ESTIMASI PARAMETER MODEL COX INGERSOLL ROSS MENGGUNAKAN METODE MAXIMUM LIKELIHOOD ESTIMATION Fy Syhftr B, Nv Styhdw, Muhlsh Novtsr Mr 3,,3 Uvrsts Tjugpur, Potk Eml korspods : fy_syhftr@yml.com Kutug yg dprolh dr sutu st fsl tu vsts dpgruh olh tgkt bug. Tgkt bug yg brubhubh scr tdk pst mybbk tgkt bug sult utuk dprdks. Plt mmbhs ttg slh stu modl prgrk tgkt bug ytu modl Cox Igrsoll Ross (CIR. Modl CIR mmprdks tgkt bug sllu brl postf. Pd modl CIR trdpt bbrp prmtr yg tdk dkthu ly. Olh kr tu, pd plt prmtr pd modl CIR dstms mgguk mtod Mxmum Lklhood Estmto (MLE. Pksr prmtr modl CIR mmbutuhk dt hstors dr tgkt bug. Dg mgguk dt tgkt bug Bk Idos mul dr Jur 006 smp dg Jur 05 dprolh l stms prmtr pd modl CIR ytu ˆ = 0,079 b ˆ = 4,679 d s ˆ = 0, 973. Kt kuc: Modl CIR, MLE, Nwto Rphso PENDAHULUAN Tgkt bug mmpuy pgruh yg ptg dlm mtuk hrg dr sutu strum vsts, sprt oblgs, shm, d ops. Kutug yg dprolh st mlkuk vsts dpgruh olh tgkt bug yg brlku pd strum vsts yg tlh dplh. Tgkt bug brubh spjg wktu yg mrupk pross stokstk. Kgt prdgg brlgsug trusmrus dlm psr kug, shgg dbutuhk sutu modl prgrk tgkt bug utuk wktu yg kotu. Dr brbg ltrtf modl tgkt bug yg d, trdpt ss postf d gtfy. Pd thu 977, Vsck mmprklk modl Vsck sbg modl suku bug stokstk prtm kly. Pd modl, prdks tgkt bug bs brl gtf, sdgk pd rlty tgkt bug tdk mugk brl gtf. Sljuty, kkurg trsbut dprbk pd sbuh modl yg dsbut modl Cox Igrsoll Ross (CIR, yg mjm prdks tgkt bug tdk gtf []. Modl k dbhs lbh ljut pd plt. Modl CIR dprklk olh Cox, Igrsoll d Ross pd thu 985. Pd modl trdpt sft m rvrso yg mrupk kcdrug dr tgkt bug utuk kmbl muju rtrt jgk pjg dr tgkt bug. Dg dy sft, prgrk tgkt bug k muju sutu lvl rtrt tgkt bug yg dsbut m rvrso lvl. Ktk tgkt bug tgg, koom cdrug mlmbt d prmt krdt dr pmjm cdrug rdh. Sbg dmpky, tgkt bug k turu. Sblky. Ktk tgkt bug rdh, k trjd kcdrug ky prmt krdt dr pmjm shgg
dmpky tgkt bug k cdrug k. Jk pross k d turuy tgkt bug trjd trus mrus, mk dlm jgk pjg tgkt bug k brd dsktr m rvrso lvl []. D dlm modl CIR trdpt bbrp prmtr yg prlu dstms hgg ddptk sutu stms yg mdkt dt sbry. Bbrp mtod yg dpt dguk utuk mgstms prmtr tr l Mxmum Lklhood Estmtor (MLE, Ordry Lst Squr (OLS, mtod Mom d ly. Pd plt dguk MLE. Mtod mmbrk hsl stms yg bk bg prmtr, trutm pbl smply bsr. METODOLOGI Plt brup stud ltrtur yg dmul dg mmpljr tor probblts, prsm dfrsl, prsm dfrsl stokstk d mtod Mxmum Lklhood Estmto (MLE. Plt dmul dg mgguk prsm modl CIR. Kmud, brdsrk dt tgkt bug Bk Idos mul dr Jur 006 smp dg jur 05, prmtrprmtr pd modl CIR dstms mgguk mtod MLE. Shgg k dprolh l stms utuk prmtr modl CIR. Prmtrprmtr trsbut ytu voltlts s, rtrt jgk pjg dr tgkt bug (m rvrso lvl, d kcpt pross utuk kmbl muju m rvrso lvl b. HASIL DAN PEMBAHASAN Pd thu 985 Cox, Igrsoll d Ross mmprklk modl tgkt bug Cox Igrsoll Ross (CIR. Modl mmprdks tgkt bug sllu brl postf. Modl CIR dpt dytk sbg brkut []: ( b( ( s ( ( dr t = r t dt + r t db t ( dg dr( t mujukk prubh tgkt bug, s mujukk voltlts, mujukk rtrt jgk pjg dr tgkt bug (m rvrso lvl, b mujukk kcpt pross utuk kmbl muju m rvrso lvl, d B( t mujukk grk Brow.
D dlm modl tgkt bug CIR d tg prmtr yg tdk dkthu d hrus dstms, ytu, b d s. Pd plt prmtrprmtr trsbut dstms mguk mtod Mxmum Lklhood Estmto (MLE. Sblum mgstms prmtr dg mgguk fugs MLE dbutuhk probblty dsty fucto (pdf dr modl CIR. Pdf dr modl CIR dpt dttuk dr m d vrs. Brdsrk modl tgkt bug pd Prsm ( d pross Ito, dprolh m d vrs modl tgkt bug CIR, ytu d bt ( ( ( ( 0 E r t = + r ( bt Vr( r( t = + ( r( 0 + r( 0 b b b s s s bt ( 3 Dsumsk modl CIR brdstrbus Norml tu dpt dtuls mjd bt s s bt s bt N ( r( 0, ( r( 0 r( 0 + + + b b b. Olh kr tu, pdf dr tgkt bug rt ( pd slg [ ut, ] dg u< t dlh b( t u ( rt ( ( ( ru ( + f( r( t = xp s s b( t u s s s b( t u b( t u s b( t u p ( ru ( ru ( ( ru ( ru ( + + + + b b b b b b b( t u ( rt ( ( ru ( = xp s s s s b( t u b( t u b( t u s s b( t u p p( ru ( p ru ( ( ru ( ru ( + + + + b b b b b b ( 4 Prsm (4 dguk utuk mmbtuk prsm Lklhood ytu b( ( ( r ( ( r ( L(, b, s = xp ( ( ( ( s s b( ( s = s s b s b b( ( p p( r ( p r ( ( r ( r ( + + + + b b b b b b Õ ( 5 Dg mgguk Prsm (5, prmtr, b d s dcr dg mmksmumk fugs Lklhood Lbs (,,. Sljuty, fugs (,, Lbs dmodfks k dlm btuk ( Lbs (. Dg mmksmumk l ( Lbs (,, k mgkbtk (,, l,, mksmum. Dg dmk Prsm (5 mjd Fugs l ( (,, bd ( r( ( r( l L(, b, s = l å s s s s b s b b s = D D = D bd p p( r( p r( ( r( r( + + + + b b b b b b Lbs mjd å ( 6 Lbs pd Prsm (6 dsbut jug fugs LogLklhood.
Utuk mmprolh stms prmtr, b d s, fugs LogLklhood dturuk skl trhdp tp prmtr yg k dstms. Hsl stms dpt dprolh dg ( l Lbs,, mylsk prsmprsm. ( l Lbs,, = 0, b b b bd l L(, b, s D D ( r ( ( r ( + + = å bd å = bd s s bd s bd ( r ( r ( = + + + ( r ( + r ( b b b = 0 d ( l Lbs,, s = 0 bd bvt ( ( r ( ( r ( (. + + l å s s s s b s b bd s = D D bd p p( r ( p r ( + ( r ( + r ( + + b b b b b b b s s b s D b ( r ( ( r ( V V + b b b + l = s s bd s bd s s bd s bd p + p( r ( + p r ( + ( r ( + r ( b b b b b b å ( 7 bv bv bv bv bv L(, b, s ( bv( 4r ( bvr ( ( ( r ( + ( r ( ( r ( ( r ( = å b b b b b V V V V = ( r ( + ( r ( + ( s + ( r ( s + s ( r ( b V b b ( r( l V V r( ( r( s s b s V b p p( r ( p r ( V + + b b b bv bv s + ( r ( s + ( r ( s ( ( d l b V r ( ( r ( s s b s V b p p( r ( p r ( V + + b b b b b s + ( r ( s + ( r ( s bd L(,, ( r ( ( r ( b s = å = s s s bd s bd ( r ( r ( + + b b b ( ( 8 V V bd ( r ( ( r ( l + å 3 3 3 = s s s s b s b bd s D D bd = p p( r ( p r ( + ( r ( + r ( + + b b b b b b å ( 9 Prsm (7, (8 d (9 mrupk fugs mplst. Utuk mmprolh l stms ˆ, b ˆ d s ˆ dguk mtod Nwto Rphso. Pd plt mtod NwtoRphso dlkuk dg btu softwr R. Dg mgguk mtod Nwto Rphso dprolh l stms prmtr modl CIR yg dytk dg ots ck ( + = sˆ, dm ˆ, b ˆ d s ˆ brturutturut dlh stms dr, b d s. Brkut dlh lgkhlgkh mtod NwtoRphso: k ( + = ˆ, bk ( + = bˆ d
. Ttuk l wl utuk ˆ, b ˆ d s ˆ ytu 0 0, b d b. Htug l b d c scr trtf, dg rumus 0 s. l L l L l L l L k k l L l L l L l L b b, k = 0,,, K,. k k c k c k l L l L l L l L c. Lgkh k du dulg hgg dkhdk. d. Stlh mlwt lgkh tg, plh prmtr k k b b TOL k k c k c k b c k k k k ( + = ˆ, bk ( + = bˆ d ck ( + = sˆ. mmuh tolrs TOL yg pd trs yg trkhr sbg stms dr Brdsrk dt tgkt bug Bk Idos mul dr Jur 006 smp dg Jur 05 dprolh l stms prmtr pd modl CIR ytu ˆ = 0,079 b ˆ = 4,679 d s ˆ = 0,973. KESIMPULAN Prmtrprmtr pd modl CIR dstms mguk mtod Mxmum Lklhood Estmto (MLE. Brdsrk dt tgkt bug Bk Idos mul dr Jur 006 smp dg Jur 05 dprolh l stms prmtr pd modl CIR ytu ˆ = 0,079 b ˆ = 4,679 d s ˆ = 0,973. DAFTAR PUSTAKA. Flpovc, Dmr, 009, TrmStructur Modls, SprgrVrlg, Nw York. Hull, J.C., 0, Optos, Futurs d Othr Drvtvs, Eghth Edto, Prtc Hll, Nw Jrsy.
3. Jog, P. D. d Hllr, G.Z., 008, Grlzd Lr Modls for Isurc Dt, Cmbrdg Uvrsty Prss, Cmbrdg.