 100 90 80 70 60 0.5 0.6 0.7 0.8 δ t T 3 i ( σ i ˆb iˆt 9
3 ˆb iˆt σ i 140 130 120 110 b it 100 90 80 70 60 0.3 0.4 0.5 0.6 σ i 4 T B it ˆb iˆt 10
4 ˆb iˆt B it 140 130 120 110 b it 100 90 80 70 60 100 102 104 106 108 B it B it ˆb iˆt δ T B it 2.2.2 i ŝ i 5 9 11
5 δ T ŝ i 5 ŝ i δ T 20% 18% 16% s i 14% Review No Review 12% 10% 0.2 0.3 0.4 0.5 0.6 0.7 δ T δ T ŝ i δ T δ T 2 δˆt ŝ i 6 ŝ i δˆt δˆt 12
6 ŝ i δˆt 20% 19% s i 18% Review No Review 17% 16% 0.5 0.6 0.7 0.8 δ t δˆt 2 δˆt ˆb iˆt δˆt ŝ i b iˆt δ T δˆt ŝ i 7 σ i ŝ i 13
7 ŝ i σ i 35% 30% 25% 20% s i 15% Review No Review 10% 5% 0% 0.30 0.35 0.40 0.45 0.50 0.55 0.60 σ i σ i σ i σ i 6 σ i ˆb iˆt 2 8 B it ŝ i 14
8 ŝ i B it 35.0% 30.0% 25.0% s i 20.0% Review No Review 15.0% 10.0% 100 102 104 106 108 110 B it B it ŝ i δ T ŝ i B it B it q i = Q i0 ŝ i 9 15
9 ŝ i q i 35% 30% 25% s i 20% Review No Review 15% 10% 150 160 170 180 190 200 210 q i q i 5 9 16
3 i = 1,...,n N A A N = n i j G i j T G ji = G i j * 12 {x} + = max{x,0} G + i j = {G i j} +, G i j = { G i j} + T i Q it + Γ i j j N Γ i j i j G + i j j Γ i j = G + i j T Γ i j < G + i j 3.1 1 T Q it + Γ i j < D i + G i j j N j N i 2 * 13 j i j *12 505 *13 17
λ i j = 3 G+ i j G + h j h N ˆt T t L t (Q it = (1 δ t Q it K t, t = ˆt,T δ t K t 3 ˆt i Lˆt (Q iˆt ρ e ρ(t ˆt Lˆt (Q iˆt T 3.1 T ˆt Dˆt T D T T Eisenberg and Noe (2001 Suzuki (2002 Rogers and Veraart (2013 Cont and Minca (2014 4 Cont and Minca, 2014 ˆt Dˆt T D T 1. D (0 T = /0 * 14 *14 D (0 T = /0 Nishide et al. (2014 greatest clearing payment vector D (0 T = N \ Dˆt 18
2. {D (k T } {Γ(k i j } D (k+1 T = { i N \ Dˆt Q it + Γ (k i j < D i + G i j j N j N } Γ (k i j G i j 0 0 G i j > 0 { } λ i j e ρ(t ˆt Lˆt (Q jˆt D j + h Γ (k 1 h j { } λ i j L T (Q jt D j + h Γ (k 1 h j + Γ (k i j = G i j + j Dˆt j D (k 1 ˆT 3. D ( k+1 T = D ( k T D (k T D T = D ( k+1 T Γ i j = Γ ( k+1 i j Γ(k i j k 0 D (k T n n ˆt i ( ( R it = 1 {i DT }D i + 1 {i DT } L T (Q it + Γ i j (3.1 j N 3.2 ˆt A iˆt = ʈt [ e ρ(t ˆt R it ] Lˆt (Q iˆt = (1 δˆt Q iˆt K iˆt (3.2 (3.2 ˆt ˆt i {Qˆt } = {(Q 1ˆt,...,Q nˆt } least clearing payment vector 19
i j ( G i j > 0 ˆt Q jˆt A iˆt i G i j 3.3 (n = 2 {Q it } (2.1 2 (Ŵ 1,Ŵ 2 η G 21 0 2 1 3.3.1 T 1 2 Γ T 21(Q 1T = {(1 δ T Q 1T K T D 1 } + ˆt 1 2 Γˆt 21(Q 1ˆt = {e ρ(t ˆt ((1 δˆt Q 1ˆt Kˆt D 1 } + G 21 Γ T 21 (D 1 + G 21 = 0 (1 δ T G 21 δ T D 1 + K T (3.3 Γ T 21 = 0 (3.3 * 15 4 T *15 T 1 G 21 3 G 21 700/3 233 (3.3 20
Dˆt = /0 D T = /0 {(Q 1T,Q 2T R+ Q 2 1T D 1 + G 21,Q 2T D 2 G 21 } D T = {1} {(Q 1T,Q 2T R+ Q 2 1T < D 1 + G 21,Q 2T D 2 } D T = {2} {(Q 1T,Q 2T R+ Q 2 1T D 1 + G 21,Q 2T < D 2 G 21 } D T = {1,2} {(Q 1T,Q 2T R+ Q 2 1T < D 1 + G 21,Q 2T < D 2 } (3.4 Dˆt = {1} D T = /0 {Q 2T R 1 + Q 2T + Γˆt 21 (Q 1ˆt D 2 } D T = {2} {Q 2T R 1 + Q 2T + Γˆt 21 (Q 1ˆt < D 2 } (3.5 Dˆt = {2} D T = /0 {Q 1T R+ Q 1 1T D 1 + G 21 } D T = {1} {Q 1T R+ Q 1 1T < D 1 + G 21 } Dˆt = {1,2} D T = /0 ˆt T 10 10 Dˆt = /0 T q 2 D T = {1} D 2 D T = /0 D T = {1,2} D 2 G 21 D T = {2} O D 1 + G 21 q 1 (B 1T = D 1 + G 21 (B 2T = D 2 G 21 D 2 21
Dˆt = /0 T (3.1 2 1. D T = /0 R 1T = D 1, R 2T = D 2 2. D T = {1} R 1T = (1 δ T Q 1T K T, R 2T = D 2 3. D T = {2} R 1T = D 1, R 2T = (1 δ T Q 2T + G 21 K T 4. D T = {1,2} R 1T = (1 δ T Q 1T K T, R 2T = (1 δ T Q 2T K T 3.3.2 ( 1 B 1T = D 1 + G 21 ˆt 2 A 1ˆt =e ρ(t ˆt D 1 Φ 1 (d 1,T ˆt (Q 1ˆt;D 1 + G 21 + (1 δ T Q 1ˆt Φ 1 ( d + 1,T ˆt (Q 1ˆt;D 1 + G 21 e ρ(t ˆt K T Φ 1 ( d 1,T ˆt (Q 1ˆt;D 1 + G 21 (2.3 ˆψ 1 (b;d 1 + G 21 = 0 ˆb 1ˆt (D 1 + G 21 Q 1ˆt ˆb 1ˆt (D 1 + G 21 Q 1ˆt < ˆb 1ˆt (D 1 + G 21 22
3.3.3 2 ˆt 2 1 2 1 ˆt T 2 ˆt 3.5 B 2T = D 2 Γˆt 21 (Q 1ˆt 2 ˆt 2 A 2ˆt =e ρ(t ˆt D 2 Φ 1 (d 2,T ˆt (Q 2ˆt;D 2 Γˆt 21 (Q 1ˆt + (1 δ T Q 2ˆt Φ 1 ( d + 2,T ˆt (Q 2ˆt;D 2 Γˆt 21(Q 1ˆt e ρ(t ˆt K T Φ 1 ( d 2,T ˆt (Q 2ˆt;D 2 Γˆt 21 (Q 1ˆt 2 (2.3 ˆψ 2 (b;d 2 Γˆt 21 (Q 1ˆt = 0 ˆb 2ˆt Q 2ˆt ˆb 2ˆt (D 2 Γˆt 21 (Q 1ˆt Q 2ˆt < ˆb 2ˆt (D 2 Γˆt 21 (Q 1ˆt Γˆt 21 (Q 1ˆt ˆt 2 1 ˆt 1 ˆt (3.4 T 2 Q 2T 1 Q 1T A 2ˆt A 2ˆt (Q 1ˆt,Q 2ˆt =e ρ(t ˆt ʈt [1 {DT =/0} {D T ={1}}D 2 + 1 {DT ={2}}{(1 δ T Q 2T + G 21 K T } ] + 1 {DT ={1,2}}{(1 δ T Q 2T K T } 23
=e ρ(t ˆt D 2 ( ˆPˆt {Q 1T D 1 + G 21,Q 2T D 2 G 21 } + ˆPˆt {Q 1T < D 1 + G 21,Q 2T D 2 } + (1 δ T Q 2ˆt ( P 2ˆt {Q 1T D 1 + G 21,Q 2T < D 2 G 21 } + P 2ˆt {Q 1T < D 1 + G 21,Q 2T < D 2 } e ρ(t ˆt K T ( ˆPˆt {Q 1T D 1 + G 21,Q 2T < D 2 G 21 } + ˆPˆt {Q 1T < D 1 + G 21,Q 2T < D 2 } + e ρ(t ˆt G 21 ˆPˆt {Q 1T D 1 + G 21,Q 2T < D 2 G 21 } (3.6 P 2 * 16 P 2 (A = Ê[ e ρt Q 2T Q 20 (3.6 1 A ] A 2ˆt (Q 1ˆt,Q 2ˆt [ =e ρ(t ˆt D 2 Φ 2 (d 1,T ˆt (Q 1ˆt;D 1 + G 21,d 2,T ˆt (Q 2ˆt;D 2 G 21 ;η ] +Φ 2 ( d 1,T ˆt (Q 1ˆt;D 1 + G 21,d 2,T ˆt (Q 2ˆt;D 2 ; η [ + (1 δ T Q 2t Φ 2 (d η 1,T ˆt (Q 1ˆt;D 1 + G 21, d + 2,T ˆt (Q 2ˆt;D 2 G 21 ; η ] +Φ 2 ( d η 1,T ˆt (Q 1ˆt;D 1 + G 21, d + 2,T ˆt (Q 2ˆt;D 2 ;η [ e ρ(t ˆt K T Φ 2 (d 1,T ˆt (Q 1ˆt;D 1 + G 21, d 2,T ˆt (Q 2ˆt;D 2 G 21 ; η ] +Φ 2 ( d 1,T ˆt (Q 1ˆt;D 1 + G 21, d 2,T ˆt (Q 2ˆt;D 2 ;η + e ρ(t ˆt G 21 Φ 2 (d 1,T ˆt (Q 1ˆt;D 1 + G 21, d 2,T ˆt (Q 2ˆt;D 2 G 21 ; η (3.7 * 17 ( ( log Q d η 1,t (Q;B = B + ρ σ i 2(1 2η 2 t σ i t *16 Black-Scholes 1 P 1 {P K} P *17 (3.3 24
ˆt 2 (3.7 A 2ˆt ˆφ 2 (Q 1ˆt,Q 2ˆt = A 2ˆt (Q 1ˆt,Q 2ˆt Lˆt (Q 2ˆt (3.7 Q 2ˆt Q 1ˆt 2 1 Q 1ˆt G 21 ˆt 1. 1 ˆψ 1 (Q 1ˆt ;D 1 + G 21 0 2. 1 1 (ψ 1 (Q 1ˆt ;D 1 + G 21 < 0 2 (i ˆψ 2 (Q 2ˆt ;D 2 Γˆt 21 (Q 1ˆt 0 2 Dˆt = {1} (ii ˆψ 2 (Q 2ˆt ;D 2 Γˆt 21 (Q 1ˆt < 0 2 Dˆt = {1,2} 3. 1 1 ( ˆψ 1 (Q 1ˆt ;D 1 + G 21 0 2 (i ˆφ 2 (Q 1ˆt,Q 2ˆt 0 2 Dˆt = /0 (ii ˆφ 2 (Q 1ˆt,Q 2ˆt < 0 2 Dˆt = {2} 25
3.3.4 0 1 2 0 P 1 1 P 1 =e ρt D 1 Φ 2 ( d 1ˆt (q 1; ˆb 1ˆt (D 1 + G 21,d 1T (q 1;D 1 + G 21 ;κ ( + (1 δ T q 1 Φ 2 d + 1ˆt (q 1; ˆb iˆt (D 1 + G 21, d 1T + (q 1;D 1 + G 21 ; κ e ρt ( K T Φ 2 d 1ˆt (q 1; ˆb 1ˆt (D 1 + G 21, d1t (q 1;D 1 + G 21 ; κ ( + (1 δˆt q 1 Φ 1 d + 1ˆt (q 1; ˆb 1ˆt (D 1 + G 21 e ρ ˆt ( Kˆt Φ 1 d 1ˆt (q 1; ˆb 1ˆt (D 1 + G 21 q i = Q i0 2 0 ] P 2 = e ρ ˆt Ê 0 [1 { ˆφ2 (Q 1ˆt,Q 2 ˆT 0} A 2ˆt (Q 1ˆt,Q 2ˆt + 1 { ˆφ2 (Q 1ˆt,Q 2 ˆT <0} Lˆt (Q 2ˆt 2 ˆφ 2 (Q 1ˆt,Q 2ˆt = A 2ˆt (Q 1ˆt,Q 2ˆt Lˆt (Q 2ˆt P 2 4 ˆt η = 0.3 G 21 = 20 3 i = 1,2 0 2 Q 20 + G 21 = 180 G 21 = 10 q 2 = Q 20 = 170 * 18 *18 Γˆt 21 = 0 26
1 D 1 + G 21 2 1 2 (1 (2 2 2 4.1 2 11 η 2 ˆb 2ˆt Dˆt = /0 Q 1ˆt = 180 27
11 η ˆb 2ˆt 120 115 b 2t 110 105 G21=0 G21=10 G21=20 G21=30 100 95 0 0.1 0.2 0.3 0.4 0.5 0.6 η η ( 2 (Q 2 1 2 G 21 Q 1ˆt = 180 B 1T = 100 ( 12 1 ˆt 2 ˆb 2ˆt 28
12 ˆt Q 1ˆt ˆb 2ˆt 120 115 b 2t 110 105 G21=0 G21=10 G21=20 G21=30 100 95 150 160 170 180 190 200 210 Q 1t 12 Q 1ˆt 2 Q 1ˆt = 150 G 21 b 2ˆt G 21 1 η 13 Q 1ˆt = 120 29
13 η ˆb 2ˆt ( 135 130 b 2t 125 120 G21=0 G21=10 G21=20 G21=30 115 110 0 0.1 0.2 0.3 0.4 0.5 0.6 η 11 ˆb 2ˆt η ˆt 1 Q 1ˆt η 2 Q 2T 1 Q 1T 2 ( Γ 21 < G 21 2 η G 21 η 12 Q 1t = 150 13 η G 21 bˆt G 21 G 21 ( G 21 10 G 21 ( G 21 10 30
4.2 2 ŝ 2 14 Q 1 Q 2 η 2 ŝ 2 14 η ŝ 2 20% 19% 18% s 2 17% Review No Review 16% 15% 0 0.1 0.2 0.3 0.4 0.5 0.6 η 2 (Q 2T 1 Q 1T 2 1 G 21 31
15 G 21 2 ŝ 2 15 G 21 ŝ 2 20% 19% s2 18% Review No Review 17% 16% 0 5 10 15 20 25 30 G 21 G 21 2 ( G 21 2 G 21 13 2 Q 2T + Γ T 21 G 21 2 1 Γ T 21 2 ŝ 2 G 21 ( 32
2 Q 2 G 21 0 2 q 2 = Q 20 = 120 2 16 q 2 η ŝ 2 16 G 21 ŝ 2 ( 70% 65% s 2 60% Review No Review 55% 50% 0 0.1 0.2 0.3 0.4 0.5 0.6 η η ŝ 2 ( 15( G 21 2 2 η 33
2 G 21 ŝ 2 17 17 G 21 ŝ 2 ( 75% 70% 65% 60% s 2 55% Review No Review 50% 45% 40% 0 5 10 15 20 25 30 G 21 15 2 G 21 G 21 15 ( ŝ 2 G 21 5 34
2 A 1 P i [ ] P i =Ê 0 1 {Qiˆt ˆb iˆt } e ρt D i Φ 1 (d i,t ˆt (Q iˆt [ ] + Ê 0 1 {Qiˆt ˆb iˆt } e ρ ˆt (1 δ T Q iˆt Φ 1 ( d + i,t ˆt (Q iˆt [ ] Ê 0 1 {Qiˆt ˆb iˆt } e ρt K T Φ 1 ( d i,t ˆt (Q iˆt ] + Ê 0 [1 {Qiˆt <ˆb iˆt } e ρ ˆt {(1 δˆt Q iˆt Kˆt } (A.1 (A.1 4 Q i0 = q i Q iˆt = q i exp {(ρ σ i 2 } ˆt + σ i Ŵ 2 iˆt (A.2 35
x = ( logq iˆt ρ σ i 2 2 ˆt σ i ˆt = W iˆt ˆt (A.3 ˆP ϕ 1 (x Ê 0 [1 {Qiˆt <ˆb iˆt (B it } Ê 0 [1 {Qiˆt <ˆb iˆt (B it } Q iˆt {Q iˆt < ˆb iˆt } = {x > d i,t (q i,; ˆb iˆt } ] ] ( = d i,ˆt (q i;ˆb iˆt (B it ϕ 1(xdx = Φ 1 d i,ˆt (q i; ˆb iˆt (B it = d i,ˆt (q i;ˆb iˆt (B it q ie (ρ σ2 i2 ˆt+σ i ˆtxϕ1 (xdx =q i e ρ ˆt d i,ˆt (q i;ˆb iˆt (B it ϕ 1 =q i e ρ ˆt Φ 1 ( d + i,ˆt (q i; ˆb iˆt (B it ( x + σ i ˆt dx (A.1 4 ( (1 δˆt q i Φ 1 ( d + i,ˆt (q i, ˆB iˆt K iˆt e ρ ˆt Φ 1 d i,ˆt (q i, ˆB iˆt (A.1 1 (A.2 (A.3 κ = ˆt/T Ê 0 [1 {Qiˆt ˆB iˆt } Φ 1 d i,t ˆt (Q iˆt;b it = d i,ˆt (q i;b it κx 1 κ 2 ( ] d d i,t ˆt (Q iˆt;b it = i,ˆt (q i;ˆb iˆt (B it ( d i,t (q i ;B it ϕ 1 (xφ 1 dx 1 κ 2 =Φ 2 (d i,ˆt (q i; ˆb iˆt (B it,d i,t (q i;b it ;κ (A.1 2 (A.3 x x = x + σ i ˆt d + i,t ˆt (Q iˆt;b it = d+ (q i ;B it + κ x 1 κ 2 36
] Ê 0 [1 {Qiˆt ˆB iˆt } e ρ ˆt Q iˆt Φ 1 ( d + i,t ˆt (Q iˆt d i,ˆt = (q i;ˆb iˆt (B it e ρ ˆt q i e (ρ σ2 i2 ( ˆt+σ i ˆtxΦ1 d + i,t (q i ;B it + κ x 1 κ 2 ( d + i,ˆt =q (q i;ˆb iˆt (B it d + i,t (q i ;B it + κ x i ϕ 1 ( xφ 1 1 κ 2 =q i Φ 2 (d + i,ˆt (q i; ˆb i,ˆt (B it, d + i,t (q i;b it ; κ d x ϕ 1 (xdx (A.1 3 2 37
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