Black-Scholes goes hypergeometric

Transcription

1 Opion pricing modls l Cuing dg Black-Schols gos hyprgomric Claudio Albans, Giuspp Campolii, Pr Carr and Alxandr Lipon inroduc a gnral pricing formula ha xnds Black-Schols and conains as paricular cass mos analyically solvabl modls in h liraur, including h quadraic and h consan lasiciy of varianc modls for Europan-syl and barrir opions. In addiion, larg familis of nw soluions ar found, conaining as many as svn fr paramrs I has bn known sinc h 97s ha Black-Schols (973 pricing formulas ar a spcial cas of mor gnral familis of pricing formulas wih mor han jus h volailiy as an adjusabl paramr. Th lis of h classical xnsions includs affin, quadraic and consan lasiciy of varianc (CEV modls. Ths modls admi up o hr adjusabl paramrs and hav bn usd o solv pricing problms for quiy, forign xchang, inrs ra and crdi drivaivs. In a sris of working paprs, h auhors hav rcnly dvlopd nw mahmaical chniqus ha allow much furhr xnsions. Svral nw familis of pricing formulas, wih up o svn adjusabl paramrs in h saionary driflss cas, ar obaind whil aaining addiional flxibiliy in h gnral im dpndn cas. Th formulas xnd o barrir opions and hav a similar srucur o h Black-Schols formulas, h mos noabl diffrnc bing ha rror funcions (or cumulaiv normal disribuions ar rplacd by (conflun hyprgomric funcions, i, h spcial ranscndnal funcions of applid mahmaics and mahmaical physics. L dno a gnric financial obsrvabl ha w know is driflss. Exampls ar h forward pric of a sock or forign currncy undr h forward masur, a Libor forward ra or a swap ra wih appropria choic of numrair ass. Black or Black-Schols formulas ar obaind by posulaing ha h im voluion of obys a sochasic diffrnial quaion of h form: whr ( = is linar and W is a sandard Brownian or Winr procss. In his cas, pricing formulas of calls and pus for boh plain vanilla and barrir opions can b wrin in xac analyical form in rms of h rror funcion. Inrsingly, quadraic volailiy modls wih: also allow for pricing formulas ha rduc o h valuaion of an rror funcion. This is bcaus quadraic volailiy modls can b rducd o a Winr procss by mans of a simpl masur chang and variabl ransformaion of h form: whr h undrlying x follows: dx ( d = dw ( = + + = x = dw Th ransformaion (3 applis undr a pricing masur whr asss ar valud in rms of a suiably dfind numrair g = g(x,. or a rviw of chang in numrair mhods in pricing hory, w rfr o Gman, El Karoui & Roch (995, Schrodr (999 and Borodin & Salminn (996. Boh funcions (x and g(x, can b drivd xplicily for any choic of paramrs,,. ( ( (3 (4 Quadraic volailiy modls ar h only saionary, driflss modls for which h combinaion of a non-linar ransformaion of h form (3 and a chang of numrair rducs h problm o a Winr procss x as h on in (4. Carr, Lipon & Madan ( considr whhr i is possibl o rlax h condiion of saionariy and find mor gnral procsss wih drif and volailiy boh dpndn on calndar im and ha sill rduc o h Winr procss. Thy find ha h gnral soluion admis as many as im-dpndn funcions. A rlad lin of rasoning lading o xnsions of h Black-Schols formula sars from h obsrvaion ha h CEV modls wih sa-dpndn volailiy spcifid as follows: = _ wih consans θ and, rducs o h Bssl procss: by mans of a non-linar ransformaion combind wih a masur chang. Lipon ( drivs gnral rducibiliy condiions o h mor gnral procsss: which ar solvabl for β =,, /. Th cas β = is h lognormal (or affin modl lading o h Black-Schols formula. Th wo cass β =, / corrspond o wll-known solvabl shor-ra modls, namly h Vasick and h Cox-Ingrsoll-Ross (CIR, 985 modls. Albans & Campolii (a, b find a gnral soluion o h rducibiliy condiions of Carr, Lipon & Madan ( for saionary, driflss procsss. In his aricl, w summaris our findings by prsning h gnral soluion formula, and illusra is us in a fw paricular cass. Sinc h original drivaion is somwha lnghy, w rfr o our ohr paprs for a consruciv drivaion. Gnral pricing formula W prsn a gnral pricing formula for h modls, which ar solvabl by h rducion mhod. Assum ha h sa variabl x has a drif λ(x for which on can find h pricing krnl for h procss: Th pricing krnl is h funcion u(x, ; x,, which solvs h forward Kolmogorov quaion in h firs pair of argumns and h backward Kolmogorov (i, Black-Schols quaion in h scond pair. Th lar quaion can b wrin as follows: ( x ( dx =λ d + x dw β = λ +λ + dx x d x dw dx =λ x d + x dw u ( x,;x, + uxx x,;x, +λ x ux x,;x, = +θ (5 (6 (7 (8 (9 DECEMBER RISK 99

2 Cuing dg l Opion pricing modls. Exampls of local volailiy funcions (/ for h quadraic modl Lognormal (local volailiy (% for, wih final im condiion a rminal im = givn by u(x, ; x, = δ(x x (a Dirac dla funcion. Th procsss in (7 ar xampls of analyically solvabl modls for which on can calcula h pricing krnl. Solvabl pricing modls can b consrucd saring from a soluion v(x, of h Black-Schols quaion in (9 wih an arbirary final im condiion a =. Th Laplac ransform of such a funcion: ρ = ρ ( ˆv x, v x, d ( is usually rfrrd o as h im-indpndn Grn s funcion and saisfis a scond-ordr ordinary diffrnial quaion wih Dirac dla funcion sourc rm δ(x x. L us considr h homognous par of his quaion as givn by: ( x ρˆv ( x, ρ + ˆv xx ( x, ρ + λ( x ˆv x ( x, ρ = ( W find ha funcions v^(x, ρ solving his quaion can b akn as h lmnary building blocks o consruc solvabl pricing modls for h spac procsss. W hrfor call v^(x, ρ h gnraing funcion. Armd wih a soluion v^(x, ρ, w dfin a volailiy funcion ( and an invribl monoonic ransformaion = (x and is invrs x = X( such ha: X( λ( sds ( X ( xp ( s ( ( = ˆv X, ρ wih arbirary consan and whr: dx ( ( =± ( x ( (3 Th wo signs corrspond o ihr monoonic incrasing or monoonic dcrasing ransformaions. Th frdom in choosing h sign givs ris o wo familis of soluions ha ar diffrn in h gnral cas. As is vrifid in h appndix, h procss: ρ g = (4 ˆv x, ρ ( can b rgardd as a forward pric procss and, undr h masur wih g as a numrair, h sa variabl x drifs a ra λ(x. nc, h pricing krnl U(, ;, for h ovrlying forward pric a im can b valuad in closd form as h xpcd rward from a limi burfly sprad conrac wih dla funcion payou: g U,;, ( = E δ( ( x (5 g d condiional on h pric having valu a iniial im =. r, h xpcaion is calculad assuming ha g is h numrair and ha h sa variabl x drifs a ra λ(x. Th final formula for h pricing krnl in spac is rlad o h krnl in h undrlying x spac as follows: ( X ( ˆvX, ( ( ρ ρ U(,;, = u( X(,;x (, (6 ˆv X, ρ Ignoring discouning, a Europan-syl call opion wrin on h forward pric a currn im =, sruck a K and mauring a im = T can b pricd in his modl by calculaing h following ingral: ρt ˆv( x, ρ C( K,T; ( ( ( = dx x K u( x,t;x (, XK (7 ˆv X, ρ Barrir and lookback opions can b handld by modifying h undrlying krnl in x-spac o accoun for h appropria boundary condiions. This is accomplishd by mans of ihr ingral rprsnaions or ignfuncion xpansion mhods, i, Grn s funcion mhods ha ar sandard in h hory of Surm-Liouvill quaions. S Davydov & Linsky (999 for a discussion in an opion pricing conx. our familis of solvabl modls Th cas β = is h usual lognormal (or affin modl. Of inrs hr ar h ohr familis wih β =, / in quaion (7. This provids four xampls of our mhodology o gnra xacly solvabl modls. If β = and λ =, w rcovr h Winr procss wih consan drif, which is radily ransformd ino a driflss Winr procss and hus suppors only quadraic volailiy funcions in spac, including h lognormal Black-Schols modl as a spcial subcas. If β = and λ, hn h krnl for x [ λ /λ, can b wrin in rms of hyprbolic rigonomric funcions and h gnraing funcion solvs rmi's quaion. If β = / and λ =, hn h pricing krnl for h sa variabl is xprssd in rms of modifid Bssl funcions as follows: Th gnraing funcion is: λ ( ( ( λ + x ( x / x 4 xx u( x,;x, = I λ x / 8ρx 8ρx ˆv( x, ρ = x qi λ qk λ + (8 (9 wih arbirary consans q, q. r I (z is h modifid Bssl funcion of ordr and K (z is h associad McDonalds funcion. In his cas, w obain wo familis (on for ach choic of sign in (3 of xac soluions wih six adjusabl paramrs. Th cas β = / and λ < givs h pricing krnl for h sa variabl x corrsponding o ha of h shor ra CIR modl, and can sill b xprssd in rms of modifid Bssl funcions as follows: λ λ x u( x,;x, = c x xp c x + x I c x x λ λ λ ( whr c λ /( (λ. or a drivaion, s Giorno al (986 and Kn (978. Th gnral soluion of quaion ( rducs o Whiakr s quaion and gnraing funcions hav h gnral form: λ / λx / ˆv( x, ρ = x qw λ k,m x qm λ k,m x ( + for arbirary consans q, q. r W k, m ( and M k, m ( ar Whiakr funcions ha can also b xprssd in rms of conflun hyprgomric funcions or in rms of Kummr funcions (Abramowiz & Sgun, 97. This consrucion givs ris o a dual family wih svn fr paramrs (i, ρ, RISK DECEMBER

3 λ, λ,, q, q and an addiional consan of ingraion for h mapping from x spac o forward pric spac, whr: ( Th svn-paramr family ha rducs o h CIR modl has a local volailiy funcion dfind on ihr a fini inrval or on a half lin, and bhavs asympoically as h CEV volailiy on on sid and as a quadraic modl on h ohr. This hybrid shap allows for a gra dal of flxibiliy in rproducing obsrvd volailiy skws. Thr is a way of gaining a visual undrsanding of h gomric maning of h svn paramrs ha prhaps ovrsimplifis h picur, bu is inriguing. Th allowd shaps, whn confind o a fini inrval, can b rgardd as hybrids bwn h quadraic and h CEV modl. Th suppor of h volailiy funcion can b ihr a fini or an infini inrval. On on sid of h inrval, h volailiy bhavs asympoically as ha of a CEV modl. On h ohr sid of h inrval, h volailiy bhavs as in a quadraic modl. unchback shaps wih a local minimum and a local maximum ar possibl. Th svn paramrs singl ou h inrval ndpoins, h blow up or dcay ra a on nd and h locaion of h local minimum and h local maximum. This rprsnaion is an ovrsimplificaion, as h minimum and maximum disappar in crain paramr rangs whil only inflcion poins prsis. Th inflcion poins also disappar in ohr paramr rangs. nc, our svnparamr modl suppors a varid zoology of skws, smils, frowns and smirks. I also suppors boh cass wih, and wihou, absorpion. Addiional xnsions ar possibl. or insanc, on can apply a drminisic im chang and sill rain solvabiliy. W rfr o forhcoming aricls for a discussion of his and ohr rlad opics. Rdiscovring xac soluions in h liraur W show ha h known xac soluions in h liraur, namly quadraic and CEV modls, can all b rdiscovrd as paricular cass of our gnral formula for h Bssl family whr w mak us of h abov soluions o h undrlying x spac procss wih β = /, λ = and λ λ. Wihou loss of gnraliy, w can fix =. W spcialis furhr o h cas whr: which lads o a procss for h forward pric wih volailiy: a ( = ( ( ( X I λ ρ X (3 (4 whr x = X( is h invrs of h funcion in quaion (3. In his family, a and ρ ar posiiv, _ is arbirary and λ >. Th funcion (x maps h half lin x [, ino (, _ ], whr (x is a sricly monoonically incrasing funcion wih d(x/dx = ((x/(x. This soluion rgion can b invrd so ha [ _,. This is accomplishd by ihr rplacing a by a in quaion (3 or by applying a linar chang of variabls ha maps ino _. In his spcial cas, w mak us of h gnraing funcion in quaion (9, wih h choic q =, and formula (6 rducs o: ρ ( X( + X( / U,;, ( = a 3 X ( I λ ( ρ X ( XX ( (5 I λ I ρ X λ λ ρ λ k = +, m= λ ( ( ρ x ( ρ K ( x = a I x λ λ W no ha his dnsiy ingras xacly o uniy in spac (i, no absorpion.. Exampls of local volailiy funcions (/ for h CEV modl (θ = 3 Lognormal (local volailiy (% Quadraic volailiy modls. Pricing krnls for quadraic volailiy modls ar radily obaind as a subs of h abov gnral family wih h spcial choic of paramr λ = 3. Afr making h subsiuion _ and sing a = ( _ = /π h ransformaion funcion (x bcoms: (6 whr >. r, w assum ha _ > =. Th invrs ransformaion X( is givn by: (7 and h volailiy funcion ( is obaind by insrion ino quaion (4 whil using h Bssl funcion of ordr /: Insring h xprssion (7 ino (5, on obains h pricing krnl: ( ( ( K x / ( x = + = + π I x / xp x ( = ( + ( X / log / = ( /8 U,;, ( = π ( ( φ ( +φ( / φ φ sinh (8 (9 whr φ( log(( = /( _. In h spcial cas of a volailiy funcion wih a doubl roo, i: = (3 ( h pricing krnl is calculad by aking h limi as = _, and on finds: ( U,;, = π 3 (3 ( ( / ( + ( / Lognormal modls. Th pricing krnl for h lognormal Black-Schols modl wih ( = is a paricular cas of h abov formula for h quadraic modl. Th drivaiv wih rspc o of h quadraic volailiy funcion in (8, valuad a = _, is. Taking h limi = (or = _ <<, whil holding h ohr paramrs fixd, DECEMBER RISK

4 Cuing dg l Opion pricing modls 3. Exampls of local volailiy funcions (/ for h CIR family of solvabl modls Lognormal (local volailiy (% on obains ( = ( _. Th pricing krnl in (9 givs h krnl for h lognormal modl in h limi =, i: U,;, ( = π (3 xp log( / ( / CEV modl. Th CEV modl is rcovrd in h limiing cas as ρ. Assum λ > and l θ > b dfind so ha λ = θ +. Th ransformaion = (x: has invrs x = X( givn by: for any consan _. Th volailiy funcion for his modl is: (33 (34 (35 In h limi ρ, h Laplac ransform v^(x(, =, which implis ha h numrair chang is rivial in his cas. Th pricing krnl can b valuad by subsiuion ino h gnral formula (6 and, afr collcing rms, i urns ou o b: = θ θ ( U,;, ( = 3 + θ ( θ θ ( ( / ( ( + ( θ = + x x ( = X θ ( ( I (36 This formula was drivd in h cas θ >, for which h limiing valu = _ is no aaind and h dnsiy is asily shown o ingra o uniy (i, no absorpion occurs and h dnsiy also vanishs a h ndpoin = _. W no ha h sam formula solvs h forward pricing quaion for θ <, lading o h sam Bssl quaion of ordr ±(θ. In h rang θ <, howvr, h propris of h abov pricing krnl ar gnrally mor subl. In paricular, on can show ha h dnsiy ingras o uniy for all valus θ < /, hnc no absorpion occurs for θ (, /. Th boundary condiions for h dnsiy can b shown o b vanishing a = _ (i, pahs do no aain h lowr ndpoin for all θ <. In conras, for θ (, / h dnsiy bcoms singular a h lowr ndpoin = _ (hnc his corrsponds o h cas ha h dnsiy has an ingrabl singulariy for +θ θ θ which pahs can also aain h lowr ndpoin, bu ar no absorbd. or h spcial cas of θ = /, h formula givs ris o absorpion. (No ha for h rang θ ( /, h abov pricing krnl is no usful sinc i givs ris o a dnsiy ha has a non-ingrabl singulariy a = _. In his cas, howvr, anohr soluion ha is ingrabl is obaind by only rplacing h ordr (θ by (θ in h Bssl funcion. Th lar soluion for h dnsiy dos no ingra o uniy and hnc givs ris o absorpion, which can b usful o pric opions in a crdi sing. Th spcial cas of θ = givs a non-zro consan valu a h lowr ndpoin, and rcovrs h Winr procss wih rflcion and no absorpion on h inrval [ _, wih: (37 Barrir opions Th original moivaion of wo of us, Claudio Albans and Giuspp Campolii, as w ngagd in his projc, was o sramlin h drivaion of pricing formulas for barrir opions for our class of financial nginring masr sudns. Th gnral xprssion for h pricing krnl givs in fac a simpl drivaion of pricing formulas for barrir opions, by allowing a rducion o sandard Brownian moion in x spac. Considr, as an xampl, a down-and-ou opion wih barrir a = wihin h Black-Schols modl wih ( =. This rducs o h driflss Winr procss wih volailiy (x =, by mans of h ransformaion whr: wih invrs = (x = x/. Spcialising quaion (6 givs: (38 U,;, ( = xp log ( / ux,;x ( (, (39 8 Th rgion x (, maps ino (,. A barrir locad a = corrsponds o = (x = x /, so x = X( = ( / log. Th uppr rgion [, maps ino x [x,. Th x-spac krnl wih absorbing boundary condiion a x = x is obaind by h mhod of imags, as: ( xx /4 ( x+ xx /4 u( x,;x, = (4 4π Insring his krnl ino h gnral pricing formula in (39 immdialy givs h pricing krnl in spac: whr U(, ;, is h barrir-fr pricing krnl: U,;, ( = xp log ( / / π (4 (4 Ignoring discouning, a down-and-ou call mauring a im T and sruck a K > has h pric a im = givn by h ingral: (43 whr is h currn forward pric of mauriy T. This ingral can b valuad in rms of cumulaiv normal disribuion funcions as follows: whr: U,;, π ( = + DO DO ( = ( ( C,K,T du,t;, K = ( C,K,T N d /K KN d /K ( ( ( ( N d / K + K / N d / K d ( x = X = / log ( x / + + (44 (45 and d (x = d (x T. No ha, sinc h risk-nural drif is absn, log x + T = T ( log / log / U (,;, = U(,;, xp / / RISK DECEMBER

5 pricing formulas ar mor compac whn wrin on forward prics insad of sock prics. In h mor gnral cas of h ohr solvabl modls, on can also obain analyic closd-form soluions for various xoic payous, including barrir opions. Basd on our gnral rsuls, h drivaion of pricing formulas is sraighforward and will b prsnd lswhr. Claudio Albans and Giuspp Campolii ar a Mah Poin and h Univrsiy of Torono. Pr Carr is a snior consulan and visiing profssor a h Couran Insiu of Nw York Univrsiy. Alxandr Lipon is in h forx produc dvlopmn group a Dusch Bank in Nw York. Claudio Albans was suppord in par by h Naional Scinc and Enginring Council of Canada. W hank Dilip Madan, Sphan Lawi, Vadim Linsky and Andri Zavidonov for discussions. Any rmaining rrors ar our own Commns on his aricl may b posd on h chnical discussion forum on h Risk wbsi a hp:// Appndix r w vrify h main formula, quaion (6. Considr a gnric pricing masur whr h procss for x obys h quaion: dx =µ ( x d +( x dw (46 for som drif µ(x. Thn, by Iô s lmma, h procss g dfind in (4 saisfis h quaion: ˆ v x ˆ vx ˆ v xx g dg = ρµ + gd + gdw (47 ˆv ˆv ˆ v whr: g ˆv x = (48 ˆv is dfind as h lognormal volailiy of g. Subsiuing quaion (: ˆ v (49 xx =ρ ˆ v λ ˆ vx ino h abov sochasic diffrnial quaion, w find: dg µλ g ( g = + d + g dw (5 g To dmonsra ha g dfins a forward pric procss, considr his quaion in h original forward masur whr h forward pric follows a maringal procss. Thn, using Iô s lmma on h mapping x = X( and quaion ( w arriv a quaion (46 wih drif: ( d dx( ( d ( x µ ( x = = (5 d d d ( Exprssing all funcions in rms of x, w hn hav: d µ ( x = = x x (5 dx whr ((x is h volailiy funcion for h forward pric. nc, by subsiuion h drif of g in h forward masur is: Equaion ( givs: ( µλ g g x ˆvx ˆvx + = λ + x + ˆv ˆv (53 x x λ v ˆx = (54 ˆv Subsiuing ino (53, w find ha h drif of g undr h forward masur vanishs. Nx, considr h masur having g as numrair. Undr his pricing masur h pric of risk q g = g. Indd, by Iô s lmma, i is known ha if on changs from a masur in which any ass A has a drif r (i, h risk-fr ra in h risk-nural masur or zro in h forward masur ino a nw masur wih g as numrair, hn h drif of A in his nw masur is µ A = r +q g A, whr q g = g is h pric of risk and A h lognormal volailiy of A. nc, in changing from h forward masur ino h masur having g as numrair µ A = g A. Th choic A = g givs µ g = ( g and: ( g g g g dg =µ gd + gdw = gd + gdw (55 Comparison wih quaion (5 shows ha h drif µ of h procss x is λ, as sad. This implis ha h rprsnaion (6 for h pricing krnl is corrc. W rfr h radr inrsd in gaining furhr insigh ino our main formula o our aricl (Albans & Campolii, b. Thr w provid a dirc parial diffrnial quaion proof, which is mor labora and fully consruciv, and is no basd on h abov sochasic analysis argumn. REERENCES Abramowiz M and I Sgun, 97 andbook of mahmaical funcions Applid Mahmaical Sris 55, Naional Burau of Sandards, Washingon Albans C and G Campolii, a Exnsions of h Black-Schols formula Working papr, Albans C and G Campolii, b Nw familis of ingrabl diffusions Working papr, Black and M Schols, 973 Th pricing of opions and corpora liabiliis Journal of Poliical Economy 8, pags Borodin A and P Salminn, 996 andbook for Brownian moion Springr-Vrlag Carr P, A Lipon and D Madan, Th rducion mhod for valuing drivaiv scuriis Working papr, April Cox J, J Ingrsoll and S Ross, 985 A hory of h rm srucur of inrs ras Economrica 53, pags Davydov D and V Linsky, 999 Th valuaion and hdging of barrir and lookback opions for alrnaiv sochasic procsss Submid for publicaion, availabl a hp://usrs.ims.nwu.du/linsky Gman, N El Karoui and J Roch, 995 Changs in numrair, changs in probabiliy masur and opion pricing Journal of Applid Probabiliy, Jun Giorno V, A Nobil, L Ricciardi and L Sacrdo, 986 Som rmarks on h rayligh procss Journal of Applid Probabiliy 3, pags Kn J, 978 Som probabilisic propris of bssl funcions Annals of Probabiliy 6, pags Lipon A, Inracions bwn mahmaics and financ: pas, prsn and fuur Risk Mah Wk, Nw York, Novmbr Schrodr M, 999 Changs in numriar for pricing fuurs, forwards and opions Rviw of inancial Sudis (5, winr, pags,43,63 DECEMBER RISK 3