SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG

Ths paprs ar producd by th Judg Institut of Managmnt Studis, Univrsity of Cambridg. Th paprs ar circulatd for discussion purposs only and thor contnts should b considrd prliminary. Not to b quotd without th author s prmission. SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. DEMPSTER and S.S.G. HONG Cntr for Financial Rsarch Judg Institut of Managmnt Studis Univrsity of Cambridg Email: mahd2@cam.ac.uk & gh10006@hrms.cam.ac.uk Plas addrss nquiris about th Judg Institut Working Papr Sris to: Publications Scrtary Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Tl: 01223 339636 Fax: 01223 339701

Sprad Option Valuation and th Fast Fourir Transform M.A.H. DEMPSTER and S.S.G. HONG Cntr for Financial Rsarch Judg Institut of Managmnt Studis Univrsity of Cambridg Email: mahd2@cam.ac.uk & gh10006@hrms.cam.ac.uk July 2000 Abstract W invstigat a mthod for pricing th gnric sprad option byond th classical two-factor Black-Schols framwork by xtnding th fast Fourir Transform tchniqu introducd by Carr & Madan 1999) to a multi-factor stting. Th mthod is applicabl to modls in which th joint charactristic function of th undrlying assts forming th sprad is known analytically. This nabls us to incorporat stochasticity in th volatility and corrlation structur a focus of concrn for nrgy option tradrs by introducing additional factors within an affin jump-diffusion framwork. Furthrmor, computational tim dos not incras significantly as additional random factors ar introducd, sinc th fast Fourir Transform rmains two dimnsional in trms of th two prics dfining th sprad. This yilds considrabl advantag ovr Mont Carlo and PDE mthods and numrical rsults ar prsntd to this ffct. 3

1 Introduction Sprad Options ar drivativs with trminal payoffs of th form: [S 1 T ) S 2 T )) K] +, whr th two undrlying procsss S 1, S 2 forming th sprad could rfr to asst or futurs prics, quity indics or dfaultabl) bond yilds. Thr is a wid varity of such options tradd across diffrnt sctors of th financial markts; for xampl, th crack sprad and crush sprad options in th commodity markts [16, 22], crdit sprad options in th fixd incom markts, indx sprad options in th quity markts [10] and th spark lctricity/ful) sprad options in th nrgy markts [9, 18]. Thy ar also applid xtnsivly in th ara of ral options [23] for both asst valuations and hdging a firm s production xposurs. Dspit thir wid applicability and crucial rol in managing th so-calld basis risk, hdging and pricing of this class of options rmain difficult and no consnsus on a thortical framwork has mrgd. Th main obstacl to a clan pricing mthodology lis in th lack of knowldg about th distribution of th diffrnc btwn two non-trivially corrlatd stochastic procsss: th mor varity w injct into th corrlation structur, th lss w know about th stochastic dynamics of th sprad. At on xtrm, w hav th arithmtic Brownian motion modl in which S 1, S 2 ar simply two Brownian motions with constant corrlation [19]. Th sprad in this cas is also a Brownian motion and an analytic solution for th sprad option is thus availabl. This, howvr, is clarly an unralistic modl as it, among othr things, prmits ngativ valus in th two undrlying prics/rats. An altrnativ approach to modlling th sprad dirctly as a gomtric Brownian motion has also provn inadquat as it ignors th intrinsic multi-factor structur in th corrlation btwn th sprad and th undrlying prics and can lad to svr misspcification of th option valu whn markts ar volatil [13]. Going on stp furthr w can modl th individual prics as gomtric Brownian motions in th spirit of Black and Schols and assum that th two driving Brownian motions hav a constant corrlation [17, 20, 22]. Th rsulting sprad, distributd as th diffrnc of two lognormal random variabls, dos not possss an analytical xprssion for its dnsity, prvnting us from driving a closd form solution to th pricing problm. W can howvr invok a conditioning tchniqu which rducs th two dimnsional intgral for computing th xpctation undr th martingal masur to a on dimnsional intgral, thanks to a spcial proprty of th normal distribution: conditional on a corrlatd random variabl a normal random variabl rmains normally distributd. As w dvlop a stochastic trm structur for volatilitis and corrlations of th undrlying procsss, w mov out of th Gaussian world and th conditioning tchniqu no longr applis. Furthrmor, a ralistic modl for asst prics oftn rquirs mor than two factors; for xampl, in th nrgy markt, random jumps ar ssntial in capturing th tru dynamics of lctricity or oil prics, and in th quity markts, stochastic volatilitis ar ndd. Intrst rat modls such as th CIR or affin jump-diffusion modls [11] frquntly assum mor than two factors and non-gaussian dynamics for th undrlying yilds. Howvr, th computational tims using xisting numrical tchniqus such as Mont Carlo or PDE mthods incras dramatically as diffusion modls tak ths issus into account. 4

In this papr w propos a nw mthod for pricing sprad options valid for th class of modls which hav analytic charactristic functions for th undrlying asst prics or markt rats. This includs th Varianc Gamma VG) modl [15], th invrs Gaussian modl [3] and numrous stochastic volatility and stochastic intrst rats modls in th gnral affin jump-diffusion family [1, 4, 6, 14, 21]. Th mthod xtnds th fast Fourir transform approach of Carr & Madan [5] to a multi-factor stting, and is applicabl to options with a payoff mor complx than a picwis-linar structur. Th main ida is to intgrat th option payoff ovr approximat rgions bounding th non-trivial xrcis rgion, analogous to th mthod of intgrating a ral function by Rimann sums. As for th Rimann intgral, this givs clos uppr and lowr bounds for th sprad option pric which tnd to th tru valu as w rfin th discrtisation. Th FFT approach is suprior to xisting tchniqus in th sns that changing th undrlying diffusion modls only amounts to changing th charactristic function and thrfor dos not altr th computational tim significantly. In particular, on can introduc factors such as stochastic volatilitis, stochastic intrst rats and random jumps, providd th charactristic function is known, to rsult in a mor ralistic dscription of th markt dynamics and a mor sophisticatd framwork for managing th volatility and corrlation risks involvd. W giv a brif rviw of th FFT pricing mthod applid to th valuation of a simpl Europan option on two assts in Sction 2. In Sction 3 our pricing schm for a gnric sprad option is st out in dtail. Sction 4 dscribs th undrlying modls implmntd for this papr and prsnts computational rsults to illustrat th advantag of th approach and th nd for a non-trivial volatility and corrlation structur. Sction 5 concluds and dscribs currnt rsarch dirctions. 2 Rviw of th FFT Mthod To illustrat th application of th fast Fourir Transform tchniqu to th pricing of simpl Europan styl options in a multi-factor stting, in this sction w driv th valu of a corrlation option as dfind in [2] following th mthod and notation of [5] in th drivation of a Europan call on a singl asst. A corrlation option is a two-factor analog of an Europan call option, with a payoff of [S 1 T ) K 1 ] + [S 2 T ) K 2 ] + at maturity T, whr S 1, S 2 ar th undrlying asst prics. Dnoting striks and asst prics by K 1, K 2, S 1, S 2 and thir logarithms by k 1, k 2, s 1,, our aim is to valuat th following intgral for th option pric: [ C T k 1, k 2 ) := E Q rt [ ] S 1 T ) K 1 + [S ] 2 T ) K 2 ]+ k 1 k 2 rt s 1 k 1 ) k 2 ) q T s 1, )d ds 1, 1) whr Q is th risk-nutral masur and q T, ) th corrsponding joint dnsity of s 1 T ), T ). 5

Th charactristic function of this dnsity is dfind by φu 1, u 2 ) := E Q [ xpiu1 s 1 T ) + iu 2 T )) ] = iu 1s 1 +u 2 ) q T s 1, )d ds 1. As in [5, 8], w multiply th option pric 1) by an xponntially dcaying trm so that it is squar-intgrabl in k 1, k 2 ovr th ngativ axs: c T k 1, k 2 ) := α 1k 1 +α 2 k 2 C T k 1, k 2 ) α 1, α 2 > 0. W now apply a Fourir transform to this modifid option pric: ψ T v 1, v 2 ) := iv 1k 1 +v 2 k 2 ) c T k 1, k 2 )dk 2 dk 1 = α 1+iv 1 )k 1 +α 2 +iv 2 )k 2 rt s 1 k ) 1 k ) 2 q T s 1, )d ds 1 dk 2 dk 1 R 2 k 2 k 1 s2 s1 = rt q T s 1, ) α 1+iv 1 )k 1 +α 2 +iv 2 )k 2 s 1 k ) 1 k ) 2 dk 2 dk 1 d ds 1 R 2 rt q T s 1, ) α1+1+iv1)s1+α2+1+iv2)s2 = R 2 α 1 + iv 1 )α 1 + 1 + iv 1 )α 2 + iv 2 )α 2 + 1 + iv 2 ) dds 1 rt φ T v1 α 1 + 1)i, v 2 α 2 + 1)i ) = α 1 + iv 1 )α 1 + 1 + iv 1 )α 2 + iv 2 )α 2 + 1 + iv 2 ). 2) Thus if th charactristic function φ T is known in closd form, th Fourir transform ψ T of th option pric will also b availabl analytically, yilding th option pric itslf via an invrs transform: C T k 1, k 2 ) = α 1k 1 α 2 k 2 2π) 2 iv 1k 1 +v 2 k 2 ) ψ T v 1, v 2 )dv 2 dv 1. Invoking th trapzoid rul w can approximat this Fourir intgral by th following sum: C T k 1, k 2 ) α 1k 1 α 2 k 2 2π) 2 m=0 n=0 whr 1, 2 dnot th intgration stps and iv 1,mk 1 +v 2,n k 2 ) ψ T v 1,m, v 2,n ) 2 1, 3) v 1,m := m N 2 ) 1 v 2,n := n N 2 ) 2 m, n = 0,..., N 1. 4) Rcall that a two-dimnsional fast Fourir transform FFT) computs, for any complx input) array, { X[j 1, j 2 ] C j 1 = 0,..., N 1 1, j 2 = 0,..., N 2 1 }, th following output) array of idntical structur: Y [l 1, l 2 ] := N 1 1 j 1 =0 N 2 1 j 2 =0 2πi N 1 j 1 l 1 2πi N 2 j 2 l 2 X[j 1, j 2 ], 5) 6

for all l 1 = 0,..., N 1 1, l 2 = 0,..., N 2 1. In ordr to apply this algorithm to valuat th sum in 3) abov, w dfin a grid of siz N N, Λ := { k 1,p, k 2,q ) : 0 p, q N 1 }, whr and valuat on it th sum k 1,p := p N 2 )λ 1, k 2,q := q N 2 )λ 2 Γk 1, k 2 ) := m=0 n=0 iv 1,mk 1 +v 2,n k 2 ) ψ T v 1,m, v 2,n ). Choosing λ 1 1 = λ 2 2 = 2π N givs th following valus of Γ, ) on Λ: Γk 1,p, k 2,q ) = = m=0 n=0 m=0 n=0 = 1) p+q iv 1,mk 1,p +v 2,n k 2,q ) ψ T v 1,m, v 2,n ) [ ] 2πi m N/2)p N/2)+n N/2)q N/2) N ψ T v 1,m, v 2,n ) m=0 n=0 2πi mp+nq)[ ] N 1) m+n ψ T v 1,m, v 2,n ). This is computd by th fast Fourir transform of 5) by taking th input array as X[m, n] = 1) m+n ψ T v 1,m, v 2,n ), m, n = 0,..., N 1. Th rsult is an approximation for th option pric at N N diffrnt log) striks givn by C T k 1,p, k 2,q ) α 1k 1,p α 2 k 2,q 2π) 2 Γk 1,p, k 2,q ) 2 1 0 p, q N. 3 FFT Pricing of th Sprad Option 3.1 Pricing a Sprad Option with Rimann Sums Lt us now considr th pric of a sprad option, givn by V K) := E Q [ rt [ S 1 T ) S 2 T ) K ] ] + = rt s 1 K ) q T s 1, )d ds 1 = log s 1 +K) rt s 1 K ) q T s 1, )d ds 1, whr th xrcis rgion is dfind as { } := s 1, ) R 2 s 1 K 0. Transforming th option pric with rspct to th log of th strik K no longr givs th sam kind of simpl rlationship with th charactristic function as in 2) of th prvious sction 7

as a consqunc of th simpl shap of th xrcis rgion of th corrlation option. If th boundaris of ar mad up of straight dgs, an appropriat affin chang of variabls can b introducd to mak th mthod in th prvious sction applicabl. This will not work for th pricing of sprad options for which th xrcis rgion is by natur non-linar s Figur 1). Exrcis Rgion Figur 1: Exrcis rgion of a sprad option in logarithmic variabls Notic howvr from abov that th FFT option pricing mthod givs N N prics simultanously in on transform, that is, intgrals of th payoff ovr N N diffrnt rgions. By subtracting and collcting th corrct pics, w can form tight uppr and lowr bounds for an intgral ovr a non-polygonal rgion analogous to intgrating by Rimann sums. Mor spcifically, w considr th following modifid xrcis rgion: { λ := s 1, ) [ } 1 2 Nλ, 1 2 Nλ) R s 1 K 0 and construct two sandwiching rgions λ out of rctangular strips with vrtics on th grid of th invrs transform s Figur 2 and 3). Tak as bfor an N N qually spacd grid Λ 1 Λ 2, whr Λ 1 := { k 1,p } := { p 1 2 N) λ 1 R 0 p N 1 } Λ 2 := { k 2,q } := { q 1 2 N) λ 2 R 0 q N 1 } For ach p = 0,..., N 1, dfin { k 2 p) := min k2,q Λ 2 k 2,q k 1,p+1 K } 0 q { k 2 p) := max k2,q Λ 2 k 2,q k 1,p < K }, 0 q 8

th -coordinats of th lowr dgs of th rctangular strips, p := [k 1,p, k 1,p+1 ) [k 2 p), ) p := [k 1,p, k 1,p+1 ) [k 2 p), ). Putting ths togthr w obtain two rgions bounding λ : := p=0 p, := p=0 p. 3 Exrcis Rgion 2 1 0-1 -2-3 -4-6 -5-4 -3-2 -1 0 1 2 3 Figur 2: Construction of th boundary of th approximat rgion Sinc λ and th sprad option payoff is positiv ovr λ, w hav a lowr bound for its intgral with th pricing krnl ovr this rgion: V K) := rt s 1 K ) q T s 1, )d ds 1 λ rt s 1 K ) q T s 1, )d ds 1. 6) Establishing th uppr bound is a trickir issu sinc th intgrand is not positiv ovr th ntir rgion. In fact, th payoff is strictly ngativ ovr \ λ by th dfinition of λ. To ovrcom this, w shall pick som ɛ > 0 such that { } s 1, ) R 2 s 1 K ɛ. 9

0 Exrcis Rgion -0.5-1 -1.5-2 -2.5-4 -3.5-3 -2.5-2 -1.5-1 -0.5 Figur 3: Approximation of th xrcis rgion with rctangular strips W thn hav [ V K) = rt s 1 K ɛ) ) ] q T s 1, )d ds 1 ɛ q T s 1, )d ds 1 [ λ λ rt s 1 K ɛ) ) ] q T s 1, )d ds 1 ɛ q T s 1, )d ds 1 [ = rt s ) 1 q T s 1, )d ds 1 K ɛ) q T s 1, )d ds 1 ɛ q T s 1, )d ds 1 ]. 7) By braking 6) and 7) into two componnts w can obtain ths bounds by intgrating s 1 ) q T s 1, ) and th dnsity q T s 1, ) ovr and, using th fast Fourir Transform mthod dscribd in th prvious sction. St Π 1 := s ) 1 q T s 1, )d ds 1 Π 2 := q T s 1, )d ds 1 Π 1 := s ) 1 q T s 1, )d ds 1 Π 2 := q T s 1, )d ds 1. Equations 6) and 7) can now b writtn as rt [ Π 1 KΠ 2 ] V K) rt [ Π 1 K ɛ)π 2 ɛπ 2 ]. 8) 3.2 Computing th Sums by FFT W now dmonstrat in dtail how to comput, by prforming two fast Fourir transforms, th four componnts Π 1, Π 2, Π 1, Π 2 in th approximat pricing quations 8) and hnc th sprad option prics across diffrnt striks. In fact, if on only wishs to approximat th option pric from blow, a singl transform is sufficint.) This is st out xplicitly for Π 1 10

blow and th othr thr cass follow similarly. Π 1 := := = whr p=0 p=0 s 1 ) q T s 1, )d ds 1 = [ k 1,p k 2 p) p=0 s 1 ) q T s 1, )d ds 1 Π 1 k 1,p, k 2 p)) Π 1 k 1,p+1, k 2 p)), p s 1 ) q T s 1, )d ds 1 k 1,p+1 k 2 p) Π 1 k 1, k 2 ) := s ) 1 q T s 1, )d ds 1. k 1 k 2 As bfor w apply a Fourir transform to th following modifid intgral: π 1 k 1, k 2 ) := α 1k 1 +α 2 k 2 Π 1 k 1, k 2 ) α 1, α 2 > 0. for a simpl rlationship with th charactristic function: χ 1 v 1, v 2 ) := = = iv 1k 1 +v 2 k 2 ) π 1 k 1, k 2 )dk 2 dk 1 α 1+iv 1 )k 1 +α 2 +iv 2 )k 2 k 2 s 1 ) q T s 1, ) s2 s 1 ) q T s 1, )d ds 1 ] k 1 s 1 ) q T s 1, )d ds 1 dk 2 dk 1 s1 α 1+iv 1 )k 1 +α 2 +iv 2 )k 2 dk 2 dk 1 d ds 1 = s ) 1 q T s 1, ) α 1+iv 1 )s 1 +α 2 +iv 2 ) α 1 + iv 1 )α 2 + iv 2 ) dds 1 = φ T v1 α 1 i, v 2 α 2 + 1)i ) φ T v1 α 1 + 1)i, v 2 α 2 i ) α 1 + iv 1 )α 2 + iv 2 ) Discrtising as in th prvious sction with 9). 10) λ 1 1 = λ 2 2 = 2π N v 1,m := m N 2 ) 1 v 2,n := n N 2 ) 2, 11) w now hav via an invrs) Fast Fourir transform valus of Π 1, ) on all N N vrtics of th grid Λ 1 Λ 2 givn by Π 1 k 1,p, k 2,q ) = α 1k 1,p α 2 k 2,q 2π) 2 α 1k 1,p α 2 k 2,q 2π) 2 m=0 n=0 = 1)p+q α 1k 1,p α 2 k 2,q 2π) 2 2 1 iv 1k 1,p +v 2 k 2,q ) χ 1 v 1, v 2 )dv 2 dv 1 iv 1,mk 1,p +v 2,n k 2,q ) χ 1 v 1,m, v 2,n ) 2 1 m=0 n=0 2πi mp+nq)[ ] N 1) m+n χ 1 v 1,m, v 2,n ) and hnc th valus of th 2 p rquird componnts in 9). Rpating th sam procdur for th othr componnts in 6) and 7) givs th bounds for th sprad option valu V K). 11

4 Numrical Prformanc 4.1 Undrlying Modls Prvious works on sprad options hav concntratd on th two-factor Gomtric Brownian motion GBM) modl in which th risk-nutral dynamics of th undrlying assts ar givn by ds 1 = S 1 r δ 1 )dt + σ 1 dw 1 ) ds 2 = S 2 r δ 2 )dt + σ 2 dw 2 ), whr E Q [dw 1 dw 2 ] = ρdt and r, δ i, σ i dnot th risk-fr rat, dividnd yilds and volatilitis rspctivly. Working with th log prics, s i := log S i, on has th following pair of SDEs: ds 1 = r δ 1 1 2 σ2 1)dt + σ 1 dw 1 d = r δ 2 1 2 σ2 2)dt + σ 2 dw 2. W shall now xtnd this modl to includ a third factor, a stochastic volatility for th two undrlying procsss. whr ds 1 = r δ 1 1 2 σ2 1ν)dt + σ 1 ν dw 1 d = r δ 2 1 2 σ2 2ν)dt + σ 2 ν dw 2 dν = κµ ν)dt + σ ν ν dw ν, E Q [dw 1 dw 2 ] = ρ dt E Q [dw 1 dw ν ] = ρ 1 dt E Q [dw 2 dw ν ] = ρ 2 dt. This is a dirct gnralisation of th singl-asst stochastic volatility modl [14, 21] and is considrd for th cas of corrlation options in [2]. Applying Ito s lmma and solving th rsulting PDE, on obtains an analytical xprssion for its charactristic function: [ φ sv u 1, u 2 ) := E Q xp iu 1 s 1 T ) + iu 2 T ) )] [ 2ζ1 θt ) = xp iu 1 s 1 0) + iu 2 0) + 2θ θ γ)1 θt ) + u j r δ j )T κµ [ 2θ θ γ)1 θt ) σν 2 2 log 2θ j=1,2 ) ν0) ) + θ γ ) T ] ], 12) whr ζ := 1 2 [ σ 2 1 u 2 1 + σ 2 2u 2 2 + 2ρσ 1 σ 2 u 1 u 2 ) + i σ 2 1 u 1 + σ 2 2u 2 ) ] γ := κ i ρ 1 σ 1 u 1 + ρ 2 σ 2 u 2 ) σν θ := γ 2 2σ 2 νζ. 12

Notic that as w lt th paramtrs of th stochastic volatility procss approach th limits κ, µ, σ ν 0, ν0) 1, th thr-factor stochastic volatility SV) modl dgnrats into th two-factor GBM modl and th charactristic function simplifis to that of a bivariat normal distribution: [ φ gbm u 1, u 2 ) = xp iu 1 s 1 0) + iu 2 0) + ζ T + ] u j r δ j )T. W shall us ths two charactristic functions to comput th sprad option prics undr th GBM and SV modl. In th formr cas th prics computd by th FFT mthod ar compard to th analytic option valu obtaind by a on dimnsional intgration basd on th conditioning tchniqu. This fails whn w introduc a stochastic volatility factor and thus a Mont Carlo pricing mthod is usd as a bnchmark for th SV modl. Prics ar also compard for th two diffusion modls. Givn a st of paramtr valus for th SV modl, on can comput from th charactristic function th man and covarianc matrix of s 1 T ), T ) undr th stochastic volatility assumption. W can thn infr for ths th paramtr valus of th two-factor GBM modl ndd to produc th sam momnts. Option valus may thn b computd and compard to th thr factor SV prics. Th cod is writtn in C++ and includs th fast Fourir Transform routin FFTW th Fastst Fourir Transform in th Wst), writtn by M. Frigo and S.G. Johnson [12]. Th xprimnts wr conductd on an Athlon 650 MHz machin running undr Linux with 512 MB RAM. 4.2 Computational Rsults Tabl 1 documnts th sprad option prics across a rang of striks undr th two factor Gomtric Brownian motion modl [22], computd by thr diffrnt tchniqus: on-dimnsional intgration analytic), th fast Fourir Transform and th Mont Carlo mthod. Th valus for th FFT mthods shown ar th lowr prics, computd ovr, rgions that approach th th tru xrcis rgion from blow and ar thrfor all lss than th analytic pric in th first column. 80000 simulations wr usd to produc th Mont Carlo prics and th avrag standard rrors ar rcordd in brackts at th bottom. Not that if on is only intrstd in computing prics in th two factor world, it is not actually ncssary to discrtis th tim horizon [0, T ] as was don hr. Sinc w know th trminal joint distribution of th two asst prics ar bivariat normal, thy can b simulatd dirctly and on singl tim stp is sufficint. Howvr, th point of this xrcis is to acquir an intuition into how th computational tim and accuracy varis as on changs th undrlying assumptions, sinc th introduction of xtra factors into a modl invitably involvs gnrating th whol paths of ths factors. Th avrag rrors of th two mthods ar computd and rcordd in Tabl 2. First w not that intgrating ovr from blow is mor accurat than ovr, as on can xpct from th lss straightforward procdur for constructing th uppr bound. For N = 1024 th lowr bound has an rror of roughly on basis point, whras N = 2048 taks us wll blow this j=1,2 13

Tabl 1: Prics computd by altrnativ mthods undr th 2-factor GBM modl Analytic Fast Fourir Transform Mont Carlo No. Discrtisation N Tim Stps Striks K 512 1024 2048 4096 1000 2000 0.0 8.513201 8.509989 8.511891 8.512981 8.513079 8.500949 8.516613 0.4 8.312435 8.311424 8.311995 8.312370 8.312385 8.300180 8.315818 0.8 8.114964 8.113877 8.114304 8.114901 8.114916 8.102730 8.118328 1.2 7.920790 7.919520 7.920173 7.920712 7.920741 7.908614 7.924135 1.6 7.729903 7.728471 7.729268 7.729810 7.729852 7.717831 7.733193 2.0 7.542296 7.540686 7.541637 7.542185 7.542242 7.530322 7.545496 2.4 7.357966 7.356278 7.357288 7.357830 7.357901 7.346038 7.361136 2.8 7.176888 7.175080 7.176185 7.176734 7.176818 7.164956 7.180054 3.2 6.999052 6.997200 6.998345 6.998881 6.998979 6.987070 7.002243 3.6 6.824451 6.822477 6.823721 6.824259 6.824371 6.812353 6.827700 4.0 6.653060 6.651047 6.652306 6.652852 6.652976 6.640874 6.656364 0.018076) 0.018184) S 1 0) = 96 δ 1 = 0.05 σ 1 = 0.1 S 2 0) = 100 δ 2 = 0.05 σ 2 = 0.2 r = 0.1 T = 1.0 K = 4.0 ρ = 0.5 Not: 80000 simulations hav bn usd in th Mont Carlo mthod rror lvl. From Tabl 3 thy tak 4.28 and 18.46 sconds rspctivly, clarly outprforming th Mont-Carlo mthod. For th sam lvl of accuracy, on would rquir simulations far mor than 80000, which alrady tak 304.95 sconds 606.40 sconds for th cas of 2000 tim stps) to gnrat. Although th Mont Carlo cod mployd uss no varianc rduction tchniqu othr than antithtic variats and its spd could b significantly improvd, th mthod is still unlikly to bat th FFT mthod in prformanc. Tabl 2: Accuracy of altrnativ mthods for th 2-factor GBM modl: Error in b.p. Fast Fourir Transform Mont Carlo Numbr of Numbr of Tim Stps Discrtisation Lowr Uppr Simulations 1000 2000 512 4.44 25.60 10000 129.15 0.051839) 70.81 0.050949) 1024 1.13 13.90 20000 22.34 0.036225) 40.67 0.035899) 2048 0.32 7.20 40000 7.44 0.025737) 7.63 0.025733) 4096 0.10 3.65 80000 18.34 0.018076) 4.94 0.018184) S 1 0) = 96 δ 1 = 0.05 σ 1 = 0.1 S 2 0) = 100 δ 2 = 0.05 σ 2 = 0.2 r = 0.1 T = 1.0 K = 4.0 ρ = 0.5 A clos xamination of Tabl 3 rvals th ral strngth of th FFT mthod. As w introduc a stochastic volatility factor, th Mont Carlo tchniqu nds to gnrat this valu at ach tim stp, which is thn multiplid with th incrmnts dw 1, dw 2 of th Brownian 14

Tabl 3: Computing Tim of Altrnativ Mthods Fast Fourir Transform Numbr of 10 Striks 100 Striks Discrtisation GBM SV GBM SV 512 1.04 1.11 1.10 1.20 1024 4.28 4.64 4.48 4.83 2048 18.46 19.54 18.42 19.74 4096 74.45 81.82 76.47 81.27 Mont Carlo: 1000 Tim Stps Numbr of 10 Striks 100 Striks Simulation GBM SV GBM SV 10000 38.2 144.87 41.95 151.75 20000 76.22 288.09 83.81 303.31 40000 152.5 576.25 168.48 606.53 80000 304.95 1152.9 335.20 1212.76 Mont Carlo: 2000 Tim Stps Numbr of 10 Striks 100 Striks Simulation GBM SV GBM SV 10000 75.57 287.41 79.83 295.21 20000 157.28 574.18 159.08 590.23 40000 303.37 1149.25 317.49 1184.32 80000 606.40 2298.37 636.33 2359.05 motions to giv th asst pric in th nxt priod. As indicatd across th columns this incrass th computational tim by almost a factor of 4. Rcalling th FFT mthod dscribd in th prvious sction, w notic that only a diffrnt charactristic function is substitutd whn mor factors ar includd, and th transform rmain two dimnsional. Comparing th tims for th GBM and SV modls, w obsrv only a 5 to 9 prcnt incras and falling as w incras th discrtisation numbr. Th xtra computing tim is du to th mor complx xprssion of th charactristic function with a largr st of paramtrs. For both mthods howvr, incrasing th numbr of striks dos not rsult in dramatic incrass in th computational tims. Tabl 4 shows th sprad option prics for diffrnt striks undr th thr factor SV modl. Th Mont Carlo prics with a discrtisation of 2000 tim stps oscillat around thos computd by th FFT mthod. Sinc w obsrv that in th two factor cas th rrors of th Mont Carlo mthod rmain high vn for 80000 simulations, mor xprimnts nd to b 15

conductd for a conclusiv judgmnt on this point. Tabl 4: Prics computd by altrnativ mthods undr th 3-factor SV modl Fast Fourir Transform Mont Carlo No. of Discrtisations No. of Simulations Striks K 512 2048 10000 20000 40000 80000 2.0 7.546895 7.543618 7.514375 7.567536 7.572211 7.523968 2.2 7.451878 7.452998 7.421861 7.475093 7.479742 7.431489 2.4 7.357703 7.363377 7.330142 7.383470 7.388080 7.339813 2.6 7.264298 7.274876 7.239209 7.292616 7.297218 7.248961 2.8 7.171701 7.186990 7.149234 7.202571 7.207191 7.158919 3.0 7.079987 7.099819 7.060043 7.113303 7.117954 7.069687 3.2 6.989008 7.013731 6.971625 7.024808 7.029515 6.981272 3.4 6.898826 6.928373 6.884026 6.937119 6.941875 6.893664 3.6 6.809471 6.843671 6.797246 6.850283 6.854984 6.806859 3.8 6.720957 6.759903 6.711328 6.764275 6.768886 6.720859 4.0 6.633232 6.676768 6.626221 6.679076 6.683587 6.635661 0.052702) 0.036984) 0.025739) 0.018206) r = 0.1 T = 1.0 ρ = 0.5 S 1 0) = 96 δ 1 = 0.05 σ 1 = 0.5 ρ 1 = 0.25 S 2 0) = 100 δ 2 = 0.05 σ 2 = 1.0 ρ 1 = 0.5 ν0) = 0.04 κ = 1.0 µ = 0.04 σ ν = 0.05 Not: 2000 tim stps hav bn usd for th Mont Carlo simulation. Finally, Figur 5 plots th diffrnc in th sprad option valus undr th 3-factor stochastic volatility modl and th 2-factor gomtric Brownian motion modl. Undr th SV modl, knowing th charactristic function of s 1,, w can calculat thir mans and covarianc matrix, which can thn b usd as th implid paramtrs r δ i and σ i, i = 1, 2, and ρ for th GBM modl. W rpat th procdur for diffrnt valus of ρ 1, ρ 2, th corrlation paramtrs btwn th Brownian motions W i, i = 1, 2 driving th asst prics and W ν driving th stochastic volatility factor ν. Whn ρ 1, ρ 2 ar high, a larg incrmnt W ν in 12) is mor likly to induc simultanously larg valus of W i, i = 1, 2, and dν. This incrass th volatilitis of both s 1 and and hnc th sprad and th sprad option valu. Compard with th two factor GBM modl, th SV modl of 12) obviously xhibits a richr structur for th sprad option valu which can b usd by tradrs with forward viws on th trm structurs of volatilitis and corrlations of th componnts of th sprad [16]. 5 Conclusions and futur dirctions W hav dscribd and implmntd an fficint mthod of computing, via a construction of suitabl approximat xrcis rgions, th valu of a gnric sprad option undr modls for which th charactristic function of th two undrlying asst prics is known in closd form. This taks us wll byond th two factor constant corrlation Gaussian framwork found in th 16

Figur 4: Pric Diffrnc btwn SV Modl and th GBM Modl with Implid Paramtrs xisting litratur, which is commonly assumd only for its tractability. In particular, on can now pric sprad options undr many multi-factor modls in th affin jump-diffusion family. For xampl, an indx sprad option in th quity markts can b pricd undr stochastic volatility modls. Spark and crack sprad options in th nrgy markt can now b valud with asst pric spiks and random volatility jumps, with major implications for trading, as wll as for asst and ral option valuation. Furthrmor, switching btwn altrnativ diffusion modls only amounts to substituting a diffrnt charactristic function for th undrlying prics/rats, laving th dimnsion of th transform and th summation procdur unchangd. As mor factors ar introducd mor tim is dvotd to th inxpnsiv valuation of th mor complx charactristic function, but not to th fast Fourir Transform algorithm. This significantly cuts down th incras of computational tims xpctd whn on applis th gnric PDE or Mont Carlo approachs to such a high dimnsional option pricing problm. Th computational advantag of th approach is dmonstratd with numrical xprimnts for both th two factor gomtric Brownian motion and th thr factor stochastic volatility modls. Pric diffrntials btwn th modls as on varis th paramtrs of th volatility procss confirm th significanc of a non-trivial corrlation structur in th modl dynamics. 17

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