T HE IMPACT OF A SCRIP

Transcription

1 H IMPAC OF A SCRIP DIVIDND ON AN QUIY FORWARD Grma Brhart XAIA Ivstmt GmbH Sostraß Müch Grmay grmabrhart@xaiacom Ja-Frdrik Mai XAIA Ivstmt GmbH Sostraß Müch Grmay ja-frdrikmai@xaiacom Dat: August 7 Summary W cosidr a quity forward cotract o a stock which pays a dividd durig th forward liftim Furthrmor th stock owr is assumd to hav th right to opt for ithr cash or scrip dividd I th lattr cas th stock owr rcivs th dividd i th form of additioal shars ad th umbr of shars to b rcivd dpds o th avrag stock pric i a crtai tim priod [ ] h dcisio btw scrip or cash must b mad by th stock owr at som tim poit durig th avragig priod i Withi a Black-Schols-typ stup w driv a closd formula for th fair strik pric of such a quity forward cotract i dpdc o th stock volatility paramtr σ > It is furthrmor outlid how a appropriat choic for σ might b rtrivd from obsrvd vailla stock optios spcially if is closr to th d of th avragig tha to th bgiig it is dmostratd how th optioality for th stock owr has a o-gligibl valu which lowrs th forward quity strik Itroductio W cosidr a quity forward cotract whos udrlyig stock pays a dividd durig th liftim of th forward cotract Morovr th stock owr is assumd to hav th right to dcid btw rcivig th dividd i cash or i th form of w shars (scrip dividd) hroughout w cosidr fiv tim poits 3 a dtrmiistic short rat r(t) usd for discoutig cash flows ad a stock pric {St }t dfid as a adaptd procss o a filtrd probability spac (Ω F {Ft } P) Furthrmor it is possibl to ld/ borrow th stock i th marktplac ad δ(t) dots th dtrmiistic rpo margi for a istataous rpo agrmt at t so that th rpo margi for a rpo agrmt ovr a ldig priod [t ] R is giv by t δ(s) ds For furthr backgroud rgardig rpo agrmts th itrstd radr is rfrrd to Brhart Mai () h tim poit is th maturity of th quity forward h tim poit is th x-dividd dat for a dividd paymt mad at 3 his mas that somo who buys th stock bfor is titld to rciv th dividd ad whovr buys th stock aftr is ot Morovr this mas that th stock pric will xhibit a jump at h avrag shar pric ovr th priod [ ] is

2 dotd by A = X Stk t < t < < t = k= whr th discrt avragig is daily ad th tk corrspod to busiss days btw ad (possibly icludig ad ) It is assumd that at a stock owr who is titld to rciv th dividd must dcid whthr h wishs to rciv cash or scrip dividd at 3 I th first cas h simply rcivs a cash amout D pr shar which has b dtrmid arlir by th compay s maagmt I th lattr cas th umbr of shars to b rcivd is dtrmid as D/A for th avragig pri od [ ] i at 3 th stock holdr rcivs D/A shars which rsults i a payoff that is worth Payoff(3 ) = D A S3 at 3 Figur summarizs th cosidrd scrip dividd timli today x-dividd dat = start of avragig t= dcisio scrip/cash dividd paymt d of avragig forward maturity 3 Fig : Illustratio of th cosidrd timli Rmark (Furthr dividds bfor?) It is th xcptio rathr tha th rul that thr ar mor tha o dividd paymt dats withi th liftim of a quity forward cotract bcaus th lattr td to b short-datd typically hrfor w do ot addrss this cas i th prst articl Ayway sic our approach rquirs th dividd cash amout D to b kow i advac which is ot th cas i rality for dividds i th far futur iclusio of furthr dividds ad log forward maturits is o-trivial i gral h rst of this articl is orgaizd as follows Sctio prsts th thortical drivatio of th fair forward strik pric ivolvig a xpctd valu which is ot asy to comput Sctio 3 shows how to valuat th drivd formula i practic assumig th stock pric aftr to follow a Black-Schols-typ modl with a crtai volatility paramtr σ > Sctio idicats how obsrvd markt prics for vailla stock optios might b usd i ordr to imply th volatility paramtr to b usd i th drivd formula Sctio 5 cocluds ypically D is aoucd by th compay maagmt prior to th x dividd dat

3 hortical drivatio Rcall our fiv tim poits 3 from Figur W assum that ow at tim t = w ar log o quity forward cotract with maturity which is a agrmt to buy o stock at tim at a pr-dtrmid strik pric which w dot by F ( ) At maturity this cotract has th payoff S F ( ) Hc big log this quity forward w hav a log xposur i th stock at which w aim to offst by short-sllig th stock via th followig rpo agrmt: w borrow th stock i th marktplac ovr th priod [ ] ad immdiatly sll it ow at t = Sic thr is a dividd paymt durig th ldig priod w assum that th ldr is compsatd accordigly at th d of th cotract i a way w mak prcis i th squl Now at t = w rciv th stock from th ldr for which w hav to post as collatral with th ldr th cash amout C := S ad th ldr is commitd to pay th rpo rat r δ to us o this collatral Sic th ldr ows th stock h may dcid at whthr h wishs to rciv cash or scrip dividd Util 3 th iitial collatral amout C is assumd to rmai ualtrd i otioal howvr this may chag aftr 3 (i) I cas th ldr opts for a cash dividd th collatral is rducd by D pr shar at 3 so that th collatral cash amout w gt back from th ldr at quals Ccash = C D 3 I rtur w hav to giv back to th ldr th stock at maturity of th rpo agrmt (ii) I cas th ldr opts for a scrip dividd th collatral is assumd to rmai ualtrd i otioal ovr th full ldig priod ad at maturity of th rpo agrmt w gt back from th ldr th collatral valu Cscrip = C I rtur w hav to giv + D/A shars back to th ldr Cosqutly at tim our portfolio cosistig of quity forward ad rpo agrmt ithr has th valu F ( ) + Ccash i cas (i) of a cash dividd or th valu F ( ) + Cscrip D S A i cas (ii) of a scrip dividd akig ths cosidratios ito accout th stock owr assumd to prfr mor to lss 3

4 should opt for th cash dividd if ad oly if at tim w hav h S i D F A h S i F 3 A R 3 i h S F A Cscrip Ccash () h last quality follows from th towr proprty of coditioal xpctatio ad th fact that [S F ] = S Sic w tr ito our portfolio at tim t = with zro iitial cost arbitrag pricig thory suggsts that th risk-utral xpctatio ovr th portfolio valu at should qual zro his yilds a quatio ivolvig F ( ) which implis a modl-fr formula for th forward strik pric takig ito accout th scrip dividd optioality: F ( ) = S D h S R 3 max A io F () Lt us mak a coupl of rmarks rgardig Formula () h formula simplifis massivly i cas that it is alrady kow at tim t = whthr th ldr will opt for cash or scrip dividd I th cash cas o simply has to rplac th apparig maximum by th valu xp R 3 r(s) δ(s) ds rsultig i th traditioal formula rlatig stock pric ad forward strik I th scrip cas th maximum ds to b rplacd by [S /A ] Similarly wh th stock owr must dcid btw scrip or cash alrady at t = h would mak this dcisio by takig th xpctatios o both sids of () which has th ffct that th coditioig o F ad hc also th xpctatio aroud th maximum disappars i () Modl-fr bouds If o is ot willig to mak ay furthr simplifyig assumptios th formula () is difficult to valuat without rsortig to tim-cosumig Mot Carlo gis Howvr w ca boud F ( ) from abov by a simplr-to-comput formula ad from blow i a spcial situatio his is brifly outlid i th squl (a) Uppr boud: Formula () is difficult to valuat i gral howvr Js s iquality applid to th covx fuctio

5 x 7 max{x c} with a costat c implis that F ( ) S D h S R 3 io max A Clarly this boud is attaid with quality i th spcial cas wh th stock pric is modld dtrmiistically so it caot b improvd without furthr modlig assumptios h sharpss of this boud improvs th smallr th variac of th ivolvd coditioal xpctatio [S /A F ] is h lattr is v zro if = ad th log rturs of th stock pric procss ar assumd to hav idpdt icrmts (g i th Black-Schols modl or othr Lévy modls) Huristically th variac is also small whvr w hav th situatio that = I gral howvr thr is a positiv variac (b) Lowr boud: Udr th asssumptio that = i th scrip or cash -dcisio must b mad o th x dividd dat it is possibl to driv a usful lowr boud for F ( ) his is o logr possibl howvr if > costitutig a major diffrc o b mor spcific w assum that is icludd i th avrag i t = Furthrmor it is rasoabl to assum that aftr th x-dividd dat udr a risk-utral masur th ormalizd stock pric procss may b writt as R +t S +t = Yt S t [ ] for som martigal {Yt }t [ ] with Y = Kpig this assumptio i mid togthr with th fact that th fuctio x 7 x/(κ + κ x) is cocav i x for arbitrary ogativ costats κ κ a itratd applicatio of Js s iquality implis th almost sur iquality R S F P R tk A k= his implis th (quit sharp) modl-fr lowr boud F ( ) S ( max R 3 D R P k= R tk ) (3) 3 Approximativ valuatio v though w hav drivd th gral modl-fr formula () for F ( ) i Sctio as wll as lowr ad uppr bouds i Subsctio thir valuatio is still difficult or ot applicabl at all (i cas of th lowr boud) h goal of th prst sctio 5

6 is to show how this computatio ca b do approximatly I ordr to procd w itroduc th otatio i h S X := F A h valuatio of F ( ) sstially boils dow to th computatio of [max{c X }] with a costat c > his is difficult i gral bcaus th probability distributio of X is ukow v i stadard modls for th stock pric procss I ordr to b abl to procd w thrfor mak th followig two covit assumptios: (A) W assum that th stock pric procss aftr th x-dividd dat is giv by th followig Black-Schols-typ modl with a volatility paramtr σ > : S +t = S R +t σ t/+σ Wt t [ ] whr {Wt }t dots a stadard Browia motio (A) h radom variabl X is assumd to b logormal with th first two momts big m (σ) := S A S m (σ) := m (σ) + A Giv (A) ad (A) th followig Black-Schols-typ formula is valid for F ( ) i dpdc of σ whr Φ dots th cumulativ distributio fuctio of a stadard ormally distributd radom variabl: F ( ) = S ( D R 3 m (σ)! m (σ) log m (σ) + r(s) δ(s) ds r log + m (σ) Φ m (σ) (σ) log mm(σ) R!) m (σ) R log m (σ) 3 r(s) δ(s) ds 3 r + Φ m (σ) log m (σ) s () Cocludig w ar lft with th task of computig m (σ) ad m (σ) For thir computatio w apply furthr logormality assumptios which ar xplaid ad whos iducd rrors ar ivstigatd i th subsqut paragraph Udr assumptio (A) th radom variabl X is ithr logormal or is its scod momt qual to m (σ) Howvr th logormality assumptio is rathr stadard s g Dufrs () ad th assumptio o th scod momt itrpolats btw kow margial cass i th ss that m (σ) is corrct if = ad if = 6

7 3 h computatio of m (σ) R r(s) δ(s) ds) S /S w may dfi a quivalt masur P P o (Ω F ) by dfiig dp := L dp Udr P Girsaov s horm implis that {W t }t whr W t := Wt σ t is a stadard Browia motio Cosqutly with th otatio S +t := xp(σ t) S +t it follows that R S L S r(s) δ(s) ds = A A R R S = = A A From th logormal radom variabl L := xp( whr dots xpctatio with rspct to P ad A X S tk X Rtk r(s) δ(s)+ σ ds+σ Wtk := = S k= k= hus w hav simplifid th computatio of th iitial xpctatio valu by gttig rid of th umrator (ad th stock valu S ) h distributio of th radom variabl A is a sum of (dpdt) logormals which is difficult to hadl i thory Fortuatly sums of logormals appar i various applicatios rlatd to fiac ad isurac Cosqutly thr xist may approximativ approachs to tackl this challgig task fficitly s g Dufrs () for a ovrviw ad furthr rfrcs O promit asatz is to mak th followig covit3 assumptio: (A3) : th radom variabl A has a logormal distributio with paramtrs µ ad σ udr P akig for gratd assumptio (A3) a momt-matchig tchiqu is applid i ordr to dtrmi µ ad σ O th o had this implis that th first ad scod momt of A ar giv by xp(µ + σ /) ad xp( µ + σ ) rspctivly O th othr had th first ad scod momts of A ca b computd xplicitly yildig th quatio systm µ + σ = µ + σ = X Rtk +σ (tk ) =: f (σ) k= X r(s) δ(s)+ σ ds+ R t` r(s) δ(s)+ σ ds k`= 3 R tk σ 3 mi{tk t` }+max{tk t` } =: f (σ) his assumptio is thortically wrog but ca b viwd as a rasoabl approximatio s Dufrs () 7

8 Solvig ths two quatios for th ukows µ ad σ w obtai udr our log-ormality assumptio (A3) that S m (σ) = = R A R (A3) = R A µ + σ µ + σ 3 (µ + σ ) R f (σ) = f (σ)3 = (5) his provids a simpl-to-implmt closd formula for m (σ) Figur visualizs this approximativ formula Black Schols proxy modl fr valu m(σ) σ i % Fig : Visualizatio of Formula (5) for varyig valus of th paramtr σ wh r(t) δ(t) 3 = ad = 7 h spcificatios of this xampl ar chos to coicid with a SX5 xampl w carry out i Sctio whr a modl-fr valuatio of [S /A ] is dmostratd h formula for m (σ) coicids with th valu obtaid from th modl-fr asatz of Sctio for σ % whr th lis i th plot itrsct Rmark 3 (Durig th avragig priod?) W hav s how to comput Yt := [S /A Ft ] for t = withi a Black-Schols sttig Wh viwd as a stochastic procss ovr tim th valu Yt rmais costat for all t [ ] by th idpdt icrmts of th ivolvd Browia motio A similar computatio ca also b applid for t ( ) wh a part of th avrag i th domiator is alrady obsrvd By xactly th sam logic as abov o obtais th followig approximativ formula for ths t: Z Yt = R R St t x / π dx c + log(c ) log(c )/+ log(c ) log(c ) x 8

9 whr c := X Stk k : tk t c := St X R tk t c := St X R tk t +σ (tk t) k : tk >t Rt σ r(s) δ(s)+ ds+ t ` r(s) δ(s)+ σ ds k` : tk t` >t σ 3 mi{tk t` }+max{tk t` } t h rmaiig o-dimsioal itgral agaist th stadard ormal distributio must b valuatd umrically Figur 3 shows o path of th procss Yt t [ ] withi th cosidrd Black-Schols modl sttig It is obsrvd that Yt is idtically costat o [ ] ad [ ] St Y t 5 3 tim Fig 3: Visualizatio of o path of th stochastic procss Yt := [S /A Ft ] for t [ ] with = = ad = 6 with r(t) δ(t) ad S = h computatio was carrid out via th logormal approximatio ad as dscribd i Rmark 3 3 How svr is th rror iducd by assumptio (A3)? W xami by mas of a Mot Carlo cas study th accuracy of th logormal approximatio with assumptio (A3) Rgardig th Mot Carlo simulatio otic that th ivolvd radom vctor (St St S ) ca asily b simulatd xactly bcaus it rsults as a fuctio of a ( + )-dimsioal ormal distributio I ordr to simulat th lattr w simulatd th idpdt ormal radom variabls rsultig from th icrmts of th ivolvd Browia motio o th itrvals [ t ] [t t ] [t ] W furthrmor icludd a small variac rductio tchiqu by ot oly usig th simulatd icrmts of {Wt } but also th os of { Wt } (atithtic variats tchiqu) 9

10 s g (Mai Schrr p 55ff) I total w usd millio sampls 5k iid sampls basd o th paths of {Wt } ad aothr st of 5k sampls rsultig from th paths of { Wt } It is obsrvd from Figurs ad 5 that th approximatio is quit good but bcoms wors with icrasig σ I particular th logormal approximatio tds to slightly ovrstimat th valu of th scrip dividd i our xampl = days Mot Carlo ( Mio rus) Logormal approximatio 998 m(σ) σ i % 5 Fig : Computatio of m (σ) wh r(t) δ(t) ad for varyig valus of th volatility paramtr σ W assum = ad = /365 corrspodig to a avragig priod of two wks 35 3 umbr of occurcs m(σ) (via Mot Carlo with Mio rus) i % Fig 5: Histogram (basd o 5 ralizatios) of th distributio of a crud Mot Carlo stimator for m (σ) with spcificatios as i Figur ad σ = % h rd dottd li rprsts th valu of th logormal approximatio

11 3 Computatio of m (σ) With a aalogous logic to th computatio of m (σ) o may also driv a approximatio for m (σ) which rsults i a v logr ad astir formula Bcaus of th grat aalogy with th computatio of m (σ) w mrly sktch th drivatio i th squl First of all with th hlp of Girsaov s horm o ca show that S A = R (r(s) δ(s))+σ ds hi Z (6) whr X Z= R tk Rt r(s) δ(s)+ 3 σ ds+ ` r(s) δ(s)+ 3 σ ds k`= σ (Wtk +Wt` ) Dotig by < x x x3 x >k th k -th largst of th four umbrs x x R k = th first ad scod momt of Z ar computd as follows: X [Z] = R tk r(s) δ(s)+ 3 σ ds+ R t` r(s) δ(s)+ 3 σ ds k`= (3 mi{tk t` }+max{tk t` } ) X Y k`mν= ( ) {k`mν} [Z ] = σ σ R t( ) r(s) δ(s)+ 3 σ ds! 8 σ <tk t` tm tν > +3 <tk t` tm tν >3 +5 <tk t` tm tν > +7 <tk t` tm tν > Makig th furthr covit assumptio that Z has a logormal distributio it follows that [Z ] = [Z ]/[Z]3 Hc with th giv first two momts it follows from (6) that S m (σ) = m (σ) + A R (r(s) δ(s))+σ ds [Z ] m (σ) + [Z]3 whr th scod quality is oly a approximatio bcaus Z is ot logormal Figur 6 visualizs a umric xampl W lt ( ) = ( ) th currt stock pric quals S = 3 th aoucd cash dividd amout quals D = ad r(t) 3 δ(t) 5 Morovr it is assumd that thr ar = quidistat avragig tim poits i th priod [ ] icludig ad hs umbrs ar ispird by a ral-world xampl for a stock for which th lvl of th rpo margi δ xcds th lvl of th short trm itrst rat δ by far h followig obsrvatios ar mad If thr was o scrip dividd altrativ at all (rgular cash dividd) th strik pric would b ivariat wrt σ If th scrip dividd was madatory (o cash dividd) th strik pric would b icrasig i σ his obsrvatio might lad to th wrog ituitio that th cash altrativ is prfrrabl i ay cas i th prst xampl Howvr wh th stock owr may opt btw scrip or cash th

12 8 9 forward basis F() S 3 5 scrip or cash optioality madatory scrip madatory cash σ (i %) 5 Fig 6: Visualizatio of F ( ) i dpdc of σ strik pric is show to b dcrasig i σ idicatig that th dcisio btw scrip or cash at th d of th avragig priod ( = i this xampl) costituts a o-gligibl valu to th stock owr which affcts th forward strik pric sstially Modl-fr approximatio basd o rplicatio Assumig = th prst sctio prsts a approximatio for th xpctd valu [S /A ] which is basd o a modl-fr approach Dotig th valu obtaid via this asatz by c for a momt o might back out a volatility paramtr σ to b usd i th approach of th aformtiod sctio as th (uiqu) solutio of th quatio m (σ) = c Assumig high ucrtaity about th appropriat lvl of σ to choos i Formula () this valu might b a pragmatic choic also i th cas > h ida is to approximat A (S + S )/ i w oly cosidr th dpoits of th avragig priod With this approximatio tak for gratd it is ough to comput S S + S = f (S ) with f : R+ R+ x 7 x/(s + x) As i particular f C (R+ ) it holds that Z S f (x) = f (S )+f (S ) (x S ) + f (y) (y x)+ dy Z + f (y) (x y)+ dy S f (x) whr = S /(S + x) f (x) = S /(S + x)3 s Brd Litzbrgr (978); Gr Jarrow (987); Nachma (988) Itroducig th followig otatio for calls ad puts R h i (S y)+ h i R Put(y) = r(s) ds (y S )+ Call(y) = r(s) ds

13 w ca xprss th xpctatio via out-of-th-moy calls ad puts o do so w hav to rplac th fuctio f by th giv rprstatio ad xchag th itgral ad th xpctatio usig olli s thorm yildig S S + S R = + r(s) ds Z S R S Put(y) dy + (S + y)3 r(s) ds Z S (7)! Call(y) dy (S + y)3 Rplacig th itgrals by Rima sums usig xistig markt quots for calls ad puts a modl-fr valu for th scrip dividd may b xtractd W provid a xmplary us of Formula (7) wh {St }t is th SX5 Idx = corrspods to 6-May- ad to 6May- W simply igor th dividd paymts du i this priod as th aim is oly to illustrat th procdur W assum r(t) δ(t) 3 It ca b obsrvd that wh usig oly xistig markt quots th fuctio f might ot b approximatd sufficitly xact as ca b s i Figur 7 For that raso w (itrpolat ad) xtrapolat th xistig optio prics o do so w comput th implid volatility smil (itr- ad) xtrapolat th volatilitis ad covrt thm back to prics Figur 8 visualizs th obsrvd implid volatility smil whr oly liquid outof-th-moy optios hav b usd o comput it w usd th Black-Formula with th forward valu implid by th put-call parity Othrwis th volatility smil would xhibit a larg kik du to th fact that th currt spot pric ad th forward valu diffr sigificatly (bcaus of th igord dividd paymts) Howvr for our rsults it would ot mak a big diffrc as w oly us it for th itrpolatio h rsultig approximativ valu for th xpctatio [S /A ] is corrspodig to σ % i Formula (5) 5 Coclusio It has b show how to dtrmi a fair strik pric for a quity forward cotract i th prsc of a scrip dividd optio for th holdr of th udrlyig stock A modl-fr formula for this strik pric has b drivd Morovr it has b show how th formula could b valuatd approximatly usig logormal approximatios withi a Black-Schols sttig Rfrcs G Brhart J-F Mai O covxity adjustmts for stock drivativs du to stochastic rpo margis XAIA hompag articl () D Brd R Litzbrgr Prics of stat cotigt claims implicit i optio prics Joural of Busiss 5 (978) pp 6 65 D Dufrs h log-ormal approximatio i fiacial ad othr computatios Advacs i Applid Probability 36:3 () pp RC Gr RA Jarrow Spaig ad compltss i markts with cotigt claims Joural of coomic hory (987) pp 3

14 58 tru payoff fuctio f implicitly usd approximatio Fig 7: Visualizatio of th implicitly usd approximatio of f if oly obsrvd markt quots ar cosidrd It is obsrvd how th approximatio suffrs outsid th rag of obsrvd strik prics xtrapolatd IV smil : proxy σ = 953% 55 currt idx (AM) rquird σ i proxy obsrvd IVs 5 5 IV i % strik Fig 8: Visualizatio of obsrvd volatility smil h dottd valus hav b obsrvd ad all visualizd (partially xtrapolatd) valus hav b usd i Formula (7) i ordr to dtrmi th valu for th scrip dividd h proxyσ of approximatly % corrspods to th paramtr σ that has to b pluggd ito th formula drivd i th subsqut sctio i ordr to obtai th sam valu as th currt modl-fr asatz J-F Mai M Schrr Simulatig Copulas Sris i Quatitativ Fiac Imprial Collg Prss Lodo ()

15 D Nachma Spaig ad compltss with optios Rviw of Fiacial Studis 3 (988) pp