わが国株式市場のモデルフリー・インプライド・ボラティリティ

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1 IMES DISCUSSION PAPER SERIES Discussion Paper No. 9-J-1 INSTITUTE FOR MONETARY AND ECONOMIC STUDIES BANK OF JAPAN

3 IMES Discussion Paper Series 9-J * = BSIV MFIV MFIV MFIV 5 MFIV (1) MFIV () MFIV (3) MFIV 1 (4) MFIV VIX JEL classification: G1 G13 G14 * yoshihiko.sugihara@boj.or.jp

4 1 implied volatility : IV = = Black-Scholes implied volatility : BSIV = BSIV BSIV = model free implied volatility : MFIV MFIV 199 Neuberger [199] Dupire [199] Demeterfi et al. [1999] Britten-Jones and Neuberger [] Jiang and Tian [5, 7] Carr and Lee [7a] MFIV MFIV BSIV 5 MFIV MFIV 1

5 1 MFIV VIX VIX MFIV 1 VIX 1 Jiang and Tian [7] VIX MFIV 3 Jiang and Tian [7] MFIV MFIV Nishina et al. [6] Maghrebi [7] [8] Nishina et al. [6] Maghrebi [7] VIX 1 MFIV MFIV [8] VIX 1 MFIV Jiang and Tian [7] MFIV 1 HV HV HV 3 VIX S&P5 1 VIX 6 VIX CBOE [9]

6 [8] Jiang and Tian [7] MFIV VIX MFIV 1 MFIV [8] Jiang and Tian [7] 1 6 MFIV MFIV MFIV 1 MFIV 3 4 MFIV 1 MFIV 3 MFIV 1 t S t t = σ{s u ; u t} S t ds t /S t S t t t T S t,t S t,t = 1 T T t t ( dsu S u ) = 1 T T t t 1 S u d[s, S] u. (1) 3

7 1/(T t) 4 S t,t t S t,t S t,t MFIVar MFIVol σ MFIVar σ MFIVol σ MFIVar = E Q [ S t,t ], () [ ] σ MFIVol = E Q S t,t. (3) Q MFIVar MFIVol σ MFIVol σ MFIVar, (4) MFIVar MFIVol MFIVar VIX MFIVol 3 MFIVar MFIVol MFIVar () K τ (= T t) P (τ, K) C(τ, K) t τ B(t, T ) 4 4

8 1 5 σ MFIVar(t, T ) ( F (t,t ) P (τ, K) dk + τb(t, T ) K F (t,t ) ) C(τ, K) dk. (5) K F (t, T ) t T F (t, T ) = S t /B(t, T ) (5) OTM OTM (5) MFIVol Carr and Lee [7a,b] 6 τ σ MFIVol π P (τ, F (t, T )) + C(τ, F (t, T )) τ F (t, T )B(t, T ) + 1 B(t, T ) π 8τF (t, T ) + K>F (t,t ) K<F (t,t ) I 1 (ln I (ln K F (t,t ) K F (t,t ) ) I 1 ( ln K K ) I ( ln K K K F (t,t ) K F (t,t ) ) P (τ, K)dK ) C(τ, K)dK. (6) 5 (5)

9 I ν ν 1 I ν (x) = = 1 ( x ) ν+n n!γ(ν + n + 1) n= 1 ( x ) ν π πγ(ν + 1 ) cosh(x cos θ) sin ν θdθ, (ν >.5) Γ Fitz and Gatheral [5] (6) ATM BSIV 3 BSIV MFIV MFIV BSIV MFIV BSIV BSIV BSIV MFIV 1 3 MFIV (5) (6) MFIV MFIV (5) (6) Jiang and Tian [7] VIX 6

10 1 BSIV MFIV BSIV MFIV Jiang and Tian Jiang and Tian [7] BSIV BSIV = (5) (6) MFIV BSIV F (t, T ) 7 BSIV 3 BSIV 7 (5) F (t, T ) 1 BSIV F (t, T ) 7

11 1 MFIVar 被積分関数値 1.4E-5 1.E-5 1.E-5 8.E-6 6.E-6 4.E-6.E-6 フォワードATM.E+ 11, 13, 15, 17, 19, 1, 権利行使価格 ( 円 ) MFIVar ATM MFIVar OTM (5) (6) MFIV Jiang and Tian [7] K θ(> ) K i = F (t, T )e iθ (i = i max, i max + 1,, i max ), (7) 8

12 [K min = F (t, T )e i maxθ, K max = F (t, T )e i maxθ ] i max ϵ(> ) [ C(τ, Ki ) i max = arg max i> Ki (5) σ MFIVar(τ) erτ τ + erτ τ (6) j=1 π e rτ σ MFIVol (τ) {C(τ, K ) + P (τ, K )} τ K e + rτ π 1 K i 1 K I (ln i 8τK i= i max + e rτ π i max K j K j 1 8τK j=1 < ϵ and P (τ, K i) K i ] < ϵ, (8) { K i 1 K i P (τ, Ki ) + P (τ, K } i 1) K i= i max i Ki 1 i max { K j K j 1 C(τ, Kj ) + C(τ, K j 1) Kj Kj 1 + I (ln }, (9) ) ) K i K K I 1 (ln i K P (τ, K i ) K i Ki ) ) K i 1 K K I 1 (ln i 1 K P (τ, K i 1 ) K i 1 Ki 1 ) ) Kj Kj I 1 (ln K I (ln K C(τ, K j ) K j Kj ) ) Kj 1 Kj 1 I 1 (ln K I (ln K + C(τ, K j 1 ) K j 1 Kj 1, MFIV F (t, T ) (9) (1) Jiang and Tian [7] (1) 9

13 θ =.1 ϵ = VIX MFIV VIX CBOE S&P5 VIX VIX MFIV σ VIX = [ (T t)b(t, T ) i P (τ, K i ) K Ki i + i> ] C(τ, K i ) K Ki i 1 ( ) F (t, T ) 1, T t K (11) K i (i =, ±1, ±,... ) K i K i = (K i+1 K i 1 )/ K F (t, T ) VIX MFIV ATM (11) (5) VIX (5) 3 1 VIX VIX 3 1 MFIVar VIX 3 7 VIX 1% % VIX 199 1

14 VIX VIX 指数 (%) S&P VIX 指数 8 7 ( 参考 )S&P5 指数 ( 右軸 ) /1 /1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 月 VIX CBOE S&P5 S&P VIX S&P5 VIX VIX VIX VIX VIX MFIV MFIV 9 MFIV

15 1 1 1 LIBOR MFIV OTM 7 1 5, MFIV 5 MFIV LIBOR MFIV

16 3 OTM 5 銘柄数 ( 権利行使価格数 ) 3 年 4 年 5 年 6 年 7 年満期までの期間 ( ヶ月 ) MFIV 6 7 MFIV MFIV C(τ, K) [max(, Se δτ Ke rτ ), Se δτ ] P (τ, K) [max(, Ke rτ Se δτ ), Ke rτ ] (1) BSIV 13

17 4 MFIV 米イラク攻撃 (MFIV) 5% 45% 4% 35% 3% 5% % 15% 中国抗日デモライブドア問題 MFIVol MFIVar ( 参考 ) 日経平均株価 ( 右軸 ) 中東情勢の悪化北朝鮮テホトン発射上海発世界同時株安サブプライム問題 ( 株価 ) 19, 17, 15, 13, 11, 9, 1% 7, 3/4 3/1 4/4 4/1 5/4 5/1 6/4 6/1 7/4 7/1 月 MFIV MFIV MFIV MFIV 17 MFIV 1% 5% 1 MFIV 4 5 MFIV 1% 5% MFIV MFIV 7 MFIV 5% 17 MFIVol MFIVar (4) MFIV ATM BSIV ATM BSIV MFIVol 14

18 MFIV MFIVar , , , , , , 18 MFIVol , , , , , , 18 % MFIV MFIV MFIVar MFIVol VIX MFIV VIX 15

19 5 BSIV 5% 5% 4% BSIV MFIVar MFIVol 4% BSIVs MFIVar MFIVol 3% 3% % % 1% マネネス ( 原資産価格権利行使価格 1) 1% ターム ( ヶ月 ) BSIV MFIV BSIV MFIV BSIV BSIV MFIV BSIV MFIV MFIVol MFIVar BSIV BSIV MFIV MFIV BSIV MFIV BSIV ATM BSIV ±1% BSIV MFIV BSIV BSIV 16

20 6 MFIVar MFIVol ATM BSIV 6% 5% 4% 3% MFIVol MFIVar ATM の BSIV % 1% 4/1 4/15 4/9 5/13 5/7 6/1 6/4 7/8 7/ 8/5 8/19 9/ 9/16 日 BSIV ±1% BSIV 1 integrated realized volatility: IRV IRV (1) σ IRV (t, T ) = S t,t (13) IRV S t,t t T N i t i (i = 1,,, N; t = t, T = t N ) i RV σ RV (t i) IRV σ IRV(t, T ) 1 T t N σrv (t i ) (14) MFIV () (3) MFIV IRV i=1 17

21 σ IRV σ MFIVar σ IRV σ MFIVol 1 IRV MFIV t IRV 1 MFIV 1 T t 1 t = σ IRV (t (T t), t) σ MFIVar (t, T ) t = σ IRV (t (T t), t) σ MFIVol (t, T ) 1 MFIV IRV t = σ IRV (t, T ) σ MFIVar (t, T ) t = σ IRV (t, T ) σ MFIVol (t, T ) 19 MFIV IRV MFIV IRV VIX S&P5 Carr and Wu [7] Bollerslev and Zhou [7] 1 σ MFIVar > σ IRV 5 IRV MFIV RV 18

22 7 (1) () 観測時点プレミアム実現プレミアム.5. 観測時点プレミアム実現プレミアム確率密度.15.1 確率密度プレミアム (% ポイント ) プレミアム (% ポイント ) 1 7 1, % % 1% 6% Hansen and Lunde [4] RV

23 4 BSIV BSIV MFIV MFIV MFIV 8 8 (1) MFIVar () MFIVol.16 確率密度.14 確率密度ヶ月物 3 ヶ月物 6 ヶ月物ヶ月物 3 ヶ月物 6 ヶ月物 % 15% % 5% 3% 35% 4% 45% 5% MFIVar の水準 1% 15% % 5% 3% 35% 4% 45% 5% MFIVol の水準

24 MFIV MFIVar 1 1 MFIVar 13% 1% 6% 8% % 4% MFIVol MFIVar 1

25 9 MFIVar 35% ライブドア問題米金融政策不透明感ゼロ金利政策解除イスラエル軍レバノン攻撃 3% 5% % 15% 1% 6/1 6/ 6/3 6/4 6/5 6/6 6/7 6/8 6/9 6/1 6/11 6/1 5% 上海発世界同時株安米サブプライム問題 45% 4% 35% 3% 5% % 15% 1% 7/1 7/ 7/3 7/4 7/5 7/6 7/7 7/8 7/9 月月 MFIVar 1 3 6

26 1 (1) MFIVar () MFIVol.4 確率密度.8 確率密度.3. 順カーブ逆カーブそれ以外.6.4 順カーブ逆カーブそれ以外.1. 1% 14% 18% % 6% 3% 34% 38% 4% 46% 5% 1% 14% 18% % 6% 3% 34% 38% 4% 46% 5% % Litterman and Scheinkman [1991] Heston [1993] 1 CEV Gatheral [8] 5 MFIV MFIV 3

27 11 (1) MFIVar () MFIVol a. 4 x x b c #1 # # #1 # #3 c. #1 MFIV MFIV 1 RV RV MFIV 4

28 σ MFIVar(t, T ) = α 1 + σ MFIVol (t, T ) = α + m ξ i σrv (t i ) + ε 1, (15) i= m ζ i σ RV (t i ) + ε. (16) i= α 1, α ξ i, ζ i (i =,, m) RV ε 1, ε MFIV 3 1 RV 3 t RV m = 1 RV 1 MFIV 4 α i RV ξ i, ζ i α ξ i, ζ i i = 5 1 i = 6 ξ i, ζ i m = 5 (15) (16) RV t ξ i, ζ i 5% 3 RV MFIV RV MFIV 1 3 (15) (16) α 1, α ξ i, ζ i 1 m =

29 4 ξ m = MFIVar (ξ i ) MFIVol (ζ i ) (i ) α 5.18 (8.93) 17.8 (6.68) (6.15) 3.5 (.3) 5.55 (.9) 6.6 (.33) 6.6 (5.7) (4.8) 4.58 (3.95).76 (.6) 1.94 (.5) 1.55 (.8) (5.88) 5.58 (4.4) (4.5) 1.77 (.7) 1.4 (.5).95 (.9) 3.57 (5.9).8 (4.43) 15.3 (4.8) 1.39 (.7) 1.1 (.6) 1.3 (.9) 3.75 (5.98) (4.48) (4.1) 1.1 (.7).94 (.6).79 (.3) (5.98) (4.48) 1.58 (4.1).96 (.7).7 (.6) 1.7 (.3) 5.8 (5.98) (4.47) 9.68 (4.1).84 (.7).54 (.6).5 (.3)* (5.98)* 6.96 (4.47)* 8.9 (4.1). (.7)*.4 (.6)*.13 (.3)* (6.)* 8. (4.51)* 5.7 (4.15)*.48 (.7)*.54 (.6).18 (.3)* (6.)* 8. (4.5)* 9.51 (4.15).51 (.7)*.4 (.6)*.36 (.3)* 9.98 (6.) (4.51) 7.3 (4.15)*.76 (.7).63 (.6).5 (.3)* (6.3) 1.43 (4.51) 8.1 (4.16).74 (.7).55 (.6).4 (.3)* (6.) 11.8 (4.51) 7.86 (4.15)*.71 (.7).7 (.6).61 (.3) (6.3) 1.1 (4.51) 8.64 (4.16).51 (.7)*.7 (.6)*.74 (.3) (6.6) 9.79 (4.54) 7.31 (4.18)*.63 (.8).55 (.6).66 (.3) (6.7) 1.35 (4.54) 9.6 (4.19).4 (.8)*.4 (.6)*.53 (.3)* (6.9) 1.46 (4.56) 11.5 (4.).64 (.8).48 (.6)*.86 (.3) (6.9)* 9.35 (4.56) 7.1 (4.)*.9 (.8)*.6 (.6).38 (.3)* (6.1) 1.33 (4.56) (4.1).55 (.8).56 (.6).57 (.3)* (6.3)* 1.3 (4.51) 8.46 (4.16).9 (.7)*.49 (.6)*.7 (.3)* (6.)* 7.67 (4.49)* 7.37 (4.13)*.3 (.7)*.33 (.6)*.4 (.9)* 1.47 (5.8)* 8.33 (4.36)* 7.31 (4.1)*.4 (.6)*.45 (.5)*.64 (.9) * 95% ξ i % ζ i % 1 ξ i, ζ i t m = 5 (1) ξ i t MFIVar () ζ i t MFIVol ゼロである - 5% 有意水準ゼロではないヶ月 3 ヶ月 6 ヶ月ラグの日数 (i 日 ) 1 ヶ月 3 ヶ月 6 ヶ月ゼロである -4-5% 有意水準ゼロではないラグの日数 (i 日 ) ξ i, ζ i t 5% 6

30 MFIV 3 VIX VDAX VDAX DAX VIX MFIVar VIX VDAX 3 6 MFIVar Granger 5 MFIV MFIV 4 VDAX VDAX VIX 7

31 13 MFIVar VIX VDAX 5 (%) 4 3 わが国 MFIVar 米国 VIX ドイツ VDAX 1 3/4 3/1 4/4 4/1 5/4 5/1 6/4 6/1 7/4 ( 月 ) VIX CBOE VDAX 14 MFIVar VIX VDAX (1) () 1.9 相関係数.8 相関係数日 MFIVar と米 VIX( 日本基点 ).8.7 日 MFIVar と米 VIX( 日本基点 ) 日 MFIVar と独 VDAX( 日本基点 ) 独 VDAX と米 VIX( ドイツ基点 ).6.4. 日 MFIVar と独 VDAX( 日本基点 ) 独 VDAX と米 VIX( ドイツ基点 ).6-5 日 -4 日 -3 日 - 日 -1 日日 +1 日 + 日 +3 日 +4 日 +5 日日数差日 -4 日 -3 日 - 日 -1 日日 +1 日 + 日 +3 日 +4 日 +5 日日数差 MFIV VIX VDAX 15 1 VIX VDAX VDAX VIX CBOE VDAX 8

32 5 MFIVar VIX VDAX 1 5.3* 89.97* 6.47* 6.67* 31.5* 8.* 6.39* 1.18* 11.6* 68.86* 48.65* 7.86* F 1% VIX CBOE VDAX MFIV t MFIVar t 1 (t 1 > t) t (t > t 1 ) 5 σmfivar(t, t 1, t ) = E t S t1,t, (17) MFIVar MFIVar t 1 t MFIVar t MFIVar E t1 S t1,t = E t S t1,t + ε t1 t, (18) ε t1 t N(µ t1 t, (t 1 t)v ). µ t1 t (t 1 t)v t t 1 t 1 (18) 5 MFIVol MFIVar 9

33 (18) (18) MFIVar (18) t T m n T = t + mn (1) S t,t+nm = 1 n 1 S t+im,t+im+m, (19) n t i= E t S t,t+nm = σmfivar(t, t + nm) = 1 n 1 E t S t+im,t+im+m, () n (18) Campa and Chang [1995] Mixon [7] σmfivar (t, t + m) 1 n 1 [ σ n MFIVar (t + im, t + im + m) σmfivar(t, t + m) ] i= i= = α + β[σ MFIVar(t, t + nm) σ MFIVar(t, t + m)] + ε, (1) ε = n 1 i=1 ε im (1) α =, β = 1 MFIVar ε ε im Newey and West [1987] t F 6 m, n 6 α, β α β m 1 t m, n α = 6 Andrews [1991] AR(1) 3

34 6 1 m n α β F (.14) (.914) (.14) (.1151) (.1) (.141) ** (.9) (.111) (.7) (.16) *.436 (.6) (.889) α, β α = β = 1 5% F α = β = 1 1% Newey and West [1987] (m, n) = (3, ) β = 1 F (m, n) = (, 3) α = β = MFIV 6 MFIV 4 MFIV MFIV (1) MFIV 31

35 () (3) (4) 5 1 (5) BSIV (6) MFIV 1 (7) MFIV VIX VDAX 1 (8) MFIV MFIV VIX MFIV MFIV 3 BSIV 3 BSIV MFIV MFIV MFIV MFIV 3

36 1 MFIVar 1 F (t, T ) = x d = σ(t, )dw t + (e x 1)[µ(dx, dt) ν(dx, t)dt], (A-1) 1 dw t σ(t, ) t x µ(x, t) = 1 x = ln( / ) x t µ(x, t) ν(x, t) J t = t F s (e x 1)µ(dx, ds), (A-) Ẽ s µ(dx, dt) = Ẽsν(dx, t)dt, ( s < t), T - Ẽ t [ ] = Ẽ[ ] = σ(f s ; s t) ν(x, t) ν(, t) =, (e x 1)ν(x, t)dx <, t [, ), (A-3) MFIVar σ MFIVar(t, T ) ( Ft P (τ, K) dk + τb(t, T ) K C(τ, K) K ) dk, (A-4) 33

37 f( ) x f(y) (y > ) f(y) = f(x)+f (x)(y x)+ x f (K)(K y) + dk + f( ) x, y y f(y) f(x) = 1{y x} f (u)du 1{x > y} x y [ f(y) = f(x) + 1{y x} f (x) + x 1{x > y} u x x = f(x) + f (x)(y x) + 1{y x} = f(x) + f (x)(y x) + 1{y x} = f(x) + f (x)(y x) + x y ] f (v)dv du [ f (x) y x y x f (v) x u y v x f (K)(y K) + dk. (A-5) x y ] f (v)dv du f (u)du, dudv + 1{x > y} (y v)f (v)dv + 1{y < x} (y v) + f (v)dv + v = K (A-5) x x y x (v y) + f (v)dv, y v f (v) dudv y (v y)f (v)dv 3 f(f T ) V t = B(t, T )Ẽtf(F T ) B(t, T ) 34

38 C(τ, K), P (τ, K) V t = f( )B(t, T ) + Ft B(t, T )Ẽt f (K)P (τ, K)dK + f (K)C(τ, K)dK. (A-6) [ ln F ] Ft T P (τ, K) C(τ, K) = dk + dk, (A-7) K K (A-5) x = t T - Ẽ t f(f T ) = f( ) + f ( )(ẼtF T ) +Ẽt Ft f (K)(K F T ) + dk + Ẽt f (K)(F T K) + dk, T - V t = f( )B(t, T ) + Ft f (K)P (τ, K)dK + f (K)C(τ, K)dK, (A-6) f(x) = ln x (A-6) Ẽ t ln F T = ln Ft P (τ, K) dk K C(τ, K) K dk, (A-7) 35

39 4 MFIVar MFIVar (5) σ MFIVar(t, T ) = MFIVar (A-3) ( Ft P (τ, K) dk + (T t)b(t, T ) K C(τ, K) K ) dk + ε, ( (dfs ) ) 3 ε o. (A-8) F s σ MFIVar(t, T ) = = = 1 T tẽt 1 T tẽt 1 T tẽt T t T t T t ( dfs F s ( 1 F s ) ) d[f, F ] s [ σ (s, )ds + ] (e x 1) µ(dx, ds). ln F s d ln F s = df s F s 1 σ (s, )ds + σ (s, ) σ MFIVar(t, T ) = T tẽt + 1 T tẽt [ ln(f s e x ) ln F s F ] s ex F s µ(dx, ds) F s [ ln F T T t ] + T df s T tẽt t F s ( e x 4e x + x + 3 ) µ(dx, ds). T - 36

40 1 (A-7) σ MFIVar = = (T t)b(t, T ) + 1 T tẽt [ Ft T (T t)b(t, T ) (A-4) t [ Ft P (τ, K) C(τ, K) dk + K K ( e x 4e x + x + 3 ) µ(dx, ds) P (τ, K) dk + K C(τ, K) K ] dk ] dk + ε, ε 3 e x = 1 + x + x / + o(x 3 ) ε = 1 T tẽt = 1 T tẽt 1 T tẽt T t T t T t ( e x 4e x + x + 3 ) µ(dx, ds) { (1 + x + x + o(x 3 )) 4(1 + x + x / + o(x 3 )) + x + 3 } µ(dx, ds) o(x 3 )µ(dx, ds). (A-8) 37

41 MFIVol Carr and Lee [7b] MFIVol σ MFIVol (t, T ) = E Q 1 T t T t ( ) dsu S u F t, (A-9) τ σ MFIVol π P (τ, ) + C(τ, ) τ B(t, T ) 1 π + B(t, T ) 8τ + K< I K> I 1 ( ln ( ln ) K I 1 (ln K K ) K I (ln K K ) K ) K P (τ, K)dK C(τ, K)dK, (A-1) 1 q 1 q = π 1 e sq s ds, s (A-11) ( ) 1 π = Γ = e s s ds. π = [ ] (1 e s ) 1 e s ds = s s + 1 e s ds = s 3 1 e s ds. s 3 38

42 s = qt (A-11) 1 e qt π = qdt = 1 (qt) 3 1 e qt dt. q t 3 σ W t T d = σ(t, )dw t, (A-1) t T X T X T X T = X T = λ λ T p ± (λ) = 1 df s = ln F T, (A-13) ( ) dfs. (A-14) t F s T t F s ( 1 ± ) 1 8λ, (A-15) σ W Ẽ t e λ X T = Ẽte p± (λ)x T. (A-16) x, y C 1, u(x, y) u du = u x dx + 1 u u d[x, x] + x y dy, 39

43 X t dx t = d 1 d X t, x = X t, y = X t du(x t, X t ) = x u(x t, X t )dx t + 1 x u(x t, X t )d X t + y u(x t, X t )d X t = 1 { 1 x u(x t, X t )d + x + y 1 } u(x t, X t )d X t, x { } 1 u x + u y 1 u x =, u T - p, λ u(x, y) = e px+λy 1 p + λ 1 p =, (A-15) p ± (λ) p = p ± (λ) u = e p± (λ)x+λ X X t = X t = Ẽ t e p± (λ)x T +λ X T = e p± (λ)x t +λ X t = 1. Ẽ 1 1 T - σ W X X (A-16) 3 (A-11) q = X T (A-16) σ W Ẽ t X T = Ẽt [ 1 ] 1 e p(λ)x T π λ. (A-17) λ θ ± = 1 p ± θ ± = θ ± [ 1 Ẽ t X T = Ẽt π ± ] θ ± (λ) 1 ex T p ± (λ) λ dλ λ. (A-18) 4

44 θ ± (λ) = 1 ( ) 1 1. (A-19) 1 8λ (A-16) Ẽte p (λ)x T = Ẽte p+ (λ)x T ) Ẽ t e λ X T = Ẽte p+ (λ)x T + θ (λ) (Ẽt e p (λ)x T Ẽte p+ (λ)x T θ ± (λ)e p± (λ)x T, = Ẽt ± (A-11) q = X T Ẽ t X T = = = = [ 1 1 e λ X T π Ẽt λ λ 1 1 Ẽte λ X T π λ dλ λ ± θ± (λ) Ẽt 1 π 1 π Ẽt ± ] dλ ± θ± (λ)e p± (λ)x T λ λ θ ± (λ) 1 ep± (λ)x T λ λ dλ, dλ ( ± θ ± (λ) = 1 (A-18) 4 Carr and Lee [7b] ) (A-17) (A-18) σ W σ W (A-17) (A-18) Carr and Lee [7b] (A-18) (A-17) (A-18) MFIVol 4 MFIVol MFIVol (A-1) 41

45 α e αx T (A-6) f f(x) = x α Ft Ẽ t FT α = Ft α + α(α 1) P (τ, K)K α dk + α(α 1) α = p ± (λ) α(α 1) = λ [ (FT ) ] p ± (λ) C(τ, K)K α dk, Ẽ t e p± (λ)x T = Ẽt [ Ft = 1 λ ( P (τ, K) K K ) p ± (λ) C(τ, K) dk + K ( K ) ] p ± (λ) dk. MFIVol (A-18) σ MFIvol = 1 τ Ẽ t X T = = 1 τπ Ft ± dλ θ± (λ) λ w(k) P (τ, K)dK + [ Ft ( P (τ, K) K K + ) p ± (λ) dk C(τ, K) K w(k) C(τ, K)dK, ( K ) ] p ± (λ) dk w(k) K k = ln(k/ ) w(k) = = = = 1 K τπ 1 K τπ e k ± ( K F t e k τπ e k θ ± (λ) λ ( K ) p± (λ) dλ ) 1 ± Ft e k (M N). τπ ( cosh λ ( ) θ ± ± 1 8λ (λ) K dλ λ k 1 8λ ) dλ ( sinh k ) 1 8λ dλ λ(1 8λ) 4

46 M u = 1 8λ, v = 8λ 1 M = = 1/8 = ( cosh 1 k 1 8λ λ ) dλ cosh k 1 8λ du + λ u cosh ku 1 u du + 1/8 ( cosh i k 8λ 1 λ ) v cos kv 1 + v dv. 1 L 1 (z) 1 1 I 1 (z) 1 L 1 (z) = 4 u cosh(uz) π du 1 u = I 1 (z) 4 ( ) u π 1 cos(uz)du, 1 + u M = π I 1 ( ) k + cos(ku)du, δ(x) dk du cos(ku) = lim u sin(ku) dk = arctan k k dv + = π, ϕ(x); ϕ(x) M ( x) dkϕ(k) du cos(ku) πϕ() = lim u {ϕ(k) ϕ()} sin(ku) dk k. y = ku, u = 1/ε L > [ L lim ε + L + ] (ϕ(yε) L L ϕ()) sin(y) y dy L M( arctan L + π) + lim ε L {ϕ(yε) ϕ()} sin(y) y dy. L 1 L 43

47 ε lim sup ε L y L = lim =. sup ε L y L L ϕ(yε) ϕ() L ϕ(yε) ϕ() arctan L sin(y) dy y M M = cos(ku)du = πδ(k), π I 1 ( ) k + πδ(k). N L (z) I (z) L (z) = π k N = = = 1 = I (z) π sinh(uz) du 1 u ( sinh k ) 1 8λ dλ λ(1 8λ) 1 sinh ku du + 1 u ( ) π k I, (k ) k = sinh = N =. w(k) k sin(uz) du, (z ) 1 + u sin kv 1 + v dv w(k) = 1 { ( ) ( ) } π K τ e k k k 4δ(k) + I 1 I 1(k ), (k ) 44

48 k < w(k) = { π 8τK 3 { π 8τK 3 I 1 (ln π τ I (ln ) K I (ln K )}, (K > ) 1 δ(k), (K = F K t ) ) K I 1 (ln K )}. (K < ) K = k = dk = dk/k σ MFIVol K= = = = = = Ft π τ K w(k) P (τ, K) K { { π τ π τ π τ (A-1) dk K + K w(k) C(τ, K) dk K K dkδ(k) C(τ, e k ) + e k e k + dkδ(k) C(τ, e k ) { C(τ, Ft ) + 1 e k + P (τ, ) + C(τ, ), B(t, T ) dkδ(k) C(τ, } e k ) e k dkδ(k)(1 e k ) dkδ(k)(1 e k ) ( C(τ, ) = C(τ, ) B(t, T ) = C(τ, F ) t) + P (τ, ). B(t, T ) } } 45

49 3 T t T HV HV HV 7 HV HV 1 t T (> ) V VOS (t, T ) ] B(t, T )E [ S Q t,t V VOS (t, T ). (A-) (A-) 1 (A-) 7 Carr and Lewis [4] 46

50 V VOS (t, T ) V VOS (t, T ) = E Q [ S t,t ] = σ MFIVol (t, T ), (A-1) MFIVol V VAS (t, T ) V VAS (t, T ) = E Q [ S t,t ] = σ MFIVar(t, T ), (A-) MFIVar MFIV (5) (6) 1 OTM OTM (5) σmfivar(τ) wi P P (τ, K i ) + wi C C(τ, K i ). i:k i F (τ) i:k i >F (τ) (A-3) [ e wi C = wi P r(τ)τ K = τk i ], (A-4) [a] a K K 47

51 w C i, w P i ATM [K i, K i+1 ) wat P M = er(τ)τ ( K i ), τft wat C M = er(τ)τ (K i+1 ). τft (A-5) ATM (5) C(τ, K) = P (τ, K) + Se δτ Ke rτ x σ MFIVar(t, T ) [ x P (τ, K) τb(t, T ) K dk + + τ x C(τ, K) K ( 1 x + ln x ] dk ), (A-6) x (A-6) MFIVol (6) σ MFIVol w S [P (τ, F (t, T )) + C(τ, F (t, T ))]+ w i P P (τ, K i )+ w i C C(τ, K i ). i:k i <F (τ) i:k i >F (τ) (A-7) 48

52 [ w S 1 = [ w i P = π [ w i C = π ], π F (t,t ) τ [ 8τKi 3F (t,t ) I (ln 8τK 3 i F (t,t ) [ I 1 (ln K i F (t,t ) K i F (t,t ) ) ( I 1 ln ) ( I ln K i F (t,t ) K i F (t,t ) )] ] K, ] )] K.. (A-8) w S ATM ATM w P i K < F (t, T ) w C K > F (t, T ) (A-9) w P, w C w S ATM 49

53 Andrews, D.W.K., Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation, Econometrica, 59(3), pp , Bollerslev, T. and H. Zhou, Expected Stock Returns and Variance Risk Premia, Finance and Economics Discussion Series, Division of Research and Statistics and Monetary Affairs, Federal Researve Board, Washington D.C., 11, 7. Britten-Jones, M. and A. Neuberger, Option Prices, Implied Price Processes and Stochastic Volatility, Journal of Finance, 55, pp ,. Campa, J.M and P.H.K. Chang, Testing the Expectations Hypothesis on the Term Structure of Volatility in Foreign Exchange Options, Journal of Finance, 5(), pp , Carr, P. and R. Lee, Realized Volatility and Variance: Options via Swaps, Journal of Risk, pp , 7a. Robust Replication of Volatility Derivatives, Discussion Paper, Bloomberg LP and University of Chicago, 7b. Carr, P. and K. Lewis, Corridor Variance Swaps, Risk, 17(), pp. 67 7, 4. Carr, P. and L. Wu, Variance Risk Premiums, Review of Financial Studies, (3), pp , 7. CBOE, The CBOE Volatility Index - VIX, Chicago Board Options Exchange, 9. Demeterfi, K., E. Derman, M. Kamal, and J. Zou, More Than You Ever Wanted To Know About Volatility Swaps, Quantitative Strategies Research Notes, Goldman Sachs, March, Dupire, B., Arbitrage Pricing with Stochastic Volatility, Societe Generale, 199. Fitz, P. and J. Gatheral, Valuation of Volatility Derivatives as an Inverse Problem, Quantitative Finance, 5, pp , 5. 5

54 Gatheral, J., Further Development in Volatility Derivatives Modeling, Briefing Paper, Quant Congress USA 8, 8. Hansen, P.R. and A. Lunde, Realized Variance and IID Market Microstructure Noise, Brown University, Department of Economics, 4. Heston, S., A Closed Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, The Review of Financial Studies, 6, pp , Jiang, G.J. and Y.S. Tian, The Model-Free Implied Volatility and Its Information Content, Review of Financial Studies, 18, pp , 5. Extracting Model-Free Volatility from Option Prices: An Examination of the VIX Index, Journal of Derivatives, Spring, pp. 35 6, 7. Litterman, R. and J. Scheinkman, Common Factors Affecting Bond Returns, Journal of Fixed Income, 1(1), pp , Maghrebi, N., An Introdustion to the Nikkei 5 Implied Volatility Index, Kinzai Riron, Wakayama Economic Review, 336, pp , 7. Mixon, S., The Implied Volatility Term Structure of Stock Index Options, Journal of Empirical Finance, 14, pp , 7. Neuberger, A., Volatility Trading, Working Paper, London Business School, 199. Newey, W.K. and K.W. West, A Simple Positive Semi-Definite Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 55, pp , Nishina, K., N. Maghrebi, and M. Kim, Stock Market Volatility and the Forecasting Accuracy of Implied Volatility Indices, Discussion Papers in Economics and Business, Graduate School of Economics and Osaka School of International Public Policy, No.6-9, 6. 51

わが国株式市場のモデルフリー・インプライド・ボラティリティ

わが国株式市場のモデルフリー・インプライド・ボラティリティ BSIV MFIV MFIV MFIV 5 MFIV 1 MFIV MFIV 3 MFIV 1 4 MFIV VIX E-mail: yoshihiko.sugihara@boj.or.j / /1.4 73 1. imlied volatility; IV Black-Scholes imlied volatility; BSIV BSIV BSIV 1 model free imlied volatility;