On Justifications for the ad hoc Black-Scholes Method of Option Pricing

Transcription

1 On Jusfcaons for he ad hoc Black-Scholes Mehod of Opon Prcng Jerey Berkowz Deparen of Fnance Unversy of Houson May 23, 2009 Absrac: One of he os wdely used opon valuaon procedures aong praconers s a verson of Black-Scholes n whch pled volales are soohed across srke prces and aures. A growng body of eprcal evdence suggess ha hs ad hoc approach perfors que well. I has prevously been argued ha such a procedure works because aouns o a sophscaed nerpolaon ool. We show ha hs s he case n a foral, asypoc sense. In addon, we conduc soe sulaons whch allow us o exane he porance of he saple sze, he order of he polynoal, and he recalbraon frequency n conrolled sengs. We also apply he ABS approach o daly S&P 100 ndex opons o show ha he procedure ouperfors he Black-Scholes forula n valung acual opon prces ou-of-saple. Acknowledgeens: I a graeful o Na-fu Chen, Peer Chrsoffersen, Jeff Fleng, Krs Jacobs, Phlppe Joron, Praveen Kuar, Bruce Mzrach, Ma Prsker, Suar Turnbull and Rober Whaley who provded any helpful suggesons. Address correspondence o Bauer College of Busness, 334 Melcher Hall, Unversy of Houson, Houson, TX , (713) , jberkowz@uh.edu. Any errors or naccuraces are solely he responsbly of he auhor.

2 On Jusfcaons for he ad hoc Black-Scholes Mehod of Opon Prcng Absrac: One of he os wdely used opon valuaon procedures aong praconers s a verson of Black-Scholes n whch pled volales are soohed across srke prces and aures. A growng body of eprcal evdence suggess ha hs ad hoc approach perfors que well. I has prevously been argued ha such a procedure works because aouns o a sophscaed nerpolaon ool. We show ha hs s he case n a foral, asypoc sense. In addon, we conduc soe sulaons whch allow us o exane he porance of he saple sze, he order of he polynoal, and he recalbraon frequency n conrolled sengs. We also apply he ABS approach o daly S&P 100 ndex opons o show ha he procedure ouperfors he Black-Scholes forula n valung acual opon prces ou-of-saple. 1

3 1. Inroducon One of he os wdely used opon-valuaon echnques aong raders and oher praconers s an ad hoc procedure n whch Black-Scholes (1973) pled volales are soohed across srke prces and aures and hen plugged back no he Black-Scholes forula. Ths ad hoc Black-Scholes approach has becoe soehng of a benchark for evaluang he forecas accuracy of opon prcng odels because of s conssenly pressve eprcal perforance. Duas, Fleng and Whaley (1998), for exaple, fnd ha he ad hoc Black-Scholes (ABS) odel forecass opon prces ou-of-saple ore accuraely han Black- Scholes and he deernsc volaly funcon (DVF) odel of Deran and Kan (1994) and Rubnsen (1994). Brand and Wu (2002) focus on cross-seconal forecas accuracy and show ha he ad hoc approach ouperfors DVF odels n valung FTSE 100 ndex opons. Heson and Nand (2000) show ha he ad hoc approach copares favorably wh a close-for GARCH opon prcng odel. Chrsoffersen and Jacobs (2004) fnd ha, wh daly updang of paraeers, he ABS odel ouperfors Heson s (1993) heorecal odel boh n and ou of saple. A he sae e, several auhors have suggesed an explanaon for why he ad hoc approach perfors so well. Davs (2001) noes ha he soohng of pled volales ay be vewed as a sophscaed nerpolaon ool used o ensure ha an opon s prced conssenly wh he arke prces of oher raded opons. In hs sense, esaon of an pled volaly surface s analogous o fng he observed se of opon prces self. Davs (2001) and Fglewsk (2002) argue ha hs knd of nerpolaon echnque can be parcularly useful for valung opons whch are llqud or whch do no rade on an exchange. An plcaon of hs arguen s ha oher nerpolaon schees slar o he ABS approach should provde slar approxaon accuracy. For exaple, Davs (2001) argues ha any nerpolaon approach wh a reasonably sooh ap fro volaly o prce should gve slar answers. Slarly, Hull (2008) saes ha f raders swched fro Black-Scholes o anoher plausble odel, hen arguably dollar prces would no change apprecably. The ncreenal conrbuon of he presen paper s o foralze he nuve arguens of Davs (2001), Fglewsk (2002) and Hull (2008). We show ha he ABS procedure can be 2

4 used o provde arbrarly accurae asypoc approxaons of a rue bu unknown opon prcng forula. Ths s esablshed hrough a farly sple applcaon of he Weersrass Approxaon Theore, whch saes ha any connuous funcon can be approxaed arbrarly well by a suable polynoal. We also confr ha he Black-Scholes forula does no play a crucal role n he nerpolaon schee n an asypoc sense and could ndeed be replaced by a wde varey of fales of connuous funcon. As long as he odel s used as an nerpolaon ool and s recalbraed frequenly, he approxaon arguen holds. In addon, we conduc soe sple Mone Carlo sulaons o ge a feel for he knd of polynoals requred o reasonably f opon prces n an envronen where he Black-Scholes forula s no applcable. The sulaons also allow us o exane he porance of he saple sze, he order of he polynoal, and he recalbraon frequency n conrolled sengs. Lasly, we apply he ABS approach o daly S&P 100 ndex opons o show ha he procedure ouperfors he Black-Scholes forula n valung acual opon prces ou-of-saple. Before oulnng he prary resuls, we brefly descrbe he ABS odel closely followng Duas, Fleng and Whaley (1998). Traders collec Black-Scholes pled volales fro a cross-secon of opons on a sngle underlyng asse of neres. Ipled volales are soees avalable hrough coercal daa servces or ay be nvered fro observed opon prces. The cross-secon of pled volales s regressed agans a polynoal funcon of he opon srke prces and opon aures, IV (1) σ n = β + β1srken + β2mauryn + ε n Fed values 0. σˆ n fro hs regresson do no have a sensble nerpreaon a any gven e, here should be one volaly pled by opons on he sae underlyng asse. Neverheless, equaon (1) fors he bass of he ad hoc Black-Scholes approach. Suppose we wsh o value an opon wh srke K and aury T, gven a cross-secon of observed opon prces on he sae underlyng asse. The opon s valued usng he Black- Scholes forla (2) BS( K, T, S, r, ˆ ( K, T )) σ n 3

5 where S s he prce of he underlyng asse, r s he rsk-free rae, and volaly s se equal o he fed value ˆ σ ( K, T ) fro regresson (1). Forula (2) dffers fro Black-Scholes only n n ha each opon has s own volaly a free paraeer whch capures devaons fro Black-Scholes across concurren opons. Regresson equaon (1) s connually re-esaed as new daa becoes avalable, ypcally once or wce daly. Ths resuls n a new pled volaly surface and new fed values. Traders hus ake he Black-Scholes odel -- conceved as a sac odel -- and connually recalbrae by updang he unknown paraeer. To undersand he effcacy of hs approach, suppose here s soe rue, bu unknown, opon prcng forula. The pled volaly surface s f o a large cross-secon of observed opon prces. As he cross-secon grows large, he surface becoes an ncreasngly accurae approxaon of he rue bu unknown odel. When plugged no he Black-Scholes forula, he resulng opon prces are ncreasngly accurae esaes of he rue opon prces. Whle he above procedure can be used o approxae o he cross-secon a a gven, here s no reason o expec he approxaon o hold over e. Ths s reeded by frequen re-calbraon of he volaly surface, whch produces connually new approxaons. We show ha f daa generang process s a sochasc volaly odel or a sochasc volaly odel wh jups, one-sep ahead forecas errors approach zero as he calbraon becoes ore frequen and he forecas horzon approaches zero. As a byproduc of our analyss, we show ha our asypoc arguen can be ade vald for Aercan opons f he volales are nvered fro Aercan opons. However, he asypoc error does no necessarly dsappear for Aercan opons f he pled volales are calculaed fro European opons. Ths ay help elucdae soe recen eprcal fndngs. In parcular, Hull and Suo (2003) fnd ha he ABS odel based on European opons does no necessarly perfor well n prcng oher knds of opons. The reander of he paper s organzed as follows. Secon 2 descrbes he conex and he cross-seconal forecasng proble. Secon 3 exends he basc fraework o e-seres forecasng. Secon 4 descrbes a sple sulaon sudy and he resuls are appled o a se of S&P 100 ndex opons n Secon 5. Secon 6 concludes. 4

6 2. Cross-Seconal Opon Prcng Suppose he rue opon prcng forula s an unknown funcon, (3) C = φ K, T, S, r, σ ) ( where K s he srke, T s he e o aury, S s he prce of he underlyng, r s he nsananeous rsk free rae and σ s he volaly. We assue ha opon prces are connuous n he arguens, (K, T, S, r,σ ). Ths knd of odel can easly be generalzed furher o nclude oher coneporaneous varables bu equaon (3) already ncludes os coon heorecal odels. In realy he rue opon prcng funcon s unknown and us be replaced wh a heorecal valuaon odel, ˆ φ ( ), such as Black-Scholes. A he sae e, curren volaly s unobserved. Ths eans he forecasng proble nvolves odelng he opon prcng funcon sulaneously wh esang one of he arguens -- volaly. The proble caused by volaly beng unobserved s closely relaed o he odel specfcaon proble. If he odel were known, he volaly could be nvered fro jus a sngle observed opon prce. Unforunaely, he odel s no known. Denoe he pled volaly calculaed fro a parcular opon, say ( K, T ), as (4) ˆ 1 σ = φ ( C, K, T, S, r ). ˆ,, Gven a cross-secon of N opon values C,..., } a e, he esaed { 1, C N, volales nvered fro Black-Scholes dsplay a srk or sle paern across srke prces and across aures. Such pled volales should be dencal across concurren opon conracs. Snce hey are no, he Black-Scholes odel s clearly sspecfed. In order o value an unobserved N+1 s opon, we us specfy boh ˆ φ ( ) and he volaly σˆ such ha: (5) C ˆ( φ ξ, ˆ σ ),, where for noaonal splcy ξ = ( K, T, S, r ). Exchanges, for exaple, ofen requre, such valuaons so ha hey can se argn requreens. Broker-dealers slarly need o value opon conracs for whch arke prces are unavalable. Snce any oher opon prces are 5

7 observed, hs s essenally a cross-seconal forecasng proble. More generally, we ay be neresed n forecasng opon values a fuure daes as well. For exaple, a large coercal banks raders forecas he value of her opon posons a a daly frequency (Berkowz, O Bren (2002)). In he presen paper, we characerze an approach o probles of hs for coonly referred o as he ad hoc Black-Scholes (ABS) odel. Duas, Fleng and Whaley (1998) consder specfcaons such as ˆ σ = β + β K + β T + β K + β T + ε (6) = β Z + ε where β = ( β 0,..., β4)' are unknown paraeers, Z are he explanaory varables andε are resduals. The pled volales are calculaed fro Black-Scholes and he coeffcens n (6) are esaed by leas squares. Fed values fro (6) are plugged back no Black-Scholes o ge he predced value: (7) C = ˆ( φ K, T, S, r, ˆ β Z ) ˆ, where βˆ s he leas squares esae. The h order polynoal generalzaon of (6) s gven by ˆ σ = β0 + β1k β K β T + β + K T ε (8) = β Z + ε 2 where agan he lef-hand sde are he pled volales nvered fro N observed opons and Z s he vecor of explanaory varables. We can now delneae a se of assupons under whch he above procedure yelds arbrarly accurae forecass as he nuber of regressors grows large. process We begn by sang wo echncal assupons regardng he rue daa generang Assupon 1. Opon values φ ( ) are connuous n all arguens on a copac se. Assupon 2. The e volaly s a connuous funcon of soe, possbly nfne, se of varables, Z. Forally, s a funcon σ =σ (Z ). 6

8 We assue ha Z s observable bu we do no ake any such assupon on he funconal for of σ ( ). We also ake he usual assupon ha he unknown paraeer vecor θ s esaed by ordnary leas squares. Assupon 3. The pled volaly equaon (8) s esaed by nzng he su of squared prcng errors ˆ = (9) θ arg n ( C φ K, T, S, r, β Z )) 2 θ N = 1, ˆ( The nuber of observed opons, N, obvously has o exceed he nuber of paraeers o be esaed: N 2. Under hese condons, he ABS procedure yelds an arbrarly accurae f of he e- se of conngen clas. Ths s rue regardless of he unknown funconal for of he rue odel.. Theore 1. Under assupons 1-3, he ad hoc odel provdes arbrarly accurae opon valuaons n-saple, ˆ C, ˆ( φ K, T, S, r, β Z ) 0 unforly n he arguens K, T, S, r ) as. ( Proof. See appendx Econocally, Theore 1 eans he approach provdes arbrarly accurae opon values whn he cross-seconal saple. Suppose, for exaple, ha ε s $.10. An pled volaly surface esaed wh a polynoal of degree a leas s guaraneed o value he observed opons used o calbrae he odel as well as any unobserved concurren opons whn 10 cens. I s no necessary for he researcher o know he rue funcon C = φ K, T, S, r, σ ). ( The procedure wll work as long as he rue opon prce obeys soe basc soohness condons. As he nuber of observed opons ncreases, he axu prcng error approaches zero. In parcular, f here were an unled nuber of opons observed a e, he pled volaly surface can be f o an unboundedly hgh degree polynoal. When plugged no a 7

9 valuaon odel lke Black-Scholes, hs ranslaes no unled flexbly n fng he rue opon prcng relaon. In hs sense, he resul s an asypoc arguen. In cases of praccal neres, he order of he polynoal and he saple sze N are fne nubers. However, eprcal sudes have generally found ha even quadrac polynoals prce opons well so ha he nuber of paraeers n (8) s ypcally que sall, n he range of 5 or 6 (e.g., Duas, Fleng and Whaley (1998), Chrsoffersen and Jacobs (2004)). The sulaon resuls n secon 4 of he presen paper appear o confr hs. In ers of saple szes, he evdence fro he Mone Carlo n secon 4 suggess ha prcng accuracy s vrually unchanged beyond saple szes of only abou N = 64. For he any of he ore wdely raded underlyng asses, hs s well whn he range of avalable opon prces a any gven e. For exaple, n he case of equy ndex opons such as S&P 500 ndex opons suded n Chrsoffersen and Jacobs (2004) or he S&P100 ndex opons suded n he presen paper, he avalable saple sze s ypcally ore han 80 a any gven e. One of he neresng plcaons of he heore s ha he opon prcng forula plays only a rval role. Suppose we wsh o use a odel, ψ ˆ( ), oher han Black-Scholes. All ha s needed s ha he odel fulfll he sae basc soohness assupon and no be degenerae n any of s arguens. Under hese condons, we can apply he Weersrass Theore whch s he essenal sep n he proof of Theore 1. If used n conjuncon wh a suffcenly flexble pled volaly surface -- as n coon pracce -- wheher raders use Black-Scholes or soe oher opon valuaon odel s no asypocally poran, precsely as argued by Davs (2001) and Hull (2008). If suffcen daa s avalable and f he echncal assupons are no unreasonable approxaon of realy, he predced opon prce converges o he rue prce. Aercan Opons If he volaly surface s calculaed fro Aercan opons, he ad hoc approach wll also succeed n large saples when appled o Aercan opons. However, f volales are pled fro Aercan opons, he prcng error does no necessarly vansh asypocally. 8

10 Ths ay shed soe lgh on soe recen eprcal fndngs of Hull and Suo (2003) who fnd ha he ABS approach based on European opons does no necessarly prce oher knds of opons accuraely. To show hese clas aheacally, wre he rue bu unknown Aercan opon prce A A, (, (10) C = φ ξ, σ ). Suppose he volaly surface s esaed fro observed Aercan opons: (11) ˆ A β = arg n C ˆ( φ ξ, β Z ) A θ,, where βˆ A s he leas squares esae. When plugged no Black-Scholes, hs yelds he odel ˆ ˆ( φ ξ, β Z ) and Theore 1 apples. In parcular, ogeher wh he connuy assupons,, A A (12) ˆ C ˆ( φ ξ, β Z ) 0 unforly n he arguens as.,, A In pracce, however, he volaly surface used for all ypes of opons s he sae one used for, and calculaed fro, European opons. The acual esaon procedure s hus: ˆ β = arg n C θ, ˆ( φ ξ, β Z, ) where by C, s he European opon and βˆ he OLS esae. The prcng error s herefore gven A C ˆ(, ˆ φ ξ β Z,, ) and n general ˆ β ˆ β, so ha Theore 1 does no hold. Ths s rue even f an unled A nuber of European opons are observed because βˆ wll no ypcally converge o asypocally. βˆ A even 3. Te-Seres Forecasng In pracce, he ad hoc odel s used no only o prce opons or oher dervaves concurrenly bu also o forecas fuure values of opons. Ths pracce s parcularly poran 9

11 n rsk anageen conexs. For exaple, he ABS approach s ofen eployed o forecas he rsk coponen of opons n calculang porfolo value-a-rsk. In hs secon we consder he proble of forecasng opon values one-sep ahead when he rue opon prcng funcon s unknown and volaly s unobserved. We begn by consderng a fxed observaon nerval, h. A each e, our objecve s o forecas he opon value C + h,. In pracce, raders ofen need o forecas or hedge her opon posons a he daly or wh even hgher frequency. Below, we wll consder wha happens f we allow he forecas nerval o shrnk o zero. A each e, he volaly surface ˆ σ ( Z ) s re-esaed and he one-sep ahead forecas s gven by ˆ( φ ξ,, ˆ( σ Z )) whch we denoe ˆ φ, for splcy. The e forecas error s (13) e(h,, ) = φ + h φ, + h., ˆ I wll be convenen o decopose he error φ φ no + h ˆ,, + h (14) [ φ φ ] + [ φ φˆ ], + h,,, where he frs er n square brackes represens he change n he opon value and s beyond he conrol of he user. The second er s he e- odelng error. In order o sudy he properes of hese wo coponens of he forecas error, we wll requre wo addonal assupons. Le y = (ξ,, σ ) for noaonal splcy. Assupon 4. The y are generaed by a connuous e dffuson process, dy = μ ( y ) d + Ω( y ) dw where μ y ) and Ω y ) are dfferenable drf and dffuson funcons and dw s a vecor ( ( Brownan oon. Assupon 5. The conngen cla possesses a fne hea, ) φ( y <. Assupons 4 and 5 ensure ha he value of he conngen cla wll self be a dffuson process. Fro Io s lea, he value φ ( y ) s he soluon o he sochasc 10

12 dfferenal equaon, 2 (15) dφ = φ + + Ω Ω + Ω μ ( y, ) φ φ j ρj d φ dw y y j y j y where dy, d + Ωdw, = μ and ρ j s he nsananeous correlaon beween dw, and dw j,. Gven assupons 4 and 5, he forecas error (14) can be wren as + h (16) e( h,, ) φ( ξ, σ ) ds + [ φ( ξ, σ ) ˆ( φ ξ, ˆ( σ Z ))] = +h whch s bounded fro below by he frs er φ ( ξs, σ s brackes, s he odel sspecfcaon error. s s ) ds. The second er, n square We now consder he behavor of he ABS approach as he re-calbraon nerval, h, approaches zero. Ths gh be he ore relevan paradg, for exaple, f he copuaonal coss of frequen recalbraon are sall enough o recalbrae he odel any es per day. In ovng o he e-seres conex, we ake use of he usual ean-square erc,, Φ = 1 2 ( Φ )2 E where E Φ = E Φ I ] s he condonal expecaon a e. [ We wll show ha as h 0, he forecas error converges o zero, C ˆ φ 0. Ths, + h s because he lower error bound wll self converge o zero as h 0 under very general condons. Theore 2. As he calbraon nerval shrnks o zero, he connual recalbraon odel provdes arbrarly accurae forecass of opon prces, ˆ C ˆ( φ K, T, S, r, β Z ) 0 as and h 0. Proof. See appendx, + h Ths resul shows ha he forecas errors can be ade of order o(h). As he recalbraon 11

13 nerval shrnks o zero, he error shrnks o zero. Of course, over a fxed horzon he error s +h φ. If he opon valuaon odel beng recalbraed s Black-Scholes, hen ˆ φ ( ) s ( ξs, σ s ) ds connuous and possesses a fne hea. Bu he aheacal developen above obvously holds n uch ore general sengs. Connuy and a fne hea ogeher wh an underlyng dffuson are suffcen properes n heselves. Ths explans Hull (2008) observaon ha, under soe condons, he prcng of opons s no very sensve o he parcular odel used. Wha s requred are soe basc soohness properes and frequen recalbraon of he odel. 3b. Mxed Jup-Dffuson Models In hs secon we consder replacng he assupon ha he underlyng varables are a pure dffuson process wh he followng daa-generang process. Assupon 4b. Suppose he underlyng asse value s descrbed by a xed jup-dffuson odel, ds = ( μ λ g) d + Ω( S ) dw + g dj where g s a rando varable wh fne varance and ean g and dj s a Posson jup process wh nensy λ. As before dw represens a sandard Brownan oon and μ(s ), Ω(S ) are real, connuous drf and dffuson funcons. Applyng a generalzaon of Io s lea o xed jup-dffuson processes we have ha he value of a conngen cla φ(s, ) s descrbed by he followng sochasc dfferenal equaon, (17) 1 dφ = ( μ λ g) φs + & φ + φssω( S ) d + φsω( S ) dw + 2 [ φ( S + g, ) φ( S, ) ] dj. Theore 3. Under assupons 1-3, 4b and 5, he connual recalbraon odel provdes arbrarly accurae forecass of opon prces, ˆ C ˆ( φ K, T, S, r, β Z ) 0 as and h 0., + h 12

14 Proof. See appendx The proof s very slar o ha of heore 2. We agan fnd ha he one-sep ahead forecass gven by he ABS odel becoe arbrarly accurae as he re-calbraon frequency shrnks o zero. However, hese resuls coe wh an poran cavea. Snce he accuracy of he forecass s of order o(h), he error can be ade arbrarly sall only over he ypcally shor, recalbraon perod. For exaple, suppose paraeers are recalbraed daly n order o acheve a parcular ean-squared error level ha s olerable. Over longer horzons such as a onh, hese forecas errors wll neverheless accuulae beyond he daly olerance level. 4. Sulaon sudy: Sochasc Volaly For llusraon, we conduc a odes Mone Carlo sudy o ge a feelng for he quanave behavor of he ABS ehod n praccal suaons. We assue ha he ABS odel s used bu ha he rue underlyng process dsplays e-varyng volaly. In parcular, we follow Baksh, Cao and Chen (1997), Jones (2002), Chrsoffersen and Jacobs (2004) and any ohers n exanng a well-known daa generang process based on he Heson (1993) sochasc volaly odel. Suppose he prce of a sock, S, s gven by he followng ean-reverng sochasc volaly process, (18) ds = μsd + σsdz1, (19) dσ = κ( α σ) d+ η σdz2, where dz 1, and dz 2, are Wener processes wh consan correlaon ρ and (μ, α, κ, η) are fxed scalar paraeers. For he sulaons, we follow Heson (1993) and se μ =.12, α =.01, κ =2 and he volaly of volaly η =.11. In order o generae soe skewness n he process we se he correlaon ρ = -0.6, whch s ypcal of he values found when he process s forally esaed (e.g., Jones (2002)). We se he rsk-free rae o.05. The nal sock prce S 0 = 41, and he nal volaly s equal o he uncondonal ean of he volaly process. 13

15 Heson (1993) shows ha he prce of a call opon exprng a e T s gven by (20) where φ( KTSr,,, ) = SP Ke P rt ( ) 1 2 slog K 1 1 e f j ( S, T, σ, ) Pj = + Re ds 2 π s 0 and f j are characersc funcons. We use he correc forula (20) o predc he value of a se of 4 call opons wh expraons of T = 130 days and 160 days, and srkes of K = 40 and K = We consder four versons of he ABS odel: ABS 1: ˆ σ β + β K + β T + ε = ABS 2: ˆ σ = β + β K + β T + β K + β T + ε ABS 3: ˆ σ β + β K + β T + β K + β T + β K T + ε = ABS 4: ˆ σ = β + β K + β T + β K + β T + β K + β T + ε Specfcaons 1 hrough 3 are conssen wh he odels consdered by Duas, Fleng and Whaley (1998) and Chrsoffersen and Jacobs (2004). Model 4 s added o exane wheher hgher order ers han quadrac can prove odel perforance. The paraeers are esaed by leas squares fro fve dfferen saple szes rangng fro N = 16 o 81. For he salles saple sze N = 16, we generae a saple of opons for esaon by creang a grd wh srke prces akng 4 equspaced values fro 38 o 41, and aures akng 4 equspaced values 100 o 180 days. For saple sze N=25, he grd conans 5 dfferen srke prces fro 38 o 41 and 5 dfferen aures fro 100 o 180 days, and so on up o saple sze 81. For each saple, we use he ABS odels and he correc opon prcng forula o value a se of 4 opons whch are no n he saple. By coparng hese predced values o he rue opon prces ha should hold, we can calculae cross-seconal ou-of-saple prcng errors. We aggregae he 4 prcng errors no a scalar quany by akng her roo ean-squared error. The resuls are hen averaged across 1000 Mone Carlo sulaons and are shown n

16 Table 1. One would expec esaes of he pled volaly surface, and hence he ABS odel, o prove wh he saple sze N. Indeed hs s he case. However, here are vrually no reducons n RMSE beyond a saple sze of abou 64. The RMSE across he four versons of he ABS odel dsplays a generally U-shaped paern ndcang ha he ncreasngly coplexy of odels 1-3 proves perforance. The quadrac ABS 2 odel leads o an proveen over he lnear odel ABS 1. Addonally, he ncluson of an neracon er generally reduces RMSE furher han odel 2. However, odel ABS 4 whch has hgher order ers han quadrac are worse n ers of RMSE and hence he exra ers do no prove he cross-seconal f. Clearly wheher he addon of a parcular hgher-order er wll prove he f or no s soehng ha has o be checked on a case by case bass. In Table 2, we dsplay he RMSE fro he e seres ou-of-saple forecas errors. The op panel, labeled 10 days, shows he 10-day-ahead forecas errors. There s a subsanal proveen n perforance n gong fro he lnear odel ABS 1 o he quadrac odel, ABS 2, bu agan a deeroraon wh he addon of he hgher order ers n ABS 4. In hs case he perforance of he esaed odels s vrually unchanged beyond a saple sze of N=36. Gven ha he ABS odels ha perfor bes here are quadrac, he degrees of freedo accuulae quckly wh N. In order o prove he f, appears ore poran o ge he rgh polynoal order, whch eprcally need only be quadrac, raher han accuulang observaons beyond abou 30 or 40. The boo panel shows he RMSE fro ½-day ahead forecass. The paerns rean he sae, hough he RMSE s obvously uch saller for all odels. Ths s conssen wh Theore 2 whch shows he accuracy of he forecas s of order o(h). Neverheless, s neresng o noe one of he eprcal resuls ha eerge. In hs parcular sulaon envronen here are sgnfcan reducons n RMSE when changng fro a 5-day o 1-day recalbraon nerval on he order of 20 percen. However, furher ncreases n he recalbraon frequency fro 1 day o a half-day only reduces prcng error by less han 3 percen for os of he odels and os saples szes consdered here. 15

17 5. Applcaon o Equy Index Opons In hs secon, we apply he odels used n he sulaon o sudy her accuracy n prcng S&P100 call opons. We chose hs daa based on several consderaons. Frs, he se of opons wren on hs ndex are aong he os acvely raded conracs. Second, he daly dvdend dsrbuons are avalable for he ndex. Thrd, he S&P100 ndex opons are Aercan-syle opons and hence by usng hs daa we copleen he leraure on he European-syle counerpar, S&P 500 ndex opons, whch have been suded exensvely by Baksh, Cao, and Chen (1997) and Duas, Fleng, and Whaley (1998) aong any ohers. Our saple conans repored prces of S&P 100 ndex call opons raded on he Chcago Board Opons Exchange (CBOE) over he perod January 2, 2005 hrough Deceber 31, 2007 for a oal of 753 radng days. The prce of he opon s se equal o he average of he bes bd and bes ask a he close of each radng day. We exclude only daa for whch here s a ssng bd or ask quoe, or ssng srke prce or aury. Ths yelds a oal of 211,276 usable call opon prces. 1 Ths saple s que large prarly due o he wde range of srke prces. In our saple, here are ofen ore han 60 dfferen srke prces and 5 dfferen opon aures observed on a sngle day. The opon's e o expraon s easured as he nuber of calendar days beween he rade dae and he expraon dae. To suarze he opon prcng daa, we dvde he saple no several caegores accordng o oneyness and e o expraon. Observaons are paroned no 6 degrees of oneyness, rangng fro 10% ou-of- he-oney (OTM) o 30% n-he-oney. For each of hese caegores, we spl he daa no 3 aury lenghs, rangng fro less han 60 days o over 180 days. The resuls are shown n Table 3. For each of he 18 oneyness-aury caegores, we repor he average opon prce, he sandard devaon and he nuber of observaons. As expeced, he ypcal paern eerges n whch average call prces ncrease onooncally wh oneyness and wh aury. The daa dsplay a very rch degree of varaon, wh he os nhe-oney calls havng an average prce ore han 1000 es ha of he os OTM opons. I s also neresng o noe whle he OTM call opons are worh only abou $0.12 on average for 16

18 aures less han 2 onhs, hey average over $14 for aures greaer han 6 onhs. For opons deep n he oney, ncreases n aury have a uch saller effec on call prces han on slar OTM opons. Ths s also a well-esablshed paern found, for exaple, n Baksh, Cao and Chen (1997). Frs we exane he ably of he ABS odels o prce concurren opon observaons cross-seconally. We calculae he Black-Scholes pled volales usng he neres raes pled by consan aury zero-coupon bonds aken fro he IvyDB OponMercs daabase. Gven he pled volales, each day we esae he ABS odels descrbed n Secon 5 usng he saple of opons avalable ha day excep for 4 arge opons. The arge opons are gven by he opons wh he 4 lowes srke prces ha day. The ABS odels are hen used o value hese 4 opons cross-seconally. The resuls are copared o he prce pled by he Black-Scholes forula whch uses he esaed saple volaly of he underlyng ndex over he saple perod. For each odel and each opon, we calculae he square roo of he averaged squared prcng error, where he average s aken over he 4 arge opons. Ths daly error s hen averaged over all days n he saple o ge he average RMSE repored n Table 4. The dollar RMSE are shown n Panel A and he errors n percenage ers n Panel B. The ypcal Black- Scholes prcng error s n he range of $2.73 or abou 18 percen. Ths s roughly coparable n percenage ers o he agnude of error ypcally found for S&P 500 ndex opons (e.g. Duas, Fleng and Whaley (1998)). The able also ndcaes ha every verson of he ABS odel delvers lower RMSE prcng errors. Agan, hs s o be expeced gven ha he ABS odel allows for varaon n he volaly paraeer whereas he Black-Scholes forula does no. For every arge opon ha we ry o value, he ABS 3 odel has he salles errors. I s neresng o noe ha hs odel, wh quadrac ers and an neracon er bu no hrdorder ers, s becong soehng of a sandard specfcaon aong praconers and acadecs sudyng he ABS odel (e.g., Chrsoffersen and Jacobs (2004)). I s also noeworhy ha hs s precsely he odel whch had he lowes prcng errors n he sulaons 1 We also nvesgaed he prcng of S&P 100 pus and fnd he resuls o be qualavely slar. 17

19 n Secon 4. Ths suggess ha he relave accuracy of hs parcular specfcaon ay be soewha robus o hese radng envronens. We nex exane he opon prcng errors of he 4 ABS odels ou-of-saple wh 1 day, 2 day, 5 day and 10 day recalbraon nervals. A 1 day recalbraon nerval eans ha a he close of each day, he cross secon of opons s used o value an opon radng he nex day. The resuls are shown n Table 5. Panel A repors he edan dollar prcng errors over all days n he saple. Each day he aggregae prcng error for a gven odel s gven by he square roo of he edan squared error. The resuls ndcae ha all 4 of he ABS odels ouperfor he Black-Scholes odel by a sgnfcan argn. Agan we confr ha he ABS 3 odel generally s os accurae aong he ad hoc odels. To gauge he econoc sgnfcance of he lower valuaon errors fro he ABS odel relave o Black-Scholes, Panel B repors he prcng errors n percenage ers. The dfference beween he ABS 3 odel n parcular and Black-Scholes s que large n he sense of beng generally beyond he 5 percen range assocaed wh ransacon coss (e.g., Baksh, Cao and Chen (1997)). Ths s rue for every recalbraon nerval we consdered. However, he relave proveen n he ABS odels s greaes when he recalbraon nerval s relavely shor a 1 or 2 days. 6. Conclusons Many arke akers and raders have ressed usng cung-edge opon prcng forulas. The os wdely used valuaon procedure aong praconers s a verson of Black- Scholes wh ad hoc adjusens and frequen updang of paraeers. A growng body of eprcal evdence suggess ha he ad hoc approach perfors que well. I has been argued prevously ha such a procedure should work well because he ad hoc approach serves as a sophscaed nerpolaon ool. Our ncreenal conrbuon s o show ha wh suffcenly frequen updang, he ad hoc approach provdes forecass wh errors ha vansh n he l. Ths provdes a rgorous, albe asypoc, bass for undersandng why he ad hoc approach o opon prcng does so well. Our heorecal resul also explans he observaon ade by prevous auhors ha, under 18

20 soe condons, he prcng of opons s no errbly sensve o he parcular odel used. As long as he odel s used as an nerpolaon ool and s recalbraed frequenly, he approxaon arguen holds. I s he frequen recalbraon, raher han he underlyng odel, ha s of prary porance n hgh frequency opon valuaon. Eprcally, we fnd n a se of sulaons ha our ypcal ABS odels subsanally ouperfor he Black-Scholes forula. The bes perforng odels requre only lnear and quadrac ers, so ha only a few paraeers need o be esaed. Conssen wh hs, we fnd ha saple szes beyond abou 64 do no sgnfcanly reduce prcng errors any furher. Lasly, we conduced a odes applcaon of he ABS odels sudy o hree years worh of S&P 100 ndex opons. The sae lnear-quadrac pled volaly funcons ha perfored bes n he Mone Carlo appear o do bes when appled o hs real daa. Our fndngs should no, however, be aken as an arguen agans developng heorecally-derved opon prcng forulas. Undoubedly, he rearkable heorecal advances beng ade wll resul n analycal forulas ha donae any curren pracce. As heorecal proveens are acheved, praconers wll adop he possbly agan wh ad hoc adjusens. 19

21 References Baksh, C., Cao, C. and Z. Chen (1997), Eprcal Perforance of Alernave Opon Prcng Models, Journal of Fnance, 52, Baes, D. (1996), Jups and Sochasc Volaly: Exchange Rae Processes Iplc n Deusche Mark Opons, Revew of Fnancal Sudes, 9, Berkowz, J. and J. O Bren (2002), How Accurae are Value-a-Rsk Models a Coercal Banks, Journal of Fnance, 57, Black, F. and Scholes, M. (1973), The Prcng of Opons and Corporae Lables, Journal of Polcal Econoy, 81, Chrsoffersen, P. F. and K. Jacobs (2004), The Iporance of he Loss Funcon n Opon Prcng, Journal of Fnancal Econocs, 72, Davs, M. (2001). Maheacs of Fnancal Markes. In B. Engqus and W. Schd (Eds.), Maheacs Unled 2001 and Beyond, pp Sprnger: New York. Davs, P. J. (1975). Inerpolaon and Approxaon. New York: Dover Publcaons. Deran, E., and I. Kan (1994), Rdng on he Sle, Rsk, 7, Duan, J. (1995), The GARCH Opon Prcng Model, Maheacal Fnance, 5, Duas, B., Fleng, F. and R. Whaley (1998), Ipled Volaly Funcons: Eprcal Tess, Journal of Fnance, 53, Heson, S. L. (1993), A Closed-For Soluon for Opons wh Sochasc Volaly wh Applcaons o Bond and Currency Opons, Revew of Fnancal Sudes, 6, Heson, S. and S. Nand (2000), A Closed-For GARCH Opon Prcng Model, Revew of Fnancal Sudes, 13, Hull, J. C. (2008). Opons, Fuures & Oher Dervaves (7 h ed.). Prence Hall: Upper Saddle Rver, NJ. Hull, J. and W. Suo (2003), A Mehodology for Assessng Model rsk and s Applcaon o he Ipled Volaly Funcon Model, Journal of Fnancal and Quanave Analyss, 37(2), p

22 Melno, A. and S. Turnbull (1990), Prcng Foregn Currency Opons wh Sochasc Volaly, Journal of Econoercs, 45, Meron, Rober C. (1976), Opon Prcng when Underlyng Sock Reurns are Dsconnuous, Journal of Fnancal Econocs, 3, Rubnsen, M. (1994), Ipled Bnoal Trees, Journal of Fnance, 49,

23 Appendx Proof of Theore 1. We wll show ha ˆ C φ ( ξ, ˆ( σ Z )) 0 (21) unforly n ξ, Z as. Fro assupon 1, φˆ () s connuous and we can apply he (ulvarae) Weersrass heore: j ˆ( φ ξ σ Z =, ˆ( )) α jξ ˆ β Z = where by assupon 3, β arg n ( C φ K, T, S, r, β Z )) 2 j 1 ˆ = ˆ(. θ Usng assupons 1 and 2, he rue bu unknown funcon C = φ( ξ, σ ( Z )) can be wren as ~ φ ( ξ, σ ( Z )) = φ ( ξ, Z ) (23) ~ where φ ( ) s unknown bu s connuous n boh arguens. Usng (22), he prcng error φ ( ξ, Z ) ˆ( φ ξ, ˆ( σ Z )) s gven by ~ ~ j φ ξ Z (, ) α jξ β j= 1 ˆ Z (22). (24) Fro Theore of Davs (1975) follows ha, for any ε >0, we can fnd a suffcenly large such ha ~ (, ) ˆ j φ ξ Z α jξ β Z < ε (25) j= 1 ~ unforly n ξ, Z. Tha s, φ ( ) can be unforly approxaed by a polynoal of suffcenly hgh degree. I follows edaely ha ~ l n (, ) j φ ξ Z α jξ β Z β = 1 as shown, for exaple n Davs (1975, p.108). Proof of Theore 2. Wre he forecas error as + h e( h,, ) = φ( ξs, σ s ) ds + [ φ( ξ, σ ) ˆ( φ ξ, ˆ( σ Z ))] 2 2 = 0, (26) 22

24 + h + h = μφ, sds + Ω φ, s dw s + [ φ( ξ, σ ) ˆ( φ ξ, ˆ( σ Z ))]. (27) Fro Theore 1, l φ( ξ, σ ) ˆ( φ ξ, ˆ( σ Z )) =0 and so l l e( h,, ) h 0 splfes o l h 0 + h + h μ φ, sds + Ω φ, sdws whch s he desred resul. 2 + h + h 1/ 2 = l E ( ) + Ω ( ) 0 μ φ y ds φ y dw h = l Oh ( ) = 0 h 0 Proof of Theore 3. Fro equaon (17), φ + h + h + h 1 + k = φ + μ s λs g) φs + φssω( S s ) ds + φsω( S s ) dws + 2 [ φ( S + ] s g s ) ( S s ) ( φ In hs case he forecas error s gven by [ φ( ξ, σ ) ˆ( φ ξ, σ ( Z ))] + h e( h,, ) = φ( ξs, σ s ) ds + [ φ( ξ, σ ) ˆ( φ ξ, σ ( Z ))] + h + h + h = μ ds μ dws [ φ( S s g s ) φ( S s )] dj φ + Ω φ + + s + dj. (28) Fro assupon 4b he expresson n he frs parenhess s O(h) so ha forecas error s agan [ φ( ξ, σ ) ˆ( φ ξ, σ ( Z ))] O( h) +. The reander of he proof s precsely as n he proof of Theore 2. s. 23

25 Table 1. The ad hoc odel n he presence of Sochasc Volaly: Cross-seconal roo ean-squared-error Model Saple sze N =16 N = 25 N = 36 N = 64 N = 81 ABS ABS ABS ABS Noes: Daa s generaed fro he bvarae sochasc volaly process (18)-(19). Four versons of he ABS odel are used o predc he value of 4 call opons wh expraons of T = 130 and 160 days, and srkes of K = 40 and K = The square roo of MSE s calculaed over 1000 Mone Carlo sulaons. 24

26 Table 2. The ad hoc odel n he presence of Sochasc Volaly: One-sep ahead roo ean squared error Model Saple Sze Recalbraon nerval, h N = 16 N = 25 N = 36 N = 64 N = days ABS ABS ABS ABS Heson days ABS ABS ABS ABS Heson day ABS ABS ABS ABS Heson Half day ABS ABS ABS ABS Heson Noes: Daa s generaed fro he bvarae sochasc volaly process (18)-(19). Four versons of he ABS odel are used o predc he value of 4 call opons wh expraons of T = 130 and 160 days, and srkes of K = 40 and K = The square roo of MSE s calculaed over 1000 Mone Carlo sulaons. 25

27 Table 3. Saple Properes of he S&P 100 Index Opons Moneyness S/K Days-o-Expraon < Suboal OTM <0.9 $0.12 $0.40 $14.89 (0.14) (0.71) (13.77) {3871} {4484} {6991} {15,346} $1.77 $6.26 $39.97 (3.01) (6.41) (21.40) {29014} {14800} {10083} {53,897} ATM $30.00 $38.76 $74.38 (15.89) (14.59) {29722} {13128} {9425} {52,275} $79.63 $84.57 $ (15.18) (15.47) (20.84) {18680} {10179} {7964} {36,823} ITM $ $ $ (15.13) (15.63) (19.95) {7708} {6955} {6583} {21,246} 1.30 $ $ $ (36.74) (35.76) (37.95) {8143} {9948} {13559} {31,650} Suboal {97163} {59504} {54609} {211276} The repored nubers are he average bd-ask dpon prce, he sandard devaon shown n parenheses and he oal nuber of observaons (n braces for each oneyness-aury caegory. The saple perod exends fro January 2, 2005 o Deceber 31, 2007 for a oal of 211,276 call opons. S denoes he spo S&P 100 ndex level and K s he exercse prce. OTM, ATM and ITM denoe ou-of-he oney, a-he-oney and n-he-oney, respecvely. 26

28 Table 4. Valung S&P 100 Index Opons: Cross-Seconal Prcng Errors fro he ad hoc Models Model Opon 1 Opon 2 Opon 3 Opon 4 Mean Panel A. Dollar Error Black-Scholes $2.73 $2.79 $2.65 $2.58 $2.69 ABS 1 $1.85 $2.12 $1.96 $1.78 $1.93 ABS 2 $1.91 $2.11 $2.04 $1.94 $2.00 ABS 3 $1.60 $1.79 $1.67 $1.61 $1.67 ABS 4 $1.89 $2.01 $1.98 $1.93 $1.95 Panel B. Percenage Error Black-Scholes ABS ABS ABS ABS Table repors RMSE for he Black-Scholes odel as well as he 4 versons of he ABS odel descrbed n Secon 4. Each day he odels are esaed fro he avalable se of S&P 100 ndex call opons. The odels are hen used o predc he value of 4 arge opons crossseconally. The RMSE s hen calculaed as he square roo of he ean squared prcng error across he 4 arge opons. Panel A repors prcng errors n dollars, Panel B repors he errors as a percenage of he rue call opon prce. 27

29 Table 5. Valung S&P 100 Index Opons: Ou-of-Saple Prcng Errors Panel A. Dollar Prcng Errors Recalbraon nerval, h=10 days Recalbraon nerval, h=5 days Recalbraon nerval, h=2 days Recalbraon nerval, h=1 day Model Opon 1 Opon 2 Opon 3 Opon 4 Mean Black-Scholes $6.91 $6.88 $7.04 $7.12 $6.99 ABS 1 $5.81 $5.48 $5.66 $5.87 $5.71 ABS 2 $5.56 $5.34 $5.58 $5.54 $5.51 ABS 3 $5.33 $5.02 $5.30 $5.27 $5.23 ABS 4 $5.67 $5.39 $5.35 $5.74 $5.58 Black-Scholes $5.75 $6.03 $5.95 $5.41 $5.78 ABS 1 $4.63 $4.92 $4.97 $4.58 $4.78 ABS 2 $4.40 $4.42 $4.67 $4.40 $4.47 ABS 3 $4.30 $4.22 $4.44 $4.05 $4.25 ABS 4 $4.45 $4.62 $4.46 $4.49 $4.50 Black-Scholes $4.72 $4.91 $4.71 $4.65 $4.74 ABS 1 $3.70 $4.11 $4.11 $3.75 $3.92 ABS 2 $3.40 $3.67 $3.77 $3.62 $3.62 ABS 3 $3.47 $3.43 $3.41 $3.38 $3.42 ABS 4 $3.52 $3.48 $3.64 $3.71 $3.58 Black-Scholes $4.36 $4.33 $4.04 $4.13 $4.21 ABS 1 $2.97 $3.56 $3.36 $3.22 $3.27 ABS 2 $2.99 $3.28 $3.22 $3.21 $3.18 ABS 3 $2.65 $3.10 $2.97 $2.98 $2.93 ABS 4 $2.88 $3.32 $3.09 $3.12 $

30 Panel B. Percenage Prcng Errors Recalbraon nerval, h=10 days Recalbraon nerval, h=5 days Recalbraon nerval, h=2 days Model Opon 1 Opon 2 Opon 3 Opon 4 Mean Black-Scholes ABS ABS ABS ABS Black-Scholes ABS ABS ABS ABS Black-Scholes ABS ABS ABS ABS Recalbraon Black-Scholes nerval, h=1 day ABS ABS ABS ABS Table repors RMSE for he Black-Scholes odel as well as he 4 versons of he ABS odel descrbed n Secon 5. Each day he odels are esaed fro he avalable se of S&P 100 ndex call opons. The odels are hen used o predc he value of 4 arge opons over forecas horzons of 10 days, 5 days, 2 days and 1 day. The RMSE s hen calculaed as he square roo of he edan squared percenage prcng error across he 4 arge opons. 29