Cere for Pracical Quaiaive Fiace No. 16 Closed Formula for Opios wih Discree Divideds ad is Derivaives Carlos Veiga, Uwe Wysup Ocober 2008 Auhors: Prof. Dr. Uwe Wysup Carlos Veiga Frakfur School of Frakfur School of Fiace & Maageme Fiace & Maageme Frakfur/Mai Frakfur/Mai u.wysup@frakfur-school.de c.m.veiga@frakfur-school.de Publisher: Frakfur School of Fiace & Maageme Phoe: +49 (0 69 154 008-0 Fax: +49 (0 69 154 008-728 Soemasr. 9-11 D-60314 Frakfur/M. Germay
Closed Formula for Opios wih Discree Divideds ad is Derivaives Carlos Veiga c.m.veiga@frakfur-school.de Tel.: +49 (069 154008-771 Fax.: +49 (069 154008-4771 Frakfur School of Fiace & Maageme Cere for Pracical Quaiaive Fiace Soemasraße 9-11, 60314 Frakfur am Mai Uwe Wysup u.wysup@frakfur-school.de Tel.: +49 (069 154008-719 Fax.: +49 (069 154008-4719 Frakfur School of Fiace & Maageme Cere for Pracical Quaiaive Fiace Soemasraße 9-11, 60314 Frakfur am Mai The auhor wishes o hak Milleium bcp ivesimeo, S.A. for he fiacial suppor beig provided durig he course of his PhD. sudies. 1
Absrac We prese a closed pricig formula for Europea opios uder he Black Scholes model ad formulas for is parial derivaives. The formulas are developed makig use of Taylor series expasios ad by expressig he spaial derivaives as expecaios uder special measures, as i Carr, ogeher wih a uusual chage of measure echique ha relies o he replaceme of he iiial codiio. The closed formulas are aaied for he case where o divided payme policy is cosidered. Despie is small pracical relevace, a digial divided policy case is also cosidered which yields approximaio formulas. The resuls are readily exesible o ime depede volailiy models bu o so for local-vol ype models. For compleeess, we reproduce he umerical resuls i Vellekoop ad Nieuwehuis usig he formulas here obaied. The closed formulas preseed here allow a fas calculaio of prices or implied volailiies whe compared wih oher valuaio procedures ha rely o umerical mehods. Key words: Equiy opio, discree divided, hedgig, aalyic formula Coes 1 Iroducio 1 1.1 Moivaio.......................................... 1 1.2 Descripio of he Problem................................ 1 1.3 Lieraure Review...................................... 2 2 Closed Formula 3 2.1 Geeral Derivaio..................................... 5 2.2 Call uder he Black Scholes Model........................... 7 2.3 The Greeks......................................... 11 3 Resuls 12 4 Coclusio 13 Appedices 14 A Codiio s Verificaio 14 B Derivaio of he Two Divideds Formula 17 C Aleraive o he Black Scholes Formula 18 D Mahemaica c Code 20 Refereces 22 2
1 1 Iroducio 1.1 Moivaio The moivaio o reur o his issue is he fac ha wheever a ew produc, model or valuaio procedure is developed, he problem ha arises wih discree divideds is dismissed or overlooked by applyig he usual approximaio ha rasforms he discree divided io a coiuous sream of divided paymes proporioal o he sock price. Afer all ha has bee said abou he way o hadle discree divideds, here are sill srog reasos o jusify such a approach. We recall here he reasos ha uderlie he use of his mehod by he majoriy of marke paricipas ad pricig ools currely available. We choose he word mehod o refer o his procedure because we believe i o be more suiable ha model. The reaso for his is ha, if oe cosiders wo opios wih differe mauriies, wrie o he same uderlyig sock, his mehod implies wo differe diffusio price processes for he same uderlyig sock uder he same measure. This is aurally a ureasoable choice o model he uderlyig sock price, especially because i would admi arbirage opporuiies. The drivers behid he huge populariy of his mehod are mosly due o he (i racabiliy of he valuaio formulas, (ii applicabiliy o ay give model for he uderlyig sock, ad (iii he preserved coiuiy of he opio price whe crossig each divided dae. However, i has some sigifica drawbacks. Firs ad foremos he iaccurae pricig i produces has o be meioed. Furhermore, he error grows larger as he divided dae is farher away from he valuaio dae. This is exacly he opposie behavior of wha oe would expec from a approximaio a larger period of ime bewee he valuaio dae ad he divided dae meas ha he opio valuaio fucios are smooher, ad hus should be easier o approximae. The oher side of he iaccurae pricig coi is he fac ha his mehod does o provide a hedgig sraegy ha will guaraee he replicaio of he opio payoff a mauriy. To sum up, o umerical procedure based o his mehod reurs (or coverges o he rue value of he opio, whaever he model. I sill seems like he advaages ouweigh he drawbacks sice i is he mos widely used mehod. A example may help o demosrae his. Cosider a sochasic volailiy model wih jumps. Now cosider he valuaio problem of a America syle opio uder his model. The complexiy of his problem is such ha a rigorous reame of discree divideds, i.e. a modificaio of he uderlyig s diffusio o accou for ha fac, would reder he model iracable. 1.2 Descripio of he Problem The problem arises due o he fac ha he diffusio models, like he Black-Scholes (BS model, are o loger a accepable descripio of he sock price dyamics whe he sock pays discree divideds. The risks ha occur i his coex are maily he poeial losses arisig from icorrec valuaio ad ieffecive hedgig sraegy. We address boh of hese issues i his paper. The mos aural exesio o he a diffusio model o accou for he exisece of discree divideds is o cosider he same diffusio, for example, ds = S (rd + σdw, (1 ad add a egaive jump wih he same size as he divided, o he divided payme dae as S D = S D, (2 D
2 where S is he sock price, r is he cosa ieres rae, σ is he volailiy ad W is a sadard Browia moio. S refers o he ime immediaely before he divided-payme mome, D, D ad S D o he mome immediaely afer. There are some commo objecios o his formulaio hough. A firs objecio may be he assumpio ha he sock price will fall by he amou of he divided size. This objecio is maily drive by he effecs axes have o he behavior of fiacial ages ad hus marke prices. We will o cosider his objecio i his paper ad hus assume model (2 o be valid. A secod objecio may be ha he divided payme dae ad amou are o precisely kow uil a few mohs before heir payme. We also believe his o be he case, bu o cosider i i a proper fashio, he model would grow sigificaly i complexiy. Our goal is raher o devise a simple variaio ha ca be applied o a wide class of models, ha does o worse he racabiliy of he model ad ha produces accurae resuls. Fially, oe ca argue ha he model admis egaive prices for he sock price S. This is i fac rue ad ca easily be see if oe akes he sock price S o be smaller ha D a ime D D. A simple soluio o his problem is o add a exra codiio, for example replace (2, S D = S D D if S > D. (3 D However, i mos pracical applicaios, his is o of grea imporace sice he vas majoriy of he compaies ha pay divideds, have divided amous ha amou o a small fracio of he sock price, i.e. less ha 10% of he sock price, rederig he probabiliy assiged o egaive prices very small. For his reaso we may drop his codiio wheever i would add sigifica complexiy. I he ex secio we review he exisig lieraure o his subjec ad he reasos ha uderlie he use of he mehod mos popular amog praciioers. We he ur o develop he formulas i secio 2 ad i secio 3 we reproduce he umerical resuls i 19]. Secio 4 cocludes. 1.3 Lieraure Review Here we shorly review he lieraure o modificaios of sock price models o cope wih he discree divided paymes. Mero 12] (1973 aalyzed he effec of discree divideds i America calls ad saes ha he oly reaso for early exercise is he exisece of uproeced divideds. Roll 15](1977, Geske 9](1979, ad Baroe-Adesi ad Whaley 1](1986 worked o he problem of fidig aalyic approximaios for America opios. Joh Hull 11] i he firs ediio of his book, 1989, esablishes wha was o be he mos used mehod o cope wih discree divideds. The mehod works by subracig from he curre asse price, he e prese value of all divideds o occur durig he life of he opio. The reasos for is populariy ad accepace were he facs ha i would preserve he coiuiy of he opio price across he divided payme dae ad ha i could cope wih muliple divideds. Oe ca show ha his formulaio is exac oly if he divided is paid immediaely afer he valuaio dae. O he oher ed of he specrum, Musiela ad Rukowsky 13] (1997 propose a model ha adds he fuure value a mauriy of all divideds paid durig he lifeime of he opio o he srike price. Agai oe ca show ha his formulaio is exac oly if he divided payme happes jus before he opio maures. To balace hese wo las mehods, Bos ad Vadermark 5] (2002 devise a mehod ha divides he divideds i ear ad far ad subracs he ear divideds from he sock price ad adds he far divideds o he srike price. This mehod performs beer ha he previous bu i is o exac, especially i he case of opios o a-he-moey. A mehod ha cosiders a coiuous geomeric Browia moio wih jumps a he divided payme daes is aalyzed i deail by Wilmo 20] (1998 by meas of umerical mehods. Berger ad Klei 2](1998 propose a o-recombiig biomial ree
3 mehod o evaluae opios uder he jump model. Bos e al. 4](2003 devise a mehod ha adjuss he volailiy parameer o correc he subracio mehod saed above. Haug e al. (2003 review exisig mehods performace ad pay special aeio o he problem of egaive prices ha arise wihi he coex of he jump model ad propose a umerical quadraure scheme. Björk 3] (1998 has oe of he cleares descripios of he discree divideds problem for Europea opios ad provides a formula for proporioal divideds. Shreve 16] (2004 also saes he resul for proporioal divideds. Recely, Vellekoop ad Nieuwehuis 19] (2006 describe a modificaio o he biomial ree mehod o accou for discree divideds preservig he crucial recombiig propery. 2 Closed Formula We sar by developig a arbirage argume ha will eable us o address he exisece of discree divideds i a uified approach, irrespecive of he opio ype ad model. Here we resric ourselves o he large se of models ha display he Markov propery. Our argumes are equivale o wha ca be foud i he lieraure, see Björk 3] for a clear ad cocise descripio. Cosider a opio wih a mauriy T o he uderlyig S, whose value a ime 0 is V (S 0, 0. We ow assume ha a uexpeced divided, of size D, is paid a he dae D, wih 0 < D < T. The value of he opio a he mome D is chaged because he uderlyig value has decreased by he size of he divided. More precisely, he opio value chage correspods o he followig differece V = V, D V D, D. (4 D Tha is, he differece bewee he value of he opio akig he uderlyig price jus before ad afer he divided payme. This differece measures he impac of he chage i he opio price wih respec o a discree chage (eiher accoued for or prediced i he value of he uderlyig. From he poi of view of he buyer/seller of such a opio, his chage i value may come as a uexpeced profi or loss, depedig o he specific opio oe cosiders. This profi or loss was o prediced by he hedgig sraegy, or could i be so because ay profis or losses from he hedgig sraegy should, by cosrucio, be refleced o he iiial price of he opio. This iclusio, i ur, would elimiae heir occurrece. We ow cosider he differece V as a payoff profile of a virual derivaive corac. This ew corac, ogeher wih he opio V, for which he exisece of divideds was igored, form a special kid of porfolio whose aggregae value is he value of he opio whe he exisece of he discree divided is cosidered. The special propery of his porfolio is ha oe of is compoes ca exis by hemselves. For Europea syle opios, his special propery is o a cosrai sice every claim i he porfolio has a lifespa from 0 o T. The same is o rue for America syle or barrier opios ha ca cease o exis before ime T. We are ow ready o sae he followig lemma. Lemma 2.1. Le V be radom variables defied o he filered probabiliy space (Ω, F, P, F. Le V be he arbirage-free price of a opio o a sock ha pays a discree divided D a ime D accordig o he policy I A, where A = {ω :he divided is paid}. The, for each ω Ω, ] V D, D (ω = V, D D V (ω + D, D D (ω V, D D (ω I A. (5
4 Proof. If ω A, he he divided policy deermies he acual payme of he divided, ad we have V D, D (ω = V D, D D (ω = V D, D (ω. (6 The opposie case, if ω / A, i.e., whe he divided policy deermies ha he divided is o paid, we have Thus we have verified ha (5 holds for each ω Ω. V D, D (ω = V, D D (ω = V D, D (ω. (7 Wih lemma 2.1 we have esablished he value of he derivaive V a ime D as a fucio of S a ime D, i.e., jus before he divided is paid. We ca ow cosider V D, D as a payoff of a claim maurig a D. Furhermore, he uderlyig asse process S, wih 0 < < D is jus he Black-Scholes model as i (1 sice here are o divided paymes durig his ime ierval. I his seig, he valuaio of he opio a ime 0, V 0, 0, is jus he discoued expecaio uder he risk-eural measure wih he process S give by (1. We ca hus sae he followig corollaries: Corollary 2.1. For Europea syle opios we have V D 0, 0 = ( e r( D 0 E Q 0 V S D (, D + V 0, 0 + e r( D 0 E Q 0 (V ( S D V ( S D ( D, D V ( D, D V S D S D, D I A ] =, D I A ], where E Q 0 X] sads for he expecaio uder he risk-eural measure Q, wih respec o he σ- algebra F 0, of he radom variable X. Corollary 2.2. For America syle opios we have ] V D 0, 0 = max E Q τ T 0 e r(τ0 Φ D (S τ, 0, D wih ad Φ (S τ Φ D (S τ = ( V S D ( V ( ( (, D + V S D, D V D S D S D if τ < D, D I A if τ = D, ], D = max E Q τ T D e r(τd Φ (S τ, (9 D,T ad Φ(x is he payoff of he opio if he sock price is a x, T D,T is he se of all soppig-imes akig values i he ierval ( D, T ad T 0, D he correspodig se for he ierval ( 0, D. Sice Lemma 2.1 does o impose ay codiios o he form of V D D, D, we are allowed o use i several imes o accou for several divided paymes, ieraig bewee he lemma ad he releva corollary for he opio a had. This procedure will be used i he followig secios o obai he pricig formula. (8
5 2.1 Geeral Derivaio I his secio we will focus o Europea syle opios. We will develop a geeral approach ad ideify he ecessary codiios for is validiy. I he ex secio, we provide a example of a call opio uder he Black-Scholes model which does saisfy he validiy codiios. The correspodig proofs are give i appedix A. As iduced by he lemma above, he approach o he divided problem should sar by argeig he las divided before mauriy, ad he proceed by movig backwards i ime uil he firs divided. Therefore, le here be divideds D i ad payme daes i, wih i = 1,, ad 0 < 1 < < < T. We will keep he oaio S o refer o he sock price a he ime jus before he i h divided i payme bu we will ideify i wih i o lighe oaio, sice hey are equal i he limi. Las Divided Before Mauriy From corollary 2.1 we ca sae ha C 1, 1 = C ( S 1, 1 + e r( 1 E Q 1 (C D, C, I A ], (10 where C sads for he price of he derivaive claim igorig he exisece of he divided D ad C is he price of he same claim bu ackowledgig he exisece of he divided D. I his form, ad for mos models, i is impossible o fid a closed formula soluio for he expecaio above. We hus choose o replace he differece ha i coais by is correspodig Taylor series expasio. The ecessary codiio for he exchage is he followig: Codiio 2.1. Le C, be a fucio ifiiely differeiable wih respec o S, ad le he correspodig ifiie Taylor series expasio be coverge. We hus have, ] E Q 1 (C D, C, I A = E Q (D i ( ] 1 C i S, I A, (11 wih C i referrig o he i h derivaive of he claim price wih respec o S 1. Furhermore, i would be helpful o ierchage he expecaio ad summaio i equaio (11. For his ierchage, he followig codiios have o hold: Codiio 2.2. Le (D i ( C i S, I A φ 1 To be cohere wih he oaio used below, we should wrie C (i isead of C i. The abbreviaed oaio was chose o lighe he formulas below ha make repeaed use of hese fucios.
coverge uiformly o ay ierval (a, b S, S D ], where φ deoes he disribuio fucio of S give F 1, ad may eiher E Q (D i ( ] 1 C i S, I A be fiie. or E Q (D i ( ] 1 C i S, I A This is eough o sae he geeral expressio for a call wih oly oe divided payme 6 C 1, 1 =C ( S 1, 1 + e r( 1 (D i ( ] E Q 1 C i S, I A. (12 Havig he covergece propery fulfilled for he series, oe ca ow rucae he series o a give η ha is goig o deped o he specific problem a had. Thus, we proceed wih a rucaed series C 1, 1 =C ( S 1, 1 + e r( 1 η (D i E Q 1 C i ( S, I A ], (13 wih η deermied i ligh of he specific problem a had, based o a crieria of added coribuio of he erms beyod he η erm. From he Las Divided o he Firs From corollary 2.1 ad lemma 2.1 we ca sae ha ( C 1 2, 2 = C S2, 2 + e r( 1 2 (14 ( ] E Q 2 C D 1, 1 C (S 1, 1 I A1, 1 where C sads for he price of he same derivaive claim igorig he exisece of he divided D 1, ad C 1, is price, ackowledgig ha fac. Agai, we would like o replace he differece by is correspodig Taylor series expasio. Cosequely, we have o impose ha C also fulfills codiio 2.1. Codiio 2.3. Le C, 1 1 ad le he correspodig ifiie Taylor series expasio be coverge. We he ge ( E Q 2 C 1 E Q (D 1 j 2 j! j=1 be a fucio ifiiely differeiable wih respec o S 1 D 1, 1 C 1 C j ( S, 1 1 ], 1 I A1 = (15 I A1.
7 Rewriig (14 wih (13 ad (15, we ge C 1 2, 2 = C ( S 2, 2 + η e r(2 E Q (D i ( ] 2 C i S, I A + ( e r( 1 2 E Q (D 1 j j 2 j! j=1 S j C 1 e r( 1 η (D i ( S, 1 + 1 E Q 1 C i ( S, I A ] I A1 ]. ( We resolve he derivaive of C S, 1 o esablish he erm for he direc effec of he 1 ( ] divided D 1, ad he derivaive wih respec o E Q 1 C i S, I A for he combied effec of boh divideds, C 1 2, 2 = C ( S 2, 2 + (16 η e r( 2 (D i ( ] E Q 2 C i S, I A + e r( 1 2 (D 1 j E Q j! 2 C ( ] j S, 1 I A1 + 1 e r( 2 E Q 2 j=1 (D 1 j η (D i j! j=1 ( ( ] j E Q 1 C i S, I A S j 1 I A1. We ca ow ierpre he coribuio of each erm o he fial resul. The firs erm ses he sarig poi as a call assumig o divideds. The firs wo summaios accou for he impac of he exisece of he divideds D ad D 1. The hird measures he combied effec of boh divideds. Oe ca also oice ha he umber of erms doubles wih each ew divided ha is added. Oe divided yielded wo erms ad wo divideds yielded four erms. The discou facor applied o each erm correspods o he dae of he las divided payme cosidered by ha erm. For furher divideds, he same procedure applies. 2.2 Call uder he Black Scholes Model I his secio we will apply he geeral approach o he case of a Europea call uder he Black Scholes model. Before we sar derivig he formula we cosider a simplificaio of he problems (13 ad (16. If he ses A i have a form differe from Ω, i.e., he divideds are paid i all saes of he world, equaio (13 becomes much more complex ad equaio (16 is ceraily oo complex o solve i closed form. Keepig formulas as simple as possible is he a exra argume i favor of assumig A i = Ω, addig o hose discussed i secio 1.2. Uforuaely, his assumpio raises a furher exra problem ulike hose discussed i secio 1.2. Eve hough he Black Scholes model does
{ ideed saisfy he codiios deermied i secio 2.1 if A i = ω : S > D i }, i fails o do so if i A i = Ω. The proofs are leghy ad ca be foud i appedix A. All proofs rely o he resuls by Esrella 7], who showed ha a Taylor series expasio of he Black Scholes formula, wih respec o he sock price S, coverges for a radius { of S iself. } Sill, give he fac ha he se ω : S D i has a very small probabiliy i almos all i realisic scearios, we believe ha he meioed simplificaio is a accepable compromise. As a less radical } simplificaio, we cosider also he possibiliy of keepig oly A = {ω : S > D wih regard o he divided payme closes o mauriy. This modificaio ca be applied by replacig formula (20 below by he formulas (57 ad (58 i appedix C. I fac, i he case of oly oe divided payme, his aleraive keeps he problem (13 uchaged ad hus is o subjec o ay of he argumes developed above. Wih he referred simplificaio, A i = Ω, he problem (16 is reduced o C 1, = C, + (17 η e r( (D i E Q C ( ] i S, + η 1 e r( 1 j=1 η 1 e r( Formulas ad Resuls j=1 (D 1 j j! (D 1 j j! E Q η ( ] C j S, 1 + 1 ( ( ] (D i E Q j E Q 1 C i S, S j. 1 We ake he Black Scholes model as i equaio (1 ad recall he formula for call prices, C, = S N(d + Ke r(t N(d, (18 d ± = log S K σ2 + (r ± 2 (T σ T, (19 where S is he curre price of he uderlyig sock, he curre ime, K he srike price, ad T he mauriy dae of he opio. N(x deoes he cumulaive fucio of he sadard ormal disribuio. Sice we are goig o rely heavily o he derivaives of his fucio, i is useful o meio here as well, he formula for a i h derivaive of he call price followig Carr 6] C i, = S i i S 1 (i, j δ j, (20 j=1 δ j = S N (d + + Ke r(t N (d σ j2 H h (d ( T h, h=0 σ T where N (x deoes he probabiliy desiy fucio of he sadard ormal disribuio, S 1 (i, j he Sirlig umber of he firs kid ad H i (d are Hermie polyomials. Also from Carr 6] we eed he resul ha allows us o express he derivaives of call prices, i he BS model, wih respec o S, i he form of expecaios. 8
9 I geeral, he spacial derivaives ca be calculaed by he followig expecaio ] C i, = e (r+ 1 2 iσ2 (i1(t E Si f (i (S T (21 where he operaor E Si idicaes ha he expecaio is calculaed from he diffusio ds = S ( (r + iσ 2 d + σdw i,, (22 ad where W i, is a sadard Browia moio uder he measure S i, which has S i as umeraire. f (i (S T is he i h derivaive of he payoff fucio wih respec o S T. For example, he dela of a call C 1, is he value, a ime, of a derivaive payig I ST >K uis of he uderlyig sock S, expressed also i uis of S, C 1, = E S I ST >K]. We shall also eed expecaios of he derivaives observed a a give ime, such as E Q C i, ], wih T. (23 By (21 ad makig use of he ower propery for codiioal expecaios, we have C i, ] ] = e (r+ 1 2 iσ2 (i1(t E Si f (i (S T. (24 E Si To compue (23 we sill eed he followig Proposiio 2.1. Le v i, = E Si f (i (S T ], he C i, ] ( = e (r+ 1 2 iσ2 (i1(t v i e iσ2 ( S,. (25 E Q Proof. From (21, E Q C i, ] ]] = E Q e (r+ 1 2 iσ2 (i1(t E Si f (i (S T ]] = e (r+ 1 2 iσ2 (i1(t E Q E Si f (i (S T. (26 The diffusios wih respec o which he expecaios E Q ad E Si are ake, respecively equaios (1 ad (22, differ oly i he drif erm sice W i, ad W are Browia moios uder heir respecive measures. Thus, o chage he measure from S i o Q, oe ca compesae he differe drif by chagig he sarig value codiio such ha he soluios of he diffusios are equal. Le, S,Q ad S,S i be he iiial codiios for diffusios (1 ad (22 respecively. The, S = S,Q e S = S,S ie (r σ2 2 (r+iσ 2 σ2 2 If we ow make S,S i = S,Q e iσ2 ( he, uder S i, S = S,Q e ( +σw uder Q, ad (27 ( +σw i, uder S i. (28 (r σ2 2 ( +σw i,. (29 This soluio is ow equivale, i a weak sese, o (27, ( sice he disribuio of S is, i boh cases, a log-ormal disribuio wih parameers µ LN = r σ2 2 ( ad σ LN = σ.
10 Figure 1: Illusraio of he weak covergece argume ha relaes wo expecaios of he same radom variable uder differe measures. The equaliy of disribuios is eough o sae he followig equaliy of expecaios, E Q X ] = E Si X ]. (30 S,S i=s,q e iσ2 ( Figure 1 illusraes he relaioship bewee wo expecaio of he same radom variables uder he measures Q ad S i. Recallig equaio (26, we have E Q C i, ] = e (r+ 1 2 iσ2 (i1(t E Q = e (r+ 1 2 iσ2 (i1(t E Si = e (r+ 1 2 iσ2 (i1(t E Si ]] E Si f (i (S T E Si f (i (S T ]] S,S i=s,q e iσ2 ( f (i (S T ] S,S i=s,q e iσ2 (, where we agai used he ower propery o prove he proposiio. Combiig equaio (21 ad proposiio 2.1 we obai E Q C i, ] = e (r+ 1 2 iσ2 (i1( C i ( e iσ2 ( S,. (31 We ca ow proceed o sae he formula. Closed Formula Despie he series covergece wihi he referred radius, Esrella 7] meios also sigifica isabiliy if oe akes a low order approximaio. We will be reurig o his issue whe we cosider a specific example. For a call opio wih oly oe divided payme durig he ime uil mauriy, we refer o he case of equaio (12 ha we recall here, η C, = C, + e r( (D i E Q C i ( S, ].
11 We ow use equaio (31 ad cacel he discou facor e r( o ge C, =C, + (32 η (D i ( e (r+ 1 2 (i1σ2 i( C i e iσ2 ( S,, wih C i as i (20, ad C as i (18. The derivaive order η is he ecessary order o obai covergece wih respec o each specific opio valuaio. For opios wih wo divided paymes before mauriy, we recall equaio (16, which raslaes o C 1, = C, + (33 η 1 (D 1 i ( e (r+ 1 2 (i1σ2 i( 1 C i e iσ2 ( 1 S, + η 1 j=1 (D 1 j j! η (D i ( d C i+j e iσ2 ( 1 (i+jσ 2 ( 1 S,, i he Black-Scholes model, wih { d = exp (r + 12 (i 1σ2 i( 1 (r + 12 (i + j 1σ2 (i + j( 1 } iσ 2 j( 1. Agai here, he derivaives orders η, η 1 are he ecessary orders o obai covergece wih respec o each specific opio valuaio. A sep-by-sep derivaio of formula (33 ca be foud i appedix B. 2.3 The Greeks A closed formula for he derivaive 2 of he opio price of arbirary order is a sraighforward applicaio of he chai rule. Thus, for he g h derivaive of he call price wih oe discree divided payme we have C(S g, =C g, + (34 η (D i e (r+( ( 1 2 (i1+gσ2 i( C i+g e iσ2 ( S,. 2 The derivaives of he opio price are usually called Greeks because Greek alphabe leers are commoly used o deoe hem.
12 Similarly, for he call wih wo divided paymes we have C g 1, = C(S g, + (35 η 1 (D 1 i e (r+( ( 1 2 (i1+gσ2 i( 1 C i+g e iσ2 ( 1 S, + η 1 j=1 (D 1 j j! η (D i ( d g C i+j+g e iσ2 ( 1 (i+jσ 2 ( 1 S,, wih { d g = exp ( r + ( r + ( 1 (i 1 + g σ 2 i( 1 (36 2 ( 1 (i + j 1 + g σ 2 (i + j( 1 2 iσ 2 j( 1 }. The derivaives of he call price wih respec o oher variables ha he spaial variable S ca be obaied as a fucio of he laer. I paricular C σ = σs2 C 2, (T, C r = ( S C 1, C, (T, C = rc, rs C 1, 1 2 S2 C 2,. Furher deails ca be foud i Carr 6] ad i Reiß ad Wysup 14]. 3 Resuls For ease of referece we reproduce he resuls saed i Vellekoop ad Nieuwehuis 19] for Europea call opios wih seve discree divided paymes. The model parameers are se a S 0 = 100, σ = 25% ad r = 6%. Furhermore he sock will pay oe divided per year, wih each divided oe year afer he previous, of amou 6, 6.5, 7, 7.5, 8, 8 ad 8 for he firs seve years respecively. We cosider hree differe scearios of divided sream paymes refereced by he payme dae of he firs divided 1, se a 0.1, 0.5 ad 0.9. Wih respec o he call opio specificaios, we cosider hree differe opios, all wih seve years mauriy, wih srikes of 70, 100 ad 130. The calculaios repored i Table 1 were performed akig a secod order approximaio for each of he divided paymes, i.e., η 1,..., η 7 = 2. This approximaio order proved o be very effecive i his case, producig errors of 0.01 i he wors cases whe compared o he resuls repored i Vellekoop ad Nieuwehuis 19]. For referece we also repor he values ha he wo mos frequely used mehods i pracice produce uder Modified sock price ad Modified srike price. I should be oed ha he values hese mehods produce differ raher srogly from he closed formula approximaio. To illusrae he imporace of several erms prese i he formula, we aggregae hem by he umber of divideds ivolved i each erm. We hus have direc erms, combied effec erms of wo divideds, ad combied effec erms of a umber of divideds up o (i his case seve.
13 The derivaives of he call price are also displayed sarig wih he firs wo spaial derivaives, dela ad gamma, ad followed by he derivaives wih respec o σ, ad r, vega, hea ad rho. All calculaios were performed i Mahemaica c usig he rouies saed i appedix D. Table 1: Europea call, σ = 25%, r = 6%, S 0 = 100, T = 7 1 = 0.1 1 = 0.5 1 = 0.9 Srikes 70 100 130 70 100 130 70 100 130 Call wih o divideds 56.5642 42.5839 31.9696 56.5642 42.5839 31.9696 56.5642 42.5839 31.9696 Terms wih divideds direc effec, combied effec of 2 divs,..., combied effec of divs. -36.6298-30.6458-24.7611-35.4202-29.4067-23.5880-34.2355-28.2015-22.4556 4.0040 5.0787 5.1579 4.1083 5.0173 4.9599 4.1882 4.9289 4.7451 1.0166 0.5521 0.1572 0.9272 0.4264 0.0547 0.8234 0.3053-0.0336-0.0335-0.1165-0.1114-0.0752-0.1257-0.1024-0.1078-0.1258-0.0889-0.0358-0.0148-0.0021-0.0311-0.0080 0.0021-0.0236-0.0018 0.0048 0.0002 0.0017 0.0013 0.0018 0.0018 0.0010 0.0027 0.0015 0.0006 0.0003 0.0001 0.0000 0.0002 0.0000-0.0001 0.0001 0.0000-0.0001 Toal opio price 24.8862 17.4394 12.4114 26.0752 18.4890 13.2968 27.2117 19.4905 14.1419 Opio price repored i Vellekoop ad Nieuwehuis 19] 24.90 17.43 12.40 26.08 18.48 13.29 27.21 19.48 14.13 Modified sock price mehod 20.1576 12.3709 7.7556 20.1576 12.3709 7.7556 20.1576 12.3709 7.7556 Modified srike price mehod 30.7358 23.1768 17.5976 30.7358 23.1768 17.5976 30.7358 23.1768 17.5976 Spaial derivaives dela (per ce, gamma (per e housad. 70.6821 56.0090 43.8271 71.1645 56.9270 44.9643 71.6629 57.8120 46.0412 60.6487 73.2547 75.1486 57.4643 70.0730 72.6798 54.6009 67.2527 70.5040 Oher derivaives vega, hea ad rho. 35.5295 42.9144 44.0239 33.6640 41.0505 42.5776 31.9866 39.3989 41.3030-1.5950-1.6643-1.5629-1.5071-1.5999-1.5234-1.4263-1.5411-1.4874 112.0643 104.7648 90.6232 105.6951 101.1355 89.0311 99.7581 97.7093 87.4769 I ay sceario of he case beig aalyzed, he umber of evaluaios of he Black-Scholes pricig formula or ay of is derivaives amous o 2187. To udersad how he differe erms coribue o he oal umber of evaluaios, Table 2 breaks i dow by erms aggregaed by he umber of divided paymes hey have. I he simples case of oly oe divided payme uil mauriy, Table 2 would be much differe. The call wih o divideds plus wo derivaives would yield a oal of oly hree erms. Naurally, oe should expec ha differe cases may require differe orders of approximaio. Problems wih larger idividual divideds, wih divideds very close o mauriy or i he presece of very low volailiies should require a higher approximaio order. The issue a sake is how smooh he fucio beig approximaed is - he smooher he fucio he lower he required approximaio order. 4 Coclusio Deparig from he well kow behavior of he opio price a he divided payme dae, we approximae i by Taylor series expasio ad successfully maipulae i o arrive a a closed form pricig formula.
14 Table 2: Number of Calls o he Black Scholes pricig formula or is derivaive. Number of erms Number of parcels Number of evaluaios of BS wih divideds ( (1 i each erm (2 or is derivaive 7 η 1... η (1x(2 0 1 1 1 1 7 2 14 2 21 4 84 3 35 8 280 4 35 16 560 5 21 32 672 6 7 64 448 7 1 128 128 Toal: 2187 We do so by makig use of he resuls from Carr 6] ad by a appropriae reparamerizaio of he Black Scholes sochasic differeial equaio. The procedure is described for opios o socks wih up o wo divided paymes ad is applied o a example wih seve divided paymes. Formulas for more divided paymes may be developed usig he same procedure. By he same meas we also derive he derivaives, of ay order, of he opio price wih respec o he uderlyig asse; ad from hose he remaiig derivaives wih respec o oher parameers i he model. The resuls show ha a low approximaio order is eough o aai reasoable resuls i mos problems observed i pracice. The exesio of his resuls o oher models ha he Black Scholes model is lef for fuure research. Tha is he case also for he exesio of his approach o muli-asse opios. Appedices A Codiio s Verificaio Codiios Verificaio Codiio 2.1 is easily saisfied bu i his a he limiaios ha his mehod will have. I reads: Le C, be a fucio ifiiely differeiable o S, ad le he correspodig ifiie Taylor series expasio be coverge. Proof. From equaio (18 oe ca see ha he call price C, is he differece of producs of ifiiely differeiable fucios if < T ad hus saisfies he codiio. If = T, he N(x urs io a sep fucio ad hus is o loger differeiable. Thus, he firs codiio deermies ha S R +. The covergece of C, was proved by Esrella 7] for shifs D such ha S < D < S. Thus, he ierval o which he fucio is differeiable ad whose series is coverge is S < D < S, (37 which icludes he releva ierval for divided shifs 0, S. Codiio 2.2 reads: le ( (D i C i S, D ], ad may E Q 1 I A φ coverge uiformly o ay ierval (a, b S, S ( ] (D i C i S, I A or ( ] EQ (D i 1 C i S, I A be
15 fiie. Wih φ he disribuio fucio of S give F 1. Proof. The covergece ierval of he produc of he fucios is he iersecio of boh iervals. The covergece of C, has already bee esablished i (37 ad so we oly eed o verify he covergece properies of I A ad φ. Fucio I A is rivially coverge eiher o D < S or o D > S. Wih respec o φ, i is well kow ha i he Black Scholes model, he disribuio of S is logormal ad hus φ is he composiio of log ad he ormal probabiliy desiy fucio. Apar from he cosas, he ormal desiy exp { (x µ 2 /(2σ 2 } / 2πσ ca by described as he successive composiio of e x ad x 2. Boh fucios are coverge everywhere, so is he ormal desiy. The composiio wih log resrics he covergece radius o ha of he log fucio, i.e., (0, 2S. I erms of shifs D wih respec o S we have S < D < S. Thus o resricios mus be imposed o he domai of covergece foud i (37. For he fial saeme of he codiio, i suffices o show ha E Q ( ] (D i 1 C i S, I A is fiie. Sice he series ( (D i C i S, coverges o C D, C, ad C, is fiie for every S as log as < T, he codiio is verified. Fially, codiio 2.3 saes: le C, 1 be a fucio ifiiely differeiable o S 1 1 ad le he correspodig ifiie Taylor series expasio be coverge. Proof. The differeiabiliy of( C is deermied by ha of C, proved above wih respec o codiio 2.1, ad (Di E Q 1 C i S, I A ]. For he differeiabiliy of he las, we oe, by (31, ( ha i is deermied by ha of CI i A S,, which is saisfied by way of (57 ad (58. The covergece of C, 1 is deermied by ha of C, deermied i he proof of 2.1, 1 ad by ha of he series ( η ( ] (D 1 j j (D i E Q 1 C i S, I A j! S j. (38 1 j=1 The correspodig covergece ierval of C wih respec o shifs D 1 is he followig The remaiig erm (38 ca be resaed makig use of (31 (D 1 j j=1 j! η (D i S < D 1 < S 1. (39 1 j ( e (r+ 1 2 iσ2 (i1( 1 C i I A (e iσ2 ( 1 S 1, 1 S j 1,
16 which he yields, e r( 1 η (D e (r+ 1 i 2 (i1σ2 ( 1 ( j D 1 e iσ2 ( 1 j! j=1 C i+j I A (e iσ2 ( 1 S, 1. (40 1 Now, for each i, he series above is he Taylor series expasio of he i h derivaive of C IA, or C i I A, muliplied by he cosa (D e (r+ 1 2 (i1σ2 ( 1 i. The cosa does o affec he covergece of C i I A. The covergece of CI i A is assured by he same argumes ha yield he covergece of he call price fucio foud i Esrella 7]. Followig he same reasoig, we ake oe specific derivaive o perform he aalysis, i = 3, sice i carries he same covergece propery of ay oher derivaive. We oe he ha CI 3 A is a produc of cosas, ψ 1, ψ 2 ad of divisio by S 3. Sice he divisio ad he cosas do o aler he covergece, we eed o check he fucios ψ 1 ad ψ 2. Agai, we have muliplicaive cosas ad he followig facors ha deped o S: N (d ad H h (d. Boh of he facors ca be obaied by successive composiio of some or all of he fucios e x, x 2 ad log(x ad hus he covergece radius shall be he iersecio of he covergece radius of all of hese fucios. Sice he expoeial ad he square fucios are coverge everywhere, he radius of covergece of CI i A is deermied by he covergece of log(x ad ha is x or, i ( our case, S. The composiio of C i 1 I A wih f S = e iσ2 ( 1 S 1 does o chage he ( ( 1 ( covergece radius sice log f = log S 1 iσ 2 ( 1 ad hus log S 1 is sill 1 he fucio ha deermies he covergece ierval. Furhermore, he shifs D 1 e iσ2 ( 1 are smaller ha D 1 for ay i. Thus, we ca ake he followig covergece ierval for he Taylor series expasio of CI i A for all i, S < D 1 < S 1. (41 1 Thus, he fulfillme of codiios ecessary for he developme of a closed formula have redered he followig resricios o he size of divideds: S < D < S ad S < D 1 < S 1. 1 These resricios overlap wih he divided payme policy ha deermies divided paymes if 0 D < S or 0 D 1 < S ad hus are fulfilled by cosrucio. 1
17 B Derivaio of he Two Divideds Formula For wo opios wih wo divideds o mauriy, we fall back o he case of equaio (16 ha we recall C 1 2, 2 = C ( S 2, 2 + e r( 2 (D i E Q 2 C i, ] + e r( 1 2 (D 1 j E Q j! 2 C j ( ] S 1, 1 + j=1 ( e r( 2 (D 1 j (D i E Q j E Q 1 C i, ] j! 2 S j. 1 j=1 (42a (42b (42c (42d The evaluaio of he erms (42a, (42b ad (42c are doe as i he oe divided case (32. The soluio for he erm (42d is obai by he followig procedure: by equaio (31 E Q 2 E Q 2 ( j E Q 1 C i, ] S j = (43 1 ( ( j e (r+ 1 2 iσ2 (i1( 1 C i e iσ2 ( 1 S 1, 1 S j, (44 1 by akig he cosa c 1 = e (r+ 1 2 iσ2 (i1( 1 ou of he expecaio ad by resolvig he derivaive, c 1 E Q 2 C i+j ( e iσ2 ( 1 S 1, 1 e iσ2 j( 1 ], (45 by akig he cosa c 2 = e iσ2 j( 1 ou of he expecaio ad equaio (30 c 1 c 2 E Si+j 2 C i+j ( e iσ2 ( 1 S 1, 1 ] S2,Si+j, (46 S 2,S i+j = S 2,Qe (i+jσ2 ( 1 2, (47 by equaio (21 ad by akig ou of he expecaio he cosa c 3, wih c 3 = e (r+ 1 2 (i+jσ2 (i+j1(t 1 ] c 1 c 2 c 3 E Si+j 2 E Si+j 1 f (i+j (S T S1,Si+j ] S2,S i+j, (48 S 1,S i+j = S 1,Qe iσ2 ( 1, (49 S 2,S i+j = S 2,Qe (i+jσ2 ( 1 2, (50