The real value of stock

Transcription

1 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 Black-Scholes fraework o value hese exoc derivaves ad explore issues wih hedgig, as well as providig a iuive explaao for soe of he resuls obaied I ergers ad acquisios (M&A), he acquirer ofe wishes o pay he cosiderao i sock, usig a exchage rao bewee he arge copay s shares ad is ow shares. he delay bewee agreee ad closure of he rasaco eas ha he aural volaliy of he acquirer s sock price, perhaps exacerbaed by aoucee of he rasaco, will affec he value of each share paid. he arge ad acquirer us carefully egoae a pricig echais o allocae his risk bewee he, ad ay choose a echais o vary he exchage rao o copesae. he sae cosideraos apply o hose srucured producs where he cosiderao is paid i sock, ad o cerai adaory-coverble bods, bu i his arcle we discuss M&A rasacos for ease of exposio. If he exchage rao is fixed a agreee of he rasaco, he arke value of he cosiderao will vary wih he acquirer s sock price. I his case, boh arge ad acquirer assue he risk ihere i he sock price, sice he cosiderao could be worh ore or less a closure ha a agreee. he acquirer does, however, have he beefi of kowig he uber of shares payable fro he ouse. ha ca help, for exaple, wih he calculao of earigs accreo or diluo. Wih a floag exchage rao, he value of he cosiderao is fixed, ad he exchage rao varies o offse he chages i he acquirer s sock price. ha way, he arge copay reduces is exposure o ay weakess i he sock price bewee agreee ad closure of he rasaco, ad he acquirer assues his risk isead. Coversely, if he sock price rises, he arge copay will iss ou o he exra value, while he acquirer will beefi. he acquirer s sock is usually valued by akig a arihec average of he arke prices for a period (ypically 3 radig days) jus before he rasaco closes. his ore accuraely reflecs he value of he sock o he arge copay s shareholders. he he floag exchage rao ca be calculaed as a agreed fixed value (for exaple, $ per share) divided by his average price. A iside collar uses a floag exchage rao, subjec o upper ad er liis. his eas ha if he average sock price lies ouside a cerai rage (he collar liis ), he exchage rao will be fixed, ad he value of he cosiderao will chage proporoally wih he sock price. Wih a ouside collar, he exchage rao is fixed whe he average sock price lies i a specified rage; oherwise, he exchage rao floas, ad he value payable is fixed. his eliiaes he dowside risk for he arge copay, bu exposes he acquirer o he risk of havig o pay a very large uber of shares if is sock price is weak. Collars are a ipora feaure of US M&A rasacos ad are icreasigly so for ieraoal oes. Ieresgly, qualiave facors, such as geographic regio, appear o be os releva i deeriig he use of a collar srucure. Quaave facors, such as he volaliy of he acquirer s sock price, igh be expeced o play a larger role i he choice of he pricig echais, bu daa aalyses ofe do o suppor such hypoheses. Merger arbirageurs ad srucurers eed o udersad very closely he issues affecg he valuao ad hedgig of a collar, while ohers will beefi fro udersadig valuao cosideraos eve if hey do o hedge heselves. Collar srucures are, as we have see, exoc derivaves o he acquirer s sock price. I his arcle, we use esablished echiques of aheacal fiace o show how o value ad hedge hese derivaves. We cosider a iside collar, bu he ehods also apply o ouside collars. We look osly a a odel ha uses a geoeric average for he acquirer s sock price, because his als a aalyc soluo. Siple uerical ehods provide he ecessary adjuses for a arihec-average versio. Oce we ve derived he valuao forula i he geoeric-average odel, we break i dow io wo key valuao copoes ad aalyse he i isolao. We fid approxiaos o hese wo copoes, o give a iuive udersadig of risk.e 89

2 A sadard iside collar srucure ixed quay of sock a axiu rao r high C/r high wha drives he valuao. ially, we look briefly a a special case where he prices used o calculae he average are chose a rado fro a se rage of radig days. A siple collar srucure We sar wih he sadard Black-Scholes fraework. Le S be he acquirer s sock price a e, where he curre e is =. Le be he volaliy of he sock ad ρ he average risk-free ieres rae o e. Black-Scholes heory provides ha we ay fid a probabiliy easure P such ha he discoued sock-price process is a argale ad we ay wrie he sock price as: S Collar liis = S e ixed value of sock a headlie value C Collar value P Exchage rao r(a) Average sock price A ( ρ ) W where W is a sadard Browia oo uder he argale easure P. Noe ha we are igorig divideds i his odel, bu oe ca adjus for divideds if required, provided here are o divideds i he averagig period. he payou fro he collar a auriy = is: = C/r ixed quay of sock a iiu rao r P S r A where A is he average sock price ad r is a fuco represeg he exchage rao give by he collar ers. Norally, we defie a iside collar usig hree cosas: he headlie value of he collar, C, ad he liis o he exchage rao r, r ad r high. he, seg B = C/A, we have: r if < r r ( A) = B if r B r high () r if > r high high igure shows he value of he sock payable a auriy, ad he exchage rao, for a sadard iside collar. he payou srucure resebles ha of a opo collar ( r calls ad shor r high pus). he graph assues ha he fial sock price S equals he average sock price A, bu his assupo i geeral will o hold. Usig a arihec average for A precludes a aalyc valuao of his payou. herefore, we look isead a he equivale collar where A is a geoeric average, so ha we ca value i aalycally. he value will be very close o ha of he arihec-average collar. he we ca use Moe Carlo echiques o esae very accuraely he differece i value bewee he wo collars. Addig his esaor o he value of he geoeric-average collar gives a corol variae esaor for he value of he arihec-average collar, which has a precisio of aroud.% for oly, pahs i he Moe Carlo siulao. We shall herefore use he defiio: = A A S... S where he cosas are:, he legh of he averagig period;, he uber of reaiig days i he averagig period; A, he geoeric average so far (or oe if = ); all es <... <, he reaiig days i he averagig period. Derivao of he value of a geoeric-average collar ro sadard Black-Scholes heory, we kow ha he prese value of a derivave is he expeced discoued payou uder he argale easure P, here e ρ EP. o evaluae his, we firs defie a ew easure, P _, equivale o P, such ha W _ := W is a Browia oo uder he ew easure. A direc applicao of Girsaov s heore 3 gives: W EP E S r A E e S r A where E _ is he expecao uder he ew easure, P _. All he rado variables excep r(a) cacel ou o he righ-had side of his equao, ad we ay siplify i o: EP ES Er A () = = = Nex, observe ha we ay rewrie A ad B as: where: A = A S e µ Z µ B = CA S e Z µ = ρ i i i Z = W ~ N, ( ) uder P = i i i We see ha A ad B have oral disribuos uder he ew easure, ad his akes i relavely sraighforward o evaluae he expeced payou give by equao (). Whe he exchage rao akes he for give i equao (), we ca decopose i as fols: r A B r B B r ( high ) = ( ) We see ha r is coposed of he rao B, wih a pu o B a srike r, ad a shor call o B a srike r high. he valuao forulas are he jus he usual Black-Scholes forulas for pus ad calls. Specifically, he expeced values uder easure P _ are respecvely: Chaper 3 of Baxer & Reie (996) provides a iuive preseao of he aheacs i he Black-Scholes odel. Hull s sadard exbook (997) offers a ore praccal perspecve o he odel his is he corol-variae Moe Carlo ehod. or a alerave descripo, see Hull (997), pages or a iuive descripo of Girsaov s heore, see chaper 3.4 of Baxer & Reie (996). or a full saee ad proof, see chaper 3.5 of Karazas & Shreve (99) 9 Risk Jue 6

3 where: ( r ) = Φ E r B r r Φ E( B r ) = Φ high r high rhigh Φ r high = EB = CA S exp µ = Var B = ( he variace beig akeuder P) he dela ad oher Greeks ca also be derived as for he usual Black-Scholes forulas. Valuao of a collar he expeced value of he collar payou P depeds o a uber of facors. We spli he value io he folig copoes: he headlie ers, as specified wih he rasaco. his is he cosa C. he averagig effec. he cosiderao is i sock, o i cash, ad he average sock price cao be realised wih ceraiy. his effec easures he differece bewee C ad he value of a collar wih he siple floag exchage rao r(a) = B. We shall fid ha his collar value is always less ha C. he collar effec. he exchage rao is cosraied o a se rage, so he collar value ca rise above or fall be he headlie ers. his effec derives fro he calls ad pus o he rao B ha we discussed above. Noe ha a call (pu) o B correspods o a pu (call) o he average sock price A. he selee effec. he exchage rao is fixed over he course of he averagig period, whereas he collar is priced for a laer auriy a e. A sall aou of ieres will accrue i he ierveig e or he sock ay pay a divided. his effec is icorporaed wihi he collar effec ad averagig effec bu soees eeds o be separaed. igure illusraes he wo os ipora copoes, he collar effec ad he averagig effec. I ay siuaos, oe of hese will prevail. So, a e before he averagig period, he collar effec will be os sigifica. Close o he averagig period, wih he sock price bewee he collar liis, he averagig effec is os sigifica. here will be circusaces i which boh will be sigifica ad hey are hard o separae. he collar effec Valuao ad approxiao. As we observed above, he collar effec jus derives fro pus ad calls o he rao B, ad we gave a aalyc valuao for hese i equaos (3). We also observed ha hese pus ad calls o he rao B correspod o calls ad pus o he (3) he collar effec ad averagig effec Exchage rao r(a) Collar effec pu o rao B, call o average A uderlyig sock. Here, we use approxiaos o show ha he valuao forulas for he calls ad pus o he sock reseble he valuao forulas for vailla opos, alhough i is o clear wha auriy dae o apply o he vailla opos. o illusrae his, cosider he call opo o B a srike r high, before he averagig period sars (so ha = ). Le K = C/r high be he correspodig srike price for he sock. Usig equao () ad he secod equao (3), we obai: ρ e E S B r high = r K µ exp high Φ K µ S K µ S Φ S irs, observe ha for he case =, =, his reduces o he sadard forula for r high pu opos wih srike price K. Nex, suppose isead ha is very large ad he es i are evely spread hrough a averagig period of legh d :=. he we ay use he approxiaos 4 : d 3 µ ρ d i i Pure averagig effec Rao B = C/A Collar effec shor call o rao B, shor pu o average A If we furher suppose ha ρ = ρ for each i he we ca reduce he righ-had side of equao (4) o: (4) (5) ( ) S ( d) d ( K r e K ) ρ d d ρ 6 Φ high d 3 S ( d) ρ ( d K ) S Φ d 3 4 he firs of hese holds asypocally as, whereas he secod oly requires he es i o be evely spaced risk.e 9

4 3 Hedgig he averagig effec i he base-case sceario 4 Hedgig he averagig effec wih 3% price drop o day 8 Sock price (l-hs) ial realised payou $99.8 ial rao.955 Realised payou (l-hs) Sock price (l-hs) ial rao.8 ial realised payou $97.56 Hedge (r-hs)...8 $ 6 4 Hedge (r-hs).6.4 $ 6 4 Realised payou (l-hs) Day Day We see ha his forula has a siilar for o he sadard Black- Scholes forula for he value of r high vailla pu opos, sruck a K. he volaliy i he Black-Scholes forula ay reasoably be ake o be ad he ieres rae o be ρ. he auriy dae of he vailla opos is, however, o clear fro he forula above, which uses differe es fro wihi he averagig period. Nowihsadig his, we ay hik of he sesivies of he collar effec as siilar o hose of r high shor pu opos sruck a C/r high ad r call opos sruck a C/r. Hedgig. Hedgig he collar effec is relavely easy. he corolvariae Moe Carlo ehod gives precise Greeks for dyaic hedgig wih he sock or wih opos o he sock. Soe e before he averagig period, he hedgig will be siilar o ha for vailla opos, as above. Oce he averagig period has sared, i will orally be reasoably clear wheher he collar will give a fixed quay of sock or a fixed value of sock. I a M&A rasaco, arbirageurs hedgig posios i he arge sock will orally sell calls ad buy pus o he acquirer s sock (or use equivale dyaic hedgig). Noe, however, ha he oal dela of he opos used for he hedge (or he uber of shares sold i dyaic hedgig) will geerally be saller ha he uber of shares ha would be sold o hedge a all-sock rasaco of he sae value. he averagig effec Valuao ad approxiao. We show here ha $ worh of sock, reaed as a derivave, is always worh less ha $, if here is ay volaliy a all i he sock price. he reaso is egave gaa i hedgig he averagig effec, which we ca hik of iuively as fols. Hedgig he averagig effec ivolves sellig sock o every day of he averagig period, so ha he oal sock sold balaces he sock received a auriy. Cosider wha happes if he sock price drops sharply par way hrough he averagig period. he, i suddely becoes appare ha he average price will be er ha previously hough ad he uber of shares receivable will be higher ha previously hough. So we have sold as uch sock as we eed for he hedge ad he sock price is ow er, so he hedge loses oey. A siilar effec occurs if he sock price rises suddely. he value of he averagig effec is jus he expeced loss o hedgig. o look a his copoe, we cosider he exchage rao r(a) = B before he averagig period sars (so ha = ). his e, equao () gives: ρ ρ e E ( S B e S B C ) = E = E exp µ = exp ρ C exp ( i ) i i he firs er o he righ-had side is a average discou facor. he differece bewee his ad he full discou facor o e gives rise o he selee effec. I oher words, you ca lock i he value of he collar by hedgig durig he averagig period, ad receive ieres o he proceeds ul e. he las er is slighly less ha oe, showig he egave gaa propery of he averagig effec. If is relavely large (which oe igh use i ay case as a approxiao whe he average price is a volue-weighed average price), we ca use he approxiaos (5) ad coclude ha he righ-had side of equao (6) will approxiae o: exp ρ C exp ( i ) i which is easy o calculae. Noe ha his resul would be differe for a arihec-average collar. 5 he Risk profile seco be copares he wo ypes of collar i oe parcular sceario. Hedgig. Cosider ow how equao (6) would look durig he averagig period (ha is, for < ). We have: e ρ S ( S B ) = A i i E exp ρ C exp ( i ) i Differeag he righ-had side wih respec o S gives he dela: 5 or he arihec-average collar, we fid epirically ha usig a /6 er i equao (7) isead of / gives a beer approxiao. We have aalysed he accuracy of his approxiao i various scearios. he averagig effec does o deped o he spo price ad collar liis. We also fid ha he e o copleo has a egligible ipac o he forward value of he averagig effec. he ai deeriig facors are he volaliy of he sock ad he legh of he averagig period. Of hese, he laer is os sigifica. he approxiao o he averagig effec is wihi 6% of he Moe Carlo esae, across a rage of volalies, ad for averagig periods er ha five days. he overall approxiao o he value of he collar is accurae o a few basis pois (6) (7) 9 Risk Jue 6

5 5 Disribuo of realised payou fro hedgig a arihec-average collar 6 Disribuo of realised payou fro hedgig a geoeric-average collar Probabiliy desiy Oe sd dev $. Mea $99.8 Oe sd dev $. Probabiliy desiy Mea $99.9 Oe sd dev $.4 $99.4 $99.5 $99.6 $99.7 $99.8 $99.9 $ $99.4 $99.5 $99.6 $99.7 $99.8 $99.9 $ C ρ A e i i S exp i ( ) i he firs er here is he proporo of he averagig period ha has elapsed ad i sees aural ha we should weigh he dela by his. he ex er is a esae of he fial rao, assuig ha he sock price S rises wih ieres raes, which is cosise wih expeced levels uder he argale easure P. he las er, as i equao (6), is slighly less ha oe ad reflecs he egave gaa of he averagig process. I is salles whe =, as i equao (6), ad icreases as decreases. I is very close o oe whe =. he hedgig o each day of he averagig period will reflec he icrease i he above dela as decreases. We see fro his ha a good approxiae hedgig ehod is o esae he fial rao o each day of he averagig period ad o esure we have hedged a proporo of his, correspodig o he proporo of he averagig period ha has passed. I parcular, afer he las day of he averagig, our hedge will correspod o he exac exchage rao payable uder he collar. A siilar ehod will work for a arihec-average collar. Alhough we cao derive a aalyc forula as above, we ca use he corol-variae Moe Carlo ehod o obai sufficie accuracy. his is epirically siilar o hedgig he expeced fial rao i proporo o he uber of days of averagig ha have passed. Reeber ha he Black-Scholes odel uderesaes he likelihood of exree oves i he sock price, so he expeced cos of hedgig should be higher ha he value i his odel. I pracce, hedgig a collar ca have a sigifica arke ipac, because i is largely coceraed i he averagig period. As a exaple, Geeral Elecric s acquisio of Aersha used a collar wih a -day averagig period. I ha period, he average daily volue of Geeral Elecric s sock was ore ha double he level of he previous hree ohs, apparely caused by arbirageurs lockig i he average price used i he collar. Risk profile. I pracce, we igh oly hedge oce o each day of he averagig period, by sellig sock a he closig price or he day s volue-weighed average price, depedig o which is used for he average. his eas ha he loss o hedgig ay be ore or less ha ha calculaed previously. I is o possible o ake a profi o hedgig i his way. o exaie he risk profile of his hedgig ehod, we use a separae, siplified odel for a arihec-average collar. Here, we siulae daily sock prices i he averagig period, accordig o he disribuos specified i he Black-Scholes odel. he we hedge he averagig effec as oulied above ad rack he cuulave realised payou fro he hedgig o each day of he averagig period. We use he folig sceario for he siulao: headlie ers of $, a -radig-day averagig period, a iial sock price of $ ad volaliy of 4%. I his sceario, he value of he averagig effec is $.. We illusrae a ypical siulaed hedgig profile i figure 3. Observe ha he realized payou rises seadily o close a $99.8, very ear he headlie ers of $ less he value of he averagig effec. Also, he fial hedge rao equals he fial exchage rao i he collar. Now le us cosider he sceario of a 3% crash halfway hrough he averagig (wih everyhig else as i he previous exaple). he hedgig profile is show i figure 4. he hedge rao jups o day as he sock crashes, ad he realised payou oly reaches $ AKE CHARGE O YOUR RISKS AgeaRisk is a powerful sofware ool for risk aalysis ad siulao. Ulike spreadshee-based soluos, AgeaRisk s visual odels are quick ad easy o build ad help you couicae ad corol your risks effecvely. Now coercially available i deskop ad eerprise versios. id ou ore ad dowload a rial versio a: risk.e 93

6 Nex, we cosider how he fial realised payou varies aroud he ea of $99.8 i oral arke codios. We ru housads of siulaos like he oe above ad plo he disribuo of he fial realized payous. igure 5 shows he resuls. he ea coes ou a $99.8 as expeced, bu he sadard deviao is relavely high a $.. Also, here is a sigifica skew i he disribuo owards he er payous. I is isrucve o copare his wih a collar ha uses a geoeric average isead of a arihec average. he resuls for he geoericaverage collar are show i figure 6. Noce firs ha he ea shorfall is half wha i was before. Secod, he disribuo is uch gher, wih a saller dowward skew. his is because a geoeric average copesaes uch beer for falls i he sock price ha he arihec average ha is orally used. I fac, i our siulao, for ay give sequece of prices, he realised payou fro hedgig a geoeric collar is always a leas as uch as ha fro hedgig a arihec oe. Rado-average collars Mos collars fi io he fraework we have used so far, or have addioal feaures ha ca be icorporaed io his fraework, such as ulple collar srike prices, forward-sarg srike prices or paral cash aleraves. Oe ieresg feaure ha is soees foud is rado averagig. I his case, o all he days i he averagig period are used o give prices for he average. Isead, soe of he days are chose a rado a he ed of he averagig period, ad oly hose days prices go io he averagig calculao. his has wo iplicaos. irs, before he averagig period, he acipaed volaliy of he average will be greaer ha before, reflecg he addioal radoess. his eeds o be icorporaed io he hedgig of he collar effec. Secod, durig ad a he ed of he averagig period, here will be radoess largely fro he rado seleco. ha eas ha hedgig he averagig effec is ore ucerai ha before. Alhough he ea realized payou should be siilar o before, he deviao aroud he ea will be greaer. Suppose he ha isead of usig all prices i he averagig period, we choose a fixed uber k of he a rado (wih all subses beig equally likely) ad fix he exchage rao usig he arihec average of hese. Suppose he prices chose are X,..., X k ad heir arihec ea is X _. As a firs sep, assue he prices S i he averagig period are fixed, ad cosider jus he iplicaos of he rado seleco. Defie S _ ad V as he arihec ea ad variace of all he S, respecvely. We fid: EX = S k Var X = V k If is very large copared wih k, he variace approxiaes o V/k. his would be he variace if he X i were idepede, idecally disribued rado variables wih variace V. Now rea he S as rado variables ad le S _ ad V be he ea ad variace of he ES isead of he S. he we have: EX = S k Var X = Cov S S Var S k, i j i i j i k V k ( ) Sice S is a sock-price process, as defied a he begiig, we ca evaluae he righ-had side of he secod equao. o approxiae he variace, firs assue ha ieres raes are sall, so ha V. Also assue ha we are close o he averagig period ad ha he days of he averagig period are evely spaced. Wihou hese assupos, he variace will be slighly higher. he we obai he approxiao: k Var X S k k ( ) 3 Observe, o he oe had, ha if k =, his approxiaely gives he variace of he sock o he iddle of he averagig period, as oe igh expec. O he oher had, if k =, his approxiaely gives he variace o oe hird of he way hrough he averagig period. Now ha we have a esae of he variace of he rado average, we ca esae he variace of collar payou. Coclusios Collars are a ipora feaure foud i M&A rasacos ad srucured producs. If a collar values he sock payable wih a geoeric average, we ay use a chage of easure o express he collar payou siply, i ers of oral rado variables. his leads direcly o aalyc forulas for he value of he collar ad he hedgig paraeers. If, as is usually foud i pracce, he collar uses a arihec average, we ay use a corol-variae Moe Carlo echique o esae hese quaes o a high degree of accuracy. he wo key copoes of he collar value (apar fro he headlie ers) are he collar effec ad he averagig effec. he collar effec accous for he fac ha he exchage rao becoes fixed oce he sock price reaches cerai levels. his effec is approxiaely equivale o ha of vailla pus ad calls, alhough i is o clear wha auriy oe should assue for hese vailla opos. he averagig effec accous for he floag exchage rao, ad reflecs he fac ha he cosiderao is paid i sock isead of cash. his effec reduces he collar value ad depeds largely o he volaliy of he sock ad he legh of he averagig period. A siple hedgig odel illusraes he possible coss of hedgig he averagig effec, ad shows i oe sceario ha he expeced cos is wice as large for a arihec-average collar as for a geoeric-average collar. While he above approach will be able o hadle ay varias of he collar srucure ha ofe appear i pracce, i cao hadle rado averagig. We have offered soe his for easurig he addioal risk ha collars wih his feaure pose for ivesors. Ahoy Pavlovich is a direcor i Deusche Bak s Equiy Derivaves Sraegy Group. He would like o express his graude o wo aoyous referees ad o his colleagues for heir helpful coes ad suggesos. he opiios or recoedaos expressed i his arcle are hose of he auhor ad are o represeave of Deusche Bak as a whole. or queries relag o his arcle, please coac Mike Beli. Eail: ike.beli@db.co Refereces Baxer M ad A Reie, 996 iacial calculus a iroduco o derivave pricig Cabridge Uiversiy Press Hull J, 997 Opos, fuures ad oher derivaves hird edio, Prece Hall Ieraoal Karazas I ad S Shreve, 99 Browia oo ad sochasc calculus Secod edio, Spriger-Verlag 94 Risk Jue 6