Heavy-tailed modeling of CROBEX

Transcription

1 Heavy-tailed modelig of CROBEX DANIJEL GRAHOVAC, PhD* NENAD ŠUVAK, PhD* Article** JEL: C15, C, C51 doi: /fitp * The authors would like to thak two aoymous referees for their helpful commets ad suggestios. ** Received: Jue 1, 015 Accepted: October 11, 015 The article was submitted for the 015 aual award of the Prof. Dr. Marija Hažeković Prize. Daijel GRAHOVAC J. J. Strossmayer Uiversity of Osijek, Departmet of Mathematics, Trg Ljudevita Gaja 6, Osijek, Croatia dgrahova@mathos.hr Nead ŠUVAK J. J. Strossmayer Uiversity of Osijek, Departmet of Mathematics, Trg Ljudevita Gaja 6, Osijek, Croatia suvak@mathos.hr

2 41 Abstract Classical cotiuous-time models for log-returs usually assume their idepedece ad ormality of distributio. However, owadays it is widely accepted that the empirical properties of log-returs ofte show a specific correlatio structure ad deviatio from ormality, i most cases suggestig that their distributio is 39 (4) (015) heavy-tailed. Therefore we suggest a alterative cotiuous-time model for logreturs, a diffusio process with Studet s margial distributios ad expoetially decayig autocorrelatio structure. This model depeds o several ukow parameters that eed to be estimated. The tail idex is estimated by the method based o the empirical scalig fuctio, while the parameters describig mea, variace ad correlatio structure are estimated by the method of momets. The model is applied to the CROBEX stock market idex, meaig that the estimatio of parameters is based o the CROBEX log-returs. The quality of the model is assessed by meas of simulatios, by comparig CROBEX log-returs with the simulated trajectories of Studet s diffusio depedig o estimated parameter values. Keywords: log-retur, heavy-tailed distributio, Studet s distributio, diffusio process, geometric Browia motio 1 INTRODUCTION CROBEX is the official stock market idex of the Zagreb Stock Exchage, first published o September 1, The idex is based o the free float market capitalizatio ad icludes the stocks of 5 compaies. CROBEX serves as the mai idicator for the Croatia stock market ad closely describes the ecoomic treds of the coutry. Like ay other stock market idex, CROBEX ca be studied as the value (price) of a risky asset at some time poit. A realistic modelig of a risky asset price through time is of great practical importace, especially i the risk assessmet ad pricig of fiacial derivatives. For this purpose, the values of the fiacial asset i some time iterval ca be cosidered as a realizatio of some stochastic process (P t, 0 t T). May differet classes of processes have bee proposed as models for (P t ), both i discrete ad cotiuous time (see e.g. Tsay, 010 for a overview). Istead of the price process P t, t = 0, 1,..., T, the fiacial time series data are usually ivestigated through the log-returs of the origial series, that is: Pt Rt = l, t = 1,, T. P t 1 The classical model for the price of the risky asset is a geometric Browia motio (GBM), also kow as the Black-Scholes model. If the asset value is assumed to follow the GBM model, the the log-returs R t, t = 1,..., T, form a sequece of idepedet ormally distributed radom variables. This feature of the log-returs is owadays cosidered urealistic for may fiacial data. Cotrary to the GBM model, the log-returs of may risky assets exhibit very weak correlatio, but are far from idepedet. Moreover, the distributio of log-returs has tails much heavier tha Gaussia, thus showig that extreme evets are more probable tha

3 i the GBM model. Models takig this ito accout use the so-called heavy-tailed distributios which have tail probabilities decayig at ifiity as slow as the power fuctio. The importace of heavy-tailed distributio lies i the fact that they ca realistically quatify the probabilities of extreme evets. Such evets are especially importat i fiacial modelig while igorig the possibility of large fluctuatios ofte leads to a severe uderestimatio of risk. More details o these ad other stylized facts of log-returs ca be foud i Cot (001). I this paper we aalyze the historic data of the CROBEX idex. First, we show that the log-returs of CROBEX exhibit behavior characteristic for a risky asset. I particular, we preset evidece that the uderlyig distributio of log-returs is heavy-tailed ad far from Gaussia. Awareess of such property is of great importace i risk assessmet. I the ext step, we claim that the distributio of logreturs ca be successfully modeled with the Studet s t-distributio, which is heavy-tailed. May empirical studies have cofirmed this for other fiacial data (see e.g. Hurst ad Plate, 1997). Sice this makes a stadard GBM model for the asset price iappropriate, i sectio 3. we propose a ew model for the log-returs based o the Studet s diffusio process. Diffusio processes have bee successfully used before i fiacial modelig (see Bibby ad Sørese, 1996; ad Rydberg, 1999). The model proposed here uses a statioary solutio of the diffusio stochastic differetial equatio which has Studet s margial distributio. Not oly is the distributio modeled more realistically, but the depedece structure is also allowed to be more complex, sice the costructed Studet s diffusio process exhibits a form of weak depedece. I sectio 4 we estimate the parameters of the proposed model by usig some recetly itroduced techiques. We tackle the statistically challegig problem of estimatig the tail idex of the log-returs which is the mai parameter of ay heavy-tailed distributio. It is worth metioig that the estimatio method used is o-parametric, i the sese that the tail idex parameter is estimated without a assumptio of the particular form of the uderlyig distributio. Estimatio of other parameters of the model is also coducted with a brief discussio o the asymptotic properties of the estimators used. The quality of the proposed model is assessed by the meas of simulatios. Sectio 5 cotais some cocludig remarks ad possible improvemets with some guidelies ad idicatios of further applicatios of the results (4) (015) The heavy-tailed ature of the CROBEX idex has bee addressed so far i several refereces. I Žiković ad Pečarić (010), left ad right-had CROBEX tails are fitted to separate geeralized Pareto distributios, ad such a model proved to be successful i forecastig some risk measures. The same distributio is advocated i Arerić, Lolić ad Galetić (01). Sice the method we use here i assessig the tail idex is o-parametric, it provides a more robust estimate. Most frequetly traded stocks icluded i the CROBEX were modeled by GARCH process with Studet s iovatios i Arerić, Juru ad Pivac (007), cofirmig the heavy-tailed structure of CROBEX. A similar model has bee cosidered i Miletić ad Miletić (015) for CROBEX ad other stock idices of the Cetral

4 414 ad Easter Europea capital markets. O the global level, there is a iexhaustive list of papers dealig with modelig of stock market idices. Comprehesive empirical studies aalyzig distributio of log-returs were doe i Gray ad Frech (1990), Hurst ad Plate (1997), Jodeau ad Rockiger (003); see also Tsay (010) ad refereces therei. 39 (4) (015) THE CLASSICAL MODEL AND HEAVY-TAILED DISTRIBUTIONS I empirical fiace, the classical model refers to the price process of the risky asset modeled with the GBM. GBM is a cotiuous time stochastic process St = S t Wt t T + 0ex α σ σ, 0. Here S 0 is the iitial price ad (W t, 0 t T) deotes the stadard Browia motio o [0, T], that is a process with statioary idepedet icremets, cotiuous sample paths ad such that W t is Gaussia (ormal), W t ~ N (0, t). The parameter α > 0 ca be iterpreted as the expected rate of retur ad parameter σ > 0 as volatility, ad therefore oe of the idicators of the riskiess of the asset. Istead of the price process P t, t = 0, 1,..., T, the fiacial time series data is usually ivestigated through the log-returs of the origial series. More precisely, log-retur at time t is defied as Pt Rt = l, t = 1,, T. P t 1 Log-returs are scale idepedet quatities ad ca usually be plausibly modeled as a statioary sequece. Moreover, there is o loss of iformatio as kowig the log-returs values ad the iitial price P 0 gives the price at time T by the equatio: T PT = P0 exp Rt (1) t= 1 The advatage over the usual returs (P t P )/P is that R is additive i the sese t-1 t t of (1) ad usually statioary. If the asset value is assumed to follow the GBM model, the the log-returs are R t = α σ + Wt t =,,, σδ 1 T () where W t = W t W t-1 are oe-step icremets of the Browia motio. This meas that R t, t = 1,..., T, is a sequece of idepedet ormally distributed radom variables, more precisely

5 R t ~ α σ, σ. 415 As discussed i the Itroductio, it is highly ulikely to ecouter this property i may fiacial data ad there is a eed for usig heavy-tailed distributios. Heavy-tailed distributios are of cosiderable importace i modelig a wide rage of pheomea i fiace ad may other fields of sciece. Promiet examples of such distributios are Pareto distributio, stable distributio ad Studet s t-distributio. Distributio of some radom variable X is said to be heavy-tailed with idex α > 0 if its tail probabilities decay as a power law, i.e. Lx ( ) P( X > x) = α, x where L(t), t > 0 is a slowly varyig fuctio, that is, L(tx)/L(x) 1 as x, for every t > 0. I particular, this implies that E X q < for q < α ad E X q = for q > α. The parameter α is called the tail idex ad measures the thickess of the tails. The lower the value of α is, the more probable are extreme values of X. This way extreme evets ca be modeled ad these evets are usually the most importat as they ca geerate great profit but also, more importatly, catastrophic loss. O the other had, the usual Gaussia distributio has tail probabilities that decay expoetially fast as ~e -x / whe x. For this reaso, Gaussia distributio is iadequate for modelig pheomea that ca exhibit extreme behavior. 39 (4) (015) Pioeerig work i applyig heavy-tailed models to fiace was doe by B. Madelbrot i Madelbrot (1963) where stable distributios have bee advocated for describig fluctuatios of cotto prices. Stable distributios allow for tail idex value 0 < α <, which is owadays cosidered a urealistically small value for most time series data. A richer modelig ability is provided by the Studet s t-distributio, which allows for arbitrary tail idex parameter..1 THE CROBEX LOG-RETURNS The data cosidered i this paper cosist of 54 closig values of the CROBEX stock market idex collected i the period from Jauary 3, 005 util December 31, 014. The time series of values is show i figure 1(a). This ad all other figures i the paper are made from the publicly available CROBEX data with Wolfram Mathematica software. The log-returs of the CROBEX idex are show i figure 1(b). From the appearace of the plot, it seems plausible to model the log-returs as a realizatio of the statioary sequece of radom variables R 1,..., R T with T = 53. As the first step of the aalysis, we ivestigate the uderlyig distributio of the sequece R 1,..., R T. For this purpose, a histogram is plotted i figure (a) ad oe

6 416 ca see that it has sharper peak ad tails heavier tha Gaussia distributio. These characteristics are kow as the stylized facts of asset returs ad are commo for almost all data of this type. Heavy-tails of the uderlyig distributio are cofirmed with the QQ-plot of ormal quatiles o the x-axis with respect to the empirical quatiles o the y-axis (figure (b)). The left ed of the patter is below 39 (4) (015) the referece lie ad the right ed of the patter is above the lie which idicates tails heavier tha Gaussia. Further evidece of the heavy-tailed ature of CROBEX will be give i subsectio 4.1 where the tail idex will be estimated. Figure 1 CROBEX data i period Jauary 3, 005 December 31, 014 (a) CROBEX values (b) Log-returs Figure Distributio aalysis of CROBEX log-returs (a) Histogram of log-returs (b) QQ-plot of log-returs STUDENT S DIFFUSION AS MODEL FOR TIME EVOLUTION OF LOG-RETURNS GBM, the classical model for stock prices ad values of the stock market idices, implies both idepedece ad ormality of distributio of log-returs (equatio ()). The log-returs o fiacial markets usually are ot i correspodece with these demads, i.e. over a log time period they ofte show a specific correlatio structure ad a deviatio from ormality. I most cases their distributio exhibits heavy tails ad for CROBEX this was idicated i sectio. A atural heavytailed geeralizatio of the Gaussia distributio is provided by Studet s t-distributio, which we ow itroduce.

7 3.1 STUDENT S DISTRIBUTION Studet s distributio represets a atural choice for modelig the distributio of log-returs, because of its heavy-tailed characteristics ad still close relatioship with the ormal distributio. I order to capture more iformatio from the realized log-returs, we use Studet s distributio with three parameters: shape parameter υ>0 (also called the umber of degrees of freedom), scale parameter δ>0, locatio parameter μ R. This distributio, usually deoted as T(υ, δ, μ), is defied by the probability desity fuctio ν + 1 Г x f ( x μ )= 1+ ν δ π Г δ ν + 1 x, (3) where Γ( ) deotes the classical gamma fuctio (see Abramowitz ad Stegu, 1964). If υ is a iteger, the T(υ, υ, 0) coicides with the usual t-distributio widely used i statistics. For large values of the parameter υ Studet s T(υ, δ, μ) distributio behaves approximately like the ormal distributio. Probability desity fuctios (PDFs) of stadard ormal distributio ad Studet s distributios with zero mea, uit variace ad υ = 3, υ = 7 ad υ = 9 degrees of freedom are plotted i figure 3(a), while the right tails of all four PDFs are plotted i figure 3(b) (4) (015) Figure 3 PDFs of stadard ormal distributio ad Studet s distributios with zero mea, uit variace ad various degrees of freedom ν (a) Normal ad Studet s PDFs (b) Right tails of ormal ad Studet s PDFs Stadard ormal distributio Studet s distributio (ν=7) Studet s distributio (ν=3) Studet s distributio (ν=11)

8 418 The left ad the right-had tails of Studet s T(υ, δ, μ) distributio (3) decrease like x -ν-1, i.e. this distributio is heavy-tailed ad the tail idex correspods to degrees of freedom, that is α = υ. I particular, momets of order greater tha υ do ot exist. The cetral momet of order exists uder the restrictio < υ, N, ad it is give by the followig expressio: 39 (4) (015) E ( R ER [ ]) 1 δ + 1 ν ν = Г Г Г, = k 1 π, k, 0, = k where R is the radom variable with Studet s T(υ, δ, μ) distributio, i.e. R~T(υ, δ, μ). We will be maily iterested i its expectatio ad variace: (4) δ E[ R]= μ, ν > 1; Var( R)=, ν >. ν The model developed i this paper uses Studet s T(υ, δ, μ) distributio as the margial distributio of the statioary sequece R t, t = 1,..., T of log-returs. Studet s distributio is heavy-tailed ad thus fits the usual empirical properties for the log-returs distributio. Additioal parameters μ ad σ allow more flexibility i modelig as they describe the mea ad the variace whe υ >, i.e. whe mea ad variace exist. Whe υ, Studet s distributio reduces to ormal distributio ad the log-returs would have the same distributio as i the stadard GBM model. That distributios of log-returs ca ofte be fitted extremely well by Studet s distributio has bee cofirmed i may empirical studies; see Hurst ad Plate (1997), Heyde ad Liu (001), Heyde ad Leoeko (005) ad refereces therei. (5) 3. STUDENT S DIFFUSION MODEL Here we propose a model that geeralizes the model (1) for log-returs ad icorporates Studet s distributio (3). The obvious step i this directio would be to replace W t, t = 1,..., T i (1) by a sequece of idepedet radom variables with Studet s distributio. However, this would imply idepedece of log-returs which is a urealistic property for the asset returs. A more advaced model ca be built by takig (R t, 0 t T) to be a statioary diffusio process such that R t ~T(υ, δ, μ) for some parameters υ, δ ad μ ad which exhibits the so-called β-mixig depedece structure with the expoetially decayig rate, meaig that the coefficiet which i some way describes the depedece i the process (R t, 0 t T) teds to zero expoetially fast. This type of mixig implies aother type of mixig called α-mixig or strog mixig, which is more frequetly used i the studies of the depedece structures of stochastic processes. For more geeral details o mixig theory we refer to Bradley (005), ad Abourashchi ad Vereteikov (010). Such a choice will allow (R t, 0 t T) to have the expoetially decayig autocorrelatio fuctio ρ(t) = Corr(R s, R s+t ), 0 s < s + t T, ad heavy-tailed margial distributio.

9 A statioary diffusio process with prescribed margial distributio ca be costructed as a solutio of a particular stochastic differetial equatio (see Bibby, Skovgaard ad Sørese, 005). For ν > Studet s diffusio is a stochastic process satisfyig the stochastic differetial equatio 419 dr = θ μ R dt t ( ) + t θδ Rt μ + dwt, t T, ν 1 1 δ 0 where (W t, 0 t T) is a stadard Browia motio ad θ > is the so-called autocorrelatio or depedece parameter appearig i the autocorrelatio fuctio (6) 39 (4) (015) θt ρ()= t Corr ( R, R )= e, 0 s < s+ t T. (7) s s+ t Moreover, if Studet s diffusio starts from Studet s T(υ, δ, μ) distributio, i.e. if R 0 ~T(υ, δ, μ), the R t ~T(υ, δ, μ) for all t [0, T] ad the process (R t, 0 t T) is said to be strictly statioary (all of its fiite-dimesioal distributios are ivariat to time-shifts). For R t iterpreted as the log-retur at time t, equatio (6) could be iterpreted i view of the chage of the log-retur i a small time iterval [t, t + t], 0 t < t + t T: R ν 1 1 θδ Rt μ t+ t Rt = θ ( μ Rt) t+ + ( Wt+ t Wt δ ). (8) Equatio (8) relates the log-retur R t + t at time (t + t) to the log-retur R t at time t takig ito accout the icrease of time t ad the chage of the value of the Browia motio (W t + t W t ), i.e. its icremet betwee time poits t ad (t + t). Therefore, it could be iterpreted as follows: the log-retur R t + t could be obtaied from the historical log-retur R t by addig to it the icrease of time t with the factor θ(μ R t ) ad the icremet of the Browia motio (W t + t W t ) betwee time poits t ad (t + t) with the factor θδ ν 1 1 μ + R t δ, (9) where both factors deped o the historical log-retur R t, parameters ν, δ ad μ of the Studet s T(υ, δ, μ) distributio ad the autocorrelatio parameter θ. Factor (9) ca be uderstood as coditioal variace of the Gaussia iovatio term (W t + t W t ), coditioally o the past values of R t. This structure resembles the discrete time coditioal heteroscedasticity models, e.g. ARCH models. It is wellkow that there is a itimate relatio betwee GARCH models ad diffusios processes (see e.g. Forari ad Mele, 000 ad refereces therei).

10 39 (4) (015) 40 4 FITTING THE CROBEX LOG-RETURNS TO STUDENT S DIFFUSION I view of the empirical properties of log-returs of the risky asset preseted i sectio ad remarks o the suitability of Studet s distributio for modelig the margial distributio of log-returs preseted i sectio 3, Studet s diffusio (6) seems to be a plausible stochastic model for log-returs. I this sectio CROBEX log-returs will be fitted to Studet s diffusio. More precisely, we derive estimators of parameters ν, δ, μ ad θ ad calculate their values based o the CROBEX data. 4.1 ESTIMATION OF UNKNOWN PARAMETERS The parameter estimatio problem will be treated i three separate but depedet steps. First, the parameter ν will be estimated as the tail idex parameter by usig the method based o the empirical scalig fuctio recetly itroduced i Grahovac et al. (015). This estimated value of ν will be treated as the kow value of this parameter i the estimatio of parameters μ ad δ by the classical method of momets (see Serflig, 1980). Fially, the autocorrelatio parameter θ will be estimated by the geeralized method of momets based o Pearso s sample correlatio fuctio (see Leoeko ad Šuvak, 010) Estimatio of parameter ν The shape parameter ν correspods to the tail idex of Studet s distributio. Extreme value theory provides may methods for estimatig the ukow tail idex (see Embrechts, Klüppelberg ad Mikosch, 1997 for a overview). Here we will use a ovel approach itroduced i Grahovac et al. (015), based o the so-called empirical scalig fuctios. Suppose that we are give a zero mea sample X 1, X,..., X, comig from some statioary heavy-tailed sequece with strog mixig property with a expoetially decayig rate. Partitio fuctio of this sample is defied as S q 1 (, t) = / t t / t i= 1 j= 1 X ( i 1) t + j q, where q > 0 ad 1 t. Usig this defiitio the empirical scalig fuctio at the poit q based o the poits s i (0,1), i = 1,..., N, ca be defied by The estimatio method is based o the asymptotic behavior of τ. Oe ca show N, that for each q > 0, whe, N, τ (q) teds i probability to N,

11 q α if q α α 1 q > α α, τ α ( q)= q 0< q α α > q ( α q) ( α+ 4q 3αq) + if q > α α >, 3 α ( q) where α is the tail idex. This implies that the shape of the empirical scalig fuctio depeds o the value of the tail idex. Sice τ (q) ca be easily computed N, from the sample, this provides iformatio o the ukow tail idex. The asymptotic form τ is plotted i figure 4(a). For the heavy-tailed samples the empirical N scalig fuctio will approximately have the shape of the broke lie. The break of the lie occurs at poit α. The limitig case α correspods to o heavytailed distributios ad the scalig fuctio would be a straight lie q/ (dotted i figure 4(a)). This way it is possible to detect heavy tails i data. Estimatio ca be doe by fittig the empirical scalig fuctio to its asymptotic form. Takig some poits q i (0, q max ), i = 1,..., M, tail idex ca be estimated as M α = mi α ( 0, ) ( τ N, (qi) τ α ( qi)). (10) i= (4) (015) More details o the method ca be foud i Grahovac et al. (015). It is importat to ote that the estimatio does ot deped o the particular form of the uderlyig distributio ad the oly assumptio is that the sample comes from the class of heavy-tailed distributios, which i particular also icludes Studet s distributio. The empirical scalig fuctio computed o the sample of CROBEX log-returs R 1,..., R with = T = 53 is show i figure 4(b). A clear departure from the lie q/ cofirms that the log-returs are heavy-tailed. The scalig fuctio has a shape of the broke lie ad breaks at aroud value 5. Computig the estimator by equatio (10) gives the value α = The estimated value appears as a break i the plot of the scalig fuctio i figure 4(a). The plot of τ for α = 4.87 (dashed i N figure 4(b)) almost coicides with the empirical scalig fuctio, cofirmig the quality of the estimate. Therefore, the estimated value of the shape parameter υ is υ = 4.87 ad as such is cosistet with may other studies that suggest that the tail idex value of the asset returs is betwee 3 ad 5 (see e.g. Hurst ad Plate, 1997).

12 4 Figure 4 Scalig fuctios (a) Asymptotic form (b) Empirical for CROBEX data τ α (q) τ (q) 39 (4) (015) α q q τ (q) τ (q) q 4.1. Estimatio of parameters μ ad δ The problem of estimatio of the locatio parameter μ R ad the scale parameter δ > 0 is approached by assumig that υ is equal to its estimated value Suppose that R 1,..., R is a radom sample of log-returs. Parameters μ ad δ will be estimated by the classical method of momets i which estimators are obtaied as solutios of the system of equatios relatig the theoretical momets to the correspodig empirical momets. Sice μ ad δ are parameters of the margial distributio of Studet s diffusio (6), estimators will be obtaied by relatig the expectatio E[R t ] = µ ad the secod momet E[R t ] = δ /(υ 1) + μ to the first ad the secod empirical momets 1 = 1 R R R = k Rk (11) k = 1 k = 1 respectively. Solutios of this system of equatios with respect to the ukows δ ad μ are the estimators of these parameters: ( R R ) = δ = ν 1 ( ν 1) 1 1 ( ) Rk k 1 μ 1 = R = R k k = 1 = k = 1 R k (1) (13) Computig the values of estimators δ ad μ based o the CROBEX log-returs results i estimated values of parameters δ ad μ of Studet s diffusio. Estimated values of all three parameters of the margial distributio of Studet s diffusio are give i table 1. Table 1 Estimated values of parameters υ, δ ad μ of Studet s diffusio Parameter υ δ μ Estimated value

13 4.1.3 Estimatio of parameter θ Autocorrelatio parameter θ > 0 is estimated by the geeralized method of momets based o the empirical autocorrelatio fuctio. Autocorrelatio fuctio (ACF) of Studet s diffusio is well defied if υ > 0, so its existece is assured by the estimated value 4.87 of the parameter υ (see table 1). 43 The empirical couterpart of the autocorrelatio fuctio (7) is give by the absolute value of Pearso s sample correlatio fuctio 39 (4) (015) (14) where the term i the umerator represets the empirical covariace of radom variables R s ad R t+s, while the term i the deomiator represets the product of the empirical stadard deviatios of radom variables R s ad R t+s, 0 s < s + t T. Pearso s sample correlatio fuctio (or empirical ACF), plotted i figure 5 for lags t = 0, 1,..., 30, shows autocorrelatio i the time series of log-returs for small values of lag t ad suggests the expoetial decay of autocorrelatios with respect to the lag t. Notice that after lag 10 the estimated correlatios stabilize ear zero, so the majority of the expoetially decayig autocorrelatio structure of Studet s diffusio is cotaied i these first few correlatios. For alterative methods of estimatio of the autocorrelatio parameter based o a small umber of lags we refer to Forma (005). Figure 5 Empirical ACF (dotted) ad theoretical ACF (solid) for θ= For fixed t, the method of momets estimator for θ is derived by solvig, with respect to the ukow parameter θ, the equatio that relates the empirical auto-

14 44 correlatio fuctio (14) to the theoretical autocorrelatio fuctio ρ(t) = e -θt give i (7). The estimator is give by the followig expressio: 39 (4) (015) 1 θ() t = l ρ () t. (15) t Notice that for each lag t we obtai a sigle estimate of θ. Sice the majority of the autocorrelatio structure is held by the first 10 lags, to obtai just oe estimate of the parameter θ we calculate the value of the estimator θ (t) for t = 1,..., 10 ad the fial estimate 0.91 for θ is obtaied as the mea of these 10 values (see table ). The theoretical autocorrelatio fuctio ρ(t) = e -θt for θ = 0.91 is also plotted i figure 5. Table Estimatio of parameter θ of Studet s diffusio Lag t Value of θ (t) Estimated value of θ 0.91 Figure 6 Empirical ACF for the squared log-returs Beside plottig the theoretical ad the empirical ACF, the usual graphical method for explorig the depedece structure of log-returs is the ACF of their squares, which are ofte used as the volatility approximatios. From ice theoretical properties of Studet s diffusio ad the methodology based o its eigefuctios (orthogoal Routh-Romaovski polyomials, see Leoeko ad Šuvak, 010) it follows that the autocorrelatio fuctio of the squared log-returs, which is well defied for υ > 4, is give by: Corr ( Rs Rs+ t )= 4 δ ( ν 1) θt ν e ν 1 4μ δ θt + e ( ν ) ( ν 4) ν, 0 s < s+ t T. (16) 4 δ ( ν 1) 4μ δ + ( ν ) ( ν 4) ν

15 Autocorrelatio fuctio (16) is expoetially decayig fuctio of the lag t. From the empirical ACF of squared CROBEX log-returs, estimated by Pearso s sample correlatio fuctio (14) for squared data ad plotted i figure 6, we see that it correspods to the theoretically suggested expoetial decay NOTES ON THE ASYMPTOTIC BEHAVIOR OF PARAMETER ESTIMATORS I this sectio we briefly discuss the two most commoly aalyzed asymptotic properties of estimators cosistecy ad asymptotic ormality of the estimators θ, δ, μ of parameters θ, δ, μ. Geerally, estimator k, where emphasizes its depedece o the umber of observatios, is a cosistet estimator of the ukow parameter k if the probability that the absolute deviatio of k from k ca be made arbitrarily small by choosig large eough. Furthermore, estimator k is asymptotically ormal if for large the stadardized estimator k has a approximately stadard ormal distributio. For more details o these properties of estimators we refer to Serflig (1980) Cosistecy It is well kow that the first ad the secod empirical momets R ad give i (11) are cosistet estimators of the first ad the secod theoretical momets ad that the Pearso sample correlatio fuctio ρ (t) give by (14) is a cosistet estimator of the autocorrelatio fuctio ρ(t) = Corr(R s, R s+t ), (see Serflig, 1980). Sice estimators μ, δ ad θ are cotiuous trasformatios of the estimators R ad ad ρ(t), from the cotiuous mappig theorem (see Serflig, 1980) it follows that μ, δ ad θ are cosistet estimators of parameters μ, δ ad θ, respectively. 39 (4) (015) 4.. Asymptotic ormality Estimators μ ad δ are cotiuous trasformatios of estimators R ad which are kow to be asymptotically ormal due to the β-mixig property of Studet s diffusio (see Leoeko ad Šuvak, 010). Therefore, accordig to the deltamethod (see Serflig, 1980) it follows that for υ > 4 the bivariate estimator (μ, δ ) is also asymptotically ormal, i.e. where υ is supposed to be the kow value of the tail idex of margial distributio of Studet s diffusio ad δ, μ ad θ are cosistet estimators of parameters δ, μ ad θ give i (1), (13) ad (15), respectively. The covariace matrix (υ, δ, μ, θ ) could be represeted as D D τ, where

16 46 ad the elemets of the matrix are as follows: σ 11 δ = + 1, ν θ e 1 39 (4) (015) σ σ 1 μδ = ν δ + 1 = σ θ 1, e 1 4 4δ ( ν 1) = ( ν ) ( ν 4) e 1 θ ν ν 1 8 μ δ + ν 1 e θ δ ( ν ) + 4μ δ ( ν )( ν 4) δ ( ν 4) +. ( ν ) ( ν 4) For more details o the methodology of aalysis of asymptotic properties of some of these estimators ad calculatio of explicit form of the covariace matrix (υ, δ, μ, θ ) we refer to Leoeko ad Šuvak (010). 4.3 INFERENCE ON THE QUALITY OF THE MODEL The quality of the model is examied by simulatios. To obtai some objective idicators that relate CROBEX log-returs ad Studet s diffusio as the stochastic model for them, we simulated 1000 idepedet sample paths of this process by usig estimated values of parameters υ, δ, μ ad θ (see tables 1 ad ). Studet s diffusio ca be simulated usig the so-called Milstei scheme for simulatig paths of solutios of stochastic differetial equatios (see Iacus, 009 for more details). The legth of each simulated sample path coicides with the umber of observed CROBEX log-returs. Several of these sample paths are plotted i figure 7. I this settig for each time poit t we deal with the sample of 1000 simulated data that will be used to describe the CROBEX log-retur at t by the sample percetiles. More specifically, for each time poit t we calculate the 5 th percetile, the lower quartile (5 th percetiles), media, upper quartile (75 th percetile) ad 95 th percetile of the sample of 1000 data simulated for this exact time poit. The values of these sample percetiles for each t, together with the time series of CROBEX log-returs, are plotted i figure 8. Furthermore, we estimate probabilities that CROBEX log-returs fall outside the iterquartile iterval ad the iterval betwee the 5 th ad the 95 th percetiles: 13.99% of CROBEX log-returs fall outside the iterquartile iterval we ca say that the probability that the CROBEX log-retur is smaller tha the lower quartile ad greater that the upper quartile is estimated to be ,.46% of CROBEX log-returs fall outside the iterval betwee the 5 th ad the 95 th percetiles we ca say that the probability that the CROBEX log-

17 retur is smaller tha the 5 th percetiles ad greater that the 95 th percetiles is estimated to be These probabilities idicate that the simulated paths of Studet s diffusio with parameters υ = 4.87, δ = 0.05, μ = ad θ = 0.91 capture the time evolutio of the historical values of CROBEX log-returs quite well. Moreover, there are o values of CROBEX log-returs that fall outside the iterval betwee the miimal ad the maximal simulated value of Studet s diffusio at time t. Figure 7 Simulated trajectories of Studet s diffusio 39 (4) (015) Figure 8 CROBEX log-returs ad percetiles of 1000 simulated trajectories of Studet s diffusio CROBEX log-returs 5 th ad 95 th percetiles 5 th ad 75 th percetiles

18 48 5 CONCLUSION I this paper we itroduced statioary Studet s diffusio as a model for log-returs of stock prices or values of a stock market idex. The model captures some mai features characteristic for log-returs of risky assets, maily heavy-tailed margial distributio ad otrivial depedece structure. The parameters of the 39 (4) (015) diffusio process provide flexibility i fittig the model to data. Here we cocetrated o fittig the CROBEX log-returs to the proposed model. Our aalysis shows that CROBEX, as well as may other stock market idices, exhibits heavy tails. This importat fact must always be take ito accout i ay serious risk aalysis. Simulatios of Studet s diffusio process show that it ca realistically model risky asset returs ad reproduce may of their features, like volatility clusterig, meaig that periods of low volatility are followed by periods of high volatility, idicatig partial predictability of volatility fluctuatios. The mai purpose of the proposed model is ot to forecast future values; rather it is tailored for a quality risk assessmet. Computatio of some risk measures, like e.g. value at risk (VaR), i the cotext of the proposed model has ot bee cosidered i this work. However, from the estimated parameters of statioary Studet s distributios, it is easy to compute the VaR as the quatile of this distributio. From the compariso with the ormal distributio made i figure 3, it is clear that these estimates will ted to give more pessimistic, although realistic predictios. We also did ot cosider the problem of optio pricig which would require a more detailed approach. Optio pricig problem ca be approached through Mote Carlo simulatios but could also be built o the kow expressios for the trasitio desity of the Studet s diffusio. The model proposed is flexible eough to cover a wide rage of heavy-tailed data. Some extesios of the model may iclude cosiderig Studet processes with prescribed depedece structure. A large class of such models has bee proposed i Heyde ad Leoeko (005). Aother possible extesio of the model would be a diffusio process usig the so-called skewed Studet s distributio which allows for o-symmetry of the tails through a additioal skewess parameter.

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