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 +, (
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 0 0 0 0 (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 5 3 3 ( 5 + + 6 + 3 9 3 ( 3 ( ( 6 3. E b + + + + + + +
3 Doubly sochsic models wih symmeric GARCH errors 09 + ( + 5 + 6 3 + 9 b 3 ( 3( 6 3 3 5 3 ( 5. K 3 Proo. See deils in [8]. ( ( 6. φ +φ +φ ( ( E( b E( y ( ( ( 3 + 6 +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 3 + 6 +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 ( (
0 Muhmmd Sherz ( ( z ( αz ( φ +φ +φ ( (( φ+ + φ E( y + E( b E( y + 3zE( + 6φ E( b E( y + 3 + 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 ( 5 6 3 9 3 ( 3 ( ( 6 3 3 ( z + + + φ 6 φ + + + + + + + φ 6 3 ( 3( 3 5 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 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
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 ( 5 6 3 9 3 ( 3 ( ( 6 3 + + + φ 6 φ + + + + + + + 3 5 3 3 φ6 3 ( 3( ( ( K, K is kurosis o he process ollows he deiniion. ( y ( y (
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
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 5 3 3 ( 5 + + 6 + 3 9 ( + + 3( + +( + 6 + + 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. 50 63, 005.. BOLLERSLEV, T., Generlized uoregressie condiionl heeroscedsiciy, Journl o Economerics, 3, pp. 307-37, 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. 987 008, 98. 5. GEORGE, L., Compuionl innce: Numericl mehods or pricing inncil insrumens, Buerworh-Heinemnn, Elseie, 00. 6. JURGEN, F., WOLFGANG, H.K., CHRISTIAN, H.M., Sisics o Finncil Mrkes: An Inroducion, Springer, 0. 7. NICHOLLS, D., QUINN, B.G., Rndom Coeicien Auoregressie Models, An Inroducion: Lecor Noes in Sisics, Springer, New York, 98. 8. PEIRIS, S., THAVANESWARAN,A., APPADOO,S.S., Doubly sochsic models wih GARCH models, Applied Mhemics Leers,, pp. 768 773, 0. 9. SHIRYAEV, A.N., Probbiliy, nd ed.,springer-verlg, New York Inc, 005. 0. 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