Multivariate Realized Stochastic Volatility Models with Dynamic Correlation and Skew Distribution: Bayesian Analysis and Application to Risk Managemen

Size: px

Start display at page:

Download "Multivariate Realized Stochastic Volatility Models with Dynamic Correlation and Skew Distribution: Bayesian Analysis and Application to Risk Managemen"

なおももき
4 years ago
Views:

1 Multivariate Realized Stochastic Volatility Models with Dynamic Correlation and Skew Distribution: Bayesian Analysis and Application to Risk Management Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

2 , Stochastic Volatility (MSV ), ( ), (skew )., ( ), Bayesian (MCMC, ), MSV. MSV, Dow 5,,, Realized Volatility,, MSV. 1,. [Issue 1] 2, forward-looking. [Issue 2] 3. [Issue 3] Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

3 Skew : Skew t Skew t, Azzalini and Capitanio [2] skew. Skew t k 1 z R k PDF f z : Skew t PDF z Skt(ξ, Ω, α, ν) f z (z; ξ, Ω, α, ν) = 2ψ(z; ν)ψ ( ( ) 1 ) α (diag(ω)) 1 ν + k 2 2 (z ξ) ; ν + k q z + ν q z = (z ξ) Ω 1 (z ξ) α ( R k ): skew. α = 0, Skew t t. ν ( R >0 ): t. ψ( ; ν): ν t PDF. Ψ( ; ν): ν t CDF. ξ ( R k ): location. Ω ( R k k ):. (1) Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

4 Skew : GH Skew t Generalized Hyperbolic (GH) Skew t, Aas [1] skew., (GIG) (Normal variance-mean mixture). GH Skew t z GHS(ξ, Ω, β, ν, δ) z = ξ + vβ + vlw v GIG ( ν ) 2, δ, 0 w N(0, I) β ( R k ): skew. β = 0, GH Skew t t ( non-central & scaled). δ ( R >0 ): scale 1. L ( R k k ): s.t. Ω = LL. (2) 1 Nakajima [8],, δ = ν, v GIG( ν/2, ν, 0) d = IG(ν/2, ν/2) (IG(, ): ). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

5 MSV : Cholesky MSV (CSV) CSV (Lopes et al. [7]), t y t ( R k ) Σ t Cholesky A t, H t (s.t. Σ t = A 1 t H t (A t) 1 ). CSV y t = µ + A 1 t H 1 2 t e t (i.e. y t N(µ, Σ t )) H 1 2 t = diag(exp( h 1,t 2 ),..., exp(h k,t 2 )) a..... A t = 21,t a k1,t... a kk 1,t 1 (3) volatility, AR(1) : Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

6 MSV : Cholesky MSV (CSV) (cont.) CSV (cont.) h t+1 = µ (h) + Φ (h) (h t µ (h) ) + η t a t+1 = µ (a) + Φ (a) (a t µ (a) ) + ξ t e t I Q 0 η t N 0, Q S V ξ t h t = (h 1,t,..., h k,t ) a t = (ã 1,t,..., ã p,t ) = (a 21,t,..., a kk 1,t ) (4) p:. Φ ( ), Q, S, V 2. CSV, Σ t,, (k + p ). 2 Q = diag({ρ i σ i }), S = diag({σi 2}), Φ ( ) = diag({ϕ ( )i }), V = diag({υi 2 }) (i = 1,..., k) ρ i i leverage. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

7 MSV : Wishart MSV (WSV) Σ t,, Wishart (W (ν, Σ), Σ R k k ). Wishart scale Σ t, WASP (Jin & Maheu [5] ). WASP y t N(µ, Σ t ) Σ t W (ν, Σ t 1 ) ν Σ t = B (0) + B (1) Σ t (5) B (0) ( R k k ):. B (1) ( R k k ): rank-1 (B (1) = bb, b R k ). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

8 MSV : skew Nakajima [8], MSV skew : CSV : y t = µ + A 1 t H 1 2 t z t WASP : y t = µ + L t z t (s.t. L t L t = Σ t ) skew z t, skew. z t Skt(m (st), I, α, ν (st) ) or z t GHS(m (ghs), I, β, ν (ghs), ν (ghs) ) 3. 3 m (st), m (ghs), E[z t] = 0. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

9 MSV : Realized Volatility MSV volatility /, volatility Realized Volatility (RV), MSV : RV t = ψ (RV ) + h t + u (RV )t u (RV )t N(0, σ (RV ) I) RC t = ψ (RC) + a t + u (RC)t u (RC)t N(0, σ (RC) I) (6) ψ ( ), u ( )t ( R k or R p ):. RV t ( R k ), RC t ( R p ): Barndorff et al. [3] Realized Kernel RK t ( R k k ) Cholesky L t (s.t. L t L t = RK t ),. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

10 :, Dow 5 4 ( :%) MSV. y i,t = 100 log(p i,t/p i,t 1). p i,t: i (i {1,..., 5}), t (t {1,..., T }). y t, RV t, RC t 5 T = 1, 000, 2. (i) (ii) Table: {y i,t }(%) (i) i asset mean sd skewness kurtosis 1 JPM IBM MSFT XOM AA Table: {y i,t }(%) (ii) i asset mean sd skewness kurtosis 1 JPM IBM MSFT XOM AA JP Morgan (JPM), IBM (IBM), Microsoft (MSFT), Exxon Mobil (XOM) and American Express (AA). 5 Realized Kernel, (i), Oxford-Man Institute s Realized Library website ( (ii),. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

11 : MCMC MSV., Stan 6 Hamiltonian Monte Carlo p(θ {y t}) 10,000, 5, : CSV µ( )i N(0, 100); (1/2)(ϕ ( )i + 1) Beta(20, 1.5). σ 2 ( )i Gamma(20, 0.01); υ 2 ( )i Gamma(20, 0.01). WASP B(0) W (ν, I); b i N(0, 100) (b i b) 8. ν Gamma(2, 1). αi N(0, 1); β i N(0, 1) ν( ) Gamma(2, 1).,, skew, leverage (ρ) (µ) 0. 6 Stan Development Team , identification, b1 Gamma(2, 1). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

12 : - WAIC Table: WAIC (i) (ii) # model MSV Skew RV WAIC S.E.(WAIC) WAIC S.E.(WAIC) 1 CSV CSV no no RCSV CSV no yes RCSV-Skt CSV Skew t yes RCSV-GHS CSV GHS yes RWASP WASP no yes RWASP-Skt WASP Skew t yes RWASP-GHS WASP GHS yes RCSV Skew t,. RWASP, Skew t, GH Skew t skew. RV, CSV RCSV, ( ), 9. 9, RV, RV RV, WAIC. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

13 : - WASP, CSV, B (0), B (1) 10,. skew (α, β), RCSV-GHS 11, RCSV-Skt, RWASP-Skt, RWASP-GHS, skew., CSV,, 1 (y 1,t), skew.,, WSV., RV CSV ; RV, (ψ (RV ) )., (ϕ ( ) ) 0.9 1, %. 11 GH Skew t, skew. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

: - (Σ t), RWASP-Skt,,.,, ([Issue 1] )., RCSV-Skt,,, WSV. 1.0 1.0 0.5 0.5 corr 0.0 corr 0.0-0.

0 2006 2007 2008 2009 2010 date 2014 2015 2016 2017 2018 date model historical (20-day)

14 : - (Σ t), RWASP-Skt,,.,, ([Issue 1] )., RCSV-Skt,,, WSV corr 0.0 corr date date model historical (20-day) historical (60-day) RCSV-Skt RWASP-Skt Figure: (asset 1 & 2) Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

15 : Bayesian MCMC, MSV {y t} T +N f t=t +1 (N f : ),. : p({y t} T +N f t=t +1, θ {yt}t t=1). m = 1 M, Bayesian M N f {ŷ t} MSV A : ( ) Step 1: MCMC p {y t} Tm+N f t=tm+1, θ (A) {y t} Tm. t=(m 1)N f +1 (T m+1 = T m + N f, θ (A) : MSV A.) Step 2: {ŷ t} := {E[y t]} Tm+N f t=tm+1.,, 1. 2 A, forward-looking. 3., ([Issue 2] ). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

: Bayesian (cont.) (i), N f = 5, M = 100 12. C-CSV-GHS, (A t A in (3)).. Table: { 1 k i ŷi,t}, VaR ES model mean sd skewness kurtosis 99% VaR 97.5% ES C-CSV-GHS 0.00 0.07 0.05 3.12-0.17-0.16 CSV 0.

16 : Bayesian (cont.) (i), N f = 5, M = C-CSV-GHS, (A t A in (3)).. Table: { 1 k i ŷi,t}, VaR ES model mean sd skewness kurtosis 99% VaR 97.5% ES C-CSV-GHS CSV CSV-GHS model count C-CSV-GHS CSV CSV-GHS value Figure: { 1 k i ŷi,t} 12,, MCMC (VI). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

17 : Bayesian {y t}, {y (obs)s }, {y (mis)u }, Bayesian,, ([Issue 3] ). : p({y (mis)u }, θ {y (obs)s })., i, y i,t (t = 1,..., T ) p mis (i)(ii), Bayesian., MSV A, p({y i,u}, θ (A) {y i,t}, {y i,s}). p mis = 0.3,, ( seed ) 15,, 10, , I-CSV ( SV ), C-CSV ( CSV ), CSV, AR(1). Table: Mean Squared Errors (asset 2 ) asset model MSE S.E.(MSE) mean ˆk ˆk < 0.7 proportion 2 AR(1) I-CSV C-CSV CSV , MCMC, VI. VI, Yao et al. [10], ˆk (ˆk < 0.7, VI ). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

18 , 2 MSV (CSV WSV ), skew (Skew t GH Skew t ). skew MSV, WAIC, skew. CSV,, idiosyncratic skew, WSV., WSV,, CSV., MSV, Bayesian,,. Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

19 [1] Kjersti Aas and Ingrid Hobæk Haff. The generalized hyperbolic skew Student s t-distribution. In: Journal of Financial Econometrics 4.2 (2006), pp [2] Adelchi Azzalini and Antonella Capitanio. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution. In: Journal of the Royal Statistical Society: Series B (Statistical Methodology) 65.2 (2003), pp [3] Ole E Barndorff-Nielsen et al. Multivariate realised kernels: consistent positive semi-definite estimators of the covariation of equity prices with noise and non-synchronous trading. In: Journal of Econometrics (2011), pp [4] Andrew Gelman et al. Bayesian Data Analysis. Chapman and Hall/CRC, [5] Xin Jin and John M Maheu. Modeling realized covariances and returns. In: Journal of Financial Econometrics 11.2 (2012), pp [6] Alp Kucukelbir et al. Automatic Differentiation Variational Inference. In: The Journal of Machine Learning Research 18.1 (2017), pp [7] HF Lopes, RE McCulloch, and R Tsay. Cholesky stochastic volatility models for high-dimensional time series. In: Discussion papers (2012). [8] Jouchi Nakajima. Bayesian analysis of multivariate stochastic volatility with skew return distribution. In: Econometric Reviews 36.5 (2017), pp [9] Mike West and Jeff Harrison. Bayesian forecasting and dynamic models. Springer Science & Business Media, [10] Yuling Yao et al. Yes, but Did It Work?: Evaluating Variational Inference. In: arxiv preprint arxiv: (2018). Dai Yamashita (Hitotsubashi ICS) MSV Models w/ DC & Skew: Bayesian Analysis & App. to Risk Management March 15, / 19

fiúŁÄ”s‘ê‡ÌŁª”U…−…X…N…v…„…~…A…•‡Ì ”s‘ê™´›ß…−…^†[…fiŠ‚ª›Âfl’«

fiúŁÄ”s‘ê‡ÌŁª”U…−…X…N…v…„…~…A…•‡Ì ”s‘ê™´›ß…−…^†[…fiŠ‚ª›Âfl’« 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