SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

Size: px
Start display at page:

Download "SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM"

Transcription

1 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

2 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 & Plas addrss nquiris about th Judg Institut Working Papr Sris to: Publications Scrtary Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Tl: Fax:

3 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 & 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

4 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

5 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

6 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 iv 1 )α 2 + iv 2 )α iv 2 ) dds 1 rt φ T v1 α 1 + 1)i, v 2 α 2 + 1)i ) = α 1 + iv 1 )α iv 1 )α 2 + iv 2 )α 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

7 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 ) 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

8 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

9 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 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

10 0 Exrcis Rgion 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

11 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) 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

12 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 σ2 1)dt + σ 1 dw 1 d = r δ 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 σ2 1ν)dt + σ 1 ν dw 1 d = r δ 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 σ 2 2u ρσ 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

13 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 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

14 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 ) ) 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: simulations hav bn usd in th Mont Carlo mthod rror lvl. From Tabl 3 thy tak 4.28 and 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 sconds 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 ) ) ) ) ) ) ) ) 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

15 Tabl 3: Computing Tim of Altrnativ Mthods Fast Fourir Transform Numbr of 10 Striks 100 Striks Discrtisation GBM SV GBM SV Mont Carlo: 1000 Tim Stps Numbr of 10 Striks 100 Striks Simulation GBM SV GBM SV Mont Carlo: 2000 Tim Stps Numbr of 10 Striks 100 Striks Simulation GBM SV GBM SV 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 simulations, mor xprimnts nd to b 15

16 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 ) ) ) ) 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

17 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

18 On possibl dirction to nrich th volatility and corrlation structur furthr is to assum a four factor modl with two corrlatd stochastic volatility procsss [7]. Th calibration issu also rmains to b rsolvd in dtail, whr th focus of concrn will b an fficint procdur for backing out a implid corrlation surfac from obsrvd option prics. Rfrncs [1] Bakshi, G. and Z. Chn 1997). An altrnativ valuation modl for contingnt claims. Journal of Financial Economics 44 1) [2] Bakshi, G. and D. Madan 2000). Spanning and drivativ-scurity valuation. Journal of Financial Economics [3] Barndorff-Nilsn, O. 1997). Procsss of normal invrs Gaussian typ. Financ and Stochastic [4] Bats, D. 1996). Jumps and stochastic volatility: Exchang rat procss implicit in Dutschmark options. Rviw of Financial Studis [5] Carr, P. and D. B. Madan 1999). Option valuation using th fast Fourir transform. Th Journal of Computational Financ 2 4) [6] Chn, R. and L. Scott 1992). Pricing intrst rat options in a two-factor Cox- Ingrsoll-Ross modl of th trm structur. Rviw of Financial Studis [7] Clwlow, L. and C. Strickland 1998). Implmnting Drivativs Modls. John Wily & Sons Ltd. [8] Dmpstr, M. A. H. and J. P. Hutton 1999). Pricing Amrican stock options by linar programming. Mathmatical Financ 9 3) [9] Dng, S. 1999). Stochastic modls of nrgy commodity prics and thir applications: man-rvrsion with jumps and spiks. Working papr, Gorgia Institut of Tchnology, Octobr. [10] Duan, J.-C. and S. R. Pliska 1999). Option valuation with co-intgratd asst prics. Working papr, Dpartmnt of Financ, Hong Kong Univrsity of Scinc and Tchnology, January. [11] Duffi, D., J. Pan and K. Singlton 1999). Transform analysis and asst pricing for affin jump-diffusions. Working papr, Graduat School of Businss, Stanford Univrsity, August. [12] Frigo, M. and S. G. Johnson 1999). FFTW usr s manual. MIT, May. [13] Garman, M. 1992). Sprad th load. RISK 5 11) [14] Hston, S. 1993). A closd-form solution for options with stochastic volatility, with applications to bond and currncy options. Rviw of Financial Studis

19 [15] Madan, D., P. Carr and E. Chang 1998). Th varianc gamma procss and option pricing. Europan Financ Rviw [16] Mbanfo, A. 1997). Co-movmnt trm structur and th valuation of crack nrgy sprad options. In Mathmatics of Drivativs Scuritis. M. A. H. Dmpstr and S. R. Pliska, ds. Cambridg Univrsity Prss, [17] Parson, N. D. 1995). An fficint approach for pricing sprad options. Journal of Drivativs 3, Fall, [18] Pilipovic, D. and J. Wnglr 1998). Basis for boptions. Enrgy and Powr Risk Managmnt, Dcmbr, [19] Poitras, G. 1998). Sprad options, xchang options, and arithmtic Brownian motion. Journal of Futurs Markts 18 5) [20] Ravindran, K. 1993). Low-fat sprads. RISK 6 10) [21] Scott, L. O. 1997). Pricing stock options in a jump-diffusion modl with stochastic volatility and intrst rats: Applications of Fourir invrsion mthods. Mathmatical Financ 7 4) [22] Shimko, D. C. 1994). Options on futurs sprads: hdging, spculation, and valuation. Th Journal of Futurs Markts 14 2) [23] Trigorgis, L. 1996). Ral Options - Managrial Flxibility and Stratgy in Rsourc Allocation. MIT Prss, Cambridg, Mass. 19

(Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for the European Normal Black Scholes Formula) (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

More information

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

More information

QUANTITATIVE METHODS CLASSES WEEK SEVEN

QUANTITATIVE METHODS CLASSES WEEK SEVEN QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.

More information

The Matrix Exponential

The Matrix Exponential Th Matrix Exponntial (with xrciss) 92.222 - Linar Algbra II - Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial

More information

The Normal Distribution: A derivation from basic principles

The Normal Distribution: A derivation from basic principles Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn

More information

Lecture 3: Diffusion: Fick s first law

Lecture 3: Diffusion: Fick s first law Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th

More information

A Note on Approximating. the Normal Distribution Function

A Note on Approximating. the Normal Distribution Function Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS 25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

More information

14.3 Area Between Curves

14.3 Area Between Curves 14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

More information

Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk. When speaking about forward or futures contracts, basis risk is the market Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

More information

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13) con 37: Answr Ky for Problm St (Chaptr 2-3) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc

More information

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book. Rsourc Allocation Abstract This is a small toy xampl which is wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

More information

Architecture of the proposed standard

Architecture of the proposed standard Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th

More information

New Basis Functions. Section 8. Complex Fourier Series

New Basis Functions. Section 8. Complex Fourier Series Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ral-valud Fourir sris is xplaind and formula ar givn for convrting

More information

AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 2008 Scoring Guidelines AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

More information

C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Reports with SAS C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000 hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails

More information

Adverse Selection and Moral Hazard in a Model With 2 States of the World

Adverse Selection and Moral Hazard in a Model With 2 States of the World Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

Traffic Flow Analysis (2)

Traffic Flow Analysis (2) Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplete 2-Port Vector Network Analyzer Calibration Methods Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

More information

A Derivation of Bill James Pythagorean Won-Loss Formula

A Derivation of Bill James Pythagorean Won-Loss Formula A Drivation of Bill Jams Pythagoran Won-Loss Formula Ths nots wr compild by John Paul Cook from a papr by Dr. Stphn J. Millr, an Assistant Profssor of Mathmatics at Williams Collg, for a talk givn to th

More information

Gold versus stock investment: An econometric analysis

Gold versus stock investment: An econometric analysis Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 268-8662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag -7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin

More information

Genetic Drift and Gene Flow Illustration

Genetic Drift and Gene Flow Illustration Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

More information

The price of liquidity in constant leverage strategies. Marcos Escobar, Andreas Kiechle, Luis Seco and Rudi Zagst

The price of liquidity in constant leverage strategies. Marcos Escobar, Andreas Kiechle, Luis Seco and Rudi Zagst RACSAM Rv. R. Acad. Cin. Sri A. Mat. VO. 103 2, 2009, pp. 373 385 Matmática Aplicada / Applid Mathmatics Th pric of liquidity in constant lvrag stratgis Marcos Escobar, Andras Kichl, uis Sco and Rudi Zagst

More information

Exponential Growth and Decay; Modeling Data

Exponential Growth and Decay; Modeling Data Exponntial Growth and Dcay; Modling Data In this sction, w will study som of th applications of xponntial and logarithmic functions. Logarithms wr invntd by John Napir. Originally, thy wr usd to liminat

More information

Asset set Liability Management for

Asset set Liability Management for KSD -larning and rfrnc products for th global financ profssional Highlights Library of 29 Courss Availabl Products Upcoming Products Rply Form Asst st Liability Managmnt for Insuranc Companis A comprhnsiv

More information

Question 3: How do you find the relative extrema of a function?

Question 3: How do you find the relative extrema of a function? ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

More information

Foreign Exchange Markets and Exchange Rates

Foreign Exchange Markets and Exchange Rates Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

More information

Principles of Humidity Dalton s law

Principles of Humidity Dalton s law Principls of Humidity Dalton s law Air is a mixtur of diffrnt gass. Th main gas componnts ar: Gas componnt volum [%] wight [%] Nitrogn N 2 78,03 75,47 Oxygn O 2 20,99 23,20 Argon Ar 0,93 1,28 Carbon dioxid

More information

Performance Evaluation

Performance Evaluation Prformanc Evaluation ( ) Contnts lists availabl at ScincDirct Prformanc Evaluation journal hompag: www.lsvir.com/locat/pva Modling Bay-lik rputation systms: Analysis, charactrization and insuranc mchanism

More information

Module 7: Discrete State Space Models Lecture Note 3

Module 7: Discrete State Space Models Lecture Note 3 Modul 7: Discrt Stat Spac Modls Lctur Not 3 1 Charactristic Equation, ignvalus and ign vctors For a discrt stat spac modl, th charactristic quation is dfind as zi A 0 Th roots of th charactristic quation

More information

Exotic Electricity Options and the Valuation. Assets. April 6, 1998. Abstract

Exotic Electricity Options and the Valuation. Assets. April 6, 1998. Abstract Exotic Elctricity Options and th Valuation of Elctricity Gnration and Transmission Assts Shiji Dn Blak Johnson y Aram Soomonian z April 6, 1998 Abstract This papr prsnts and applis a mthodoloy for valuin

More information

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among

More information

the so-called KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through

the so-called KOBOS system. 1 with the exception of a very small group of the most active stocks which also trade continuously through Liquidity and Information-Basd Trading on th Ordr Drivn Capital Markt: Th Cas of th Pragu tock Exchang Libor 1ÀPH³HN Cntr for Economic Rsarch and Graduat Education, Charls Univrsity and Th Economic Institut

More information

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

Closed-form solutions for Guaranteed Minimum Accumulation Benefits

Closed-form solutions for Guaranteed Minimum Accumulation Benefits Closd-form solutions for Guarantd Minimum Accumulation Bnfits Mikhail Krayzlr, Rudi Zagst and Brnhard Brunnr Abstract Guarantd Minimum Accumulation Bnfit GMAB is on of th variabl annuity products, i..

More information

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List. Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

Theoretical aspects of investment demand for gold

Theoretical aspects of investment demand for gold Victor Sazonov (Russia), Dmitry Nikolav (Russia) Thortical aspcts of invstmnt dmand for gold Abstract Th main objctiv of this articl is construction of a thortical modl of invstmnt in gold. Our modl is

More information

Policies for Simultaneous Estimation and Optimization

Policies for Simultaneous Estimation and Optimization Policis for Simultanous Estimation and Optimization Migul Sousa Lobo Stphn Boyd Abstract Policis for th joint idntification and control of uncrtain systms ar prsntd h discussion focuss on th cas of a multipl

More information

Rural and Remote Broadband Access: Issues and Solutions in Australia

Rural and Remote Broadband Access: Issues and Solutions in Australia Rural and Rmot Broadband Accss: Issus and Solutions in Australia Dr Tony Warrn Group Managr Rgulatory Stratgy Tlstra Corp Pag 1 Tlstra in confidnc Ovrviw Australia s gographical siz and population dnsity

More information

e = C / electron Q = Ne

e = C / electron Q = Ne Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

More information

Solutions to Homework 8 chem 344 Sp 2014

Solutions to Homework 8 chem 344 Sp 2014 1. Solutions to Homwork 8 chm 44 Sp 14 .. 4. All diffrnt orbitals mans thy could all b paralll spins 5. Sinc lctrons ar in diffrnt orbitals any combination is possibl paird or unpaird spins 6. Equivalnt

More information

Production Costing (Chapter 8 of W&W)

Production Costing (Chapter 8 of W&W) Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary

More information

7 Timetable test 1 The Combing Chart

7 Timetable test 1 The Combing Chart 7 Timtabl tst 1 Th Combing Chart 7.1 Introduction 7.2 Tachr tams two workd xampls 7.3 Th Principl of Compatibility 7.4 Choosing tachr tams workd xampl 7.5 Ruls for drawing a Combing Chart 7.6 Th Combing

More information

Parallel and Distributed Programming. Performance Metrics

Parallel and Distributed Programming. Performance Metrics Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

Modern Portfolio Theory (MPT) Statistics

Modern Portfolio Theory (MPT) Statistics Modrn Portfolio Thory (MPT) Statistics Morningstar Mthodology Papr May 9, 009 009 Morningstar, Inc. All rights rsrvd. Th information in this documnt is th proprty of Morningstar, Inc. Rproduction or transcription

More information

A Multi-Heuristic GA for Schedule Repair in Precast Plant Production

A Multi-Heuristic GA for Schedule Repair in Precast Plant Production From: ICAPS-03 Procdings. Copyright 2003, AAAI (www.aaai.org). All rights rsrvd. A Multi-Huristic GA for Schdul Rpair in Prcast Plant Production Wng-Tat Chan* and Tan Hng W** *Associat Profssor, Dpartmnt

More information

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 08-16-85 WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 Summary of Dutis : Dtrmins City accptanc of workrs' compnsation cass for injurd mploys; authorizs appropriat tratmnt

More information

Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost

Improving Managerial Accounting and Calculation of Labor Costs in the Context of Using Standard Cost Economy Transdisciplinarity Cognition www.ugb.ro/tc Vol. 16, Issu 1/2013 50-54 Improving Managrial Accounting and Calculation of Labor Costs in th Contxt of Using Standard Cost Lucian OCNEANU, Constantin

More information

Introduction to Finite Element Modeling

Introduction to Finite Element Modeling Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

More information

Category 7: Employee Commuting

Category 7: Employee Commuting 7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

REPORT' Meeting Date: April 19,201 2 Audit Committee

REPORT' Meeting Date: April 19,201 2 Audit Committee REPORT' Mting Dat: April 19,201 2 Audit Committ For Information DATE: March 21,2012 REPORT TITLE: FROM: Paul Wallis, CMA, CIA, CISA, Dirctor, Intrnal Audit OBJECTIVE To inform Audit Committ of th rsults

More information

On the moments of the aggregate discounted claims with dependence introduced by a FGM copula

On the moments of the aggregate discounted claims with dependence introduced by a FGM copula On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula - Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval - Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,

More information

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a high-lvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End

More information

Global Sourcing: lessons from lean companies to improve supply chain performances

Global Sourcing: lessons from lean companies to improve supply chain performances 3 rd Intrnational Confrnc on Industrial Enginring and Industrial Managmnt XIII Congrso d Ingniría d Organización Barclona-Trrassa, Sptmbr 2nd-4th 2009 Global Sourcing: lssons from lan companis to improv

More information

Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange

Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity

More information

Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi

Gas Radiation. MEL 725 Power-Plant Steam Generators (3-0-0) Dr. Prabal Talukdar Assistant Professor Department of Mechanical Engineering IIT Delhi Gas Radiation ME 725 Powr-Plant Stam Gnrators (3-0-0) Dr. Prabal Talukdar Assistant Profssor Dpartmnt of Mchanical Enginring T Dlhi Radiation in absorbing-mitting mdia Whn a mdium is transparnt to radiation,

More information

A Project Management framework for Software Implementation Planning and Management

A Project Management framework for Software Implementation Planning and Management PPM02 A Projct Managmnt framwork for Softwar Implmntation Planning and Managmnt Kith Lancastr Lancastr Stratgis Kith.Lancastr@LancastrStratgis.com Th goal of introducing nw tchnologis into your company

More information

Development of Financial Management Reporting in MPLS

Development of Financial Management Reporting in MPLS 1 Dvlopmnt of Financial Managmnt Rporting in MPLS 1. Aim Our currnt financial rports ar structurd to dlivr an ovrall financial pictur of th dpartmnt in it s ntirty, and thr is no attmpt to provid ithr

More information

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY

SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY 1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically

More information

ME 612 Metal Forming and Theory of Plasticity. 6. Strain

ME 612 Metal Forming and Theory of Plasticity. 6. Strain Mtal Forming and Thory of Plasticity -mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.

More information

A Theoretical Model of Public Response to the Homeland Security Advisory System

A Theoretical Model of Public Response to the Homeland Security Advisory System A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial

More information

Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing

Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing Uppr Bounding th Pric of Anarchy in Atomic Splittabl Slfish Routing Kamyar Khodamoradi 1, Mhrdad Mahdavi, and Mohammad Ghodsi 3 1 Sharif Univrsity of Tchnology, Thran, Iran, khodamoradi@c.sharif.du Sharif

More information

Lecture 20: Emitter Follower and Differential Amplifiers

Lecture 20: Emitter Follower and Differential Amplifiers Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.

More information

Section 7.4: Exponential Growth and Decay

Section 7.4: Exponential Growth and Decay 1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009

Abstract. Introduction. Statistical Approach for Analyzing Cell Phone Handoff Behavior. Volume 3, Issue 1, 2009 Volum 3, Issu 1, 29 Statistical Approach for Analyzing Cll Phon Handoff Bhavior Shalini Saxna, Florida Atlantic Univrsity, Boca Raton, FL, shalinisaxna1@gmail.com Sad A. Rajput, Farquhar Collg of Arts

More information

Sigmoid Functions and Their Usage in Artificial Neural Networks

Sigmoid Functions and Their Usage in Artificial Neural Networks Sigmoid Functions and Thir Usag in Artificial Nural Ntworks Taskin Kocak School of Elctrical Enginring and Computr Scinc Applications of Calculus II: Invrs Functions Eampl problm Calculus Topic: Invrs

More information

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman

Cloud and Big Data Summer School, Stockholm, Aug., 2015 Jeffrey D. Ullman Cloud and Big Data Summr Scool, Stockolm, Aug., 2015 Jffry D. Ullman Givn a st of points, wit a notion of distanc btwn points, group t points into som numbr of clustrs, so tat mmbrs of a clustr ar clos

More information

STATEMENT OF INSOLVENCY PRACTICE 3.2

STATEMENT OF INSOLVENCY PRACTICE 3.2 STATEMENT OF INSOLVENCY PRACTICE 3.2 COMPANY VOLUNTARY ARRANGEMENTS INTRODUCTION 1 A Company Voluntary Arrangmnt (CVA) is a statutory contract twn a company and its crditors undr which an insolvncy practitionr

More information

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:

More information

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives.

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives. Volum 3, Issu 6, Jun 2013 ISSN: 2277 128X Intrnational Journal of Advancd Rsarch in Computr Scinc and Softwar Enginring Rsarch Papr Availabl onlin at: wwwijarcsscom Dynamic Ranking and Slction of Cloud

More information

10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases.

10/06/08 1. Aside: The following is an on-line analytical system that portrays the thermodynamic properties of water vapor and many other gases. 10/06/08 1 5. Th watr-air htrognous systm Asid: Th following is an on-lin analytical systm that portrays th thrmodynamic proprtis of watr vapor and many othr gass. http://wbbook.nist.gov/chmistry/fluid/

More information

Simulated Radioactive Decay Using Dice Nuclei

Simulated Radioactive Decay Using Dice Nuclei Purpos: In a radioactiv sourc containing a vry larg numbr of radioactiv nucli, it is not possibl to prdict whn any on of th nucli will dcay. Although th dcay tim for any on particular nuclus cannot b prdictd,

More information

Constraint-Based Analysis of Gene Deletion in a Metabolic Network

Constraint-Based Analysis of Gene Deletion in a Metabolic Network Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany

More information

Data warehouse on Manpower Employment for Decision Support System

Data warehouse on Manpower Employment for Decision Support System Data warhous on Manpowr Employmnt for Dcision Support Systm Amro F. ALASTA, and Muftah A. Enaba Abstract Sinc th us of computrs in businss world, data collction has bcom on of th most important issus du

More information

LABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACK-BOX

LABORATORY 1 IDENTIFICATION OF CIRCUIT IN A BLACK-BOX LABOATOY IDENTIFICATION OF CICUIT IN A BLACK-BOX OBJECTIES. To idntify th configuration of an lctrical circuit nclosd in a two-trminal black box.. To dtrmin th valus of ach componnt in th black box circuit.

More information

Statistical Machine Translation

Statistical Machine Translation Statistical Machin Translation Sophi Arnoult, Gidon Mailltt d Buy Wnnigr and Andra Schuch Dcmbr 7, 2010 1 Introduction All th IBM modls, and Statistical Machin Translation (SMT) in gnral, modl th problm

More information

Factorials! Stirling s formula

Factorials! Stirling s formula Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

More information

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

More information

Free ACA SOLUTION (IRS 1094&1095 Reporting)

Free ACA SOLUTION (IRS 1094&1095 Reporting) Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 279-1062 ACA Srvics Transmit IRS Form 1094 -C for mployrs Print & mail IRS Form 1095-C to mploys HR Assist 360 will gnrat th 1095 s for

More information

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final. Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

An Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images

An Adaptive Clustering MAP Algorithm to Filter Speckle in Multilook SAR Images An Adaptiv Clustring MAP Algorithm to Filtr Spckl in Multilook SAR Imags FÁTIMA N. S. MEDEIROS 1,3 NELSON D. A. MASCARENHAS LUCIANO DA F. COSTA 1 1 Cybrntic Vision Group IFSC -Univrsity of São Paulo Caia

More information

Essays on Adverse Selection and Moral Hazard in Insurance Market

Essays on Adverse Selection and Moral Hazard in Insurance Market Gorgia Stat Univrsity ScholarWorks @ Gorgia Stat Univrsity Risk Managmnt and Insuranc Dissrtations Dpartmnt of Risk Managmnt and Insuranc 8--00 Essays on Advrs Slction and Moral Hazard in Insuranc Markt

More information

Key Management System Framework for Cloud Storage Singa Suparman, Eng Pin Kwang Temasek Polytechnic {singas,engpk}@tp.edu.sg

Key Management System Framework for Cloud Storage Singa Suparman, Eng Pin Kwang Temasek Polytechnic {singas,engpk}@tp.edu.sg Ky Managmnt Systm Framwork for Cloud Storag Singa Suparman, Eng Pin Kwang Tmask Polytchnic {singas,ngpk}@tp.du.sg Abstract In cloud storag, data ar oftn movd from on cloud storag srvic to anothr. Mor frquntly

More information

Efficiency Losses from Overlapping Economic Instruments in European Carbon Emissions Regulation

Efficiency Losses from Overlapping Economic Instruments in European Carbon Emissions Regulation iscussion Papr No. 06-018 Efficincy Losss from Ovrlapping Economic Instrumnts in Europan Carbon Emissions Rgulation Christoph Böhringr, Hnrik Koschl and Ulf Moslnr iscussion Papr No. 06-018 Efficincy Losss

More information

Relationship between Cost of Equity Capital And Voluntary Corporate Disclosures

Relationship between Cost of Equity Capital And Voluntary Corporate Disclosures Rlationship btwn Cost of Equity Capital And Voluntary Corporat Disclosurs Elna Ptrova Eli Lilly & Co, Sofia, Bulgaria E-mail: ptrova.lnaa@gmail.com Gorgios Gorgakopoulos (Corrsponding author) Amstrdam

More information

Deer: Predation or Starvation

Deer: Predation or Starvation : Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,

More information

Expert-Mediated Search

Expert-Mediated Search Exprt-Mdiatd Sarch Mnal Chhabra Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA chhabm@cs.rpi.du Sanmay Das Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA sanmay@cs.rpi.du David

More information

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D 24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

More information

Simulation of a Solar Cell considering Single-Diode Equivalent Circuit Model

Simulation of a Solar Cell considering Single-Diode Equivalent Circuit Model Simulation of a Solar Cll considring Singl-Diod Equivalnt Circuit Modl E.M.G. Rodrigus, R. Mlício,, V.M.F. Mnds and J.P.S. Catalão, Univrsity of Bira Intrior R. Font do Lamiro, - Covilhã (Portugal) Phon:

More information

User-Perceived Quality of Service in Hybrid Broadcast and Telecommunication Networks

User-Perceived Quality of Service in Hybrid Broadcast and Telecommunication Networks Usr-Prcivd Quality of Srvic in Hybrid Broadcast and Tlcommunication Ntworks Michal Galtzka Fraunhofr Institut for Intgratd Circuits Branch Lab Dsign Automation, Drsdn, Grmany Michal.Galtzka@as.iis.fhg.d

More information

SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII

SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII SAMPLE QUESTION PAPER MATHEMATICS (4) CLASS XII 6-7 Tim allowd : 3 hours Maimum Marks : Gnral Instructions: (i) All qustions ar compulsor. (ii) This qustion papr contains 9 qustions. (iii) Qustion - 4

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scic March 15, 005 Srii Dvadas ad Eric Lhma Problm St 6 Solutios Du: Moday, March 8 at 9 PM Problm 1. Sammy th Shar is a fiacial srvic providr who offrs loas o th followig

More information

ICES REPORT 15-01. January 2015. The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712

ICES REPORT 15-01. January 2015. The Institute for Computational Engineering and Sciences The University of Texas at Austin Austin, Texas 78712 ICES REPORT 15-01 January 2015 A locking-fr modl for Rissnr-Mindlin plats: Analysis and isogomtric implmntation via NURBS and triangular NURPS by L. Birao da Viga, T.J.R. Hughs, J. Kindl, C. Lovadina,

More information

Throughput and Buffer Analysis for GSM General Packet Radio Service (GPRS)

Throughput and Buffer Analysis for GSM General Packet Radio Service (GPRS) Throughput and Buffr Analysis for GSM Gnral Packt Radio Srvic (GPRS) Josph Ho, Yixin Zhu, and Sshu Madhavapddy Nortl Ntorks 221 Laksid Blvd. Richardson, TX 7582 E-Mail: joho@nortlntorks.com Abstract -

More information

Financial Mathematics

Financial Mathematics Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

More information

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore Intrnational Journal of mrging Tchnology and dvancd nginring Wbsit: www.ijta.com (ISSN 2250-2459, Volum 2, Issu 4, pril 2012) n road outlin of Rdundant rray of Inxpnsiv isks Shaifali Shrivastava 1 partmnt

More information

MONEY ILLUSION IN THE STOCK MARKET: THE MODIGLIANI-COHN HYPOTHESIS*

MONEY ILLUSION IN THE STOCK MARKET: THE MODIGLIANI-COHN HYPOTHESIS* MONEY ILLUSION IN THE STOCK MARKET: THE MODIGLIANI-COHN HYPOTHESIS* RANDOLPH B. COHEN CHRISTOPHER POLK TUOMO VUOLTEENAHO Modigliani and Cohn hypothsiz that th stock markt suffrs from mony illusion, discounting

More information