A Bayesian Analysis of a Jump-Diffusion Model with double exponential distribution Senniang Chen Dr. Yu, Major Advisor Department of Statistics, Iowa State University September 8, 2011 1 Outline Introduction The model, MCMC method Simulation results Empirical results Conclusion 2 Introduction Three stylized behaviors: stochastic volatility, leverage effect, and jumps Continuous-time model AJD (Affine Jump-Diffusion) model Let Yt =log(St) is the log price. νt is the stochastic volatility dY dt dB y dJ y t t t t d ( ) dt dB t t t t where(Byt , Bνt )are two standard Brownian motions with Corr(Byt , Bνt )=ρdt 3 Jumps in the return Compound Poisson process Finite activity jumps Advantage: capture the three stylized behaviors Disadvantage: fail to capture many small movements Levy jump process proposed by Li, Wells, and Yu (2008) Infinity activity jumps Advantage: capture many small movements Double exponential jumps proposed by Kou(2002) Finite activity jumps Advantage: more flexible than compound Poisson process Li, H., Wells , M. T., and Yu , C. L., 2008, A Bayesian Analysis of Return Dynamics with Levy Jumps, The Review of Financial Studies,21:2345-2378 Kou, S.G., 2002,A Jump-Diffusion Model for Option Pricing, Management Science, 48:1086-1101 4 Goals of the project Propose a AJD model with double exponential jumps Develop a MCMC method model parameters latent variables Whether our model can capture the many small movements in the real data 5 AJD Model with Double Exponential Jumps First-order Euler discretized version of the model Y Y y J y t t 1 t t 1 t 1 t 1 t ( t ) t t 1 y y ~ N(0,1), ~ N(0,1), Corr( , ) t 1 t 1 t 1 t 1 y I( I 1) I(I 1)( ), t 1 t 1 t 1 t 1 t 1 1, w.p. p1 ~ exp( ) t 1 I 0, w.p. p 0 t 1 ) ~ exp( 1, w.p. p t 1 1 J 6 Eraker, B., Johannes M., and Polson N., 2003, The impact of jumps in equity index volatility and returns, Journal of Finance, 58:1269-1300 The model Observations {Yt}Tt=0 Model parameters Θ={μ, κ, θ, ρ, σν,η+, η-, p1 , p-1 , p0} Latent variables {Xt}Tt=0 volatility {νt}Tt=0 Jump time {It}Tt=0, positive jump size {ξ+t }Tt=0 and negative jump size {ξ-t}Tt=0 Stochastic 7 Challenge of Likelihood Challenge: high dimensional latent variables p( | Y) p(, X | Y)dX Bayes’ rule p(Θ,X|Y)∝ p(Y|X,Θ)× p(X| Θ)× p(Θ) The model is Markov p(, X | Y) t 0 p( y t 1 y t | X t , ) t 0 p(X t 1 X t | ) p() T 1 T 1 8 Markov Chain Monte Carlo Method Clifford-Hammersley theorem p(Θ,X|Y)↔ p(Θ|X,Y) and p(X|Θ,Y) Given the initial values Θ(0) and X (0) Θ(1) ~ p(Θ|X (0),Y) X(1) ~ p(X|Θ (1),Y) … … {Θ(g) , X(g) }Gg=0 The posterior Chain will converge to p(Θ,X|Y) 1 G ˆ (, X̂) g 1 ((g ) , X (g ) ) G 9 MCMC Method in Our Case Joint distribution p(, , I, | Y) t 0 p( y t 1 y t | t , J t , ) t 0 p(( t 1 t ), I t , t | ) p() T 1 T 1 Posterior distributions of parameters and latent variables Given Θ(0) , ν(0), ξ(0) and I(0), we can draw Θ(1) ~ p(Θ| ν (0) , ξ(0), I(0),Y) ν(1) ~ p(V|Θ (1), ξ(0), I(0), Y) I(1) ~ p(J|Θ (1), ν(1), ξ(0), Y) ξ(1) ~ p(J|Θ (1), ν(1), I(1), Y) … … {Θ(g) , ν(g) , I(g) , ξ(g)}Gg=0 ˆ , ˆ , Î, ˆ ) 1 G ((g ) , ( g ) , I( g ) , ( g ) ) ( G g 1 10 Priors of the Model Parameters μ~N(f,F2) θ~N (g,G2)I(θ>0) κ ~N (k,K2)I(κ >0) η+~ IG(α+, β+), and η- ~IG(α-, β-), where IG represents inverse gamma distribution (p1,p0,p-1) ~ Dirichlet(d1,d0,d-1) 11 Priors of the Model Parameters According to Jacquier, Polson, and Rossi (1994) , are reparameterized as , , where , 2 2 (1 ) and the jo int priors are 1 ) | ~ N(0, 2 . ~ Gamma ( w , W ) Jacquier, E., Polson, N., and Rossi, P., 1994, Bayesian Analysis of Stochastic Volatility Models, Journal of Business & Economic Statistics,12:371-389 12 Joint Distribution 1 1 p(, V, I, , Y) exp{ 21 1 T 1 t 0 T 1 t 0 2 t 1 2ty1 t 1 } t I ( I t 1 1) 1 p 2 2 y 2 t 1 p I ( I t 1 0 ) 0 p I ( I t 1 1) 1 t 1 t 1 1 exp{ }p() Y Y y J y t t 1 t t 1 t 1 , AJDDE : t 1 t ( t ) t t 1 y y where ~ N(0,1), ~ N(0,1), Corr ( , ) and t 1 t 1 t 1 t 1 y J I( I 1) I(I 1)( ), t 1 t 1 t 1 t 1 t 1 1, w.p. p1 ~ exp( ) t 1 I 0, w.p. p0 t 1 1, w.p. p t 1 ~ exp( ) 1 13 Joint Distribution p(, V, I, , Y) t 0 1 T 1 t 1 2 exp{ 2 1 y 2 y 2 t 1 t 1 t 1 t 1 } 2 21 T 1 I ( I 1) I ( I 0 ) I ( I 1) t 1 t 1 1 t 1 t 1 t 1 exp{ } p p p p ( ) 1 0 1 t 0 f 2 k 2 f 2 1 p() exp exp exp{ } exp 2 2 2 2F 2F 2K 1 exp{ d1 d 0 d 1 } p 1 p 0 p 1 1 1 exp{ } 2 w 1 exp{ W } 14 Posterior Distribution of μ 2 1 f T 1 y 2 y p( | Θ , V, I, , Y) exp [( t 1 ) 2t 1 t 1 ] 2 t 0 2 2 ( 1 ) 2 F Define C t 1 y t 1 y t J ty1 ; D t 1 t 1 t t ; then, D t 1 C f 1 1 T 1 1 T 1 t 1 2 exp [ ] 2 2 t 0 2 2 t 0 t 2F (1 ) t F 2(1 ) | ~ N( s 1 T 1 , 1 ), s w w (1 2 ) t 0 C t 1 D t 1 t f 1 1 T 1 , w F2 (1 2 ) t 0 t F2 15 Posterior Distribution of θ g 2 2 1 T 1 y p( | , V, I, , Y) exp 2t 1 t 1 t 1 exp 2 t 0 2 2 ( 1 ) 2 G define C t 1 y t 1 y t J ty1 ; D t 1 t 1 t t ; then, 2 1 1 2 1 g T 1 T 1 C t 1 D t 1 exp 2 2 t 0 2 2 2 t 0 2 t 2G t G (1 ) 2(1 ) Thus | ~ N( s , 1 ) I(0,), w w 2 1 g 1 1 T 1 C t 1 D t 1 T 1 s , w (1 2 ) t 0 2 t G2 (1 2 ) t 0 2 t G 2 16 General method of Sampling from a Truncated Distribution For a typical truncated distribution x ~ f(x)×I(a<x<b), we can do sampling following these two steps below. Draw u~ U[0,1] Compute x=F-1(F(a)+(F(b)-F(a))×u), where F(∙) is the cdf of f(x) 17 Sampling from a Truncated Distribution b x~f(x)×I(a<x<b), fx(x)=f(x)/c , a<x<b and c= a f ( x )dx . Suppose u~ U[0,1] and ξ=F-1( F(a)+(F(b)-F(a))*u). Then P ( x ) P(F(a ) (F(b) F(a )) u F( x )) P((F(b) F(a )) u F( x ) F(a )) F( x ) F(a ) ) F(b) F(a ) F( x ) F(a ) F( x ) F(a ) F(b) F(a ) c 1 1 f x ( x ) F' ( x ) f ( x ) c c P(u 18 Posterior Distribution of κ k 2 2 1 T 1 y p( | , V, J, Y) exp 2t 1 t 1 t 1 exp 2 t 0 2 2 ( 1 ) 2 K define C t 1 y t 1 y t J ty1 ; D t 1 t 1 t ; then, 2 1 1 2 1 k T 1 t T 1 C t 1 D t 1 t 2 exp 2 t 0 2 2 2 t 0 2 t 2K t K 2(1 ) (1 ) Thus | ~ N( s , 1 ) I(0,) w w 1 k 1 1 T 1 C t 1 D t 1 T 1 t s , w t (1 2 ) t 0 2 t K2 (1 2 ) t 0 2 t K2 2 19 Posterior Distribution of (ρ,σν) p ( , | ( , ) , J , V , Y ) 1 1 2 2 1 exp T exp{ 1 2 1 2 t 0 ( vt 1 ) 2 2vt 1} T 1 1 w 1 W exp a 2 * 1 ( ) 1 * 1 b b exp exp 2 2 b ~ IG ( * , 1 * ) | ~ N ( a , ) b b 1 T 1 2 a2 where b 2 t 0 C ; a t 0 C t 1D t 1 ; w T / 2; W t 0 D t 1 2 2b y t 1 y t J ty1 t ( t ) C t 1 ; D t 1 t 1 ; t t T 1 2 t 1 T 1 We draw from IG( * , 1 2 2 * * ), then is sampled from N ( a , b * b ) 20 Posterior Distribution of (η+,η-) t 1 1 p( | 5 , J, V, Y) t 0 exp T Thus, T 1 follows IG(T , t 0 t 1 ) T 1 t 1 1 p( | 5 , J, V, Y) t 0 exp T 1 T 1 Thus, exp T 1 1 t 0 t 1 exp ( ) 1 1 1 exp T 1 t 0 t 1 exp ( ) follows IG(T , t 0 t 1 ) T 1 21 Posterior Distribution of (p1,p-1,p0) p(p1 , p 1 , p 0 | 6 , V, J, Y) p1d1 p d0 0 p d11 t 0 p1I ( I t1 1) p 0I ( I t1 0) p I(1I t1 1) T 1 I ( I t 1 1) d 0 I ( I t 1 0 ) d 1 I ( I t 1 1) t 0 p1 p 0 t 0 p 1 t0 d1 T 1 T 1 T 1 Thus (p1 , p 1 , p 0 ) ~ Dirichlet (D1 , D 1 , D 0 ) D1 d1 Tt 01 I(I t 1 1), D 0 d 0 Tt 01 I(I t 1 0), D 1 d 1 Tt 01 I(I t 1 1) To sample (p1 , p 1 , p 0 ), we use following steps: simulate( x1 , x 1 , x 0 ) seperately from Gamma (D1 , C), Gamma (D 1 , C), and Gamma (D 0 , C), where C is any positive number . p1 x0 x1 x 1 , p 1 , p0 x1 x 1 x 0 x1 x 1 x 0 x1 x 1 x 0 Devroye, Luc 1986, Non-Uniform Random Variate Generation, Page 594 22 Posterior Distribution of Jump Times It+1 1 T 1 I ( I t 1 1) I ( I t 1 0 ) I ( I t 1 1) y y 2 p(I t 1 | , V, Y) exp 2 p p0 p 1 t 1 t 1 t 1 1 2 t 0 2(1 ) p1 exp u1 ; i 1 S p exp u 1 p(I t 1 i | ) 1 ; i 1 S p 0 exp u 0 ; i 0 S 1 2 where u1 C 2 , where C y y 1 t 1 1 t 1 t t 1 ; 2 2(1 ) 1 2 where u 0 C 2 0 t 1 , where C 0 y t 1 y t ; 2(1 2 ) 1 2 where u 1 C 2 , where C y y 1 t 1 1 t 1 t t 1 ; 2 2(1 ) where S p1 exp u1 p 0 exp u 0 p 1 exp u 1 23 Posterior Distribution of Jump Sizes p ( t 1 1 T 1 | , V, Y ) exp 2 t 1 ty1 ty1 2 t 0 2(1 ) p ( t 1 1 T 1 y y | , V, Y ) exp 2 t 1 t 1 t 1 2 t 0 2(1 ) The posterior distribution of ξ t+1 2 t 1 exp t 1 exp are following : t 1 and t 1 + ξ t+1and 2 D t 1 t 1 2 C t 1 1 s 1 N ( , ), s , w ; when I t 1 1 2 2 w w t 1 | ~ t 1 t 1 t 1 exp , when I t 1 1 D t 1 t 1 2 C t 1 1 s , 1 ), s N ( , w ; when I t 1 1 w w t 1 | ~ t 1 2 t 1 2 t 1 exp , when I t 1 1 24 C t 1 y t 1 y t and D t 1 t 1 t t Posterior Distribution of Stochastic Volatility νt For 0 t 1 T, the posterior of t 1 is 2y t 1 t 1 t 1 p( t 1 | , Y, J) ~ exp 2 2(1 ) where y t 1 2 y t 1 y t J ty1 ( t ) , t 1 t 1 t t t Jo int distribution : p(, V, I, , Y) t 0 T 1 y 2 2y 1 t 2 t 2 t 2 exp t 2 2 2(1 ) t 1 1 t 1 2 exp{ 2 1 y 2 2 y 2 t 1 t 1 t 1 t 1 } 2 2 1 T 1 I ( I 1) I ( I 0 ) I ( I 1) t 1 t 1 1 t 1 t 1 t 1 exp{ } p p p p ( ) 1 0 1 t 0 25 Posterior Distribution of Stochastic Volatility νt For 0 t 1 T, the posterior of t 1 is 2y t 1 t 1 t 1 p( t 1 | , Y, J) ~ exp 2 2(1 ) where y t 1 2 y 2 2y 1 t 2 t 2 t 2 exp t 2 2 2(1 ) t 1 2 y t 1 y t J ty1 ( t ) , t 1 t 1 t t t 1 and the sec ond exp onential part. 0 For t 1 0, the posterior of 0 depends on For t 1 T, the posterior of T depends on the first exp onential part. 26 Metropolis-Hastings Method To sample θ from f(θ), given θ(t) at tth iteration, θ(t+1) is generated as following: Sample θ* from g(θ|θ(t)) R(θ*, θ(t))= g(θ(t)|θ*)/g(θ*|θ(t)) × f(θ*)/f(θ(t)) ( t 1) * , with prob. (* , ( t ) ) * (t) * (t) (t) , ( , ) min{ R ( , ),1} * (t) , with prob. 1 ( , ) If θ*= θ(t)+N(0,σ2), σ2 >0, then we call it normal random walk sampling 27 Simulation Study Setup To examine whether MCMC method can accurately estimate model parameters and latent variables 50 sample paths with length T=1000 True parameters μ=0.05, θ=0.9, κ=0.25, ρ=0.25, σν=0.4, p0=0.6, p1=0.1, p-1=0.3,η+=10, η-=30 10K iterations for each paths, the estimates of the parameters are the means of the last 5K iterations. 28 Hyperparameters The hyperparameters are listed below μ ~ N(0,1) θ ~ N(0,1)I(θ >0) κ ~ N(0,1)I(κ >0) (p1, p0, p-1) ~ dirichlet(0.5, 0.1, 0.4) η+~IG(1,1) η- ~ IG(1,3) (ρ, σν) is transformed to (ων, φν) 1 ) | ~ N(0, , 2 . 2 2 (1 ) ~ Gamma (1,120) 29 Initial Values From the model, J y (Yt 1 Yt ) t y t 1 t 1 E( J y ) (Yt 1 Yt ) t 1 change of y 50 change of y 0 I (t 0) -50 -100 1, when ( y t 1 y t ) 5 1, when ( y t 1 y t ) 10 0, otherwise -150 -200 0 200 400 600 800 1000 time Figure 1 Plots of the changes of y 30 Initial Values (η+)(0) is the mean of the initial positive jump sizes and (η-)(0) is the mean of the initial negative jump sizes The initials of jump intensities and sizes are estimated as I((I ) 1) (0) t 0 p1 T 1 T (0) I ( ( I t ) 1) (0) t 0 p 1 T 1 T (0) ( 0) t 0 I((I t ) 0) p0 T 1 T (0) t ( t ) ( 0) ( t ) ( 0) y t 1 y t ( 0) , when I t 1 ( 0) exp ( ) , otherwise y t 1 y t ( 0) , when I t 1 ( 0) exp ( ) , otherwise 31 Simulation Results of 50 sample paths μ κ ρ θ συ p1 p-1 p0 η+ η- True 0.05 0.25 -0.4 0.9 0.25 0.1 0.6 0.3 10 30 Monte Carlo Mean 0.0406 0.257 -0.297 1.194 0.297 0.0932 0.613 0.293 10.515 30.678 RMSE 0.0357 0.0646 0.1711 0.469 0.0913 0.0177 0.0290 0.0185 1.713 2.155 Monte Carlo STD 0.0345 0.0642 0.1370 0.366 0.0727 0.0163 0.0171 0.0257 1.634 2.046 Table 1 This table reports simulation results on the accuracy of MCMC estimators of model parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η- } for 50 sample paths 32 Simulation Results of one sample path μ κ ρ θ συ p1 p-1 p0 η+ η- True 0.05 0.25 -0.4 0.9 0.25 0.1 0.6 0.3 10 30 Initial 0.0181 0.0784 -0.3906 2.0525 0.3024 0.0560 0.7300 0.2150 18.7409 38.7680 Mean 0.0566 0.2146 -0.1454 0.8452 0.2404 0.0856 0.5988 0.3157 13.1253 27.9566 RMSE 0.0392 0.0511 0.2871 0.0716 0.0204 0.0177 0.0173 0.0224 3.4746 2.6295 Std 0.0387 0.0368 0.1326 0.0556 0.0180 0.0103 0.0173 0.0159 1.5186 1.6551 Table 2 This table reports simulation results on the accuracy of MCMC estimators of model parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η-} for the one sample path. 33 chain plot of , initial=0.0784, mean=0.2146, std=0.0368, RMSE=0.0511 chain plots of , initial=0.0181, mean=0.0566, std=0.0287, RMSE=0.0392 chain plot of , initial=-.03906, mean=-0.1454, std=0.1326, RMSE=0.2871 0.8 0.3 2.5 0.6 0.1 0.4 -0.2 1.5 2 0 0 2000 4000 6000 iteration time 8000 10000 -0.4 0.5 x 0 -0.6 0 2000 chain plot of ,initial=0.3024,mean=0.2404, 8000 10000 x x 0.3 0.7 0.2 0.6 p1 p0 0.25 0.5 0.15 0.4 4000 6000 iteration time 8000 8000 10000 0 2000 4000 6000 iteration time 8000 0 10000 0.2 0.4 x0 4000 6000 iteration time 8000 x 30 30 + 40 0.3 x 0.2 0.1 2000 4000 6000 iteration time 8000 10000 0 2000 4000 6000 iteration time 8000 x 20 10 10 0 2000 4000 6000 iteration time 8000 10000 0 0 2000 4000 6000 iteration time 8000 10000 std=0.0159,RMSE=0.0224 0.5 0.05 2000 chain plot of p-1, initial=0.2150,mean=0.3157, 0.25 0.15 0 chain plot of +, initial=18.7409,mean=13.1253, std=1.5186,RMSE=3.4746 40 0 4000 6000 iteration time 0.1 10000 chain plot of -,initial=38.768,mean=27.9566, std=0.16551,RMSE=2.6295 20 2000 chain plot of p1,initial=0.0560,mean=0.0856, std=0.0103,RMSE=0.0177 0.8 2000 0 std=0.0173, RMSE=0.0173 0.35 - 4000 6000 iteration time chain plot of p0, initial=0.73,mean=0.5988, std=0.0103, RMSE=0.0177 0 1 x 0.2 -0.1 0.1 x 0 p-1 x 0.2 -0.2 chain plot of , initial=2.0525,mean=0.8452, std=0.0556, RMSE=0.0716 0.2 10000 Figure 2 Chain plots of the model parameters for one simulation path. 34 10000 volatility comparison of the true volatility and the estiamated volatility 4 true volatility the estimated volatility 3 2 1 0 0 200 400 600 800 1000 time Figure 3 Comparison of the true volatility variables and the mean of the last 100 iterations of estimated volatility comparison of the true positive jumps and the estimated jumps 50 true positive jumps estimated positive jumps -50 negative jump postive jump 40 comparison of the true negative jumps and the estimated negative jumps 0 30 20 -100 -150 true negative jumps estimated negative jumps 10 0 -200 0 200 400 600 time 800 1000 0 200 400 600 800 1000 time Figure 4 Comparison of the true jumps and the mean of the last 100 iterations of estimated jumps 35 Level of the S&P 500 Index from 01/02/1990 to 12/31/2010 S&P 500 Index 800 750 700 650 600 1990 1995 2000 time 2005 2010 log change of S&P 500 Index Empirical study log change of the S&P 500 Index from 01/02/1990 to 12/31/2010 15 10 5 0 -5 -10 1990 1995 2000 time 2005 2010 Figure 5 Level and log changes of the S&P 500 index from January 2, 1990, to December 31, 2010. mean volatility 0.0236 1.1707 skewness kurtosis min max -0.1987 11.8688 -9.4695 10.9572 Table 3 This table summarizes statistics of the continuously compounded returns of the daily return S&P 500 index return form 01/02/1990 to 12/31/2010 36 Initial Values μ(0)=0.05, θ(0)=1.8, κ(0)=0.05,ρ(0)=-0.4, and (σν) (0) =0.1 (νt) (0) =VIX2/252, for t=1,2,…,T From Figure 5, we have I (t 0) 1, when ( y t 1 y t ) 2 1, when ( y t 1 y t ) 2.5 0, otherwise The other initials are defined the same way as simulation studies. 37 Empirical Studies μ κ ρ θ συ p1 p-1 p0 η+ η- Initial 0.05 0.05 -0.4 1.8 0.1 0.0349 0.0221 0.953 3.000 3.731 Mean 0.0328 0.0734 -0.4022 1.6494 0.297 3.828*10-4 3.967*10-4 0.9992 1.791 3.573 Std 0.0114 0.006 0.0473 0.0727 5.912*10-4 7.989*10-4 0.001 0.0754 0.814 2.663 Table 4This table reports simulation results on the accuracy of MCMC estimators of model parameters Θ={μ, κ, ρ, θ, συ, p1 , p-1 , p0, η+, η-} for the empirical studies. 38 chain plot of , mean=0.0734, standard error=0.006 0.03 p-1 0.08 0.06 0.02 0.04 0.01 0.02 0 0 1 2 3 iteration time 4 5 0 1 4 2 3 iteration time 4 15 10 5 0 5 4 0 1 x 10 x 10 chain plot of , mean=1.6494, standard error=0.0754 3.5 20 postive jump size + 0.1 0 chain plot of +,mean=1.791,std=0.814 chain plot of p-1, mean=3.9670*10-4, standard error=7.9670*10-4 0.04 0.12 2 3 iteration time 4 5 4 x 10 chain plot of -,mean=3.573, std=2.664 chain plot of p0, mean=0.9992, standard error=0.0010 80 1 negative jump size - 3 0.98 p0 2.5 2 0.96 1.5 0.94 1 0 1 2 3 iteration time 4 0 1 5 2 3 iteration time 4 60 40 20 5 0 4 x 10 0 1 4 x 10 2 3 iteration time 4 5 4 x 10 chain plot of , mean=0.0328,standard error= 0.0114 0.15 chain plot of , mean=-0.4022, standard error=0.0473 chain plot of p1, mean=3.8288*10-4,standard error=5.9115*10-4 0.05 0.1 0 -0.1 0.04 -0.2 0.05 0 0.02 -0.4 0.01 -0.05 -0.1 -0.3 p1 0.03 0 0 1 2 3 iteration time 4 5 4 x 10 chain plot of , mean=0.3730,standard error=0.0120 -0.5 0 1 2 3 iteration time 4 5 4 x 10 -0.6 -0.7 0 1 2 3 iteration time 4 5 4 x 10 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 iteration time x 10 Figure 6 Chain plots of the model parameters for the empirical studies. The mean and the RMSE of the model parameters are the mean and RMSE of the last 20K iterations. 4 39 30 comparision of the implied volatility and the estimated volatility implied volatility the estimated volatility volatility t 25 20 15 10 5 0 1990 1995 2000 time 2005 2010 Figure 7 Comparison of the true volatility variables and the estimated volatility the positive jumps the estimated negative jumps 1.5 0 positive jumps negative jumps -0.5 -1 -1.5 -2 1 0.5 -2.5 -3 1990 1995 2000 time 2005 2010 0 1990 1995 2000 time 2005 2010 Figure 8 The estimated positive and negative jumps 40 Ability to Capture Many Small Jumps ˆ Ĵ ty1 y t 1 y t ˆ t The model residuals ˆ model is correctly specified. y t 1 follow N(0,1) if the kernel density plots 0.7 kernel density estimates of the residuals N(0,1) 0.6 0.5 0.4 0.3 0.2 0.1 0 -4 -3 -2 -1 0 1 2 3 4 5 Figure 9 Kernel density plot and QQ plot of the residuals 41 Conclusions MCMC methods have very good performance in estimating model parameters and high dimensional latent variables AJD model with double exponential jumps is inadequate of modeling the many small jumps in S&P 500 data 42 Thank you! 43