THE HAWKES PROCESS AND TIME-VARYING JUMP INTENSITY IN FINANCIAL TIME SERIES. Maciej Kostrzewski

Transcription

1 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 THE HAWKES PROCESS AND TIME-VARYING JUMP INTENSITY IN FINANCIAL TIME SERIES Macej Kosrzews Absrac News mgh rgger arrvals of jumps n fnancal me seres. The Bayesan JD(M)J model s appled o deec jumps. The Bayesan framewor, founded upon he dea of laen varables and compuaonally faclaed wh Marov Chan Mone Carlo mehods, enables he deecon of jumps and he analyss of her frequency. The presened mehodology s llusraed wh emprcal sudes employng boh smulaed and real-world daases. A very nuve observaon s made, namely ha hgher poseror probables of jumps are nferred durng he perods of hgher absolue values of reurns. A seres of wang mes beween wo consecuve jumps s also of neres n he sudy. Perods of no jumps alernang wh he ones of frequen jumps confrm he exsence of jump cluserng. The above resuls may promp one o apply Hawes processes o model he momens when jumps occur. The resuls of he maxmum lelhood esmaon of Hawes process, agan, ndcae he jump cluserng phenomenon. Informaon crera pon o a major superory of he models feaurng a sochasc nensy of jumps. Key words: jump cluserng, Bernoull jump-dffuson model, Hawes processes, Bayesan nference, maxmum lelhood esmaon JEL Code: G17, C58 Inroducon Numerous sudes ndcae ha many fnancal me seres feaure drasc occasonal movemens (referred o as jumps), alhough, obvously, wheher o classfy a gven observaon as a one feaurng a jump or no ypcally hnges on some arbrary defnon of a jump self. One of he mos common group of models employed n modelng me seres ha 743

2 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 conss of ypcal (and connuous) changes and, smulaneously, allow for abnormal occasonal shfs (jumps) s he class of jump-dffuson processes. In he curren research, a very specfc nsance of hese s under consderaon, namely he Jump-Dffuson wh M Jumps (henceforh JD(M)J) process, developed by (Kosrzews, 1a, 1b, 13a), and derved by dscrezng some jump-dffuson process. To esmae he parameers of he JD(M)J model and o nfer abou jumps, we resor o Bayesan sascal framewor, equpped wh he MCMC mehods The very erm jump cluserng que analogous o he one of volaly cluserng, pervadng he GARCH and SV leraure (see, e.g. (Osewals, Ppeń, 4), (Pajor, 9)) means ha jump arrvals (or wang mes beween wo consecuve jumps) end o cluser,.e. f a gven jump arrves n a shor me snce he prevous one, hen, mos possbly, anoher jump wll follow soon, oo. Jump clusers have already been dscussed n he fnancal economercs leraure see, e.g., (Kngh and Sachell, 1998), (Maheu and McCurdy, 4), (Lee, 1). The man dea s based on he assumpon of a sochasc jump nensy whch follows, e.g., a self-excng process. In wha follows, a very common (hough sll arbrary) rule of classfyng a gven observaon as a jump s adhered o, accordng o whch a daa pon s dagnosed as a jump f he poseror probably of a jump exceeds.5. Then, a seres of wang mes beween wo consecuve jumps s formed, enablng one o examne he jump cluserng phenomenon. Furhermore, he seres of me momens of jumps s f (va a maxmum lelhood approach) wh a Hawes process, whch provdes a mehod of analyzng he me-varably of jump nensy. The laer also faclaes he deecon of jump clusers. Such an (eclecc, ndeed) approach s employed n hs sudy o denfy jumps and her clusers n he case of boh smulaed and real-world daases. For relevan defnons and properes of he Hawes processes we refer he reader o (Hawes, 1971a, 1971b). The non-bayesan esmaon of he models founded upon hese processes s dscussed by (Ogaa, 1978), (Daley and Vere-Jones, 3), whereas he Bayesan approach s exposed by (Rasmussen, 13). The Hawes processes are appled o many felds ncludng sesmology, socology, neuroscence and ohers, also ncludng fnancal me seres analyss (see (A-Sahala, Cacho-Daz and Laeven, 13); (Maneesoonhorn, Forbes and Marn, 14)). The conrbuon of he paper resdes n performng an analyss of me-varyng jump frequency and desgnng a vsual mehod of deecng he jump cluserng phenomenon. The use of he proposed mehodology of deecng jumps, for some me seres under 744

3 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 consderaon, s parcularly jusfed n he conex of selng wheher he jump cluserng phenomenon should or need no be explcly ncorporaed no he srucure of some common jump-dffuson models. 1 The JD(M)J model Consder a sandard Wener process W W, a Posson process N N wh a consan nensy parameer, and he sequence of ndependen random varables Q j j 1 Q. Le us assume ha W, N and Q are muually ndependen. Fnally, S S denoes he prce process of some rsy asse. The logarhm of soluon of he equaon: I mgh be shown ha: S s governed by a jump-dffuson process ha consues he d 1 ln S d dw QdN. S ln S 1 W W Q,. N N 1 The process s bul of wo componens: he (pure) dffuson par 1 W W, represenng connuous varaons n he seres, and he (pure) N N 1 jump componen, Q, reflecng abnormal (exreme) movemens n reurns. The connuous prce behavor beween jumps s descrbed by he geomerc Brownan moon, W, whle he arrval rae of jumps s descrbed by he homogeneous Posson process, N, and he jump magnudes by Q. where S The dsrbuon of logarhmc raes of reurn, ln exp f x, 745 S, s an nfne mxure: (1)! f are some denses. Snce he seres gven by (1) s nfne, he densy s nracable. Therefore, consder he followng fne approxmaon of (1): exp M ()!! f exp f. The approxmaon resrcs he number of jumps over any me nerval o M. The case of

4 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 M ndcaes no jumps over nerval. To oban he condonal daa densy (gven he vecor of parameers ) he approxmaon s normalzed. Furher consderaons are resrced o he dscree me framewor. Tme seres S 1, x x n s comprsed of 1 x ln x,..., observed a mes,.... Moreover, S 1 s a fxed me nerval beween followng observaons. The specfcaon, ermed he JD J model, s defned by assumng a normal dsrbuon for Q j : 1 1 x Q x exp f, and seng M. The model s used o model seres of daly Q j Q Q 1 logarhmc raes of reurn of S&P1 ndex, ndcang 5. The choce of M = s explaned n (Kosrzews, 13a). 1, The Bayesan JD()J model p x, p x p where A Bayesan sascal model s defned by he jon densy:, x s he observed daa, s he vecor of unnown parameers, x x,..., 1 x n samplng densy and p x of gven daa x. p s he p s he pror densy. The nference ress upon he poseror densy Under he JD()J specfcaon he process depends on fve unnown parameers,,, Q, Q, where R,,,1,,. When we analyze me seres whch s beleved o be a rajecory of some JD()J process hen one does no really now f a gven daa pon observaon has been generaed by he pure dffuson or he jump-dffuson componen. In oher words, one canno deermne 1 Q s whch componen of he seres n (),.e. ;, Q,,1, responsble for he observaon. To manage he problem we nroduce laen varables Z such ha Z,1, and PZ j wj where 1,...,n and j,1, Z,..., 1 Z n. The value Z means no jump a (an nerval). The values Z 1 and Z mply ha a jump occurs a (an nerval) and s value dsrbuon s Q, Q,, and ; Q ; Q respecvely. In oher words, he evens Z 1 and Z correspond o a smaller and larger jump (n erms of absolue value), respecvely. By means of Z one can deec mes of jumps. Now, he Bayesan JD(M)J model enhanced wh laen varables Z s gven by: 746

5 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 p x,, Z px, Zp, Z. Formally, he occurrence of a jump s equvalen o he even Z. Unforunaely, one does no observe Z, bu he poseror probably of a jump, PZ x, can be evaluaed for each 1,..., n. Le us assume ha a jump occurs a he -h perod f he probably P Z x exceeds an arbrarly chosen value of. 5. The resulng seres conssng of zeros and ones correspondng o such s ha PZ x. 5 and Z x. 5 P, respecvely, s furher employed n sudyng he jump cluserng phenomenon. Poseror characerscs of all unnown quanes are calculaed va he Marov Chan Mone Carlo (MCMC) mehods, combnng he Gbbs sampler, he ndependence and he sequenal Meropols-Hasngs algorhms, as well as he accepance-rejecon samplng (Gamerman and Lopes, 6). The deals on he pror srucure nroduced no he model, he adoped MCMC mehods and he resuls of esmaon could be fnd n (Kosrzews, 13a). 3 The one-dmensonal Hawes model Le us consder he smple pon process N n me doman. The smple pon process N mgh be specfed by condonal nensy E N, hsory of he process N up o and N,. The condonal nensy 1 lm, where represens he, s he number of pons (jumps) n he nerval s nerpreed as he nsananeous rae of occurrence of evens a me. A well-nown example of he smple pon process s he (pon) Posson process. The homogeneous Posson process couns evens ha occur a a consan rae, whereas he non-homogeneous one couns evens ha occur a a varable (me-dependen) rae E. An expeced number of evens (jumps) over a fne nerval,t N, T d T. 747 s The Hawes process s he smple pon process. The one-dmensonal Hawes-ype cluser model (Daley and Vere-Jones, 3) for he mes of evens (jumps) consdered n he paper s an example of he classcal lnear Hawes process specfed by he condonal nensy : g snds g, (3)

6 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 1 cz where g z K a 1 z e s usually referred o as he excng funcon. Moreover, a, c. These nequales ensure ha he condonal nensy s posve. The parameer represens he bacground rae of occurrence,.e. he nensy f here have been no pas evens. The parameers a and c conrol he level of cluserng. K represens he order of he excng funcon. Noe ha s a sochasc process. The nensy of he process depends on he enre hsory and s self-excng. The process has he cluserng propery whch s a consequence of he self-excaon feaure. Such processes are broadly employed n he leraure on, e.g., he occurrence of earhquaes. I has been shown ha under general condons, he maxmum lelhood esmaes of smple pon processes are conssen and asympocally normal (see: (Ogaa, 1978), (Daley and Vere-Jones, 3)). 4 Examples In hs secon we llusrae he mehodology presened above. Two daases are under consderaon: a smulaed and a real-world one. In boh cases we perform Bayesan esmaon of he model n queson by means of he auhor s own algorhms programmed n R (R Core Team, 13), whereas he maxmum lelhood esmaes of he Hawes model s parameers are obaned va he R pacage pproc (Peng, ). 4.1 Smulaon case sudy The seres of n = 1 random daa pons generaed from he unform dsrbuon over an nerval [,3] s under consderaon. Smulaed values are perceved as me momens of jumps occurrng over hree years. Noe ha hs mples an average of hry jumps per year. Accordng o he way he daa are generaed, he jumps do no manfes hemselves n clusers. Below we examne he resuls of fng (va he maxmum lelhood esmaor) he smulaed seres wh he Hawes process under K = 1 (see Table 1). A close-o-zero value of he esmae of ndcaes no jump cluserng. Wha s more, ˆ 33. 9, whch s close o he acual expeced number of jumps per year. Tab. 1: The MLE esmaes of he Hawes process for K = 1 and smulaon daa 748

7 4 4 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 θ a c ˆ MLE e Source: own elaboraon. Inroducng he parameers esmaes no (3) yelds he expresson of jumps condonal nensy: e. Snce he summaon erm n hs formula s close o zero, he condonal nensy of jumps s almos consan: We also esmaed a model for K = (.e. he homogeneous Posson process). Accordng o he values of he Aae nformaon creron calculaed for boh specfcaon: for K = 1, and for K =, he daa suppors he smpler model srucure. Fgure 1 presens he me momens of jumps (op) and he correspondng values of () wh ndcaed momens of jumps (boom). Fg. 1: The me momens of jumps (op) and he values of he condonal nensy (wh ndcaed momens of jumps; boom) under K = 1 (smulaed daa) Source: own elaboraon S&P1 Index To llusrae he mehodology presened n he paper, we also analyze a seres of daly logarhmc raes of reurn on he S&P1 Index over he perod from March 5, 1984 hrough July 8, The seres has already been employed by (Honore, 1998), who fs wh he Bernoull jump-dffuson model by means of he maxmum lelhood mehod, as well as by (Kosrzews, 13a). Quoaons on he S&P1 Index have been downloaded from (EconSas, 1). 749

8 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 In Table values of he Aae nformaon creron are presened for models feaurng dfferen values of K{, 1,, 4}. The resuls suppor he Hawes process wh K = 1. Table 3 dsplays he MLE esmaes of he Hawes process parameers (under K = 1) along wh her sandard errors. 75

9 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 Tab. : Aae nformaon creron values for varous K and he seres of daly logarhmc raes of reurn on he S&P1 Index K AIC Source: own elaboraon. Tab. 3: The MLE esmaes and sandard errors of he Hawes process parameers under K = 1 and he reurns on he S&P1 Index Source: own elaboraon. a c Esmaes (MLE) Sandard errors Inroducng he parameers esmaes no (3) yelds he expresson of jumps condonal The esmae ˆ. 83 represens he bacground nensy: e. rae of occurrence,.e. he nensy f here have been no pas evens. The assessed value of a: a ˆ 1.61 ndcaes ha mmedaely afer a jump, he condonal nensy ncreases by abou 1.61 evens per day and mples jump cluserng. Noe ha n longer perods whou jumps he condonal nensy ˆ. 83. The MLE esmae of he consan nensy under he homogeneous Posson process ( ˆ ) s wce as large as ˆ. A larger value of ˆ may resul from he occurrence of perods feaurng hgher nensy of jumps, hese beng no ncorporaed n he homogeneous Posson process. Fg. : Logarhmc raes of reurn on he S&P1 Index (op), he me momens of jumps (mddle) and he values of he condonal nensy (wh ndcaed momens of jumps and a doed lne represenng he value of ˆ ; boom) under K =

10 4 6 8 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 Source: own elaboraon Fgure plos he modeled seres of logarhmc reurns on he S&P1 Index, he me momens of jumps (mddle) and he values of he condonal nensy (wh ndcaed momens of jumps and a doed lne represenng he value of ˆ ; boom) under K = 1. One can easly observe perods of no jumps alernang wh he ones of frequen jumps, clearly ndcang he me-varably of jump s nensy and he jump cluserng phenomenon. Fnal remars Two major conclusons can be drawn from he research. Frsly, he jumps may manfes hemselves n clusers (he jump cluserng phenomenon). Secondly, as follows from he laer may be emprcally more jusfable o allow for he sochascy of he jump nensy, nsead of resrcng he nensy o be consan hroughou. I s worh nong ha even hough he JD()J model smlarly as some oher common specfcaons does no accoun for any dependence srucure n he occurrence of jumps, s sll nformave (n he conex of deecng jump clusers) o nspec he seres of me momens when jump occur and he seres of mes elapsed beween consecuve jumps, for sll can exhb paerns suggesve of cluserng. Specfcally, va he approach proposed n he sudy has been shown ha he seres of reurns on he S&P1 exhb clusers of jumps. The jump frequency analyses of some oher me seres, no menoned here, also suppor he me-varyng nensy of jumps and confrm he exsence of jump cluserng. Naurally, such resuls nclne one o relax he assumpon of a consan jump nensy and o allow for some (sochasc) me-varably of hs parameer. Acnowledgemens Suppor by he Faculy of Managemen, Cracow Unversy of Economcs s acnowledged. Useful commens and remars by anonymous referee are hghly apprecaed. The auhor would also le o han Łuasz Kwaows for he language verfcaon of he manuscrp. 75

11 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 References A-Sahala, Y., Cacho-Daz, J., & Laeven, R.J.A. (13). Modelng Fnancal Conagon Usng Muually Excng Jump Processes. Worng Paper. Rereved Aprl 1, 14, from hp:// Daley, D. J., & Vere-Jones, D. (3). An Inroducon o he Theory of Pon Processes: Volume I: Elemenary Theory and Mehods ( ed.). New Yor: Sprnger. EconSas: GDP Indusral Producon Money Supply Inflaon US Japan Chna. Rereved 1, from hp:// Gamerman, D., & Lopes H.F. (6). Marov Chan Mone Carlo. Sochasc Smulaon for Bayesan Inference. Boca Raon: Chapman & Hall/CRC. 6. Hawes, A.G. (1971a). Pon Specra of Some Muually Excng Pon Processes. Journal of he Royal Sascal Socey, Seres B, vol. 33, Hawes, A.G. (1971b). Specra of Some Self-Excng and Muually Excng Pon Processes. Bomera, vol. 58, Honore, P. (1998, January 4). Pfalls n Esmang Jump-Dffuson Models. Rereved Aprl 8, 14, from hp://dx.do.org/1.139/ssrn Kngh, J.L., & Sachell, S. (7). GARCH Processes Some Exac Resuls, Some Dffcules and a Suggesed Remedy. [In:] Kngh, J. L., & Sachell, S. (eds.) (7). Forecasng Volaly n he Fnancal Mares. Amserdam: Elsever. Kosrzews, M. (1a). Bayesan Prcng of he Opmal-Replcaon Sraegy for European Opon n he JD(M)J Model. Dynamc Economerc Models, vol. 1, Kosrzews, M. (1b). On he Exsence of Jumps n Fnancal Tme Seres. Aca Physca Polonca B, vol. 43, Kosrzews, M. (13a). Bayesan Inference for he Jump-Dffuson Model wh M Jumps. Communcaons n Sascs Theory and Mehods. DOI: 1.18/ Lee, S.S. (1). Jumps and Informaon Flow n Fnancal Mares. Revew of Fnancal Sudes, vol. 5, Maheu, J.M., & McCurdy, T.H. (4). News Arrval, Jump Dynamcs, and Volaly Componens for Indvdual Soc Reurns. The Journal of Fnance, vol. 59, Maneesoonhorn, W., Forbes, C., & Marn, G. (14, January 16). Inference on Self-Excng Jumps n Prces and Volaly usng Hgh Frequency Measures. Worng Paper. Rereved Aprl 1, 14, from hp://arxv.org/pdf/ v1.pdf 753

12 The 8 h Inernaonal Days of Sascs and Economcs, Prague, Sepember 11-13, 14 Ogaa, Y. (1978). The Asympoc Behavour of Maxmum Lelhood Esmaors for Saonary Pon Processes. Annals of he Insue of Sascal Mahemacs, 3, Osewals J., & Ppeń M. (4). Bayesan Comparson of Bvarae ARCH-TYPE Models for he Man Exchange Raes n Poland, Journal of Economercs, vol. 13, Pajor A. (9). Bayesan analyss of he Box-Cox ransformaon n Sochasc Volaly models. Dynamc Economerc Models, vol. 9, Peng R.D. (). Mul-Dmensonal Pon Process Models n R. Rereved Aprl 8, 14, from hp:// R CORE TEAM. R: a language and envronmen for sascal compung. The R Foundaon for Sascal Compung, 13, hp:// Rasmussen, J.G. (13). Bayesan Inference for Hawes Processes. Mehodology and Compung n Appled Probably, vol. 15, Conac Macej Kosrzews Cracow Unversy of Economcs ul. Raowca 7, Kraów, Poland macej.osrzews@ue.raow.pl 754