Heterogeneous Basket Options Pricing Using Analytical Approximations

Transcription

1 Canada Research Chair in Risk Managemen Working paper 6- Heerogeneous Baske Opions Pricing Using Analyical Approximaions Georges Dionne, Geneviève Gauhier, Nadia Ouerani and Nabil Tahani * February 6 Absrac This paper proposes he use of analyical approximaions o price an heerogeneous baske opion combining commodiy prices, foreign currencies and zero-coupon bonds. We examine he performance of hree momen maching approximaions: inverse gamma, Edgeworh expansion around he lognormal and Johnson family disribuions. Since here is no closed-form formula for baske opions, we carry ou Mone Carlo simulaions o generae he benchmark values. We perform a simulaion experimen on a whole se of opions based on a random choice of parameers. Our resuls show ha he lognormal and Johnson disribuions give he mos accurae resuls. Keywords: Baske Opions, Opions Pricing, Analyical Approximaions, Mone Carlo Simulaion. JEL classificaion: C5, C6, G, G3. Résumé Ce aricle raie de la arificaion des opions paniers héérogènes en présence de aux d'inérê sochasiques. Nous proposons d'uiliser les approximaions analyiques pour arifer une opion panier héérogène qui combine à la fois des denrées, des aux de change e des obligaions zérocoupon. Nous avons comparé la performance de rois approximaions basées sur la echnique d appariemen des momens : la disribuion gamma inverse, l expansion Edgeworh auour de la disribuion lognormale e la famille de disribuions de Johnson. Puisqu il n exise pas de soluion analyique au prix de l opion panier, nous avons uilisé les simulaions de Mone Carlo pour générer les valeurs de référence. Deux expériences de simulaion on éé menées, l'une basée sur une analyse de sensibilié locale e l'aure, plus générale, basée sur un choix aléaoire de paramères. Nos résulas monren que les approximaions lognormale e Johnson son rès précises e beaucoup plus performanes que l'approximaion gamma inverse. Mos clés : Opions panier, arificaion d opion, approximaion analyique, simulaion Mone Carlo. Classificaion JEL : C5, C6, G, G3. * Georges Dionne, Canada Research Chair in Risk Managemen, HEC Monréal, Geneviève Gauhier, Managemen Science Deparmen, HEC Monréal, Nadia Ouerani, IESEG School of Managemen, Lille and Nabil Tahani, School of Adminisraive Sudies, York Universiy. The auhors acknowledge he financial suppor of he Insiu de Finance Mahémaiques de Monréal, (IFM ), he Naional Science and Engineering Research Council of Canada (NSERC) and he Fonds Québécois de Recherche sur la Sociéé e la Culure, (FQRSC). This paper exends N. Ouerani's Ph.D. hesis on which Phelim P. Boyle, Michel Denaul, and Pascal François have made very helpful suggesions. We also hank Moshe A. Milevsky and Chris Robinson for heir commens. Previous versions of his paper have been presened a he 8 h Annual Ausralasian Finance and Banking Conference (Sydney, December 5) and he Norhern Finance Associaion Meeing (Vancouver, Ocober 5).

2 Inroducion The use of derivaive securiies in risk managemen aciviies emerged in he early 99s and has evolved rapidly since. They now are he mos imporan ool in nancial risk managemen. According o he 998 Wharon school survey of nancial risk managemen by US non- nancial rms, over 5% of he responding rms use derivaives producs o hedge heir exposure, 83% of hem use derivaives o hedge foreign-exchange risk, 76% o hedge ineres-rae risk, and 56% o hedge commodiy-price risk. In he curren economic siuaion, many non- nancial insiuions such as gold-mining rms, energy companies or airlines companies may face di eren nancial risks simulaneously and hence look for he mos e cien way o hedge heir porfolio. For example, gold-mining rms may be exposed o di eren ypes of risk: commodiy risk, which can include uncerainy abou he price of heir primary produc, gold, as well as heir by-producs such as silver and copper; since hese rms sell heir producs in oher counries, hey are exposed o currency risk; and heir ineres rae risk exposure is primarily relaed o heir xed-rae and variable-rae deb. Though all hese markes are very acive and very liquid so ha a rm could hedge all is di eren risk exposures separaely, i would be more ineresing o adop a porfolio approach because i allows he rm o accoun for he correlaions beween hese di eren nancial markes and hedge simulaneously a grea variey of di eren nancial risks. To aain his goal, baske opions are an e cien insrumen o use. Baske opions are a ype of exoic opion whose payo depends on he value of a baske of asses. They o er he exibiliy of being able o include virually any kind and any number of asses, and hence can suiably respond o he speci c needs of a rm in hedging is risk exposures. A risk manager may have oher incenives in using baske opions: depending on he design of each opion, hey are usually cheaper han a porfolio of sandard opions. In pracice, baske opions are raded over-he-couner and are designed speci cally o mee he needs of he buyer. For hese reasons, he liquidiy premium required by he counerpar may annihilae some of he advanages coming from he correlaion srucure of he baske. The pricing of baske opions is more challenging han ha of sandard opions because here is no explici analyical soluion for he densiy funcion of a weighed sum of correlaed asses. Several approaches are proposed in he lieraure o price baske opions. They can be caegorized as follows:. Numerical mehods: Mone Carlo simulaions [Barraquand (995), Pellizzari ()] and Quasi-Mone Carlo [Dahl () and Dahl and Benh ()], and mulinomial rees [Rubinsein (994), Wan()],. Upper and lower bounds: Curran (994), Vanmaele, Deelsra and Liinev (4), Laurence and Wang (4) and Deelsra, Liinev and Vanmaele (4), 3. Analyical approximaions: Genle (993), Huynh (994), Milevsky and Posner (998a, 998b, 999), Posner and Milevsky (999), Ju (), Daey, Gauhier and Simonao (3) and Brigo, Mercurio, Rapisarda and Scoi (4). Financial managemen, Vol. 7, No. 4, winer 998.

3 However, all hese papers are based on wo main hypohesis. They assume consan ineres raes and homogeneous baske opions; ha is, all he asses underlying he baske are of he same naure and hus hedge he same ype of risk. Unforunaely, his homogeneiy in baske asses does no always mach a rm s needs, and wih a growing diversi caion in invesors porfolios, heerogeneous baske opions become a very e cien ool o hedge muliple risk exposures simulaneously. This aricle proposes analyical approximaions o price heerogeneous baske opions, consising of commodiies, foreign currencies and zero-coupon bonds, wih sochasic ineres raes, and compares he accuracy and he performance of hese approximaions for di eren ses of parameers. Three disribuions based on he momen maching echnique will be used o approximae he baske densiy funcion: inverse gamma disribuion, Edgeworh expansion around he lognormal disribuion and Johnson disribuion. I is found ha he Edgeworh-lognormal and Johnson approximaions perform beer han he inverse gamma approximaion. Our conribuions are: () o design an arbirage-free framework in which he domesic and he foreign ineres raes, he exchange rae, he commodiy price and he convenience yield are sochasic, () o compue he momens of he heerogeneous baske under he T forward measure, (3) o show ha some of he exising analyical approximaions may be used in his very general seing and (4) quanify he approximaion errors. The nex secion presens he pricing model under he forward measure. Secion 3 derives he inverse gamma, he Edgeworh-lognormal and he Johnson approximaions. Secion 4 compares he performance of he hree approximaions. Secion 5 compues he hedging raios and Secion 6 concludes. Pricing of a European heerogeneous baske opion Le us consider a non- nancial rm looking for an alernaive way o simulaneously hedge is di eren nancial risk exposures: commodiy price risk, exchange rae risk and ineres rae risk. In he following, S denoes he ime value of he commodiy, is is coninuously-compounded convenience yield a ime, C is he value a ime of one uni of he foreign currency expressed in he domesic currency, P (; T ) is he ime value of a zero-coupon bond paying one uni of he domesic currency a ime T, P (; T ) is he ime value of a foreign zero-coupon bond paying one uni of he foreign currency a ime T. We assume ha he rm s nancial asses dynamics are given by he following sochasic di erenial equaions (SDE hereafer) under he hisorical In his seing, he convenience yield and he commodiy price share he same source of risk o ensure he marke compleeness. For more deails, see Dionne, Gauhier and Ouerani (6). 3

4 probabiliy measure P : ds = S h ( s ) d + s dw () i ; (a) d = ( )d + dw () ; (b) h i dc = C c d + c dw () ; (c) ;T dp (; T ) = P (; T ) r ;T ;T d ;T dw (3) ; T ; (d) ;T "!! # dp (; T ) = P (; T ) r ;T ;T ;T ;T d ;T dw (4) ; T (e) The expressions for he bond prices are obained from he domesic and he foreign insananeous forward raes: df (; T ) = ;T d + ;T dw (3) ; T ; (f) df (; T ) = ;T d + ;T dw (4) ; T ; (g) in which he volailiy parameers ;T and ;T are deerminisic funcions of ime and mauriy. Consequenly, ;T = R T ;s ds, ;T = R T ;sds and r = f (; ) ; r = f (; ) are respecively he coninuously-compounded domesic and foreign risk-free spo ineres raes. We do no specify he drif erms ;T and ;T since hey do no appear in he pricing formulae. The four-dimensional Brownian moion W () ; W () ; W (3) ; W (4) : is consruced on a lered probabiliy space (; F; ff : g ; P ) wih he correlaion srucure: Corr P W (i) ; W (j) = i;j ; for each i; j = f; ; 3; 4g and > : We consider an European call opion ha gives he rm he righ o buy a baske consising of he commodiy, he domesic zero-coupon bond and he foreign zero-coupon bond expressed in domesic currency unis, a a srike price. The payo a mauriy T is given by X B = max [B T ; ] ; where he baske value a ime is B = w S + w P (; T ) + w 3 C P (; T ), T T ; T, w ; w and w 3 correspond o he numbers of shares invesed respecively in he commodiy S T, he domesic zero-coupon bond P (T; T ) and he foreign zero-coupon bond expressed in domesic currency unis C T P (T; T ). Since he proposed marke model is complee, 3 Harrison and Pliska (98) allows he pricing of any coningen claim as he expecaion of he discouned payo under he risk-neural measure Q. Consequenly, he price of he European call opion a ime is given by: V B = E exp Q r v dv max (B T ; ) F = P (; T )E Q T [max (B T ; )j F ] = P (; T ) Z + 3 The derivaion of he unique risk-neural measure is available upon reques. max (x ; ) v (x) dx; () 4

5 where v (x) is he rue (unknown) densiy funcion of he baske value B T under he T forward measure Q T. This forward measure has a Radon-Nikodym derivaive wih respec o Q denoed : The associaed Q maringale is given by: dq T dq = E Q dqt dq F = exp R T P (; T ) r v dv : The SDEs sais ed by he baske underlying asses under he T -forward measure Q T is given in Appendix A and is srong soluion is, for any T, s S T = S exp r u u T = e (T ) + + s C T = C exp r u e (T r u P (T; T ) = P (; T ) exp P (T; T ) = P (; T ) exp R T s s u) du + ru c r u r u where W c = cw () ; W c() ; W c(3) ; W c (4) is a QT i;j. s ;3 u;t e (T ) du + s ;3 e (T u) dw c u () c ;3 u;t du + c u;t + u;t u;t dw c u () e (T dw c u () u) u;t du du u;t dw c u (3) u;t + c ;4 u;t + 3;4 u;t u;t R T u;t dw c u (4) Brownian moions wih Corr Q T du cw (i)! (3a) (3b) (3c) (3d) (3e) ; W c (j) = The evaluaion of he baske call opion is complicaed by he absence of a closed-form equaion for he densiy funcion v(x) in Equaion (). Among he di eren approaches proposed in he baske opions lieraure, we nd some numerical echniques such as Mone Carlo, Quasi-Mone Carlo and laice-based mehods, he upper and lower bound compuaions, 4 and some analyical approximaions. Laice-based approaches are widely used for opions on a single asse. They are, exponenially complicaed and compuaionally expensive for opions on muliple asses. For example, our hreeasse baske opion needs (n + ) 3 erminal nodes on an n-sep rinomial ree. On he oher hand, Mone Carlo and Quasi-Mone Carlo mehods can be used for muli-asses opions and are less ime-consuming han laice-based approaches. The esimaes can be as accurae as needed a a compuaional cos however: o improve he accuracy of an n-pah simulaion by one half, one needs o simulae 4n pahs and hus needs 4 imes more compuing ime. A praciioner migh be ineresed in slighly less accurae bu very fas mehods such as analyical approximaions. These mehods approximae he unknown baske densiy funcion wih an alernaive and easy-o-compue disribuion. In he nex secions, we will exend hree well-known 4 The upper and lower bounds mehods are no useful in our case since he marke model proposed is complee and hus a unique opion price can be compued. 5

6 analyical approximaions o heerogeneous baske opions: he inverse gamma disribuion, he Edgeworh expansion around he lognormal disribuion and Johnson disribuion. 3 Analyical approximaions In order o apply hese momen maching-based approximaions, we need o calculae he rs four momens of he weighed sum underlying he European opion under he T forward measure Q T. We adop he following noaions: n (h) = n (h) = Z + Z + x n h (x) dx (4a) x (h) n h (x) dx; (4b) n (h) and n (h) represen respecively he n h non-cenered and cenered momens of he densiy funcion h fv; ag ; where h = v corresponds o he exac densiy of he baske value under he forward measure while h = a corresponds o he approximae densiy. The rs four cumulans of disribuion h, ha is, he mean, he variance, he skewness and he kurosis are de ned as: (h) = (h) (5a) (h) = (h) (5b) 3 (h) = 3 (h) (5c) 4 (h) = 4 (h) 3 (h) : (5d) Lemma For any posiive ineger n, he n h non-cenered momen of he rue disribuion of he weighed sum B T ; under he T -forward measure Q T, is n (v) = E Q T [BT n ] = E Q T [(w S T + w P (T; T ) + w 3 C T P (T; T )) n ] nx kx n! = j! (k j)! (n k)! wj j) (n k) w(k w 3 E Q T k= j= h S j T (P (T; T )) (k j) C n k T (P (T; T )) n ki : Noe ha he log normal disribuion of S T, P (T; T ), C T, and P (T; T ) under he forward measure allows he compuaion of he las expecaion of Lemma from following ideniy: E [exp ( + Z)] = exp + where Z N (; ) : Deails are given in Appendix B. 3. Inverse gamma approximaion In his secion, we use he inverse gamma disribuion o approximae he sum of correlaed lognormal variables. This approximaion was rs used by Milevsky and Posner (998a, 998b) o price Asian and baske opions. In fac, a nie sum of correlaed lognormal variables converges asympoically o an inverse gamma variable. Under an inverse gamma disribuion for he underlying 6

7 baske, an European baske call opion has a closed-form soluion ha looks like a Black and Scholes (973) (B&S hereafer) formula: Vgamma B = P (; T ) (v) G ; G ; ; (6) where G ( j ; ) is he cumulaive funcion of a gamma disribuion wih parameers (; ). These parameers are deermined by maching he rs wo momens of he exac and he approximae disribuions o obain: = (v) (v) (v) (v) and = (v) (v) (v) : (7) (v) Mahemaical deails of he pricing formula are provided in Appendix C. 3. Edgeworh expansion around he lognormal disribuion We now presen an analyical approximaion based on a generalized Edgeworh expansion around he lognormal disribuion. This approach, inroduced by Jarrow and Rudd (98) in opion pricing, subsiues an unknown densiy funcion v () wih a Taylor-like expansion around an easy-o-use densiy funcion denoed a (). Noice, however, ha Edgeworh expansions usually lead o a funcion which is no a rue densiy funcion. Baron and Denis (95) derive some condiions on he hird and fourh momens of he unknown disribuion o guaranee ha he approximaion obained wih a runcaed Edgeworh expansion is posiive and unimodal. Moreover, Ju () poins ou ha he Edgeworh expansion may diverge for some parameer values, which consequenly can give incorrec prices for high volailiy and long mauriy opions. However, in his paper we do no have his problem in he applicaion of he Edgeworh expansion. Following Huynh (994) who uses his approach for baske opions, we will use an Edgeworh expansion of order 4 and we will mach he rs and second momens of he exac and he lognormal disribuions. Under his approximaion, an European baske call opion can be obained as a sor of Black and Scholes price adjused for he excess skewness and he excess kurosis from he lognormal densiy: where V B log normal = P (; T ) V 3 (v) 3 (a) 3! da ( ) dx + 4 (v) 4 (a) 4! d a ( ) dx ; (8a) V = (v) N (d ) N (d ) ; (8b) d = d + ; (8c) d = ln () ; (8d) = ln (v) ln (v) + (v) ; (8e) r = ln + (v) ( (v)) ; (8f) a () is he densiy funcion of a log-normal disribuion (see Equaion (9) in Appendix D), and N () represens he cumulaive funcion of he sandard normal disribuion. The hird and fourh 7

8 momens of he lognormal disribuion needed for he Edgeworh expansion depend only on he rs and second momens of he exac disribuion and are given by: 3 (a) = (v) 3 (v) and 4 (a) = (v)6 : (8g) (v)8 Deails abou he Edgeworh approximaion formula are given in Appendix D. 3.3 Johnson approximaion Johnson (949) proposes a family of densiy funcions, obained via a ransformaion of a sandard normal variable, ha can be used o approximae unknown disribuions. Le Z be a sandard normal variable and X be a random variable wih an unknown densiy funcion, Johnson (949) suggess he following ransformaions beween Z and X: X " Z = + ; (9a) X = " + Z ; (9b) where and are he shape parameers of he Johnson disribuion, is he scale parameer, " is he hreshold parameer, and () is one of he following Johnson funcions: L(x) = ln (x) ; (Lognormal sysem) U (x) = ln x + p x + ; (Unbounded sysem) x B (x) = ln. (Bounded sysem) x The choice of he sysem and ing parameers provides a grea exibiliy in adjusing he curve o mach he rs four momens of he unknown disribuion. We use he lognormal ( L ) and he unbounded ( U ) sysems ha are common in he lieraure. We apply he Hill, Hill and Holder (976) algorihm, based on he rue skewness and kurosis of he baske disribuion, o deermine which of he Johnson sysems ( L or U ) should be used in he approximaion. Unlike approximaions obained wih a runcaed Edgeworh expansion, approximaions based on Johnson (949) sysems correspond o rue densiy funcions wih a perfec mach of he rs four momens. Following Posner and Milevsky (999), we subsiue he unknown disribuion of he underlying baske wih he lognormal and he unbounded Johnson funcions where he sysem parameers are calculaed by maching he four momens. Under a Johnson densiy funcion, a European baske call opion can be priced as: V B Johnson = P (; T ) = P (; T ) Z + (x ) (x) dx Z + = P (; T ) (v) + = P (; T ) (v) + x (x) dx Z + Z KB Z KB 8 Z x Z x (x) dx (y) dy dx Z KB (x ) (x) dx (y) dy dx : ()

9 The hird line in he equaion is obained using an inegraion by pars. Milevsky and Posner (999) show ha he las double inegral, involving a Johnson densiy funcion, can be compued as follows:. Lognormal sysem L : X = " + exp Z Z KB Z x (y) dy dx = ( ") N. Unbounded sysem U : X = " + sinh Z Z KB Z x where q = + sinh " : 5 KB " + ln KB " exp N + ln (y) dy dx = ( ") N (q) + exp exp exp 4 Performance analysis of approximaions N q exp N : () q + ; (3) In his secion, we analyze he performance of he hree approximaions presened previously. We will use prices obained by Mone Carlo simulaions as benchmarks since here is no closed-form soluion for baske opions. Following Barraquand (995), we will apply a variance reducion echnique based on he maching of he rs and second momens. This will ensure ha sample mean and variance are equal o he heir heoreical counerpars. The Mone Carlo baske opion price combined wih he variance reducion echnique is given by: V B = N NX P (; T ) max Bi;T i= ; where B i;t = B i;t B p (v) b S + (v) ; B i;t is he ime T baske value obained wih sample pah i; and B = P m m i= B i;t and S b = q P m m i= B i;t B are respecively he sample baske mean and sandard deviaion and m is he number of simulaed pahs. Our performance sudy will be conduced wih wo analyses. In he rs sensiiviy analysis, we will compare he baske opion price obained wih he analyical approximaions o he Mone Carlo price obained wih,, pahs repeaed for di eren mauriies, di eren moneynesses and di eren levels for he baske volailiy. The second sensiiviy analysis is a more deailed analysis based on works done by Broadie and Deemple (996). I compues he opion prices over a wide range of parameers chosen randomly from a realisic se of values in order o generalise our 5 Noice ha sinh (x) = exp(x) exp( x) and hus sinh (x) = ln x + p x + = U (x): 9

10 previous resuls independenly of he model s parameers. For each combinaion, we compare he prices obained wih he approximaions and Mone Carlo simulaions. Table presens he se of parameers used in he rs analysis. These values are based on esimaions using real daa on gold prices, CAD/USD exchange rae and Canadian and American 3-monh zero-coupon bonds. Alhough we have posiive and negaive correlaions in he se of parameers, he volailiy of he baske will increase when individual asses volailiies increase. Table : Parameers used in he rs sensiiviy analysis Domesic risk-free rae, f (; ) = f for any.6 Foreign risk-free rae, f (; ) = f for any.5 Commodiy drif, s.5 Exchange rae drif, c.4 Commodiy volailiy, s.5 Exchange rae volailiy, c.6 Domesic insananeous forward rae volailiy, ;T = for any and T. Foreign insananeous forward rae volailiy, ;T = for any and T. Convenience yield volailiy,.3 Convenience yield mean reversion parameer,. Convenience yield long-run mean,.5 Commodiy and exchange rae correlaion, ;. Commodiy and domesic insananeous forward rae correlaion, ;3 -. Commodiy and foreign insananeous forward rae correlaion, ;4 -.5 Exchange rae and domesic insananeous forward rae correlaion, ;3.5 Exchange rae and foreign insananeous forward rae correlaion, ;4. Foreign and domesic insananeous forward raes correlaion, 3;4.85 Baske weighs (commodiy, domesic and foreign zero-coupon bonds).5;.5;.5 Iniial prices S ; C and are respecively $33; $/CAD.65;. Table presens sensiiviy analysis of he baske opion price wih respec o moneyness, which corresponds o he raio of he exercise price over he iniial value of he baske B ; and opion mauriy. The resuls show ha Edgeworh-lognormal and Johnson approximaions are much more accurae, wih relaive errors beween 6 and 3, han he inverse gamma approximaion wih a relaive error beween 4 and. 6 6 This resul is consisen wih Milevsky and Posner (998a) who showed ha he convergence resul for he inverse gamma works well when he risk-neural drif of he underlying di usion is negaive or when he correlaion marix decays quickly.

11 Table : Sensiiviy analysis w.r.. he moneyness and opion mauriy Moneyness Mone Sandard Inverse Gamma Lognormal Johnson Carlo Deviaion Approximaion Approximaion Approximaion Price Relaive Price Relaive Price Relaive error error error.75-year mauriy e *.3e e e e e e e e-3.94*.47e e e e-3.8* 9.e e e e-4.39* 4.69e e e-3 -year mauriy e *.59e e e e e e e e-3.795*.7e e e e-3.68* 6.6e e e e-4.46*.8e e e-3.5-year mauriy e-.66* 8.65e e e e-.465*.5e e e e- 8.84* 6.4e e e e e e e e * 4.77e e e-4 3-year mauriy e *.98e-.8.e e e * 4.e e e e * 4.53e e e e * 5.5e- 9.6.e e e * 6.e e e-4 The Mone Carlo benchmarks are based on 6 rajecories using momen maching as a variance reducion echnique. The esimaed sandard deviaions of he Mone Carlo prices are repored. The parameers used for he simulaion are provided in Table. The mauriy daes of he domesic and foreign bonds are respecively T = T + 9=36 and T = T + =36: Relaive errors are compued as e = V B (a) V B B =V ; where V B (a) and V B are he opion prices obained respecively wih he analyical approximaion and he Mone Carlo simulaion. The prices marked wih a sar (*) are signi canly di eren from heir Mone Carlo benchmark a a level of 5%.

12 Table 3 presens sensiiviy analysis of he baske opion price wih respec o he baske volailiy and opion mauriy, for in he money opions, B = :95. The average level volailiy corresponds o he values in Table, while high and low levels correspond respecively o an increase and a decrease of 5% in he volailiy values given in Table. Table 3: Sensiiviy analysis wih respec o he baske volailiy and opion mauriy Baske Mone Sandard Inverse Gamma Lognormal Johnson Volailiy Carlo Deviaion Approximaion Approximaion Approximaion Price Relaive Price Relaive Price Relaive error error error.75-year mauriy Low.6.63e e e e-5 Average e e e e-5 High e e e e-6 -year mauriy Low e e e e-5 Average e e e e-5 High e e e e-5.5-year mauriy Low e * 4.45e e e-6 Average e-.465*.4e e e-5 High e *.95e e e-5 3-year mauriy Low e * 7.93e e e-5 Average e * 4.e e e-5 High e * 7.7e * 5.5e e-4 The Mone Carlo benchmarks are based on 6 rajecories using momen maching as a variance reducion echnique. The esimaed sandard deviaions of he Mone Carlo prices are repored. The parameers used for he simulaion are provided in Table. The mauriy daes of he domesic and foreign bonds are respecively T = T + 9=36 and T = T + =36 and he moneyness is =B = :95 Relaive errors are compued as e = V B (a) V B B =V ; where V B (a) and V B are he opion prices obained respecively wih he analyical approximaion and he Mone Carlo simulaion. The prices marked wih a sar (*) are signi canly di eren from heir Mone Carlo benchmark a a level of 5%. The ndings are similar o hose in Table. The relaive errors presened in Table 3 are very low wih a magniude beween 6 and. For low volailiy levels, Edgeworh-lognormal and Johnson approximae prices and Mone Carlo prices are very close. As for he inverse gamma approximaion, our resuls show ha i is much less accurae wih relaive errors beween 4 and. Noice ha he accuracy of he hree approximaions decreases for longer mauriies and higher

13 volailiies, which suppors he ndings in Ju (). 7 prices increase wih mauriy and volailiy. Moreover, for all approximaions, opion The previous analysis is only local. Our ndings depend on he se of parameers used and may change if we modify hem. To con rm our conclusions, a more carefully designed numerical sudy where he parameers are randomly chosen is conduced. Following Broadie and Deemple (996), he signi can model parameers are chosen randomly from coninuous or discree uniform disribuions. These uniform disribuions are based on he esimaion of he model parameers using real daa. The correlaion beween he commodiy and he exchange rae is expeced o be posiive, so we assume ha [:; :55] : Based on Schwarz (997), he convenience yield long-run mean is posiive for gold and he mean reversion parameer is small and less han ; we assume hus ha [:5; :5] and [:5; :8]. All paramaer disribuions used in he pricing are presened in Table 4. Table 4: Parameers disribuions Domesic risk free rae, f (; ) = f for any U(.;.8) Foreign risk free rae, f (; ) = f for any U(.,.8) Commodiy drif, s U(.5,.35) Exchange rae drif, c.4 Commodiy volailiy, s U(.,.35) Exchange rae volailiy, c U(.,.5) Domesic insananeous forward rae volailiy, ;T = for any and T U(.,.6) Foreign insananeous forward rae volailiy, ;T = for any and T U(.,.6) Convenience yield volailiy, U(.,.4) Convenience yield mean reversion parameer, U(.5,.8) Convenience yield long-run mean, U(.5,.5) Commodiy and exchange rae correlaion, ; U(.,.55) Commodiy and domesic insananeous forward rae correlaion, ;3 U(.5,.5) Commodiy and foreign insananeous forward rae correlaion, ;4 U(.5,.5) Exchange rae and domesic insananeous forward rae correlaion, ;3 U(.35,.35) Exchange rae and foreign insananeous forward rae correlaion, ;4 U(.35,.35) Foreign and domesic insananeous forward raes correlaion, 3;4 U(.5,.9) Opion mauriy (in years) U{.83;.5;.5;.75; ;.5; ; 3} Moneyness U{.8;.9;.95; ;.5;.;.} Baske weighs (commodiy domesic and foreign zero-coupon bonds), U ; 4 4 ; 4 4 ; 4 Iniial prices S ; C and are respecively $33; $/CAD.65;. 4 We compare he hree previous analyical approximaions and Mone Carlo simulaion combined wih he momen maching echnique for 5 random ses of parameers. Only 474 ses of parameers provided posiive de nie correlaion marices. We also removed all baske opions 7 Ju () proposes an analyical approximaion o price Asian and baske opions based on a Taylor expansion of he raio of he characerisic funcion of he average of lognormal variables o ha of he approximaing lognormal random variable around zero volailiy. 3

14 prices wih a value lower han 5 cens. We obained 4347 ses of parameers. Firs, we examine he accuracy of each approximaion by calculaing is roo mean square error (RMSE). Second, we calculae he maximum relaive error (MRE) for each approximaion o examine he wors case scenario. More precisely, we de ne, v RMSE = u n MRE = max i nx i= V B i (a) V B i V B i V B i (a) V B i where n is he number of di eren ses of paramaers, ha is, Mone Carlo prices are obained wih ; ; pahs combined wih he momen maching echnique. V B i ;! Table 5: RMSE and MRE for he hree approximaions Inverse Gamma Lognormal Johnson Approximaion Approximaion Approximaion RMSE 9.68%.5%.56% MRE 79.47% 3.6% 4.% The resuls in Table 5 con rm hose obained wih he rs sensiiviy analysis (Tables and 3) and show ha he Edgeworh-lognormal and Johnson approximaions are much more accurae han he inverse gamma approximaion. I is also found ha, for he Edgeworh-lognormal and Johnson approximaions, a very small proporion of opions, :3% and :35% respecively, have relaive errors above 5% while for he inverse gamma approximaion, a larger proporion, 7:74%, of opions have a relaive error above 5%. A more deailed look a he resuls shows ha ou-of-he-money and high volailiy opions have he larges relaive errors, which con rms our ndings in he rs sensiiviy analysis. However, since he ou-of-he-money opions have small prices, he augmenaion of he relaive errors in hese cases may be aribued o he small denominaors. Figures, and 3 presen respecively he hisograms of he relaive errors of he inverse gamma, he Edgeworh-lognormal and Johnson approximaions. Table 6 shows a summary of descripive saisics of he relaive errors. 4

15 Figure : Hisogram of relaive errors of he inverse gamma approximaion 35 Relaive Errors Hisogram for he Inverse Gamma Disribuion Number of Observaions Relaive Errors X-axis represens relaive errors Y-axis represens he number of observaions 5

16 Figure : Hisogram of relaive errors of he Edgeworh approximaion 45 Relaive Errors Hisogram for he Edgeworh-Lognormal Disribuion 4 Number of Observaions Relaive Errors X-axis represens relaive errors Y-axis represens he number of observaions Figure 3: Hisogram of relaive errors of Johnson approximaion 45 Relaive Errors Hisogram for he Johnson Disribuion 4 Number of Observaions Relaive Errors X-axis represens relaive errors Y-axis represens he number of observaions 6

17 Table 6: Descripive saisics of relaive errors Inverse Gamma Lognormal Johnson Approximaion Approximaion Approximaion Maximum Minimum Mean Median.36.. Sandard Deviaion Nb of observaions The hisograms show ha for he Edgeworh-lognormal and Johnson approximaions, 99% of parameers ses (43 ou of 4347) have relaive errors less han %: This demonsraes ha hese approximaions are very accurae and ha hey give prices very close o Mone Carlo benchmarks. The inverse gamma approximaion has 9% (4 ou of 4347) cases where he relaive errors are less han %. We reach he same resuls by using larger and more general inervals for he parameers disribuions. To conclude his secion regarding he pricing of heeregeneous baske opions, we show ha Edgeworh expansion around he lognormal disribuion a order 4 and he Johnson disribuion are equally accurae and very accepable for praciionners. However, we sugges he use of he Edgeworh-lognormal disribuion for wo reasons: rs, i is slighly more accurae, and second, he algorihm o calibrae Johnson disribuions may no converge in a few cases, which can lead o mispriced opions. 5 Hedging raios The analyical approximaions allow for deriving he opion price in a funcional form which can also provide analyical expressions for he sensiiviies wih respec o he underlying parameers, such as delas, vegas and hea, known as he hedging raios or he Greeks. However, due o he complexiy of he momens involved in our analyical approximaions, deriving he hedging raios analyically is beyond he scope of his aricle. Insead, we can compue hem numerically as follows: Raio B App = V B(") App " V B App ; (4) where " > is a small number, V B(") App and VApp B are respecively he approximae "-disurbed and non-disurbed opion prices. As an applicaion, we compue numerically he dela wih respec o he commodiy price for di eren values of ". We base our calculaions on he parameers in Table. We consider an in-he-money call baske opion KB B = :95 wih a 9-monh mauriy. The delas are presened in Table 7. 7

18 Table 7: Baske opion Dela w.r.. he commodiy price Commodiy Inverse Gamma Lognormal Johnson Price Approximaion Approximaion Approximaion Price Dela* Price Dela* Price Dela* $ $ $ $ $ * Dela corresponds o he sensiiviy wih respec o he commodiy price and is given by Equaion (4). VApp B, presened in he rs row of he able, is he analyical baske opion price obained for a commodiy price of $33: Table 7 shows ha, for he hree approximaions, he dela is very sable over di eren values of ". Indeed, an increase of $ in he commodiy price leads o an increase of approximaely $: in he opion price. The posiiveness of dela is expeced bu is value depends on he se of parameers used. Following he same procedure, one can compue he oher sensiiviies wih respec o oher parameers of ineres. 6 Conclusion Firms can use baske opions o hedge heir exposure o di eren risks, such as commodiy risk, ineres rae risk and exchange rae risk. However, pricing his kind of opions is no an easy ask since no closed-form soluion can be derived for he baske densiy funcion. Consequenly, a sandard pricing formula such as Black and Scholes canno be derived. The main conribuion of his aricle is he comparison of he performance of hree analyical approximaions o price a heeregeneous baske opion, consising of a commodiy, a domesic and a foreign zero-coupon bonds, when ineres raes are sochasic. The hree approximaions used are he inverse gamma proposed in Milevsky and Posner (998a, 998b), he Edgeworh expansion around he lognormal disribuion as well as Johnson disribuion developped in Posner and Milevsky (999). In order o assess and compare he accuracy of he approximaions, we use wo analyses. The rs one is a local sensiiviy analysis where he parameers of he model are xed arbirarily. Our ndings show ha boh he Edgeworh-lognormal and Jonhson approximaions are very accurae while he inverse gamma approximaion is much less accurae. These resuls are con rmed wih he second analysis where 5 ses of parameers are chosen randomly from di eren uniform disribuions. I is also found ha he pricing relaive errors, compued wih Mone Carlo benchmark prices, are small and of an accepable magniude for praciionners. The accuracy of all approximaions deerioraes for ou-of-he-money as well as for high volailiy opions. This aricle considers only plain vanilla baske opions. However, exending he approach followed here o more exoic baske opions is a promising area for fuure research. 8

19 Appendices A Derivaion of he forward measure In his appendix, we will derive he model under he T forward measure. Le A i be he i h row of he marix A; where A = [a ij ] i;j=f;;3;4g is he Cholesky decomposiion of he correlaion marix of he fourdimensional P Brownian moions W = W () ; W () ; W (3) ; W (4) ; and e B corresponds o he vecor of independen Brownian moions under Q. The SDEs sais ed by all underlying asses under Q can be wrien as: 8 ds = S h (r ) d + s A d e B i (5a) d = ( ( s r ) )d + A db e (5b) h s dc = C (r r ) d + c A db e i (5c) dp (; T ) = P (; T ) hr d ;T A 3 db e i (5d) h dp (; T ) = P (; T ) r + ;T c ;4 d ;T A 4 db e i (5e) R T The risk-neural measure Q corresponds o a numeraire equal o he domesic bank accoun exp r u du ; while he T forward measure Q T corresponds o a numeraire equal o he domesic zero-coupon bond wih mauriy dae T, P (; T ). Using Girsanov heorem, we have ha: d b B = d e B + ;T A 3d; where b B are independen Brownian moions under Q T. The correlaion srucure is he same under Q and Q T ; and he previous SDEs can be rewrien as: h ds = S r 3 s ;T d + s dw c i () d = ( ( s r ) 3 ;T )d + dw c() h s dc = C r r 3 c ;T d + c dw c i () h dp (; T ) = P (; T ) r + ;T ;T d ;T dw c i (3) h dp (; T ) = P (; T ) r + ;T c ;4 + 3;4 ;T ;T d ;T d c W (4) where W c = cw () ; W c() ; W c(3) ; W c (4) = AB: b The srong soluion exiss and is given by he sysem of equaions (3). i (6a) (6b) (6c) (6d) (6e) B Derivaion of momens This appendix gives he deailed calculaion of he rs four momens of he baske value a mauriy T under he T forward measure Q T. Using he srong soluion of he model under he forward-measure Q T, 8 The deails of he compuaion are available from he auhors upon reques. 9

20 one can wrie where Therefore, wih where S (T ) = P (T ) = CP (T ) = S T = S exp S (T ) + P (T; T ) = exp P (T ) + 4X i= C T P (T; T ) = C exp CP (T ) R T R T 4X i= 4X i= S;i () d c W (i) P;i () d c W (i)!! CP ;i () d c W (i) e (T u) s f (; u) du e T T + s s e T s s + s u;t du + s + ;3 R T u;t T S; () = s f (; u) du f (; u) du S;3 () = ;T s P;3 () = ;T ;T ; CP ; () = c ; CP ;3 () = ;T ; CP ;4 () = ;T ; n = R T e (T u) R u v;u v;t v;u dv du e (T u) s du u;t c ;3 u;t du + c ;4 e (T ) ; u;t du; Z f T (; u) du u;t + u;t du u;t du + 3;4 u;t u;t du; e (T v) ;v dv; S; () = S;4 () = P; () = P; () = P;4 () = CP ; () = ; nx kx k= j= n! j! (k j)! (n k)! wj j) w(k E Q T (n k) w 3 E Q T h S j T (P (T; T )) k j C n k T (P (T; T )) n ki " = S j Cn k (P (; T )) n k E Q T exp (T ) + = S j Cn k (P (; T )) n k exp (T ) +! ; c T ; h S j T (P (T; T )) k j C n k T (P (T; T )) n ki 4X 4X i= i= `= 4X i;` (T ) = j S (T ) + (k j) P (T ) + (n k) P (T ) i () d c W (i)!# i () ` () d i () = j S:i () + (k j) P:i () + (n k) P :i (), i f; ; 3; 4g :!

21 Table 8 presens he heoreical rs four momens, calculaed as explained previously, and hose obained by Mone Carlo simulaion. This ensures ha our heoreical formulas give he exac momens. Table 8: Comparison of heoreical and simulaed momens Order Theoreical Momens Simulaed Momens e e e e+8 Mean Variance C Inverse gamma approximaion This appendix shows how we obain he pricing formula using he inverse gamma approximaion. The densiy funcion of a gamma random variable X of parameers (; ) ; X s G (; ) ; is given by: x x exp g (x j; ) = ; x () where > ; > and () is he Gamma funcion. Proposiion Le X be a gamma random variable wih parameers (; ) : Then, he random variable Y = X ; follows an inverse gamma disribuion, Y s G R (; ) ; and is densiy funcion is given by: y g R (y j; ) = exp y g y j; = y + ; () y > ; ; > : (7) Proposiion The non-cenered momens of he random variable Y s G R (; ) are given by: n (g R ) = n ; n = ; ; 3; ::: (8) ( ) ( ) ::: ( n) We price he baske opion by approximaing he sum of lognormal variables by an inverse gamma disribuion. We mach he wo rs momens (v) = (g R ) = ( ) and (v) = (g R ) = ( ) ( ) ; o ge he wo parameers of he inverse gamma densiy given a line (7). Using he inverse gamma densiy,

22 he opion price P (; T ) R + max (x ; ) v (x) dx is approximaed by: Z + Vgamma B = P (; T ) (x ) g R (xj ; ) dx Z + = P (; T ) (x ) x g x ; dx Z = P (; T ) g (yj ; ) dy y = P (; T ) leading o Equaion (6). ( ) Z g (yj ; ) dy Z since y g (yj ; ) = g (yj ; ) ( ) = P (; T ) G ( ) ; from he change of variable y = x! G g (yj ; ) dy ; D Edgeworh-lognormal expansion This appendix shows how we obain he pricing formula using an Edgeworh expansion around he lognormal disribuion. Maching he rs wo momens, he lognormal densiy used is given by a (x) = p! x exp ln x where and are de ned a lines (8e) and (8f) respecively. Following Jarrow and Rudd (98), he unkown baske densiy funcion can be approximaed by: v (x) = a (x) + (v) (a) d a (x)! + ( 4 (v) 4 (a)) + 3 ( (v) (a)) 4! 3 (v) 3 (a) dx 3! d 3 a (K) dx 3 (9) d 4 a (x) dx 4 + (x) ; () where (x) is an error erm and i (h) ; i = f; ; 3; 4g are he rs four cumulans of he densiy funcion h = fv; ag de ned by he sysem of Equaions (5). Jarrow and Rudd (98) sae ha, in general, here is no bound on he error erm resuling from an Edgeworh expansion. Consequenly, he error does no necessarily decrease wih he expansion s order. Given ha he wo rs momens are he same for he rue densiy and for he approximaed densiy, Equaion () becomes v (x) = a (x) 3 (v) 3 (a) d 3 a (K) 3! dx (v) 4 (a) 4! d 4 a (x) dx 4 + (x) : () The baske opion price P (; T ) R + max (x ; ) v (x) dx can hus be approximaed by: Z + Vlog B normal = P (; T ) (x ) a (x) Using he fac ha Z + (x ) dj a dj (x) dx = dxj 3 (v) 3 (a) d 3 a (x) 3! dx (v) 4 (a) 4! dx j B a ( ) for j ; d 4 a (x) dx 4 dx:

23 we obain V B log normal = P (; T ) Z + (x ) a (x) dx 3 (v) 3 (a) da 3! dx () + 4 (v) 4 (a) d a 4! dx () : () Noice ha he rs inegral in Equaion () is very similar o a Black and Scholes price. References [] Barraquand, J., 995. "Numerical Valuaion of High Dimensional Mulivariae European Securiies". Managemen Science, 4,, [] Baron, D. E. and Dennis, K. E. R., 95. "The Condiions Under Which Gram-Charlier and Edgeworh Curves are Posiive de nie and Unimodal," Biomerika, 39, [3] Baxer, M. and Rennie, A., 996. "Financial Calculus, An Inroducion o Derivaive Pricing". Cambridge, New York. [4] Black, F. and Scholes, M., 973. "The Pricing of Opions and Corporae Liabiliies". The Journal of Poliical Economy, 8, 3, [5] Brigo, D., Mercurio, F., Rapisarda, F. and Scoi, R., 4. "Approximaed Momen-Maching for Baske-opions Pricing". Quaniaive Finance, 4, -6. [6] Broadie, M. and Deemple, J., 996. "American Opion Valuaion: New Bounds, Approximaions and a Comparison of Exising Mehods". The Review of Financial Sudies, 9, 4, -5. [7] Carmona, R. and Durrleman, V., 3. "Pricing and Hedging Mulivariae Coningen Claims", Working paper, Princeon Universiy. [8] Casellacci, G. and Siclari, M.J., 3. "Asian Baske Spreads and Oher Exoic Averaging Opions". Energy Power Risk Managemen. [9] Curran, M., 994. "Valuing Asian and Porfolio Opions by Condiioning on he Geomeric Mean Price". Managemen Science, 4,, [] Dahl, L.O.,. "Valuaion of European Call Opions on Muliple Underlying Asses by Using a Quasi- Mone Carlo Mehod. A Case wih Baskes from Oslo Sock Exchange". In Proceedings AFIR,, [] Dahl, L.O. and Benh, F.E.,. "Valuaion of Asian Baske Opions wih Quasi-Mone Carlo Techniques and Singular Value Decomposiion". Pure Mahemaics, 5, -. [] Daey, J.Y., Gauhier, G. and Simonao, J.G., 3. "The Performance of Analyical Approximaions for he Compuaion of Asian-Quano-Baske Opion Prices". The Mulinaional Finance Journal, 7,, [3] Deelsra, G., Liinev, J. and Vanmaele, M., 4. "Pricing of Arihmeic Baske Opions by Condiioning". Insurance: Mahemaics and Economics, 34,, -3. [4] Dionne, G., Gauhier, G. and Ouerani, N., 6. "Baske Opions on Heerogeneous Underlying Asses". Working paper, HEC Monréal (forhcoming). 3

24 [5] Flamouris, D. and Giamouridis, D., 4. "Valuing Exoic Derivaives wih Jump Di usions: The Case of Baske Opions". Working Paper, Ahens Universiy of Economics and Business. [6] Genle, D., 993. "Baske Weaving". Risk, 6, 6, 5-5. [7] Harrison, J.M. and Pliska, S.R., 98. "Maringales and Sochasic Inegrals in he Theory of Coninuous Trading". The Sochasic Processes and Their Applicaions,, 6-7. [8] Hill, I., Hill, R. and Holder, R., 976. "Fiing Johnson Curves by Momens". Applied Saisics, 5,, 8-9. [9] Huynh, C.B., 994. "Back o Baskes". Risk, 5, [] Jarrow, R. and Rudd, A., 98. "Approximae Opion Valuaion for Arbirary Sochasic Processes". Journal of Financial Economics,, 3, [] Johnson, N.L., 949. "Sysems of Frequency Curves Generaed by Mehods of Translaion". Biomerika, 36, [] Ju, N.,. "Pricing Asian and Baske Opions via Taylor Expansion". The Journal of Compuaional Finance, 5, 3, [3] Laurence, P. and Wang, T.H., 4. "Wha s a Baske Worh?". Risk, 7,, [4] Milevsky, M.A. and Posner, S.E., 998a. "Asian Opions, he Sum of Lognormals and he Reciprocal Gamma Disribuion". The Journal of Financial and Quaniaive Analysis, 33, 3, 49-4 [5] Milevsky, M.A. and Posner, S.E., 998b. "A Closed-Form Approximaion for Valuing Baske Opions". The Journal of Derivaives, 5, 4, [6] Milevsky, M.A. and Posner, S.E., 999. "Anoher Momen for he Average Opion". Derivaives Quarerly, [7] Pellizzari, P.,. "E cien Mone Carlo Pricing of European Opions Using Mean Value Conrol Variaes". Decisions in Economics and Finance, 4, 7-6 [8] Posner, S.E. and Milevsky, M.A., 999. "Valuing Exoic Opions by Approximaing he SPD wih Higher Momens". The Journal of Financial Engineering, 7,, 9-5. [9] Rubinsein, M., 994. "Reurn o Oz". Risk, 7, [3] Vanmaele, M., Deelsra, G. and Liinev, J., 4. "Approximaion of Sop-Loss Premiums Involving Sums of Lognormals by Condiioning on Two Variables". Insurance: Mahemaics and Economics, 35, [3] Vyncke, D., Goovaers, M. and Dhaene, J., 3. "An Accurae Analyical Approximaion for he Price of a European-Syle Arihmeic Asian Opion", Working paper, K.U. Leuven, Dep. of Applied Economics, Belgium [3] Wan, H.,. "Pricing American-Syle Baske Opions by Implied Binomial Tree", Working paper, Haas School of Business, Universiy of California a Berkeley 4