Acta Unverstats Carolnae. Mathematca et Physca L. Jarešová: EWMA hstorcal volatlty estmators Acta Unverstats Carolnae. Mathematca et Physca, Vol. 51 (2010), No. 2, 17--28 Persstent URL: http://dml.cz/dmlcz/143654 Terms of use: Unverzta Karlova v Praze, 2010 Insttute of Mathematcs of the Academy of Scences of the Czech Republc provdes access to dgtzed documents strctly for personal use. Each copy of any part of ths document must contan these Terms of use. Ths paper has been dgtzed, optmzed for electronc delvery and stamped wth dgtal sgnature wthn the project DML-CZ: The Czech Dgtal Mathematcs Lbrary http://project.dml.cz
2010 ACTA unverstats carolnae mathematca et physica VOL. 51, no. 2 EWMA Hstorcal Volatlty Estmators Luca JareŠová Praha Receved May 17, 2010 Revsed August 30, 2010 In ths paper dfferent types of hstorcal volatlty estmators based on open-hgh-lowclose (OHLC) values are studed. The estmators are broken down to the man buldng blocks and the correlaton structure of these buldng blocks wth tme dependent varance (volatlty squared) s nvestgated. The buldng blocks are estmated from the equty ndex (SPX n USA and DAX n Germany) and compared wth a volatlty ndex (VIX n USA and VDAX n Germany) whch stands as a proxy for volatlty, because the values of the volatlty process are n general not avalable. In an emprcal study t s observed that both the autocorrelaton functon of varance and the cross-correlaton functons of buldng blocks wth the varance decrease exponentally wth the same degree. Ths dependence can be explaned as exponentally decreasng amount of nformaton and t naturally leads to use of exponentally decreasng weghts n hstorcal estmators. The proposed EWMA style estmators have hgher predctng power over the commonly used estmators and n predcton beat the very popular GARCH(1,1). Introducton The word volatlty s used most often n fnance as a measure of varablty n the changes of asset prces. Hgher volatlty means n general hgher probablty of bgger losses, so volatlty s drectly lnked to the rsk of the asset. The recent development n fnancal markets ncreased the mportance of tme dependent volatlty modelng and new and new more sophstcated models are proposed to better represent the realty. The volatlty process s n general assumed to be drectly unobservable and consequently the estmaton of these models s very dffcult. Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc 2000 Mathematcs Subject Classfcaton. 62M10, 91B82, 91B84 Key words and phrases. Volatlty, equty ndex, volatlty ndex, GARCH model, EWMA model E-mal address: jaresova@karln.mff.cun.cz 17
In the last decades the contnuous volatlty models attracted much attenton because of the theoretcally nce and elegant stochastc calculus. For an overvew about the volatlty modelng and measurement see [1]. Unfortunately the practcal part of these contnuous tme models s developng much more slowly and t s mostly necessary to dscretze these models for estmaton, snce the observed values are avalable for an analyss only as daly tme seres. Of course more detaled data exsts, for example n the form of hgh-frequency tck-by-tck data, but ths data s not avalable to everybody and ts analyss requres more computer power as t s very technologcally demandng. 18 Daly Fnancal Data The asset prce process s denoted by S t, t 0. A wdely used model for S t s the Generalzed geometrc Brownan moton gven by the followng equaton ds t = µ t S t dt + σ t S t dw t, (1) where the drft µ t s the expected return on the asset, the volatlty σ t measures the varablty around µ t and W t s the standard Brownan moton. In the standard Geometrc Brownan moton (GBM) t s assumed that µ t = µ and σ t = σ (the parameters are constant) and t holds for T > t that ln(s T ) = ln(s t ) + (µ σ 2 /2)(T t) + σ T tɛ (2) where ɛ s the standard normal random varable. From equaton (2) t follows that the logarthmc returns r(t, T) = ln(s T /S t ) are normally dstrbuted. The asset prce S t s only observed durng the tme when the market s open and the daly fnancal data are often avalable n the form of OHLC (open-hgh-low-close) values. For the tradng days = 0,...,N we denote the mornng openng prce by O, the evenng closng prce by C, the daly lowest prce by L and the daly hghest prce by H. These values are for many assets and ndces avalable to everybody, snce they can be downloaded for example from Yahoo Fnance 1 by usng free software envronment R [8] and a Rmetrcs 2 package fimport. The daly log-returns r 1,...,r N are computed from the closng prces C as r = ln(c /C 1 ). (3) The tme dependent yearly volatlty wll further be denoted by σ and t s mostly estmated by the standard devaton of daly log-returns multpled by the scalng factor 3. The yearly varance s defned as h = σ 2. 1 http://fnance.yahoo.com/ 2 https://www.rmetrcs.org/ 3 It s common to consder the annualzed daly volatlty, so the standard devaton has to be scaled by the square root of the number of tradng days n a year. We assume that a year has tradng days.
A more sophstcated wdely used approach for tme-dependent volatlty modelng n dscrete tme s the GARCH (Generalzed AutoRegressve Condtonal Heteroskedastcty) model. Ths model uses only daly closng data. The descrpton of these models can be found n almost every fnancal tme seres lterature, for example n [10]. A lot of generalzatons of the GARCH were derved, summary of them s n [2] (more than 100 models). Hstorcal Volatlty Estmators In ths secton the man hstorcal volatlty estmators wdely used by practtoners are ntroduced. The formulas for hstorcal estmators are taken from the Quant Equaton Archve http://www.stmo.com/eqcat/4. Hstorcal Close-to-Close Volatlty. Ths smplest estmator s equal to the standard devaton of log-returns scaled to one year gven by the formula σ cc = N 1 (r r) 2 or σ cc = N (r r) 2, (4) where r = 1 N N r. The drft of asset prces estmated by r s usually very small, so sometmes the followng formulas are used σ cc = N 1 r 2 or σ cc = N r 2. (5) Hstorcal Hgh-Low Volatlty (Parknson). Ths estmator uses only the hghest and lowest daly values and s gven by σ p = 4N ln(2) ( ln H ) 2. (6) L Hstorcal Open-Hgh-Low-Close Volatlty (Garman and Klass). Ths estmator uses all OHLC values and s gven by σ gk = N 1 2 ( ln H ) 2 ( (2 ln 2 1) ln C ) 2 L O (7) Hstorcal Open-Hgh-Low-Close Volatlty (Garman and Klass, Yang Zhang extenson). Ths estmator s currently the preferred verson of OHLC volatlty estmator and t dffers from the prevous estmator only by the term (ln(o /C 1 )) 2 whch takes 19
nto account the openng jump (the change from the closng prce yesterday C 1 to the openng prce today O ). ( σ gkyz = N ln O ) 2 + 1 ( ln H ) 2 ( (2 ln 2 1) ln C ) 2 C 1 2 L O (8) Hstorcal Open-Hgh-Low-Close Volatlty (Rogers Satchell). The last estmator uses all OHLC values and s gven by [ σ rs = ln H ln H + ln L ln L ] (9) N C O C O Buldng Blocks Hstorcal Volatlty Estmators. The mentoned hstorcal volatlty estmators have the followng buldng blocks: A = {ln(c /C 1 ) 2 } = {r 2 } represents the daly squared close-to-close changes. B = {ln(h /L ) 2 } represents the daly squared extreme changes. C = {ln(o /C 1 ) 2 } represents the squared openng jumps. D = {ln(c /O ) 2 } represents the squared tradng daly changes. E = {ln(h /C ) ln(h /O )} s based on the frst term of Rogers Satchell estmator. F = {ln(l /C ) ln(l /O )} s based on the second term of Rogers Satchell estmator. The buldng blocks A, B, C, D, E and F wll be used as an nput n the correlaton emprcal study. 20 Proxy for the Unobserved Volatlty All varables except for volatlty are drectly observable n the market. Volatlty can be ndrectly observed n the market through the opton prces, snce the hgher volatlty the hgher prces of plan vanlla optons. We can use the observed opton market prce and the Black-Scholes (BS) opton prcng formula (for more nformaton about opton prcng see [5] or [11]) to calculate the volatlty that yelds a theoretcal value of the opton equal to the observed market prce. Ths volatlty s called the mpled volatlty and t n general depends (n contrast to the BS model where the volatlty s assumed to be constant) on the strke prce K and the expraton of the opton T t (tme to maturty). The collecton of all such mpled volatltes wth respect to the strke prce and tme to maturty s known as the volatlty surface. For more understandng of volatlty surfaces look n [4] or [9]. Impled volatlty s a very good fnancal ndcator of the fear n the market snce t often sgnfes fnancal turmol. Some exchanges have transformed ths nformaton nto volatlty ndces. The most known ndex s the VIX ndex of the CBOE (Chcago Board Optons Exchange) launched n 2003 (data begns 1990). The CBOE utlzes a wde varety of strke prces of optons on the S&P 500 ndex (SPX ndex, the core ndex for U.S. equtes) n order to obtan the estmator of 30-day expected volatlty.
More detals about the calculaton are descrbed n the methodology [3]. VDAX- NEW ndex (launched n 2005 as a successor for VDAX launched n 1996, data begns 1992) s an analogous ndex of the Deutsche Börse n Germany based on the prces of optons on the German DAX ndex. The volatlty ndces have become very popular and ther number s ncreasng. The volatlty ndex can nowadays even be traded through exchange traded futures (snce 2004) and exchange trade optons (snce 2006). The volatlty ndces were used n the emprcal study [6], where the mpled volatlty, GARCH volatlty and hstorcal volatlty were compared. In ths paper we wll use the volatlty ndex value as a proxy for an unobserved volatlty process σ. Further we defne Y = {σ 2 /} = {h /} as the actual one day varance 4. Auto- and Cross-Correlaton Study Now we wll take the data A, B, C, D, E, F and Y and nvestgate ther correlaton relatonshps. The used data are from 3. 1. 2000 to 2. 4. 2010. To make the results more relevant, the analyss s made on two dfferent markets. In the US market the equty ndex SPX and the volatlty ndex VIX are used. In the Germany market the equty ndex DAX and the volatlty ndex VDAX are used. TABLE 1. Prce ndex SPX, volatlty ndex VIX Y A B C D E F Y 1.00 0.53 0.71 0.13 0.60 0.52 0.45 A 0.53 1.00 0.81 0.04 0.67 0.19 0.20 B 0.71 0.81 1.00 0.11 0.70 0.49 0.66 C 0.13 0.04 0.11 1.00 0.08 0.01 0.05 D 0.60 0.67 0.70 0.08 1.00 0.47 0.24 E 0.52 0.19 0.49 0.01 0.47 1.00 0.19 F 0.45 0.20 0.66 0.05 0.24 0.19 1.00 The estmated correlaton matrces of the data are n the tables 1 and 2. It s very nterestng, that the correlatons are n both tables very smlar. 4 Volatlty s n the market quoted as a value per annum smlarly as nterest rates. We need to dvde the squared value by the number of workng days per annum to obtan the scaled daly values. 21
TABLE 2. Prce ndex DAX, volatlty ndex VDAX Y A B C D E F Y 1.00 0.48 0.68 0.23 0.55 0.51 0.45 A 0.48 1.00 0.74 0.17 0.66 0.22 0.24 B 0.68 0.74 1.00 0.26 0.66 0.61 0.66 C 0.23 0.17 0.26 1.00 0.31 0.24 0.19 D 0.55 0.66 0.66 0.31 1.00 0.40 0.28 E 0.51 0.22 0.61 0.24 0.40 1.00 0.23 F 0.45 0.24 0.66 0.19 0.28 0.23 1.00 SPX DAX correlaton wth Y 0.0 0.2 0.4 0.6 0.8 1.0 A B C D E F Y correlaton wth Y 0.0 0.2 0.4 0.6 0.8 1.0 A B C D E F Y 250 200 150 100 50 0 250 200 150 100 50 0 tme shft (days) tme shft (days) Fgure 1: Cross-Correlaton Functons The autocorrelaton functon of Y and the cross-correlaton functon of A, B, C, D, E, F wth Y are graphed by sold lnes n the fgure 1. The dashed lnes n the fgure represent the ftted exponental functons. It can be seen, that the exponental functons ft the auto- and cross-correlaton functons very well. It means that the lnear dependence on the past values decreases exponentally as the tme goes on. It s very obvous, that these functons look very smlar n both markets. It ponts to the hypothess, that these patterns are propertes of the real tme dependent varance process and every good volatlty model should reflect these propertes as well. 22
GARCH model and EWMA model GARCH models are nowadays very popular. Process {r n }, n Z s a (strong) GARCH(p, q), f E[r n F n 1 ] = 0 (the condtonal mean s unpredctable) and Var[r n F n 1 ] = σ 2 t (the condtonal varance s tme dependent), where q p σ 2 n = ω + α rn 2 + β j σ 2 n j (10) and Z n = r n /σ n are..d. random varables. F n 1 denotes the σ-algebra generated by the hstorcal returns, α and β j are real coeffcents. Suffcent condton for σ 2 n 0 s ω, α,β j 0. In the GARCH(1, 1) model the condtonal varance has the form j=1 σ 2 n = ω + αr 2 n 1 + βσ2 n 1. (11) The EWMA (Exponentally Weghted Movng Average) model of volatlty s known very well thanks to the RskMetrcs 5 techncal report [7] and has been used n many practcal applcatons. The EWMA volatlty s defned as σ 2 n = (1 λ) λ j 1 (r n j r) 2 = (1 λ)(r n 1 r) 2 + λσ 2 n 1 (12) or, when r s small, as j=1 σ 2 n = (1 λ) λ j 1 rn 2 j = (1 λ)r2 n 1 + λσ2 n 1, (13) j=1 where λ s a decay factor. Typcal values of the decay factor are close to one (values 0.94 and 0.97 are recommended n [7]). In the case when we have N hstorcal observatons, the EWMA estmator of volatlty can be computed as (1 λ) σ E,n = 1 λ N λ 1 (r n r) 2 (14) or σ E,n = (1 λ) 1 λ N λ 1 rn 2, (15) where n s a tme ndex. Ths estmator can be expressed n the followng way σ E,n = w (λ)rn 2, (16) where the weghts w (λ) = (1 λ)λ 1 = λ 1 1 λ N N sum to 1. Snce the weghts are equal to λ 1 1/N for λ = 1, the EWMA estmator s equal to the hstorcal close-to-close volatlty 5 http://www.rskmetrcs.com/ 23
estmator for λ = 1. The man advantage of ths estmator s, that t s very smple and that t does not need any addtonal software, because t can be computed even n Excel. EWMA Style Estmators The observed pattern of exponentally declnng amount of nformaton wth the tme left can be used to mprove all hstorcal volatlty estmators by ntroducng exponental weghts n the same way as n the classcal close-to-close EWMA estmator. It means that an estmator of the form σ 2 = (buldng blocks) n (17) N s changed to an EWMA-style estmator by weghts w σ 2 EWMA = w (λ) (buldng blocks) n, (18) where λ [0, 1] and w (λ) = λ 1 λ 1 (1 λ)λ 1 N = λ 1 1 λ N. (19) weght 0.00 0.02 0.04 0.06 equal weghts lambda = 0.93 lambda = 0.94 lambda = 0.95 lambda = 0.96 lambda = 0.97 lambda = 0.98 120 100 80 60 40 20 0 tme lag (days) Fgure 2: Comparson of EWMA weghts wth the equal weghts n classcal hstorcal estmators It s clear that N w (λ) = 1 for all λ and that EWMA style estmator s equal to the classcal estmator for λ = 1. The EWMA weghts w (λ) gve more mportance on the more recent observatons. The tme structure of weghts can be seen n the 24
Fgure 2, where the horzontal lne represents the equal weghts n the classcal estmators correspondng to λ = 1 and the other lnes represent the EWMA weghts for dfferent values of decay factor λ. Emprcal Study In the emprcal study [6] the actual volatty was estmated for t equal to dates from 3.5.2002 to 24.4.2009. For each t the three year hstory tll t was taken to estmate the volatlty. The man goal of the performance study was to evaluate the performance of GARCH(1,1) n forecastng volatlty. The results of ths study are shown n the Table 3. The GARCH(1,1) was estmated by usng the envronment R wth the Rmetrcs lbrary fgarch. TABLE 3. Coeffcent of determnaton n the lnear model where the actual values of volatlty ndex are regressed on the estmated values and constant term Index prevous ndex value GARCH(1,1) GARCH(1,1) forecast 3M hst. volatlty SPX 97.47% 88.43% 89.21% 84.24% DAX 98.23% 86.71% 87.44% 83.36% In ths secton a smlar emprcal study s performed, where the classcal hstorcal volatlty estmators are compared to the EWMA style estmators wth the decay factor λ = 0.96. Recall that an EWMA style estmator wth decay factor 1 s equal to classcal hstorcal estmator. In the followng GARCH denotes the GARCH forecast from the last year emprcal study (was the best performng), EWMA(λ) denotes the EWMA volatlty estmator, PARK(λ) denotes the EWMA-style Parknson estmator, GK(λ) denotes the EWMA-style Garman Klass estmator, GKYZ(λ) denotes the EWMA-style Garman Klass (Yang Zhang) estmator, RS(λ) denotes the EWMA-style Rogers Satchell estmator, ndex yesterday denotes the prevous value of the volatlty ndex and λ s the decay factor. Models were estmated for t equal to dates from 12. 4. 2002 to 2. 4. 2010. Estmators wth λ = 0.96 are based on two year hstory and estmators wth λ = 1 are based on three months hstory, snce usng of longer hstory makes the estmaton worse. To evaluate the performance of these estmators we agan consder a lnear model y = a + bx + e, where a and b are constant and e s an error term. The estmated volatlty values are the explanatory data x = {x 1,...,x n } and the values of volatlty 25
ndex on the correspondng days are the dependent data y = {y 1,...,y n }. The coeffcent of determnaton R 2 s n ths stuaton equal to Cor(x, y) 2 and t wll be used as an overall performance measure (the hgher value of R 2 the better estmator). Further the followng performance measures are consdered 6 bas A = mean(x y) bas R = mean(x/y 1) sd A = Var(x y) sd R = Var(x/y 1) MS E A = mean((x y) 2 ) MS E R = mean((x/y 1) 2 ). Absolute measures are denoted wth the subscrpt A and the relatve measures are denoted wth the subscrpt R. The bas s measured by bas A and bas R and the standard devaton s measured by sd A and sd R. The measures MS E A and MS E R (mean squared error) take nto account both bas and varance n the same tme. The lower absolute value of these measures the better estmator. The performance results are shown n the Table 4. The best volatlty predctor s the prevous value of volatlty ndex. The problem wth ths predctor s, that we do not have volatlty ndex for many assets. Comparng the predcton propertes of GARCH model wth classcal hstorcal estmators, we fnd out, that based on the coeffcent of determnaton s the GARCH model n all cases better. When we compare the coeffcents of determnaton of the classcal estmators wth the EWMA style estmators, we fnd out, that the EWMA style estmators are not just n all cases better than the classcal estmators, but they performed even better than the GARCH estmators. The absolute and relatve measures gve us addtonal nformaton about the propertes of the estmators. For example n the case of SPX ndex we can see that the GK(0.96) performed better than PARK(0.96) even f the coeffcent of determnaton s the same. Concluson In ths paper the auto- and cross- correlaton structure of varance wth the buldng blocks of some open-hgh-low-close hstorcal volatlty estmators was nvestgated. It was emprcally shown through the correlaton structure, that the lnear dependence decreases exponentally and as a consequence new EWMA style hstorcal estmators based on the open-hgh-low-close values were proposed. The EWMA style estmators of volatlty were n an emprcally study compared wth the GARCH predcton and wth the classcal estmators. The EWMA style estmators were n all cases better than the classcal estmators. A very nterestng result s, that the EWMA estmators were n all cases better than the GARCH predcton, 6 x y := {x 1 y 1,...,x n y n }, x/y 1:= {x 1 /y 1 1,...,x n /y n 1}, x 2 := {x1 2,...,x2 n}, mean(x) = x = 1 n n x, Var(x) = 1 n n 1 (x x) 2, Cor(x, y) = 1 n n 1 (x x)(y ȳ). Var(x) Var(y) 26
TABLE 4. Performance of the estmators. The absolute measures are n the same unts as the volatlty (n percent), the sgn % s omtted n absolute measures (.e. 1 s for example the change from volatlty 20% to 21%), whereas the relatve measures are stated as percent of volatlty (.e. 1% s for example the change from volatlty 20% to 20.2%) SPX ndex R 2 bas A sd A MS E A bas R sd R MS E R GARCH 87.8% 3.0 4.3 5.2 16.5% 13.7% 21.4% EWMA(1) 83.6% 2.8 5.0 5.7 15.8% 17.5% 23.6% EWMA(0.96) 91.0% 2.8 3.8 4.7 16.6% 14.7% 22.2% PARK(1) 83.3% 5.6 4.4 7.1 27.4 % 13.4% 30.5% PARK(0.96) 91.4% 5.6 3.2 6.5 28.0% 10.7% 29.9% GK(1) 83.4% 0.4 5.7 5.7 0.1% 18.8% 18.8% GK(0.96) 91.4% 0.4 4.5 4.5-0.8% 15.1% 15.1% GKYZ(1) 83.5% 0.7 5.8 5.8 1.0% 19.0% 19.1% GKYZ(0.96) 91.5% 0.6 4.6 4.7 0.2% 15.3% 15.3% RS(1) 82.5% 7.0 4.7 8.4 33.2% 12.1% 35.3% RS(0.96) 90.7% 7.0 3.7 7.9 33.7% 9.8% 35.1% ndex yesterday 97.4% 0.0 1.8 1.8 0.2% 5.9% 5.9% DAX ndex R 2 bas A sd A MS E A bas R sd R MS E R GARCH 86.8% 0.7 5.1 5.1 5.3% 16.0% 16.8% EWMA(1) 83.4% 0.5 5.3 5.3 4.1% 18.2% 18.7% EWMA(0.96) 91.0% 0.5 4.2 4.2 5.0% 14.7% 15.5% PARK(1) 84.4% 3.7 4.5 5.8 18.2% 15.8% 24.1% PARK(0.96) 91.4% 3.8 3.4 5.0 18.9% 13.2% 23.0% GK(1) 84.1% 3.4 7.1 7.8 10.4% 21.5% 23.9% GK(0.96) 91.2% 3.3 6.2 7.0 9.5% 18.1% 20.4% GKYZ(1) 84.0% 4.4 7.4 8.6 15.1% 21.7% 26.4% GKYZ(0.96) 91.3% 4.4 6.5 7.8 14.2% 17.8% 22.7% RS(1) 84.9% 4.4 4.3 6.2 21.0% 15.2% 25.9% RS(0.96) 91.2% 4.5 3.3 5.5 21.6% 13.0% 25.2% ndex yesterday 98.2% 0.0 1.5 1.5 0.1% 4.7% 4.7% even n the case of the smplest EWMA close-to-close volatlty estmator, whch s actually a specal case of GARCH model wth fxed parameters. Ths ponts to some possble neffcences n the estmaton of the GARCH model or to some possble msspecfcatons n the GARCH model. The postve thng s, that t s probably not necessary to buy software to estmate the GARCH model n order to predct the volatlty, snce the smple EWMA volatlty estmator works even better. 27
Acknowledgement Ths work was partally supported by GACR grant No 201/09/0755. References [1] Andersen, T. G., Bollerslev, T. and Francs X. Debold, F. X.: Parametrc and nonparametrc vola tlty measurement. In Yacne Aït-Sahala, Y. and Lars Peter Hansen, L. P., edtors, Handbook of Fnancal Econometrcs: Volume 1 Tools and Technques, pages 67 137, North-Holland, 2010. [2] Bollerslev, T.: Glossary to ARCH (GARCH), Techncal Report 49, CREATES, 2008. Avalable at ftp://ftp.econ.au.dk/creates/rp/08/rp08 49.pdf. [3] CBOE and Goldman Sachs, The CBOE volatlty ndex - VIX. Calculaton methodology s avalable at http://www.cboe.com/mcro/vx/vxwhte.pdf. [4] Gatheral, J.: The Volatlty Surface. John Wley and Sons, 2006. [5] Hull, J. C.: Optons, Futures, and Other Dervatves (7 th Ed.). Prentce Hall, 2008. [6] Jarešová, L.: Volatlty modellng. In Safrankova J. and Pavlu J., edtors, WDS 09 Proceedngs of Contrbuted Papers: Part I Mathematcs and Computer Scences, pages 148 153. Matfyzpress, 2009. [7] J. P. Morgan / Reuters. RskMetrcs techncal document, 1996. Avalable at http://www.rskmetrcs.com/publcatons/techdocs/rmcovv.html. [8] R Development Core Team. R: A Language and Envronment for Statstcal Computng. R Foundaton for Statstcal Computng, Venna, Austra, 2006. [9] Rebonato, R.: Volatlty and Correlaton, John Wley and Sons, 2004. [10] Tsay, R. S.: Analyss of Fnancal Tme Seres, Wley-Interscence, 2005. [11] Wlmott, P.: Paul Wlmott On Quanttatve Fnance, John Wley and Sons, 2006. 28