Modeling of Tradeable Securities with Dividends

Transcription

1 Modeling of Tradeable Securiies wih Dividends Michel Vellekoop 1 & Hans Nieuwenhuis 2 April 7, 26 Absrac We propose a generalized framework for he modeling of radeable securiies wih dividends which are no necessarily cash dividends a fixed imes or coninuously paid dividends. In our seup he dividend processes are only required o be semi-maringales. We give a definiion of self-financing replicaion which incorporaes dividend processes, and we show how his allows us o ranslae sandard resuls for he pricing and hedging of derivaives on asses wihou dividends o he case of asses wih dividends. We hen apply his framework o analyze and compare he differen assumpions ha have been made in earlier dividend models. We also sudy he case where we have uncerain dividend daes, and we look a securiies which are no equiy-based such as fuures and credi defaul swaps, since our weaker assumpions on he dividend process allow us o consider hese oher applicaions as well. Keywords: Financial modeling, Dividends, Fuures, Credi Defaul Swaps. 1 Inroducion The pricing heory for derivaives on non-dividend paying socks is well undersood nowadays a a concepual level, see for example Duffie (21) and Musiela and Rukowski (1997). In his paper we will clarify sock-price models wih dividends, and show ha our more general framework can also be used for securiies oher han socks, such as fuures and cerain credi derivaives. By inroducing he economic concep of a radeable we are able o reduce models wih dividends o models wihou dividends. In he sandard Black-Scholes model of opion pricing (Black and Scholes (1973)) wih one sock and one bank accoun wih fixed ineres rae r > he sock and bank accoun are considered o be basic radeables. For he sock i is clear ha one can rade i and as our bank accoun is equivalen o a zero coupon bond on a fixed ime inerval [, T, i is also clear ha we may view our bank accoun as a radeable. I is assumed ha here are no ransacion coss, ha shorselling is allowed and ha he producs are perfecly divisible. For he marke paricipans we assume ha hey possess a perfec memory, an assumpion ha is refleced in he use of he concep of filraion. Given he assumpions above, every produc ha can be made from hese wo basic radeables by a reasonable self-financing sraegy (o be defined in a precise manner laer on) is a radeable as well. Inroducing dividends in such models means ha he ex-dividend price process of he sock can no longer be hough of as a radeable. Indeed i is clear ha nobody would like o inves in such an asse wihou receiving he dividend sream. I is herefore useful o invesigae how dividends can be incorporaed ino models for markes of radeables in a consisen manner. Quie ofen, cerain assumpions have been made concerning he dividend paymens which seem o have been specificly designed o simplify he compuaion of sandard European opion prices. The racabiliy of he Black-Scholes model is based upon he fac ha he asse prices 1 Corresponding Auhor, Financial Engineering Laboraory, Universiy of Twene, P.O. Box 217, 75 AE Enschede, The Neherlands, m.h.vellekoop@mah.uwene.nl 2 Deparmen of Economics, Universiy of Groningen, The Neherlands, j.w.nieuwenhuis@eco.rug.nl 1

2 follow Geomeric Brownian Moions, which leads o explici closed-form formulas for sandard opions such as European calls and pus. Taking dividends proporional o he sock price on he dividend dae reains his propery, and i has herefore been a popular choice for academic models. Bu many praciioners, i.e. opion marke makers, acually prefer o model dividends as fixed amouns of cash, which are no dependen on he sock price jus before he dividends are paid. If one coninues o assume ha in beween dividend daes sock prices follow Geomeric Brownian Moion, hen he lognormal disribuion no longer describes fuure values of he sock, and in general no closed-form pricing formulas can be derived anymore. Therefore, differen assumpions have been made for dividends in he lieraure which ry o remedy his problem. In he Escrowed Model for dividends, for example, i is assumed ha he asse price minus he presen value of all dividends o be paid unil he mauriy of he opion follows a Geomeric Brownian Moion. In he Forward Model one assumes ha he asse price plus he forward value of all dividends (from pas dividend daes o oday) follows a Geomeric Brownian Moion. For boh models one can use he original Black-Scholes pricing formulas for European-syle calls and pus, when adjused values for he srike or he curren sock price are insered in hese formulas. Even more imporanly, one can sill use he powerful numerical mehod of binomial ree pricing o price opions ha can be exercised before mauriy (American and Bermudean opions). Bu he assumpions in hese models may lead o inconsisencies, since hey assume differen dynamics for he underlying process when differen opions on he same underlying are considered simulaneously, and his may even lead o arbirage opporuniies in he marke, see Beneder and Vors (22) and Frishling (22). The reason for his is obvious: differen asse price process dynamics are assumed for producs up unil he firs dividend dae. One can fix his by changing he definiion o an assumpion ha he asse price minus he presen value of all dividends o be paid in he fuure follows a Geomeric Brownian Moion (an Adjused Escrowed Model). Bu his would mean ha he prices of opions will depend on he dividends which are being paid afer he opions have expired. This is unsaisfacory as well, since his means ha a rader would have o adjus he price of a wo-year opion once his view on he five-year dividend predicion changes. All his exemplifies he need for a consisen framework o model cash dividends. In his paper we define such a consisen modeling framework o handle dividends. The dividend sream process and he ex-dividend sock price process can be freely specified and we hen show how radeable securiies (i.e. he sock process which include dividends) can be generaed. We noe ha in a very ineresing recen paper by Korn and Rogers (Korn and Rogers (24)) he same problem is being reaed. Their soluion is o define he sock price o be he ne presen value of all fuure dividends. They model he (discree) dividend process direcly and hen derive he sock price from his. I urns ou ha under heir assumpions, he dividends are proporional o he sock price on he dividend dae, if i is assumed ha dividends are announced before ha dae. We do no make ha assumpion in our model: ex-dividend sock prices and dividend values can be specified independenly in our seup, since many marke makers prefer models in which i is possible o specify in advance he exac discree dividend amoun ha will be paid. The Escrowed, Forward, Korn-Rogers and oher dividend models will be compared in examples given a he end of his paper. This paper consiss of hree pars. In he firs par we will briefly discuss coninuous dividends of finie variaion o ge some inuiion for he more general case, which is reaed in he second par. As menioned earlier, our emphasis on creaing radeable securiies from he (no radeable) ex-dividend process and he dividends can also be applied o he modelling problem for oher securiies, such as fuures and credi defaul swaps. This will be invesigaed in he las par. 2 Coninuous Dividends of Finie Variaion We assume given a filered probabiliy space (Ω, F, (F ) [,T, P) where he filraion (F ) [,T is he usual one associaed wih a given sandard Brownian Moion W : Ω [, T R wih T > a given fixed ime-horizon. Throughou he paper, all filraions we use are assumed o saisfy he 2

3 usual condiions. We use he noaion R + = {x R x } and R ++ = {x R x > }. We assume ha he adaped càdlàg sochasic process S : Ω [, T R ++ describes he price of one uni of sock ex-dividend. The adaped sochasic process δ : Ω [, T R + saisfying E δ u du < represens a coninuous sream of dividend paymens. The inerpreaion of his process is as follows. Assume you own x shares of sock during he ime inerval, + ɛ wih ɛ infiniesimally small, hen you will receive a ime + ɛ he amoun of money xδ ɛ as dividend. I goes wihou saying ha in our conex x may be negaive or nonineger as well. We will no assume ha S (he ex-dividend price process of he sock) is a radeable in he marke. In fac, we will need o consruc he radeable S from S. Informally we would reason as follows. Suppose we sar a ime zero wih x = 1 uni of sock. When we are a ime [, T we have x socks. Le ɛ > be small. A ime + ɛ we receive ɛx δ money unis which we immediaely inves in sock so x +ɛ = x + ɛx δ S +ɛ Now le us define S = x S. We are hen inclined o view S as he price process of a radeable. We have or S +ɛ = x +ɛ S +ɛ = x S +ɛ + x δ ɛ = x (S + S +ɛ S ) + x δ ɛ S +ɛ S = x (S +ɛ S ) + x δ ɛ Formally we herefore proceed as follows. We are looking for a predicable adaped sochasic process x : Ω [, T R wih x = 1 almos surely and an adaped sochasic process S : Ω [, T R such ha he following equaions are saisfied simulaneously: where S = S + x u d(s u + D u ) (1) S = x S (2) D = defines he cumulaive dividend process, and where we assume he above equaions o be welldefined, i.e. S + D is a semi-maringale. Our economically moivaed inuiion says ha in an arbirage-free marke model here is precisely one predicable adaped process x such ha he equaions above are saisfied, and we will now show ha his inuiion is correc. Theorem 2.1. Assume ha S + D is a coninuous semi-maringale. Then here exiss a unique process x such ha he equaions (1)-(2) above are saisfied. δ u du Proof. We assumed ha S + D is a semi-maringale and as D is a semi-maringale as well i follows ha S is also a semi-maringale. From he general heory of sochasic inegraion i follows ha S is a semi-maringale oo. As S is assumed o be sricly posiive and càdlàg i 3

4 follows from Io s lemma and a localizaion argumen (see for example Proer (23)) ha 1/S is a semi-maringale and hence x mus be a semi-maringale oo. We assumed ha S + D is coninuous and since D is coninuous, S has o be coninuous oo. Bu hen [S, x = S x = S x S u dx u S u dx u + x u ds u x u dd u by equaion (1)-(2) so his shows ha S udx u, and hence x, has finie variaion. Applying Io s rule and using he fac ha x has finie variaion we find and combining his wih d S = S dx + x ds we find ha d S = x ds + x dd dx = x S dd As D is non-decreasing, so is x. From he general heory of sochasic differenial equaions i follows ha dx = x S dd x = 1 has a unique srong predicable soluion on [, T (see for example Proer (23)). In fac which complees he proof. x = e δu Su du Noe ha a proof of exisence for he process x could easily be seled using he las few equaions, bu i is he uniqueness which is of paricular ineres here. Also noe ha we assumed here ha he semi-maringale S + D is coninuous and ha he cumulaive dividend process D is of finie variaion. In he nex secion, where we discuss he more general case, we will show ha we do no need o make hese assumpions o show uniqueness and exisence of radeables S in a more general seup. 3 General Dividend Processes We would now like o be able o define on he same probabiliy space, bu wih a filraion which need no necessarily be generaed by a Brownian Moion, an asse process which may pay a discree (i.e. cash) dividend equal o D on ime D, T[ where D F D and such ha S D D > (P a.s.) where S again describes he ex-dividend process. In fac, we would even like o consider cases where an asse pays boh coninuous and discree dividends, or even more generally, where he cumulaive dividend process is jus assumed o be a semi-maringale. Le V, S and be adaped càdlàg ex-dividend price processes for asses V, S and B which are sricly posiive and le D V, DS and D be he corresponding càdlàg adaped cumulaive 4

5 dividend processes (which are no necessarily posiive), such ha V + D V, S + D S and + D are all semi-maringales. We will assume ha D S = D B = D V = hroughou he paper. The asse B will ofen represen a bank accoun in his seup. We would like o define he noion of replicabiliy i.e. he idea ha he price process of a cerain asse V can be mimicked by rading in oher asses. Definiion 3.1. We say ha an asse V can be replicaed using asses S and B iff here exis adaped and predicable processes φ S and φ B such ha for all [, T V = φ S S + φ (3) d(v + D V ) = φs d(s + D S ) + φ d( + D ) (4) where he firs equaion for = should be read as V = φ S S + φ B B (i.e. wihou aking lef-hand side limis). Noe ha for coninuous processes wihou dividends we find he classical definiion of replicaion back in (3)-(4), bu he lefhand side limis in he firs equaions are an imporan difference compared o he case wihou dividends. Indeed we can no longer say ha V = φ S S + φ as in he usual formulaions in he absence of dividends, bu insead V V = φ S (S S ) + φ ( ) which is of course a reformulaion of (3) since X X = X. If we define hen dv + dd V V φ S = ψ S V /S (5) φ = ψ V / (6) = ψ S ds + dd S S + ψ for cerain predicable adaped processes ψ S and ψ B such ha ψ S + ψ = 1 d + dd The inerpreaion is ha he rae of reurn of V (which equals he difference in value based on changes in boh he ex-dividend price and he dividends, divided by he price before any dividends have been paid ou) is based on percenages invesed in asses S and B. Working wih percenages guaranees in an inuiive manner ha we only consider sraegies which do no necessiae cash wihdrawal or injecion, i.e. i is a convenien way o define self-financing sraegies. However, our definiion above is slighly more general in he sense ha i allows he price processes becoming zero for cerain imes as well. Throughou he paper we will assume D B = i.e. our bank accoun does no pay dividends (or coupons), only ineres. Noe ha we have assumed ha S + D S is a semi-maringale bu we have no assumed i o be coninuous, as we did in Theorem 2.1. Theorem 3.1. Le S + D S and B be semi-maringales saisfying he condiions saed above. Then here exiss a unique asse price process Ṽ wih DṼ and Ṽ = S ha can be replicaed wih φ S 1. To prove Theorem 3.1, we need he following resul which is saed and proven in Jaschke (23). 5

6 Theorem 3.2. Le H be a semi-maringale and le Z be a semi-maringale wih Z = and Z 1, for all R +. Then he soluion of he equaion X = H + + X s dz s is given by X = H E H s d( 1 E s ) (7) E = e Z 1 2 [Z,Zc + (1 + Z s )e Zs (8) <s and his soluion is unique. Proof of Theorem 3.1. Since we wan φ S 1 we define φ saisfy = (Ṽ S )/, so Ṽ should dṽ = d(s + D S ) + Ṽ S d (9) We define A = Ṽ S, hen da = dd S + A d so if we ake H = D S and Z = d/, we can apply Theorem 3.2 o prove he exisence and uniqueness of he process A and hence of he process Ṽ = A + S. Indeed, subsiuion in (8) gives E = e + dbu 1 2 [ dbu, dbu c = e + d(ln Bu) so according o (7) he process <s <s (1 + + B s B s (1 + Bs Bs B s ) = /B d )e dbu + Ṽ = S + A = S + D S + D S u d( 1 ) (1) saisfies our requiremens. The inerpreaion of he resul proven in he Theorem is of course ha i should be possible o inves our dividend sream in he bank accoun and by doing so end up wih a process which no longer pays any dividends. We will denoe he process V consruced in he Theorem by S B in he sequel. Since X Y = X dy + Y dx +[X, Y for all semi-maringales X and Y, we can rewrie he formula (1) derived in he Theorem as follows: S = S + D S = S D S u d( 1 ) dd S u + [D S, B 1 (11) 6

7 The necessiy of he las bracke erm o compensae for he fac ha paid ou cashflows and he bank accoun may have nonzero covariaion was already noed in Norberg and Seffensen (25). If we assume ha B is coninuous and of finie variaion hen we simply find S = S + + dd S u. In he special case for jus one cash dividend D S = D1 [D,T(), we can reduce his o S = S + 1 [D,T() D D Noe ha if we work on a Brownian filraion, hen S+D S and V +D V are coninuous processes, so D S = S and he imes a which D S and S are disconinuous hus have o coincide. In general we would have for adaped processes X on his filraion ha X = X X = X + D X and he firs replicaion equaion (3) would hen boil down o V + D V = φ S (S + D S ) + φ ( + D ) which is he classical noion of a gains process o model dividend, and has been inroduced earlier in he lieraure, see for example Duffie (21). This may seem a naural alernaive choice for he firs equaion in our definiion of replicaion, bu i will no generalize in a nice way when we use oher filraions han hose generaed by Brownian Moion, since we will see laer ha on filraions which are no lef-coninuous we may no always have ha D S = S. On such filraions our definiion is herefore differen from he one in Duffie (21). Theorem 3.3. Le S + D S and B be semi-maringales saisfying he condiions saed above. Then here exiss a unique asse price process V wih D V and V = S such ha V can be replicaed using S only, i.e. such ha φ B. This asse price process V = S can, ogeher wih B, replicae S B. Proof. We are looking for a process V such ha D V, wih φ B. Bu his las assumpion implies ha φ S = V /S so we need o prove ha here exiss a unique process V such ha wih V = S, so dv = V S d(s + D S ) (12) V u V = S + d(s u + Du S + S ) u We can hus apply Theorem 3.2 again wih Z = d(s u+d S u ) S u and H S o prove exisence and uniqueness. Finally, he asse Ṽ = SB can be replicaed using V = S and B since Ṽ = φ S S + φ dṽ = φ S d S + φ d if we ake φ S = S / S and φ = (Ṽ S )/, as can be seen from (12) and (9). Noe ha in he special case of a single discree dividend D a one paricular ime D, i.e. when D S = D1 [D,T(), we can simplify he expression for he process V based on formula (7) 7

8 considerably. In fac, we hen find ha V = S wih S = S e + d(su+du S ) S 1 u 2 [ + d(su+du S ) S, u + d(su+d S u ) S u = S e + d(ln Su) 1 [ D,T() (ln S ) (1 + D1 [D,T() ) D = S + 1 [D,T() S( D ) S S D c (1 + D1 [D,T() S D D1 )e [ D,T () S D and his process indeed saisfies he requiremens. Noe ha his expression represens our economic inuiion of wha happens when we reinves dividend proceeds in he underlying asse S. To check ha his expression models he correc behavior, firs noice ha < D : S = S S = S (13) = D : SD = S D + D S D S D S D = S D (14) > D : S = S (1 + D S D ) S = S (1 + D S D ) (15) We can now show direcly ha V = S saisfies (12). Indeed, we have for < D ha d S = ds = ds + dd S = S S d(s + D S ) as required, where we have used (13) and he definiion of D S. For = D we find, using (14) and finally, for > D so we are done. d S = ds + dd S = S S d(s + D S ) d S = ds + D S D ds = (1 + D S D )d(s + D S ) = S S d(s + D S ) The approach aken in he proof of Theorem 3.3 formalizes he idea ha we could reinves dividend payous in he asse which pays he dividends, insead of he approach aken in he previous Theorem, where he dividend proceeds were invesed in he bank accoun. The unique processes S and S ha we have creaed and which do no conain any dividends, can now be used for replicaion purposes, so he original ex-dividend process S and is dividend process D S have become superfluous in his sense: Corollary 3.1. If an asse V can be replicaed using he asses S and B, hen i can be replicaed using he asses S and B. If an asse V can be replicaed using he asses S and B, hen i can be replicaed using he asses S B and B. Proof. If an asse V is replicaed using S and B we may wrie V = φ S S + φ (16) d(v + D V ) = φ S d(s + D S ) + φ d (17) 8

9 bu using (12) we can rewrie his as V = φ S S S S + φ d(v + D V ) = φs S S d S + φ d so aking φ S = φ S S / S shows he firs resul. The second resul follows when we use (12) o rewrie (16)-(17) in he form V = φ S S + (φ φ S S B S ) d(v + D V ) = φ S ds + (φ φ S so replicaion is possible in his case as well. S B S )d We now consider an arbirage-free marke wih he asses ( S, B) in i. We know ha here exiss a measure Q, equivalen o our original measure P, such ha S/B is a maringale under Q. Definiion 3.2. We say ha V is he price process of a radeable asse iff 1. I can be replicaed using S and B 2. The process D(V ) is a maringale under Q, where 1 D(V ) = V + DV B D V B 1 Due o he corollary proven above, we migh as well have required ha V can be replicaed using S B and B. We noed before in (11) ha we may rewrie D(V ) as D(V ) = V B + B 1 DV + [D V, B 1 bu we prefer he noaion used in he definiion since i does no involve a bracke. The main poin of he definiion given above is ha we would like D o be a maringale, and no jus a local maringale. Tha i is a local maringale is already guaraneed by he firs par of he definiion, as he following resul shows. This represenaion heorem is he main resul of he paper, which shows how he usual maringale represenaion heory for asses wihou dividends carries over o our more general case. Theorem 3.4. If an asse price process V can be replicaed using S and hen here exiss an adaped predicable process φ such ha ( ) S dd(v ) = φ d Proof. We apply Io s rule for (no necessarily coninuous) semi-maringales which saes ha for wice coninuously differeniable funcions f : R n R and semi-maringales X on R n we have n n n f f(x ) f(x ) = x i (X s )dxs i f 2 x i x j (X s )d[x i, X j c s i=1 + + [ f(x s ) f(x s ) <s i=1 j=1 n i=1 + f x i (X s ) X i s 1 We use he common noaion Z = X Y for a process Z saisfying dz = X dy. 9

10 In paricular, for f(x, y) = x/y we find ha d X = dx X dy d[x, Y c Y Y Y Y Y 2 + X d[y, Y c Y 3 + ( ) ( Y X Y X ) Y Y Y Y If V can be replicaed using S and B, i can be replicaed using S and B by he previous corollary, so here exis φ S and φ B such ha where we have used he fac ha D S = D B. Bu hen d V + D V = d(v + D V ) D V d 1 φ S d S V = φ S S + φ (18) d(v + D V ) = φ S d S + φ d (19) V + D V d d[v + DV, B c B 2 ( ) ( B (V + D + ) V (V + D V ) ) [ = D V db B 2 + d[b, ( )( ) Bc B B B 3 + = φ S [ d S S d d[ S, B c B 2 ( ) ( B S φ S S ) + S B 3 d[b, B c We sum he hree expressions o calculae dd(v ) and collec erms: dd(v ) φ S d S = d V + D V = d(v + D V ) d[v + DV, B c D V d 1 φ S d S B 2 V + D V d D V d B 2 [ d S φ S S d [ φ S + V + D V B 3 d[b, B c D V d[b, B c B 3 ( ) [ B (V + D V ) (V + D V ) D V + V + D V B 3 d[b, B c d[ S, B c B 2 + S d[b, B c B 3 φ S S + φ S S We subsiue (18)-(19) and ge 1 [ = (φ S d S + φ d) (V + D V )d + D V d φ S ( d S S d ) B [ B 3 d[v + D V, B c + V d[b, B c + φ S d[ S, B c φ S S d[b, B c ( ) [ B V (V + D V ) φ S S + φ S S = 1 B 2 [ φ V + φ S S d + B [ B 3 d[v + D V, B c + φ S d[ S, B c + φ d[b, B c ( ) ( ) B 1 [ + (V φ S B S ) (V + D V ) + φ S S 1

11 and we see his is zero by using (18)-(19) again, and using he fac ha (19) implies ha (V + D V ) = φ S S + φ This complees he proof. We have hus proven ha asse price processes V ha can be consruced in a self-financing manner using sock and he bank accoun, inheri he local maringale propery from he underlying asses: if he discouned version of S is a local maringale under Q, hen so is D(V ), he properly discouned version of V and is dividend process D V. This will allow us o apply he usual heory for opion pricing in arbirage-free markes wihou dividends. Noe ha we allow radeables here o have dividend processes. Alernaively we could say ha V is a radeable whenever D V and V B is a Q-maringale, bu we will see in he applicaions of he nex secion ha his would be oo resricive for many financial applicaions. Since D(V ) is a Q-maringale we have ha E Q [D(V ) F s = D(V ) s and aking limis s we find ha E Q [ D(V ) F =. So when B is coninuous and of finie variaion we mus have ha E Q [ V + D V F = This expression immediaely shows ha on lef-coninuous filraions (such as hose generaed by Brownian Moion) where F = F, we mus have ha V = D V since boh V and D V are adaped. Bu if he underlying filraion is no lef-coninuous his is no longer necessary, even if cash dividend paymens are announced in advance (i.e. when D V is F -measurable). We hen only know ha E Q [ V F = D V so he jump in he ex-dividend process of a radeable does no necessarily cancel he jump due o a dividend paymen. This was already noed in Heah and Jarrow (1988) and Baauz (22). In he las paper an asse price model is formulaed in which D V = D1 D wih D and D deerminisic, and V D = D + Y (V D D) for a sochasic variable Y wih suppor 1, 1[ and such ha E Q [Y F D =. This provides a nice example of a racable dividend model where V D V. 4 Examples We will now show how he framework developed so far can be applied o differen ypes of securiies. In all he differen producs we consider he key noion ha we will use is he fac ha if an asse V is radeable in an arbirage-free and complee marke on a filered probabiliy space (Ω, F, P, (F ) R+ ), hen here exiss a unique equivalen maringale measure Q such ha he process D(V ) is a maringale under Q. Throughou his secion he processes B will be of finie variaion and coninuous, so [D V, B 1 and D(V ) being a Q-maringale hen leads o V = E Q [ V T B T + dd V u F Noe ha his expression has a nice inerpreaion: he curren price of a radeable can be seen as he price of a derivaive which represens he sum of he ex-dividend price a a laer dae and all he cashflows paid ou by he radeable unil ha dae, afer all hese have been properly discouned. (2) 11

12 4.1 Equiy Dividend Models: Deerminisic processes for Dividends In he firs secion of his paper we menioned some differen approaches o handle he incorporaion of dividends in equiy price processes. As we explained here, he Escrowed Model for dividends assumes ha he (cumulaive) dividend process is deerminisic and ha asse price minus he presen value of all dividends o be paid unil he mauriy of he opion follows a Geomeric Brownian Moion. This means ha V = S dd S u is a Geomeric Brownian Moion, and if i is also a radeable, i mus be a Q-maringale afer discouning, so ( S T ) 1 dd S S T 1 u = dd S u e σudw Q u 1 2 σ2 u du B for some deerminisic process σ : R + R + and W Q a Brownian Moion under Q. The sandard European Call opion has a payoff (S T K) + which under Q can be wrien as B T ( ) V e σudw Q u 1 + T 2 σ2 u du K This shows ha he original Black-Scholes formula can be used o calculae he Call Opion price B T E Q [(S T K) + F, if one insers a differen saring value for he asse price process: insead of he Black-Scholes formula wih curren asse price S we now use a Black-Scholes formula wih curren asse price V. In he Forward Model, he he asse price plus he forward value of all dividends (from pas dividend daes o oday) is assumed o follow a Geomeric Brownian Moion, so V = S + ddu S is a Geomeric Brownian Moion, and since i has o be a radeable as well we find ha and he European Call payoff can be wrien as B T S 1 + ddu S = S e σudw u Q 1 2 σ2 u du B ( V e σudw Q u 1 2 T T σ2 u du [ dd S u + K so we see ha his ime we can use he original Black-Scholes formula wih a differen srike: insead of he srike K we need o inser he srike K + ddu S ino he Black-Scholes formula for European Calls, and inser V insead of S for he curren asse price. 4.2 Korn-Rogers Model: Bounded Variaion processes for Dividends In he model of Korn and Rogers, sochasic dividends are paid a dividend imes which are known a priori while he ex-dividend asse price process S equals he condiional expecaion, under he equivalen maringale measure, of he sum of all (discouned) fuure dividends, so S = E Q [ dd S u F ) + 12

13 In he case reaed by Korn and Rogers, he filraion is generaed by a Lévy process. We define V = S + ddu S which implies ha where V [ = E Q dd S u D = F + dd S u = E Q [D F dd S u. is assumed o be a well-defined finie sochasic variable, which is inegrable wih respec o Q. I is hus immediaely clear ha in his model S is auomaically a radeable. Korn and Rogers le he process D S have he specific form D S = i=1 1 i X i wih X an exponeniaed Lévy process and he imes i deerminisic. Obviously, D S is of bounded variaion in ha case. 4.3 Fuures: Îo-processes for Dividends A fuures conrac is an exchange-raded sandardized conrac which gives he holder he obligaion o buy or sell a cerain commodiy (or anoher financial conrac) a a cerain dae in he fuure, he delivery dae, for a price specified on ha day, he selemen price. I should be conrased wih a forward conrac, which gives he holder he obligaion o buy or sell a a dae in he fuure for a price specified oday bu paid or received a he fuure dae (oday s forward price for he commodiy or underlying conrac). Forwards are concepually easier bu more complicaed in pracice, since i assumes ha a buyer and a seller agree on cash being paid oday and delivery aking place a a fuure dae. If one wans o buy he commodiy on a specific delivery dae in he fuure one can obain a fuure conrac, a zero coss oday. Today s fuures price for ha delivery dae ells you for wha price you will obain he commodiy a ha ime, bu insead of paying ha amoun righ now (which you would do if you had aken ou a forward conrac) you pay nohing now. Insead, you open a bank accoun on he exchange, he so-called margin accoun. From now unil he delivery dae (or unil he firs dae before ha dae on which you ge rid of he fuure) you will receive every day, afer he new fuures price for your commodiy and your delivery dae has been specified, he difference beween he new fuures price and he previous day s fuures price, if his difference is posiive. When his difference is negaive, he corresponding amoun i aken from your accoun. The ne effec of his is ha you end up paying he fuures price a which you obained your conrac in he marke: you pay he fuures price on he delivery dae (which mus equal he price of he commodiy on ha dae, of course) bu you have been compensaed on a daily basis if ha price is higher han he fuures price a which you go in. On he oher hand, if he fuures price on he delivery dae is lower, you have acually paid ha difference by he daily adjusmens before ha dae. The fuures conrac has herefore hree essenial elemens: Going long or shor any number of fuures conracs is free a all imes Wih every fuure conrac we ener, we can associae a margin accoun in which he differences beween he curren and previous fuures price is being paid (if we are long one conrac) or wihdrawn (if we are shor one conrac). This margin accoun earns ineres. We will use hese hree elemens as he basis of a definiion of a fuures price. 13

14 Definiion 4.1. We call m : Ω [, T R he fuures price process associaed wih delivery of asse S a ime T if he following holds: m is a semi-maringale and m T = S T For all bounded previsible processes ψ he following process M is a radeable: { db dm = M M = + ψ dm (21) Noice ha delivery involves he ex-dividend price, and no he price of he radeable. We will use he noaion M ψ for he process M o remind ourselves ha i depends on he process ψ. Noe ha he process ψ in he definiion above has he inerpreaion of a fuures rading sraegy: ψ represens he number of fuures conracs in our posiion a ime. Our definiion reflecs he fac ha we may ener he fuures marke a any ime a zero coss. Wha we do is o inves he proceeds of he fuures sraegy ψ ino he margin-accoun M which earns he riskfree rae. This approach is differen from he usual one (see for example Bjork (24)) where margin accouns are never aken ino accoun explicily. The only excepion we know of is he work of Duffie and Sanon (1992) in which he margin accoun is menioned direcly. Our reamen here is inspired by he paper by Pozdnyakov and Seele on he maringale framework for fuures pricing, Pozdnyakov and Seele (24), bu our definiion differs from heirs. We only impose ha m is such ha M ψ /B is a Q-maringale on [, T (i.e. ha M ψ is a radeable in economic parlance) and we do no need o impose any regulariy condiions on m from he sar. Anoher difference wih he approach in Pozdnyakov and Seele (24) is ha we inroduce a whole collecion of radeables from he very beginning and his is compleely in line wih he fac ha one may ener a fuures conrac a any ime in real life. The following wo resuls are hen immediae: Theorem 4.1. The margin accoun process can be replicaed using a zero ex-dividend process wih pays coninuous dividends equal o he fuures price. Proof. Taking φŝ = ψ, φ = M / and Ŝ =, DŜ = m replicaes V = M wih D V =, see equaions (3)-(4). Theorem 4.2. We have for all [, T ha m saisfies m = E Q [S T d[ E Q [S T F, B u F Proof. From he proof of Theorem 3.1. we see ha we can solve (21) for M. In fac we have ha ( ) ( ) M dm d = ψ + d[m, B 1 = ψ ( dm d[m, B ) By he definiion of radeable, D(M) mus be a Q-maringale, bu since M pays no dividends, his means ha M/B mus be a Q-maringale. If we ake ψ = for all, we hus have ha E Q [m T m and since m T = S T we hus find he resul m = E Q [S T d[m, F = d[m, F Wrie m = E Q [S T F A hen A has finie variaion, so [m, B = [E Q [S T F, B and he resul follows. 14

15 In many models i is assumed ha B is coninuous and of finie variaion, and in his case we ge he well-known resul ha m = E Q [S T F. More ineresing is he case where he bank accoun B has quadraic variaion. Le S and B be driven by Brownian Moions V and W wih correlaion coefficien ρ in a marke ha is compleed by addiional asses, i.e. under Q (he maringale measure for numeraire B) we have ds /S d / = rd + σ S dv = rd + σ B dw for known consans r, σ S and σ B. Then he fuures price m equals m = E Q [S T = S e r(t ) = S e r(t ) d[e Q [S T F s, B s F B s d[s s e r(t s), B s s B s e r(t s) σ S S s σ B B s ρ ds B s = S e r(t ) ρσ S σ B e rt E Q [ e rs S s F ds = S e r(t ) ρσ S σ B e rt e rs S e r(s ) ds = S e r(t ) [1 ρσ S σ B (T ) Noe ha he fuures price may hus become negaive for posiively correlaed S and B processes when he ime o mauriy is large! In pracice, of course, he ineres rae earned on a fuures margin accoun is usually fixed or cerainly of finie bounded variaion. In ha case, negaive fuure prices will no occur for posiive asse price processes S. 4.4 Credi Defaul Swaps: Sopped Jump Process for Dividends We now consider credi defaul swaps, as an example of a filraion for he dividend process which is no a Brownian filraion. In paricular, we would like o derive a rading sraegy which allows us o hedge a posiion in credi defaul swaps using defaulable coupon bonds. We will use he same seup as Bielecki, Jeanblanc, and Rukowski (25). Define on a probabiliy space (Ω, F, Q) a sricly posiive sochasic variable τ and E = 1 τ p() = E Q (1 E ) = Q(τ > ) and le (F ) R+ be he filraion generaed by he process E, hen τ is obviously a sopping ime wih respec o his filraion. We assume ha p is a coninuous funcion on R +. In a marke wih a deerminisic bond process B which is coninuous and of finie variaion, we define a credi defaul swap wih mauriy T for he defaul even τ as an asse S such ha S T =, D S = A( τ T) + I(τ)E where A and I are deerminisic coninuous funcions which represen a (cumulaive) amoun paid as long as here is no defaul, and an amoun received upon defaul respecively. We assume ha A is differeniable as well. The process M = E + (1 E u ) dp(u) p(u) (22) 15

16 is a Q-maringale on (F ) R+, since a direc compuaion verifies ha E Q [M M s F s = for all s < : E Q [ s E Q [E E s F s = (1 E u ) dp(u) p(u) F s = E Q [1 s<τ F s = 1 s<τ E Q [1 s<τ = 1 s<τ (p(s) p()) 1 s<τ E Q [1 s<τ (1 E u ) dp(u) = 1 s<τ E Q [1 s<τ 1 τ τ = 1 s<τ s v s s s p(u) dp(u) p(u) dp(v) + p() dp(u) p(u) + 1 s<τ1 τ> s s dp(u) p(u) dp(u) p(u) = 1 s<τ(p() p(s)). If we wan S o be a radeable, equaion (2) hen gives for he ex-dividend price: [ [ S = E Q S T T ddu S T + B T F = E Q (1 E u )da(u) + I(u)dE u F Using sandard resuls his can be rewrien as [ 1 τ T τ> S = E Q E Q 1 τ> 1 τ> da(u) + I(τ) 1 <τ T B τ = L K() (23) wih L = (1 E )/p() and K() = p(u) da(u) I(u) dp(u) a deerminisic coninuous process. The Q-maringale S/B + B 1 D S can be represened in erms of M. We have dl = d(1 E ) p() (1 E )dp() p() 2 and we find for τ, since S = L K(), = dm p() = dm (1 E ) p() = L dm d S + dds = d(l K()) + (1 E )da() + I()dE = K()dL + L dk() + (1 E )da() + I()dE = K()L dm + I() (de + (1 E ) dp() p() ) = 1 (I() S )dm (24) We can use his o calculae how he Credi Defaul Swap can be hedged wih a defaulable bond. We define he defaulable bond P wih mauriy T and known coupon paymens C i a known imes i T for i = 1...n as a radeable wih P T =, D P = n C i 1 i 1 τ>i. Similar calculaions as above hen give he ex-dividend price as [ P = E Q P T T dd P u + B T F 1 E = p() 16 i=1 n i=1 p( i ) C i 1 <i i

17 and his allows us o wrie n P = L v i (1 R i ), DP = wih i=1 n p( i )L i R i i=1 v i = C i p( i )/i, R i = 1 i Bu hen d P + ddp = n i=1 v i d [ (1 R)L i + L i R i and d [ (1 R)L i + L i R i so = dl + d[(l i L )R i = dl + L i R i d(l R i ) = dl + L i R i [L dr i + R i dl + L R i L R i L R i R i L = dl + L i R i Ri dl L R i = (1 R )dl + (L i L ) R i = (1 R )L dm + = 1 i 1 τ dm p() d P + ddp = n i=1 C i p( i ) i p() 1 i 1 τ dm (25) and equaions (24)-(25) hus show ha if we wan o replicae he Credi Defaul Swap using defaulable coupon bonds, he amoun of bonds per swap o hold in our porfolio equals for τ φ = I() S 1 p() {i: i } C i p( i ) i 4.5 Uncerain Dividend Daes: Jump-diffusion Processes The resuls of he previous subsecion concerning dividend processes generaed by jumps a random imes can also be used o model uncerainy in dividend daes. Suppose we have an exdividend sock price process S and a dividend of known magniude D which will be paid ou a he unknown dividend dae τ which has a known disribuion p under an equivalen maringale measure Q, i.e. D S = DE, E = 1 τ, Q(τ > ) = p() As before, we also have a bank accoun given by = B e r. Since we would like our model o have some racabiliy, we would like he process S o be adaped o (F ) R+, he filraion generaed by he process E and a sandard Brownian Moion W. We define he process V = S + and his should be a Q-maringale afer discouning if we wan he sock o be a radeable. Noice ha we use a model here which is similar o he Forward model we menioned in subsecion 4.1 because he alernaive, he Escrowed model which uses V = S ddu/b S u, is no longer adaped when he cumulaive dividend process D S is sochasic. If V/B is a Q-maringale, hen by dd S u 1. 17

18 predicable represenaion heorems (see for example Proer (23)) here mus exis predicable processes A and J such ha d(v / ) = A dw + J dm where M has been defined in he previous subsecion. Since (V / ) = J M i is clear ha J = D/ so we find or d S = d V dds = A dw + D dm D de = A dw + D (1 E ) dp() p() d( S τ B D 1 dp(u) p(u) ) = A dw To arrive a a racable formula we choose A now in such a way ha we can solve his SDE, which is cerainly he case if he lefhand side becomes he incremen of a lognormal process, i.e. we ake S τ B D so he ex-dividend price hen becomes 1 dp(u) p(u) = ( S )e σw B τ S = S e (r 1 2 σ2 )+σw + D 1 2 σ2 dp(u) p(u) If he maringale measure Q is unique, a vanilla call wih payoff (S T K) + based on he ex-dividend price mus hen have he price E Q [(S T K) + F B T = B ( E Q [ B T (S D τ dp(u) p(u) )e(r 1 2 σ2 )(T )+σ(w T W ) + D ) + T τ B T dp(u) p(u) K F If one assumes ha he processes W and E are independen, his leads o a European opion pricing formula ha can be wrien in erms of inegrals of he Black-Scholes call opion formula over differen srike and sock values. 5 Conclusions We have shown how dividends can be modeled consisenly in arbirage-free markes by he inroducion of radeable securiies wihou dividends ha can be replicaed using underlying asses wih dividends. We believe ha our definiion of wha replicaion should mean in he presence of dividends provides a naural concep for he modelling of dividends, as winessed by he many differen examples given in he previous secion. The las example given here (where he dividend daes are uncerain) shows ha we need o be careful when defining a model for he ex-dividend process if we wan he combinaion of ex-dividend and dividend processes o be a radeable in an abirage-free marke: i is obvious ha when dividends are presen, he ex-dividend process canno be a maringale under an equivalen maringale measure afer discouning. Bu once radeables have been defined in a proper manner by reinvesing dividend proceeds, Theorem 3.4 shows ha pricing and hedging problems can be addressed using he well-known ools of maringale represenaion heorems in sochasic calculus. We herefore believe ha our resuls may be of some ineres when designing hedging sraegies for financial producs which include dividends or when designing hedging sraegies ha use securiies ha have dividend payoffs hemselves in he hedge. 18

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