Estimating Correlated Jumps and Stochastic Volatilities

Transcription

1 Insttute of Economc Studes, Faculty of Socal Scences Charles Unversty n Prague Estmatng Correlated umps and Stochastc Volatltes ří Wtzany IES Workng Paper: 35/ Electronc copy avalable at:

2 Insttute of Economc Studes, Faculty of Socal Scences, Charles Unversty n Prague [UK FSV IES] Opletalova 6 CZ-, Prague E-mal : es@fsv.cun.cz Insttut ekonomckých studí Fakulta socálních věd Unverzta Karlova v Praze Opletalova 6 Praha E-mal : es@fsv.cun.cz Dsclamer: The IES Workng Papers s an onlne paper seres for works by the faculty and students of the Insttute of Economc Studes, Faculty of Socal Scences, Charles Unversty n Prague, Czech Republc. The papers are peer revewed, but they are not edted or formatted by the edtors. The vews expressed n documents served by ths ste do not reflect the vews of the IES or any other Charles Unversty Department. They are the sole property of the respectve authors. Addtonal nfo at: es@fsv.cun.cz Copyrght Notce: Although all documents publshed by the IES are provded wthout charge, they are lcensed for personal, academc or educatonal use. All rghts are reserved by the authors. Ctatons: All references to documents served by ths ste must be approprately cted. Bblographc nformaton: Wtzany,. (). Estmatng Correlated umps and Stochastc Volatltes IES Workng Paper 35/. IES FSV. Charles Unversty. Ths paper can be downloaded at: Electronc copy avalable at:

3 Estmatng Correlated umps and Stochastc Volatltes ří Wtzany* *Unversty of Economcs, Prague E-mal: November Abstract: We formulate a bvarate stochastc volatlty jump-dffuson model wth correlated jumps and volatltes. An MCMC Metropols-Hastngs samplng algorthm s proposed to estmate the model s parameters and latent state varables (jumps and stochastc volatltes) gven observed returns. The methodology s successfully tested on several artfcally generated bvarate tme seres and then on the two most mportant Czech domestc fnancal market tme seres of the FX (CZK/EUR) and stock (PX ndex) returns. Four bvarate models wth and wthout jumps and/or stochastc volatlty are compared usng the devance nformaton crteron (DIC) confrmng mportance of ncorporaton of jumps and stochastc volatlty nto the model. Keywords: jump-dffuson, stochastc volatlty, MCMC, Value at Rsk, Monte Carlo EL: C, C5, G Acknowledgements: The research has been supported by the Czech Scence Foundaton grant no. 4/9/73 Market Rsk and Fnancal Dervatves

4 . Introducton A number of emprcal studes confrmed that fnancal asset returns are not normal and exhbt fat tals (leptokurtc dstrbuton). Many models gong beyond the standard geometrcal Brownan dffuson model have been proposed n order to accommodate the emprcal facts. The most promnent are jump-dffuson models (see e.g. Cont, Tankov, 4 for a revew), models wth stochastc volatlty (see Shephard, 4 for selected papers), or models combnng both features,.e. jump-dffuson models wth stochastc volatlty, or even models wth jumps n volatlty. Modelng of portfolo returns or valuaton of varous multasset dervatves requres generalzng of the models nto multvarate settng. We are gong to consder a bvarate jump-dffuson model wth stochastc volatltes ncorporatng possble correlaton of jump occurrence, jump sze, and of the stochastc volatltes. The man goal of the paper s to propose an MCMC estmaton method that wll be tested on artfcal and real world data. We have chosen the two most mportant Czech fnancal markets tme seres, namely a seres of FX (CZK/EUR) exchange rates and of the stock market (PX ndex) returns. ont modelng of the two seres mght be mportant for and an asset manager exposed to the Czech stock market and the exchange rate, or n case of certan dervatves (e.g. quanto) modelng. In both cases, the goal s to model the dstrbuton of future returns of a portfolo exposed to both factors. For example, nspectng the development of the markets n 4- (Fgure 4 - Fgure 6) t seems that the returns of the tme seres exhbt many jumps and perods of low volatlty that are followed by perods of hgh volatlty or vce versa. Although the sample correlaton of returns turns out to be almost zero, the volatltes appear to move n the same drecton (Fgure 6). The questons are: How do the two markets jump and how volatle are the stochastc volatltes? Is the non-gaussan behavor explaned by jumps or rather by stochastc volatlty? Are the jumps n the two markets correlated n terms of occurrence and sze? And moreover, are the stochastc volatltes correlated? Our estmaton methodology s based on the MCMC (Markov Chan Monte Carlo) approach followng acquer et al. (7) and ohannes, Polson (9). The frst break-through applcaton of the Bayesan methods for the analyss of stochastc volatlty models has been made n acquer et al. (994). The authors appled the MCMC algorthm to estmate

5 parameters as well the latent states of the stochastc volatlty model on the US stock return data. The estmaton method s shown to outperform other known estmaton approaches, as the Method of Moments or the Quas-Maxmum Lkelhood Estmator. Snce then extensve research on applcaton of Bayesan methods on stochastc volatlty models have appeared (see e.g. Shephard, 4). ohannes, Kumar (999) estmate state dependent jump models (on US stock data) n whch arrval ntensty and jump szes depend on a gven state varable ncludng lagged jumps. Eraker et al. (3) examne stochastc volatlty models ncorporatng jumps n returns and volatlty usng US stock ndces returns. Eraker (4) utlzes n addton stock ndex opton date and allows the dffuson and volatlty processes beng correlated. The contrbuton of ths paper s n specfcaton and estmaton of a bvarate model wth correlated stochastc volatltes and jumps. ohannes, Polson (9) consder a multvarate verson of Merton s jump-dffuson model where jumps occur at the same tmes for all processes and the jump szes have a multvarate dstrbuton. Inspectng the tmes of probable jumps from the MCMC estmaton of two unvarate jump-dffuson models appled to the two consdered return seres we have noted that the jump tmes overlap only partally. So, n our specfcaton we have two correlated Posson processes and correlated jump szes (f the jumps occur at the same tme). Regardng the correlaton of stochastc volatltes we may agan frstly nspect the mean stochastc volatltes gven by the MCMC algorthm appled to the two processes separately. Snce the stochastc volatlty resduals do not ndcate any sgnfcant correlaton but show a strong correlaton n levels of the stochastc volatltes we propose a bvarate stochastc volatlty jump-dffuson model wth possble Granger causalty between the stochastc volatltes as our full model specfcaton (see Asa et al., 6 or Yu, Meyer, 6 for an overvew of multvarate stochastc volatlty models). The computatonally dffcult estmaton of the model s based on our generalzaton of the method proposed frstly n acquer et al. (994). The proposed bvarate jump-dffuson wth stochastc volatltes wll be frstly tested on artfcally generated data. The goal of the test s to demonstrate that the estmaton procedure yelds acceptable results wth respect to generatng parameters, n partcular that t s able to dentfy exstence of jumps, stochastc volatltes, and ther correlatons. The methodology wll be fnally appled to estmate and compare four bvarate models on the FX and stock returns seres, specfcally: the ordnary dffuson model, the jump-dffuson model, the dffuson model wth stochastc volatlty, and the jump-dffuson model wth stochastc volatlty. Performance of the models wll be compared usng the devance nformaton crteron (DIC) generalzng accordng to Spegelhalter et al () the Akake nformaton crteron (AIC) and that s applcable to models wth a large number of latent state varables. Importance of the choce of an approprate model wll be llustrated calculatng VaR for varous tme horzons and confdence levels.

6 . Methodology In ths secton we are gong to gve a bref overvew of the relevant stochastc models. We also outlne the key elements of the MCMC methods and ther mplementaton n case of the stochastc models under consderaton. Fnally we shortly descrbe the classcal and Bayesan VaR estmaton methodology.. Stochastc Asset Prce Models The most tradtonal (related to the Black-Scholes formula) contnuous-tme fnancal model s the geometrc Brownan moton descrbed by the stochastc dfferental equaton (SDE) () ds Sdt Sdz where St () s an asset prce, ts drft, ts volatlty, and dz the Wener process ncrement (see e.g. Shreve, 4). The equaton can be smplfed applyng the Ito s lemma nto a generalzed Wener process equaton () log / d S dtdz. The left hand sde of () can be nterpreted as the log return over a tme nterval of the length dt. In practce, a tme (Euler) dscretzaton s used n order estmate the parameters from an observed fnancal tme seres or n order to generate future returns. In case of the equaton () the Euler dscretzaton takes the smple form (3) r ň, ň ~ N(,), where r ln S / S s the log return over a regular tme nterval of length t, / t, and t. Accordng to (3), observed returns should have a normal dstrbuton. However, many studes demonstrate that the returns have, typcally, a leptokurtc dstrbuton, n partcular fat tals. Consequently, the geometrc Brownan moton s proposed to be generalzed n varous drectons, n partcular, allowng for jumps (see Cont, Tankov, 4) and stochastc volatlty (see Shephard, 4). The jump-dffuson SDE can be wrtten n the log-return form as d log S ( / ) dtdz d, (4) where the jump term, d ZdN, has a normally dstrbuted N(, ) jump-sze component and a component gven by the Posson countng process N wth ntensty. Essentally, ths component adds a mass to the tals of the returns dstrbuton. We consder a tme dscretzaton where at most one jump can happen over a sngle tme step: (5) r ň Z ň ~ N(,), Z ~ N(, ), ~ Bern( ). Accordng to Eraker et al (3) ths assumpton does not ntroduce any bas n the parameter estmates. 3

7 Stochastc volatlty models allow varance V or log-varance to evolve accordng to an SDE,.e. the constant volatlty n () or (4) s replaced by the stochastc volatlty V. For example the Heston s (993) model sets dv ( Vd ) t V VdzV. Followng acquer et al (994) and ohannes, Polson (9) wll we rather consder the logvarance SDE: (6) dl g V ( logvdt ) dz. o V V In the dscrete settng, wth h logv, the equaton takes the form of AR() model: ( ) V, V h h (,) h t tň ň N. V Rearrangng the terms the equaton can be wrtten as V (7) h h ň, where t, t, and V t. If two or more fnancal assets s to be model than possble correlatons needs to be consdered. For example f S and S are two asset prces followng () wth approprate ndexes we have to admt possblty of correlated dz and dz. If the prces follow the jumpdffuson process (4) then we have to admt a correlaton between dand d. Fnally, n V V case of stochastc volatlty (6) we mght consder a correlaton between dz and dz, but V V also a mutual correlaton between dz and dz, and the correlaton between dz and dz. Due to the mean-revertng form of (6) we should also consder a possble correlaton between logv and logv that could be captured ntroducng lnv nto the SDE for lnv and vce versa. A nonzero coeffcent s then nterpreted as Granger causalty form one asset varance to another. An analyst that needs to model the dstrbuton of future returns of a portfolo or of a dervatve payoff dependng on two or even more assets stands n front of a dffcult task: to choose an optmal model and at the same tme to estmate n a feasble way ts parameters usng hstorcal or currently observable data. Due to ncreasng complexty of the models we wll focus on the MCMC Bayesan estmaton and model comparson approach.. Markov Chan Monte Carlo (MCMC) The Bayesan MCMC samplng algorthm has become a strong and frequently used tool to estmate complex models wth multdmensonal parameter vectors, ncludng latent state varables. Examples are fnancal stochastc models wth jumps, stochastc volatlty processes, models wth complex correlaton structure, or swtchng-regme processes. For a more complete treatment of MCMC methods and applcatons we refer reader for example to ohannes, Polson (9), Rachev et al. (4), or Lynch (). 4

8 MCMC provdes a method of samplng from multvarate denstes that are not easy to sample from drectly, by breakng these denstes down nto more manageable unvarate or lower dmensonal multvarate denstes. To estmate a vector of unknown parameters,..., k from a gven dataset, where we are able to wrte down the Bayesan margnal p, j, data p data, the MCMC but not the multvarate densty denstes j Gbbs sampler works accordng to the followng generc procedure:. Assgn a vector of ntal values to,...,. Set j j. j j j. Sample p(,..., k,data). j j j j 3. Sample p(, 3,..., k, data). and set j. k j j j j k+. Sample p(,,...,, data) and return to step. k k k Accordng to the Clfford-Hammersley theorem the condtonal dstrbutons p, j, data j fully characterze the jont dstrbuton data p and moreover under mld condtons the Gbbs sampler dstrbuton converges to the target jont dstrbuton (ohannes, Polson, 3). The condtonal probabltes are typcally obtaned applyng the Bayes theorem to the lkelhood functon and a pror densty, e.g. j (8) j j j j j p,..., k, data L data,,..., k pror,..., k We wll often use unnformatve prors,.e.. pror and assume that the parameters are ndependent. In order to apply the Gbbs sampler the rght hand sde of the proportonal relatonshp needs to be normalzed,.e. we need to be able to ntegrate the rght hand sde j j wth respect to condtonal on,..., k. Useful Gbbs samplng dstrbutons are unvarate or multvarate normal, Inverse Gamma or Wshart, and the Beta dstrbuton. If y y,..., yt s an observed seres assumng that y N, wth unknown parameters and then d Inverse gamma probablty dstrbuton densty functon wth the shape parameter and scale parameter s IG x;, x exp( / x) where s the Gamma functon. The mean of x s ( ) 5 gven by

9 (9) T T p( y, ) L( y, ) p( ) ( y;, ) exp T y y e xp ;, T T ( y ) usng the unnformatve pror p( ) and () p (, ) T ( y, ) L( y, ) p( ) y ; ( y ) ( y ) T T exp IG ;, usng the pror p( ) / equvalent to the unnformatve pror p(log ). Hence the Bayesan dstrbutons for and can be obtaned by the Gbbs sampler teratng (9) and (). The pror dstrbutons are often specfed n order to mprove convergence but not to nfluence (sgnfcantly) the fnal results, typcally a wde normal dstrbuton conjugate pror dstrbuton for and a flat nverse gamma dstrbuton for. If the seres s multvarate normal then the dstrbutons are generalzed to multvarate normal and nverse Wshart (Lynch, 7). A multvarate dscrete-tme dffuson process (3) s n fact equvalent to a multvarate normal return seres model wth d r N( μ, ), where r ( r,,..., r, m) ' s the vector of returns on m assets observed at tme, μs a vector of means, and a covarance matrx. The margnal dstrbutons are () p T μ r, μ; r, T T and tr () ( Tm, ;, )/ p r μ IW T S exp S Where IW ; T, S matrx, and the mproper pror denotes the nverse Wshart dstrbuton, p( ) m T S ( r μ)'( r μ) s the scale analogous to the unvarate case has been used. for and the varance s ( ) ( ) get and. 6 for. Alternatvely, gven and we

10 If b b,..., bt s a bnary seres where b Bern( ) d, then can be sampled usng the beta dstrbuton: (3) T b b n Tn b) ( b ) ( ) ) ) Beta( ; p( L p ( ( n, T n) wth the unnformatve pror p( ). Generally, the beta dstrbuton Beta( x;, ) would be a conjugate pror where and can be nterpreted as pror successes and falures. If the ntegraton on the rght hand sde of (8) s not analytcally possble (whch wll be also our case) then the Metropols-Hastngs algorthm can be used. It s based on a rejecton samplng algorthm. For example n step the dea s frstly to sample a new proposal value j j j of and then accept t or reject t (.e. reset : ) wth approprate probablty so that, ntutvely speakng, we rather move to the parameter estmates wth hgher correspondng lkelhood values. Specfcally, step s replaced wth a two step procedure: (4) j j j j. A. Draw from a proposal densty q(,,..., k, dt a a), j B. Accept wth the probablty mn R,, where j j j j j j j,..., k, data q,,..., k, data j j j j j j j,..., k, data q,,..., k, data p R. p In practce the step B s mplemented by samplng a u U(,) from the unform j dstrbuton and acceptng f and only f u R. It s agan shown (see ohannes, Polson, 3) that under certan mld condtons the lmtng dstrbuton s the jont dstrbuton p data of the parameter vector. Note that the lmtng dstrbuton does not depend on the proposal densty, or on the startng parameter values. The proposal densty and the ntal estmates only make the algorthm more-or-less effcent. A popular proposal densty s the random walk,.e. samplng by (5). j j N(, c) The algorthm s then called Random Walk Metropols-Hastngs. The proposal densty s n ths case symmetrc,.e. the probablty of gong from j to s the same as the probablty j of gong from to j j (fxng the other parameters), and so the second part of the fracton n ( ) The pdf of the beta dstrbuton s Beta( x;, ) x ( x) ( ) ( ) be expressed as and the varance. ( )( ) for x. The mean can 7

11 the formula (4) for n step B cancels out. Consequently, assumng non-nformatve pror, the acceptance or rejecton s drven just by the lkelhood rato L data,,..., R. L data.. j j j k j j j,,., k Another popular approach we shall use s the Independence Samplng Metropols-Hastngs j algorthm where the proposal densty q does not depend on j (gven the other parameters). The acceptance probablty rato (4) s slghtly smplfed but note that the proposal denstes do not cancel out. In order to acheve effcency the shape of the proposal densty q should be close to the shape of the target densty p whch s known only up to a normalzng constant. Typcally, estmatng complex stochastc models, we need to estmate the parameter vector wth a few model parameters, and a vector wth a large number of state varables X (typcally proportonal to the number of observatons). We know that p(, X data) p(data, X) p( X, ) and so we may estmate teratvely the parameters and the state varables: p( X,data) p(data, X) p( X ) p( ), p( X,data) p(data, X) p( X) p( X). The parameters and state varables are sampled step by step, or n blocks, often combnng Gbbs and Metropols-Hastngs samplng..3 Unvarate jump-dffuson stochastc volatlty model The man goal of ths secton s to propose an MCMC estmaton algorthm for a bvarate jump dffuson stochastc volatlty model. However, t wll be useful to outlne frstly the unvarate jump-dffuson model and the extended model wth stochastc volatlty. It s then easer to defne the samplng steps of the bvarate model estmaton algorthm. Moreover, gven two fnancal seres, t s useful to estmate frstly the unvarate models separately on the two seres. Snce the latent varables,.e. jumps and stochastc volatltes, are also estmated, the output can be used to make a prelmnary analyss of correlatons between the jumps and stochastc volatltes. Let us frstly consder the dscrete-tme jump dffuson model (5) wth constant volatlty. Gven the sequence of observed returns d ata={ r ;,..., T}, the parameters and latent state varables to be estmated are:,,,, Z.,, In ths case we may use the pure Gbbs MCMC algorthm:. Sample reasonable ntal values,,,,, Z, () () () () () () (). Sample Z ( Z;, ) f, and ( g) ( g ) ( g ) ( g ) 8

12 Z ( r; Z, ) ( Z;, ) f. ( g) ( g ) ( g ) ( g ) ( g ) ( g ) 3. Sample {,}, Pr[ ] p /( p p ), where ( g ) p ( g) ( g) ( g) = ( ;, ) (- ), p ( r; Z, ) r ( g) ( g) ( g) 4. Sample to (9) and (). ( g) ( g), based on the normally dstrbuted tme seres r Z accordng ( g ) ( g ) 5. Sample () based on Bernoull dstrbuted bnary tme seres accordng to (3). ( g) ( g) 6. Sample, based on the normally dstrbuted tme seres and (). ( g ) Z accordng to (9) Secondly let us consder a jump-dffuson model wth stochastc volatlty followng the equaton (6). The dscrete tme specfcaton s: r Vň Z V (6) logv logv ň V ňň, ~ N(,), Z ~ N(, ), ~ Bern( ),d In ths case we need to estmate not only the latent state varables Zbut, also the vector of latent stochastc varances V. The MCMC estmaton unfortunately requres applcaton of the Metropols-Hastngs snce the condtonal dstrbuton for the varance V (condtonal on the other varances and parameters) s not a known one. It follows from (6) and the Bayes Theorem that: (7) pv ( ), rz,, ) pr ( V, Z ) pv ( V ) p( V V ( V,,,,, ). Here the frst part of the rght hand sde of (7) s nverse gamma n V : (8) pr (,,, ) ;,.5 exp.5( ) V Z r Z V V r Z / V But the remanng two factors are lognormal 3 n V : log V (logv; log V, ),.e. ( g ). p( V V, ) V exp (logv log V ) /( ), 3 Lognormal probablty densty fcton wth parameters and s gven by LN( x;, ) exp.5(l ogx ) / x exp( / ). It s useful to note that the mean of x s and the varance s (ex p( ) ) exp( ). 9

13 and smlarly log V ; lo (log g, ) V V,.e. ( V, ) exp (log V logv) /( ) pv n terms of V. It s easy to verfy that the product of the two lognormal dstrbutons s proportonal to the lognormal dstrbuton wth the correspondng normal dstrbuton mean and standard devaton: (9) ( ) (log V lo V ) / g, /. In order to obtan a proposal dstrbuton acquer et al (994) suggest replacng the lognormal dstrbuton wth an nverse gamma dstrbuton fttng the frst two moments. It s confrmed emprcally that the choce of a proposal dstrbuton wth a shape closely mmckng the orgnal dstrbuton s of key mportance snce hgh dmensonalty of the varance state varable vector makes convergence of the MCMC algorthm dffcult. The product of two nverse gamma dstrbuton densty functons s an nverse gamma dstrbuton densty functon, hence combnng the nverse gamma dstrbuton (8) and the ftted nverse gamma dstrbuton we fnally obtan the proposal densty functon: () q,,, ) IG,5 ) Z ( V V( ), rz V;,( exp(.5 ).5( r ), exp( ) where. exp( ) The proposal densty s used n the Metropols/Hastng algorthm wthn a new block, e.g. followng the step 3 n the MCMC procedure for the jump-dffuson processes. Ths bloc updates all the varances V,,..., T. For V and VT the formula (9) needs to modfed slghtly snce V and VT are not known. The dffuson volatlty s obvously replaced by the square root of the latest estmate of the varance V and we also need to add a new MCMC step for the AR() coeffcents, and (for example followng the step that updates V ). The coeffcents, can be sampled wth a bvarate normal dstrbuton and wth the nverse gamma dstrbuton. The extended MCMC algorthm s n detal descrbed as follows:. Sample reasonable ntal values,,,,,,, V, Z, () () () () () () () () () (). Sample Z ( Z;, ) f, and ( g) ( g ) ( g ) ( g ) Z ( r; Z, V ) ( Z;, ) f. ( g) ( g ) ( g ) ( g ) ( g ) ( g ) 3. Sample {,}, Pr[ ] p /( p p ), where ( g )

14 () p r V ( g) ( g) ( g) = ( ;, ) (- ) p ( r; Z, V ) ( g) ( g) ( g), 4. Sample new stochastc varances V for,..., T usng Metrols-Hastngs (4) wth ( g ) the proposal densty gven by () 5. Sample new stochastc volatlty autoregresson coeffcents ( g ) ( g, ), ( g ) from h logv for,..., T usng the Bayesan lnear regresson model (Lynch, 7): ( g ) () ˆ ˆ... β ( X'X) Xy, e ˆ = y - Xβ,where X, h... ht y, h... ht ( g ) n ee ˆˆ ' ( ) IG,, ( g) ( g) ( ) (, )'; ˆ g β X'X,,( ) ( ). ( g) g 6. Sample based on the normally dstrbuted tme seres r Z ( g ) V,.e. p 7. Sample ( ) ( g ) r Z rz V. T ( g) ( g) T T ( g) ( g) ( g) ( g),,, ;, ( g) ( g) ( g) V V V () based on Bernoull dstrbuted ( g ) wth varances bnary tme seres accordng to (3). 8. Sample and (). ( g) ( g), based on the normally dstrbuted tme seres Z accordng to (9) ( g ).4 Bvarate jump-dffuson stochastc volatlty model Our ultmate goal s to study relatonshp between returns, jumps, and volatltes of two related fnancal seres. Frst, we may estmate ndependently the parameters and latent varables, Z, Vand, Z, Vof (6) for two gven seres of returns r and r. Snce we also get estmates of the latent varables,.e. jump tmes, jump szes, and varances, we may nspect ther relatonshp. For example we may analyze the overlap of probable jump tmes of the two processes,.e. of the sets { P[, ]. 5} and { P[, ]. 5}. Smlarly the relatonshp between mean volatltes can be analyzed. Snce h, logv, s specfed as a normal varable by the model, the mean volatlty should be expressed by, exp( h, / ) where h, s the MCMC estmaton of the mean of logv,. The correlaton between the seres ;,...,, T and ;,...,, T (resduals or levels) ndcates what the correlaton between the stochastc volatltes mght be.

15 However, to gve a consstent answer we need to specfy a full bvarate stochastc volatlty jump dffuson model: (3) r V ň Z,,,,, logv logv logv ň V,,,, r V ň Z,,,,, logv logv logv ň V,,,, V ň, ň, V ~ N,, ~ N, V, ň ň, V Z, Z ~ N, Z, Z ~ Bern( ), ~ Bern( ), co rr,,,,, The model does not take nto account a possble correlaton between the dffuson and logvarance resduals, but t ncorporate possble Granger causalty between the two log-varance processes. For example f then hgh level of logv (Granger) causes logv to become larger. Our prelmnary analyss of the data we ntend to study ndeed ndcates that the V V volatlty resduals ň, and ň, are not correlated, but logv, and logv, are correlated. We also compare the results of the stochastc volatlty jump-dffuson model aganst the restrcted bvarate jump-dffuson model wth constant volatltes nspectng n partcular the jump probabltes and the jump correlatons estmated wth and wthout stochastc volatltes. Correlated jumps In order to ncorporate correlated jumps and jump szes nto the MCMC algorthm gven n Secton.3 we need to modfy the steps and 3 gven as follows: Sample the new jump szes (omttng the upper ndces g and g ) where,,,, y ' y ( Z ( Z p Z r p r Z p Z exp.5 )' ) Z Z )' ) exp.5( Z m V ( Z m r, Z,, r,, Z,, y r Z, r, Z,, (4) V, V, V,, VV,, V, Z Z V, m V ( r )., and

16 Next we need to resample the correlated jump tmes. Snce by defnton p where p Pr[, &, ] we have ( )( ) Pr[ ] ( )( ),, and Pr[ ] ( ),,. Smlarly to () (agan omttng the upper ndces g and g ) we set p= r;, Z,,, Pr[,, ], p = r; Z,, Z,,, Pr[,, ], and sample, {,}, Pr[, ] p /( p p ). Fnally we need to add two addtonal steps re-samplng the correlatons and Z. The jump sze correlaton can be n fact sampled n one step wth jump sze volatltes usng the nverse Wshart dstrbuton () as the seres Z s assumed to be bvarate normal. Regardng we use the random walk Metropols-Hastngs step and the relatonshp T,, p(,,, ) L(,,, ) p(, ), where p(,) ( ) ( ), p(, ) p(,), p(,) p(,), p(,) p(,). Regardng the dffuson resduals correlaton coeffcent t can be easly re-sampled usng the random walk Metropols-Hastngs: T, ;, r,,z,v, r,,z,v p r wth the bvarate normal densty functon and the notaton (4). Correlated stochastc volatltes Correlaton between stochastc volatltes can be captured by correlated resduals and by correlated levels of the log volatltes expressed n the VAR() model (3). In order to smplfy the notaton set hk, logvk,, ( k,;,..., T). Re-samplng the varance V, condtonal on the other latent varables and parameters MCMC needs to take nto 3

17 account ts relatonshp to r, and to V,, V, gven by the VAR equaton for h,, but also to V, and, V gven by the equatons for h, and h,. Applyng the Bayes theorem: pv (, V,( ), V,, r, Z, ) pr ( V,, Z, ) pv ( V, V, ) pv ( V, V, ) pv ( V, V, ).,,,,,,,,,,,,, The frst probablty dstrbuton on the rght hand sde s agan nverse gamma n V,, whle the remanng ones are lognormal. The product of the three lognormal denstes s lognormal wth the correspondng normal dstrbuton mean and varance: (5) ( h h ) ( h h ) ( h h ),,,,,,, ( ) (). The Metropols-Hastngs proposal densty for updated V, s then gven by (). The second process varancev, s resampled analogously. Fnally, gven V and V (omttng the upper ndex g ) we need to update the coeffcents,,, and,,,. We shall use the Bayesan lnear regresson model as n the unvarate case:... ˆ, ˆ β ( X'X) Xy e ˆ = y - Xβ,where X h,... h, T, h,... h y, T, h,... h, T n 3 eˆˆ ' e IG,,,,, ; βˆ, ( X'X)., Smlarly we proceed for,,,..5 Model Comparson and Value at Rsk Estmatons In order to compare the dfferent models we wll use the devance nformaton crteron (DIC) of Spelgelhalter et al (). It s shown to generalze the Akake nformaton crteron (AIC) that s not approprate to compare stochastc volatlty or jump-dffuson models (Yu, Meyer, 6). In order to calculate AIC one needs to specfy the number of free parameters. In case of the stochastc volatlty and jump-dffuson models the number of addtonal latent varables estmated has an order of T. On the other hand, snce the varables are not ndependent t s not clear what should be the number of free varables. Lkewse AIC, DIC has two components, a term that measures goodness-of-ft and a penalty term for ncreasng model complexty: 4

18 DIC D p D. The frst term s defned as the posteror mean of the devance D( ) log L(dat a ),.e. D E [ D( )] data. Here ncludes all the model parameters and estmated state varables. Snce the lkelhood functon L(data ) s known the measure s estmated just averagng the lkelhood over ( g ) MCMC sample estmates of. The second term measurng the effectve number of parameters s defned as the dfference between the posteror mean of the devance and the devance evaluated at the estmated parameters ˆ,.e. p DD ˆ. D The parameters are, n our case, estmated as means of the sampled values dscardng an ntal perod of the MCMC samplng. The stochastc volatltes are estmated as ˆ k, exp ( hk, / ) where hk, s the mean of sampled values. In case of jump occurrence we set ˆ f and only f the mean of sampled jump ndcators s larger than.5,.e. the k, probablty of jump occurrence s estmated to be larger than 5%. The jump sze s agan estmated as the sample mean of sampled jumps. Other classcal classcal Bayesan goodness-of-ft measures to be mentoned are the Bayes factors or margnal lkelhood. We follow Yu and Meyer (6) who recommend the DIC measure that s shown to have a consstent performance wth respect to the two standard measures and s relatvely easy to compute. Value at Rsk (VaR) Fnally we ntend to llustrate dfference between the estmated bvarate models calculatng VaR for dfferent tme horzons and confdence levels. If the true data were generated by a smple process, e.g. the pure dffuson one, then the estmatons yelded by more general models (ncludng jumps and/or stochastc volatltes) calbrated on the same dataset should be smlar. However f the data contan true jumps and/or stochastc volatlty, then the VaR estmatons wll probably sgnfcantly dffer and the choce of model becomes mportant. Formally, Value at Rsk, VaR( N, ) s defned as X q [ X] where q [ X] denotes the - quantle of the random varable X modelng the market value of a portfolo, N (busness) days to the future. The varable X can be also defned as the future return relatve to the ntal nvestment. We wll consder a smple foregn stock nvestment where the domestc currency value of the DC FC portfolo s V SV, S s the exchange rate measurng foregn currency FC n terms of FC domestc currency DC, and V s the foregn currency value. The domestc currency logreturn over a tme horzon can be expressed as the sum r r r of the exchange DC S FC rate 5

19 log-return and the foregn currency stock return. Thus, n order to get the dstrbuton and DC S FC quantles of r we need to model the jont dstrbuton of r and r. The smplest and wdely used (parametrc normal VaR) method calculates the varances and the covarance of r S FC DC S FC and r, combnes them to get the standard devaton of r r r, and multples t wth a standard normal dstrbuton quantle to get an estmate of VaR(, ). To get N day estmate the value s scaled by N. We wll use the estmaton wth an EWMA (exponentally movng weghted average) covarance matrx as a benchmark value. Gven a specfc stochastc model and ts parameters we may smulate the returns r,..., T rtnone to N days ahead, and so the compounded N-days log-return r N rt. Note that n case of the stochastc volatlty model (6) the ntal values must also nclude an estmate of the last varance V T. The bvarate model jump-dffuson model wth stochastc volatlty model (3) allows us to smulate the jont dstrbuton of r S FC and r over an N days horzon. The Bayesan estmatons n fact yelds an emprcal dstrbuton of the parameters approxmated by a sequence of MCMC smulated values,..., K (dscardng an ntal samplng perod). The Value at Rsk estmated as above s then condtonal on the estmate VaR N, ˆ. Snce the parameters are uncertan and modfed ˆ,.e. n fact we get parameters mght nfluence sgnfcantly the VaR measure we should rather calculate a VaR estmate ncorporatng the parameter uncertanty. Ths can be easly done n the Bayesan framework samplng frstly from,..., K and then smulatng the returns gven the parameter vector. The dstrbuton of smulated returns can be used to obtan the Bayesan Value at Rsk denoted as VaR N, data,.e. VaR condtonal on the data used to estmate the parameters rather than on a specfc vector of pont estmates. 3. Emprcal Study The man goal of ths secton s to apply the bnomal jump-dffuson model stochastc volatlty model, ts submodels (bnomal dffuson, jump-dffuson, and stochastc volatlty), and the proposed estmaton method to the FX and stock market tme seres returns. However, before dong so we are gong to test the estmaton procedures on artfcally generated data n order to demonstrate that the model s able to dentfy or reject jumps and stochastc volatlty parameters. The MCMC algorthm and all calculatons have been mplemented n Matlab. 3. An Emprcal Test We are gong to sample T = returns accordng to the bvarate jump-dffuson stochastc volatlty model (3) wth several sets of the model parameters (and startng wth ntal log- 6

20 varances logv, logv, 7 ). The proposed MCMC estmaton method s then appled to the generated data and estmated parameters are compared to the orgnal parameters that were used to generate the data. The true parameters are expected to le wth the Bayesan confdence ntervals. The upper part of Table shows parameters of the frst artfcally generated bvarate process. There are no jumps, there s no correlaton between the dffuson resduals, and no correlaton between the varances. The lower part of the table shows the MCMC estmates wth standard devatons n parenthess. The MCMC procedure has been run 3 tmes and frst estmatons have been dscarded. Fgure shows, for example, relatvely fast convergence of the frst stochastc volatlty equaton coeffcents and. Generated jump-dffuson process wth stochastc volatlty,, Generated jump-dffuson process wth stochastc volatlty,, Correlatons Estmated jump-dffuson process wth stochastc volatlty,,. (4.477e-4) Z.6 (.5) -.4 (.4).893 (.8) (.595).9794 (.69). (.66) Estmated jump-dffuson process wth stochastc volatlty,,.3 (4.66e-4).45 (.5).34 (.4) (.4) (.63) Estmated correlatons.33 (.6). (.59) -.56 (.43).475 (.9).764 (.554).973 (.74) Z.33 (.38) Table. Generated bvarate stochastc volatlty process (upper table) and the MCMC estmated parameters. The estmated stochastc volatlty parameters are consstent wth the true parameters, ndeed the estmated coeffcents and are not sgnfcantly dfferent from zero and so Granger causalty would not be detected. The estmated means and seem to dffer from the true values sgnfcantly, but we have to take nto account the error caused by the data generaton process tself possbly correlated to a relatvely large volatlty. The mean of the frst generated return seres turns out to be.3 and of the second.9 n lne wth the estmatons. The zero jump ntenstes fall wthn the 95% confdence ntervals around the 7

21 estmated values. In fact no jumps wth probablty larger than.5% are dentfed by the sampled jump probabltes (averagng the jump ndcator k, ) see Fgure Fgure. Convergence of the coeffcents and Fgure. Sampled jump probabltes Table shows the generatng and estmated parameters of a process wth nonzero jump ntenstes, wth a postve jump occurrence correlaton, but stll wth zero varance correlaton. The estmates are agan more-or-less consstent wth the true parameters. Fgure 3 tres to compare the smulated jumps and ther absolute szes wth the estmated jump probabltes. The pont s that f a smulated jump sze s too small than the jump can be ex post hardly dentfed and the estmated jump probablty s low. Ths may explan the dfference between the true jump probablty.3and the estmated value.37 (.7). The precson of the jump ntensty estmates should mprove wth a larger number of observatons. Generated jump-dffuson process wth stochastc volatlty,, Generated jump-dffuson process wth stochastc volatlty

22 ,, Correlatons.36 Z Estmated jump-dffuson process wth stochastc volatlty,,.6 (4.876e-4).37 (.7).344 (.39).737 (.68) -.94 (.55).9836 (.53).9 (.53) Estmated jump-dffuson process wth stochastc volatlty,, 4.775e-4 (4.574e-4).9 (.5) -.36 (.3) (.44) (.859) Estmated correlatons.6 (.6) -. (.8) e-4 (.43).46 (.37).49 (.574).9588 (.) Z.794 (.97) Table. Generated bvarate jump-dffuson stochastc volatlty process (upper table) and the MCMC estmated parameters Fgure 3. Blue bars below the x-axs show the smulated jumps and ther absolute sze, red bars above the x-axs show the MCMC estmated jump probabltes (left process, rght process ) Fnally Table 3 shows parameters and estmates of a bvarate jump-dffuson process wth correlated stochastc volatltes. We have stll set some parameters to zero, e.g or, n order to check that he estmaton s able to confrm or reject sgnfcance of the ndvdual parameters. 9

23 Generated jump-dffuson process wth stochastc volatlty,, Generated jump-dffuson process wth stochastc volatlty,, Correlatons.348 Z Estmated jump-dffuson process wth stochastc volatlty,,.8 (5.6e-4).39 (.33).56 (.45).589 (.9) -.93 (.63).973 (.9).57 (.5) Estmated jump-dffuson process wth stochastc volatlty,,.3 (5.543e-4).88 (.5) -.8 (.75) (.99) (.93) Estmated correlatons.57 (.74).8 (.99).79 (.4).36 (.64) (.454).936 (.86) Z.676 (.54) Table 3. Generated bvarate jump-dffuson process (correlated) stochastc volatlty model (upper table) and the MCMC estmated parameters. Durng mplementaton of the models many emprcal tests have been performed. We have shown only a few to llustrate a good performance of the proposed estmaton algorthm. 3. FX and Stock Market Data and Emprcal Results Our data set conssts of CZK/EUR exchange rates and the Czech PX stock ndex values from Sep, 4 to Feb, (Fgure 4). Note that the CZK/EUR s quoted as the nverse of the normal EUR/CZK exchange rate,.e. as a drect quotaton of CZK n terms of EUR from the perspectve of an EUR nvestor. The perod of strong declne of the stock ndex and deprecaton of CZK/EUR corresponds obvously to the fnancal crss n 7-9.

24 .44 CZK/EUR PX Fgure 4. CZK/EUR exchange rate and PX stock ndex value Sep, 4 to Feb,. Tme seres of daly returns (Fgure 5) and days movng wndow volatlty (Fgure 6) vsually shows many jumps and overlappng perods of relatvely hgh volatlty..5 CZK/EUR daly returns.5 PX daly returns Fgure 5. CZK/EUR exchange rate and PX stock ndex daly returns.45 CZK/EUR and PX volatlty Fgure 6. CZK/EUR (blue lne) and PX (red lne) days movng wndow volatlty Standard tests, e.g. arque-bera test, reject normalty of both seres at the.% confdence level (Table 4). The Person s correlaton of the two return seres s slghtly negatve -.% wth p-value.6,.e. the correlaton does not sgnfcantly dffer from zero.

25 CZK/EUR daly returns PX daly returns Mean.5949e e-4 Standard devaton.49.7 Skewness Kurtoss arque-bera statstc e+3.56e+4 p-value <. <. Table 4. CZK/EUR exchange rate and PX stock ndex daly returns descrptve statstcs It seems obvous that the normal VaR based on the assumpton of normalty and no correlaton would underestmate the true rsk of a combned CZK/EUR and PX nvestment. In order to mprove our ablty to predct the rsk we are gong to estmate and compare the four bvarate models descrbed n Secton. Model : Bvarate pure dffuson model (Dff) The smplest model we consder s the bvarate dffuson model. Table 5 shows MCMC estmated coeffcents based on 3 teratons when the frst have been dscarded. The results are n lne wth the descrptve statstcs gven n Table 4. The correlaton between the two seres does not sgnfcantly dffer from zero n the context of ths model. k FX returns (k=).5345e-4 (.97e-4).49 (8.6e-5) PX returns (k=) 4.6e-4 (4.5433e-4).7 (.647e-4) -.9 (.65) Table 5. Estmated parameters for the CZK/EUR and PX pure dffuson bvarate model Model : Bvarate jump-dffuson model (D) Table 6 shows MCMC estmates of the bvarate jump-dffuson model (3) wth correlated jumps, but wth constant volatltes. We have agan used 3 MCMC smulatons, nonnformatve prors, and dscarded the frst ones. The ntal means and correlatons were set to, ntal dffuson volatltes to. and jump standard devatons to.. Fgure 7 shows relatvely fast convergence of the jump correlaton to a surprsngly hgh level over 5% and the smulated kernel smoothed densty. Note that the dffuson correlaton and the jump-sze correlaton Z do not sgnfcantly dffer from. k

26 CZK/EUR jump-dffuson process.486e-4 (9.8678e-5) e-4 (3.99e-4).3 (.4e- 4). (3.354e- 4).737 (.8) e-4 (7.37e-4 ) PX jump-dffuson process.549 (.44) Correlatons.95 (8.33e- 4) -.38 (.5).353 (.7) -.4 (.337).54 (.675).86 (.896) Table 6. Estmated parameters for the CZK/EUR and PX jump-dffuson models wth constant volatlty Z.8 MCMC convergence 6 MCMC densty of Fgure 7. Convergence and the Bayesan densty of the jump correlaton Probably the most nterestng fndng s that the probabltes of jumps are relatvely hgh, over 5% for both return seres. However, the outcome should not be so surprsng lookng at the very hgh kurtoss (Table 4) of both seres. The jump-dffuson model n fact decomposes the dstrbuton nto a mx of two normal dstrbutons, one wth the lower standard devaton and the other wth a hgher (more than three tmes) standard devaton. Whle CZK/EUR mean jump sze does not sgnfcantly dffer from zero, the mean PX jump sze s negatve showng that the stock ndex tends to jump down rather than up, as one would expect. The hgh probabltes of jumps contradct to our ntuton of jumps beng rare events. The MCMC estmatons gve us also smulated dstrbutons of jump occurrences and jumps szes that allow us to analyze the jumps n more detal. Each run of the MCMC smulaton samples specfc jumps tmes and jump szes. Hence for each day we may calculate the emprcal probablty of jump and dentfy days where a jump probably happened,.e. where the probablty of jump s larger than 5%. For those days t makes sense to nspect the mean value of the smulated jump szes (condtonal on the jump occurrence). The results are shown n Fgure 8. It s obvous that the jumps are not dstrbuted evenly beng clustered especally n 3

27 the fnancal crses perod. We wll see that the jump clusterng s essentally fltered out n the model wth stochastc volatlty. EUR/CZK jump probabltes.4 EUR/CZK expected jump szes PX jump probabltes. PX expected jump szes Fgure 8. CZK/EUR and PX returns jump probabltes and mean jump szes Model 3: Bvarate jump-dffuson wth stochastc volatlty (D SV) Before estmatng the bvarate stochastc volatlty model we frstly mplement the unvarate model (6) for both return seres n order to nspect the relatonshp between the two latent stochastc volatltes tme seres. The estmated parameters and ther standard devatons based on 3 MCMC smulatons are shown n Table 7. Fgure 9 shows for the sake of llustraton relatvely fast convergence of the coeffcent n case of the PX return process. The parameter means and standard devatons n Table 7 are based on the last 5 smulatons dscardng the frst 5. Note that by ntroducng the stochastc volatlty nto the model the probabltes of jumps have been sgnfcantly reduced to less than 3% and the jump sze standard devaton went up. The hgh value of the stochastc volatlty (log-varance) autocorrelaton coeffcent, almost 99% for CZK/EUR and almost 98% for PX, shows a hgh persstence of stochastc volatltes that s n lne wth other emprcal studes on US stock market data (e.g. acquer et al, 994, Eraker et al, 3). The volatlty of the stochastc volatlty,.e. the coeffcent, around 3% for CZK/EUR and over % for PX, s also n the range estmated on US data..856e-4 (6.374e-5). (.9e-4) CZK/EUR unvarate jump-dffuson process wth stochastc volatlty.84 (.83) -.66e-4 (.4).7 (.8) -.5 (.545).9893 (.48) PX unvarate jump-dffuson process wth stochastc volatlty.37 (.68). (.79).47 (.66).33 (.93) (.63).978 (.69).9 (.47) Table 7. Estmated parameters for the CZK/EUR and PX unvarate jump dffuson models wth stochastc volatlty 4

28 PX MCMC convergence 7 PX densty Fgure 9. Convergence and MCMC smulated densty of the parameter for PX returns The latent stochastc volatltes are sampled at each MCMC smulaton run and we get a dstrbuton for each partcular day. In order to nvestgate the relatonshp between the CZK/EUR and PX volatltes we use the mean estmates, specfcally gven by the equaton ˆ exp ( h / ), where h s the MCMC mean of normally dstrbutedlogv sampled values. Fgure shows that the mean stochastc volatlty for both seres copy well the pattern of the observed returns. The fgures also explan why many jumps dentfed n the constant volatlty model have been fltered out n the stochastc volatlty model..5 CZK/EUR mean stochastc volatlty.5 PX mean stochastc volatlty Fgure. CZK/EUR and PX returns (blue bars) and mean stochastc volatltes (red lnes) The mean stochastc volatlty seres and mean estmated coeffcents,, and may be used V to obtan the resduals of the two seres efx, and e V PX,. The Pearson s correlaton of the resduals s relatvely low 5.7% and not sgnfcant at % confdence level, however the correlaton of hfx, and hpx, comes out hghly sgnfcant 6.66% as ndcated vsually by Fgure. Consequently, havng parsmony n mnd, t s reasonable to specfy the bvarate jump-dffuson model wth correlated stochastc volatltes just through the mutual Granger causalty and wthout correlated volatlty resduals,.e. accordng to the model (3). The model s estmated by the methodology has been outlned n Secton.4 and the resultng estmates are shown n Table 8. We have run agan 3 smulatons, used the last, and dscarded the frst. Fgure shows relatvely fast convergence of the nterestng 5

29 coeffcent. The sgnfcant postve value.35 proves there s a Granger dependence of the PX stochastc volatlty on the CZK/EUR stochastc volatlty. On the other hand the coeffcent reflectng the causalty n the opposte drecton turns out not to be sgnfcant. Ths corresponds to our ntuton: the FX market s more lqud and closely lnked to the global markets whle the Czech stock market lqudty s relatvely low and n a sense behnd the global FX market. It can be verfed that the VAR() process estmated coeffcents mply a hgh Pearson s correlaton around 58% close to our fndng based on the unvarate models. CZK/EUR jump-dffuson process wth stochastc volatlty,,.588e-4 (5.994e-5).9 (.57) -9.8e-4 (.57).7 (.6) (.97).969 (.5) PX jump-dffuson process wth stochastc volatlty,,.3 (.89e-4).7 (.48) -.99 (.43).447. (.6) (.87) Correlatons.86 (.87). (.36).35 (.9) (.67).638 (.97).473 (.3593).958 (.78) Table 8. Estmated parameters for the bvarate jump dffuson models wth Granger related stochastc volatltes Z.68 (.74).35 MCMC convergence 45 densty Fgure. MCMC convergence and densty of the coeffcent reflectng the Granger dependence of the PX stochastc volatlty on the CZK/EUR stochastc volatlty Fnally we may nspect the behavor of jumps. It s nterestng to note that the jump probablty fell to.3% n case of CZK/EUR and.% n case of PX returns. Fgure shows the MCMC emprcal daly jump probabltes. For both seres there s only one day wth jump probablty hgher than 5% and a few days wth jump probabltes over %. The stochastc volatlty component has clearly fltered out the clusterng and the jumps seem to be dstrbuted evenly. There s stll a postve jump occurrence correlaton.6 and t s nterestng to note that n ths case the jump sze correlaton s postve.47 (though not hghly sgnfcant) meanng that f there s a concdence of jumps on both markets than the jumps probably go n the same drecton. 6

30 Lookng at the jump analyss n the jump-dffuson models wth and wthout stochastc volatlty t s obvous that jump dentfcaton strongly depends on the stochastc model chosen..7 CZK/EUR jump probabltes.7 PX jump probabltes Fgure. CZK/EUR and PX returns jump probabltes n the context of the bvarate stochastc volatlty model Model 4: Bvarate dffuson model wth stochastc volatlty (SV) Snce the estmated jump probabltes n the prevous models have been low and not hghly sgnfcant Table 9 gves, for the sake of completes, estmates of the bvarate stochastc volatlty model wthout jumps. It s nterestng to note that the estmated coeffcents do not dffer sgnfcantly from the results n Table 8 (wth jump parameters mssng).,,.7474e-4 (5.6e- 4).3 (.3) dffuson process FX wth stochastc volatlty (.95),,.979 (.) dffuson process PX wth stochastc volatlty -. (.867) Correlatons.4 (.77).75 (.393).387 (.).95 (.5) -.37 (.57) - - Table 9. Estmated parameters for the bvarate stochastc volatlty model wthout jumps 3.3 Comparson of the Models.564 (.5) The four estmated models can be compared usng the devance nformaton crteron (DIC) measure shown n Table. The last two columns gve the goodness-of-ft and the model complexty measures. For example, n case of the bvarate dffuson model pd equals to 5 (after roundng to unts),.e. to the number of the parameters estmated, as expected. It s more dffcult to nterpret the p values n case of the other models. The hgh complexty of D the Model (D) could be explaned by exstence of many jumps where the sze needs to be 7 Z

31 n addton estmated. The hgh value of (D SV) s, however, slghtly puzzlng. p D for the Model 4 (SV) compared to the Model 3 An absolute dfference n DIC over s already consdered mportant (Spegelhalter et al,, Asa et al, 6), and so the models wth jumps and/or stochastc volatlty strongly outperform, n terms of DIC, the pure dffuson model. The best ranked model s the Model (D) followed by the Model 3 (D SV). The dfference between the DIC of D and DSV mght be dsappontng from the perspectve of the stochastc volatlty modelng and the estmaton effort. However the D model estmates many jump szes sgnfcantly mprovng the goodness-of-ft wth respect to the estmaton dataset, but the jump szes estmates do not have any added value for predctons of future returns (dstrbutons). Ths s not the case the DSV where the last estmated stochastc volatlty s used to predct future volatltes and return dstrbutons. The ablty of the dfferent models to predct future dstrbutons (n partcular VaR) could be compared usng a back-testng procedure. Ths s unfortunately unfeasble at the moment snce one MCMC estmaton takes more than one or two hours on a relatvely powerful desktop computer and back-testng would requre repeatng the procedure hundreds or thousands of tmes. DIC D p D Model (Dff) Model (D) Model 3 (D SV) Model 4 (SV) Table. Devance nformaton crteron of the four estmated bvarate models VaR Estmatons We are gong to show VaR estmatons gven by the four tested models and by a benchmark model to llustrate mportance of the model choce. Table provdes a comparson of the estmates VaR measures for the sum of lognormal returns rczk / EUR rpx calculated usng the dfferent models and by the methodology descrbed n Secton.5.. The returns could be nterpreted for example as returns on a PX stock ndex nvestment from the perspectve of a EUR based nvestor as of February,. The VaR measure has been calculated n,, and 3 days horzon, on the 95% and 99% probablty level, and condtonal on the pont parameter estmates (Table 5 -Table 9 and the least mean varance for the SV models) or on the correspondng MCMC parameter dstrbutons. The last lne gves EWMA VaR estmates used frequently n practce, for the sake of comparson wth our models. It s based on exponentally weghted movng average (EWMA) covarance matrx wth the weght set at.97. The number of Monte Carlo smulatons has been set to 4. Snce the jump-dffuson (D) model s usng constant volatltes, t s comparable rather to the dffuson model (Dff), correspondng to a standard normal VaR estmate. On the other hand the jump-dffuson model wth stochastc volatlty (DSV) starts wth the last 8