ON THE RATE OF CONVERGENCE

Transcription

1 ON THE RATE OF CONVERGENCE OF BINOMIAL GREEKS SAN-LIN CHUNG WEIFENG HUNG HAN-HSING LEE* PAI-TA SHIH This study ivestigates the covergece patters ad the rates of covergece of biomial Greeks for the CRR model ad several smooth price covergece models i the literature, icludig the biomial Black Scholes (BBS) model of Broadie M ad Detemple J (1996), the flexible biomial model (FB) of Tia YS (1999), the smoothed payoff (SPF) approach of Hesto S ad Zhou G (000), the GCRR-XPC models of Chug SL ad Shih PT (007), the modified FB-XPC model, ad the modified GCRR-FT model. We prove that the rate of covergece of the CRR model for computig deltas ad gammas is of order O(1/), with a quadratic error term relatig to the positio of the fial odes aroud the strike price. Moreover, most smooth price covergece models geerate deltas ad gammas with mootoic ad smooth covergece with order O(1/). Thus, oe ca apply a extrapolatio formula to ehace their accuracy. The umerical results show that placig the strike price at the ceter of the tree seems to We thak the commets of Keg-Yu Ho, Yaw-Huei Wag, Chug-Yig Yeh, ad the editor Bob Webb. We are particularly grateful for the suggestios ad commets of a aoymous referee. The fiacial support of Natioal Sciece Coucil of Taiwa ad the research assistace of Wei-Che Tsai are ackowledged. *Correspodece author, Graduate Istitute of Fiace, Natioal Chiao Tug Uiversity, No Uiversity Road, Hsichu 300, Taiwa. Tel: #57076, Fax: , hhlee@mail.ctu.edu.tw Received October 009; Accepted July 010 Sa-Li Chug ad Pai-Ta Shih are at the Departmet of Fiace, Natioal Taiwa Uiversity, Taipei, Taiwa. Weifeg Hug is at the Departmet of Fiace, Feg-Chia Uiversity, Taichug, Taiwa. Ha-Hsig Lee is at the Graduate Istitute of Fiace, Natioal Chiao Tug Uiversity, Hsichu, Taiwa. The, Vol. 31, No. 6, (011) 010 Wiley Periodicals, Ic. Published olie September 7, 010 i Wiley Olie Library (wileyolielibrary.com)..0484

2 O the Rate of Covergece of Biomial Greeks 563 INTRODUCTION ehace the accuracy substatially. Amog all the biomial models cosidered i this study, the FB-XPC ad the GCRR-XPC model with a two-poit extrapolatio are the most efficiet methods to compute Greeks. 010 Wiley Periodicals, Ic. Jrl Fut Mark 31:56 597, 011 Biomial methods, developed by Cox, Ross, ad Rubistei (1979, CRR thereafter), are well kow for their flexibility ad efficiecy i calculatig optio prices. Oe stream of the literature modifies the lattice or tree type to improve the accuracy ad efficiecy for computig optio prices. The pricig errors i the biomial models are maily due to distributio error ad oliearity error (see Figlewski & Gao, 1999, for thorough discussios). Withi the literature, there are may proposed solutios that reduce the distributio error ad/or oliearity error. For example, Broadie ad Detemple (1996) ad Hesto ad Zhou (000) modified biomial models by replacig the biomial prices oe period prior to the ed of the tree by the Black Scholes values, or by smoothig payoff fuctios at maturity, ad the computig the rest of biomial prices as usual. The other improved lattice approaches iclude Omberg (1988), Leise ad Reimer (1996), Figlewski ad Gao (1999), Tia (1999), ad Chug ad Shih (007). 1 I this study, we focus o recet biomial models whose biomial optio prices coverge to the true value mootoically ad smoothly. I other words, the pricig errors of these biomial optio prices are of the same sig ad decrease at a kow rate as the umber of time steps () icreases. Thus, their accuracy for pricig optios ca be ehaced usig the stadard Richardso extrapolatio techique. Specifically, the biomial models ivestigated i this study iclude the biomial Black Scholes (BBS) model of Broadie ad Detemple (1996), the flexible biomial model (FB) of Tia (1999), the smoothed payoff (SPF) approach of Hesto ad Zhou (000), the GCRR-XPC models of Chug ad Shih (007), the modified FB-XPC model, ad the modified GCRR-FT model. 3 1 Omberg (1988) developed a family of efficiet multiomial models by applyig the highly efficiet Gauss- Hermite quadrature to the itegratio problem (e.g. N(d 1 ) i the Black Scholes formula) preseted i the optio pricig formulae. Leise ad Reimer (1996) modified the sizes of up- ad dow-movemets by applyig various ormal approximatios (e.g. the Camp Paulso iversio formula) to the biomial distributio derived i the mathematical literature. Figlewski ad Gao (1999) proposed the so-called adaptive mesh method, which sharply reduces oliearity error by addig oe or more small sectios of fie high-resolutio lattice oto a tree with coarser time ad price steps. Tia (1999) ad Chug ad Shih (007) added a stretch parameter (l) ito the CRR model to fie-tue the lattice structure so as to efficietly price optios. Biomial models with the smooth ad mootoic covergece property are the most accurate oes i the recet literature because their accuracy ca be substatially improved by applyig the Richardso extrapolatio techique (Chag, Chug, & Stapleto, 007). 3 These models will be reviewed i Sectio. We thak a aoymous referee for suggestig the FB-XPC model i which the lattice is set up i a way that the strike price is at the ceter of the fial odes.

3 564 Chug et al. Although the above models have bee widely applied to price optios, their covergece patters ad rates of covergece for calculatig hedge ratios are ot kow. 4 Actually, the asymptotic property of biomial Greeks as icreases is ot well studied eve for the stadard CRR model. To fill the gap i the literature, we apply the exteded tree method proposed by Pelsser ad Vorst (1994) to calculate Greeks uder the CRR model ad these smooth price covergece models. 5 We first prove that by usig the exteded tree method, the rate of covergece of the CRR model for computig delta ad gamma is of order O(1/), with a quadratic error term relatig to the positio of the fial odes aroud the strike price (see Theorem 1 of this study for details). We the show that the rates of covergece of the biomial Greeks uder these six smooth price covergece models are also of the order O(1/). Moreover, our umerical results idicate that most smooth price covergece models ca also geerate biomial deltas ad gammas with mootoic ad smooth covergece. Thus, oe ca apply the extrapolatio formula to ehace the accuracy of these hedge ratios. Amog all the biomial models cosidered i this study, the FB-XPC ad the GCRR-XPC model are the most efficiet methods for the calculatio of deltas ad gammas for Europea ad America optios whe a two-poit extrapolatio formula is used. The results suggest that placig the strike price at the ceter of the tree ca ehace the accuracy substatially. The rest of the study is orgaized as follows. Sectio briefly reviews the biomial models cosidered i this study. The rates of covergece of applyig the exteded tree method to the CRR model, the SPF model, ad the GCRR- XPC model are proved i Sectio 3, alog with a discussio of the extrapolatio formula. Sectio 4 presets the umerical results of various biomial models for the evaluatios of deltas ad gammas. Sectio 5 cocludes the study. REVIEWS OF THE BINOMIAL OPTION PRICING MODELS We assume that the Black Scholes ecoomy holds ad thus optios ca be valued as if the ivestors are risk-eutral. I other words, the optios are priced 4 The oly exceptio is Chug ad Shackleto (00) who show the covergece patters of the BBS model for computig hedge ratios. 5 Numerical differetiatio formulae (e.g. [C(S S) C (S S)] S, where C(x) is the biomial optio price whe the uderlyig asset price is x) ca also be applied to these biomial models for the calculatios of Greeks. However, by usig umerical differetiatio formulae, it eeds to costruct two biomial trees to calculate delta ad to costruct three biomial trees to calculate gamma. Thus, it takes more time to calculate Greeks. Besides, the umerical results show that the coverget patters of biomial Greeks usig umerical differetiatio formulae are ot smooth ad mootoic. Thus, we do ot apply umerical differetiatio formulae i this study.

4 O the Rate of Covergece of Biomial Greeks 565 uder the risk-eutral measure where stock price follows a geometric Browia motio give by ds S (r q) dt s dz where S is the stock price, r is the risk-free rate, q is the divided yield, ad s is the istataeous volatility of S. The biomial optio pricig model was first developed by Cox et al. (1979) ad Redlema ad Bartter (1979). Cosider the pricig of a optio maturig at time T. I a -period biomial model, the time to maturity [0, T] is partitioed ito equal time steps t T/. If the stock price is S i this period, the it is assumed to jump either upward to us with probability p or dowward to ds with probability 1 p i the ext period, where 0 d e (r q) t u ad 0 p 1. The biomial model is completely determied by the jump sizes u ad d ad the risk-eutral probability p. I the traditioal CRR model, the followig three coditios are utilized to determie u, d, ad p. psu (1 p)sd Se (r q) t, (1) pu (1 p)d pu (1 p)d s t, () ud 1. The first two coditios are used to match the mea ad variace of the stock price i the ext period, ad the third coditio is imposed arbitrarily by CRR. With these three coditios, oe ca easily determie the biomial parameters as follows: u e s t, d e s t, ad p (e (r q) t d) (u d). Accordig to the Cetral Limit Theorem, 6 the discrete distributio of the asset price uder the CRR model will coverge to its cotiuous-time limit (i.e., Black Scholes model). I other words, as t S 0, the price distributio of the CRR model coverges to a logormal distributio, l S T S d N al S 0 ar q s b T, stb. (3) As a result, the optio prices calculated by the biomial model will also coverge to the Black Scholes price. 6 The followig coditio should be satisfied to apply the Cetral Limit Theorem: p ƒ u e (r q) t ƒ 3 (1 p) ƒ d e (r q) t 3 ƒ as t T S 0. p(u e (r q) t ) (1 p)(d e (r q) t ) 1.5 S 0

5 566 Chug et al. The Biomial Models with Mootoic ad Smooth Covergece Property Although the CRR model is a well kow ad widely used model for valuig optios, the optio prices calculated from it usually coverge to the Black Scholes price i a wavy erratic fashio. I order to ehace the accuracy of the biomial optio prices, several articles i the recet literature have developed various ways to overcome the wavy erratic problem embedded i the CRR model. Six importat biomial models with mootoic ad smooth covergece property are discussed ad applied i this study. Mootoic covergece is desirable because more time steps do guaratee more accurate prices. Moreover, smooth covergece is also advatageous because the stadard Richardso extrapolatio ca be used to ehace the accuracy. The first model is the Biomial Black ad Scholes (BBS) model proposed by Broadie ad Detemple (1996). The BBS model is idetical to the CRR model except that at oe time step before optio maturity the Black Scholes formula replaces the usual cotiuatio value. Broadie ad Detemple (1996) showed that the optio prices obtaied from the BBS model coverge to the Black Scholes formula smoothly ad mootoically. Thus, they suggested usig the Richardso extrapolatio to ehace the accuracy. The secod model is the flexible biomial (FB) model of Tia (1999) where a with a tilt parameter (l) is itroduced to alter the shape ad spa of the biomial tree. Specifically, the parameters of the FB model are as follows: u e s t ls t, d e s t ls t, p e(r q) t d. (4) u d With a positive tilt parameter, l 0, the up movemet is larger tha the correspodig up move i a stadard CRR tree. Cosequetly, the cetral odes depict a upward slopig lie. The resultig tree spa is thus shifted upward. The exact opposite is true for biomial trees with a egative tilt parameter, l 0. Tia (1999) first proved that his FB model coverges to its cotiuoustime couterparts (i.e., the Black Scholes model) for ay value of the tilt parameter. 7 He the suggested the way to select a particular tilt parameter that ehaces the rate of covergece of biomial prices to their cotiuous-time limit. This is doe by selectig a tilt parameter such that a ode i the tree coicides exactly with the strike price at the maturity of the optio. 8 Thus, the formula for determiig the tilt parameter is give by: l l(x S) (j 0 )s t s, t 7 However, the tilt parameter must satisfy the followig iequality i order to have oegative probability : ƒ l (r q) s ƒ 1 s t. 8 A similar idea has bee applied by Leise (1998) i his SMO model. (5)

6 O the Rate of Covergece of Biomial Greeks 567 where j 0 [l(x S) l(d 0 )/l(u 0 d 0 )], [.] deotes the closest iteger to its argumet, X is the strike price, u, ad d 0 e s t 0 e s t. Usig the FB model with the above tilt parameter, Tia (1999) showed that mootoic ad smooth covergece is possible for pricig stadard Europea ad America optios, ad extrapolatio methods are used to ehace the accuracy. Whe biomial models are applied to price Europea ad America optios, Leise ad Reimer (1996) ad Chug ad Shih (007) foud that placig the strike price at the ceter of the tree ca improve the efficiecy substatially. Thus, we also adjust the FB model i a way that the strike price is placed at the ceter of the fial odes. This model is called the FB-XPC model i this study. Specifically, the parameters of the FB-XPC model are chose as follows: 9 u e s t ls t, d e s t ls t, p e(r q) t d u d ad l l(x S) s T The fourth model is the smoothed payoff (SPF) approach of Hesto ad Zhou (000). They first showed that the accuracy or rate of covergece of the biomial model depeds crucially o the smoothess of the payoff fuctio. Hesto ad Zhou (000) the developed a approach to smooth the payoff fuctio. Ituitively, if the payoff fuctio at sigular poits ca be smoothed, the biomial recursio might be more accurate. They defied the smoothed payoff fuctio G(x) as follows: G(x) 1 x x g(x y) dy, x where g(x) is the payoff fuctio. This is a rectagular smoothig of g(x). The smoothed fuctio, G(x), ca be easily computed aalytically for most payoff fuctios used i practice. For example, the smoothed payoff fuctio of a Europea put optio with a strike price X ca be derived as follows: 0, l S T l X s t e s t e s t G(S T ) µ X S T, s t l S T l X s t X(s t l(s T X)) S T e s t X, s t s t l S T l X s t (7) (6) 9 It is straightforward to verify that the chose parameters satisfy Su / d / X ad thus the strike price is ideed placed at the ceter of the fial odes.

7 568 Chug et al. Applyig the smoothed payoff fuctio G(x), istead of g(x), to the biomial model yields a rather surprisig ad iterestig result. The biomial optio prices of the SPF approach coverge at the O(1/) rate to its cotiuous-time limit ad this covergece is smooth ad mootoic as the umber of time steps icreases. The fifth model explored i this study is the geeralized CRR (GCRR) model proposed by Chug ad Shih (007). I the GCRR model, the jump sizes ad probability of goig up are as follows: u e ls t, d e 1 l s t, ad p e(r q) t d (8) u d where l 0 is a stretch parameter that determies the shape (or spaig) of the biomial tree. The CRR model is obviously a special case of our GCRR model with l 1. Whe t S 0, the GCRR biomial optio prices also coverge to the Black Scholes formulae for plai vailla Europea optios. Geerally speakig, the rate of covergece of the GCRR model for pricig optios is of order O(1 ) whe l This rate of covergece is iferior to other biomial models such as the BBS model ad the FB model. Thus, we cosider two approaches to adjust the GCRR model. First, we adopt the idea of Tia (1999) by selectig a particular parameter l such that a ode i the tree coicides exactly with the strike price at the maturity of the optio. Let the iitial value of l be oe, which represets the CRR model. We ca determie the ode closest to the strike price X, (, j 0 ), by solvig the followig equatio: S(e s t ) h (e s t ) h X j 0 [h] where [# ] deotes the closest iteger to its argumet. To esure that ode (, j0 ) coicides exactly with the strike price X, a ew l is selected such that: S(e ls t ) j 0 ae 1 l s t b j 0 X Thus, the formula for determiig l is give by: l l(x S) (l(x S)) 4j 0 ( j 0 )s t. (9) j 0 s t This model is essetially a GCRR model with a fie-tued parameter l closest to oe ad thus is called GCRR-FT model i this study Please refer to Theorem of Chug ad Shih (007). 11 I other words, i GCRR-FT model, the parameter l is chose to be closest to oe such that a ode i the tree coicides exactly with the strike price at the maturity of the optio. It is straight forward to show that l approaches oe as approaches ifiity.

8 O the Rate of Covergece of Biomial Greeks 569 The secod adjustmet of the GCRR model, suggested by Chug ad Shih (007), is the GCRR-XPC model, where the lattice is set up i a way that the strike price is at the ceter of the fial odes, to price optios. I particular, the biomial optio prices of the GCRR-XPC model ot oly coverge at a high order (of order O(1 )) but also coverge smoothly ad mootoically to the Black Scholes formula. I summary, we cosider six smooth covergece biomial models, icludig BBS, SPF, FB, FB-XPC, GCRR-FT, ad GCRR-XPC. Calculatig Hedge Ratios with Biomial Models Greeks (or hedge ratios) are the sesitivity of the optio price with respect to the chage of the uderlyig risk factors. Deote the uderlyig asset price ad the optio price at time i t ad state j (i.e. the umber of up-movemets from time 0) as S i,j ad C i,j, respectively. Hull (006) suggested that the estimates of delta ad gamma ca be obtaied as follows: ˆ 0C 0S (C 1,1 C 1,0 ) (S 1,1 S 1,0 ) ˆ 0 C 0S (C, C,1) (S, S,1) (C,1 C,0) (S,1 S,0). 0.5(S, S,0 ) (10a) (11a) It should be oted that the delta (gamma) estimate obtaied from the Hull (006) method actually represets the delta value at time t( t) because it is calculated from optio prices at time t( t). To overcome this problem, this study applies the exteded tree method of Pelsser ad Vorst (1994) to calculate delta ad gamma. I the exteded biomial tree, we build the biomial tree startig from two time steps prior to time 0. Figure 1 illustrates the exteded biomial tree. 1 The estimates of delta ad gamma i the exteded tree method ca ow be obtaied as the followig: ˆ ˆ (C 0,1 C 0, 1 ) (S 0,1 S 0, 1 ) (C 0,1 C 0,0 ) (S 0,1 S 0,0 ) (C 0,0 C 0, 1 ) (S 0,0 S 0, 1 ). 0.5(S 0,1 S 0, 1 ) (10b) (11b) 1 Note that to esure that the stock price at ode (0, 0) equals the iitial stock price S 0, i.e. S 0,0 S 0, i our study we choose S, 1 S 0 (ud) as show i Figure 1. Therefore, for CRR model, BBS model, ad SPF model, S, 1 S 0 because ud 1. However, for GCRR-FT model, GCRR-XPC model, FB model, ad FB-XPC model, S, 1 does ot equal S 0 sice ud does ot equal oe.

9 570 Chug et al. S 0,1 S,3 S 1,0 S 1, S, 1 S 0,0 S S, S 0 0 ud S 1, 1 S 1,1 S,1 S 0, 1 t t 0 t t C 0,1 C, C 1,0 C 1, C, 1 C 0,0 C, C 1, 1 C 1,1 C,1 C 0, 1 t t 0 t t FIGURE 1 The exteded biomial tree. THE RATE OF CONVERGENCE OF THE BINOMIAL GREEKS AND THE EXTRAPOLATION FORMULA Whe the covergece patter of the biomial Greeks is mootoic ad smooth, oe ca apply the extrapolatio techique to ehace the accuracy. However, the extrapolatio formula depeds o the rate of covergece of the biomial model. Thus, we will first discuss the rate of covergece of the CRR model ad the six biomial models with mootoic ad smooth covergece property whe they are used to calculate hedge ratios. 13 The Rate of Covergece of the Biomial Deltas I the followig, we formally derive the rate of covergece of the CRR, the SPF, ad the GCRR-XPC biomial models for computig deltas whe the umber 13 Note that the rate of covergece of the cosidered six biomial models for pricig optios is of order O(1 ). Please see Hesto ad Zhou (000) ad Chug ad Shih (007) for the proofs of the SPF model ad the GCRR-XPC model, respectively. However, to the best of our kowledge, the rate of covergece of these six biomial models for computig deltas ad gammas is ukow i the literature.

10 O the Rate of Covergece of Biomial Greeks 571 S x 1 Su m 1 d (m 1) () l (S x 1 /X) l (S x 1 /S x ) X S x Su m d m FIGURE Defiitio of e(). of time steps icreases. 14 Note that the proof is based o the applicatio usig the exteded tree method of Pelsser ad Vorst (1994). 15 To show the covergece rate for the CRR model, we eed to defie a variable, e(), to quatify the positio of the fial odes aroud the strike price. Let S x Su m d m be the closest ode at maturity date ad smaller tha the strike price, i.e., m is the greatest iteger which satisfies Su m d m X. The variable e() is defied i equatio (1) as the logarithmic distace betwee the strike price ad the ode above it, ormalized with respect to the lattice size l(u d) (see Figure ), i.e., e() l(s x 1 X) l(s x 1 S x ) l(s x 1 X), l(u d) (1) where S x 1 Su m 1 d (m 1) is the price of the fial ode just above S x. Havig defied the positioig variable e(), we first preset the rate of covergece of the CRR model for calculatig deltas i Theorem 1 ad the prove this theorem i Appedix A1. Theorem 1. Let ˆ,CRR be the -period CRR biomial delta of a stadard Europea put optio usig the exteded tree method ad BS is the true delta. Therefore, ˆ,CRR BS e d 1 f(e()) O(1 ) p (13) 14 Sice the BBS method ca be regarded as oe way to smooth the payoff fuctio at maturity date, its rate of covergece for computig deltas is the same as that of the SPF method. Moreover, the FB model of Tia (1999) has the same covergece rate as that of the CRR model because both models differ oly by a amout of ls t, which is egligible i compariso to s t whe t is small. For simplicity, the proofs for the BBS model, the GCRR-FT model, the FB model, ad the FB-XPC model are ot show i this study. 15 Moreover, the rate of covergece of biomial gammas is of the same order as that of biomial deltas. The derivatios for gammas are omitted here ad ca be obtaied from the authors upo request.

11 57 Chug et al error X 45 X FIGURE 3 A graph of error agaist e() for delta estimates of Europea puts uder the CRR model. The parameters used i this figure are: S 40, s 0., r 0.06, T 0.5, ad q 0. () where d f(e()) d ( e() e() 1 l (S X ) (r q s ) T, ), ad st d d 1 st. Theorem 1 idicates that, although usig the CRR model with a exteded tree to calculate delta has the covergece rate of O(1 ), the covergece patter may be oscillatory because the relative positio of the strike price betwee two most adjacet odes (i.e. e()) chages as the umber of time steps icreases. I additio, if d 0 (d 0), the umerical error of delta estimates achieves the highest (lowest) value whe e() 0.5. Figure 3 plots the error agaist e() for the delta estimates of Europea puts with strike prices equalig 35(d 0) ad 45(d 0). The result is as expected that the maximum (miimum) errors occur whe e() 0.5 whe d 0 (d 0). Our fidig echoes the umerical observatio of Widdicks, Adricopoulos, Newto, ad Duck (WAND, 00) who foud that the pricig error of the CRR model is related to the positioig of the fial odes aroud the strike price ad the error reaches its maximum whe e() 0.5 (see their Figure 5 for details). 16 This study cotributes to the literature by provig that the delta error of the CRR model is also related to the positioig of the fial 16 WAND (00) defied a positioig variable ( ) as the distace betwee the strike price ad the ode at or above it, ormalized with respect to the lattice size, i.e. () (S x 1 X) (S x 1 S x ). It is ot difficult to verify that e() ad () is close to each other, especially whe is large.

12 O the Rate of Covergece of Biomial Greeks 573 odes aroud the strike price. Thus, oe ca follow WAND (00) to select to give a costat e() i the CRR model, to produce mootoic ad smooth covergece of delta estimates, ad to eradicate the error through extrapolatio. I additio, the rates of covergece of applyig exteded tree models to compute delta estimates uder SPF ad GCRR-XPC models are show i Theorems ad 3, respectively. The detailed proofs of these theorems are give i Appedix A ad A3. Theorem. Let ˆ,SPF be the -period SPF biomial delta of a stadard Europea put optio usig the exteded tree method ad BS is the true delta. The, ˆ,SPF BS O(1 ). (14) Theorem 3. Let ˆ,GCRR XPC be the -period GCRR-XPC biomial delta of a stadard Europea put optio usig the exteded tree method ad BS is the true delta. Thus, ˆ,GCRR XPC BS O(1 ). (15) The Two-Poit Extrapolatio Formula It is foud from Theorems ad 3 that the rates of covergece of delta estimates uder SPF ad GCRR-XPC models are of order O(1/). Furthermore, if their covergece patters are mootoic ad smooth, oe ca apply a extrapolatio formula to ehace the accuracy of the delta estimates. I what follows we derive a two-poit extrapolatio formula whe the covergece order is O(1 ). Let e() be the error of the -step biomial model for computig delta, i.e.: e() ˆ BS (16) where ˆ is the -step biomial delta ad BS is the Black Scholes delta. Defie the error ratio as: r() e( ) e(). (17) The error ratio is a measure of the improvemets i accuracy as the umber of time steps doubles. Give the fact that all the biomial models cosidered i this study have the covergece order O(1 ), the error ratios uder these models coverge to as the umber of time steps icreases. Thus, the Black Scholes delta ca be approximated usig a two-poit extrapolatio formula as follows: ~ BS() ˆ ˆ. (18)

13 574 Chug et al. It is ot difficult to show that the error after applyig the extrapolatio ca be writte as: e ~ () ~ BS() BS ( r())e(). (19) Thus, the error with extrapolatio critically depeds o the differece betwee ad the error ratio. If the error ratio r() is withi the rage of (1, 3), the absolute error with extrapolatio (i.e., ƒ e ~ () ƒ ) is smaller tha the absolute error without extrapolatio (i.e., e() ). Otherwise, applyig the extrapolatio method to the biomial deltas will icrease rather tha decrease the errors. NUMERICAL RESULTS AND ANALYSIS I the followig umerical aalysis, we will ivestigate the covergece patter, the rate of covergece, ad the umerical efficiecy of various biomial models for computig deltas ad gammas. Covergece Patter ad Covergece Rate of Various Biomial Models We first illustrate the covergece patters of various biomial models cosidered i this study. Figure 4 shows the covergece patters of the cosidered seve biomial models, icludig the CRR model, the BBS model, the FB model, the FB-XPC model, the SPF method, the GCRR-FT model, ad the GCRR-XPC model, for computig delta ad gamma of a Europea put optio. The parameters used i Figure 4 are as the followig: the asset price is 40, the strike price is 41, the volatility is 0.3, the maturity of the optio is oe moth, the risk-free rate is 0.06, ad the divided yield is 0. The error is defied as the differece betwee the biomial estimate ad the closed-form solutio uder the Black Scholes model. I Figure 4 the delta errors are plotted agaist the umber of eve time steps ragig from 10 to Similarly, Figure 5 shows the covergece patters of the cosidered seve models for computig delta ad gamma of a America put optio with the same parameters as those of Figure 4. The true delta ad gamma of the America put is calculated usig the exteded tree of the GCRR-XPC model with a two-poit extrapolatio ( 10,000). It is apparet from Figures 4 ad 5 that the Europea ad America delta ad gamma estimates uder the CRR model coverge smoothly but ot 17 I order to place the strike price exactly i the ceter ode of the fial odes for the FB-XPC model ad the GCRR-XPC model, the umber of time steps is chose to be eve umbers oly.

14 O the Rate of Covergece of Biomial Greeks Pael A: Delta estimates of a Europea put 0.01 Europea delta error GCRR-FT GCRR-XPC FB FB-XPC CRR BBS SPF Number of time steps Pael B: Gamma estimates of a Europea put Europea gamma error GCRR-FT GCRR-XPC FB FB-XPC CRR BBS SPF Number of time steps FIGURE 4 Covergece patters of delta ad gamma estimates of a europea put. This figure shows the covergece patters of the GCRR-FT model, the GCRR-XPC model, the FB model, the FB-XPC model, the CRR model, the BBS model, ad the SPF model for computig Europea deltas ad gammas. The parameters used i this figure are: S 40, X 41, s 0.3, r 0.06, T 1 1, ad q 0.

15 576 Chug et al Pael A: Delta estimates of a America put 0.01 America delta error GCRR-FT GCRR-XPC FB FB-XPC CRR BBS SPF Number of time steps Pael B: Gamma estimates of a America put America gamma error GCRR-FT GCRR-XPC FB FB-XPC CRR BBS SPF Number of time steps FIGURE 5 Covergece patters of delta ad gamma estimates of a America put. This figure shows the covergece patters of the GCRR-FT model, the GCRR-XPC model, the FB model, the FB-XPC model, the CRR model, the BBS model, ad the SPF model for computig America deltas ad gammas. The parameters used i this figure are: S 40, X 41, s 0.3, r 0.06, T 1 1, ad q 0.

16 O the Rate of Covergece of Biomial Greeks 577 mootoically to the accurate values. 18 O the cotrary, the delta ad gamma estimates uder the other six models geerally coverge smoothly ad mootoically (except for the GCRR-FT model ad the FB model) 19 to the accurate values. Therefore, various extrapolatio techiques are applicable to these models to ehace the accuracy. We also aalyze the covergece patters of delta ad gamma estimates of Europea ad America put optios across differet degrees of moeyess. For deep-i-the-moey America put optios, the probability of early exercise is high ad itrisic value (time value) accouts for a much larger (smaller) fractio of the price of the America put. As itrisic value is model idepedet, the cosidered seve models provide very similar delta ad gamma estimates of the America put optios. O the cotrary, for deep-out-of-the-moey put optios, the probability of early exercise is low ad the early exercise premium o the America put is small. The America put is thus priced more closely to the correspodig Europea put, ad therefore the covergece patters of Europea ad America put optios are quite similar. 0 For the sake of brevity, these results are ot reported here but are available upo request. Next we umerically ivestigate the covergece rates of various biomial models for computig Europea deltas ad gammas. The error ratio defied i Equatio (17) is reported to measure the improvemets i accuracy as the umber of time steps doubles. Tables I ad II report the error ratios of the BBS model, the FB model, the FB-XPC model, the SPF method, the GCRR-FT model, ad the GCRR-XPC model for computig delta ad gamma of a Europea put optio, respectively. It is clear from Tables I ad II that the errors of the Europea delta ad gamma estimates uder the BBS model, the FB model, the FB-XPC model, the SPF method, the GCRR-FT model, ad the GCRR-XPC model are almost exactly halved whe the umber of time steps doubles, i.e. the error ratio almost equals two. Therefore, the rates of covergece of these models are clearly of order O(1/), which are cosistet with our theoretical proofs. O the cotrary, while the covergece rates of the GCRR-FT ad the FB model are of order O(1/) for the computatio of deltas, their covergece rates for calculatig gamma are ot stable as the umber of time steps icreases Because we plot the figures oly usig eve umber of time steps, the covergece patter of the CRR model geerally follows a smooth way. However, whe both odd ad eve umbers of time steps are used, the CRR model coverges i a wavy, erratic fashio. 19 Nevertheless, whe the umber of time steps icreases, the covergece patters of the GCRR-FT model ad the FB model becomes more smooth ad mootoica. 0 We thak the referee for providig the isightful explaatios of our results for deep-i-the-moey ad deep-out-of-the-moey put optios. 1 I the ureported umerical results, we fid that the errors of the America delta ad gamma estimates uder the BBS model, the FB model, the FB-XPC model, the SPF method, the GCRR model, ad the GCRR-XPC model are also almost exactly halved whe the umber of time steps doubles.

17 578 Chug et al. TABLE I Delta Estimates ad Error Ratios for a Europea Put Optio GCRR-FT GCRR-XPC Time Steps Delta Error Error Ratio Delta Error Error Ratio , , , FB FB-XPC , , , BBS SPF , , , True delta Notes. This table reports Europea delta estimates, delta errors, ad error ratios from the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models as described i the text by usig exteded method. The parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0., the maturity of the optio is six moths, the risk-free rate is 0.06, ad the divided yield is 0. The umber of time steps starts at 0 ad doubles each time subsequetly. Moreover, Tables I ad II idicate that the error ratios of the FB-XPC model ad the GCRR-XPC model for calculatig deltas ad gammas also coverge mootoically to a costat as S q. As a result, repeated extrapolatio techiques ca be used to further reduce the delta ad gamma errors of the FB-XPC ad the GCRR-XPC model. Withi all the biomial models cosidered i this study, the FB-XPC model ad the GCRR-XPC model are the oly models that coverge so smoothly that oe ca apply repeated extrapolatio techiques. Please see Chag et al. (007) for the repeated extrapolatio formulae ad the discussios therei.

18 O the Rate of Covergece of Biomial Greeks 579 TABLE II Gamma Estimates ad Error Ratios for a Europea Put Optio GCRR-FT GCRR-XPC Time Steps Gamma Error Error Ratio Gamma Error Error Ratio , , , FB FB-XPC , , , BBS SPF , , , True gamma Notes. This table reports Europea gamma estimates, gamma errors, ad error ratios of the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models as described i the text by usig the exteded tree method. The parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0., the maturity of the optio is six moths, the risk-free rate is 0.06, ad the divided yield is 0. The umber of time steps starts at 0 ad doubles each time subsequetly. Numerical Efficiecy of Various Biomial Models for the Computatios of Deltas ad Gammas After cofirmig the covergece patter ad covergece rate of various biomial models, we ext ivestigate the umerical efficiecy of these biomial models for computig deltas ad gammas. To have a comprehesive aalysis, we choose a large set of optios (43 optios). The 43 parameter sets are

19 580 Chug et al. draw from the combiatios of X {35, 40, 45}, T {1/1, 4/1, 7/1}, r {3%, 5%, 7%}, q {%, 5%, 8%}, s {0., 0.3, 0.4}, ad S 40. We report the results for differet umbers of time steps {0, 40, 60, 80, 100, 500, 1000}. The accuracy measure used i this study is the root mea squared (RMS) error, defied as: m 1 RMS error B m a e i i 1 (0) where e i ĉ i c i is the error of the biomial estimate for the ith optio, c i is the true value, ad ĉ i is the estimated value from the biomial model. For the Europea optios, the true values are estimated from the Black Scholes formula ad for the America optios, the true values are estimated usig the exteded tree of the GCRR-XPC model with a two-poit extrapolatio ( 10,000). To clarify the role played by Richardso extrapolatio, we first compare the umerical efficiecy of each model o a raw basis without ay extrapolatio procedure. Paels A of Tables III ad IV report the RMS errors of Europea delta estimates ad America delta estimates respectively for various biomial models. The results idicate that the accuracy of computig deltas is similar for all biomial models cosidered i this study. Before the extrapolatio formula is applied to ehace the accuracy, the FB-XPC method performs slightly better tha the other methods from Paels A of Tables III ad IV. Similarly, Paels A of Tables V ad VI also suggest that the RMS errors of gamma estimates for various biomial models are of similar magitude. The GCRR-FT method ad the FB method perform similarly ad are slightly better tha the other methods for computig Europea ad America gamma estimates before the extrapolatio techique is used. Moreover, Paels A of Tables III ad V also cofirm that for Europea put optios, the delta ad gamma estimates of all biomial methods compared i this study coverge to the Black Scholes values at the 1/ rate. For example, whe icreases from 100 to 1,000, the RMS errors become approximately oe-teth for each method before the extrapolatio techique is applied. We ext compare the umerical efficiecy of six smooth covergece models whe a two-poit extrapolatio formula is applied. 3 Accordig to Tables III ad IV, the Europea ad America delta estimates of each method become far more accurate after the extrapolatio techique is utilized. For example, from Table III, whe 1,000, the RMS error of the FB method for calculatig Europea deltas decreases from 1.00E 4 to.94e 6 (i.e. the accuracy improves 34 times) after the extrapolatio is applied. I additio, from Table IV, whe 1,000, the RMS error of the FB method for calculatig 3 We do ot apply a two-poit extrapolatio method to the CRR model sice its covergece is oscillatory.

20 O the Rate of Covergece of Biomial Greeks 581 TABLE III RMS Errors of Europea Delta Estimates for Various Biomial Models Time steps ,000 Pael A: without extrapolatio CRR 7.47E E 03.9E E E E E 04 (0.1) (0.19) (0.7) (0.36) (0.47) (5.58) (1.00) GCRR-FT 5.01E 03.51E E E E 03.0E E 04 (0.1) (0.19) (0.7) (0.36) (0.47) (5.59) (1.01) GCRR-XPC 5.09E 03.55E E E E 03.05E E 04 (0.14) (0.1) (0.30) (0.40) (0.51) (5.75) (1.40) FB 4.95E 03.50E E E E 03.0E E 04 (0.1) (0.19) (0.7) (0.36) (0.47) (5.58) (1.00) FB-XPC 3.14E E E E E E E 05 (0.14) (0.1) (0.30) (0.40) (0.51) (5.74) (1.39) BBS 7.31E E 03.51E E E E E 04 (0.56) (1.00) (1.47) (1.93) (.4) (14.66) (37.8) SPF 8.71E E 03.99E 03.5E E E E 04 (0.15) (0.) (0.3) (0.4) (0.54) (5.91) (1.53) Pael B: with extrapolatio GCRR-FT 9.46E E 04.3E E E E 06.98E 06 (0.16) (0.5) (0.36) (0.49) (0.63) (7.5) (6.49) GCRR-XPC 1.55E E E E E E E 08 (0.17) (0.8) (0.39) (0.53) (0.69) (7.47) (6.97) FB 9.31E E 04.43E E E E 06.94E 06 (0.16) (0.5) (0.36) (0.49) (0.63) (7.4) (6.48) FB-XPC.0E E 05.30E E E E E 08 (0.17) (0.8) (0.39) (0.53) (0.69) (7.46) (6.96) BBS 6.17E E E E E E E 07 (0.83) (1.49) (.16) (.84) (3.54) (0.96) (51.74) SPF 9.E 04.41E E E E 05.19E 06 9.E 07 (0.18) (0.9) (0.40) (0.54) (0.70) (7.5) (7.7) Notes. This table reports the RMS errors of Europea delta estimates uder the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models without- ad with-extrapolatio, respectively. The 43 parameter sets are draw from the combiatios of X {35, 40, 45}, T {1/1, 4/1, 7/1}, r {3%, 5%, 7%}, q {%, 5%, 8%}, s {0., 0.3, 0.4}, ad S 40. RMS errors ad computatioal times averaged over 43 differet sets of parameters are reported. The CPU time (i secods) required to value 43 deltas is based o the ruig time o a Petium 4.8-GHz PC ad is give i paretheses. America deltas decreases from 9.47E 05 to 4.35E 06 (i.e. the accuracy improves times) after the extrapolatio is applied. Although FB-XPC is the best method for calculatig Europea deltas without extrapolatio from Pael A of Table III, GCRR-XPC method with extrapolatio improves the most (e.g. whe 1,000, the accuracy improves 1,30 times!). Amog all biomial methods, the GCRR-XPC ad the FB-XPC models with extrapolatio perform similarly ad are the best two methods for computig Europea ad America delta estimates. Tables V ad VI show the RMS errors of gamma estimates of each method for Europea ad America put optios, respectively. From Table V, it is evidet that the FB ad the GCRR-FT methods perform slightly better tha

21 58 Chug et al. TABLE IV RMS Errors of America Delta Estimates for Various Biomial Models Time Steps ,000 Pael A: without extrapolatio CRR 7.8E E 03.E E E E E 04 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.6) GCRR-FT 4.7E 03.37E E E E E E 05 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.63) GCRR-XPC 4.98E 03.49E E E E 04.00E E 04 (0.) (0.36) (0.53) (0.73) (0.96) (11.98) (46.0) FB 4.65E 03.35E E E E E E 05 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.6) FB-XPC.86E E E E E E E 05 (0.) (0.36) (0.53) (0.73) (0.96) (11.97) (46.01) BBS 7.05E E 03.4E E E 03.97E E 04 (0.64) (1.15) (1.70) (.6) (.87) (0.89) (61.90) SPF 8.39E E 03.89E 03.18E E E E 04 (0.3) (0.37) (0.55) (0.75) (0.99) (1.14) (46.15) Pael B: with extrapolatio GCRR-FT 1.13E E 04.16E E E E E 06 (0.7) (0.44) (0.66) (0.93) (1.1) (15.06) (56.5) GCRR-XPC 7.6E 04.68E E E E E 06.06E 06 (0.8) (0.47) (0.69) (0.97) (1.7) (15.8) (56.69) FB 1.1E 03 5.E 04.6E E E E E 06 (0.7) (0.44) (0.66) (0.93) (1.1) (15.05) (56.4) FB-XPC 1.4E E E E E E 06.01E 06 (0.8) (0.47) (0.69) (0.96) (1.7) (15.7) (56.68) BBS 9.17E E E E E E E 06 (0.94) (1.68) (.46) (3.8) (4.1) (8.77) (81.50) SPF 1.15E E E E E E E 06 (0.9) (0.48) (0.70) (0.98) (1.8) (15.33) (57.03) Notes. This table reports the RMS errors of America delta estimates uder the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models without- ad with-extrapolatio, respectively. The 43 parameter sets are draw from the combiatios of X {35, 40, 45}, T {1/1, 4/1, 7/1}, r {3%, 5%, 7%}, q {%, 5%, 8%}, ad s {0., 0.3, 0.4} ad S 40. RMS errors ad computatioal times averaged over 43 differet sets of parameters are reported. The CPU time (i secods) required to value 43 deltas is based o the ruig time o a Petium 4.8-GHz PC ad is give i paretheses. the other methods without ay extrapolatio procedure for computig Europea gammas. However, whe a two-poit extrapolatio is applied, the FB-XPC ad the GCRR-XPC methods perform best. As for computig America gammas, Table VI also idicates that the FB ad the GCRR-FT methods perform slightly better tha the other methods. However, whe a two-poit extrapolatio is applied, GCRR-XPC performs slightly better tha FB-XPC followed by GCRR, FB, BBS, ad SPF for computig America gammas. To facilitate a fair compariso of the umerical efficiecy of various biomial models for computig deltas, we follow Broadie ad Detemple (1996) to coduct the speed accuracy trade-off aalysis. Computatio speed is measured

22 O the Rate of Covergece of Biomial Greeks 583 TABLE V RMS Errors of Europea Gamma Estimates for Various Biomial Models Time steps ,000 Pael A: without extrapolatio CRR 9.90E E 04.99E 04.30E E E 05.0E 05 (0.1) (0.19) (0.7) (0.36) (0.47) (5.58) (1.00) GCRR-FT 7.41E E 04.4E E E 04.93E E 05 (0.1) (0.19) (0.7) (0.36) (0.47) (5.59) (1.01) GCRR-XPC 1.11E E E 04.86E 04.9E E 05.31E 05 (0.14) (0.1) (0.30) (0.40) (0.51) (5.75) (1.40) FB 7.38E E 04.4E E E 04.93E E 05 (0.1) (0.19) (0.7) (0.36) (0.47) (5.58) (1.00) FB-XPC 1.76E E E E E E E 05 (0.14) (0.1) (0.30) (0.40) (0.51) (5.74) (1.39) BBS 1.56E E E E E E E 05 (0.56) (1.00) (1.47) (1.93) (.4) (14.66) (37.8) SPF 1.98E E E E E E E 05 (0.15) (0.) (0.3) (0.4) (0.54) (5.91) (1.53) Pael B: with extrapolatio GCRR-FT 5.6E E E E E E E 06 (0.16) (0.5) (0.36) (0.49) (0.63) (7.5) (6.49) GCRR-XPC 1.30E E E E E 06.63E E 08 (0.17) (0.8) (0.39) (0.53) (0.69) (7.47) (6.97) FB 5.38E E E E E E E 06 (0.16) (0.5) (0.36) (0.49) (0.63) (7.4) (6.48) FB-XPC 8.77E E E E 06.30E E 08.18E 08 (0.17) (0.8) (0.39) (0.53) (0.69) (7.46) (6.96) BBS 1.80E E 05.35E E E E E 07 (0.83) (1.49) (.16) (.84) (3.54) (0.96) (51.74) SPF.93E E E 05.8E E E E 07 (0.18) (0.9) (0.40) (0.54) (0.70) (7.5) (7.7) Notes. This table reports the RMS errors of Europea gamma estimates uder the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models without- ad with-extrapolatio, respectively. The 43 parameter sets are draw from the combiatios of X {35, 40, 45}, T {1/1, 4/1, 7/1}, r {3%, 5%, 7%}, q {%, 5%, 8%}, ad s {0., 0.3, 0.4} ad S 40. RMS errors ad computatioal times averaged over 43 differet sets of parameters are reported. The CPU time (i secods) required to value 43 deltas is based o the ruig time o a Petium 4.8-GHz PC ad is give i paretheses. i optio prices calculated per secod ad is based o the ruig time o a Petium 4.8-GHz PC. The accuracy measure is the RMS error averaged over the 43 optios used i Table III. The overall results are give i Figure 6. Because of the extreme differeces i speed ad accuracy, the results are show o a log log scale. Note that preferred methods are i the upper-left corer. The results of Figure 6 are similar to those of Table III. For example, Figure 6 also suggests that the FB-XPC ad GCRR-XPC models with a twopoit extrapolatio domiate all the other methods for computig Europea deltas. I summary, the results suggest that placig the strike price at the ceter of the tree ca ehace the accuracy. Similar results are foud by

23 584 Chug et al. TABLE VI RMS Errors of America Gamma Estimates for Various Biomial Models Time Steps ,000 Pael A: without extrapolatio CRR 9.60E E E 04.3E E E 05.00E 05 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.6) GCRR-FT 8.0E E 04.47E E E 04.97E E 05 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.63) GCRR-XPC 1.1E E E 04.83E 04.5E E 05.6E 05 (0.) (0.36) (0.53) (0.73) (0.96) (11.98) (46.0) FB 8.00E E 04.47E E E 04.97E E 05 (0.0) (0.34) (0.50) (0.69) (0.9) (11.81) (45.6) FB-XPC.36E E E E E E E 05 (0.) (0.36) (0.53) (0.73) (0.96) (11.97) (46.00) BBS 1.53E E E E 04 3.E E E 05 (0.64) (1.15) (1.70) (.6) (.87) (0.89) (61.90) SPF 1.94E E E E E E E 05 (0.3) (0.37) (0.55) (0.75) (0.99) (1.14) (46.15) Pael B: with extrapolatio GCRR-FT 7.87E E E E E E E 06 (0.7) (0.44) (0.66) (0.93) (1.1) (15.06) (56.5) GCRR-XPC 3.34E 04.36E E E E E 06.88E 06 (0.8) (0.47) (0.69) (0.97) (1.7) (15.8) (56.69) FB 7.98E E E E E E E 06 (0.7) (0.44) (0.66) (0.93) (1.1) (15.05) (56.4) FB-XPC 1.83E E 04.53E E E E E 06 (0.8) (0.47) (0.69) (0.96) (1.7) (15.7) (56.68) BBS 5.34E E E E E E E 06 (0.94) (1.68) (.46) (3.8) (4.1) (8.77) (81.50) SPF 5.51E E E E E E E 06 (0.9) (0.48) (0.70) (0.98) (1.8) (15.33) (57.03) Notes. This table reports the RMS errors of America gamma estimates uder the GCRR-FT, GCRR-XPC, FB, FB-XPC, BBS, ad SPF models without- ad with-extrapolatio, respectively. The 43 parameter sets are draw from the combiatios of X {35, 40, 45}, T {1/1, 4/1, 7/1}, r {3%, 5%, 7%}, q {%, 5%, 8%}, ad s {0., 0.3, 0.4} ad S 40. RMS errors ad computatioal times averaged over 43 differet sets of parameters are reported. The CPU time (i secods) required to value 43 deltas is based o the ruig time o a Petium 4.8-GHz PC ad is give i paretheses. Leise ad Reimer (1996) ad Chug ad Shih (007) whe biomial models are applied to price optios. It is worth otig that all the smooth covergece models cosidered i this study are pretty good models for calculatig optio Greeks. The umerical errors (Tables I ad II) are geerally quite small for most practical applicatios. Thus geerality ad ease of implemetatio may be more importat i choosig the model for a particular applicatio. 4 For example, the FB model ad the GCRR model are flexible to compute prices ad Greeks for barrier optios, please see Tia (1999) ad Chug ad Shih (007), respectively. I terms of 4 We thak the referee for poitig out this issue.