Using a Prediction Error Criterion for Model Selection in Forecasting Option Prices

Transcription

1 Ung a Predcon Error Creron for Model Selecon n Forecang Opon Prce Savro Degannak and Evdoka Xekalak Deparmen of Sac, Ahen Unvery of Economc and Bune, 76, Paon Sree, 0434 Ahen, Greece echncal Repor no 3, March 00 Abrac he common way o meaure he performance of a volaly predcon model o ae ably o predc fuure volaly. However, a volaly unobervable, here no naural merc for meaurng he accuracy of any parcular model. Noh e al. (994 aeed he performance of a volaly predcon model by devng radng rule o rade opon on a daly ba and ung foreca of opon prce obaned by he Black & Schole (BS opon prcng formula. (An opon a ecury ha gve owner he rgh, no he oblgaon, o buy or ell an ae a a fxed prce whn a pecfed perod of me, ubjec o ceran condon. he BS formula amoun o buyng (ellng an opon when prce foreca for omorrow hgher (lower han oday marke elemen prce. In h paper, adopng Noh e al. (994 dea, we ae he performance of a number of Auoregreve Condonal Heerocedacy (ARCH model. For, each radng day, he ARCH model, eleced on he ba of he predcon error creron (PEC nroduced by Xekalak e al. (003 and uggeed by Degannak and Xekalak (999 n he conex of ARCH model, ued o foreca volaly. Accordng o h creron, he ARCH model wh he lowe um of quared andardzed one ep ahead predcon error eleced for forecang fuure volaly. A comparave udy made n order o examne whch ARCH volaly emaon mehod yeld he hghe prof and wheher here any gan n ung he PEC model elecon algorhm for peculang wh fnancal dervave. Among a e of model elecon algorhm, even margnally, he PEC algorhm appear o acheve he hghe rae of reurn. Keyword and Phrae: ARCH model, Foreca Volaly, Opon Prcng, Model elecon, Predcably, Correlaed Gamma Rao Drbuon, Predcon Error Creron Correpondng auhor. el.: ; fax: E-mal addre: exek@aueb.gr.

2 . Inroducon Noh e al. (994 condered he problem of aeng he performance of wo volaly-forecang model, he ARCH modelng baed mehod and he mpled volaly regreon mehod, by radng opon. he ARCH model provde one common condonal volaly emae for boh call and pu opon prce, whle he mpledvolaly forecang mehod provde dfferen volaly emae for call and pu opon prce. Over he Aprl 986 o December 99 perod, for Sandard and Poor 500 (S&P500 ndex opon, he ARCH model baed forecang mehod led o a greaer prof han he rule baed on he mpled volaly regreon model. In parcular, by he radng raegy baed on he ARCH model a daly prof of 0.89% wa earned, whle by he mpled volaly mehod a daly lo of.6% wa made. In h paper, he performance of a varey of ARCH model nvegaed and her condonal volaly foreca are ued o compue he volaly durng he lfe of he opon. Foreca of opon prce are calculaed ung he Black & Schole (BS opon prcng formula and volaly emaed by a e of ARCH model. Moreover, a model forecang ably rang baed elecon creron condered by Degannak and Xekalak (999, he Predcon Error Creron (PEC, condered n order o chooe he approprae ARCH model for emang he volaly of he underlyng ae reurn. he performance of he elecon creron condered n he feld of forecang fuure volaly and prcng fnancal dervave. he advanage of he mehod uggeed le n he fac ha he rader flexble a o he choce of he model a each of a equence of pon n me. In he equel, hown ha he creron uggeed ha a afacory performance n elecng he ARCH model ha generae beer volaly predcon wh applcaon n creang peculave raege wh opon. I demonraed ha over he perod from March 998 o June 000, akng no conderaon a ranacon co of $, an agen who would conder ung h model elecon algorhm could have made a daly prof of.00% from radng S&P500 ndex opon. Secon provde a bref decrpon of he BS opon prcng formula. In econ 3, he ARCH proce preened and he PEC model elecon algorhm n he conex of ARCH model decrbed. Secon 4 preen Noh e al. (994 radng rule o ae he performance of volaly-forecang mehod. In econ 5, he cah flow, from radng opon baed on a e of ARCH procee, he PEC model elecon algorhm, and a number of oher mehod of model elecon, are compued. Fnally, n econ 6, a bref dcuon of he reul provded.

3 3. Opon An opon a ecury ha gve owner he rgh, no he oblgaon, o buy or ell an ae a a fxed prce (exerce prce whn a pecfed perod of me, ubjec o ceran condon. here are wo man ype of opon: call and pu. A call opon he rgh o buy a number of hare, of he underlyng ae, a a fxed prce on or before he maury day. A pu opon a rgh o ell a number of hare, of he underlyng ae, a a fxed prce on or before he maury day. A raddle opon he purchae (or ale of boh a call and a pu opon, of he underlyng ae, wh he ame expraon day. he maury day he lae dae ha he opon can be exerced. If he opon can be exerced only on he maury day, ermed a European opon, wherea an Amercan opon can be exerced on or before he expraon day. he purchaer of a call (pu opon acqure he rgh o buy (ell a hare of a ock for a gven prce on or before me and pay for he rgh a he me of purchae. On he oher hand, he wrer of h call (pu collec boh he opon prce oday and he oblgaon o delver (buy one hare of ock n he fuure for he exerce prce, f he purchaer of he call (pu demand.. Sock and Exerce Prce Relaonhp he exerce prce of he a he money opon equal o he prce of he underlyng ae. he exerce prce of he near a he money opon approxmaely he ame a he prce of he underlyng ae. A call (pu opon ad o be n he money f exerce prce le (greaer han he curren prce of he underlyng ae. A call (pu opon ad o be ou of he money f exerce prce greaer (le han he prce of he underlyng ae.. Black & Schole Opon Prcng Formula he prcng of opon a cornerone of fnancal leraure. he BS opon prcng model a very mporan and ueful model n emang he far value of an opon. Baed on he law of one prce or no arbrage condon, he opon prcng model of Black and Schole (973 and Meron (973 ganed an almo mmedae accepance among academc and nvemen profeonal. her approach can be ued o prce any ecury whoe payoff depend on he prce of oher ecure. he man dea o creae a cole elf-fnancng porfolo raegy, whereby long poon are compleely fnanced by hor poon, whch can replcae he payoff of he dervave. Under he no-arbrage condon, he dynamc raegy reduce o a paral dfferenal equaon

4 ubjec o a e of boundary condon ha are deermned by he pecfc erm of he dervave ecury. he adjued for dvdend BS formula o prce call and pu opon a me gven he nformaon avalable a me can be preened n he followng form: where, C P d d ( τ γ τ rfτ = Se N( d Ke N( d ( τ γ τ rfτ = S e N( d Ke N( d S ln K = = d ( τ ( τ ( rf γ ( τ ( τ ( τ : he foreca prce of a call opon, a me C wh τ day o maury, τ τ (., gven he nformaon a me, ( τ : he foreca prce of a pu opon, a me, gven he nformaon a me, P wh τ day o maury, S : he daly clong ock prce a me, a a foreca of τ : he remanng lfe of he opon n day (me o maury, S, rf : he annual connuouly compounded rk free nere rae (.e. hree-monh reaury bll, ( γ : he dvdend yeld of S, ( γ = ln( V τ d S, where V ( τ d he preen τ value of all dvdend pad durng he remanng lfe of he opon, K : he exerce (or rke prce a maury day τ, N (. : he cumulave normal drbuon funcon, ( τ : he volaly durng he lfe of he opon (he average volaly from unl he maury dae gven he nformaon a me. Fgure o 6 preen he relaonhp beween he prce of he opon and he varable nvolved n he BS formula. 4

5 Fgure. Relaonhp beween Opon prce and me o maury Fgure. Relaonhp beween Opon prce and Volaly Call Opon Prce /4 8/7 8/0 8/3 8/6 8/9 9/ 9/4 9/7 9/0 9/3 9/6 9/9 9/ me o maury Fgure 3. Relaonhp beween Opon prce and Sock Prce 9/5 9/8 0/ 0/4 0/7 0/0 0/3 0/6 0/9 Pu Opon Prce % % 4% 6% 8% 0% % 4% 6% 8% 30% 3% 34% 36% 38% 40% 4% 44% 46% 48% 50% 5% Volaly Fgure 4. Relaonhp beween Opon prce and Exerce Prce 54% Call Pu 56% 58% Opon Prce Call Pu Opon Prce Cal l Pu Sock Prce Exerce Prce Fgure 5. Relaonhp beween Opon prce and Rk Free Rae 30.0 Fgure 6. Relaonhp beween Opon prce and Dvdend Yeld Call Pu Opon Prce Call Pu Opon Prce % 0.5%.0%.5%.0%.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% Rk Free Rae 7.5% 8.0% 8.5% 9.0% 9.5% 0.0% 0.5%.0%.5%.0% % 0.5%.0%.5%.0%.5% 3.0% 3.5% 4.0% 4.5% 5.0% 5.5% 6.0% 6.5% 7.0% 7.5% Dvdend Yeld 8.0% 8.5% 9.0% 9.5% 0.0% 0.5%.0%.5%.0%.3 An Example n Compung heorecal Opon Prce Conder a rader who wan o evaluae he BS heorecal prce of a European call and pu opon wh hree monh o expry. he ock prce $60, he rke prce $65, he rk free rae 8% per annum (he reurn of hree monh reaury bll, he dvdend yeld 5% per annum and he volaly 30% per annum. hu, S = 60, K = 65, τ = 0. 5, rf = 0. 08, γ = and = Compung: d d = 0.409, = 0.559, N N ( d = 0.34, N( d = 0.659, ( d = 0.88, N( d = 0.7, 0.05* *0,5 he prce of he call opon : C = 60e e 0.88 = $ and he 0.05* *0.5 prce of he pu opon : P = 60e e 0.7 = $

6 6 3. he Auoregreve Condonal Heerocedacy (ARCH Proce and he Predcon Error Creron (PEC for Model Selecon 3. he ARCH Proce Le { ( θ } y refer o he unvarae dcree me real-valued ochac proce o be predced where θ a vecor of unknown parameer and E ( y ( θ I E ( y ( θ µ ( θ denoe he condonal mean gven he nformaon e avalable a me, I. he nnovaon proce for he condonal mean, { ( θ } hen gven by ε ( θ ( θ µ ( θ ( ( y ε, = wh correpondng uncondonal varance V ( ε ( θ E ε ( θ θ, zero uncondonal mean and ( ε ( θ ε ( θ = 0 = E,. he condonal varance of he proce gven I gven by V ( ( ( y ( θ = E ε ( θ θ. he nnovaon proce, { ε ( θ } an ARCH proce n he followng form: ε ( θ = z ( θ.. d. z ~ f [ E( z = 0, V ( z = ] ( θ = g ( θ, ( θ,...; ε ( θ, ε ( θ (,...; υ, υ,...,, can be expreed a (3. where { z } a equence of ndependenly and dencally drbued random varable, E ( = 0, V ( =, (. z z f he deny funcon of z, ( θ a me-varyng, pove and meaurable funcon of he nformaon e a me, υ a vecor of predeermned varable ncluded n I and (. g could be a lnear or nonlnear funconal form a uually aumed n he ARCH leraure. A wde range of propoed ARCH model covered n urvey uch a Bollerlev e al. (99, Bollerlev e al. (994, Bera and Hggn (993, Hamlon (994 and Goureroux (997. By defnon, ε ( θ erally uncorrelaed wh mean zero, bu wh a me varyng condonal varance equal o ( θ. he condonal varance a lnear or nonlnear funcon of lagged value of he andard ARCH model aume ha (. f he normal drbuon. However, Bollerlev (987 propoed ung he uden drbuon wh an emaed kuro regulaed by he degree of freedom parameer and Nelon (99 propoed ung he generalzed error drbuon. Oher drbuon, ha have been employed, nclude he generalzed drbuon (Bollerlev e al. (994, he normal Poon mxure drbuon (Joron (988, he normal lognormal mxure (Heh (989, and a erally dependen mxure of normally drbued varable (Ca (994 or uden drbued varable (Hamlon and Sumel (994.

7 and ε and predeermned varable ncluded n I, ( υ, υ, In he equel for noaonal convenence, no explc ndcaon of he dependence on he vecor of parameer, θ, gven when obvou from he conex. Snce very few fnancal me ere have a conan condonal mean of zero, an ARCH model can be preened n a regreon form by leng ε be he nnovaon proce n a lnear regreon: = g ε I ~ ( 0, ( ( θ, ( θ,...; ε ( θ, ε ( θ,...; υ, υ,..., y = x β ε f (3. where x a k vecor of endogenou and exogenou explanaory varable ncluded n he nformaon e I and β a k vecor of unknown parameer. Uually, he condonal mean eher he overall mean or a fr order auoregreve proce. heorecally, he AR ( proce allow for he auocorrelaon nduced by dconnuou (or non-ynchronou radng n he ock makng up an ndex (Schole and Wllam (977, Lo and MacKnlay (988. Accordng o Campbell e al. (997, he nonynchronou radng are when me ere, uually ae prce, are aken o be recorded a me nerval of one lengh when n fac hey are recorded a me nerval of oher, poble rregular lengh. he Schole and Wllam model ugge he order movng average proce for ndex reurn, whle he Lo and MacKnlay model ugge an AR ( form. 3. he Mo Commonly Ued ARCH Model and her Foreca Recuron Relaon he mo commonly ued condonal varance funcon are he GARCH (Bollerlev (986, he Exponenal GARCH, or E-GARCH, (Nelon (99 and he hrehold GARCH, or ARCH, (Gloen e al. (993 funcon. In he equel, hee ARCH model are condered n he followng form: he GARCH(p,q model q p ( aε ( b = a (3.3 0 = = he EGARCH(p,q model q p ε ( ε ln = a0 a γ ( b ln( (3.4 = =

8 he ARCH(p,q model q p ( aε γε d ( b = a, (3.5 0 = = where d = f ε < 0, and d = 0 oherwe. he condonal varance for he GARCH(p,q proce may be rewren a: ( u, η, w ( v, ζ ω =,,, ζ = 0, where u = (, ε,..., ε q, η = 0, w = (,..., p, v = ( a0, a,..., a q ω = ( b,...,b p For he EGARCH(p,q proce, he condonal varance can be expreed a: ( u, η, w (, ζ ω ln = v, ( where u = (, ε,..., ε q q, η = [ ε ],...,[ ε q q ] w = ( ln,...,, v = ( a a,...,, ζ = ( γ,..., ω = ln p 0, a q,, ( γ q b,...,b p Alo, for he ARCH(p,q proce, he condonal varance can ake he form: ( u, η, w ( v, ζ ω =, where u = (, ε,..., ε q, η = ( d ε, w = (,..., p, v = ( a, 0 a,..., a q ζ = ( γ, ω = (, d = f ε < 0, and d = 0 oherwe. b,...,b p, 8 (3.6 (3.7 (3.8 In general, he condonal varance foreca recuron relaon could be preened a: ( ( ( ( ( ( ( I = E( u, η, w I ( v, ζ, ω = ( u, η, w ( v, ζ, ω ˆ E (3.9 he foreca of ε gven ha he avalable nformaon e, I, nclude only pa me nformaon a me, hould be equal o condonal expeced value. hu, ( I = 0 E ε and ( I E = ε. On he oher hand, for 0, he predcve error and quare are compued by he model wh he avalable nformaon a me. Specfcally, he foreca recuron relaonhp for he condonal varance for he GARCH(p,q model : ˆ q p ( ( a ε ( b ( ( = a0 = = (3.0a and q q p ( ( ( a ( a ε ( b ( ( ˆ. (3.0b = a0 = = for < for =

9 9 For he EGARCH(p,q model, we have (3.a ( ( ( ( ( ( ( = = = p q b a a 0 ln ˆ ln ε γ ε and ( ( ( ( ( ( ( ( ( = < = = = p q for q for b a a a 0 ln ˆ ln π ε γ ε. (3.b he correpondng recuron n he cae of he ARCH(p,q model are: (3.a ( ( ( ( ( ( = = = p q b d a a 0 ˆ ε γ ε and ( ( ( ( ( ( ( ( ( = = < = = p q for q for b d E a a a 0 ˆ γ ε. (3.b Here, ( d E denoe he percenage of negave nnovaon. Under he aumpon of normally drbued nnovaon, he expeced number of bad new, 0 < ε, equal o he expeced number of good new, 0 > ε. Noe ha over he perod from January, 995 o June, 000, he negave daly reurn on he Ahen Sock Exchange Index (ASE and on he S&P500 are 46.8% and 45.5%, repecvely. Applyng a ypcal AR(- GARCH(, model, he negave nnovaon are 49% and 5% repecvely, very cloe o he expeced value. 3.3 he PEC Model Selecon Algorhm Under he aumpon ha (. f he normal drbuon, Degannak and Xekalak (999 have condered a comparave evaluaon of wo ARCH model (.e. model A and model B, on he ba of her ably o predc he fuure value of he dependen varable and volaly foreca. For ˆ ε and ˆ denong he one-epahead predcon error and he predcon of he condonal varance a me gven he nformaon avalable a me, he predcve able of model A and B can be compared hrough eng a null hypohe ha he wo model are of equvalen predcve ably. Here, he noon of he equvalence of wo model wh repec o her predcve ably condered n Xekalak e al. (003 ene o be defned mplcly Applyng he ARCH and E-GARCH model gve mlar reul.

10 hrough her mean quared predcon error. Followng Xekalak e al. (003, he cloe decrpon of he hypohe o be eed veru H 0 : Model A and B have equal mean quared predcon error H : Model A ha lower mean quared predcon error han model B ( B ( B ( A ( A ung ˆ ε ˆ ˆ ε ˆ a a e ac. = = he null hypohe rejeced f: ( B ( B ( A ( A ˆ ε ˆ ˆ ε ˆ > CGR( k, ρ, α, (3.3 = = CGR k, a 00 α percenle of he Correlaed gamma rao (CGR where (, ρ he ( drbuon, k = and 0 ( A ( A ( B ( ( ε ˆ, ( ˆ ε B. (For more deal ˆ ˆ ρ = Cor concernng he CGR drbuon, ee Xekalak e al. (003. he ue of andardzed one-ep-ahead predcon error can alo be condered for eng he null hypohe of equvalence of everal model n her predcve ably agan he alernave ha one of he model produce beer predcon. Le u aume ha a e of canddae ARCH model avalable and he mo uable model ough for predcng condonal volaly. he ARCH model, wh he lowe value of he um ˆ ε ˆ of he mo recenly emaed = quared andardzed one-ep-ahead predcon error, can be condered for obanng a foreca of he condonal volaly a me gven any nformaon avalable a me. In he equel, an algorhm for elecng he mo uable of M canddae model a each of a ere of pon n me preened. Conder a e of M compeng ARCH procee, whch have been emaed ung a rollng ample of obervaon and for a ample of ze. he algorhm compred of he followng ep. For model m, ( m,,..., M = and for each, ( =,,..., of coeffcen emaed ung obervaon: ˆ ( m ( ( ( ( ( ( ( ( ( ( ˆ m m, ˆ, ˆ m m θ β v ζ, ˆ ω ( ( Ung he vecor of coeffcen ˆ m θ, emae he vecor: ( m ( m (, yˆ ˆ, he vecor

11 Compue ˆ ( m ( m ( y yˆ ( m z ˆ Compue zˆ = ( m ( m R. he mo uable model o foreca volaly a me he model m wh he ( m mnmum value of. he algorhm repeaed for each of a equence of pon n R me o gude one choce of he mo approprae model. 4. Forecang Opon Prce Noh e al. (994 deved radng rule o rade a he money raddle 3. If he raddle prce foreca greaer han he marke raddle prce, he raddle bough. If he raddle prce foreca le han he marke raddle prce, he raddle old,.e. If ( τ ( τ ( τ ( τ > P P C he raddle bough a me. (4. C If ( τ ( τ ( τ ( τ < P P C he raddle old a me. (4. C he raegy can be underood wh he help of he followng example: On Monday, afer he ock marke cloe 4, ueday prce of an opon ha expre on Frday, emaed. he remanng lfe of he opon 3 day, from ueday o Frday. If opon predcon prce on ueday hgher han he oberved opon prce on Monday, he opon bough n order o be old on ueday. If he predced opon prce on ueday lower han he oberved opon prce on Monday, he opon or-old n order o be bough on ueday. 3 he raddle radng ued becaue an a he money raddle Dela neural. Dela, Lambda, Gamma, hea, Vega and Rho compre he prcng enve and repreen he key relaonhp beween he ndvdual characerc of he opon and he opon prce. he opon enve are he paral dervave of he BS opon prce n relaon o each ndvdual facor ha affec he prce of he opon. Dela he change n he opon prce for a gven change n he ock prce, ha, he hedge rao. C γτ CALL = = e N( d > 0 S P γτ = = e N d < PU ( ( 0 S For example, a rader who buy one call opon wh =0.6, and ell a dfferen call opon wh =0.4, ha a ne = =0.. hu, a $ change n he ock prce creae a $0. ncreae n he combned opon poon. 4 he radng raegy aume ha here enough me o foreca he opon prce gven all he nformaon a me (clong prce of ock o a he rader o be able o decde he radng of an opon a me (before he opon marke cloe. For example, he Chcago Sock Marke cloe a 3:00 pm local me and he Chcago Board of Opon Exchange cloe a 3:5 pm local me.

12 Monday ueday Wedneday hurday 3 Frday 4 he rae of reurn from radng an opon : C P C P R =, on buyng a raddle, (4.3 C P C P C P R =, on or-ellng a raddle. (4.4 C P Noe ha he ranacon co, X, hould be aken no accoun. If h he cae, he ne rae of reurn from radng an opon gven a: X NR = R. (4.5 C P Moreover, a fler can be appled n he radng raegy, o a o rade an opon only when he dfference beween foreca and oday opon prce exceed he amoun of he fler. Noh e al. (994 appled he AR(-GARCH(, model n order o foreca he fuure volaly. hey ued an adjued for calendar effec ARCH model n he followng form: n δ y ε I = a = c 0 0 a c y ~ N n δ ε ( 0, ε b n δ, (4.6 where he varable n repreen he number of calendar day ha have elaped nce he prevou radng day. he value of n he number of non-radng day precedng he h day ncreaed by. So, for example, n equal o one on any radng day preceded by radng day, equal o 3 on a Monday, 4 on a ueday when Monday a non-radng day (e.g. a holday, ec. Moreover, he non-radng day nduce only a fracon of he volaly of radng day. So, Noh e al. (994 nroduced an exponen δ ha meaure he average neny of he varance rae per day over he nerval nce he prevou cloe, and adju he nerval varance by a facor n δ. he condonal varance foreca recuron relaon gven by:

13 3 ˆ ( ( δ ( ( ( = n a0 a ( b δ δ n n ε (4.7a ( ( n ( δ ( ( ( ˆ = n a0 a b, for =,..., τ. (4.7b δ Foreca of opon prce, on he nex radng day, are calculaed ung he BS opon prcng formula and condonal volaly foreca. he volaly durng he lfe of he opon compued a he quare roo of he average foreca condonal varance: where ˆ / τ ( τ τ, (4.8 = = ˆ denoe he predcon of he condonal varance a me gven he nformaon e avalable a me. Noh e al. (994 aeed he performance of her ARCH model for raddle wren on he S&P500 ndex over he perod from Aprl 986 o December 99 and found ha he ARCH model earn a prof of $0.885 per raddle n exce of a $0.5 ranacon co and applyng a $0.5 fler. 5. Opon Prcng Ung a Se of ARCH Procee and he PEC Model Selecon Algorhm he GARCH(, model he mo commonly ued model n fnancal applcaon. he queon ha are a h pon : Why hould one ue he mple GARCH(, model nead of ung a hgher order of GARCH(p,q model, an aymmerc ARCH model, or even a more complcaed form of an ARCH proce?. here a va number of ARCH model. Whch one hould be preferred? he volaly predcon model, whch gve he hghe rae of reurn n radng opon, hould be he preferable one. Moreover, under he aumpon ha he BS formula decrbe perfecly he dynamc of he marke ha affec he prce of he opon, he model gve he mo prece predcon of condonal volaly hould be he model ha gve he hghe rae of reurn. Unforunaely, an mporan lmaon ll reman. Even f one could fnd he model, whch predc he volaly precely, well known ha he BS formula doe no decrbe he dynamc prcng he opon perfecly 5. Bede, he rue fuure volaly unoberved, o predcon canno be compared drecly o he rue value of. Moreover, he valdy of he varance foreca depend on whch opon prcng 5 here a large number of arcle ha examne he baedne of BS formula.

14 formula ued. For example, Engle e al. (997 ued Hull and Whe (987 modfcaon o he BS formula for prcng raddle on a mulaed opon marke. However, depe lmaon he BS formula, a derved by Black and Schole (973 and Meron (973, he mo acceped European opon prcng formula among floor rader on opon exchange. In he equel, a varey of volaly predcon model are emaed ung S&P500 ock ndex daly reurn and he rae of reurn from radng raddle, baed on he volaly predcon, calculaed. he PEC model elecon algorhm ubequenly appled n order o chooe for each parcular day he mo approprae ARCH model for emang he prce of an opon. he day-by-day rae of reurn are reflecve of he correpondng predcve performance of he model. Comparng he reul, provde an ndrec comparave aemen of a radng raegy baed on opon prce foreca provded by any one of hee model for each of a number of day o he radng raegy of decdng each day on he ba of he opon prce foreca by he model eleced by he PEC algorhm a he mo approprae for ha parcular day Opon Sraege and Cah Flow Suppoe he prce of ock a me wh expraon day and exerce prce K, are S and he prce of a call and pu opon, C and P, repecvely. In erm of cah flow, he purchaer of an opon (a long opon poon alway ha an nal negave cah flow, he prce of he opon, and a fuure cah flow ha a wor zero. he wrer of he opon (a or opon poon ha an nal pove cah flow followed by a ermnal cah flow ha a be zero. A expraon day,, he call opon exerced only f S > K. hu, he cah flow, a me, of he call purchaer 6 : rf ( rf ( e C f S K max ( 0, S K e C = rf (. (5. S K e C f S > K he cah flow of he call wrer oppoe o ha of he call purchaer: e rf rf ( e C f S K max = rf (. (5. ( C ( 0, S K Moreover, he pu exerced only f pu purchaer : e C K S f S > K S < K. hu, a maury day, he cah flow of he 6 I aumed ha nveor, a me, are borrowng and lendng a he ame rk free rae, rf. hu, he rf exp. money durng he perod from o nveng wh a daly reurn of ( ( rf

15 rf ( rf ( e P f S K max ( 0, K S e P = rf (, (5.3 and he cah flow of he pu wrer : e rf ( P ( 0, K S K S e P f S < K rf ( e P f S K max = rf (. (5.4 e P S K f S < K Fgure 7 preen he prof and lo performance of buyng and wrng opon. A long raddle poon an opon raegy n whch a call and a pu of he ame exerce prce, maury and underlyng erm are purchaed. h poon called a raddle nce wll prof from a ubanal change n he ock prce n eher drecon. rader purchae a raddle under one of wo crcumance. he fr crcumance ex when a large change n he ock prce expeced, bu he drecon of he change unknown. Example nclude an upcomng announcemen of earnng, unceran akeover or merger peculaon, a cour cae for damage, a new produc announcemen, or an unceran economc announcemen uch a nflaon fgure or a change n he prme nere rae. A raddle eem a rk free radng raegy when a large change n he prce of a ock expeced. Fgure (7. he cah flow of akng long and or poon n call and pu opon. 5 However, n he real world, h no necearly he cae. If he general vew of he marke ha here wll be a bg jump n he ock prce oon, he opon prce hould reflec he ncreae n he poenal volaly of he ock. A rader wll fnd opon on he ock o be gnfcanly more expenve han opon on a mlar ock for whch no jump expeced. For a raddle o be an effecve raegy, he rader mu beleve ha bg movemen n he ock prce are lkely and h belef mu be dfferen from ha of mo of he oher marke parcpan. he econd crcumance n whch raddle are purchaed occur when he rader emae ha he rue fuure volaly of he ock wll be greaer ha he volaly ha currenly mpounded n he opon prce. Noe ha

16 alhough he long raddle ha heorecally unlmed poenal prof and lmed rk, hould no be vewed a a low rk raegy. Opon can loe her value very quckly, and n he cae of a raddle, here wce he amoun of eroon of me value a compared o he purchae of a call or pu. he oppoe of a long raddle raegy a hor raddle poon. h raegy ha unlmed rk and lmed prof poenal, and herefore only approprae for experenced nveor wh a hgh olerance for rk. he hor raddle wll prof from lmed ock movemen and wll uffer loe f he underlyng ae move ubanally n eher drecon. Fgure 8 preen he payoff of akng long and or raddle poon. A expraon day,, he cah flow of akng a long and a or raddle poon are: S e rf K e ( ( C P ( (, C P rf S K repecvely. Fgure (8. he cah flow of akng long and or raddle poon. 6 (5.5a (5.5b An Example of Sraddle radng Conder a rader who feel ha he prce of a ceran ock, currenly valued a $54 by he marke, wll move gnfcanly n he nex hree monh. he rader could creae a raddle by buyng boh a pu and a call wh a rke prce of $55 and an expraon dae n hree monh. Suppoe ha he call and he pu co are $5 and $4, repecvely. he mo ha can be lo he amoun pad, or $9, f he ock prce move o $55. If he ock prce move above $63 or below $45, he long poon earn a prof. In he cae of akng a hor raddle poon, he maxmum prof he premum receved, or $9. he maxmum lo unlmed, and he or poon wll loe f he ock prce move above $63 or below $45.

17 5. radng Sraddle Baed on a Se Of ARCH Procee = ln For ( S S y denong he connuouly compound rae of reurn from me o, where S he ae prce a me, a e of ARCH model are emaed. he condonal mean condered a a y µ = c z 0 =. ~. d. AR : h κ order auoregreve proce ( ( κ = µ z κ ( c y N( 0,, and he condonal varance regarded a a GARCH( ARCH( AR(κ GARCH( 7 (5.6 p, q, an EGARCH( p, q or a p, q funcon of he form defned n econ 3.. hu, he p, q, AR(κ EGARCH( p, q and AR(κ ARCH( p, q model are appled, for κ = 0,..., 4, p = 0,, and q =,, yeldng a oal of 85 cae 7. he daa e con of 064 S&P500 ock ndex daly reurn n he perod from March 4 h, 996 o June nd, 000. he ARCH procee are emaed ung a rollng ample of conan ze equal o 500. hu, he fr one-ep-ahead volaly predcon, ˆ, avalable a me = 500, or on March h, 998. Maxmum lkelhood emae of he parameer are obaned by numercal maxmzaon of he log-lkelhood funcon ung he Marquard algorhm (Marquard (963, a modfcaon of he Bernd, Hall, Hall and Hauman, or BHHH, algorhm (Bernd e al. (974. he qua-maxmum lkelhood emaor (QMLE ued, a accordng o Bollerlev and Wooldrdge (99, generally conen, ha a normal lmng drbuon and provde aympoc andard error ha are vald under non-normaly. he S&P500 ndex opon 8 daa were obaned from he Daaream for he perod from March h, 998 hrough June nd, 000, oally 564 radng day. Unforunaely, he daa record no avalable for all he radng day. Daa are avalable for 456 radng day and conan nformaon for he elemen prce of he call and pu opon, exerce prce, expraon dae, and he number of conrac raded. From he whole daa e, he raddle opon, whoe exerce prce cloe o he ndex level, her maury longer han 0 radng day, and her radng volume greaer han 00, are colleced for each radng day. he average and he andard devaon of he colleced S&P500 opon prce are preened n able. 7 Numercal maxmzaon of he log-lkelhood funcon, for he E-GARCH(, model, frequenly faled o converge. So he fve E-GARCH model for p = q = were excluded. 8 S&P500 ndex opon are raded on he Chcago Board Opon Exchange (CBOE.

18 A he money radng ued becaue he BS formula end o mprce deep ou of he money and deep n he money opon. Longer han 0 radng day maury raddle are ued becaue he cloer one ge o he expraon day, he more upec he heorecal BS opon prce become. rader pay le and le aenon o model generaed value a expraon approache. I common pracce among floor rader, on opon exchange, o ake wh hem o he radng floor hee wh heorecal opon prce o enable hem o make marke ha are conen wh a heorecal prcng model. Bu a expraon approache rader are more lkely o dcard her hee becaue he heorecal value become le relable. Wh one or wo day o expraon, mo rader mply make decon baed on experence and nuon. he BS model aume ha he ock prce follow a dffuon proce, an unrealc aumpon a n mo marke he underlyng conrac follow a combnaon of boh a dffuon proce and a jump proce 9. Mo of he me, ae prce change moohly and connuouly wh no gap. However, every now and hen a gap wll occur, nananeouly endng he prce o a new level. hee prce wll agan be followed by a mooh dffuon proce unl anoher gap wll occur. Snce a gap n he marke wll have greae effec on a he money opon cloe o expraon 0, hee opon ha are lkely o be mprced by he radonal BS prcng model wh connuou dffuon proce. 8 9 A varaon of he BS model, whch aume ha he underlyng conrac follow a jump dffuon proce, ha been developed. See for example Meron (976 and Becker (98. Unforunaely, he model conderably more complex mahemacally han he radonal BS model. Moreover, n addon o he fve cuomary npu, he model alo requre wo new npu: he average ze of a jump n he underlyng marke and he frequency wh whch uch jump are lkely o occur. Unle he rader can adequaely emae hee new npu, he value generaed by a jump dffuon model may be no beer, and mgh be wore, han hoe generaed by he radonal model. Mo rader ake he vew ha whaever weakne are encounered n a radonal model can be be offe hrough nellgen decon makng baed on acual radng experence, raher han hrough he ue of a more complex jump dffuon model. 0 A gap n he marke ha greae effec on a hgh Gamma opon and a he money opon cloe o expraon have he hghe Gamma. Gamma meaure he change n he Dela for a gven change n he ock prce. Gamma dencal for call and pu opon. d γτ C P e ΓCALL, PU = = = > 0 S S S πτ Gamma one meaure of he effec of nably on he opon poon (he oher Vega. I how he rk nheren n Dela. If Gamma mall, Dela no enve o change n he ock prce. If Gamma large, Dela enve o ock prce change. If Gamma 0.5 and he curren Dela 0, hen an ncreae

19 able (. and andard devaon of he S&P500 opon prce and her radng volume for he radng day colleced n he daa record ( March 998 June 000 ype of Opon radng Day Opon Prce Sandard Devaon radng Volume Sandard Devaon Call ,6 0, Pu 456 8,, Sraddle 456 6,7 8, On each radng day, for each of he 85 ARCH model, he call and pu opon prce are foreca. able, n he Appendx, preen he mean and he andard devaon of he foreca S&P500 opon prce. he ARCH foreca for boh call and pu opon are lower han he acual opon prce, whch n accordance o Noh e al. (994 reearch. Le u aume ha here are 85 rader and each rader apple an ARCH model o foreca fuure volaly and raddle prce. Each radng day, f he raddle prce foreca greaer han he marke raddle prce he raddle bough, oherwe he raddle old. For each rader, he daly rae of reurn from radng raddle for 456 day compued a n equaon (4.3 o (4.4 and preened n he econd column of able 3, n he Appendx. Accordng o he -rao, compued a rao of he mean o he andard devaon dvded by he quare roo of he radng day, all he rader acheve prof gnfcanly greaer han zero. However, he rader who follow he AR(3EGARCH(, model acheve he hghe prof. he AR(3EGARCH(, agen make 4.4 per cen per day radng for 456 day, wh a -rao of 5.3. Fgure 9 depc he cumulave reurn of he AR(3EGARCH(, agen from radng raddle on a daly bae. However, each me an agen rade a conrac ha o pay a ranacon co. akng no conderaon a ranacon co of $, whch reflec he bd ak pread, he rae of reurn would naurally be lower. able 3 preen for each rader he ne rae of reurn afer a radng co of $, a compued n equaon (4.5. However, a raonal rader wll rade raddle only when prof are predced o exceed ranacon co. So, raddle are raded only when he abolue dfference beween foreca and oday opon prce exceed he amoun of he fler, F, yeldng a ne rae of reurn of: 9 n he ock prce of $ caue he Dela o ncreae from 0 o 0.5. Now, he new Dela mean ha an ncreae n he ock prce of $ wll now ncreae he opon prce by $0.5. Becaue of he large amoun of daa, able 3, n he Appendx, decompoed no four par. Bd prce he prce ha a rader offerng o pay for he opon. Ak prce he prce ha a rader offerng o ell he opon. he ak prce hgher ha he bd prce and he amoun by whch he ak exceed he bd referred o a he bd ak pread. he exchange e he upper lm for he bd ak pread. For example, accordng o he CBOE rule, he wdh uppoed o be no more han $0.5 wde for conrac under $.00, $0.40 wde beween $ and $5, ffy cen from $5 o $0, 80 cen from $0 - $0, and above $0 can be a dollar wde. However, Exchange rule allow for doublng and even rplng he wdh dependng upon he marke condon. For a real nveor, co hgher and vare gnfcanly from broker o broker. he acual amoun charged uually calculaed a a fxed co plu a proporon of he dollar amoun of he rade,.e. from a dcoun broker he purchae of conrac of $0.000 would co $45 n common. Real common from full ervce broker are hgher.

20 NR R = C 0 X P ( τ ( τ, f C P C P, oherwe 0 > F. (5.7 Varou value for he fler are appled (.e. $.5, $.75, $.00, $.5, $.75, $3.50. Noce ha alhough he before ranacon co prof are gnfcanly greaer han zero, applyng a $ ranacon co, he prof are no gnfcanly greaer han zero for any of he agen. Accordng o able 4, he model ha acheve he hghe rae of reurn are no he ame for each fler raegy. Fgure (9. Cumulave rae of reurn of he AR(3EGARCH(, agen from radng raddle on he S&P500 ndex ( March 998 June % Cumulave Reurn 500% 000% 500% 0% 3//98 6//98 9//98 //98 3//99 6//99 9//99 //99 3//00 Dae able (4. ARCH model yeld he hghe rae of reurn from radng raddle on he S&P500 ndex ( March 998 June 000. ran. Co - Fler Model S.Dev -rao p-value radng Day oal Reurn $ $0.00 AR(3EGARCH(, 4.4% 7.75% % $.00 - $0.00 AR(3EGARCH(, 0.77% 7.% % $.00 - $.5 AR(EGARCH(, 0.90% 7.34% % $.00 - $.75 AR(0GARCH(,.06% 8.46% % $.00 - $.00 AR(0GARCH(,.0% 8.53% % $.00 - $.5 AR(0GARCH(,.35% 8.50% % $.00 - $.75 AR(4GARCH(0,.60% 7.60% % $.00 - $3.50 AR(3GARCH(0,.89% 8.% % 5.3 radng Sraddle Baed on he PEC Model Selecon Algorhm he man purpoe o examne he applcaon of he PEC algorhm of elecon of volaly model on he ba of forecang opon prce and creang radng raege ha yeld abnormal reurn. he erm abnormal reurn refer o prof ha are uncorrelaed wh he marke rae of reurn a he a he money raddle radng a

21 Dela neural 3 radng raegy. Accordng o he PEC model elecon algorhm, he mo approprae model, among a e of canddae ARCH model, o foreca volaly a each of a equence of pon n me he model wh he lowe um of he mo recenly emaed quared andardzed one-ep-ahead predcon error. Applyng he PEC model elecon algorhm, he um of quared andardzed one-ep-ahead predcon error, = zˆ, wa emaed conderng varou value for, and, n parcular, = 5(80 5. hu, aumed ha here are 6 rader and each rader apple h/her model o foreca fuure volaly and raddle prce. able 5, n he Appendx, preen for each rader, who follow he PEC model elecon raegy, he ne rae of reurn from radng raddle on a daly ba. Wh ranacon co of $, he rader ulzng he PEC algorhm wh a ample ze of = 5 acheve he hghe rae of reurn. he agen baed on he PEC(5 foreca algorhm make.00% per day radng for 39 day, wh a -rao of.95. He/he buy raddle for 09 day and make a daly prof of 4.54% and hor-ell raddle for 0 day makng a prof of 0.75% per rade. He/he elec he ARCH model preened n able 6. So, for example, he model wh AR(0 condonal mean and GARCH(0, condonal varance wa eleced 7 radng day. he elecon algorhm chooe hgher order of he condonal mean auoregreve proce for half he number of radng day. A concern he condonal varance funcon, he GARCH, E-GARCH and ARCH model are uggeed a he mo uable n he 35%, 7%, and 38% of he cae, repecvely. Conequenly, he PEC algorhm doe no appear o be noceably baed oward elecng a pecfc ype of model. In order o compare he raegy performance over he enre ample, agen are aumed o nve a he rk free rae when hey do no rade. hu, he ne rae of reurn now compued a: NR = C,f C C P C P X C P,f C P C P X rf C P rf ( τ ( τ P P C C P ( τ ( τ, oherwe P > F > F. (5.8

22 able (6. Number of ARCH model eleced by he PEC(5 algorhm for radng raddle on he S&P500 ndex wh ranacon co of $.00 and a $3.5 fler ( March 998 June 000, clafed by he ype of model condered for her condonal mean and varance. ype of Condonal Model ype of Condonal Varance Model AR(0 AR( AR( AR(3 AR(4 oal GARCH(0, GARCH(0, GARCH(, 7 9 GARCH(, GARCH(, 3 GARCH(, ARCH(0, 3 6 ARCH(0, ARCH(, ARCH(, 3 9 ARCH(, ARCH(, EGARCH(0, EGARCH(0, EGARCH(, EGARCH(, EGARCH(, oal For X = $. 00 and F = $3. 50, he rader ung he AR(3GARCH(0, foreca make a daly prof of.35% wh a correpondng andard devaon of 5.4% and a - rao of.89 (or p-value On he oher hand, he agen ha follow he PEC(5 model elecon algorhm acheve a prof of.46% per day wh a correpondng andard devaon of 5.85% and a -rao of.97 (or p-value Even margnally, he PEC(5 model elecon algorhm acheve hgher cumulave reurn han any of he ARCH foreca. Moreover, a -rao of.97 ndcae ha prof from he PEC(5 algorhm are gnfcanly dfferen from zero. hu, he PEC model elecon algorhm ha a afacory performance n elecng hoe model ha generae beer volaly predcon. h can be helpful n creang peculave raege wh opon. 5.4 radng Sraddle Baed on Oher Mehod Of Model Selecon Mo of he mehod ued n he me ere leraure for elecng he approprae model are baed on evaluang he ably of he model o decrbe he daa. 3 Dela he change n he opon prce for a gven change n he ock prce. An opon ermed dela neural when he um oal of all he pove and negave dela add up o approxmaely zero. he rae of reurn of a dela neural radng raegy ndfferen o any change n he underlyng ock prce.

23 Sandard model elecon crera uch a he Akake Informaon Creron (AIC (Akake (973 and he Schwarz Bayean Creron (SBC (Schwarz (978 have wdely been ued n he ARCH leraure, depe he fac ha her acal propere n he ARCH conex are unknown. hee are defned n erm of l ( θˆ n 3, he maxmzed value of he log-lkelhood funcon of a model, where θˆ he maxmum lkelhood emaor of he parameer vecor θ baed on a ample of ze n and θ denoe he dmenon of θ, hu: = l ( ˆ n θ θ ( ˆ θ ln( n. AIC (5.9 SBC = ln θ (5.0 In addon, he evaluaon of lo funcon for alernave model manly ued n model elecon. When he focu manly on emaon of mean, he lo funcon of choce ypcally he mean quared error. However, when he ame raegy appled o varance emaon, he choce of he mean quared error much le clear. Becaue of hgh non-lneary n volaly model a number of reearcher conruced heerocedacy adjued lo funcon. Denong he forecang varance over an N day perod meaured a day by ( N = N ˆ N =, and he realzed varance over he ame perod by ( N = N y N =, a e of acal crera o meaure he cloene of he foreca o he realzaon are preened n able 7, n he Appendx. In he equel, he ably of acal crera o elec he approprae ARCH model for forecang opon prce nvegaed. he fr four funcon have been wdely ued n he leraure (ee, e.g. Heynen and Ka (994. he heerocedacy adjued funcon were condered by Anderen e al. (999, whle he logarhmc error funcon wa ulzed by Pagan and Schwer (990. Applyng he PEC model elecon algorhm, he um of quared andardzed one-ep-ahead predcon error, = ˆ ε ˆ, wa emaed conderng varou value for. herefore, each of he model elecon crera, n able 7, wa compued conderng varou value for, and, n parcular, = 0( 080. he AIC and SBC crera were compued baed on he rollng ample of conan ze equal o 500 ha ued a each me o emae he parameer of he model. Selecng a raegy baed on any of everal compeng mehod of model elecon naurally amoun o elecng

24 he ARCH model ha, a each of a equence of pon n me, ha he lowe value of he evaluaon funcon. able 8 o 7, n he Appendx, preen he daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he model elecon mehod preened n h econ. Each of he model elecon mehod wa compued conderng varou value for he ample ze,, and, n parcular, = 0( 080. Afer ranacon co of $, he agen baed on he HASEVar, HAAEVar, HASEDev and HAAEDev crera acheve he hgher reurn. Moreover, a rader who elec he volaly foreca model accordng o he andard model elecon crera, SBC and AIC, make a cumulave prof hgher han n he cae he/he would elec ARCH model baed on he heerocedacy unadjued and logarhmc error funcon. However, n none of he cae, he daly reurn came ou o be gnfcanly dfferen from zero (accordng o he -rao of he able 8 o 8 or hgher han he reurn acheved by he PEC algorhm. Aumng ha he agen nve a he rk free rae when hey do no rade, able 8 preen he raegy performance over he enre ample. he ne rae of reurn compued accordng o equaon (5.8. he PEC model elecon algorhm, for = 5, lead o he hghe prof of.46% per day and a -rao of.97. Of he remanng model elecon crera condered n h econ, he HAAEVar elecon algorhm, for = 40, yelded he hghe daly prof (.4% wh a correpondng andard devaon of 6.% and a -rao of.65. hu, none of he model elecon algorhm condered appear o lead o daly reurn ha are hgher han he reurn aaned ung he PEC algorhm. 4

25 able (8. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he PEC model elecon algorhm and he ARCH model elecon algorhm preened n Secon 5, wh ranacon co of $.00 and a $3.5 fler. he column ample ze refer o he ample ze,, for whch he correpondng model elecon algorhm lead o he hghe rae of reurn. Agen are aumed o nve a he rk free rae when hey do no rade. he ne rae of reurn compued a n equaon ( ran. Co - Fler Model Selecon Mehod Sample ze Sand. -rao Day $.00 -$3.50 PEC =5.46% 5.85% $.00 -$3.50 AIC % 5.57% $.00 -$3.50 SBC -.06% 5.93% $.00 -$3.50 SEVar = % 6.34% $.00 -$3.50 AEVar = % 5.8% $.00 -$3.50 SEDev = % 6.9% $.00 -$3.50 AEDev = % 5.96% $.00 -$3.50 HASEVar = 0.0% 5.98% $.00 -$3.50 HAAEVar = 40.4% 6.% $.00 -$3.50 HASEDev = % 6.3% $.00 -$3.50 HAAEDev = 30.% 6.47% $.00 -$3.50 LEVar = % 5.9% Concluon Worldwde, n all he opon exchange, profeonal praccan ue he BS formula n order o prce opon. All he npu facor n he BS formula (he prce of he underlyng ae, he exerce prce of he opon, he me o maury, he rk free rae and he dvdend yeld of he underlyng ae are known excep he volaly of he ae reurn durng he lfe of he opon. he volaly of he S&P500 ndex reurn wa predced ung a e of ARCH model. Over he March 998 o June 000 perod, foreca of opon prce were calculaed ung he BS formula, ncorporang he volaly emaed by he ARCH model. he performance of volaly-forecang mehod wa aeed by devng radng rule. Moreover, a number of model elecon algorhm were appled n order o ndcae, a each of a equence of pon n me, he approprae ARCH model for emang he volaly of he underlyng ae reurn. Afer a ranacon co of $, he rader baed on he PEC model elecon algorhm acheved a prof of.00% per day. Prof ha are gnfcanly greaer han zero and hgher han any oher model elecon algorhm made. Alhough he BS formula aume ha he nananeou varance of he rae of reurn conan, ncorporang me-varyng ochac varance emaon a common

26 pracce. Our reul, whch are n agreemen wh Noh e al. (994 udy, ndcae ha marke prce can be predced even wh he ue of a mpecfed model f ae volale can be predced. 6 Reference Akake, H. (973. Informaon heory and an Exenon of he Maxmum Lkelhood Prncple. Proceedng of he econd nernaonal ympoum on nformaon heory. B.N. Perov and F. Cak (ed., Budape, 67-8 Anderen,.,. Bollerlev and S. Lange (999. Forecang Fnancal Marke Volaly: Sample Frequency v-à-v Foreca Horzon. Journal of Emprcal Fnance, 6, Becker, S. (98. A Noe on Emang he Parameer n he Jump-Dffuon Model of Sock Reurn. Journal of Fnancal and Quanave Analy, 7-40 Bera, A.K. and M.L. Hggn (993. ARCH Model: Propere, Emaon and eng. Journal of Economc Survey, 7, Bernd, E., B. Hall, R. Hall, and J. Hauman (974. Emaon and Inference n Nonlnear Srucural Model. Annal of Economc and Socal Meauremen, 3, Black, F. and M. Schole (973. he Prcng of Opon and Corporae Lable. Journal of Polcal Economy, 8, Bollerlev,. (986. Generalzed Auoregreve Condonal Heerokedacy. Journal of Economerc, 3, Bollerlev,. (987. A Condonal Heerokedac me Sere Model for Speculave Prce and Rae of Reurn. Revew of Economc and Sac, 69, Bollerlev,., R. Chou and K.F. Kroner (99. ARCH Modelng n Fnance: A Revew of he heory and Emprcal Evdence. Journal of Economerc, 5, 5-59 Bollerlev,., R.F. Engle and D. Nelon (994. ARCH Model, n Handbook of Economerc, Volume 4, ed. R. Engle and D. McFadden, Elever Scence, Amerdam, Bollerlev,. and J.M. Wooldrdge (99. Qua-maxmum Lkelhood Emaon and Inference n Dynamc Model wh me-varyng Covarance. Economerc Revew,, 43-7 Ca, J. (994. A Markov Model of Swchng-Regme ARCH. Journal of Bune and Economc Sac,, Campbell, J., A. Lo, and A.C. MacKnlay (997. he Economerc of Fnancal Marke. New Jerey. Prnceon Unvery Pre

27 Degannak, S., and E. Xekalak, (999. Predcably and Model Selecon n he Conex of ARCH Model, Ahen Unvery of Economc and Bune, Deparmen of Sac, echncal Repor, 69 Engle, R.F., A. Kane, and J. Noh (997. Index-Opon Prcng wh Sochac Volaly and he Value of Accurae Varance Foreca. Revew of Dervave Reearch,, 0-44 Gloen, L., R. Jagannahan, and D. Runkle (993. On he Relaon Beween he Expeced Value and he Volaly of he Nomnal Exce Reurn on Sock. Journal of Fnance, 48, Goureroux, C. (997. ARCH model and Fnancal Applcaon. Sprnger-Verlag, New York Hamlon, J.D., (994. me Sere Analy. New Jerey: Prnceon Unvery Pre Hamlon, J.D. and R. Sumel (994. Auoregreve Condonal Heerokedacy and Change n Regme. Journal of Economerc, 64, Heynen, R. and H. Ka (994. Volaly Predcon: A Comparon of he Sochac Volaly, Garch(,, and Egarch(, Model. Journal of Dervave, 94, Heh, D.A. (989. Modelng Heerocedacy n Daly Foregn-Exchange Rae. Journal of Bune and Economc Sac, 7, Hull, J. and A. Whe (987. he Prcng of Opon on Ae wh Sochac Volaly. Journal of Fnance, 4, Joron, P. (988. On Jump Procee n he Foregn Exchange and Sock Marke. Revew of Fnancal Sude,, Lo, A. and A.C. MacKnlay (988. Sock Marke Prce Do No Follow Random Walk: Evdence from a Smple Specfcaon e. Revew of Fnancal Sude,, 4-66 Marquard, D.W. (963. An Algorhm for Lea Square Emaon of Nonlnear Parameer. Journal of he Socey for Indural and Appled Mahemac,, Meron, R.C. (973. Raonal heory of Opon Prcng. Bell Journal of Economc and Managemen Scence, 4, 4-83 Meron, R.C. (976. Opon Prcng when Underlyng Sock Reurn are Dconnuou. Journal of Fnancal Economc, 3, 5-44 Nelon, D. (99. Condonal Heerokedacy n Ae Reurn: A New Approach. Economerca, 59, Noh, K., R.F. Engle and A. Kane (994. Forecang Volaly and Opon Prce of he S&P 500 Index. Journal of Dervave,

28 Pagan, A.R. and G.W. Schwer (990. Alernave Model for Condonal Sock Volaly. Journal of Economerc, 45, Schole, M., and J. Wllam (977. Emang Bea from Non-Synchronou Daa. Journal of Fnancal Economc, 5, Schwarz, G. (978. Emang he Dmenon of a Model. Annal of Sac, 6, Xekalak E., J. Panareo and S. Parak (003. A Predcve Model Evaluaon and Selecon Approach - he Correlaed Gamma Rao Drbuon. Sochac Mung: Perpecve from he Poneer of he Lae 0 h Cenury, (J. Panareo, ed., Lawrence Erlbaum Aocae Publher (o appear 8

29 9 Appendx able. and andard devaon of he S&P500 opon prce baed on he 85 ARCH volaly foreca for he radng day colleced n he daa record able 3. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he 85 ARCH volaly foreca able 5. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he PEC model elecon algorhm able 7. ARCH model elecon mehod appled for radng raddle on he S&P500 ndex able 8. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he SEVar model elecon mehod able 9. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he AEVar model elecon mehod able 0. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he SEDev model elecon mehod able. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he AEDev model elecon mehod able. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he HASEVar model elecon mehod able 3. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he HAAEVar model elecon mehod able 4. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he HASEDev model elecon mehod able 5. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he HAAEDev model elecon mehod able 6. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he LEVar model elecon mehod able 7. Daly rae of reurn from radng raddle on he S&P500 ndex baed on he ARCH model eleced by he AIC and SBC model elecon mehod