No. 16. Closed Formula for Options with Discrete Dividends and its Derivatives. Carlos Veiga, Uwe Wystup. October 2008


 Erik Berry
 3 years ago
 Views:
Transcription
1 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 Publisher: Frakfur School of Fiace & Maageme Phoe: +49 ( Fax: +49 ( Soemasr D Frakfur/M. Germay
2 Closed Formula for Opios wih Discree Divideds ad is Derivaives Carlos Veiga Tel.: +49 ( Fax.: +49 ( Frakfur School of Fiace & Maageme Cere for Pracical Quaiaive Fiace Soemasraße 911, Frakfur am Mai Uwe Wysup Tel.: +49 ( Fax.: +49 ( Frakfur School of Fiace & Maageme Cere for Pracical Quaiaive Fiace Soemasraße 911, 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
3 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 localvol 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 Moivaio Descripio of he Problem Lieraure Review Closed Formula Geeral Derivaio Call uder he Black Scholes Model The Greeks 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
4 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 BlackScholes (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
5 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 dividedpayme 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 BaroeAdesi 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 ahemoey. 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 orecombiig biomial ree
6 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 arbiragefree 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
7 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 BlackScholes 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 riskeural 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 riskeural 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 soppigimes 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
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 BlackScholes 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.
9 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.
10 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, 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
11 { 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, ( ( ] (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
12 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 logormal disribuio wih parameers µ LN = r σ2 2 ( ad σ LN = σ.
13 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, ].
14 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 BlackScholes 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 sepbysep 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.
15 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.
16 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 = = = 0.9 Srikes Call wih o divideds Terms wih divideds direc effec, combied effec of 2 divs,..., combied effec of divs Toal opio price Opio price repored i Vellekoop ad Nieuwehuis 19] Modified sock price mehod Modified srike price mehod Spaial derivaives dela (per ce, gamma (per e housad Oher derivaives vega, hea ad rho I ay sceario of he case beig aalyzed, he umber of evaluaios of he BlackScholes pricig formula or ay of is derivaives amous o 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.
17 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( 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 muliasse 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
18 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,
19 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
20 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
CHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING
CHAPTER 22 ASSET BASED FINANCING: LEASE, HIRE PURCHASE AND PROJECT FINANCING Q.1 Defie a lease. How does i differ from a hire purchase ad isalme sale? Wha are he cash flow cosequeces of a lease? Illusrae.
More informationConfidence Intervals for Paired Means
Chaper 496 Cofidece Iervals for Paired Meas Iroducio This rouie calculaes he sample size ecessary o achieve a specified disace from he paired sample mea erece o he cofidece limi(s) a a saed cofidece level
More informationDuration Outline and Reading
Deb Isrumes ad Markes Professor Carpeer Duraio Oulie ad Readig Oulie Ieres Rae Sesiiviy Dollar Duraio Duraio Buzzwords Parallel shif Basis pois Modified duraio Macaulay duraio Readig Tuckma, Chapers 5
More informationBullwhip Effect Measure When Supply Chain Demand is Forecasting
J. Basic. Appl. Sci. Res., (4)4743, 01 01, TexRoad Publicaio ISSN 0904304 Joural of Basic ad Applied Scieific Research www.exroad.com Bullwhip Effec Measure Whe Supply Chai emad is Forecasig Ayub Rahimzadeh
More informationUNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Katarína Sakálová
The process of uderwriig UNDERWRITING AND EXTRA RISKS IN LIFE INSURANCE Kaaría Sakálová Uderwriig is he process by which a life isurace compay decides which people o accep for isurace ad o wha erms Life
More informationThe Term Structure of Interest Rates
The Term Srucure of Ieres Raes Wha is i? The relaioship amog ieres raes over differe imehorizos, as viewed from oday, = 0. A cocep closely relaed o his: The Yield Curve Plos he effecive aual yield agais
More informationSecond Order Linear Partial Differential Equations. Part II
Secod Order iear Parial Differeial Equaios Par II Fourier series; EulerFourier formulas; Fourier Covergece Theorem; Eve ad odd fucios; Cosie ad Sie Series Eesios; Paricular soluio of he hea coducio equaio
More informationRanking of mutually exclusive investment projects how cash flow differences can solve the ranking problem
Chrisia Kalhoefer (Egyp) Ivesme Maageme ad Fiacial Iovaios, Volume 7, Issue 2, 2 Rakig of muually exclusive ivesme projecs how cash flow differeces ca solve he rakig problem bsrac The discussio abou he
More informationFORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND
FORECASTING MODEL FOR AUTOMOBILE SALES IN THAILAND by Wachareepor Chaimogkol Naioal Isiue of Developme Admiisraio, Bagkok, Thailad Email: wachare@as.ida.ac.h ad Chuaip Tasahi Kig Mogku's Isiue of Techology
More informationAPPLICATIONS OF GEOMETRIC
APPLICATIONS OF GEOMETRIC SEQUENCES AND SERIES TO FINANCIAL MATHS The mos powerful force i he world is compoud ieres (Alber Eisei) Page of 52 Fiacial Mahs Coes Loas ad ivesmes  erms ad examples... 3 Derivaio
More informationANNUITIES AND LIFE INSURANCE UNDER RANDOM INTEREST RATES
ANNUITIES AND LIFE INSURANCE UNDER RANDOM INTEREST RATES Bey Levikso Deparme of Saisical Uiversiy of Haifa Haifa, 395 ISRAEL beyl@rsa.haifa.ac.il Rami Yosef Deparme of Busiess Admiisraio Be Gurio Uiversiy,
More information4. Levered and Unlevered Cost of Capital. Tax Shield. Capital Structure
4. Levered ad levered Cos Capial. ax hield. Capial rucure. Levered ad levered Cos Capial Levered compay ad CAP he cos equiy is equal o he reur expeced by sockholders. he cos equiy ca be compued usi he
More informationON THE RISKNEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW ABSTRACT KEYWORDS 1. INTRODUCTION
ON THE RISKNEUTRAL VALUATION OF LIFE INSURANCE CONTRACTS WITH NUMERICAL METHODS IN VIEW BY DANIEL BAUER, DANIELA BERGMANN AND RÜDIGER KIESEL ABSTRACT I rece years, markecosise valuaio approaches have
More informationREVISTA INVESTIGACION OPERACIONAL VOL. 31, No.2, 159170, 2010
REVISTA INVESTIGACION OPERACIONAL VOL. 3, No., 5970, 00 AN ALGORITHM TO OBTAIN AN OPTIMAL STRATEGY FOR THE MARKOV DECISION PROCESSES, WITH PROBABILITY DISTRIBUTION FOR THE PLANNING HORIZON. Gouliois E.
More informationWhy we use compounding and discounting approaches
Comoudig, Discouig, ad ubiased Growh Raes Near Deb s school i Souher Colorado. A examle of slow growh. Coyrigh 00004, Gary R. Evas. May be used for orofi isrucioal uroses oly wihou ermissio of he auhor.
More information3. Cost of equity. Cost of Debt. WACC.
Corporae Fiace [090345] 3. Cos o equiy. Cos o Deb. WACC. Cash lows Forecass Cash lows or equiyholders ad debors Cash lows or equiyholders Ecoomic Value Value o capial (equiy ad deb)  radiioal approach
More informationExchange Rates, Risk Premia, and Inflation Indexed Bond Yields. Richard Clarida Columbia University, NBER, and PIMCO. and
Exchage Raes, Risk Premia, ad Iflaio Idexed Bod Yields by Richard Clarida Columbia Uiversiy, NBER, ad PIMCO ad Shaowe Luo Columbia Uiversiy Jue 14, 2014 I. Iroducio Drawig o ad exedig Clarida (2012; 2013)
More informationFEBRUARY 2015 STOXX CALCULATION GUIDE
FEBRUARY 2015 STOXX CALCULATION GUIDE STOXX CALCULATION GUIDE CONTENTS 2/23 6.2. INDICES IN EUR, USD AND OTHER CURRENCIES 10 1. INTRODUCTION TO THE STOXX INDEX GUIDES 3 2. CHANGES TO THE GUIDE BOOK 4 2.1.
More informationManaging Learning and Turnover in Employee Staffing*
Maagig Learig ad Turover i Employee Saffig* YogPi Zhou Uiversiy of Washigo Busiess School Coauhor: Noah Gas, Wharo School, UPe * Suppored by Wharo Fiacial Isiuios Ceer ad he Sloa Foudaio Call Ceer Operaios
More informationCapital Budgeting: a Tax Shields Mirage?
Theoreical ad Applied Ecoomics Volume XVIII (211), No. 3(556), pp. 314 Capial Budgeig: a Tax Shields Mirage? Vicor DRAGOTĂ Buchares Academy of Ecoomic Sudies vicor.dragoa@fi.ase.ro Lucia ŢÂŢU Buchares
More informationModeling the Nigerian Inflation Rates Using Periodogram and Fourier Series Analysis
CBN Joural of Applied Saisics Vol. 4 No.2 (December, 2013) 51 Modelig he Nigeria Iflaio Raes Usig Periodogram ad Fourier Series Aalysis 1 Chukwuemeka O. Omekara, Emmauel J. Ekpeyog ad Michael P. Ekeree
More informationDerivative Securities: Lecture 7 Further applications of BlackScholes and Arbitrage Pricing Theory. Sources: J. Hull Avellaneda and Laurence
Deivaive ecuiies: Lecue 7 uhe applicaios o Blackcholes ad Abiage Picig heoy ouces: J. Hull Avellaeda ad Lauece Black s omula omeimes is easie o hik i ems o owad pices. Recallig ha i Blackcholes imilaly
More informationTHE FOREIGN EXCHANGE EXPOSURE OF CHINESE BANKS
Workig Paper 07/2008 Jue 2008 THE FOREIGN ECHANGE EPOSURE OF CHINESE BANKS Prepared by Eric Wog, Jim Wog ad Phyllis Leug 1 Research Deparme Absrac Usig he Capial Marke Approach ad equiyprice daa of 14
More informationResearch Article Dynamic Pricing of a Web Service in an Advance Selling Environment
Hidawi Publishig Corporaio Mahemaical Problems i Egieerig Volume 215, Aricle ID 783149, 21 pages hp://dx.doi.org/1.1155/215/783149 Research Aricle Dyamic Pricig of a Web Service i a Advace Sellig Evirome
More informationTeaching Bond Valuation: A Differential Approach Demonstrating Duration and Convexity
JOURNAL OF EONOMIS AND FINANE EDUATION olume Number 2 Wier 2008 3 Teachig Bod aluaio: A Differeial Approach Demosraig Duraio ad ovexi TeWah Hah, David Lage ABSTRAT A radiioal bod pricig scheme used i iroducor
More informationCircularity and the Undervaluation of Privatised Companies
CMPO Workig Paper Series No. 1/39 Circulariy ad he Udervaluaio of Privaised Compaies Paul Grou 1 ad a Zalewska 2 1 Leverhulme Cere for Marke ad Public Orgaisaio, Uiversiy of Brisol 2 Limburg Isiue of Fiacial
More informationThe Norwegian Shareholder Tax Reconsidered
The Norwegia Shareholder Tax Recosidered Absrac I a aricle i Ieraioal Tax ad Public Fiace, Peer Birch Sørese (5) gives a ideph accou of he ew Norwegia Shareholder Tax, which allows he shareholders a deducio
More informationOptimal Combination of International and Intertemporal Diversification of Disaster Risk: Role of Government. Tao YE, Muneta YOKOMATSU and Norio OKADA
京 都 大 学 防 災 研 究 所 年 報 第 5 号 B 平 成 9 年 4 月 Auals of Disas. Prev. Res. Is., Kyoo Uiv., No. 5 B, 27 Opimal Combiaio of Ieraioal a Ieremporal Diversificaio of Disaser Risk: Role of Goverme Tao YE, Muea YOKOMATSUaNorio
More informationTHE IMPACT OF FINANCING POLICY ON THE COMPANY S VALUE
THE IMPACT OF FINANCING POLICY ON THE COMPANY S ALUE Pirea Marile Wes Uiversiy of Timişoara, Faculy of Ecoomics ad Busiess Admiisraio Boțoc Claudiu Wes Uiversiy of Timişoara, Faculy of Ecoomics ad Busiess
More informationConvergence of Binomial Large Investor Models and General Correlated Random Walks
Covergece of Biomial Large Ivesor Models ad Geeral Correlaed Radom Walks vorgeleg vo Maser of Sciece i Mahemaics, DiplomWirschafsmahemaiker Urs M. Gruber gebore i Georgsmariehüe. Vo der Fakulä II Mahemaik
More informationAn Approach for Measurement of the Fair Value of Insurance Contracts by Sam Gutterman, David Rogers, Larry Rubin, David Scheinerman
A Approach for Measureme of he Fair Value of Isurace Coracs by Sam Guerma, David Rogers, Larry Rubi, David Scheierma Absrac The paper explores developmes hrough 2006 i he applicaio of markecosise coceps
More informationDBIQ USD Investment Grade Corporate Bond Interest Rate Hedged Index
db Idex Developme Sepember 2014 DBIQ Idex Guide DBIQ USD Ivesme Grade Corporae Bod Ieres Rae Hedged Idex Summary The DBIQ USD Ivesme Grade Corporae Bod Ieres Rae Hedged Idex (he Idex ) is a rule based
More informationTotal Return Index Calculation Methodology
Toal Reur Idex Calculaio Mehodology Las updaed: Ocober 31, 2011 1 Impora Noice: 1. This Eglish versio is o a officially accurae raslaio of he origial Thai docume. I ay cases where differeces arise bewee
More information1/22/2007 EECS 723 intro 2/3
1/22/2007 EES 723 iro 2/3 eraily, all elecrical egieers kow of liear sysems heory. Bu, i is helpful o firs review hese coceps o make sure ha we all udersad wha his heory is, why i works, ad how i is useful.
More informationA Queuing Model of the Ndesign Multiskill Call Center with Impatient Customers
Ieraioal Joural of u ad e ervice, ciece ad Techology Vol.8, o., pp. hp://dx.doi.org/./ijuess..8.. A Queuig Model of he desig Muliskill Call Ceer wih Impaie Cusomers Chuya Li, ad Deua Yue Yasha Uiversiy,
More informationReaction Rates. Example. Chemical Kinetics. Chemical Kinetics Chapter 12. Example Concentration Data. Page 1
Page Chemical Kieics Chaper O decomposiio i a isec O decomposiio caalyzed by MO Chemical Kieics I is o eough o udersad he soichiomery ad hermodyamics of a reacio; we also mus udersad he facors ha gover
More informationA Strategy for Trading the S&P 500 Futures Market
62 JOURNAL OF ECONOMICS AND FINANCE Volume 25 Number 1 Sprig 2001 A Sraegy for Tradig he S&P 500 Fuures Marke Edward Olszewski * Absrac A sysem for radig he S&P 500 fuures marke is proposed. The sysem
More informationA panel data approach for fashion sales forecasting
A pael daa approach for fashio sales forecasig Shuyu Re(shuyu_shara@live.c), TsaMig Choi, Na Liu Busiess Divisio, Isiue of Texiles ad Clohig, The Hog Kog Polyechic Uiversiy, Hug Hom, Kowloo, Hog Kog Absrac:
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationCAPITAL BUDGETING. Capital Budgeting Techniques
CAPITAL BUDGETING Capial Budgeig Techiques Capial budgeig refers o he process we use o make decisios cocerig ivesmes i he logerm asses of he firm. The geeral idea is ha he capial, or logerm fuds, raised
More informationHYPERBOLIC DISCOUNTING IS RATIONAL: VALUING THE FAR FUTURE WITH UNCERTAIN DISCOUNT RATES. J. Doyne Farmer and John Geanakoplos.
HYPERBOLIC DISCOUNTING IS RATIONAL: VALUING THE FAR FUTURE WITH UNCERTAIN DISCOUNT RATES By J. Doye Farmer ad Joh Geaakoplos Augus 2009 COWLES FOUNDATION DISCUSSION PAPER NO. 1719 COWLES FOUNDATION FOR
More informationGeneral Bounds for Arithmetic Asian Option Prices
The Uiversiy of Ediburgh Geeral Bouds for Arihmeic Asia Opio Prices Colombia FX Opio Marke Applicaio MSc Disseraio Sude: Saiago Sozizky s1200811 Supervisor: Dr. Soirios Sabais Augus 16 h 2013 School of
More informationDAMPING AND ENERGY DISSIPATION
DAMPING AND ENERGY DISSIPATION Liear Viscous Dampig Is A Propery Of The Compuer Model Ad Is No A Propery Of A Real Srucure 19.1. INTRODUCTION I srucural egieerig, viscous, velociydepede dampig is very
More informationModelling Time Series of Counts
Modellig ime Series of Cous Richard A. Davis Colorado Sae Uiversiy William Dusmuir Uiversiy of New Souh Wales Yig Wag Colorado Sae Uiversiy /3/00 Modellig ime Series of Cous wo ypes of Models for Poisso
More informationA formulation for measuring the bullwhip effect with spreadsheets Una formulación para medir el efecto bullwhip con hojas de cálculo
irecció y rgaizació 48 (01) 933 9 www.revisadyo.com A formulaio for measurig he bullwhip effec wih spreadshees Ua formulació para medir el efeco bullwhip co hojas de cálculo Javier ParraPea 1, Josefa
More informationCOLLECTIVE RISK MODEL IN NONLIFE INSURANCE
Ecoomic Horizos, May  Augus 203, Volume 5, Number 2, 6775 Faculy of Ecoomics, Uiversiy of Kragujevac UDC: 33 eissn 2279232 www. ekfak.kg.ac.rs Review paper UDC: 005.334:368.025.6 ; 347.426.6 doi: 0.5937/ekohor30263D
More informationStudies in sport sciences have addressed a wide
REVIEW ARTICLE TRENDS i Spor Scieces 014; 1(1: 195. ISSN 999590 The eed o repor effec size esimaes revisied. A overview of some recommeded measures of effec size MACIEJ TOMCZAK 1, EWA TOMCZAK Rece years
More informationChapter 4 Return and Risk
Chaper 4 Reur ad Risk The objecives of his chaper are o eable you o:! Udersad ad calculae reurs as a measure of ecoomic efficiecy! Udersad he relaioships bewee prese value ad IRR ad YTM! Udersad how obai
More informationUnit 7 Possibility and probability
Grammar o go! Lesso Lik Lesso legh: 45 mis Aim: 1. o review he use of may, migh, could, mus ad ca o express possibiliy ad probabiliy 2. o review he use of may, migh ad could whe alkig abou possibiliy ad
More informationA Heavy Traffic Approach to Modeling Large Life Insurance Portfolios
A Heavy Traffic Approach o Modelig Large Life Isurace Porfolios Jose Blache ad Hery Lam Absrac We explore a ew framework o approximae life isurace risk processes i he sceario of pleiful policyholders,
More informationDepartment of Economics Working Paper 2011:6
Deparme of Ecoomics Workig Paper 211:6 The Norwegia Shareholder Tax Recosidered Ja Söderse ad Tobias idhe Deparme of Ecoomics Workig paper 211:6 Uppsala Uiversiy April 211 P.O. Box 513 ISSN 16536975 SE751
More informationMechanical Vibrations Chapter 4
Mechaical Vibraios Chaper 4 Peer Aviabile Mechaical Egieerig Deparme Uiversiy of Massachuses Lowell 22.457 Mechaical Vibraios  Chaper 4 1 Dr. Peer Aviabile Modal Aalysis & Corols Laboraory Impulse Exciaio
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationACCOUNTING TURNOVER RATIOS AND CASH CONVERSION CYCLE
Problems ad Persecives of Maageme, 24 Absrac ACCOUNTING TURNOVER RATIOS AND CASH CONVERSION CYCLE Pedro OríÁgel, Diego Prior Fiacial saemes, ad esecially accouig raios, are usually used o evaluae acual
More informationON THE ARITHMETIC OF COMPOUND INTEREST: THE TIME VALUE OF MONEY
II. O THE ARITHMETIC OF COMPOUD ITEREST: THE TIME VALUE OF MOEY From our everyday experieces, we all recogize ha we would o be idiffere o a choice bewee a dollar o be paid o us a some fuure dae (e.g.,
More informationHanna Putkuri. Housing loan rate margins in Finland
Haa Pukuri Housig loa rae margis i Filad Bak of Filad Research Discussio Papers 0 200 Suome Pakki Bak of Filad PO Box 60 FI000 HESINKI Filad +358 0 83 hp://www.bof.fi Email: Research@bof.fi Bak of Filad
More informationRanking of Mutually Exclusive Investment Projects
Faculy of Maageme Techology Workig Paper Series Rakig of Muually Exclusive Ivesme Projecs How Cash Flow Differeces ca solve he Rakig Problem by Chrisia Kalhoefer Workig Paper No. 3 November 27 Rakig of
More informationImplementation of Lean Manufacturing Through Learning Curve Modelling for Labour Forecast
Ieraioal Joural of Mechaical & Mecharoics Egieerig IJMMEIJENS Vol:09 No:0 35 Absrac I his paper, a implemeaio of lea maufacurig hrough learig curve modellig for labour forecas is discussed. Firs, various
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationChapter 2 Linear TimeInvariant Systems
ELG 3 Sigals ad Sysems Caper Caper Liear TimeIvaria Sysems. Iroducio May pysical sysems ca be modeled as liear imeivaria (LTI sysems Very geeral sigals ca be represeed as liear combiaios of delayed impulses.
More informationRanking Optimization with Constraints
Rakig Opimizaio wih Cosrais Fagzhao Wu, Ju Xu, Hag Li, Xi Jiag Tsighua Naioal Laboraory for Iformaio Sciece ad Techology, Deparme of Elecroic Egieerig, Tsighua Uiversiy, Beijig, Chia Noah s Ark Lab, Huawei
More informationFinancial Data Mining Using Genetic Algorithms Technique: Application to KOSPI 200
Fiacial Daa Miig Usig Geeic Algorihms Techique: Applicaio o KOSPI 200 Kyugshik Shi *, Kyougjae Kim * ad Igoo Ha Absrac This sudy ieds o mie reasoable radig rules usig geeic algorihms for Korea Sock Price
More informationHilbert Transform Relations
BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume 5, No Sofia 5 Hilber rasform Relaios Each coiuous problem (differeial equaio) has may discree approximaios (differece equaios)
More informationSERIE DOCUMENTOS BORRADORES
ISSN 244396 E C O N O M Í A SERIE DOCUMENTOS No. 58, oviembre de 24 A Jump Telegraph Model for Opio Pricig Nikia Raaov BORRADORES DE INVESTIGACIÓN Cero Ediorial Rosarisa Faculad de Ecoomía Auor del libro:
More informationDeterminants of Public and Private Investment An Empirical Study of Pakistan
eraioal Joural of Busiess ad Social Sciece Vol. 3 No. 4 [Special ssue  February 2012] Deermias of Public ad Privae vesme A Empirical Sudy of Pakisa Rabia Saghir 1 Azra Kha 2 Absrac This paper aalyses
More informationUNIT ROOTS Herman J. Bierens 1 Pennsylvania State University (October 30, 2007)
UNIT ROOTS Herma J. Bieres Pesylvaia Sae Uiversiy (Ocober 30, 2007). Iroducio I his chaper I will explai he wo mos frequely applied ypes of ui roo ess, amely he Augmeed DickeyFuller ess [see Fuller (996),
More informationRainbow options. A rainbow is an option on a basket that pays in its most common form, a nonequally
Raibow optios INRODUCION A raibow is a optio o a basket that pays i its most commo form, a oequally weighted average of the assets of the basket accordig to their performace. he umber of assets is called
More informationDistributed Containment Control with Multiple Dynamic Leaders for DoubleIntegrator Dynamics Using Only Position Measurements
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 57, NO. 6, JUNE 22 553 Disribued Coaime Corol wih Muliple Dyamic Leaders for DoubleIegraor Dyamics Usig Oly Posiio Measuremes Jiazhe Li, Wei Re, Member, IEEE,
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationForecasting Unemployment Rate in Selected European Countries Using Smoothing Methods
Ieraioal Joural of Social, Behavioral, Educaioal, Ecoomic, Busiess ad Idusrial Egieerig Vol:9, No:4, 2015 Forecasig Uemployme Rae i Seleced Europea Couries Usig Smoohig Mehods Kseija Dumičić, Aia Čeh Časi,
More informationResponse of a Damped System under Harmonic Force. Equating the coefficients of the sine and the cosine terms, we get two equations:
Respose of a Damped Sysem uder Harmoic Force The equaio of moio is wrie i he form: m x cx kx F cos ( Noe ha F is he ampliude of he drivig force ad is he drivig (or forcig frequecy, o o be cofused wih.
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More information12. Spur Gear Design and selection. Standard proportions. Forces on spur gear teeth. Forces on spur gear teeth. Specifications for standard gear teeth
. Spur Gear Desig ad selecio Objecives Apply priciples leared i Chaper 11 o acual desig ad selecio of spur gear sysems. Calculae forces o eeh of spur gears, icludig impac forces associaed wih velociy ad
More informationSwaps: Constant maturity swaps (CMS) and constant maturity. Treasury (CMT) swaps
Swaps: Costat maturity swaps (CMS) ad costat maturity reasury (CM) swaps A Costat Maturity Swap (CMS) swap is a swap where oe of the legs pays (respectively receives) a swap rate of a fixed maturity, while
More informationIndex arbitrage and the pricing relationship between Australian stock index futures and their underlying shares
Idex arbirage ad he pricig relaioship bewee Ausralia sock idex fuures ad heir uderlyig shares James Richard Cummigs Uiversiy of Sydey Alex Frio Uiversiy of Sydey Absrac This paper coducs a empirical aalysis
More informationA simple SSDefficiency test
A simple SSDefficiecy es Bogda Grechuk Deparme of Mahemaics, Uiversiy of Leiceser, UK Absrac A liear programmig SSDefficiecy es capable of ideifyig a domiaig porfolio is proposed. I has T + variables
More informationTransforming the Net Present Value for a Comparable One
'Club of coomics i Miskolc' TMP Vol. 8., Nr. 1., pp. 43. 1. Trasformig e Ne Prese Value for a Comparable Oe MÁRIA ILLÉS, P.D. UNIVRSITY PROFSSOR email: vgilles@uimiskolc.u SUMMARY Tis sudy examies e
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationEquities: Positions and Portfolio Returns
Foundaions of Finance: Equiies: osiions and orfolio Reurns rof. Alex Shapiro Lecure oes 4b Equiies: osiions and orfolio Reurns I. Readings and Suggesed racice roblems II. Sock Transacions Involving Credi
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationI. Basic Concepts (Ch. 14)
(Ch. 14) A. Real vs. Financial Asses (Ch 1.2) Real asses (buildings, machinery, ec.) appear on he asse side of he balance shee. Financial asses (bonds, socks) appear on boh sides of he balance shee. Creaing
More informationCombining Adaptive Filtering and IF Flows to Detect DDoS Attacks within a Router
KSII RANSAIONS ON INERNE AN INFORMAION SYSEMS VOL. 4, NO. 3, Jue 2 428 opyrigh c 2 KSII ombiig Adapive Filerig ad IF Flows o eec os Aacks wihi a Rouer Ruoyu Ya,2, Qighua Zheg ad Haifei Li 3 eparme of ompuer
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationValuing Bonds and Stocks
Leaig Objecives 5 Valuig Bods ad Socks 5 Copoae Fiacial Maageme e Emey Fiey Sowe 5 Udesad ypical feaues of bods & socks. Lea how o obai ifomaio abou bods ad socks. Ideify he mai facos ha affec he value
More informationEstimating NonMaturity Deposits
Proceedigs of he 9h WSEAS Ieraioal Coferece o SIMULATION, MODELLING AND OPTIMIZATION Esimaig NoMauriy Deposis ELENA CORINA CIPU Uiversiy Poliehica Buchares Faculy of Applied Scieces Deparme of Mahemaics,
More informationKyoungjae Kim * and Ingoo Han. Abstract
Simulaeous opimizaio mehod of feaure rasformaio ad weighig for arificial eural eworks usig geeic algorihm : Applicaio o Korea sock marke Kyougjae Kim * ad Igoo Ha Absrac I his paper, we propose a ew hybrid
More informationTHE IMPROVEMENT OF UNEMPLOYMENT RATE PREDICTIONS ACCURACY
THE IMPROVEMENT OF UNEMPLOYMENT RATE PREDICTIONS ACCURACY DOI: 0.867/j.pep.59 Mihaela Simioescu* Absrac: This research is relaed o he assessme of aleraive uemployme rae predicios for he Romaia ecoomy,
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationForeign Exchange and Quantos
IEOR E4707: Financial Engineering: ConinuousTime Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in
More information1. Introduction  1 
The Housig Bubble ad a New Approach o Accouig for Housig i a CPI W. Erwi iewer (Uiversiy of Briish Columbia), Alice O. Nakamura (Uiversiy of Albera) ad Leoard I. Nakamura (Philadelphia Federal Reserve
More informationIndex Manual relating to the. Solactive 3D Printing Index (SOLDDD)
Idex Maual relaig o he Solacive 3 Priig Idex (SOL) Versio.0 daed November 2h, 203 Coes Iroducio Idex specificaios. Shor ame ad ISIN.2 Iiial value.3 isribuio.4 Prices ad calculaio frequecy.5 Weighig.6 ecisiomakig
More informationDBIQ Regulated Utilities Index
db Ide Develome March 2013 DBIQ Ide Guide DBIQ Regulaed Uiliies Ide Summary The DBIQ Regulaed Uiliies Ide ( Uiliies Ide is a rulesbased ide aimig o rack he reurs ou of he regulaed uiliies secor i develoed
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationORDERS OF GROWTH KEITH CONRAD
ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus
More informationThe real value of stock
he real value of sock Collars ivolve he paye of a variable aou of sock, depedig o a average sock price. I his arcle, Ahoy Pavlovich uses he BlackScholes fraework o value hese exoc derivaves ad explore
More informationNikkei Stock Average Volatility Index Realtime Version Index Guidebook
Nikkei Sock Average Volailiy Index Realime Version Index Guidebook Nikkei Inc. Wih he modificaion of he mehodology of he Nikkei Sock Average Volailiy Index as Nikkei Inc. (Nikkei) sars calculaing and
More informationIntroduction to Statistical Analysis of Time Series Richard A. Davis Department of Statistics
Iroduio o Saisial Aalysis of Time Series Rihard A. Davis Deparme of Saisis Oulie Modelig obeives i ime series Geeral feaures of eologial/eviromeal ime series Compoes of a ime series Frequey domai aalysishe
More information