THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 15, Number 2/2014, pp

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 5, Number /0, pp. 07 DOUBLY STOCHASTIC MODELS WITH ASYMMETRIC GARCH ERRORS Muhmmd SHERAZ Uniersiy o Buchres, Fculy o Mhemics nd Compuer Science, Buchres, Romni E-mil: Muhmmd_sherz8@homil.com The mehods ddressing oliliy in compuionl innce nd economerics he been recenly repored in inncil lierure. Recenly Peiris e l. [8] he inroduced doubly sochsic oliliy models wih GARCH innoions. Rndom coeicien uoregressie sequences re specil cse o doubly sochsic ime series. In his pper, we consider doubly sochsic sionry ime series wih symmeric GARCH errors. Some generl properies o process, like rince nd kurosis re deried. Key words: GARCH processes, AGARCH-(I-(0,, doubly sochsic ime series, RCA(, RCA- MA(, Sign-RCA-MA(.. INTRODUCTION Recenly, here hs been growing deelopmen in he use o nonliner oliliy models in inncil lierure. Blck-Scholes opion pricing models hs some conrdicory ssumpions, such s consn oliliy nd normlly disribued log reurns. In mny inncil ime series, empiricl sudies reel some cs such s sock reurns nd oreign exchnge res h exhibi lepokurosis nd sochsic oliliy. Any disribuion cn be chrcerized by number o eures such s men, rince, skewness nd kurosis. The mesure o kurosis is considered s wheher he d re peked or l relie o Norml disribuion. The concep o sochsic oliliy or inncil ime series, he uoregressie condiionlly heeroscedsic (ARCH models, ws irs sudied by Engle [], nd is generlizion, he GARCH models by Bollersle []. The GARCH models ssume symmeric eecs on oliliy, h is good nd bd news he sme eec on oliliy, which is shorll o hese models. The symmeric GARCH (AGARCH by Engle nd Ng nd rious oher nonliner GARCH exensions he been proposed o cpure symmeric eecs [3, 5]. Rndom coeicien uoregressie (RCA ime series were inroduced by Nicholls nd Quinn [7] nd some o heir properies were sudied by Appdoo e l []. Thneswrn e l. [8] he lso sudied some RCA ime series nd oliliy modeling. In his pper, we derie he kurosis or rious clsses o doubly sochsic models wih symmeric GARCH innoions. Recenly Peiris e l. [8] he inroduced doubly sochsic oliliy models wih GARCH innoions. In Secion some rndom coeicien uoregressie ime series he been discussed. In Secion 3 some doubly sochsic models wih symmeric GARCH innoions re inroduced nd we derie ormul or kurosis in erms o model prmeers.. RANDOM COEFFICIENT AUTOREGRESSIVE TIME SERIES Rndom coeicien uoregressie sequences re specil cse o doubly sochsic ime series. Some rndom coeicien uoregressie ime series re s ollows: (, y φ+ b +, (

2 08 Muhmmd Sherz (, y φ+ b + +θ, ( ( y φ+ b + Φ s + +θ,, (3, i y > 0, where s 0, i y 0,, i y < 0. In ( 3 we he Rndom coeicien uoregressie (RCA(, Rndom coeicien uoregressie-moing erge (RCA-MA(, Sign-RCA-MA (, respeciely. The men, rince nd kurosis or he boe processes he been discussed in [0]. The wo necessry nd suicien condiions or second order sionriy o { y } re s ollows: b i.,, 0 b 0 N 0 0 ii. φ + b <. Here { b } nd { } re errors in he model, while φ, Φ nd ( E s, we cn clcule he kurosis. For more deils see [0]. A process o he ollowing orm ws considered in [9]: ( + + s re prmeers. Since E( s nd b b + + b,. ( Recenly Peiris e l [8] he sudied doubly sochsic model o he orm y ( φ+ b +,, b b + + b,. ( + + The momens or he model he been recenly clculed [8]. Here independen sequence o ribles wih men zero nd rince. i.,, ii. <, iii. φ +φ +φ <. N (5 is n ideniclly disribued In (5 he obsered rndom rible is modeled in wo seps. Firs, he disribuion o he obsered oucome is represened in sndrd wy nd second sep, b is reed s being isel rndom rible. THEOREM.. Consider (5 wih condiions i, ii nd iii hen we he ollowing resuls : (. E( b 0. E( b 3 ( E b 6 3 ( 3( ( ( 3 ( ( 6 3. E b

3 3 Doubly sochsic models wih symmeric GARCH errors 09 + ( b 3 ( 3( ( 5. K 3 Proo. See deils in [8]. ( ( 6. φ +φ +φ ( ( E( b E( y ( ( ( φ 7. 3 ( φ E b 6φ E b φe b φ φ E( b ( E( b 3 ( φ E( b 6φ E( b φe( b ( y 8. K 3. DOUBLY STOCHASTIC MODELS WITH ASYMMETRIC GARCH ERRORS In his secion we discuss doubly sochsic oliliy models wih symmeric GARCH errors. The ollowing heorem proides he momens o doubly sochsic models wih AGARCH- (I-(0,. THEOREM. Consider he doubly sochsic oliliy process (5 wih AGARCH (I (0, errors o he ollowing orm hen we he resuls, nd 3 s ollows: ( (, y φ+ b y + ( b b + + b, + + z ( ω+α + r ( z ( αz ( φ +φ +φ. ( ze( z ( E( b E( E( y 3 ( φ E( b 6φ E( b φe( b φ. ( ( K, K is kurosis o he process. ( y ( y ( ( (( φ+ + Proo. E b y ( ( ( ( ( ( z E ( z ( αz φ E( b φ E( b E φ y + b y + + φ b y + φ y φ + E b + E z ( (

4 0 Muhmmd Sherz ( ( z ( αz ( φ +φ +φ ( (( φ+ + φ E( y + E( b E( y + 3zE( + 6φ E( b E( y E( b E( y 6E( b E( y E( E( b E( y E( ze( + 6zE( b E( y E( + 6zφ E( y E ( 3 ( φ E( b 6φ E( b φe( b 3zE( + 6 z ( E( b +φ E( E( y E( y 3 φ E( b 6φ E( b φe( b. E b y ( ( φ + + ( ( ( z ( αz ( φ +φ +φ ω+αr + E( y ( ( ( ( 3α α { } 3 ω+α r +α + α r z ( ( ( ( { } ( E ω+α r +α + α r E, 3α α α ( ω+αr +φ z αz φ +φ +φ 6 3 ( ( 3 ( ( ( z φ 6 φ φ 6 3 ( 3( ( ( ( y K. ( THEOREM. Consider he ollowing doubly sochsic ime series sisying condiions i, ii, iii s ollows: (, y φ+ b + +θ ( b b + + b, + + where is n ideniclly disribued independen sequence o ribles wih men zero nd rince nd we he ollowing condiions i, ii nd iii hen we he resuls, nd 3 s ollows: N 0 0 i.,, 0 0

5 5 Doubly sochsic models wih symmeric GARCH errors (. ii. <, iii. φ +φ +φ <. (. ( ( ( +θ ( ( φ +φ +φ ( E( y ( ( ( E( b 3 φ E( b 6φ E( b φe( b 3 + 6θ +θ + 6 +θ +φ ( ( K, K is kurosis o he process. ( y ( y ( ( Proo.. E φ+ b y + +θ ( φ ( + ( ( + ( +θ ( E b E E ( ( +θ ( ( φ +φ +φ ( ( b +θ E φ+ ( (( φ+ + +θ E( ( φ+ b y ( ( + +θ + φ+ b y + θ + φ+ b θ y E( φ+ b y + E( +θ E( + θ E( φ+ b E( E( y + E( φ+ b E( E( y. E b y 6 6 K ( y ( ( E( y ( ( ( E( b 3 φ E( b 6φ E( b φe( b 3 + 6θ +θ + 6 +θ +φ ( ( ( ( E( y ( ( ( E( b 3 φ E( b 6φ E( b φe( b ( +θ ( ( φ +φ +φ 3 + 6θ +θ + 6 +θ +φ. THEOREM Consider he doubly sochsic oliliy process wih AGARCH (I (0, errors o he ollowing orm hen we he resuls, nd 3 s ollows: (. (, y φ+ b y + +θ ( b b + + b, + + z ( ω+α + r z ( ω+α r ( +θ ( ( z ( α φ +φ +φ

6 Muhmmd Sherz 6 (. ( ( ( ( ( ( ( ( 3 φ E( b 6φ E( b φe( b 3 E + 6θ +θ + 6 +θ E b +φ E z z ( ( K, K is kurosis o he process. ( y ( y ( ( (( φ+ + +θ Proo.. E b y ( φ ( + ( ( + ( +θ ( ( E ( E( φ+ b +θ E b E E ( z z ( +θ ( ( z ( α φ +φ +φ ( (( φ+ + +θ E( ( φ+ b y ( ( + +θ + φ+ b y + θ + φ+ b θ y 3E( ( + 6θ +θ + 6 z ( +θ ( E( b +φ E( E( y 3 φ E( b 6φ E( b φe( b. E b y ( ( + ( ( ( ( ( 3α α 3 { } E ω+α r +α + α r ( ( z z ( ( ( ( z ( ω+αr +θ ω+αr E(, E( y α α φ +φ +φ ( ( ( ( ( 3α α { } 3 + 6θ +θ ω+α r +α + α r z z ( ( r ( ( αz ( φ +φ +φ 6 +θ ω+α φ φ φ + 3 ( ( 3 ( ( φ 6 φ φ6 3 ( 3( ( ( K, K is kurosis o he process ollows he deiniion. ( y ( y (

7 7 Doubly sochsic models wih symmeric GARCH errors 3 THEOREM. Consider he doubly sochsic oliliy process wih AGARCH(I (0, errors o he ollowing orm hen we he resuls, nd 3 s ollows: ( (, y φ+ b +Φ s y + +θ ( b b + + b, + + z ( ω+α + r z ( ω+α r ( +θ ( ( αz ( Φ +Φ +Φ φ +φ +φ. ( ( ( ( ( ( ( ( ( φ E( b 6φ E( b Φ 6Φ φ + E( b 3zE + 6θ +θ + 6 z +θ E b +φ +Φ E. ( ( ( K, K is kurosis o he process. ( y ( y ( ( (( φ+ +Φ + +θ ( E ( ( φ ( +Φ ( + ( ( + ( +θ ( E( b ( ( ( z +θ ω+αr E( y ( αz ( Φ +Φ +Φ φ +φ +φ Proo.. E b s y z +θ E b E E Φ φ+ ( (( φ+ +Φ + +θ ( ( + 6θ +θ + 6( +θ ( +φ +Φ ( φ E( b 6φ E( b Φ 6Φ ( φ + E( b 3 ( ( 6 6 ( ( φ E( b 6φ E( b Φ 6Φ φ + E( b. E b s y ( ( ( E E b E ( ( ( ( E + θ +θ + +θ E b +φ +Φ E ( z z ( ( + ( ( ( ( ( 3α α 3 { } E ω+α r +α + α r E ( ( αz z ( ω+α r ( +θ ( ( z ( α Φ +Φ +Φ φ +φ +φ

8 Muhmmd Sherz 8 ( ω + αr ( ( ( ( ( 3α α { } 3 + 6θ + θ ω + αr + α + α r z ( 6 z ( +θ ( φ +Φ ( + ( αz ( Φ +Φ +Φ φ +φ +φ ( ( + + 3( + +( φ 3 3 6( φ +Φ 6 Φ φ Φ ( ( K, K is kurosis o he process ollows he deiniion. ( y ( y (. CONCLUSIONS In his pper, he kurosis o doubly sochsic models wih AGARCH-(I-(0, innoions re deried. Sisicl inerences or hese doubly sochsic models wih symmeric GARCH errors nd se spce modeling cn be iewed s specil cse or nonliner ime series. REFERENCES. APPADOO, S.S., GHAHRAMANI, M., THAVANESWARAN, A., Momen properies o someime series models, The Mhemicl Scienis, 30, pp , BOLLERSLEV, T., Generlized uoregressie condiionl heeroscedsiciy, Journl o Economerics, 3, pp , 986. CHRISTIAN, F., JEAN, Z.M., GARCH models, Jhon Wiley nd Sons, 00.. ENGLE, R.F., Auoregressie condiionl heeroscedsiciy wih esimes o he rince o U.K.inlion, Econonomeric, 50, pp , GEORGE, L., Compuionl innce: Numericl mehods or pricing inncil insrumens, Buerworh-Heinemnn, Elseie, JURGEN, F., WOLFGANG, H.K., CHRISTIAN, H.M., Sisics o Finncil Mrkes: An Inroducion, Springer, NICHOLLS, D., QUINN, B.G., Rndom Coeicien Auoregressie Models, An Inroducion: Lecor Noes in Sisics, Springer, New York, PEIRIS, S., THAVANESWARAN,A., APPADOO,S.S., Doubly sochsic models wih GARCH models, Applied Mhemics Leers,, pp , SHIRYAEV, A.N., Probbiliy, nd ed.,springer-verlg, New York Inc, THAVANESWARAN, A., APPADOO, S.S., BECTOR,C.R, Recen deelopmens in oliliy modeling nd pplicions, Journl o Applied Mhemics nd decision sciences, Aricle ID8630, pp. Receied Ocober 3, 0

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