2016/3/11
Realized Volatility RV 1 RV 1 Implied Volatility IV Volatility Risk Premium VRP
1 (Fama and French(1988) Campbell and Shiller(1988)) (Hodrick(1992)) (Lettau and Ludvigson (2001)) VRP (Bollerslev et al (2009)) VRP 3 6 (Londono (2014)) VRP
VRP VRP E P t [V t,t+1 ] VRP VRP t Et Q [V t,t+1] Et P [V t,t+1] IV RV t t + 1 2 RV t,t+1 = n r 2 i=1 t+i/n r t n n σ t plim n RV t,t+1 = Et P [V t,t+1 ] RV VRP VRP Ẹt P [V t,t+1 ] = RV t 1,t Ẹt P [V t,t+1 ] = RV t,t+1 VRP Ẹ P t [V t,t+1 ] = RV t,t+1
, RV 1996/1 2015/8, RV Man Institute Realized Library IV VIX VI 225 Table 1: RV IV Panel A: IV( ) and RV( ) 225 02421 57908-03412 06273 225RV 03147 02788 50775 402917 VI 06046 06455 66065 577505 S&P500 06657 41988-07006 12848 S&P500RV 02380 04018 73148 726261 VIX 04138 03950 32272 141319 Panel B: VRP( ) 225 VRP (1) 02593 03825 63643 538932 225 VRP (2) 02607 05360 53745 471726 S&P500 VRP (1) 01485 02166-34265 360496 S&P500 VRP (2) 01502 03590-56277 576147
RV Table 2: F p F p 27434 00427** 16662 01735 08408 04720 38899 00092** IV 65749 00002** 15669 00000** RV 23329 00740 14831 02190 IV 71026 00001** 15669 00000** IV Engle Granger RV JP = 00008 + 04380 IV JP, RV US = 00014 + 09847 IV US (6396) (30713) ( 6027) (23774) t Augmented Dicky-Fuller -48803-49119 5% ()
HAR(Heteroskedasticity Autoregressive) RSV(Realized Stochastic Volatility) Table 3: MMS RV RV RV RV RV RV RSV-HAR RSV RV HAR RSV-VAR(Vector autoregression) RV VAR RSV RV RSV-VECM(Vector Error RSV RV IV Correction Term) VECM RV DMA Dynamic Model Averaging HAR RSV DMA HAR 60 24
RSV-VECM RV IV RSV RV IV r t = µ + exp(h t /2)ϵ t, ϵ t N(0, 1), h t = ξ + ϕ(h t 1 ξ) + σ ηη t, η t N(0, 1), ( y rv) t yt iv = h 1 N(ξ, ση/(1 2 ϕ 2 )), ( ) ( ) ( ν rv ν iv + α rv y rv ) α iv (β rv β iv t 1 ) yt 1 iv ( ψ rv) ( ψ iv h t 1 + σ rv t ut rv σt iv ut iv + ( γ 11 γ 12 γ 21 ), u rv t, u iv t N(0, 1) t 1 yt 1 iv γ 22 ) ( y rv ) + r t yt rv RV yt iv IV h t r t α β β t y t = ECT t 1 γ y t
RV Figure 1: RV 2001/7 2015/7 RV x y RV RSV-VAR RSV-VECM DMA
Mincer-Zarnowitz RV t = β 0 + β 1 ˆσ t 2 + ϵ t, ( ) 2 MSE(MeanSquaredError) = 1 T RV T t=1 t ˆσ t 2, ( ) 2 HMSE(HeteroskedasticMSE) = 1 T 1 ˆσ2 t, T t=1 RV t MAE(MeanAbsoluteError) = 1 T T t=1 RV t ˆσ t 2, HMAE(HeteroskedasticMAE) = 1 T T t=1 1 ˆσ2 t RV t Table 4: 1 R 2 MSE HMSE MAE HMAE R 2 MSE HMSE MAE HMAE HAR 283% 01387 02209 00017 03922 353% 01709 04172 00013 04470 RSV 257% 01642 03051 00026 04236 353% 01574 04388 00016 06096 RSV-HAR 259% 02496 03272 00020 04779 363% 05925 18624 00019 06605 RSV-VAR 319% 01376 03855 00016 04184 336% 01228 04784 00012 04184 RSV-VECM 323% 00787 02642 00013 03424 404% 01502 05409 00014 04644 DMA 359% 00392 01487 00009 02741 475% 01581 05093 00012 04605
r t,t+h h = a(h) + b(h)vrp t + u t,t+h, h = 1, 2,, 12 Figure 2: ( ) 2001/7 2015/7 VRP (2) 1 12 x y VRP (3) VRP (1) RV
(RV IV ) IV 1 RV 1 RV IV VRP 1 IV 1 4 IV Figure 3: VRP 2001/7 2015/7 x y
( 1 ) Ariel (1987) VRP RV IV VRP (1) 3 6 Figure 4: (VRP (1), 1 23 ) 2001/7 2015/7 VRP (1) x y
( 2 ) Ariel (1987) VRP RV IV VRP (2) 3 12 6 Figure 5: (VRP (2), 1 23 ) 2001/7 2015/7 VRP (2) x y
RV RSV-VECM DMA HAR RSV RV VRP (3) RV VRP (3) VRP (1) VRP RV IV
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