Pricing correlation options: from the P. Carr and D. Madan approach to the new method based on the Fourier transform 1

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1 Economics and Businss Rviw, Vol. 4 (8, No., 8: 6-8 DOI:.8559/br.8.. Pricing corrlaion opions: from h P. Carr and D. Madan approach o h nw mhod basd on h Fourir ransform Arkadiusz Orzchowski Absrac : Pricing of opions plays an imporan rol in h financial indusry. Invsors knowing how o pric drivaiv conracs quickly and accuraly can ba h mark. On h ohr hand mark paricipans consrucing hir invsmn sragis wih h us of opions basd on chniqus ha do no assur h highs compuaional spd and fficincy ar doomd o failur. h aim of h aricl is o xnd h xising mhodology of pricing corrlaion opions basd on h Fourir ransform. h aricl sars wih a prsnaion of Carr and Madan s concp (Carr & Madan, 999. hn ohr mhods of pricing opions wih h us of h Fourir ransform ar summarizd. Finally, a nw approach o pricing drivaiv conracs is drivd and hn applid o h corrlaion opions. Kywords : Fourir ransform, pricing of opions, corrlaion opions. JEL Cods : G3, C. Inroducion h rapid dvlopmn of opion marks worldwid in rcn yars is accompanid by a ris in rading aciviis of invsors and a dvlopmn of nw, highly sophisicad, financial producs. On dirc consqunc of such a rnd is h possibiliy for buyrs and sllrs of h drivaivs o mor fficinly spcula on h movmns of inrs and xchang ras, commodiy and quiy prics, as wll as crdi raings of h issurs of bonds. As h capaciy for spculaion using drivaivs is growing, incrasd dmand for nw insrumns allowing o hdg agains diffrn aspcs of risks appars. Opions, along wih forwards and fuurs, sm o b h bs ools ha can b usd o proc mark paricipans agains ponial capial losss. On of h mos inrsing yp of opions usd for managing risks ar xoic opions whr pay-offs dpnd on Aricl rcivd 5 Fbruary 7, accpd January 8. Warsaw School of Economics, Collgium of Socio-Economics, Dparmn of Banking, ul. Wiśniowa 4, -5 Warszawa, aorzc@sgh.waw.pl.

2 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan 7 h corrlaion bwn ass prics,.g. xchang opions, quoin opions, corrlaion opions. Som of h conracs,.g. sprad opions, ar radd on xchangs (NYMEX, NYCE, CBO, ohrs, lik quoin and produc opions, can b bough or sold in h OC mark (Zhu,. h aim of h aricl is o xnd h analysis of h Carr and Madan mhod (Carr & Madan, 999 for pricing corrlaion opions in h Black and Schols framwork (Black & Schols, 973. h aricl consiss of svral scions. A h bginning h maringal mhod is applid o h valuaion of analyzd drivaivs. Nx, h mhod is modifid o h form proposd by Carr and Madan (999. Finally, alrnaiv mhods of pricing corrlaion opions, which ar basd on h Fourir ransform, ar prsnd and a nw modl for pricing of his yp of drivaivs is dvlopd.. Pricing corrlaion opions using h maringal mhod On of h mos popular mhods of pricing corrlaion opions is basd on h risk-nural concp (mhod rfrrd o as BS-M (Black & Schols, 973. h mhod is basd on h assumpion ha h pric of h undrlying insrumn follows gomric Brownian moion. In consqunc h naural logarihm of mark valu of h undrlying insrumn has normal disribuion. h join probabiliy dnsiy funcion of h wo-dimnsional variabl ha has o b analyzd for h corrlaion opions can b dscribd wih h us of an lgan formula as follows: q ( S, S πτσ σ ρ S S + r σ τ ρ S S + r σ τ S S + r σ τ S S + r σ τ + σ τσ τ σ στ ρ, ( whr: S, S ar h prics of h undrlying asss and a im, σ, σ ar h sandard dviaions of ras of rurn on h asss and, r is h risk-fr ra of rurn, ρ is h cofficin of corrlaion and τ. On dimnsional analogu of h quaion ( for h singl ass plain vanilla Europan opion can b xprssd as: lns + lns r σ τ q ( S xp, ( S σ τ σ πτ for and τ.

3 8 Economics and Businss Rviw, Vol. 4 (8, No., 8 In quaion ( h naural logarihm of h mark pric of h undrlying scuriy is rplacd by mark pric of h scuriy islf. If h pric of h insrumn bing h basis for h analyzd conracs is qual o 6, h risk-fr ra of rurn amouns o 5%, sandard dviaion of h ras of rurn is % and rlaiv im o xpiraion, i.. / changs from,5 o,9 h disribuion of h undrlying ass is prsnd on Figur..6 Dnsiy of probabiliy /,5 LN BS /,7 LN BS /,9 LN BS Pric of undrlying ass (S Figur. Disribuion of undrlying ass prics in h maringal mhod As h BS-M mhod is basd on h assumpion ha h horical valu of h Europan opion quals h valu of fuur paymns discound wih h risk-fr ra wih rspc o h maringal masur q, h following formulas for h prics of h Europan calls and pus rspcivly bcom valid: ( rτ q + C( S, E ( S K, Ω ( rτ q P( S, E ( K S, +, (3 Ω, (4 whr: q (. Ω is h probabiliy dnsiy funcion dpndn on filraion Ω, K is h srik pric of h Europan opion. Alrnaivly on can wri ha: ( C( S, ( S K q S Ω ds, (5 K K ( P( S, ( K S q S Ω ds. (6

4 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan 9 Subsiuing h analyical xprssion of probabiliy dnsiy funcion for q (S Ω allows h formulaion of h following samns: lns lns + r σ τ σ τ K S πσ τ C( S, ( S K ds, (7 lns lns + r σ τ K r σ τ S πσ τ P( S, ( K S ds. (8 Equaions (7 and (8 can b asily xndd o h wo-dimnsional cas. In h pay-off funcions of h corrlaion call and pu opions ar dfind rspcivly as (Dmpsr & Hong, : ( rτ q + + C( S, S, E ( S K, ( S K, Ω, (9 rτ q + + P( S, S, E (( K S, ( K S, Ω, ( h quaions allowing for h valuaion of h conracs can b formulad wih h us of quaion (, i..:, K K, ( C( S, S, ( S K ( S K q( S S Ω ds ds Ω K K, ( P( S, S, ( K S ( K S q( S, S ds ds whr: q (.,. Ω is h join probabiliy dnsiy funcion dpndn on filraion Ω. Unforunaly applying his procdur o h wo-dimnsional variabl in ordr o conclud abou h final shap of h pay-off funcion is highly infficin. ha is why applicaion of an alrnaiv approach sms o b ncssary.. Mhod of P. Carr and D. Madan Unlik h BS-M concp h mhod dvlopd by Carr and Madan (999 is basd on h Fourir ransform (h mhod is rfrrd o as BS-FCM. h approach applid o h valuaion of corrlaion opions consiss of svral sps. A h bginning variabls rprsning spo prics of h undrlying asss and srik prics of h opion (s quaions ( and ( ar ransformd

5 Economics and Businss Rviw, Vol. 4 (8, No., 8 according o h following schm: s lns, s lns, k lnk, k lnk. Finally, quaions (3 and (4 ar wrin as: s k s k Ω k k C( s, s, ( ( q ( s, s ds ds, (3 k s k s k k P( s, s, ( ( q ( s, s Ω ds ds. (4 Limiing h analysis only o call opions h quaion (3 can b modifid by paramrs α and α in h following way: mod C ( s, s, C( s, s,, (5 α k + αk whr: α and α ar h posiiv consans. Calculaion of h Fourir ransform of h modifid funcion C mod (s, s, allows h conclusion ha: iξk + iξk mod ( s ( + α + iξ + s ( + α + iξ ψ( ξ, ξ C s, s, dk dk q ( s, s Ω dξdξ. ( α + iξ ( α + iξ ( + α + iξ ( + α + iξ (6 Proof: iξk + iξk mod ( ψ( ξ, ξ C s, s, dk dk iξk + iξk αk + αk C( s, s, τ dk dk iξk + iξk αk + αk s k s k ( ( k k q ( s, s Ω ds ds dkdk s s s + s k q ( s, s Ω ( ( α iξ k ( α iξ + s + k k ( α + iξ + k ( α + iξ s + k k ( α + iξ + k ( α + iξ +

6 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan ( + k k k ( α iξ k α iξ dkdk dsds s ( + α + iξ + s ( + α + iξ s ( + α + iξ + s ( + α + iξ q ( s, s Ω + ( α iξ( α iξ ( α iξ( α iξ s ( + α + iξ + s ( + α + iξ s ( + α + iξ + s ( + α + iξ + ds ds ( + α + iξ ( α + iξ + α ( + iξ( α + iξ s ( + α + iξ + s ( + α+ iξ Ω α + iξ α + iξ + α + iξ ( + α + iξ q ( s, s s ds. ( ( ( (7 I is worh noing ha: iξ s + iξ s, Ω Φ( ξ, ξ q( s s ds ds (8 is h wo-dimnsional characrisic funcion for h variabls s and s. I mans ha h quaion (7 can b xprssd as: ( ξ α + i ξ α + i Φ (, ( ψ( ξ, ξ ( α + α ξ + i(α + ξ ( α + α ξ + i(α + ξ. (9 Calculaion of h invrs Fourir ransform allows h drminaion of h pric of h corrlaion call opion, i..: αk αk iξk iξk π ψ ξ ξ dξ dξ αk αk C( s, s, (, π iξk iξk Φ ( ξ ( α + i, ξ ( α + i α + α ξ + i(α + ξ α + α ξ + i(α + ξ ( ( dξ dξ. ( If h prics of h insrumns bing h basis for h analyzd conracs ar qual o 6 and 6, risk-fr ra of rurn amouns o 5%, sandard dviaions of h ras of rurn ar % and 5%, rlaiv im o xpiraion is qual o,, cofficin of corrlaion ρ and α α h payoff funcion of h corrlaion call opion is prsnd on Figur. Alhough h BS-FCM mhod can b applid o pricing corrlaion opions i is far from bing prfc. h biggs problm ha appars whn using h mhod is associad wih h ndncy of h subingral funcion from

7 Economics and Businss Rviw, Vol. 4 (8, No., 8 Payoff Pric of undrlying ass (S Pric of undrlying ass (S Figur. Pay-off funcion of h corrlaion call opion in h BS-FCM mhod quaion ( o rapid oscillaion whn ou-of-h-mony (OM nar-o-xpiraion conracs ar considrd. ha is why ohr concps for pricing opions should b xplord. 3. Alrnaiv mhods of pricing opions uilizing Fourir ransform hr ar many alrnaiv concps basd on h Fourir ransform which can b usd o valu drivaivs (Orzchowski, 6. o h bs knowldg of h auhor only wo of hm wr applid o pricing corrlaion opions. Nvrhlss almos all of hm can b good alrnaivs o h BS-M and BS- FCM mhods. Only h four mos imporan approachs ar prsnd blow. h firs mhod of pricing opions using Fourir ransform was dvlopd by Bakshi and Madan (h mhod is rfrrd o as BS-FBM (Bakshi & Madan,. In his approach h pric of h undrlying asss and h xrcis pric of h opion in quaion (5 ar convrd o logarihmic valus. hn h righ hand sid of a nw quaion is spli ino wo pars. For ach of hm Fourir ransforms ar drmind and rvrsd. Finally h horical valu of h opion may b obaind wih h us of quaion (: iξk iξk ϕ( ξ i ϕ( ξ C( S, ( S K + S dξ K dξ R ( R, ( π iξϕ π iξ whr: R is h ral par of h subingral funcion and i is h imaginary par of h complx numbr.

8 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan 3 h scond mhod of pricing opions wih h us of h Fourir ransform was drivd by Aari (h mhod is rfrrd o as BS-FA (Aari, 4. his approach bgins wih h gnralizaion of h procss rsponsibl for h sock pric movmns. Nx quaion (5 is ransformd ino h following formula: x(, r (, (, τ q ( q ( (, (, π Kπ S x dx K x dx S l l, ( whr: l K/S rτ and x(, is h unprdicabl shock in prics of h scuriy. (, Noing ha x q ( x(, dx(, allows o ra boh π and π as l probabiliy dnsiy funcions. Drmining hir Fourir and invrs Fourir ransforms lads dircly o h pric of h opion, i..: l iξl C( S, S + ( + ( π R ϕ ξ dξ i ξ + i iξl r( K + R ϕ ( ξ dξ. π iξ (3 h hird mhod of pricing opion using Fourir ransform was proposd by Bas (h mhod is rfrrd o as BS-FB (Bas, 6. his approach is basd on h modificaion of h prvious concp. As bfor quaion (5 is ransformd bu in his cas o h following form: K S q ( Ω q ( Ω S K K C( S, S K S ds K S d. (4 Nx, i is modifid and Fourir ransformd. Finally, h pric of h opion is drmind in h following way: iξk C( S, + ( π R S K ϕ ξ dξ. (5 iξ( iξ h fourh mhod of pricing opions using Fourir ransform was dvlopd by Lwis (h mhod is rfrrd o as BS-FL (Lwis,. his concp is basd on a mulipl ransformaion of h quaion (5, convrsion of h dpndn variabls o logarihmic valus and calculaion of h Fourir and invrs Fourir ransforms. Finally, h pric of h opion can b drmind wih h us of h following formula:

9 4 Economics and Businss Rviw, Vol. 4 (8, No., 8 rτ iuk S i K C( S, S R π ϕ u du. (6 u + 4 Alhough all h mhods prsnd abov can b asily applid o h valuaion of corrlaion opions (Fan & Wang, 7, h rmaining par of h aricl is dvod only o h nw mhod of pricing opions basd on h Fourir ransform and is applicaion o h valuaion of corrlaion opions. 4. Nw mhod of pricing corrlaion opions using Fourir ransform h nw mhod of pricing opions using Fourir ransform (h mhod is rfrrd o as BS-FAu consiss of svral sps. A h bginning dpndn variabls in quaion (5 ar convrd o logarihmic valus. Nx, h righ hand sid of h quaion obaind is spli ino wo pars, i..: rτ s q( Ω s rτ k q s ds k k C( S, s d ( Ω. (7 h firs par of quaion (7 is Fourir ransformd as in BS-FBM modl, i..: s s ds q ( Ω iξk ξ k s Ψ ( ( s Ω ds q dk. (8 s q ( s Ω ds s I can asily b sn ha q ( s Ω ds may b rad boh as h characrisic funcion of s assuming ξ i, i.. ϕ( i, and xpcd valu of S. I allows h ransformaion of h quaion (8 o h following form: S ϕ( ξ i Ψ ( ξ iξϕ( i. (9

10 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan 5 Proof: s s ds s ds s q( Ω q( Ω iξk ξ r k r S iξk dk S k Ψ ( dk ϕ( i ϕ( i s S iξk s S iξk s q ( s Ω ds dk q ( s Ω dk ds ϕ( i ( k ϕ i s iξk+ s S q( s Ω ds ϕ( i iξ s S ( + iξ S ϕ( ξ i q( s ds. (3 ϕ( i Ω iξ iξ ϕ( i h scond par of quaion (7 is also Fourir ransformd, i..: Proof: iξk k ϕ( ξ i Ψ ( ξ q( s Ω dsdk iξ +. (3 k iξk r Ψ ( ξ K q ( s Ω ds dk k s r iξk K q ( s Ω dk ds s iξk r K q ( s Ω ds iξ iξs r r ϕ( ξ K q ( s Ω ds K. (3 iξ iξ Invring boh Fourir ransforms allows obnion of h final pric of h opion, i..: C ( S, S iξk ϕ( ξ i d π R iξ( iξ + ξ. (33 Bfor applying quaion (33 o h valuaion of corrlaion opions i pays o ransform quaion (3 o h form:

11 6 Economics and Businss Rviw, Vol. 4 (8, No., 8 s k s k s s s s dsds k k C( s,, ( ( q ( q (, (34 whr: q ( s s is h condiional probabiliy dnsiy funcion of s givn s and q ( s is h probabiliy dnsiy funcion of s (Carmona & Durrlman, 3. I is worhy of no ha s and s ar normally disribud variabls wih known analyical probabiliy disribuion funcions. As h applicaion of h formula (33 o quaion (34 is vidn h closdform formula for h final pric of h corrlaion opion is prsnd wihou drivaion (Funou & afolong, 5. rτ rτ iξk φ( ξ i C ( s, s, S K d π R ξ iξ ( iξ + rτ iξ ( k φ ξ i S d. π R ξ (35 iξ ( iξ + I should b nod ha quaion (35 is corrc bu only undr h prviously inroducd assumpion ha ρ. Givn h sam daa as prviously h pay-off funcion of h corrlaion opion in h BS-FAu mhod can b asily drmind (s Figur 3. Payoff Pric of undrlying ass (S Pric of undrlying ass (S Figur 3. Pay-off funcion of h corrlaion call opion in h BS-FAu mhod Applying h BS-FAu mhod for h calculaion of h corrlaion opion has on advanag ovr ohr concps basd on h Fourir ransform. h spd of calculaing h final pric of h opion is grar han in ohr approachs.

12 A. Orzchowski, Pricing corrlaion opions: from h P. Carr and D. Madan 7 I is worh noing ha h prsnd mhod can b asily applid o ohr modls of pricing opions,.g. jump-diffusion modls (Kou, ; Mron, 976, pur jump modls (Carr, Gman, Madan, & Yor, ; Kirkby, 7; Madan, Carr, & Chang, 998. h biggs bnfis rsuling from h applicaion of h Fourir ransform and h BS-FAu mhod o h pricing of opions, howvr, appar in sochasic volailiy modls (Hson, 993; Hull & Whi, 987; Sin & Sin, 99. Conclusions Pricing drivaivs is an imporan issu in h financial indusry. For ha rason fficinly funcioning mhods allowing for h valuaion of h conracs consrucd on h basis of laws whos xcuion may b dmandd by on pary from h ohr aracs h paricular anion of many rsarchrs. In his aricl svral mhods of pricing opions basd on h Fourir ransform wr prsnd and wo of hm wr applid o h valuaion of corrlaion opions. h firs approach, i.. h on proposd by Carr and Madan (999 sms o b infficin du o h fac ha h subingral funcion in quaion ( bcoms highly oscillaory, spcially for h OM nar-o xpiraion opions. In consqunc, alrnaiv mhods hav o b xplord. Alhough many concps in his fild wr dvlopd, only on, proposd by h auhor, was fully drivd in h aricl. h mhod allows h pricing of h corrlaion opions mor fficinly. h main rasons for h suprioriy of h mhod ovr ohr approachs ar: a mor fficin schm of calculaing h Fourir ransform and rplacing h join probabiliy dnsiy funcion wih h wo ohr probabiliy dnsiy funcions. In consqunc a nw and br mhod basd on h Fourir ransform which can b applid o h valuaion of h corrlaion opions is proposd. Rfrncs Aari, M. (4. Opion pricing using Fourir ransforms: A numrically fficin simplificaion. Rrivd from hp://paprs.ssrn.com/sol3/paprs.cfm?absrac_ id54. doi:.39/ssrn.54 Bakshi, G., & Madan, D. (. Spanning and drivaiv scuriy valuaion. Journal of Financial Economics, 55(, doi:.6/s34-45x(995- Bas, D. (6. Maximum liklihood simaion of lan affin procsss. Rviw of Financial Sudis, 9(3, doi:.93/rfs/hh Black, F., & Schols, M. (973. h pricing of opions and corpora liabiliis. h Journal of Poliical Economy, 8(3, doi:.86/66

13 8 Economics and Businss Rviw, Vol. 4 (8, No., 8 Carmona, R., & Durrlman, V. (3. Pricing and hdging sprad opions. SIAM Rviw, 45(4, doi:.37/s Carr, P., Gman, H., Madan, D., & Yor, M. (. h fin srucur of ass rurns: An mpirical invsigaion. Journal of Businss, 75(, doi:.86/33875 Carr, P., & Madan, D. (999. Opion valuaion using h fas Fourir ransform. Journal of Compuaional Financ, (4, doi:.34/jcf Dmpsr, M. A. H., & Hong, S. S. G. (. Sprad opion valuaion and fas Fourir ransform. Rrivd from hp://cisrx.is.psu.du/viwdoc/download?doi &rprp&yppdf. doi:.7/ _ Fan, K., & Wang, R. (7. Valuaion of corrlaion opions undr a sochasic inrs ra modl wih rgim swiching. Fronirs of Mahmaics in China, (5, 3-3. doi:.7/s Funou, B., & afolong, E. (5. Fourir invrsion formulas for mulipl-ass opion pricing. Sudis in Nonlinar Dynamics & Economrics, 9(5, doi:.55/snd-4-34 Hson, S. L. (993. A closd form soluion for opions wih sochasic volailiy wih applicaions o bond and currncy opions. Rviw of Financial Sudis, 6(, doi:.93/rfs/6..37 Hull, J., & Whi, A. (987. h pricing of opions on asss wih sochasic volailiis. Journal of Financ, 4(, 8-3. doi:./j b568. Kirkby, J. L. (7. Robus barrir opion pricing by fram projcion undr xponnial Lévy dynamics. Applid Mahmaical Financ, 4(4, doi:.8/35486x Kou, S. G. (. A jump-diffusion modl for opion pricing. Managmn Scinc, 48(8, 86-. doi:.39/ssrn.4367 Lwis, A. (, A simpl opion formula for gnral jump-diffusion and ohr xponnial lvy procsss. Rrivd from hp://paprs.ssrn.com/sol3/paprs. cfm?absrac_id8. doi:.39/ssrn.8 Madan, D. B., Carr, P., & Chang, E. C. (998. h varianc gamma procss and opion pricing. Europan Financ Rviw,, doi:.3/a: Mron, R. C. (976. Opion pricing whn undrlying sock rurns ar disconinuous. Journal of Financial Economics, 3(-, doi:.6/34-45x(769- Orzchowski, A. (6. Analiza fkywności obliczniowj opcji na przykładzi modlu F. Blacka i M. Scholsa. Finans, 9(, Sin, E., & Sin, J. (99. Sock pric disribuions wih sochasic volailiy. Rviw of Financial Sudis, 4(4, doi:.93/rfs/ Zhu, J. (. Modular pricing of opions. An applicaion of Fourir analysis. Hidlbrg: Springr-Vrlag. doi:.7/