Consequences for option pricing of a long memory in volatility. Stephen J Taylor*

Transcription

1 Consequences for opion pricing of a long memory in volailiy Sephen J Taylor* Deparmen of Accouning and Finance, Lancaser Universiy, England LA1 4YX December 000 Absrac The economic consequences of a long memory assumpion abou volailiy are documened, by comparing implied volailiies for opion prices obained from shor and long memory volailiy processes. Numerical resuls are given for opions on he S & P 100 index from 1984 o 1998, wih lives up o wo years. The long memory assumpion is found o have a significan impac upon he erm srucure of implied volailiies and a relaively minor impac upon smile effecs. These conclusions are imporan because evidence for long memory in volailiy has been found in he prices of many asses. * S.Taylor@lancaser.ac.uk, The auhor acknowledges he suppor of Inquire Europe, wih paricular hanks o Bernard Dumas and Ton Vors for heir helpful advice.

2 1 1. Inroducion Long memory effecs in a sochasic process are effecs ha decay oo slowly o be explained by saionary processes defined by a finie number of auoregressive and moving-average erms. Long memory is ofen represened by fracional inegraion of shocks o he process, which produces auocorrelaions ha decrease a a hyperbolic rae compared wih he faser asympoic rae of saionary ARMA processes. Long memory in volailiy occurs when he effecs of volailiy shocks decay slowly. This phenomenon can be idenified from he auocorrelaions of measures of realised volailiy. Two influenial examples are he sudy of absolue daily reurns from sock indices by Ding, Granger and Engle (1993) and he recen invesigaion of daily sums of squared five-minues reurns from exchange raes by Andersen, Bollerslev, Diebold and Labys (000). Sochasic volailiy causes opion prices o display boh smile and erm srucure effecs. An implied volailiy obained from he Black-Scholes formula hen depends on boh he exercise price and he ime unil he opion expires. Exac calculaion of smile and erm effecs is only possible for special, shor memory volailiy processes, wih he resuls of Heson (1993) being a noable example. Mone Carlo mehods are necessary, however, when he volailiy process has a long memory. These have been evaluaed by Bollerslev and Mikkelsen (1996, 1999). The objecive of his paper is o documen he economic consequences of a long memory assumpion abou volailiy. This is achieved by comparing implied volailiies for opion prices obained from shor and long memory specificaions. I is necessary o

3 use a long hisory of asse prices when applying a long memory model and his paper uses levels of he S & P 100 index from 1984 o For his daa i is found ha a long memory assumpion has a significan economic impac upon he erm srucure of implied volailiies and a relaively minor impac upon smile effecs. Marke makers in opions have o make assumpions abou he volailiy process. The effecs of some assumpions are revealed by he prices of opions wih long lives. Bollerslev and Mikkelsen (1999) find ha he marke prices of exchange raded opions on he S & P 500 index, wih lives beween nine monhs and hree years, are described more accuraely by a long memory pricing model han by he shor memory alernaives. Thus some opion prices already reflec he long memory phenomenon in volailiy, alhough Bollerslev and Mikkelsen (1999) find ha significan biases remain o be explained. The same LEAPS conracs are also invesigaed by Bakshi, Cao and Chen (000) who show ha opions wih long lives can differeniae beween compeing pricing models, bu heir analysis is resriced o a shor memory conex. Three explanaory secions precede he illusraive opion pricing resuls in Secion 5, so ha his paper provides a self-conained descripion of how o price opions wih a long memory assumpion. A general inroducion o he relevan lieraure is provided by Granger (1980) and Baillie (1996) on long memory, Andersen, Bollerslev, Diebold and Labys (000) on evidence for long memory in volailiy, Duan (1995) on opion pricing for ARCH processes, and Bollerslev and Mikkelsen (1999) on applying hese pricing mehods wih long memory specificaions.

4 3 Secion defines and characerises long memory precisely. These characerisics are illusraed heoreically for he fracionally inegraed whie noise process and hen he empirical evidence for hese characerisics in volailiy is surveyed. The empirical evidence for he world's major markes is compelling and explanaions for he source of long memory effecs in volailiy are summarised. Secion 3 describes parsimonious volailiy models ha incorporae long memory eiher wihin an ARCH or a sochasic volailiy framework. The former framework is easier o use and we focus on using he fracionally inegraed exension of he exponenial GARCH model, ha is known by he acronym FIEGARCH. An imporan feaure of applicaions is he unavoidable runcaion of an auoregressive componen of infinie order. Empirical resuls are provided for en years of S & P 100 reurns. Secion 4 provides he necessary opion pricing mehodology. Coningen claim prices are obained by simulaing erminal payoffs using an appropriae risk-neural measure. A risk-premium erm hen appears in he simulaed process and has a nonrivial effec upon he resuls ha is ypically seen in erm srucures ha slope upwards on average. Numerical mehods ha enhance he accuracy of he simulaions are also described. Secion 5 compares implied volailiies for European opion prices obained from shor and long memory volailiy specificaions, for hypoheical S & P 100 opions whose lives range from one monh o wo years. Opions are valued on en daes, one per annum from 1989 o The major impac of he long memory assumpion is seen o be he very slow convergence of implied volailiies o a limi as

5 4 he opion life increases. This convergence is so slow ha he limi can no be esimaed precisely. Secion 6 conains conclusions.. Long memory.1. Definiions There are several definiions ha caegorise sochasic processes as having eiher a shor memory or a long memory; examples can be found in McLeod and Hipel (1978), Brockwell and Davis (1991), Baillie (1996) and Granger and Ding (1996). The fundamenal characerisic of a long-memory process is ha dependence beween variables separaed by τ ime unis does no decrease rapidly as τ increases. σ Consider a covariance saionary sochasic process { x } ha has variance and auocorrelaions τ defined by ρ, specral densiy ( ω ) f and n-period variance-raios Vn cor( x, x ), (1) ρ τ = + τ σ f ( ω ) = ρτ cos( τω ), ω > 0, () π τ = ( x x ) var n n n τ V = = 1+ n ρ τ. (3) nσ τ = 1 n

6 5 Then a covariance saionary process is here said o have a shor memory if n ρ τ τ = 1 converges as n, oherwise i is said o have a long memory. A shor memory process hen has n ρ τ τ = 1 C ( ω ) C, V C, as n, ω 0, 1, f n 3 (4) for consans C 1, C, C3. Examples are provided by saionary ARMA processes. These processes have geomerically bounded auocorrelaions, so ha C > 0 and 1 > φ > 0, and hence (4) is applicable. ρ τ τ Cφ for some In conras o he above resuls, all he limis given by (4) do no exis for a ypical covariance saionary long memory process. Insead, i is ypical ha he auocorrelaions have a hyperbolic decay, he specral densiy is unbounded for low frequencies and he variance-raio increases wihou limi. Appropriae limis are hen provided for some posiive 1 d < by τ ρ τ d 1 ( ω ) f Vn D1, D, D3, d d ω n as n, ω 0, (5) for posiive consans D 1, D, D3. The limis given by (5) characerise he saionary long memory processes ha are commonly used o represen long memory in volailiy. The fundamenal parameer d can be esimaed from daa using a regression, eiher of ( fˆ ( ω )) ln on ω or of ( ) Ebens (000). ln Vˆn on n as in, for example, Andersen, Bollerslev, Diebold and.. Fracionally inegraed whie noise

7 6 An imporan example of a long memory process is a sochasic process { y } ha requires fracional differencing o obain a se of independen and idenically disribued residuals { ε }. Following Granger and Joyeux (1980) and Hosking (1981), such a process is defined using he filer d( d 1)! d( d 1)( d ) 3! 3 ( 1 L) d = 1 dl + L L +... (6) where L is he usual lag operaor, so ha Ly = y 1. Then a fracionally inegraed whie noise (FIWN) process { y } is defined by ( L) d y = ε 1 (7) wih he ε assumed o have zero mean and variance σ ε. Throughou his paper i is assumed ha he differencing parameer d is consrained by 0 d < 1. The mahemaical properies of FIWN are summarised in Baillie (1996) and were firs derived by Granger and Joyeux (1980) and Hosking (1981). The process is covariance saionary if 1 d < and hen he following resuls apply. Firs, he auocorrelaions are given by d( d + 1) ( 1 d )( d ) d( d + 1)( d + ) ( 1 d )( d )( 3 d ),... d ρ 1 =, ρ =, ρ3 = (8) 1 d or, in erms of he Gamma funcion, wih ( 1 d ) Γ( τ + d ) ( d ) Γ( τ + 1 d ), Γ ρ τ = (9) Γ

8 7 ( 1 d ) Γ( ) ρ τ Γ d 1 τ d as τ. (10) Second, he specral densiy is d σ ε iω d σ ε ω f ( ω) = 1 e = sin, > 0, ω (11) π π so ha f Also, σ (1) π ε d ( ω) ω for ω near 0. V n n d Γ( 1 d ) ( 1+ d ) Γ( 1+ d ) as n. (13) When 1 d he FIWN process has infinie variance and hus he auocorrelaions are 1 < no defined, alhough he process has some saionariy properies for d Evidence for long memory in volailiy When reurns r can be represened as r = µ + σ u, wih σ represening volailiy and independen of an i.i.d. sandardised reurn u, i is ofen possible o make inferences abou he auocorrelaions of volailiy from he auocorrelaions of eiher r µ or ( r µ ), see Taylor (1986). In paricular, evidence for long memory in powers of daily absolue reurns is also evidence for long memory in volailiy. Ding, Granger and Engle (1993) observe hyperbolic decay in he auocorrelaions of powers

9 8 of daily absolue reurns obained from U.S. sock indices. Dacorogona, Muller, Nagler, Olsen and Pice (1993) observe a similar hyperbolic decay in 0-minue absolue exchange rae reurns. Breid, Crao and de Lima (1998) find ha specral densiies esimaed from he logarihms of squared index reurns have he shape expeced from a long memory process a low frequencies. Furher evidence for long memory in volailiy has been obained by fiing appropriae fracionally inegraed ARCH models and hen esing he null hypohesis d = 0 agains he alernaive d > 0. Bollerslev and Mikkelsen (1996) use his mehodology o confirm he exisence of long memory in U.S. sock index volailiy. The recen evidence for long memory in volailiy uses high-frequency daa o consruc accurae esimaes of he volailiy process. The quadraic variaion σ ˆ of he logarihm of he price process during a 4-hour period denoed by can be esimaed by calculaing inraday reurns r, and hen j N r, j j= 1 σ ˆ =. (14) The esimaes are illusraed by Taylor and Xu (1997) for a year of DM/$ raes wih N = 88 so ha he r, j are 5-minue reurns. As emphasised by Andersen, Bollerslev, Diebold and Labys (000), he esimae σ ˆ will be very close o he inegral of he unobservable volailiy during he same 4-hour period providing N is large bu no so large ha he bid-ask spread inroduces bias ino he esimae. Using five-minue reurns provides conclusive evidence for long memory effecs in he esimaes σ ˆ in four sudies: Andersen, Bollerslev, Diebold and Labys (000) for en years of DM/$

10 9 and Yen/$ raes, Andersen, Bollerslev, Diebold and Ebens (000) for five years of sock prices for he 30 componens of he Dow-Jones index, Ebens (1999) for fifeen years of he same index and Areal and Taylor (000) for eigh years of FTSE-100 sock index fuures prices. These papers provide sriking evidence ha ime series of esimaes σ ˆ display all hree properies of a long memory process: hyperbolic decay in he auocorrelaions, specral densiies a low frequencies ha are proporional o d ω and variance-raios whose logarihms are very close o linear funcions of he aggregaion period n. I is also seen from hese papers ha esimaes of d are beween 0.3 and 0.5, wih mos esimaes close o Explanaions of long memory in volailiy Granger (1980) shows ha long memory can be a consequence of aggregaing shor memory processes; specifically if AR(1) componens are aggregaed and if he AR(1) parameers are drawn from a Bea disribuion hen he aggregaed process converges o a long memory process as he number of componens increases. Andersen and Bollerslev (1997) develop Granger's heoreical resuls in more deail for he conex of aggregaing volailiy componens and also provide supporing empirical evidence obained from only one year of 5-minue reurns. I is plausible o asser ha volailiy reflecs several sources of news, ha he persisence of shocks from hese sources depends on he source and hence ha oal volailiy follows a long memory process. Scheduled macroeconomic news announcemens are known o creae addiional volailiy ha is very shor-lived (Ederingon and Lee, 1993), whils oher sources of

11 10 news ha have a longer impac on volailiy are required o explain volailiy clusering effecs ha las several weeks. Gallan, Hsu and Tauchen (1999) esimae a volailiy process for daily IBM reurns ha is he sum of only wo shor memory componens ye he sum is able o mimic long memory. They also show ha he sum of a paricular pair of AR(1) processes has a specral densiy funcion very close o ha of fracionally inegraed whie noise wih d = 0. 4 for frequencies ω 0. 01π. Consequenly, evidence for long memory may be consisen wih a shor memory process ha is he sum of a small number of componens whose specral densiy happens o resemble ha of a long memory process excep a exremely low frequencies. The long memory specificaion may hen provide a much more parsimonious model. Barndorff-Nielsen and Shephard (001) model volailiy in coninuous-ime as he sum of a few shor memory componens. Their analysis of en years of 5-minue DM/$ reurns, adjused for inraday volailiy periodiciy, shows ha he sum of four volailiy processes is able o provide an excellen mach o he auocorrelaions of squared 5-minue reurns, which exhibi he long memory propery of hyperbolic decay. 3. Long memory volailiy models A general se of long memory sochasic processes can be defined by firs applying he filer ( 1 L) d and hen assuming ha he filered process is a saionary ARMA(p, q) process. This defines he ARFIMA(p, d, q) models of Granger (1980), Granger and Joyeux (1980) and Hosking (1981). This approach can be used o obain long memory

12 11 models for volailiy, by exending various specificaions of shor memory volailiy processes. We consider boh ARCH and sochasic volailiy specificaions ARCH specificaions The condiional disribuions of reurns r are defined for ARCH models using informaion ses 1 I ha are here assumed o be previous reurns {, i 1} r i, condiional mean funcions µ ( I 1 ), condiional variance funcions ( I 1 ) probabiliy disribuion D for sandardised reurns z. Then he erms h and a z r µ = (15) h are independenly and idenically disribued wih disribuion D and have zero mean and uni variance. Baillie (1996) and Bollerslev and Mikkelsen (1996) boh show how o define a long memory process for h by exending eiher he GARCH models of Bollerslev (1986) or he exponenial ARCH models of Nelson (1991). The GARCH exension can no be recommended because he reurns process hen has infinie variance for all posiive values of d, which is incompaible wih he sylized facs for asse reurns. For he exponenial exension, however, ln ( h ) is covariance saionary for hen be conjecured ha he reurns process has finie variance for paricular specificaions of h. d < 1 ; i may

13 1 Like Bollerslev and Mikkelsen (1996, 1999), his paper applies he FIEGARCH(1, d, 1) specificaion : 1 d ( h ) + ( 1 φl) ( 1 L) ( 1+ ψl) g( z ), ln = α 1 (16) g ( z ) = z + γ ( z C) θ, (17) wih α, φ, d, ψ respecively denoing he locaion, auoregressive, differencing and moving average parameers of ln ( h ). The i.i.d. residuals ( z ) g depend on a symmeric response parameer γ and an asymmeric response parameer θ ha enables he condiional variances o depend on he signs of he erms z ; hese residuals have zero mean because C is defined o be he expecaion of z. The EGARCH(1,1) model of Nelson (1991) is given by d = 0. If φ =ψ = 0 and > 0 d, hen ( ) α ln is a fracionally inegraed whie noise process. In general, ln ( h ) is an ARFIMA(1, d, 1) process. Calculaions using equaion (16) require series expansions in he lag operaor L. We noe here he resuls : j d 1 L j a 1 = d, a j = a j 1, j, (18) j d j ( 1 ) = 1 a L, j= 1 d j ( 1 L)( 1 L) = 1 b L, φ b = d +, b j = a j φ a j 1, j, (19) j j= 1 d 1 j ( 1 φl)( 1 L) ( 1+ ψl) = 1 φ L, φ 1 + φ + ψ, j= 1 j 1 φ j 1 = d = b ( ψ ) + ( ψ ) b, j, φ (0) j j j k= 1 j k k h

14 13 and ha he auoregressive weighs in (0) can be denoed as φ ( d, φ, ψ ) 1 d j ( 1 φl) ( 1 L) ( 1+ ψl) = 1+ ψ L, j= 1 ψ 1 + φ + ψ, j = d ψ = φ ( d, ψ, φ ) j. Also, j j. (1) I is necessary o runcae he infinie summaions when evaluaing empirical condiional variances. Truncaion afer N erms of he summaions in (1), (0) and (19) respecively give he MA(N), AR(N) and ARMA(N, 1) approximaions : N 1 j= 1 ln( h ) = + g( z ) + ψ g( z ), ln α () j j 1 N ( h ) + φ [ ln( h ) α ] g( z ) = j j + 1 j= 1 α, (3) N ( h ) + b [ ln( h ) α ] + g( z ) ψg( z ). ln = α j j 1 + (4) j= 1 As j, he coefficiens b j and φ j converge much more rapidly o zero han he coefficiens approximaion. ψ j. Consequenly i is bes o use eiher he AR or he ARMA 3.. Esimaes for he S & P 100 index Represenaive parameers are required in Secion 5 o illusrae he consequences of long memory in volailiy for pricing opions. As ARCH specificaions are preferred for hese illusraions, a discussion is presened here of parameer values for he FIEGARCH(1, d, 1) specificaion. These parameers are esimaed from daily reurns

15 14 r for he S & P 100 index, excluding dividends, calculaed from index levels ( p p ) r. = ln 1 p as Evaluaion of he condiional variances requires runcaion of he infinie series defined by he fracional differencing filer. Here he variances are evaluaed for 1 by seing = 1000 N in equaion (4), wih ln ( ) replaced by α and g( ) h j z j replaced by zero whenever j 0. The log-likelihood funcion is calculaed for he,58 rading days during he en-year esimaion period from 3 January 1989 o 31 December 1998, which corresponds o he imes 1,1 3, 748 for our daase; hus he firs 1,0 reurns are reserved for he calculaion of condiional variances before 1989 which are needed o evaluae he subsequen condiional variances. Resuls are firs discussed and are abulaed when reurns have a consan condiional mean which is esimaed by he sample mean. The condiional variances are obained recursively from equaions (15), (17), (19) and (4). The condiional disribuions are assumed o be Normal when defining he likelihood funcion. This assumpion is known o be false bu i is made o obain consisen parameer esimaes (Bollerslev and Wooldridge, 199). Preliminary maximisaions of he likelihood showed ha a suiable value for E[ ] C = is 0.737, compared wih π for z he sandard Normal disribuion. They also showed ha an appropriae value of he locaion parameer α of ln ( h ) is -9.56; he log-likelihood is no sensiive o minor N b j j= 1 deviaions from his level because α is muliplied by a erm 1 in equaion (4) ha is small for large N. Consequenly, he resuls summarised in Table 1 are given by

16 15 maximising he log-likelihood funcion over some or all of he parameers θ, γ, φ, ψ, d. The esimaes of θ and γ provide he usual resul for a series of U.S. sock index reurns ha changes in volailiy are far more sensiive o he values of negaive reurns han hose of posiive reurns, as firs repored by Nelson (1991). When z is negaive, g( z ) = ( γ θ )( z ) γc, oherwise g( z ) ( γ + θ ) z γc =. The raio γ γ θ + θ is a leas 4 and hence is subsanial for he esimaes presened in Table 1. The firs wo rows of Table 1 repor esimaes for shor memory specificaions of he condiional variance. The AR(1) specificaion has a persisence of 0.98 ha is ypical for his volailiy model. The ARMA(1,1) specificaion has an addiional parameer and increases he log-likelihood by 3.0. The hird row shows ha he fracional differencing filer alone ( d > 0, φ =ψ = 0 ) provides a beer descripion of he volailiy process han he ARMA(1,1) specificaion; wih d = he loglikelihood increases by A furher increase of 7.8 is hen possible by opimising over all hree volailiy parameers, d, φ and ψ, o give he parameer esimaes 1 in he fifh row of Table 1. The esimaes for he mos general specificaion idenify wo issues of concern. Firs, d equals 0.57 for our daily daa which is more han he ypical esimae of The log-likelihood funcion is maximised using a complee enumeraion algorihm and hence sandard errors are no immediaely available. A conservaive robus sandard error for our esimae of d is 0.1, using informaion provided by Bollerslev and Mikkelsen (1996).

17 16 produced by he sudies of higher frequency daa menioned in Secion.3. The same issue arises in Bollerslev and Mikkelsen (1996) wih d esimaed as 0.63 (sandard error 0.06) from 9,559 daily reurns of he S & P 500 index, from 1953 o 1990; here are similar resuls in Bollerslev and Mikkelsen (1999). Second, he sum d + φ + ψ equals As his sum equals ψ 1 in equaions (1) and (), more weigh is hen given o he volailiy shock a ime han o he shock a ime 1 when calculaing ( ) h ln. This is counerinuiive. To avoid his oucome, he consrain d + φ + ψ 1 is applied and he resuls given in he penulimae row of Table 1 are obained. The loglikelihood is hen reduced by.0. Finally, if d is consrained o be 0.4 hen he loglikelihood is reduced by an addiional 8.3. The esimaes obained here for φ and ψ, namely -0.7 and 0.68 for he mos general specificaion, are raher differen o he 0.78 and given by Bollerslev and Mikkelsen (1999, Table 1), alhough he esimaes of d are similar, namely 0.59 and However, he moving-average represenaions obained from hese ses of parameers esimaes are qualiaively similar. This is shown on Figure 1 ha compares he moving-average coefficiens ψ j defined by equaion (1). The coefficiens are posiive and monoonic decreasing for he four ses of parameer values used o produce Figure 1. They show he expeced hyperbolic decay when d > 0 and a geomeric decay when d = 0. The values of b j in equaions (19) and (4) ha are used o calculae he condiional variances decay much faser. For each of he four curves shown on Figure 1, ψ 10 > and ψ 100 > whils 0 < b 10 < 0. 0 and 0 < b 100 <

18 17 The resuls repored in Table 1 are for a consan condiional mean, µ = µ. Alernaive specificaions such as µ = µ + βr 1, µ = µ 1 h and µ = µ + λ h give similar values of he log-likelihood when he volailiy parameers are se o he values in he final row of Table 1. Firs, including he lagged reurn r 1 is no necessary because he firs-lag auocorrelaion of he S&P 100 reurns equals -0.0 and is saisically insignifican. Second, including he adjusmen 1 h makes he condiional expecaion of p p p 1 1 consan when he condiional disribuion is Normal. The adjusmen reduces he log-likelihood by an unimporan 0.3. Third, incorporaing he ARCH-M parameer λ gives an opimal value of 0.10 and an increase in he log-likelihood of 1.5. This increase is no significan using a non-robus likelihood-raio es a he 5% level Sochasic volailiy specificaions Two shocks per uni ime characerise sochasic volailiy (SV) models, in conras o he single shock z ha appears in ARCH models. A general framework for long memory sochasic volailiy models is given for reurns r by r = µ + σ u (5) wih ( ) σ ln following an ARFIMA(p, d, q) process. For example, wih p = q = 1, ln 1 d ( σ ) = α + ( 1 φl) ( 1 L) ( 1+ ψl) v. (6)

19 18 This framework has been invesigaed by Breid, Crao and de Lima (1998), Harvey (1998) and Bollerslev and Wrigh (000), all of whom provide resuls for he simplifying assumpion ha he wo i.i.d. processes { u } and { } v are independen. This assumpion can be relaxed and has been for shor memory applicaions (Taylor, 1994, Shephard, 1996). Parameer esimaion is difficul for SV models, compared wih ARCH models, because SV models have wice as many random innovaions as observable variables. Breid, Crao and de Lima (1998) describe a specral-likelihood esimaor and provide resuls for a CRSP index from 196 o For he ARFIMA(1, d, 0) specificaion hey esimae d = and φ = Bollerslev and Wrigh (000) provide deailed simulaion evidence abou semiparameric esimaes of d, relaed o he frequency of he observaions. I is apparen ha he ARCH specificaion (15)-(17) has a similar srucure o he SV specificaion (5)-(6). Shor memory special cases of hese specificaions, given by d = q = 0, have similar momens (Taylor, 1994). This is a consequence of he special cases having he same bivariae diffusion limi when appropriae parameer values are defined for increasingly frequen observaions (Nelson, 1990, Duan, 1997). I seems reasonable o conjecure ha he mulivariae disribuions for reurns defined by (15)-(17) and (5)-(6) are similar, wih he special case of independen shocks { } u and { } v corresponding o he symmeric ARCH model ha has θ = 0 in equaion (17).

20 19 4. Opion pricing mehodology 4.1. A review of SV and ARCH mehods The pricing of opions when volailiy is sochasic and has a shor memory has been sudied by several researchers using a variey of mehods. The mos popular mehods commence wih separae diffusion specificaions for he asse price and is volailiy. These are called sochasic volailiy (SV) mehods. Opion prices hen depend on several parameers including a volailiy risk premium and he correlaion beween he differenials of he Wiener processes in he separae diffusions. Hull and Whie (1987) provide soluions ha include a simple formula when volailiy risk is no priced and he correlaion beween he differenials is zero. The closed form soluion of Heson (1993) assumes ha volailiy follows a square-roo process and permis a general correlaion and a non-zero volailiy risk premium; for applicaions see, for example, Bakshi, Cao and Chen (1997, 000) and for exensions see Duffie, Pan and Singleon (000). There is much less research ino opion pricing for shor memory ARCH models. Duan (1995) provides a valuaion framework and explici resuls for he GARCH(1,1) process ha can be exended o oher ARCH specificaions. Richken and Trevor (1999) provide an efficien laice algorihm for GARCH(1,1) processes and exensions for which he condiional variance depends on he previous value and he laes reurn innovaion.

21 0 Mehods for pricing opions when volailiy has a long memory have been described by Come and Renaul (1998) and Bollerslev and Mikkelsen (1996, 1999). The former auhors provide analysis wihin a bivariae diffusion framework. They replace he usual Wiener process in he volailiy equaion by fracional Brownian moion. However, heir opion pricing formula appears o require independence beween he Wiener process in he price equaion and he volailiy process. This assumpion is no consisen wih he empirical evidence for sock reurns. The assumpion is refued, for example, by finding ha θ is no zero in he funcion g( z ) ha appears in an exponenial ARCH model. The mos pracical way o price opions wih long memory in volailiy is probably based upon ARCH models, as demonsraed by Bollerslev and Mikkelsen (1999). We follow he same sraegy. From he asympoic resuls in Duan (1997), also discussed in Richken and Trevor (1999), i is anicipaed ha insighs abou opions priced from a long memory ARCH model will be similar o he insighs ha can be obained from a relaed long memory SV model. 4.. The ARCH pricing framework When pricing opions i will be assumed ha reurns are calculaed from prices (or index levels) as ( p p ) r and hence exclude dividends. A consan risk-free = ln 1 ineres rae and a consan dividend yield will also be assumed and, o simplify he noaion and calculaions, i will be assumed ha ineres and dividends are paid once

22 1 per rading period. Condiional expecaions are defined wih respec o curren and prior price informaion represened by I = { p, i 0} i. To obain fair opion prices in an ARCH framework i is necessary o make addiional assumpions in order o obain a risk-neural measure Q. Duan (1995) and Bollerslev and Mikkelsen (1999) provide sufficien condiions o apply a risk-neural valuaion mehodology. For example, i is sufficien ha a represenaive agen has consan relaive risk aversion and ha reurns and aggregae growh raes in consumpion have condiional normal disribuions. Kallsen and Taqqu (1998) derive he same soluion as Duan (1995) wihou making assumpions abou uiliy funcions and consumpion. Insead, hey assume ha inraday prices are deermined by geomeric Brownian moion wih volailiy deermined once a day from a discree-ime ARCH model. A ime ', measured in rading periods, he fair price of an European coningen claim ha has value ( p ) y + n ' + n ' a he erminal ime ' + n is given by ( p ) I ] Q ρn y ' = E [ e y' + n ' + n ' (7) wih ρ he risk-free ineres rae for one rading period. Our objecive is now o specify an appropriae way o simulae p + n ' under a risk-neural measure Q and hereby o evaluae he above condiional expecaion using Mone Carlo mehods. Following Duan (1995), i is assumed ha observed reurns are obained under a probabiliy measure P from P r I 1 ~ N( µ, h ), (8) wih

23 z r µ P = ~ i. i. d. N( 0,1) (9) h and ha in a risk-neural framework reurns are obained under measure Q from r I N( ρ δ 1 h, h ), (30) Q 1 ~ wih z * = r ( ρ δ 1 h ) h ~ Q i. i. d. N ( 0,1). (31) d e 1 p = δ Here δ is he dividend yield, ha corresponds o a dividend paymen of ( ) ρ δ per share a ime. Then E [ p I 1] = e p 1 Q and he expeced value a ime of one share and he dividend paymen is E [ p + d I 1] = e p 1, as required in a riskneural framework. Q ρ Noe ha he condiional means are differen for measures P and Q bu he funcions ( p, p,... ) h ha define he condiional variances for he wo measures 1 are idenical. Duan (1995) proves ha his is a consequence of he sufficien assumpions ha he saes abou risk preferences and disribuions. The same conclusion applies for he less resricive assumpions of Kallsen and Taqqu (1998). Opion prices depend on he specificaions for Duan (1995) and assume ha µ and h. We again follow µ = ρ δ 1 h + λ h (3) wih λ represening a risk-premium parameer. Then he condiional expecaions of r for measures P and Q differ by λ h and

24 3 * z z = λ. (33) 4.3. Long memory ARCH equaions Opion prices are evaluaed in his paper when he condiional variances are given by he ARMA(N, 1) approximaion o he FIEGARCH(1, d, 1) specificaion. From (17), (4) and (33), 1 N j * b j L ( ln( h ) α ) = ( 1+ ψl) g( z 1 ) = ( 1+ ψl) g( z 1 λ), (34) j= 1 and g ( z ) = z + γ ( z C) θ (35) wih he auoregressive coefficiens b j defined by (19) as funcions of φ and d; also C = /π for condiional normal disribuions. Suppose here are reurns observed a imes 1 ', whose disribuions are given by measure P, and ha we hen wan o simulae reurns for imes > ' using measure Q. Then ln ( ) is calculaed for h using he observed reurns, wih ln ( ) = α and ( ) = 0 1 ' + 1 * Q followed by simulaing z ~ N(0,1) and hence obaining h g for < 1, z ln +1 r and ( ) h for > '. z, [ z λ ] = π exp( λ ) + λ Φ( λ) When ~ N(0,1) disribuion funcion of z. ( 1) 1 E wih Φ he cumulaive

25 4 The expecaion of ( ) h measure P. I is differen for measure Q because ln depends on he measure 3 when λ 0. I equals α for ( / π ) * Q * [ g( z λ) ] = λθ + γ E [ z λ ] Q λ γ E λθ + (36) π when λ is small, and his expecaion is in general no zero. For a fixed ', as Q 1 [ ] N [ ( ) ] α Q * ln h I + 1 b ( + ψ ) E g( z λ), E j. (37) j= 1 ' 1 The difference beween he P and Q expecaions of ln ( h ) could be inerpreed as a volailiy risk premium. This "premium" is ypically negaive, because ypically λ > 0, θ 0 and γ > 0. Furhermore, when θ is negaive he major erm in (36) is λθ, because λ is always small, and hen he "premium" reflecs he degree of asymmery in he volailiy shocks g ( z ). The magniude of he volailiy "risk premium" can be imporan and, indeed, he quaniy defined by he limi in (37) becomes infinie 4 as N when d is posiive. A realisic value of λ for he S & P 100 index is 0.08, obained by assuming ha he equiy risk premium is 6% per annum 5. For he shor memory parameer values in he firs row of Table 1, when d = 0 and = 1000 Q N, he limi of E ( h ) [ I ] α ln ' 3 The dependence of momens of h on he measure is shown by Duan (1995, p. 19) for he GARCH(1,1) model. d 4 As ( 1 ) 1 = 0 L for d > 0, i follows from (18) and (19) ha a = j b j = 1. j= 1 j= 1 5 The condiional expecaions of r for measures P and Q differ by average value of saed value of λ. λ h and a ypical h is Assuming 53 rading days in one year gives he

26 5 equals This limi increases o 0.0 for he parameer values in he final row of Table 1, when d = 0. 4 and = 1000 ln is o N. The ypical effec of adding 0. o ( ) h muliply sandard deviaions h by 1.1 so ha far-horizon expeced volailiies, under Q, are slighly higher han migh be expeced from hisorical sandard deviaions. Consequenly, on average he erm srucure of implied volailiies will slope upwards Numerical mehods The preceding equaions can be used o value an European coningen claim a ime ' by simulaing prices under he risk-neural measure Q, followed by esimaing he expeced discouned erminal payoff a ime ' + n as saed in equaion (7). Two variaions on hese equaions are used when obaining represenaive resuls in Secion 5. Firs, he specificaion of µ can be differen for imes on eiher side of ' o allow for changes hrough ime in risk-free ineres raes and risk premia. Equaion (3) is hen replaced by µ = m 1 h = ρ δ + λ' 1 h h + λ, h, ', > '. (38) Second, because he observed condiional disribuions are no Normal whils he simulaions assume ha hey are, i is necessary o define C = C', = π, ', > '. (39)

27 6 for a consan C ' esimaed from observed reurns. An alernaive mehod, described by Bollerslev and Mikkelsen (1999), is o simulae from he sample disribuion of sandardised observed reurns. Sandard variance reducion echniques can be applied o increase he accuracy of Mone Carlo esimaes of coningen claim prices. A suggesed aniheic mehod uses one i.i.d. (0,1) { z * }, { z } and { } beween * z and N sequence { } z wih he erms * z o define he furher i.i.d. N (0,1) sequences, z chosen so ha here is negaive correlaion * * z ; his is achieved by defining ( z ) + Φ( z ) = + 1 sign( z ) Φ. The 1 four sequences provide claim prices whose average, y say, is much less variable han he claim price from a single sequence. An overall average ŷ is hen obained from a se of K values { 1 k K} parameer y CV y k,. The conrol variae mehod makes use of an unbiased esimae ŷ CV of a known, such ha ŷ is posiively correlaed wih ŷ CV. A suiable parameer, when pricing a call opion in an ARCH framework, is he price of a call opion when volailiy is deerminisic. The deerminisic volailiy process is defined by replacing all erms ln ( ), > ' + 1, h by heir expecaions under P condiional on he hisory I '. Then y CV is given by a simple modificaion of he Black-Scholes formula, whils ŷcv is obained by using he same 4K sequences of i.i.d. variables ha define ŷ. Finally, a ~ β y CV y CV wih more accurae esimae of he opion price is hen given by y = yˆ ( ˆ ) β chosen o minimise he variance of y ~.

28 7 5. Illusraive long memory opion prices 5.1. Inpus The Black-Scholes formula has six parameers, when an asse pays dividends a a consan rae, namely he curren asse price S, he ime unil he exercise decision T, he exercise price X, he risk-free rae R, he dividend yield D and he volailiy σ. There are many more parameers and addiional inpus for he FIEGARCH opion pricing mehodology described in Secion 4. To apply ha mehodology o value European opions i is necessary o specify eigheen numbers, a price hisory and a random number generaor, as follows : Conracual parameers - ime unil exercise T measured in years, he exercise price X and wheher a call or a pu opion. The curren asse price p ' S = and a se of previous prices { p, 1 < ' }. Trading periods per annum M, such ha consecuive observed prices are separaed by 1 / M years and likewise for simulaed prices { p, '< ' + n} wih n = MT. Risk-free annual ineres rae R, from which he rading period rae ρ = R / M is obained. Annual dividend yield D giving a consan rading period payou rae of δ = D / M ; boh R and D are coninuously compounded and applicable for he life of he opion conrac.

29 8 The risk premium λ for invesmen in he asse during he life of he opion, such ha one-period condiional expeced reurns are µ = ρ δ + 1 h λ h. Parameers m and λ ' ha define condiional expeced reurns during he ime period of he observed prices by 1 h λ µ = m + ' h. Eigh parameers ha define he one-period condiional variances h. The inegraion level d, he auoregressive parameer φ and he runcaion level N deermine he parameers b j (given by equaion (19)) of he AR(N) filer in equaion (34). The mean α and he moving-average parameer ψ complee he ARMA(N, 1) specificaion for ln ( h ) in equaion (34). The values of he shocks o he ARMA(N, 1) process depend on γ and θ, ha respecively appear in he symmeric funcion ( z C) γ and he asymmeric funcion θz whose oal deermines he shock erm g ( z ); he consan C is a parameer C ' for observed prices bu equals π when reurns are simulaed. K, he number of independen simulaions of he erminal asse price S T = p ' + n. A se of Kn pseudo-random numbers disribued uniformly beween 0 and 1, from which pseudo-random sandard normal variaes can be obained. These numbers ypically depend on a seed value and a deerminisic algorihm. 5.. Parameer selecions

30 9 Opion values are abulaed for hypoheical European opions on he S & P 100 index. Opions are valued for en daes defined by he las rading days of he en years from 1989 o 1998 inclusive. For valuaion daes from 199 onwards he size of he price hisory is se a '= 000 ; for previous years he price hisory commences on 6 March 1984 and '< 000. I is assumed ha here are M = 5 rading days in one year and hence exacly 1 rading days in one simulaed monh. Opion values are abulaed when T is 1,, 3, 6, 1, 18 and 4 monhs. Table liss he parameer values used o obain he main resuls. The annualised risk-free rae and dividend yield are se a 5% and % respecively. The risk parameer λ is se a 0.08 o give an annual equiy risk premium of 6% (see foonoe 5). The mean reurn parameer m is se o he hisoric mean of he complee se of S & P 100 reurns from March 1984 o December 1998 and λ ' is se o zero. There are wo ses of values for he condiional variance process because he primary objecive here is o compare opion values when volailiy is assumed o have eiher a shor or a long memory. The long memory parameer se akes he inegraion level o be d = 0. 4 because his is an appropriae level based upon he recen evidence from high-frequency daa, reviewed in Secion.3. The remaining variance parameers are hen based on Table 1; as he moving-average parameer is small i is se o zero and he auoregressive parameer is adjused o reain he uni oal, d + φ + ψ = 1. The AR

31 30 filer 6 is runcaed a lag 1000, alhough he resuls obained will neverheless be referred o as long memory resuls. The shor memory parameers are similar o hose for he AR(1) esimaes provided in Table 1. The parameers γ and θ are boh 6% less in Table han in Table 1 o ensure ha seleced momens are mached for he shor and long memory specificaions; he uncondiional mean and variance 7 of ln ( h ) are hen mached for he hisoric measure P, alhough he uncondiional means differ by approximaely 0.10 for he risk-neural measure Q as noed in Secion 4.3. Opion prices are esimaed from K = 10, 000 independen simulaions of prices { p '< ' + n}, wih n = 504. Applying he aniheic and conrol variae mehods described in Secion 4.4 hen produces resuls for a long memory process in abou 50 minues, using a PC running a 466 MHz. Mos of he ime is spen evaluaing he highorder AR filer; he compuaion ime is less han 5 minues for he shor memory 6 This filer is ( 1 0.6L)( 1 L) = 1 ( L 0.1L 0.008L L L L +...) = 1 b j L j= 1 j. Also, b 6 > b j > b j + 1 > 0 for j > 6, b 100 = , 6 b1000 = and b equals j j= 1 7 I hank Granville Tunnicliffe-Wilson for calculaing he variance of he AR(1000) process.

32 31 process. Separae seed values are used o commence he "random number" calculaions 8 for he en valuaion daes; hese seed values are always he same for calculaions ha have he same valuaion dae Comparisons of implied volailiy erm srucures The values of all opions are repored using annualised implied volailiies raher han prices. Each implied volailiy (IV) is calculaed from he Black-Scholes formula, adjused for coninuous dividends. The complee se of IV oupus for one se of inpus forms a marix wih rows labelled by he exercise prices X and columns labelled by he imes o expiry T; examples are given in Tables 5 and 6 and are discussed laer. Iniially we only consider a-he-money opions, for which he exercise price equals he forward price F ( R D)T = Se, wih IV values obained by linear inerpolaion across wo adjacen values of X. As T varies, he IV values represen he erm srucure of implied volailiy. Tables 3 and 4 respecively summarise hese erm srucures for he shor and long memory specificaions. The same informaion is ploed on Figures and 3 respecively. The IV values for T = 0 are obained from he condiional variances on he valuaion daes. The sandard errors of he abulaed implied volailiies increase wih T. The maximum sandard errors for a-he-money opions are respecively and for he shor and long memory specificaions. 8 The Excel VBA pseudo-random number generaor was used. This generaor has cycle lengh 4. Use is made of 30% of he complee cycle when K = 10, 000 and n = 504.

33 3 The en IV erm srucures for he shor memory specificaion commence beween 9.5% (1993) and 18.8% (1997) and converge owards he limiing value of 14.3%. The iniial IV values are near he median level from 1989 o 1991, are low from 199 o 1995 and are high from 1996 o Six of he erm srucures slope upwards, wo are almos fla and wo slope downwards. The shapes of hese erm srucures are compleely deermined by he iniial IV values because he volailiy process is Markovian. There are hree clear differences beween he erm srucures for he shor and long memory specificaions ha can be seen by comparing Figures and 3. Firs, he long memory erm srucures can and do inersec because he volailiy process is no Markovian. Second, some of he erm srucures have sharp kinks for he firs monh. This is paricularly noeworhy for 1990 and 1996 when he erm srucures are no monoonic. For 1990, he iniial value of 14.1% is followed by 15.6% a one monh and a gradual rise o 16.% a six monhs and a subsequen slow decline. For 1996, he erm srucure commences a 15.6%, falls o 13.6% afer one monh and reaches a minimum of 1.8% afer six monhs followed by a slow incline. The eigh oher erm srucures are monoonic and only hose for 1997 and 1998 slope downwards. Third, he erm srucures approach heir limiing value very slowly 9. The wo-year IVs range from 1.1% o 16.1% and i is no possible o deduce he limiing value, alhough 15.0% o 9 The resuls suppor he conjecure ha deermined by he hisory of observed reurns. d 1 IV ( T ) a1 + at for large T wih a

34 % is a plausible range 10. I is noable ha he dispersion beween he en IV values for each T decreases slowly as T increases, from.% for one-monh opions o 1.4% for wo-year opions. There are subsanial differences beween he wo IV values ha are calculaed for each valuaion dae and each opion lifeime. Figure 4 shows he differences beween he a-he-money IVs for he long memory specificaion minus he number for he shor memory specificaion. When T = 0 hese differences range from -1.9% (1997) o 1.5% (199), for hree-monh opions from -1.5% (1995, 1996) o.1% (1990) and for wo-year opions from -1.9% (1995) o 1.7% (1990). The sandard deviaion of he en differences is beween 1.1% and 1.4% for all values of T considered so i is common for he shor and long memory opion prices o have IVs ha differ by more han 1% Comparisons of smile effecs The columns of he IV marix provide informaion abou he srengh of he so-called smile effec for opions prices. These effecs seem o be remarkably robus o he choice of valuaion dae and hey are no very sensiive o he choice beween he shor and 10 An esimae of he consan a 1 (defined in he previous foonoe) is 16.0%. An esimae of 15.0% follows by supposing he long memory limi is 105% of he shor memory limi, based on he limi of ln ( h ) being higher by 0.1 for he long memory process as noed in Secion 4.3. The difference in he limis is a consequence of he risk premium obained by owning he asse; is magniude is mainly deermined by he pronounced asymmery in he volailiy shock funcion g ( z ).

35 34 long memory specificaions. This can be seen by considering he en values of IV = IV T, X ) IV ( T, ) obained for he en valuaion daes, for various values ( 1 X of T, various pairs of exercise prices X 1, X and a choice of volailiy process. Firs, for one-monh opions wih S = 100, X 9 and X 108, he values of IV range 1 = from 3.0% o 3.3% for he shor memory specificaion and from 3.7% o 4.0% for he long memory specificaion. Second, for wo-year opions wih X 1 = 80 and X = 10, he values of IV range from 1.8% o.0% and from 1.8% o 1.9%, respecively for he shor and long memory specificaions. Figure 5 shows he smiles for hree-monh opions valued using he shor memory model, separaely for he en valuaion daes. As may be expeced from he above remarks he en curves are approximaely parallel o each oher. They are almos all monoonic decreasing for he range of exercise prices considered, so ha a U-shaped funcion (from which he idea of a smile is derived) can no be seen. The near monoonic decline is a sandard heoreical resul when volailiy shocks are negaively correlaed wih price shocks (Hull, 000). I is also a sylized empirical fac for U.S. equiy index opions, see, for example, Rubinsein (1994) and Dumas, Fleming and Whaley (1998). Figure 6 shows he hree-monh smiles for he long memory specificaion. The shapes on Figures 5 and 6 are similar, as all he curves are for he same expiry ime, bu hey are more dispersed on Figure 6 because he long memory effec induces more dispersion in a-he-money IVs. The minima of he smiles are generally near an exercise price of 116. Figure 7 shows furher long memory smiles, for wo-year opions =

36 35 when he forward price is The parallel shapes are clear; he wo highes curves are almos idenical, and he hird, fourh and fifh highes curves are almos he same. Tables 5 and 6 provide marices of implied volailiies for opions valued on 31 December When eiher he call or he pu opion is deep ou-of-he-money i is difficul o esimae he opion price accuraely because he risk-neural probabiliy q(x ) of he ou-of-he-money opion expiring in-he-money is small. Consequenly, he IV informaion has no been presened when he corresponding sandard errors exceed 0.%; esimaes of q (X ) are less han 3%. The sandard errors of he IVs are leas for opions ha are near o a-he-money and mos of hem are less han 0.05% for he IVs lised in Tables 5 and 6. All he secions of he smiles summarised by Tables 5 and 6 are monoonic decreasing funcions of he exercise price. The IV decreases by approximaely 4% o 5% for each abulaed secion Sensiiviy analysis The sensiiviy of he IV marices o hree of he inpus has been assessed for opions valued on 31 December Firs, consider a change o he risk parameer λ ha corresponds o an annual risk premium of 6% for he abulaed resuls. From Secion 4.3, opion prices should be lower for large T when λ is reduced o zero. Changing λ o zero reduces he a-he-money IV for wo-year opions from 16.0% o 15.4% for he long memory inpus, wih a similar reducion for he shor memory inpus. Second, consider reducing he runcaion level N in he AR(N) filer from 1000 o 100. Alhough his has he advanage of a subsanial reducion in he compuaional ime i changes

37 36 he IV numbers by appreciable amouns and can no be recommended; for example, he wo-year a-he-money IV hen changes from 16.0% o 14.7%. The smile shapes on Figures 5, 6 and 7 are heavily influenced by he negaive asymmeric shock parameer θ, ha is subsanial relaive o he symmeric shock parameer γ. The asymmery in he smile shapes can be expeced o disappear when θ is zero, which is realisic for some asses including exchange raes. Figures 8 and 9 compare smile shapes when θ is changed from he values used previously o zero, wih γ scaled o ensure he variance of ln ( h ) is unchanged for measure P. Figure 8 shows ha he one-monh smile shapes become U-shaped when θ is zero, whils Figure 9 shows ha he IV are hen almos consan for one-year opions. 6. Conclusions The empirical evidence for long memory in volailiy is srong, for boh equiy (Andersen, Bollerslev, Diebold and Ebens, 000, Areal and Taylor, 000) and foreign exchange markes (Andersen, Bollerslev, Diebold and Labys, 000). This evidence may more precisely be inerpreed as evidence for long memory effecs, because here are shor memory processes ha have similar auocorrelaions and specral densiies, excep a very low frequencies (Gallan, Hsu and Tauchen, 1999, Barndorff-Nielsen and Shephard, 001). There is also evidence ha people rade a opion prices ha are more compaible wih a long memory process for volailiy han wih a parsimonious shor memory process (Bollerslev and Mikkelsen, 1999).

38 37 The heory of opion pricing when volailiy follows a discree-ime ARCH process relies on weak assumpions abou he coninuous-ime process followed by prices (Kallsen and Taqqu, 1998) and he numerical implemenaion of he heory is sraighforward. Applicaion of he heory when he volailiy process is fracionally inegraed does, however, require pragmaic approximaions because he fundamenal filer ( 1 L) d is an infinie order polynomial ha mus be runcaed a some power N. Opion prices are sensiive o he runcaion poin N, so ha large values and long price hisories from an assumed saionary process are required. The erm srucure of implied volailiy for a-he-money opions can be noably differen for shor and long memory ARCH specificaions applied o he same price hisory. Long memory erm srucures have more variey in heir shapes. They may have kinks for shor mauriy opions and hey may no have a monoonic shape. Also, erm srucures on differen valuaion daes someimes inersec each oher. None of hese possibiliies occurs for a Markovian shor memory specificaion. Long memory erm srucures do no converge rapidly o a limi as he lifeime of opions increases. I is difficul o esimae he limi for he ypical value d = 0.4. Implied volailiies as funcions of exercise prices have similar shapes for shor and long memory specificaions. The differences in hese shapes are minor in comparison o he differences in he erm srucure shapes. I is common for he shor and long memory implied volailiies o differ by more han 1% for opions on he S & P 100 index, regardless of he opion lifeime and he exercise price; if he shor memory implied is a is average level of 14% hen he long memory implied is ofen