20h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 1 6 December 2013 www.mssanz.org.au/modsim2013 Time Series Properies of Liquidaion Discoun F. Chan a, J. Gould a, R. Singh a and J.W. Yang b a School of Economics and Finance, Curin Universiy, GPO BOX U1987, Perh, Wesern Ausralia, 6845 b School of Accouning and Finance, Universiy of Wesern Ausralia, Perh, Wesern Ausralia, 6009 Email: ranjodh.singh@curin.edu.au Absrac: This paper proposes an approach for quanifying liquidiy risk. Urgen liquidaion of a porfolio will enail a liquidaion discoun. This is he marke impac discoun in value yielded by he immediae sale of he porfolio relaive o is in hand marke value calculaed from he prevailing marke condiions.the proposed approach is o firsly consruc he log liquidaion discoun rae using sock marke daa available from he order book. The behaviour of his empirical ime series is modelled and subsequenly used o predic fuure behaviour of he liquidiy risk associaed wih he porfolio. This is achieved by consrucing eigh differen sized porfolios, each corresponding o differen numbers of shares from N socks over wo ime periods (morning and afernoon). Each sock is o be liquidaed on a daily basis. The bid side order book is used o price he immediae sale of a given sock a ime. The price differenial beween he bid value and marke value of he sock is defined as he liquidaion discoun rae of he sock a ime. Replicaing his process for N socks produces a ime series of porfolio liquidaion discoun raes. Specifically, here are oal of eigh ime series which based on eigh differen scenarios, each consising of a differen number of shares for a given sock. These scenarios are represened by α which denoes differing proporions of all shares on issue for a given sock. A log ransform is applied o he series and hese are furher segmened ino wo ime periods o invesigae liquidiy behaviour over ime. This paper proposes o model he ime series properies of he log liquidaion discoun rae using he Auoregressive Fracional Inegraed Moving Average (ARFIMA) - Generalized Auoregressive Condiional Heeroskedasiciy (GARCH) model. The mean componen of he series is modelled using he ARFIMA(r, d, s) model and conains boh ARMA(r, 0, s) and ARIMA(r, 1, s) as special cases (d = 0 and d = 1 respecively). The GARCH(p, q) model is used o model he variance componen. A number of models are esed under varying lag srucures i.e. differen values of p and q. Model performance is based on a model s abiliy o forecas fuure values of log liquidaion discoun rae. The forecas accuracy is measured using he mean square error (MSE). Opimal models resuling from differing values of α over wo ime periods are idenified. The resuls indicae ha he ARFIMA(p, d, q) - GARCH(p, q) model consisenly produces he mos accurae forecass over boh ime periods. For pracical purposes a simpler model (in erms of lag srucure) is proposed. This model offers a more inuiive inerpreaion wih only a marginal loss in performance. The parameer esimaes peraining o each model are averaged over all values of α for each ime period. This produces a wo final models each corresponding o a ime period. Using hese models one can forecas (over n horizons) he variance of he log liquidaion discoun rae. This forecas is inerpreed as he fuure liquidiy risk associaed wih he porfolio. The empirical resuls sugges ha he variance converges o is long run value a a faser rae in he morning compared o he afernoon. Keywords: Liquidiy risk, fracional differencing, GARCH, ime of day effec 1215
1 INTRODUCTION Sock marke invesors are mos obviously subjec o marke risk. This is he risk ha fuure marke prices will be less favourable han oday. Addiionally, invesors will also be subjec o liquidiy risk. This is he risk ha he volume of he shares o be raded canno be immediaely ransaced a prevailing marke prices. The degree of absence of sock marke liquidiy i.e. illiquidiy poses a real cos o invesors. Previous sudies on liquidiy risk include Brennan and Subrahmanyam (1996) who invesigae he empirical relaion beween monhly sock reurns and measures of illiquidiy obained from inradaily daa. Bersimas and Lo (1998) derived porfolio ransacion sraegies ha opimize he rade-off beween marke impac cos and he risk associaed wih price volailiy. Engel e al. (2006) inroduced liquidaion value a risk o assess he price risk versus marke impac cos rade-off under various liquidaion sraegies. Aiken and Comeron- Forde (2003) and Goyenko e al. (2009) provide a summary and comparison of a wide variey of many approaches o measuring liquidiy. This paper proposes an approach for quanifying liquidiy risk; i does so by aking a perspecive of a sock invesor wih a long spo posiion in a porfolio. Urgen liquidaion of a porfolio will enail a liquidaion discoun. This is he marke impac discoun in value yielded by he immediae sale of he porfolio relaive o is in hand marke value calculaed from he prevailing marke condiions. The proposed approach is o firsly model he empirical ime series behaviour of he log liquidaion discoun rae and subsequenly use his o predic fuure behaviour of he liquidiy risk associaed wih he porfolio. This is a sep owards improved recogniion of liquidiy risk wihin analyical models of general porfolio risk. In fac, Selemens (2001) promoes he developmen of risk assessmens ha ake accoun of marke liquidiy and consideraion of how such measures could be used in he disclosure of marke risk. This paper has he following srucure: Secion 2 describes he consrucion of he liquidaion discoun rae. Secion 3 oulines he modelling echnique applied o he liquidaion discoun rae as well as he forecasing mehodology used in he paper. Secion 4 saes he daa sources used in his sudy and discusses he resuls of hfrom a pracical perspecive. Lasly, secion 5 summarizes he major findings and limiaions of his sudy. 2 LIQUIDATION DISCOUNT RATE A porfolio consiss of N company socks a ime where x = 1, 2, 3,.., N. Le S x denoe he oal number of ordinary shares on issue belonging o company x a ime. I is assumed ha his porfolio conains αs x shares of company x a ime. The α is a proporionae holding facor and as such represens he fracion of all shares on issue. An order can be hough of as an inenion o buy or sell a quaniy of shares of given company sock a a specified price. An order book ranks he buy orders (bids) from he highes o lowes bidding prices and he sell orders (asks) are ranked from he lowes o highes asking prices. Consequenly, he bids are mached wih he asks for a given company sock in an order-driven marke. This process occurs insananeously hroughou he marke s operaing hours. Le q,i x represen he quaniy of shares for a buy order a a bid price bx,i for company x a ime. A i = 1, q,1 x is he quaniy of shares for a buy order a he highes bid price b x,1 for company x a ime. Similarly, when i = 2, q,2 x is he quaniy of shares for a buy order a he second highes bid price b x,2 for company x a ime. When i = m (he maximum value of i), q,m x is he quaniy of shares for a buy order a he lowes bid price b x,m for company x a ime. The liquidaion of αs x shares will expend he bid side order book o a cerain deph. In oher words, he order is processed saring from he bes price and quaniy combinaion available hrough o he ih bes combinaion unil he number of shares o be liquidaed a ime is obained. This is represened by he following expression: q,i x αs x i=0 +1 q x,i (1) where (0, 1, 2, 3.., m) denoes he deph of he bid side order book for a given proporion (α) of x shares a ime. Noe ha he value of α direcly influences he deph i.e. a large number of liquidaed shares will deplee he bid order book o a greaer deph han a small number of shares. In cases where he number of shares o be liquidaed exceeds he quaniy of available shares, ha is he deph of he bid side order book is exhaused i is assumed ha he m + 1h bes bid price of he excess shares is zero. This is an exremely rare 1216
occurrence in he empirical daa. However, his assumpion is conservaive and as such represens he wors case scenario. The quaniy of x excess shares a ime is defined as q x,m+1 = αs x m q,i. x (2) The liquidaed value of αs x shares is given by lv α,x lv α,x = i=0 q,ib x x,i + (αs x i=0 where q x,i)b x, +1. (3) The marke value of αs x shares is denoed as mv α,x and is assumed o be equal o he highes bid price plus a small premium p. This premium is derived from he empirical daa iself. mv α,x = αs x (b x,1 + p). (4) For a given value of α a ime, he porfolio liquidaion discoun rae is expressed as L α = 1 N x=1 N x=1 lv α,x mv α,x. (5) This is simply he discoun rae beween he aggregaed liquidaion value (equaion (3)) and he aggregaed marke value (equaion (4)). For he purpose of analysis a log ransformaion is applied o equaion (5). This ransformed quaniy is denoed by l α. 3 MODELLING FRAMEWORK This sudy aemps o invesigae he ime series properies of he log liquidaion discoun rae. Preliminary analysis indicaes ha he ime series exhibis a slow decaying auocorrelaion funcion. This could poenially indicae ha he series is fracionally inegraed. Consider he following model: φ r (L)(1 L) d l,α = µ + θ s (L)ɛ (6) ɛ = η h η iid(0, 1) (7) p q h = ω + α i ɛ 2 i + β i h i. (8) Equaion (6) represens he ARFIMA(r, d, s) model where r and s denoe he order of auoregressive and moving average pars of he model respecively. The d parameer represens he fracional difference erm in he model and deermines he long run behaviour of he ime series. I is convenienly relaed o he Hurs index (d = H 1 2 ) which is a measure of he long erm memory/dependence in a ime series (refer o Mandelbro and Van Ness (1968) for more deails). The d parameer is esimaed in he modelling process. However, fixing he value of d o 0 or 1 produces he ARMA and ARIMA models respecively. Granger and Joyeux (1980) and Hosking (1981) inroduced his model. I aemps o capure he dynamics of he mean of he process. In equaion (6), µ represens he drif erm and φ r (L) = 1 φ 1 (L) φ 2 (L)... φ r (L) is he auoregressive operaor where L is he lag operaor such ha Ly = y 1. Similarly, θ s (L) = 1+θ 1 (L)+θ 2 (L)+...+θ s (L) represens he moving average operaor. Boh hese polynomials have heir uni roos ouside he uni circle and share no common roos. The fracional differencing operaor (1 L) d can be rewrien as (1 L) d d(1 d) = 1 dl L 2 2! = 1 + j=1 d(1 d)(2 d) L 3 +... (9) 3! Γ(j d) Γ(j + 1)Γ( d) Lj (10) 1217
where Γ(.) is he Gamma funcion. This model assumes ha he condiional variance is consan over ime. However, esimaing he ARFIMA model for he log liquidaion discoun rae shows ha he residuals are no consan over ime. Therefore, he variance of he process is modelled using a GARCH(p, q) model (equaion (8)) where p and q represen he order of he auoregressive and moving average pars of he model respecively. Bollerslev (1986) inroduced he GARCH model whils exending he work of Engle (1982) on ARCH models. This allows one o model a ime varying condiional variance. Hence, he ARFIMA(r, d, s) - GARCH(p, q) model is used o model he log liquidaion discoun rae. When ω > 0, α 1, α 2,..., α p > 0, β 1, β 2,..., β q 0, p α i + q β i < 1 and he esimaed value of d ( 1, 0.5), he proposed model is saionary as well as inverible. Specifically, when d (0, 0.5) he model exhibis long memory. In a pioneering sudy, Baillie e al. (1996) apply his model o analyze he inflaion rae ime series. A laer sudy by Ling (2003) shows ha he US consumer price index inflaion series is fracionally inegraed. The AFRIMA(r, d, s) - GARCH(p, q) model is used o capure he ime series behaviour of log liquidaion discoun rae for a given α and ime period. This is achieved by assessing he model s performance under varying lag srucures. For example, le r and s ake on values of 0,1,2 and 3. Similarly, p and q are assigned values of 1,2 and 3. The d parameer is eiher se o a fixed values of 0 and 1 or esimaed. Given ha α akes eigh values (0.0001% - 0.0008%) for each ime period (morning and afernoon), he number of models ha are fied is 6912 (4 r values, 3 d values, 4 s values, 3 p values, 3 q values, 8 α values and 2 ime periods). From his se, models are seleced based on heir abiliy o forecas fuure values of log liquidaion discoun raes. Given ha he daa has 1283 observaions for each value of α for he morning period, model parameers are esimaed on he firs 1000 observaions. These esimaes are hen used o generae 283 forecass. These forecass are compared o he 283 acual observaions. Le d,α denoe he difference beween he acual value and he forecas on h day for a given α i.e. d,α = l,α ˆl,α. (11) The opimal model is he one ha minimizes hese differences for all and α. The mean squared error (MSE) is used for his purpose. The MSE is defined as MSE α = 1 f j N j =N j f j+1 d,α 2 (12) where f j is he forecas horizon and N j is he oal number of observaions for a given ime period j. Similarly, he mean absolue deviaion (MAD) is also compued for comparison purposes. MAD α = 1 f j N j =N j f j+1 d,α d,α. (13) where d,α he mean value of he deviaions for a given α. The models producing he minimum MSE α across all α are seleced. The resuls from boh measures are consisen for mos values of α. The Schwarz-Bayesian Informaion Crierion (SBIC) ogeher wih sandard residual informaion is used o assess he validiy of a model fi. The SBIC is defined as SBIC = 2LL N + k lnn N where LL is he log likelihood value, N is he number of observaions (1000) and k is he number of parameers ha are esimaed. The resuling opimal models (one model for each value of α and ime period) are compared o a firs order lag model. This is done in order o gauge he increase in forecasing abiliy ha higher order models possess. I is expeced ha his comparison will assis in he model selecion process. The parameer esimaes of he seleced model are compued across all values of α and ime period. These esimaes are averaged over each ime period in order o yield wo final models i.e. one for each ime period. From a pracical perspecive, an invesor may wan o forecas (over a given horizon) he liquidiy risk based on he curren informaion. This liquidiy risk is defined as he variance of he log liquidaion discoun rae. Assuming ha he variance componen is governed by GARCH(1,1), ieraing equaion (8) over n horizons given he curren informaion yields he following resul: E[h +n I 1 ] = h + (α 1 + β 1 ) n (h h) (15) where h = ω/(1 α 1 β 1 ) and represens he uncondiional variance. For furher deails refer o Baillie and Bollerslev (1992). (14) 1218
4 RESULTS 4.1 DATA Daa for his sudy consiss of socks lised on he Ausralian Securiies Exchange (ASX). The ASX is a purely order-driven marke. In January of each year from 2006 o 2011, a value weighed porfolio of he op N = 10 socks from he ASX and S&P200 index is compiled. The number of shares on issue is exraced from he Morning DaAnalysis daabase. In addiion, morning (10:15 a.m.) and afernoon (15:45 p.m.) snapshos of he bid side order book are obained from he Securiies Indusry Research Cenre of Asia-Pacific (SIRCA) AusEquiy daabase for he period Ocober 2006 hrough o Ocober 2011. This daabase provides bid orders o a maximum deph of 20 i.e. m = 20. The value of he premium p in equaion (4) is $0.005. This is he average gap beween he highes bid and he curren share price. The values of α are 0.0001%, 0.0002%,.., 0.0008%. Given ha he maximum deph of he bid side order book is 20, increasing he value of α beyond 0.0008% more frequenly produces excess shares (refer o equaion (2) in secion 2). Hence, due o he available deph (daa) resricion he maximum value of α is se o 0.0008%. For all combinaions of α (0.0001% 0.0008%) and d [0, 1], he model wih he minimum MSE is seleced. Resuls are classified by he differencing scheme implemened. As a consequence, here are 24 opimal models for each ime period resuling from eigh values of α and hree differencing opions. For a given value of α and ime period, each model s MSE is divided by of he minimum MSE produced by he hree differencing opions. This relaive MSE allows one o rank MSE values. The resuls indicae ha he fracional differencing opion produces he minimum MSE across all values of α for he morning period. This is also he case for majoriy of he α values for afernoon period. The excepions being α values of 0.0002%, 0.0003% and 0.0004%. For hese cases, d = 1 leads by a small margin. This may sugges ha he zero differencing (d = 0) is no be enough (under differencing) and he firs difference (d = 1) leads o over differencing. Hence, fracional differencing (0 < d < 1) is deemed o be opimal. However, he resuls across all differencing opions are quie close ogeher. The esimaed value of d is always less 0.5 and his indicaes ha here is long memory presen in he daa. These values are shown in able 2. For boh ime periods, he lag srucure seems o be lile erraic for firs hree or four values of α. The remaining values of α have an idenical lag srucure (wih he excepion of α = 0.0008%). This lag srucure represens an ARFIMA(1, d, 0) - GARCH(2, 3) model. A naural quesion o ask is ha how much beer is GARCH(2,3) compared o GARCH(1,1). I can be argued ha on average one canno do beer han a GARCH(1,1). To es his proposiion, an ARFIMA(1, d, 0)- GARCH(1, 1) model is fied and benchmarked agains he opimal model for each value of α. The resuls are displayed in able 1. The MSE resuling from he proposed model is wihin four percen of he MSE of he opimal model. The simpler lag srucure offers a more inuiive inerpreaion wih only a marginal compromise in forecasing abiliy. The parameer esimaes of he proposed model are compued for all α values across boh ime periods as shown in able 2. I is eviden from he resuls ha hese esimaes are influenced by he values of α. In paricular, he drif erm µ, becomes marginally larger as he value of α increases. Some deparures from cenral values can be seen in he resuls. These are caused by ouliers in he daa series. For example, he p and q parameer esimaes for he morning periods a α = 0.0002% are differen from he remaining p and q values. The resuls also indicae ha he fracional differencing esimae (d) is largely he same across boh ime periods. The compued value of d is less han a 0.5 implying ha he series is saionary. Furhermore, his ranslaes o a Hurs index of approximaely 0.9 (defined in secion 3) which implies ha here is a relaively high degree of persisence presen in he log liquidaion discoun rae. Given ha he parameer esimaes remain relaively unchanged over differen values of α, i may be ideal o propose an overall model for each ime period. The overall esimaes are he average values of he individual esimaes across all values of α. Baes and Granger (1969) showed ha combining models will produce more superior forecass han individual models on average. The resuling models (morning and afernoon) are fied on all values of α for a given ime period. The MSE is once again used o assess he qualiy of he fi. From he resuls, i is eviden ha he MSE increases as he value of α increases. This is o be expeced since he bid side order book sars o move owards is maximum deph as he number of shares o be liquidaed increases i.e. as α increases. The afernoon models produces lower MSE values compared o he morning model. This is due o he fac ha he orders are being fulfilled more smoohly. Poenially, his could imply ha he marke is more sable in he afernoon compared o he morning. 1219
Table 1. MSE comparison beween he opimal and proposed model Morning Afernoon Opimal alpha0001.3.0.2.1 0.02675 alpha0001.0.1.2.1 0.01117 Proposed alpha0001.1.0.1.1 0.02713 alpha0001.1.0.1.1 0.01155 Opimal alpha0002.1.0.2.3 0.03771 alpha0002.1.2.2.2* 0.01716 Proposed alpha0002.1.0.1.1 0.03787 alpha0002.1.0.1.1 0.01782 Opimal alpha0003.3.2.2.1 0.04305 alpha0003.1.2.3.2* 0.02027 Proposed alpha0003.1.0.1.1 0.04368 alpha0003.1.0.1.1 0.02083 Opimal alpha0004.1.0.3.2 0.04724 alpha0004.1.3.1.1* 0.02234 Proposed alpha0004.1.0.1.1 0.04762 alpha0004.1.0.1.1 0.0227 Opimal alpha0005.1.0.1.3 0.05078 alpha0005.1.0.1.3 0.02279 Proposed alpha0005.1.0.1.1 0.05132 alpha0005.1.0.1.1 0.02308 Opimal alpha0006.1.0.1.3 0.05388 alpha0006.1.0.1.3 0.02273 Proposed alpha0006.1.0.1.1 0.0546 alpha0006.1.0.1.1 0.02306 Opimal alpha0007.1.0.2.3 0.05646 alpha0007.1.0.2.3 0.02324 Proposed alpha0007.1.0.1.1 0.05726 alpha0007.1.0.1.1 0.02361 Opimal alpha0008.1.0.2.3 0.05905 alpha0008.1.0.3.3 0.02401 Proposed alpha0008.1.0.1.1 0.05977 alpha0008.1.0.1.1 0.02442 Table 2. Parameer esimaes for morning and afernoon daa Morning α (%) µ r d ω p q 0.0001-7.93035-0.26481 0.37842 0.00058 0.01624 0.97434 0.0002-7.66334-0.27665 0.40132 0.04754 0 0.37783 0.0003-7.46893-0.25241 0.38740 0.00055 0.01202 0.98096 0.0004-7.29658-0.24724 0.38464 0.00067 0.01479 0.97710 0.0005-7.14616-0.23499 0.38435 0.00064 0.01556 0.97667 0.0006-7.01559-0.22552 0.38687 0.00055 0.01443 0.97883 0.0007-6.90785-0.22181 0.39075 0.00057 0.01495 0.97806 0.0008-6.82838-0.23730 0.40638 0.01865 0.06719 0.71947 Afernoon α (%) µ r d ω p q 0.0001-8.09875-0.14601 0.37772 0.00020 0.01953 0.97461 0.0002-7.96591-0.15864 0.37295 0.00084 0.03365 0.94928 0.0003-7.85305-0.17299 0.37768 0.00078 0.03282 0.95332 0.0004-7.74218-0.17606 0.38263 0.00021 0.02183 0.97436 0.0005-7.64380-0.18472 0.38769 0.00011 0.02058 0.97730 0.0006-7.55451-0.18912 0.39187 0.00006 0.02072 0.97782 0.0007-7.47340-0.19409 0.39628 0.00003 0.02096 0.97802 0.0008-7.39675-0.19912 0.39996 0.00001 0.02056 0.97863 4.2 LIQUIDITY RISK From equaion (15) i is eviden ha as forecas horizon increases o some large value i.e. n he condiional variance forecas approaches he uncondiional variance. The rae a which his convergence happens is dependen on quickly (α 1 + β 1 ) n 0. In his empirical sudy α 1 + β 1 0.89 for he morning period and 0.99 for he afernoon period. This implies ha he condiional variance (risk) of he morning log liquidaion discoun rae converges o is limiing case (uncondiional variance) much faser han he afernoon discoun rae. From an invesmen prospecive, i may be bes o liquidae in he afernoon since, he discoun rae in he morning converges o is maximum a a faser pace. As menioned in he previous secion, he order book s operaion is smooher in he afernoon compared o he morning. 1220
5 CONCLUSION This paper has inroduced a liquidaion discoun measure. This being he marke impac discoun in value yielded by immediae sale of he porfolio relaive o is in-hand marke value. For differenly sized porfolios of major Ausralian socks, i is found ha an ARFIMA(1, d, 0)-GARCH(1, 1) model offers an explanaion and a consisen paramerizaion for log liquidaion discoun rae. This model is used o quanify liquidiy risk in erms of he condiional variance of he log liquidaion discoun rae. The resuls sugges ha his risk changes over ime. However, hese resuls are resriced o op performing socks in he ASX over he period Ocober 2006 hrough o Ocober 2011. ACKNOWLEDGMENTS The hird auhor would like o hank he firs auhor for heir insighful suggesions and echnical experise. The hird auhor would also like o hank he second and fourh auhor for proposing he research quesion. The auhors would like o hank he wo anonymous referees for heir helpful suggesions. The auhors are graeful for he financial assisance provided by he Ausralian Research Council. REFERENCES Aiken, M. and C. Comeron-Forde (2003). How should liquidiy be measured? Pacific-Basin Finance Journal 11(1), 45 59. Baillie, R. T. and T. Bollerslev (1992). Predicion in dynamic models wih ime-dependen condiional variances. Journal of Economerics 52(12), 91 113. Baillie, R. T., C.-F. Chung, and M. A. Tieslau (1996). Analysing inflaion by he fracionally inegraed ARFIMA-GARCH model. Journal of Applied Economerics 11(1), 23 40. Baes, J. M. and C. W. J. Granger (1969). The combinaion of forecass. OR 20(4), pp. 451 468. Bersimas, D. and A. W. Lo (1998). Opimal conrol of execuion coss. Journal of Financial Markes 1(1), 1 50. Bollerslev, T. (1986). Generalized auoregressive condiional heeroskedasiciy. Journal of Economerics 31(3), 307 327. Brennan, M. J. and A. Subrahmanyam (1996). Marke microsrucure and asse pricing: On he compensaion for illiquidiy in sock reurns. Journal of Financial Economics 41(3), 441 464. Engel, R., R. Fersenburg, and J. Rusell (2006). Measuring and modeling execuion cos and risk. Working Paper. Engle, R. F. (1982). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica 50(4), pp. 987 1007. Goyenko, R. Y., C. W. Holden, and C. A. Trzcinka (2009). Do liquidiy measures measure liquidiy? Journal of Financial Economics 92(2), 153 181. Granger, C. W. J. and R. Joyeux (1980). An inroducion o long-memory ime series models and fracional differencing. Journal of Time Series Analysis 1(1), 15 29. Hosking, J. R. M. (1981). Fracional differencing. Biomerika 68(1), 165 176. Ling, S. (2003). Adapive esimaors and ess of saionary and nonsaionary shor- and long-memory ARFIMA-GARCH models. Journal of he American Saisical Associaion 98(464), 955 967. Mandelbro, B. B. and J. W. Van Ness (1968). Fracional brownian moions, fracional noises and applicaions. SIAM review 10(4), 422 437. Selemens, B. f. I. (2001). Final Repor of Mulidisciplinary Working Group on Enhanced Disclosure. Bank for Inernaional Selemens. 1221