SPEC model selection algorithm for ARCH models: an options pricing evaluation framework

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1 Applied Financial Economics Leers, 2008, 4, SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics, Ahens Universiy of Economics and Business, Greece b Deparmen of Saisics, Universiy of California, Berkeley, USA A number of single ARCH model-based mehods of predicing volailiy are compared o Degiannakis and Xekalaki s (2005) poly-model sandardized predicion error crierion (SEC) algorihm mehod in erms of profis from rading acual opions of he S&500 index reurns. The resuls show ha raders using he SEC for deciding which model s forecass o use a any given poin in ime achieve he highes profis. I. Inroducion Degiannakis and Xekalaki (2007) examined he abiliy of he sandardized predicion error crierion (SEC) model selecion algorihm o indicae he ARCH model ha generaes beer volailiy predicions wih a number of saisical evaluaion crieria. In he conex of a simulaed opions marke, Xekalaki and Degiannakis (2005) have found ha he SEC algorihm performs beer han any oher comparaive mehod of model selecion in pricing sraddles wih 1 day o mauriy. The presen manuscrip evaluaes he abiliy of he SEC algorihm in selecing a each poin in ime an accurae volailiy forecas for he remaining life of a sraddle 1 opion. The forecass of opion prices are calculaed by feeding he volailiy esimaed by he ARCH models ino he Black and Scholes (BS) opion pricing model. The obained resuls indicae ha SEC has a saisfacory performance in selecing he ARCH models ha yield beer volailiy predicions for opion pricing. II. ARCH Models For y ¼ ln(s /S 1 ) denoing he coninuously compound rae of reurn from ime 1 o, where S is he asse price a ime, a se of ARCH models are esimaed. The condiional mean is considered as a h order auoregressive process: y ¼ c 0 þ X i¼1 ðc i y i Þþz ð1þ for z i:i:d: Nð0, 1Þ, and he condiional variance is commonly regarded as one of Assumpion (i) a GARCH(p, q) funcion: 2 ¼ u0, 0, w0 ðv,,! Þ 0, ð2þ wih u 0 ¼ð1, "2 1,..., "2 q Þ, 0 ¼ 0, w0 ¼ ð 2 1,..., 2 p Þ, v0 ¼ (a 0, a 1,..., a q ), 0 ¼ 0,! 0 ¼ (b 1,..., b p ), Assumpion (ii) an EGARCH(p, q) funcion: ln 2 ¼ u 0, 0, w0 ðv,,! Þ 0, ð3þ *Corresponding auhor. sdegia@aueb.gr 1 A sraddle opion is he purchase of boh a call and a pu opion wih he same expiraion dae and exercise price. Applied Financial Economics Leers ISSN prin/issn online ß 2008 Taylor & Francis 419 hp:// DOI: /

2 420 S. Degiannakis and E. Xekalaki wih u 0 ¼ð1, " 1= 1,..., " q = q Þ, 0 ¼ð" 1 = mos appropriae model o be used for obaining a 1,..., " q = q Þ, w 0 ¼ðlnð2 1 Þ,...,lnð2 p ÞÞ, volailiy forecas for he nex poin in ime. v 0 ¼ (a 0,a 1,..., a q ), 0 ¼ (g 1,..., g q ),! 0 ¼ (b 1,..., b p ), Assumpion (iii) or as a TARCH(p, q) funcion: 2 ¼ u0, 0, w0 ðv,,! Þ 0, ð4þ IV. Measuring he Forecasing erformance wih u 0 ¼ð1, "2,..., "2 Þ, 0 1 q ¼ðd 1" 2 1 Þ, w0 ¼ ð 2,..., 2 Þ, v0 ¼ (a 1 p 0, a 1,..., a q ), 0 ¼ (g), The BS formula o price call and pu opions! 0 ¼ (b 1,..., b p ), d ¼ 1if" < 0 and d ¼ 0 oherwise. a day þ 1 given he informaion available a The predicion of he condiional variance a day day, wih days o mauriy, denoed, respecively, þ i given he informaion se available a day can by C and þ1, can be presened in he be compued as: ^ 2 þi Eð 2 þi I following form: Þ¼Eu þi, þi 0, w0 þi I Þðv, ðþ,! ðþ Þ¼ðu 0 þi þi þi, ðþ,! ðþ Þ. Thus, he C ¼ S e Nðd 1 Þ Ke rf Nðd 2 Þ ð5þ AR()GARCH(p, q), AR()EGARCH(p, q) and ¼ S e Nð d 1 ÞþKe rf Nð d 2 Þ AR()TARCH(p, q) models are applied, for 2 ¼ 0,...,4,p ¼ 0, 1, 2 and q ¼ 1, 2. lnðs =KÞþ rf þ 1=2 d 1 ¼ pffiffiffi III. The SEC Model Selecion Algorihm Assume ha a se of M candidae ARCH models is available and ha he mos suiable model is sough for predicing condiional volailiy. The ARCH model, wih he lowes value of he sum of he T mos recen esimaed squared sandardized one-sepahead predicion errors, T 1 ^"2 þ1 =^2 þ1, can be considered for obaining one-sep-ahead forecass of he condiional volailiy. Assume furher ha he M compeing ARCH processes have been esimaed using a rolling sample of n observaions. The SEC algorihm for selecing he mos suiable of he M candidae models a each of a series of poins in ime is comprised of he following seps. For model m, (m ¼ 1, 2,..., M) and for each poin in ime, ( ¼ n, n þ 1,...), he vecor of coefficiens ^ ðmþðþ ð^ ðmþðþ, ^v ðmþðþ, ^ ðmþðþ, ^! ðmþðþ Þ is esimaed using a rolling sample of n observaions. Using he d 2 ¼ d 1 þ1 p ffiffiffi, ð6þ where, S is he daily closing sock price as a forecas of S þ 1, rf is he daily risk free ineres rae, g is he daily dividend yield, K is he exercise price, N(.) is he cumulaive q normal disribuion funcion and is he volailiy during he life of he opion. If he sraddle price forecas is greaer han he marke sraddle price, he sraddle is bough. If he sraddle price forecas is less han he marke sraddle price, he sraddle is sold: þi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ1 i¼2 ^2 If C þ 4 þ C ) The sraddle is bough a ime ð7þ If C þ 5 þ C ) The sraddle is sold a ime ð8þ The rae of reurn from sraddle rading is: 8 C þ C 1 1 X, if C 1 C 1 þ þ 1 C 1 1 4F >< 1 NRT ¼ C 1 þ 1 C X þ r f, if C 1 þ 1 C 1 C 1 þ 1 4F >: 1 rf, oherwise, ð9þ vecor of coefficiens ^ðmþðþ, compue R ðmþ Tþnþ1 ¼n ^" 2ðmÞ þ1 =^2ðmÞ Tþn The mos suiable model o forecas volailiy a ime T þ n is he model m wih he minimum value of. The algorihm is repeaed for each of a sequence of poins in ime for he selecion of he R ðmþ Tþn where X denoes he ransacion cos. We assume ha he sraddles are raded only when he absolue difference beween he forecas and he acual sraddle price exceeds he amoun of he filer, F. Oherwise, agens are assumed o inves a he risk free rae.

3 Evaluaing SEC algorihm in opion pricing 421 V. Daases The daa se consiss of 1064 S & 500 sock index daily reurns in he period from 14 March 1996 o 2 June A rolling sample of consan size equal o n ¼ 500 is considered. Hence, he firs one-sepahead volailiy predicion, ^ 2 þ1, is available a ime ¼ 500, or on 11 March The use of a resriced sample size incorporaes changes in he rading behaviour more efficienly. 2 The S & 500 index opions daa were obained from he Daasream for he period from 11 March 1998 hrough 2 June 2000, oally 564 rading days. roper daa are available for 456 rading days. In order o minimize he biasedness of he BS formula, only he sraddle opions wih exercise prices closes o he index level, mauriy longer han 10 rading days and rading volume 4100 were considered. racice has shown ha he BS pricing model ends o misprice deep-ou-ofhe-money and deep-in-he-money opions, while i works beer for near-he-money opions (see, e.g. Daigler, 1994, p. 153). Also, a mauriy period of lengh no shorer han 10 rading days is considered o avoid mispricings aribuable o causes of pracical as well as of heoreical naure. VI. Resuls The day-by-day raes of reurn are reflecive of he corresponding predicive performances of he models. We have on he one hand raders who always choose o use one and he same ARCH model for heir forecass and raders who a each poin in ime choose o use he ARCH model suggesed by he SEC algorihm on he oher. There are 85 raders and each rader employs an ARCH model o forecas fuure volailiy and sraddle prices. For each rader, he daily rae of reurn from rading sraddles for 456 days is compued according o Equaion 9. 3 A ransacion cos of $2 ha reflecs he bid ask spread is considered. Various values for he filer F are applied, i.e. $0, $1.25, $1.75, $2.00, $2.25, $2.75, $3.50. For F ¼ $3.50, he rader using he AR(3)GARCH(0,2) forecass makes he highes daily profi of 1.35% wih a corresponding SD of 15.24% and a -raio of 1.89 (or p-value 0.06). Applying he SEC model selecion algorihm, he sum of squared sandardized one-sep-ahead predicion errors, T ^z2 1, was esimaed considering various values for T, and, in paricular, T ¼ 5(5)80. 4 Thus, i is assumed ha here are 16 raders each of which uses on each rading day, he ARCH model picked by he SEC algorihm o forecas volailiy and sraddle prices for he nex rading day. Wih a filer of $3.5, he rader uilizing he SEC algorihm wih T ¼ 5 achieves he highes profi of 1.46% per day wih a corresponding SD of 15.85% and a -raio of 1.97 (or p-value 0.05). Even marginally, he SEC(5) model selecion algorihm generaes higher reurns han hose achieved by any oher rader using only a single ARCH model. 5 Thus, he SEC model selecion algorihm appears o have a saisfacory performance in selecing hose models ha generae beer volailiy predicions. One migh ake he view ha he SEC algorihm would favour he model ha produces higher volailiy forecass. However, comparing he SEC algorihm wih a model selecion algorihm ha was consruced so as o selec he model wih he maximum sum of he T mos recen esimaed onesep-ahead volailiy forecass (denoed by MAXVAR) for various values of T revealed ha his is no he case. In none of he cases did he daily profis achieved by raders using MAXVAR(T) exceed he profis made by raders using SEC(T) for T ¼ 5(5)80. Only in an average of 5% of he rading days did he MAXVAR(T) algorihm pick he same models as hose picked by he SEC(T) algorihm. Considering he squared daily reurns as a proxy for he unobserved acual variance, a se of saisical crieria o measure he closeness of he forecass o he realizaions are also esimaed: Squared Error of Condiional Variance (SEVar): 2 ^ 2 y 2 þ1 ð10þ Absolue Error of Condiional Variance (AEVar): ^ 2 þ1 y 2 þ1 ð11þ 2 See for example Xekalaki and Degiannakis (2005). 3 Because of he large amoun of daa, ables wih all he ARCH models are available upon reques. 4 T ¼ a(b)c denoes T ¼ a, a þ b, a þ 2b,..., c b, c. 5 For any value for he filer, he SEC algorihm generaes he highes reurns, bu he p-value is he lowes for F ¼ $3.5. The Sharpe raios, which are available upon reques, were also calculaed giving similar resuls.

4 422 S. Degiannakis and E. Xekalaki Table 1. The ne rae of reurn, compued as in Equaion 9, from rading sraddles on he S & 500 index based on he SEC algorihm and he model selecion algorihms presened in Equaions 10 18, wih $2.00 ransacion coss and a $3.5 filer Model selecion mehod Sample size Mean -raio SEC T ¼ % 1.97 SEVar T ¼ % 0.80 AEVar T ¼ % 1.03 SEDev T ¼ % 0.97 AEDev T ¼ % 1.08 HASEVar T ¼ % 1.47 HAAEVar T ¼ % 1.65 HASEDev T ¼ % 1.18 HAAEDev T ¼ % 1.45 LEVar T ¼ % 1.00 Noes: The column iled sample size refers o he sample size, T, for which he corresponding model selecion algorihm leads o he highes rae of reurn. Squared Error of Condiional SD (SE Dev): 2 ^ y þ1 Absolue Error of Condiional SD (AEDev): ^ þ1 y þ1 ð12þ ð13þ Heeroscedasiciy Adused Squared Error of Cond. Variance (HASEVar): 1 y2 þ1 ^ 2!! 2 ð14þ Heeroscedasiciy Adused Absolue Error of Cond. Variance (HAAEVar):! 1 y2 þ1 ð15þ ^ 2 Heeroscedasiciy Adused Squared Error of Cond. S. Deviaion (HASEDev): 1 y 2 þ1 ð16þ ^ Heeroscedasiciy Adused Absolue Error of Cond. S. Deviaion (HAAEDev): 1 y þ1 ^ ð17þ Logarihmic Error of Condiional Variance (LEVar): 0! 1 ln y2 þ1 A ð18þ ^ 2 Applying he SEC model selecion algorihm, he sum of squared sandardized one-sep-ahead predicion errors, T ^"2 þ1 =^2 þ1, was esimaed considering various values for T. Therefore, each of he model selecion crieria is compued considering various values for T, and, in paricular, T ¼ 10(10)80. Selecing a sraegy based on any of several compeing mehods of model selecion naurally amouns o selecing he ARCH model ha, a each of a sequence of poins in ime, has he lowes value of he evaluaion funcion. In none of he cases, did he daily reurns come ou o be higher han he reurns achieved by he SEC algorihm. Table 1 presens he daily rae of reurns based on he ARCH models seleced by he 10-model selecion mehods. 6 The HAAEVar selecion algorihm, for T ¼ 40, yielded he highes daily profi (1.24%) wih a -raio of VII. Conclusion and Suggesions for Furher Research The resuls of our sudy showed ha he SEC algorihm ouperformed all of he single ARCH model-based mehods as well as a se of oher mehods of model selecion. This is in agreemen wih Xekalaki and Degiannakis s (2005) findings 6 Deailed ables for he daily rae of reurn from rading sraddles based on he ARCH models seleced by he 10-model selecion mehods are available upon reques.

5 Evaluaing SEC algorihm in opion pricing 423 from a comparaive sudy of ARCH model selecion algorihms performed on he basis of simulaed opions daa, who also showed ha he SEC algorihm for T ¼ 5 achieved he highes rae of reurn. The validiy of he variance forecass depends on which opion pricing formula is used. Even if one could find he model, which predics he volailiy precisely, i is well known ha he BS formula does no describe he dynamics of pricing he opions perfecly. In fuure research, he esimaion of ARCH-based opion pricing models such ha of Duan (1995) and Heson and Nandi (2000) is suggesed. The SEC algorihm does increase he volailiy predicion accuracy and can be considered as a ool in picking he model ha would yield he bes volailiy predicion. However, he SEC algorihm provides profis significanly greaer han 0 under a perfec framework of no commissions. Only he bid-ask spread was aken ino accoun. 7 Under realisic ransacion charges for a rader and marke impac coss, he daily profis are wiped ou. If someone could really gain 1.46% per rading day afer commissions, he presened resuls would make a good case for marke inefficiency or a leas for a huge emporary inefficiency. References Daigler, R. T. (1994) Advanced Opion Trading, robus ublishing Company, Chicago, USA. Degiannakis, S. and Xekalaki, E. (2005) redicabiliy and model selecion in he conex of ARCH models, Journal of Applied Sochasic Models in Business and Indusry, 21, Degiannakis, S. and Xekalaki, E. (2007) Assessing he performance of a predicion error crierion model selecion algorihm in he conex of ARCH Models, Applied Financial Economics, 17, Duan, J. (1995). The GARCH opion pricing model, Mahemaical Finance, 5, Heson, S. L. and Nandi, S. (2000) A closed-form GARCH opion valuaion model, The Review of Financial Sudies, 13, Xekalaki, E. and Degiannakis, S. (2005) Evaluaing volailiy forecass in opion pricing in he conex of a simulaed opions marke, Compuaional Saisics and Daa Analysis, 49, On average, a ransacion cos of 2% for each opion conrac was considered, or 8% (2% 4) for rading sraddles. However, he bid-ask spread could be even wider. A sraddle rader migh sipulae limi prices for boh pu and call opions o narrow i down, bu, in mos cases, he orders would remain unfilled.