Inernaonal Research Journal of Fnance and Economcs ISSN 450-887 Issue 49 (00) EuroJournals Publshng, Inc. 00 h://www.eurojournals.com/fnance.hm Fnancal Tme Seres Forecasng: Comarson of Neural Neworks and ARCH Models AK Dhamja Defence Research & Develomen Organsaon (DRDO), Delh, Inda E-mal: dhamja.ak@gmal.com; Webage: www.akdhamja.webs.com Tel: 9--39370 (O); 9-73508(R); 9-9845363(M) VK Bhalla Faculy of Managemen Sudes (FMS), Delh, Inda E-mal: vkbfms@yahoo.co.uk Absrac Ths aer comares he redcve accuracy of neural neworks and condonal heeroscedasc models lke ARCH, GARCH, GARCH-M, TGARCH, EGARCH and IGARCH, for forecasng he exchange rae seres.the Mul Layer Perceron (MLP) and Radal Bass Funcon (RBF) neworks wh dfferen archecures and condonal heeroscedasc models are used o forecas fve exchange rae me seres. The resuls show ha boh Neural nework and condonally heeroscedasc models can be effecvely used for redcon. RBF neworks do consderably beer han MLP neworks n neural neworks case. IGARCH and TGARCH fare beer han oher condonal heeroscedasc models. Neural neworks' erformance s beer han ha of condonal heeroscedascy models n forecasng exchange rae. I s shown ha neural nework can be effecvely emloyed o esmae he condonal volaly of exchange rae seres and he mled volaly of NIFTY oons. Neural nework s found o bea condonal heeroscedasc models n ou-of-samle forecasng. Keywords: Neural Nework; Heeroscedascy; Condonal Heeroscedasc Models; Exchange Rae; Predcve Accuracy.. Inroducon Durng las weny fve years many dfferen nonlnear models have been roosed n he leraure o model and forecas exchange raes. Some auhors clamed ha exchange raes are raher unredcable and so random walk model s beer redcor (Chang and Osler, 999; Meese and Rose, 990; Gencay, 999). Kuan and Lu (995) esmae and selec feedforward and recurren neworks o evaluae her forecasng erformance of fve exchange raes agans USD. The neworks erformed dfferenly for dfferen exchange rae seres. Yao and Tan (000) show ha f echncal ndcaors and me seres daa are fed o neural neworks o caure he underlyng rocess of he movemen n currency exchange raes, hen useful redcon can be made. Yang and Gradojevc (006) consruc a neural nework ha never erforms worse han a lnear model bu always erforms beer han he random walk model when redcng Canadan dollar/dollar exchange rae. Kan and Kasens (008) have successfully emloyed Neural neworks o forecas he exchange rae.
95 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) The GARCH model has been used n he as for volaly esmaon n U.S. dollar foregn exchange markes (Bal-le and Bollerslev, 989) and n he Euroean Moneary Sysem (Neely, 999). Inal sudes no exlanaory ower of ou-of-samle forecass gave ou dsaonng resuls (Wes and Cho, 995). Joron (995) found ha volaly forecass for several major currences from he GARCH model were ouerformed by mled volales generaed from he Black-Scholes oonrcng model. These sudes used he squared daly reurn as he varable o be forecas. Snce, he exchange rae may move around a lo durng he day, has been esablshed ha one can sgnfcanly mrove he forecasng ower of he GARCH model by usng sum of nraday squared reurns (Andersen and Bollerslev, 998). Ths measure s referred o as negraed or realzed volaly. Varance forecass hus obaned show ha volaly shocks are que erssen and he forecass of condonal varance converge o he seady sae que slowly. The sudes exermenng on forecasng exchange raes so far, have no ncluded he daa for he erod of he curren fnancal meldown. Ths sudy exressly uses hs daa and esablshes ha neural nework and auoregressve condonal heeroscedasc models boh can effecvely caure he long-erm non-lneares of he daa and redc successfully no he umuluous erod also.. Neural Neworks.. Neuron The basc buldng block of a neural nework s he neuron. A neuron can be reresened by a mang y: ransformng a n dmensonal nu no a real number. The neuron consss of a roagaon funcon f: and an acvaon funcon g: [0,] where g(x) akes he ouu of f(x) as argumen. Thus, a neuron can be reresened n he general form as y(x) = g(f(x; w)) () If f(x) s a olynomal, s degree s called he order of he neuron and s coeffcens are he arameers of he neuron. These neurons are assembled n layered srucure o consruc he arfcal neural nework (ANN). The heorecal framework of neural neworks menoned n hs secon has been adoed from Gacomn (003)... Arfcal Neural Neworks Arfcal neural neworks roduce he mang ø NN : and can be wren as g(y,, y m ) = ø NN (X, x n ) () Where x = (x,, x n ) T s he nu vecor and y = (y,, y m ) T s ouu vecor. A n arcular Neuron wll fre when weghed sum w x > θ =. The θ s he hreshold level for neurons o fre. Ths hreshold level can be bul no he roagaon funcon be weghng wh w 0 =. n Therefore, he roagaon funcon f ( x) = = w x θ s he weghed sum of nus. The acvaon funcon g of a neuron may assume many forms. I can be a lnear funcon or non-lnear funcon. Mos commonly used funcon s a sgmod funcon. These nerconneced neurons (Haykn, 999) can be dsosed accordng o a ceran archecure. A nework ø NN where he hreshold values (Bsho, 995) are ncororaed n he nu vecor x = (x 0,, x n ) T, x 0 = and he ouu vecor s y = (y,, y m ) T s reresened on Fgure..3. Mul Layer Perceron Neworks - MLP Neural neworks where he hdden neurons have sgmodal acvaon funcon and he ouu neurons have sgmodal or deny funcon are called Mul Layer Percerons (MLP) Neworks ø MLP :
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 96. Ths archecure consss of an nu layer, an ouu layer and k-hdden layers, each conanng j k neurons. Each -comonen of y = (y,, y m ) s released by he m-neuron a he ouu layer as a funcon of he nu x = (x,, x n ) and of he arameers w. Wrng n comac from, wh weghs on he nu vecors and d as oal number of hdden layers. Fgure : Feed-forward neural nework ø NN = = = = = k k d u d j n k x w g w j u w g j w g y 0 0 0 0 (3) Fgure shows he grah of a neural nework ø MLP, where d = 3, n =, j = 4, j = 5 and m = or ( 4 5 ) MLP. Fgure : ( - 4-5 - ) MLP ø NN
97 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00).4. Radal Bass Funcon Neworks - RBF Radal Bass Funcon (RBF) neurons are neurons n whch he roagaon funcon has he form f(x) = x - w, where x = (x,, x n ) T, w r = (w,, w n ) T are he nus and weghs. The acvaon funcon h(x) has he form of a radal symmerc funcon, commonly he gaus-san funcon. Neworks wh one hdden layer conanng r RBF neurons and ouu neurons wh n roagaon funcon f ( x) = wjx = and deny acvaon funcon g(x) = x are called RBF neworks ø RBF : wh r RBF neurons on he hdden layer. Each -comonen of he ouu y = (y,, y m )s gven by y r ( x) = w h ( x ) (4) w = The roagaon funcon calculaes how close (usng he eucldan dsance) he nu vecor x s o he vecor w r. The gaussan acvaon funcon roduces hgher values for nu vecors ha are close o he vecor w r and smaller values for nus ha are far away from. Thus he weghs form clusers n he nu sace. 3. Heeroscedascy The works of Mandelbro (963) and Fama (965) were among he frs few works ha examned he sascal roeres of sock reurns. Durng he 980s, Engle (003) worked on mrovng me-seres analyss. Sascal echnques hen n use mosly reaed volale varables, such as sock rces, as consans, even hough such varables can change sgnfcanly from day o day and week o week. Engle (003) afer observng he varance of sock reurns, develoed a sascal echnque known as ARCH (auoregressve condonal heeroscedascy), whch uses revously observed aerns of varance o redc fuure volaly. Refnemens of ARCH models are now beng used n bankng and fnance o hel deermne he rces and rsk nvolved of nvesng n socks. The heorecal framework descrbed n hs secon s adoed from Tsay (005) A unvarae sochasc rocess Y s sad o be homoscedas-c f he sandard devaons of erms are consan for all mes. Oherwse, s sad o be heeroscedasc (see Fgure 3 adoed from www.rskglossary.com). A rocess s uncondonally heeroscedasc f uncondonal sandard devaons are no consan. I s condonally heeroscedasc f condonal sandard devaons are no consan. For examle, sock or bond reurns end o be condonally heeroscedasc. These rces exhb non-consan volaly, bu erods of low or hgh volaly are generally no known n advance. New Delh elecrcy rces, on he oher hand, exhb uncondonal heeroscedascy. The rces end o have hgher volales durng he Summer han durng oher seasons. Ths s redcable, herefore he elecrcy rces exhb uncondonal heeroscedascy. If a rocess s uncondonally heeroscedasc, hen s necessarly condonally heeroscedasc. The converse s no rue. All hese noons exend o hgher dmensons. A mulvarae sochasc rocess Y s sad o be homoscedasc f s covarance marx s consan for all mes, ec.
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 98 Fgure 3: Homoscedasc vs. Heeroscedasc 3.. Srucure of a Model Le r be he log reurn of an asse a me ndex. In he volaly sudy r s eher aken as serally uncorrelaed or wh mnor lower order seral correlaons, bu s a deenden seres. The condonal mean and varance of r gven he nformaon se avalable a me as I are secfed as, μ = E( r \ I ), = Var( r \ I ) = E[( r μ ) I ] (5) Snce seral deendence of r s weak, we can say ha r follows a smle me seres model lke saonary ARMA(, q) model. The model becomes q r = = = μ + e, μ = c + φ r θ e, (6) where, and q are non-negave negers and e are nnovaons or error erms, e ~ N(0, ). Ths s he mean equaon for r. The order (, q) of an ARMA model may deend on he frequency of he reurn seres. The varance can be secfed as = Var r I ) = Var( e I ) (7) 3.. The ARCH Model ( The major assumon behnd he leas square regresson s homoscedascy.e consancy of varance. If hs condon s volaed, he esmaes wll sll be unbased bu hey wll no be mnmum varance esmaes. The sandard error and confdence nervals calculaed n hs case become oo narrow, gvng a false sense of recson. ARCH and relaed models handle hs by modelng volaly self n he model and hereby correcng he defcences of leas squares model. The ARCH models (Engle, 98; Tsay, 005), characerzed by mean and volaly equaons, are secfed as r = μ + e, e =, = α + α e (8) e = α0 + αe +, = +,, T = 0 = where, denoes he error erm and T s he samle sze. Ths s called ARCH() model. The nex se s o check he ARCH effecs by usng resduals of mean equaon. We can aly he usual Ljung-Box sascs Q() o he { e } seres (McLeod and L, 983) or aly he whe's es of sgnfcance of α = 0( =,, ) by F-sasc under he null hyohess Ho:α = = α = 0. Ths F-sasc s dsrbued as χ dsrbuon. (9)
99 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) If ARCH effecs are found sgnfcan we can use he PACF of e o deermne he ARCH order. Afer secfyng he volaly model we erform he jon esmaon of he mean and volaly e models. Lasly we evaluae he fed model and furher refne. The sandardzed resduals, ẽ = can be seen o check he adequacy of a fed ARCH model. We can evaluae he QQ los of ẽ and ẽ o check valdy of mean and varance equaons resecvely. Afer fndng he arameers of he model, redcon can be done. 3.3. The GARCH Model Bollerslev roosed a useful exenson known as he generalzed ARCH (GARCH) model. The e follows a GARCH(, q) model (Bollerslev, 986; Tsay, 005) f q, = α0 + αe + β j j = j= e = (0) max(,q) = β In addon o ARCH condons, we also have β j 0, and ( α + ) <. The consran on α +β mles ha he uncondonal varance of e s fne, whereas s condonal varance evolves over me. The α and β j are ARCH and GARCH arameers, resecvely. Smlar o ARCH models, GARCH models also observe volaly cluserng, leverage effec and heaver als. Secfyng he order of a GARCH model s no easy and only lower order GARCH models are used n mos alcaons. 3.4. The Inegraed GARCH (IGARCH) Model IGARCH models are un-roo GARCH models. An IGARCH(,q) model can be wren as q, = α0 + αe + β j j = j= q + = β j= j e = () where addonal consrans are α = and > β j > 0. 3.5. The GARCH-M Model Ofen he reurn of a secury may deend on s volaly. In hese cases, we use GARCH-M or GARCH n mean model. A GARCH(,q)-M model can be secfed as r = μ + k + e, e = 0 = q j= j = α + α e + β () j The consan arameer k s called he rsk remum arameer. 3.6. The Exonenal GARCH (EGARCH) Model An EGARCH model s secfed as e =, In( ) = α 0 + = α e + γ + e q j= λ In( j j = π ) (3)
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 00 consequenly For e ~ N(0, ) he sandardzed varable e follows a sandard normal dsrbuon and e e E( ) =. The arameers α caure he leverage effec. For good news ( > 0) π he mac of nnovaon e s e ( α + γ ) and for bad news ( < 0) e s e ( α + γ ). If α e becomes 0, ( In ) resonds symmercally o. To roduce a leverage effec α mus be negave. The fac ha he EGARCH rocess s secfed n erms of log-volaly mles ha s always osve and, consequenly, here are no resrcons on he sgn of he model arameers. 3.7. The Threshold GARCH (TGARCH) Model/GJR Model A TGARCH(, q) model as roosed by (Glosen e al., 993) can also handle leverage affecs model by assumng he followng form 0 = q j= j j = α + α e + β + γ v e (4) = where, < 0 v - = { 0, 0 (5) and α, γ, and β j are non-negave arameers sasfyng condons smlar o hose of GARCH models. I can be seen ha a osve e - conrbues o, whereas a negave e has a larger mac ( α +γ ) wh γ > 0. 4. Neural Neworks n Volaly Esmaon 4. Esmaon of Condonal Volales Neural neworks can be used o esmae he condonal volaly (Gacomn, 003; Eun and Resnck, 004; Bhalla, 008) of fnancal me seres. Consder ha a me seres wh sochasc volaly follows an AR()-ARCH() rocess wh he followng form r + = ( r, r -, r -,... r - +, X,..., X h ) + ( r, r -, r -,... r - +, X,..., X h ) + where, s..d. wh E( ) = 0, ( ) =. + Defnng = ( r r- +,,..., X h) R, z R as r + = ( ) + ( ) + We can wre ψ ( z) = θ( z) - φ ( z) where z = z = φ [ ] ( ) r + [ z = z] = θ( ) + + (6) he AR()-ARCH() rocess can be wren s (0) [ r z = z] = ψ ( ) + () (7) (8) (9)
0 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) where Usng a neural nework Φ NN o aroxmae φ(z) and Θ NN o aroxmae θ(z), we oban φ ˆ( ) = Φ ( ; wˆ ) () θ ˆ = Θ ; w (3) ( ) ( ) ˆ = w ˆ = argmn ( + - Φ ( ; w )) (4) N - ( - ( ; w )) wˆ = argmn + Θ (5) N - = An esmaor of Ψ (Z) s obaned as () ˆ() ˆ ˆ z = θ z - ( z) (6) Hardle e al. (00) used he aroach where he resduals are subsued by he samle resduals. Aroxmang he resduals hrough he samle resduals ˆ - ˆ + = r + ( ) (7) and he squared samle resduals wh a neural nework Ψ NN wh arameers wˆ = argmn ( + - ψ ( ; w) ) (8) N - = he esmaon of he condonal volaly can be wren as ˆ ( ) = ψ ( ; ŵ) (9) 4.. Esmaon of Imled Volales In her landmark aer,black and Scholes (973) gave model o deermne he rce of a call oon C a me, whch s gven by he formula = S Φ( )- Φ( ) (30) s ln + + r τ K = (3) τ = d - τ (3) where S s he so rce of he underlyng asse, he volaly of he underlyng asse rce rocess, r he rsk free neres rae, τ he me o maury, K he srke rce of he oon and Φ he cumulave dsrbuon funcon of he normal dsrbuon. The Black Scholes model assumes ha s consan over he rce rocess of a gven underlyng asse. Snce acual volaly of he underler can no be observed drecly, we use he volaly whch s mled by oon rces observed n he markes. Ths s called mled volaly and s defned as he arameer ˆ ha yelds he acually observed marke rce of a arcular oon when subsued no he Black-Scholes formula. The mled volaly of a Euroean u wh he same srke and maury can be deduced usng he u-call ary. = S - (33) - In oose o he heorecal formulaon, he mled volales are no consan. They form a volaly smle when loed agans he srke rces K a me,(hardle e al., 00) and change also accordng o he me o maury τ. Calculaon of mled volales a dfferen srkes and maures yelds a surface called mled volaly surface on a secfed grd usng a b-dmensonal kernel smoohng rocedure. A Nadaraya-Wason esmaor wh a quarc kernel s usually emloyed.
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 0 Fgure 4: Volaly Smle We can wre he deendency of he mled volaly he srke rce K or he moneyness and me o maury τ as. ˆ = f ( K, τ) (34) or K ˆ = f, τ (35) S Ths relaon beng non lnear form can be esmaed wh neural neworks, gven ha mled volales for a srke rce or moneyness and for dfferen maures are avalable o consruc he ranng se. The nework Ψ NN K ˆ = ψ, ; wˆ (36) S where w ˆ = argmn ˆ - ψ, τ; wˆ (37) n - = s used o generae mled volaly surface. 5. Exermen The exermen descrbed on hs secon comares one se ahead forecass of mes seres roduced by MLP and RBF neworks wh dfferen archecures by changng he number of neurons n he hdden layer. Fve dfferen exchange rae me seres and en dfferen archecures are used. A non lnear me deendency of sze (lag), s consdered for all he seres. The exermen uses a nework φ NN wh one hdden layer conanng h neurons o forecas he realzaons of he me seres a +, as gven n Equaon 38. + = ( r, -, r -,... r- + ) (38) Aferwards, he erformance of he forecass are evaluaed. 5.. Tme Seres Fve exchange rae me seres used are daly observaons of:brsh Pound vs US-Dollar (BPUSD) from 8//993 o 8//008, German Mark vs US-Dollar (DEMUSD) from 8//993 o 8//006, Jaanese Yen vs US-Dollar (JPYUSD) from 8//993 o 8//008, Indan Ruees vs US-Dollar (IRUSD) from 8//993 o 8//008 and Euro vs US-Dollar (EURUSD) from 5//998 o 8//008
03 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 5.. Transformaon To elmnae rend and seasonaly he me seres are ransformed by frs dfferences of logarhms. Afer hs oeraon, he me seres elemens r reresen he logarhm of he fnancal reurn of holdng a un of he currency for one erod: = log - = log - log (39) ( ) ( ) - The me seres are sl no wo ses, he ranng se and he es se:he ranng se = conans roughly 95% of he observaons,.e., = (,, 0 ), 0 = mod(0.95n) and he es se conans roughly 5% of he observaons,.e., = ( 0 +,, N) The able shows he nformaon abou he me seres and sze of subses used. Table : Tme seres and samle sze Tme Seres From To N BPUSD 8//993 8//008 506 5480 DEMUSD 8//993 8//006 45 4749 JPYUSD 8//993 8//008 506 5480 INRUSD 8//993 8//008 506 5480 EURUSD 5//998 8//008 3484 3667 5.3. Tme Deendency The rocess s modeled wh lag 5, he realzaon a + s deenden on he realzaons of he las 5- radng days. 5.4. Neworks Varous knds of arameers whch can be adjused n Neural Nework archecure are number of uns, number of hdden layers, ye of neurons, learnng raes for suervsed and unsuervsed ranng and nal weghs ec. In our ex-ermen,he RBF and MLP neworks are bul n XloRe Sofware wh one hdden layer of h neurons formng he archecure 5 h. The number h of uns on he hdden layer s ncreased from 5 o 50 n ses from 5 uns. For each archecure, he neworks are raned on he ranng ses unl a MSE from 5.0-5 or less s reached. The oher arameers are he defauls for RBF and MLP ranng quanles from he XloRe neural neworks lbrary. 5.5. Performance Measures The forecass are made on he es se = ( 0 +,, N). The k = N ( 0 + + lag) forecass are comared wh he rue realzaons. Followng erformance measures are used. 0 Normalzed Mean Squared Error (NMSE) ( ˆ ) = k = ˆ where ˆ s he varance of he ranng se (n samle uncondonal volaly) Mean Absolue Error (MAE) = - rˆ k = NMSE and MAE are he mercs used o esmae he error of redcon. (40) (4)
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 04 Funcon (SIGN) = δ k = where, + rˆ + 0 δ = { 0, To evaluae wheher he resul of he nework can be used as a radng sraegy, he fracon of redcons wh same sgn as he rue realzaons s calculaed by he merc SIGN. (4) (43) 5.6. Resuls and Dscusson 5.6.. Neural Neworks For each me seres, he resul s summarzed below. BPUSD: The RBF neworks erformed beer han he MLP for all archecures, concernng NMSE and MAE. The bes nework s a RBF wh 0 hdden uns. DEMUSD: The MLP neworks erformed beer han he RBF for all archecures (exce for 5 hdden uns), concernng NMSE and MAE. The bes nework s a RBF wh 5 hdden uns, he second bes a MLP wh 5 hdden uns. JPYUSD: The RBF neworks erformed beer han he MLP for all archecures, concernng NMSE and MAE. The bes nework s a RBF wh 0 hdden uns. INRUSD: The MLP neworks erformed beer han he RBF for all archecures, concernng NMSE and MAE. The bes nework s a MLP wh 5 hdden uns. EURUSD: The MLP neworks erformed beer han he RBF for 6 archecures whle RBF neworks erformed beer han he MLP for 4 archecures, concernng NMSE and MAE. The bes nework s a RBF wh 5 hdden uns. In all he cases he resul of he nework can be consdered as a radng sraegy, snce he value of he he funcon SIGN s greaer han 0.5. 5.6.. Condonal Heeroscedasc Models USD/GBP Seres r = -0.000065 + a; a = ; = 0.00795+ 0.9594 a- + 0.03650 USD/DM Seres r = -0.000043 + a; a = ; = 0.000888 + 0.975946 a + 0.00566 USD/JPY Seres r = 0.000045 + a; a = ; = 0.0089 + 0.949508 a- + 0.04940 USD/INR Seres r = 0.00009 + a; a = ; = 0.000000 + 0.9897 a- + 0.69468 USD/EUR Seres r = - 0.0004 + a; a = ; = 0.000545 + 0.9790 a + 0.09795 - - (44) (45) (46) (47) (48) 5.6.3. Comarson of Neural Nework and Condonal Heeroscedasc Models Comarsons of Neural Nework models and condonal heeroscedasc models (GARCH, GARCH- M, EGARCH, TGARCH/GJR and IGARCH) for all he fve exchange raes are shows n followng ables (Tables, 3, 4, 5 and 6)
05 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) Table : Comarave analyss of USD/GBP Seres Model NMSE MAE.0 SIGN Neural Nework 0.56853 0.74 0.749 GARCH(,).000 0.777 0.63689 GARCH(,)-M 0.86999 0.683 0.6799 EGARCH(,) 0.8986 0.3988 0.68674 TGARCH/GJR(,) 0.87995 0.664 0.68790 IGARCH(,) 0.80463 0.775 0.65564 Table 3: Comarave analyss of USD/DM Seres Model NMSE MAE.0 SIGN Neural Nework 0.73085 0.35033 0.5357 GARCH(,) 0.833 0.46988 0.49777 GARCH(,)-M 0.88089 0.44983 0.48563 EGARCH(,) 0.74093 0.36979 0.578 TGARCH/GJR(,) 0.76097 0.38988 0.50787 IGARCH(,) 0.8047 0.4044 0.49768 Table 4: Comarave analyss of USD/JPY Seres Model NMSE MAE.0 SIGN Neural Nework 0.740 0.37706 0.5743 GARCH(,) 0.89899 0.366 0.54 GARCH(,)-M 0.84957 0.3600 0.539 EGARCH(,) 0.79654 0.3844 0.5634 TGARCH/GJR(,) 0.8097 0.307 0.5599 IGARCH(,) 0.8374 0.3636 0.53760 Comarave analyss clearly show ha Neural nework has an edge over he condonal heeroscedasc models lke GARCH, GARCH-M, GJR (TGARCH), IGARCH, EGARCH ec for exchange rae forecasng. Neural Nework clearly ouerform oher models n erms of he merc "SIGN" conssenly. Ths shows ha uward/downward movemen of exchange rae s more accuraely redced by Neural Neworks as comared o condonal heeroscedasc models. However n some solaed cases condonal heeroscedasc models dd fare beer han Neural Neworks. These cases are menoned below Table 5: Comarave analyss of USD/INR Seres Model NMSE MAE.0 SIGN Neural Nework.095 0.484 0.69643 GARCH(,).8739 0.3084 0.577 GARCH(,)-M.8398 0.3059 0.5500 EGARCH(,).69866 0.87 0.665 TGARCH/GJR(,).6433 0.757 0.6360 IGARCH(,).6863 0.989 0.59089
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 06 Table 6: Comarave analyss of USD/EUR Seres Model NMSE MAE.0 SIGN Neural Nework 0.8844 0.38644 0.5899 GARCH(,) 0.99998 0.34930 0.48804 GARCH(,)-M 0.93987 0.3498 0.4959 EGARCH(,) 0.89833 0.3097 0.53994 TGARCH/GJR(,) 0.89986 0.344 0.5785 IGARCH(,) 0.9064 0.34999 0.4656. "MAE" merc of GJR s beer han ha of Neural Neworks n case of USD/JPY (exchange rae of Jaanese Yen) seres.. "NMSE" merc of GJR s beer han ha of Neural Neworks n case of USD/INR (exchange rae of Indan Ruee) seres. 3. "MAE" merc of EGARCH s beer han ha of Neural Neworks n case of USD/EUR (exchange rae of Euro) seres. Barrng hese hree cases, Neural Neworks erformed sgnfcanly beer han condonal heeroscedasc models. Whn he condonal heeroscedasc models, he erformance of IGARCH and TGARCH was beer han oher heeroscedasc models. 6. Esmaon of Condonal Volales Boh Neural Neworks and GARCH(,) models are used o esmae condonal volales of exchange rae seres. The resuls are found o be comarable. For US/GBP seres he esmaon of condonal volaly usng boh Neural Neworks and he GARCH(,) models are shown below. Snce Volaly can' be observed drecly, so comarson beween he chars can' be made. However snce he edge of Neural Neworks over he condonal heeroscedascy models has already been shown n case of Exchange rae forecasng, can be sad ha Neural Neworks wll fare beer n hs case oo and more rof can be made f volaly esmaes based on Neural Neworks are used for volaly radng sraeges. Fgure 5: Log reurns and condonal volales from he exchange rae Brsh Pound / US Dollar from 8//993 o 8//008. Esmaed wh RBF nework, 5 hdden uns, lag 5
07 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) Fgure 6: GARCH (,) Fed model for Brsh Pound / US Dollar Seres 7. Esmaon of Imled Volales Fnally he mled volaly surface are esmaed from he daa se 3-MAR-008, 9-SEP-008 and 3-DEC-008.da usng neural neworks. The daa se conans selemen rce of he NIFTY Oon (underlyng asse), srke rces, neres raes, mes o maury and rces from us and calls raded a he Naonal Sock Exchange (NSE), Inda on 3/03/008,9/08/008 and 3//008. The mled volaly surface esmaed hrough a MLP nework wh 5 hdden uns s shown on Fgures 7, 9 and. The mled volaly surface esmaed hrough a RBF nework wh 5 hdden uns s shown on Fgures 8, 0 and. All cures also show he mled volaly curves (red), used on he esmaon of he surface. The resuls shows volaly surfaces for re meldown, meldown and os meldown erods. The exreme volales due o fnancal meldown are clearly vsble n he fgures. Concluson The henomenons of volaly cluserng, auocorrelaon, heeroscedascy are observed n Nfy reurns. Neural Neworks do a farly good job n forecasng Exchange rae (NMSE 0.6, Sgn 0.6).RBF neworks do consderably beer han MLP neworks n hs case. 3 Condonal heeroscedasc models can be effecvely used o redc mean and volaly of NIFTY daly reurns. 4 Whn he condonal heeroscedasc models, he erformance of IGARCH and TGARCH was beer han oher heeroscedasc models. 5 Neural Neworks erformance s beer (around 0-5% mrovemen) han condonal heeroscedascy models lke GARCH, GARCH-M, EGARCH, GJR, IGARCH ec n forecasng Exchange rae. 6 Neural nework can be effecvely emloyed o esmae he condonal volaly (besdes exsng mehods of condonal heeroscedascy models lke GARCH, GARCH-M, EGARCH, TGARCH, IGARCH ec.) 7 Neural nework can be effecvely emloyed o esmae he mled volaly of he oons.
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 08 Fgure 7: Imled volaly surface esmaed usng a (-5-) MLP. Parameers: srke rces and maures. Daa: NIFTY Oon a NSE, Inda on 3/03/008 Fgure 8: Imled volaly surface esmaed usng RBF nework wh 5 hdden uns. Parameers: moneyness and maures. Daa: NIFTY Oon a NSE, Inda on 3/03/008 8 Volaly surface can be effecvely generaed afer geng he forecas of mled volaly of oons and long hs along wh exercse rce and me o maury. 9 RBF neworks do consderably beer han MLP neworks n exracng he nformaon necessary o erform a good generalzaon from he ranng se. The MLP may learn nformaon secfc o he ranng se ha has no use for generalzaon. Besdes ha, we need o consder he ossbly ha MLPs wh more han one hdden layer may generalze beer, maybe beer han RBFs. 0 The number of hdden uns used does no seem o have a sragh relaon wh he forecas erformance. Neworks wh few hdden uns erformed beer han neworks wh many hdden uns and he way around.
09 Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) Fgure 9: Imled volaly surface esmaed usng a (-5-) MLP. Parameers: srke rces and maures. Daa: NIFTY Oon a NSE, Inda on 9/08/008 Fgure 0: Imled volaly surface esmaed usng RBF nework wh 5 hdden uns. Parameers: moneyness and maures. Daa: NIFTY Oon a NSE, Inda on 9/08/008
Inernaonal Research Journal of Fnance and Economcs - Issue 49 (00) 0 Fgure : Imled volaly surface esmaed usng a (-5-) MLP. Parameers: srke rces and maures. Daa: NIFTY Oon a NSE, Inda on 3//008 Fgure : Imled volaly surface esmaed usng RBF nework wh 5 hdden uns. Parameers: moneyness and maures. Daa: NIFTY Oon a NSE, Inda on 3//008 Neural nework can smulaneously and effecvely exrac he non-lnear funconal form as well as model arameers (as oosed o condonal heeroscedascy models where he funconal form needs o be secfed for esmaon of arameers). Neural neworks rovde quanave fnance wh srong suor n roblems relaed o non-aramerc regresson. Also remarkable are he heursc consderaons nvolved on he se u of neural neworks: somemes arameers and archecures are chosen only by ral and error.
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