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1 2016/3/11

2 Realized Volatility RV 1 RV 1 Implied Volatility IV Volatility Risk Premium VRP

3 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

4 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 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

5 , RV 1996/1 2015/8, RV Man Institute Realized Library IV VIX VI 225 Table 1: RV IV Panel A: IV( ) and RV( ) RV VI S&P S&P500RV VIX Panel B: VRP( ) 225 VRP (1) VRP (2) S&P500 VRP (1) S&P500 VRP (2)

6 RV Table 2: F p F p ** ** IV ** ** RV IV ** ** IV Engle Granger RV JP = IV JP, RV US = IV US (6396) (30713) ( 6027) (23774) t Augmented Dicky-Fuller % ()

7 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

8 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

9 RV Figure 1: RV 2001/7 2015/7 RV x y RV RSV-VAR RSV-VECM DMA

10 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% % RSV 257% % RSV-HAR 259% % RSV-VAR 319% % RSV-VECM 323% % DMA 359% %

11 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

12 (RV IV ) IV 1 RV 1 RV IV VRP 1 IV 1 4 IV Figure 3: VRP 2001/7 2015/7 x y

13 ( 1 ) Ariel (1987) VRP RV IV VRP (1) 3 6 Figure 4: (VRP (1), 1 23 ) 2001/7 2015/7 VRP (1) x y

14 ( 2 ) Ariel (1987) VRP RV IV VRP (2) Figure 5: (VRP (2), 1 23 ) 2001/7 2015/7 VRP (2) x y

15 RV RSV-VECM DMA HAR RSV RV VRP (3) RV VRP (3) VRP (1) VRP RV IV

16 [1] Ariel, R A (1987) A monthly effect in stock returns, Journal of Financial Economics, Vol 18, No 1, pp [2] Bakshi, G and N Kapadia (2003) Delta-hedged gains and negative market volatility risk premium, The Review of Financial Studies, Vol 16, No 2, pp [3] Bansal, R and A Yaron (2004) Risks for the long run: A potential resolution of asset pricing puzzles, The Journal of Finance, Vol 59, No 4, pp [4] Bollerslev, T, D Osterrieder, N Sizova, and G Tauchen (2013) Risk and return: Long-run relations, fractional cointegration, and return predictability, Journal of Financial Economics, Vol 108, No 2, pp [5] Bollerslev, T and H Zhou (2006) Volatility puzzles: A simple framework for gauging return-volatility regressions, Journal of Econometrics, Vol 131, No 1, pp [6] Ng, A (2000) Volatility spillover effects from Japan and the US to the Pacific Basin, Journal of international money and finance, Vol 19, No 2, pp [7] Raftery, A E, T Gneiting, F Balabdaoui, and M Polakowski (2005) Using bayesian model averaging to calibrate forecast ensembles, Monthly Weather Review, Vol 133, No 5, pp [8] Takahashi, M, Y Omori, and T Watanabe (2009) Estimating stochastic volatility models using daily returns and realized volatility simultaneously, Computational Statistics & Data Analysis, Vol 53, No 6, pp [9] Ubukata, M and T Watanabe (2011) Market variance risk premiums in Japan as predictor variables and indicators of risk aversion, Working Paper, Hitotsubashi University

03.Œk’ì

03.Œk’ì HRS KG NG-HRS NG-KG AIC Fama 1965 Mandelbrot Blattberg Gonedes t t Kariya, et. al. Nagahara ARCH EngleGARCH Bollerslev EGARCH Nelson GARCH Heynen, et. al. r n r n =σ n w n logσ n =α +βlogσ n 1 + v n w