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 WWW.RISK.NET DECEMBER RISK 99
Cuing dg l Opion pricing modls. Exampls of local volailiy funcions (/ for h quadraic modl Lognormal (local volailiy (% 8 6 4 5 7 9 3 5 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 WWW.RISK.NET
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 (% 8 6 4 4 6 8 4 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, WWW.RISK.NET DECEMBER RISK
Cuing dg l Opion pricing modls 3. Exampls of local volailiy funcions (/ for h CIR family of solvabl modls Lognormal (local volailiy (% 8 6 4 3 4 5 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 WWW.RISK.NET
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://www.risk.n 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, www.mahpoin.ca Albans C and G Campolii, b Nw familis of ingrabl diffusions Working papr, www.mahpoin.ca Black and M Schols, 973 Th pricing of opions and corpora liabiliis Journal of Poliical Economy 8, pags 637 659 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 385 47 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 398 48 Kn J, 978 Som probabilisic propris of bssl funcions Annals of Probabiliy 6, pags 76 77 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 WWW.RISK.NET DECEMBER RISK 3