On the pricing of illiquid options with Black-Scholes formula

Transcription

1 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth Black-Scholes formula TomášTchý 1,2,MlošKopa 2,SebastanoVtal 2,3 Abstract Detectng the far, e. no-arbtrage, prce of an opton s a very nterestng and challengng task of quanttatve fnance. It results mostly from the fact that the opton payoff s nonlnear and the prce can be very senstve to the changes of underlyng factors(especally ATMoptons). Fromtheotherpontofvew,ATMvanllaoptonsareoftentradedand lqud,whledeepitmandotmoptonsaremostlyllqudandtsdffculttoestmate the model parameters. Another ssue s how to obtan the market assumptons about rskless rate relevant for the opton maturty and the future expected dvdends. In ths paper we focus on a partcular problem of extractng parameters to value optons on dvdend payng stocks va BS model usng real data from German opton market. Keywords BS formula, German opton market, llqud opton, mpled parameters opton valuaton. JEL Classfcaton: G13, G14, C14, C52 1. Introducton Optons are a qute mportant type of fnancal dervatves snce they allow to ft even very specfc fears(hedgng) and outlooks (speculaton) about the future evoluton. Due to the nonlnear payoff functon and potental hgh senstvty to changes n the nput factors, such as volatlty or even maturty, optons are very challengng also for modelng purposes. Obvously, snce the standard opton valuaton model of Black and Scholes(and Merton) was based on the assumpton of normally dstrbuted returns, the presence of skewness and kurtoss at the market complcates the stuaton sgnfcantly. A common market practce s to use the market prce as an exogenous varable to be put nto the BS formula(black and Scholes(1973), Merton(1973)). Thus, a so called mpled volatlty s obtaned, e. a number that assures that BS formula provdes the rght prce. Such mpled volatlty can subsequently be used to value 1 DepartmentofFnance,FacultyofEconomcs,VŠB-TUOstrava,Sokolská33,70121Ostrava,Czech Republc. E-mal: tomas.tchy@vsb.cz. 2 DepartmentofEconometrcs,InsttuteofInformatonTheoryandAutomatonoftheASCR,Pod Vodárenskou veží 4, Prague, Czech Republc. E-mal: kopa@karln.mff.cun.cz 3 DepartmentofManagement,EconomcsandQuanttatveMethods,UnverstyofBergamo,Vade Canana 2, Bergamo, Italy. E-mal: sebastano.vtal@unbg.t The research was supported by the Czech Scence Foundaton(GACR) under project No S, throughout the European Regonal Development Fund n the IT4Innovatons Centre of Excellence project (CZ.1.05/1.1.00/ ), the European Socal Fund(CZ.1.07/2.3.00/ ) and SP2014/16, an SGS research project of VSB-TU Ostrava. The support s greatly acknowledged. All computatons were done n Mathematca

2 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 optons, whch are not traded at the market or are not suffcently lqud, and even exotc optons, whcharenottradedatthemarketatall. However, an nexperenced user can be surprsed by several consequences of such approach. Frst, snce the llqud optons we wsh to prce have obvously at least some of the parameters dfferenttotheoptonsusedtogetthempledvolatlty, weneedtoexecutesomekndof nterpolaton and subsequent smoothng to get nce resultng smle or smrk of the volatlty curve or even surface. However, wthout respectng several mportant rules, the results mght lead to arbtrage opportunty, e. the opton prces calculated wth such volatltes mght be mutually nconsstent, see eg. Benko et al. (2007). Second, the market mostly provdes the maturty, moneyness, opton prce and mpled volatlty. But n the Black-Scholes formula there are two parameters more the nterest rate and dvdend yeld. Hence, the second ssue theusersfacedtoswhatarethemarketexpectatonsaboutthenterestrateanddvdends over the opton lfe? Incontrasttoourprevousresearchfocusedespecallyonthefrstssue,seeeg.Kopaetal. (2014a,b),nthspaperwestudythesecondssueandshowhowtoassurethatoptonprces wllbeconsstentwththemarketvewonthenterestrateaswellasexpecteddvdendyeld so that llqud optons can be prced effcently. We also address a specfc problem related to thefarotmoritmoptons. We proceed as follows. In the subsequent secton, basc theoretcal foundatons of optons are provded. We proceed wth the defnton of the opton prcng formula and the mpled volatlty approach. Fnally, an effcent procedure to prce llqud optons s suggested usng real market data of German opton market. 2. Opton termnology Optons are nonlnear types of fnancal dervatves, whch gves the holder the rght(but not the oblgaton) to buy the underlyng asset n the future(at maturty tme) at prespecfed exercse prce. Smultaneously, the wrter of the opton has to delver the underlyng asset f the holder asks. Opton analyss s subject of study of almost any book dealng wth fnancal markets and especally those focusng on quanttatve fnance. In ths secton, we mostly follow Tchý(2011, 2012). Optons can be classfed due to a whole range of crtera, such as counterparty poston (short and long), maturty tme, complexty of the payoff functon, etc. The basc features are the underlyng asset(s), whch should be specfed as precsely as possble(t s mportant manlyforcommodtes), 4 theexercseprce(k),andthematurtytme(t). Iftheoptoncanbeexercsedonlyatmaturtytme T,wecallttheEuropeanopton. By contrast,ftcanbeexercsedalsoatanytmeprorthematurtyday,e. t [0,T],werefer to t as the Amercan opton. A specal type of optons, possble to be classfed somewhere between European and Amercan optons s the Bermudan opton, whch can be exercsed at fnal number of tmes durng the opton lfe. In dependency on the complexty of the payoff functon, we usually dstngush smple plan vanlla optons(pv) and exotc optons. However, by a plan vanlla opton we generally mean callandputoptonswththemostsmplepayofffuncton. 5 Thus, for vanlla call, and forvanllaput,where (x) + max(x;0). Ψ vanlla call = (S T K) + (1) Ψ vanlla put = (K S T ) + (2) 4 Itwllbesupposedthattheunderlyngassetsanon-dvdendstockfnotstatedotherwse. 5 Sometmes,byplanvanllaoptonswemeananyoptonwhchsregularlytradedatthemarket. 808

3 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 Duetothedefntonofanopton tgvesarght,butnooblgatontomakeapartcular trade wecandeducebascdfferencesbetweentheshortandthelongposton. Whlethe payoffresultngfromthelongpostonsnon-negatve,ether 0or S T K,thepayoffofthe shortpostonwllneverbepostve,e.tsether K S T or 0.Moreover,tsobvous,thatthe longcallpayoffsnotlmtedfromabove,buttheshortpostonpayofffunctongoesonlyup totheexercseprce(underlyngassetprceszero). Itshouldbenotedthatevenfthevalue of long and short poston s dentcal, ther market prces mostly dffer due to the transacton costs(a so called bd/ask spread). Theratoofthecurrentprceoftheunderlyngassetandtheexercseprcescommonly referredtoasmoneyness,whchsequaltooneforatmoptons,hgherthanoneforitm optons, e. such optons, whch would lead to postve payoff f exercsed mmedately, and below one for OTM optons. Sometmes, t s useful to work wth log-moneyness, e. zero for ATM, postve for ITM and negatve for OTM optons, or moneyness whch uses forward prces nsteadofspotprces,e.wetakentoaccountthetmefactor. Table 1: Spot prce, exercse prce and moneyness Relaton of Vanlla call Vanlla put S T and K type log/moneyness type log/moneyness S T > K ITM > 0/ > 1 OTM < 0/ < 1 S T = K ATM = 0/ = 1 ATM = 0/ = 1 S T < K OTM < 0/ < 1 ITM > 0/ > 1 AnyfnancaloptonwthmorecomplexpayofffunctonthanstheoneofastandardEuropean(Amercan)callorputoptonsreferredtoastheexotcopton.Themajortyofexotc optons s traded outsde organzed exchanges, at so called OTC markets. However, several types of exotcs are so popular that the major dervatves exchanges have lsted them(e.g. some optons wth barrers). By contrast, many exotc optons are so unque that they are sutable only for nvestors for whom they were orgnally desgned. Thus, wthn the payoff pattern specal needs or beleves and fears of corporate or nsttutonal nvestors are respected. Ths fact decreases the lqudty sgnfcantly and also the prcng and hedgng procedure can be substantally complcated. 3. Analytcal valuaton Although the opton valuaton formula can be derved by varous approaches, such as utlzaton of rsk neutral expectatons or solvng of partal dfferental equatons, t must under the same condtons always lead to the same result. In ths secton, valuaton formulas assumng several knds of avalable nputs wll be consdered. BS model. Assumng the payoff functon of plan vanlla call(1), f T = Ψ vanlla call = (S T K) +, wegetthevaluatonformulaasfollows(bsmodelforvanllacall): 6 V vanlla call (;S,K,r,σ) = SN(d + ) e r KN(d ). (3) Smlarly, assumng the payoff functon of vanlla put(2), we get(bs model for vanlla put): V vanlla put (;S,K,r,σ) = e r KN( d ) SN( d + ). (4) 6 BlackandScholes(1973);analternatvemodelfordvdendpayngstockssduetoGeske(1978), whereas the case of FX rates was analyzed by Garman and Kohlhagen(1983). 809

4 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 In both cases above: d ± = ln S K +(r ± 1 2 σ2 ) σ. (5) Moreover, Sstheunderlyngassetprceatthevaluatontme(t)andtssupposedtofollow log-normaldstrbuton, sthetmetomaturty(e. = T t), rsthersklessratevaldover, σsthevolatltyexpectedoverthesameperod,bothperannum,and N(x)sdstrbuton functon for standard normal dstrbuton. From the formulas above, a strct relaton between prces of call and put optons s obvous (Put-Call Party): Vcall vanlla (;S,K,r,σ) +Ke r = Vput vanlla (;S,K,r,σ) +S. (6) BS model for call optons wth mpled parameters. Assume frst that therearesomeoptons V tradedatthemarket(eg.optonexchangemarket)wthsuffcent lqudty.itfollowsthattheresnoneedtoapplythebsmodelsncethecorrectprcescanbe obtanedfromthemarket, V mar. Obvously,therestllmghtbeaneedtoprceoptonswrttenonthesameunderlyngasset, 7 whcharenottradedatthemarket.forsuchpurposes,wecanusethesocalledmpledvolatlty σ mp, whch can be obtaned from the followng equalty: V mar = SN ( ln S K + r σ mp ( ) σ m 2 ( ) e r ln S K (r KN σ mp σ mp ) ) 2. (7) Here n f and σ specfestherestofoptonparameters thematurty,log-moneynessand rskless rate (; ln(s/k), r). Whle both, the maturty and moneyness are unambguously known, the precedng formula requres to estmate the rskless rate. In practce however, the market(e. opton exchange market organzers) already provdes not only the traded prce, but also the BS mpled volatlty, though, apotentalusermghtbeconfusedwhchrsklessrateshouldbeusedtoprcetheoptonswth a gven maturty. Thesolutonsobvous snceweknoweachdataneededtoapplyformula(7),butthe rsklessrate,wecandefneasocalledmpledrsklessrate, r mp V mar = SN ( ln S K + r mp (σmar σ mar ) 2) e rmp KN : ( ln S K + r mp 1 2 (σmar ) 2). (8) whereonceagan statesthematurty(therateshouldbefxedforeachmoneyness)and σ mar s the volatlty obtaned from the market. As concerns put optons, a smlar procedure can be derved. Alternatvely, one can use the put-call party(6). BSmodelforoptononadvdendpayngstock. Applyngthesame deas,whchledtothedervatonofstandardbsmodel(seeabove),onecanobtanalsots extenson for optons wrtten on stocks payng contnuous-lke dvdend yeld q(bs model for vanlla call on dvdend payng stock): σ mar V vanlla call (;S,K,r,σ,q) = e q SN(d + ) e r KN(d ). (9) 7 Iftswrttenonadfferentasset,tmustbeatleastalmostperfectlycorrelated. 810

5 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 and(bsmodelforvanllaputondvdendpayngstock): V vanlla put (;S,K,r,σ,q) = e r KN( d ) e q SN( d + ), (10) where d ± = ln S K +(r q ± 1 2 σ2 ) σ. (11) BSmodelforcalloptononadvdendpayngstockwthmpled parameters. Ifwewouldlketousethemarketprcesofsomeoptononastockforwhch we assume some dvdend payment, we need to estmate one parameter more the mpled dvdend yeld, e. whch level of dvdends was expected by the market. We proceed as follows. Frst, we obtan the market prces of avalable optons on nondvdend stocks(eg. dvdend free equty market ndex) and related market volatltes. Second stepstoestmatethenterestrates r mp.inthenextstepwetakethemarketprces VI mar and volatltes σ mar of avalable optons on dvdend-payng stocks and use them to estmate mpled dvdendyelds q mp (togetherwththemplednterestrates r mp from step two). Fnally, we canprcethetargetopton wth σ mar, r mp, q mp : V = e qmp SN(d + ) e rmp KN(d ). (12) where And smlarly for put opton. d ± = ln S K + ( r mp q mp ± 1 2 (σmar σ mar ) 2). (13) 4. Illustratve study Inthelnesbelowwewllshowtheproceduredescrbedntheprevoussectononrealmarket data. Frst, we wll consder a dvdend-less equty market ndex and later one of ts component stocks, whch s supposed to pay some dvdends. The opton data were obtaned from the ReuterstermnalforSeptember15,2011andweconsderedallcallandputoptonsonDAX ndex maturng on June 15, Onagvendaythespotprceofthendexoptonunderlyngasset(e.thevalueofthendex) was approxmately Frst we calculate the log-moneyness for all avalable exercse prces between 500 to 10,000(the nterval length s sometmes 500, sometmes 100, or even 50). Snce thelog-moneynessofcallsnverselyrelatedtotheoneofputoptons,wemultplythelatter bymnusonetogettheadjustedmonneynessforput. InFgure1weshowthemarketprcesofcall(black)andputoptons(gray)ndependency of such adjusted log-moneyness. It s apparent that most observatons are located close to moneynessofzero(e.closetoatmoptons)andtheprcesofputandcalloptonsarevery closeeachtotheotheronlyforatmoptons(moneynessszero). Smlarly,nFgure2,volatltesofcallandputoptonsprovdedbythemarketbyusngBS modelarecompared.thstme,tsclearthatthevolatltysmoreorlesssmlarforatmas wellasclosetoatmoptons,whletdvergesfordeepitm/otmoptons.itsalsoapparent that the market volatltes of ITM/OTM calls are much lower than the volatltes of related OTM/ITM puts. The most dffcult step s to estmate the rskless rate, whch should be possble unque for alloptons. AswecanseenFgure3,thevaluewhchmatchestheBSformulaandmarket prces of calls, puts and volatltes oscllate slghtly below 2% p.a. Unfortunately, t seems to be mpossble to prce deep ITM call optons snce the prces and volatltes provded by the 811

6 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 Fgure 1: DAX call and put opton market prces n dependency on absolute log-moneyness Fgure 2: DAX call and put opton market volatltes n dependency on absolute log-moneyness Fgure 3: DAX call and put opton mpled rskless rates n dependency on absolute log-moneyness 812

7 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 market would requre even negatve nterest rate to match the BS formula. Ths s especally thecaseofthecalloptonwththelowestexercseprce(500),whchsabout 10%ofthespot prceandwhchfurthermoremeansthattheoptondeltasnfactone,tsgammaszeroand theoptonvaluesnsenstvealsotothenterestrates(rhoszeroaswell). 8 Applyngsomeoptmzatonstepswegetrsklessrate r = 1.911%p.a.,anumberwhchwe wll use further to obtan the mpled dvdend yeld(e. contnuous dvdend yeld assumed by themarketforagvenstock). ThstmeweassumetheoptonsonAllanz,naturallywththesamematurtyasnthe analyssabove. FromFgure4tsapparentthatmostofthecalloptonsareOTM(negatve moneyness), whle the puts are ITM(reverted moneyness s also negatve). Ifwefocusonthevolatltyobtanedfromthemarket,seeFgure5,wecanseesgnfcant dfferences for all levels of moneyness. The reason mght be the dvdend yeld but also lmted lqudty and resultng bd/ask spread. Fnally,nFgure6weprovdethedvdendyeldestmatedbythemarket.Formostcases tsfxedjustbelow 9.7%p.a.However,wecanobservesomenstabltyforITMputs. The mplcaton for prcng of OTC optons on Allanz therefore s that the trader should ask forsuffcentbd/askspreadevenforclosetoatmoptonstocoverhsorhercostofhedgng byreversepostonandshouldbeverycarefulfordeepitmandotmoptons. 5. Concluson Ths paper was focused on extractng of parameters for prcng of llqud optons by BS model, ncludng market assumptons about the volatlty, rskless rate and dvdend yeld. Usng emprcal data(call and put opton prces and mpled volatltes) from German opton market we have estmated the so called mpled rskless nterest rate usng DAX ndex optons n thefrststep.inthesecondstepwehaveusedtheoptmzednterestrate(1.911%)toestmate thesocalledmpleddvdendyeldforcallandputoptonsonallanz,whchsoneofthe componentsofdaxequtyndex. Asanextstep,thetradercanusethesedata,e.rskless nterest rate, dvdend yeld and volatltes mpled by the market prce to valuate llqud optons on Allanz. Wthn the analyss we have specfcally stressed potental problems related to valuaton of very deep ITM call optons(unstable mpled nterest rate) and deep ITM put optons(unstable mpled dvdend yeld), as well as large dfferences n mpled volatltes for related ITM/OTM call/put optons. These real market observatons should have an mpact on bd/ask spread requred by traders. References [1] BENKO, M., FENGLER, M., HARDLE, W., KOPA, M.(2007). On Extractng Informaton Impled n Optons. Computatonal statstcs 22: [2] BLACK, F., SCHOLES, M.(1973). The prcng of optons and corporate labltes, Journal of Poltcal Economy 81: [3] GARMAN, M., KOHLHAGEN, S.(1983). Foregn currency opton values. Journal of Internatonal Money and Fnance 2: [4] GESKE, R.(1978). The prcng of optons wth stochastc dvdend yeld. Journal of Fnance 25: Delta,gammaandrhoarealsoknownastheGreekletters,asenstvtesofoptonprcesonpartcular parameters. 813

8 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 [5] KOPA, M., TICHÝ, T.(2014a). No arbtrage condton of mpled volatlty and bandwdth selecton. Anthropologst 17(3): [6] KOPA, M., TICHÝ, T., VITALI, S.(2014b). State prce densty estmaton for optons wth dvdend yelds. Workng paper. [7] MERTON, R.C.(1973). Theory of ratonal opton prcng. Bell Journal of Economcs and Management Scence 4: [8] TICHÝ, T.(2011). Lévy models Selected applcatons wth theoretcal background. SAEI Vol. 9(Seres on Advanced Economc Issues). Ostrava: VŠB-TU Ostrava. [9] TICHÝ T.(2012). Some results on prcng of selected exotc optons va subordnated Lévy models. In: Managng and Modellng of Fnancal Rsks, Ostrava, 2012, pp

9 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 Fgure 4: Allanz call and put opton market prces on dependency n absolute log-moneyness Fgure 5: Allanz call and put opton market volatltes n dependency on absolute log-moneyness Fgure 6: Allanz call and put opton mpled dvdend rates n dependency on absolute log-moneyness 815