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1 ISSN (Online) Working Paper Series Pricing and Hedging Baske Opions wih Exac Momen Maching Tommaso Palea, Aruro Leccadio and Radu Tunaru Ken Business School 1 Working Paper No. 294 February 214

2 Pricing and Hedging Baske Opions wih Exac Momen Maching Tommaso Palea a,, Aruro Leccadio b, Radu Tunaru a a Business School, Universiy of Ken, Park Wood Road, Canerbury CT2 7PE, UK, b Diparimeno di Economia, Saisica e Finanza, Universià della Calabria, Pone Bucci cubo 3C, Rende (CS), 873, Ialy Absrac Theoreical models applied o opion pricing should ake ino accoun he empirical characerisics of he underlying financial ime series. In his paper, we show how o price baske opions when asses follow a shifed log-normal process wih jumps capable of accommodaing negaive skewness. Our echnique is based on he Hermie polynomial expansion ha can mach exacly he firs m momens of he model implied-probabiliy disribuion. This mehod is shown o provide superior resuls for baske opions no only wih respec o pricing bu also for hedging. Keywords: Baske opions, Shifed log-normal jump process, Hermie polynomials, Negaive skewness, Opion pricing and hedging JEL: C18, C63, G13, G19 Corresponding auhor: URL:.palea@ken.ac.uk (Tommaso Palea) 1

3 1. Inroducion Baske opions are coningen claims on a group of asses such as equiies, commodiies, currencies and even oher vanilla derivaives. Spread opions can be concepualised as baske opions whose payoffs depend on he price differenial of wo asses. Baske opions are a subclass of exoic opions commonly raded over-he-couner in order o hedge away exposure o correlaion or conagion risk. Hedge-funds also use hem for invesmen purposes, o combine diversificaion wih leveraging. Spread opions are heavily raded on he commodiy markes, in paricular on energy markes, where several final producs are indusrially produced from he same raw maerial. From a modelling poin of view, he framework ough o be mulidimensional since baskes of 15 o 3 asses are frequenly raded. Many pricing models ha seem o work well for single asses canno be easily expanded o a mulidimensional se-up, mainly due o compuaional difficulies. Hence, in order o circumven hese difficulies, praciioners resor o classic mulidimensional geomeric Brownian moion ype models which can be easily implemened. However, by doing so, he empirical characerisics of he asses in he baske are simply ignored. In paricular, negaive skewness, which is well known o characerize equiies, canno be capured properly by hese simple models which can produce a limied range of values for skewness. Recenly, Borovkova and Permana (27) and Borovkova e al. (27, 212) have proposed a new mehodology ha can incorporae negaive skewness while sill reaining analyical racabiliy, under a shifed log-normal disribuion, by considering he enire baske as one single asse. This srong assumpion allowes he derivaion of closed-form formulae for opion pricing. Ideally, one would like he bes of boh worlds, realisic modelling and precise calculaions. In his paper, we presen a general compuaional soluion o he problem of mulidimensional models which lack closed-form formulae or models ha require burdensome numerical procedures. The shifed lognormal process wih jumps exemplifies he problem encounered wih pricing baske opions. On one hand, his disribuion is very useful o follow he dynamics of one asse, bu on he oher hand expanding his se-up o a baske of asses leads o severe compuaional problems. We circumven his problem by employing he Hermie polynomial expansion which is maching exacly he firs m momens of he model implied probabiliy disribuion. Hence, he only prerequisie of our mehod is o be able o calculae he momens of he baske in closed form. In addiion, he same echnique can 2

4 be applied for any oher similar modelling siuaions for oher models. Furhermore, our mehodology is applicable o he siuaion when some asses in he baske follow one diffusion model and oher asses follow a differen diffusion model. The aricle is srucured as follows. In Secion 2, we briefly review he mehods proposed for pricing baske and spread opions, focusing on approximaion echniques. Secion 3 conains a descripion of he coninuous-ime models we employed here. Our new mehodology is discussed in Secion 4 and he empirical resuls are presened in Secion 5. The final secion concludes. 2. Relaed Lieraure The number of papers covering baske opions and, in paricular, spread opions has increased considerably in he las hree decades. Margrabe (1978) was he firs o develop an exac formula for European spread opions when he wo asses are assumed o follow a geomeric Brownian moion. Carmona and Durrleman (23) presened an exensive lieraure review on pricing mehods for spread opions as well as inroducing a new mehod. The mehods used o price baske opions can be classified ino analyical, purely numerical and a hybrid semi-analyical class based on various expansions and momen maching echniques. Our mehod belongs o he las caegory. By analogy o early papers on pricing Asian opions, Genle (1993) proposed pricing baske opions 1 by approximaing he arihmeic weighed average wih is geomerical-average counerpar so ha a Black-Scholes ype formula could be applied. Korn and Zeyun (213) improved his approximaion using he fac ha, if he spo prices of asses in he baske are shifed by a large scalar consan C, heir arihmeic and geomeric means converge asympoically. They consider log-normally disribued asses and approximae he C-shifed disribuion by sandard log-normal disribuions. Kirk (1995) developed a echnique for pricing a spread opion by coupling he asse wih negaive weigh wih he srike price, considering heir combinaion as one asse having a shifed disribuion and hen employing he Margrabe (1978) formula for exchanging wo asses. This shif assumpion corresponds o a linear approximaion of he exercise boundary 2. The mehod in Li e al. 1 In ha paper i is assumed ha all asses in he baske have posiive weighs. 2 The exercise boundary is he minimal sandardized log-price of he firs asse ha makes he spread opion in-he-money as a funcion of he sandardized log-price of he 3

5 (28) can be considered as an exension of Kirk (1995). They derived a closed-form pricing formula for spread opions by applying a quadraic Taylor expansion of he exercise boundary. These resuls were furher exended by Li e al. (21) o he case of N asses wih posiive and negaive weighs. Venkaramanan and Alexander (211) and Alexander and Venkaramanan (212) priced spread opions and more general muli-asse opions (baske and rainbow opions) using a porfolio of compound exchange opions (CEO). Their idea was o uilise exac replicaing porfolios and hen approximae he formulae o price he CEOs. Remarkably, Venkaramanan and Alexander (211) derived an analyical formula for American spread opions using he early exercise premium approach proposed in Kim (199). Bjerksund and Sensland (211) also priced spread opions by direc use of he implied exercise boundary in Kirk (1995). When analyical formulae are difficul o find under a paricular model, i is common, in he finance indusry, o resor o Mone Carlo (MC) mehods. Conrol variae echniques for pricing baske opions are described in Pellizzari (21) and Korn and Zeyun (213). Barraquand (1995) advanced a very general framework o price mulidimensional coningen claims by Mone Carlo simulaion and quadraic re-sampling. Mone Carlo simulaion was also successfully used o price American syle baske opions by Barraquand and Marineau (1995), Longsaff and Schwarz (21) and Broadie and Glasserman (24). While Mone Carlo mehods offer a feasible soluion, he compuaional cos may be oo high even for sandard-size baskes commonly raded on he financial markes. Hence, he bulk of he lieraure on baske opion pricing graviaes around approximaion mehods ha circumven he numerical problems generaed by he high-dimensionaliy of baske models. A ypical example is he research by Li (2) who employed an Edgeworh expansion of a four-parameer skewed generalized- disribuion. Edgeworh series expansions were proposed firs by Jarrow and Rudd (1982) and Turnbull and Wakeman (1991) o price European baske opions and arihmeic Asian opions respecively. Rubinsein (1998) combined an Edgeworh expansion and a binomial ree o price American-syle opion wih pre-specified skewness and kurosis. This mehod has wo disadvanages. Firsly, he maching of skewness and kurosis is no exac given ha a rescaling of probabiliy is second asse. 4

6 necessary. Secondly, no all combinaions of skewness and kurosis can be mached because negaive probabiliies and muli-modal disribuion may resul. Levy (1992) approximaed he disribuion of a baske by maching is firs wo momens wih he momens of a log-normal densiy funcion, and consequenly a Black-Scholes pricing formula could be employed. Oher works improved he log-normal approximaion allowing for improved skewness and kurosis calibraion. The displaced diffusion inroduced by Rubinsein (1981) considers he shifed baske value as being log-normally disribued. Borovkova e al. (27), henceforh BPW, proposed a generalized log-normal approach ha is superior o he model in Rubinsein (1981) because i allows disribuions of a baske o cover negaive values and negaive skewness. Zhou and Wang (28) advocaed a mehod similar o ha of BP W, selecing he log-exended-skew-normal as he approximaing disribuion. They obained a Black-Scholes ype pricing formula where he sandard exended-skew-normal cumulaive disribuion funcion replaces he normal one. Borovkova e al. (212) exended his mehod o price Americansyle baske opions via a one-dimensional binomial ree. In an ineresing applicaion, Borovkova and Permana (27) adaped he mehod described in BP W o price Asian baske opions. Milevsky and Posner (1998) used he reciprocal gamma disribuion o approximae a posiively weighed sum of correlaed log-normal random variables. Maching he firs wo momens of he baske, hey priced European baske opions by a Black-Scholes ype formula where he normal cumulaive funcion is subsiued by he cumulaive disribuion funcion of he gamma disribuion. This mehod reurns good resuls only when he baske has a decaying correlaion srucure (similar o he one for Asian opion). Posner and Milevsky (1998) derived a closed-form pricing formula by using wo disribuions from he Johnson sysem of disribuions ha mach he firs four momens of he baske value. Asian and baske opions prices were calculaed by Ju (22) using he Taylor s expansion for he raio of he characerisic funcion of he value of he baske a mauriy o ha of he approximaing log-normal random variable. While he lieraure on pricing baske opions is large here is sparse research on calculaing he hedging parameers for baske opions. Hurd and Zhou (21) price spread opions for wo or more asses and also derive he Greek parameers by using fas Fourier ransform. The only assumpion for he underlying asse price processes is ha he characerisic funcion of he join reurn is known analyically. 5

7 3. The Modeling Framework In his paper, we consider a new process for asse prices: he shifed jumpdiffusion process. We firsly describe he sandard jump-diffusion model in Secion 3.1 ha provides he plaform for designing he shifed jump-diffusion model in Secion Jump-diffusion Model Consider he filered probabiliy space 3 (Ω, F, (F ) T, P). Le us define, on his space, he financial marke consising of Υ asses, S (i) for any i = 1,, Υ, wih dynamics given by ds (i) = (α i β i λ i )S (i) d + S (i) and he bank accoun n w j=1 γ ij dw (j) + S (i) dq (i), i = 1,, Υ (3.1) dm = rm d (3.2) ha can be used o borrow and deposi money wih coninuously compounded ineres rae r, assumed consan over ime. Equaion (3.1) describes a jump-diffusion process where α i is he expeced { } rae of reurn on he asse i, W (j) are n w muually independen Wiener { } processes, Q (i) are independen compound Poisson processes formed { } from some underlying Poisson processes wih inensiy λ i and Y (i) j N (i) represening he jump ampliude of he j-h jump of N (i) for any for any i = 1,, Υ are independen and i = 1,, Υ. The jumps Y (i) j idenically disribued random variables wih probabiliy densiy funcion f (i) (y) : [ 1, + ) [, 1] and expeced value under he physical measure β i = E[Y (i) ] = + 1 yf (i) (y)dy. Moreover, jumps for differen asses are independen. Applying sandard Io s rule for jump processes (see Shreve, 24, Chap ), i is possible o derive a closed-form soluion for he SDEs in (3.1) 3 The conens and noaion in his subsecion benefi from (Shreve, 24, chap. 11.5). 6

8 as: S (i) = S (i) e (α i β i λ i 1 nw 2 j=1 γij)+ 2 N l (i) nw j=1 γ ij W (j) l=1 (Y (i) j + 1), i = 1,, Υ. (3.3) The marke given by (3.1) and (3.2) is arbirage free if and only if here exiss θ = [θ 1,, θ nw ], β = [ β1,, β Υ ] and λ = [ λ 1,, λ Υ ] solving he sysem of marke price of risk equaions α i β i λ i r = n w j=1 γ ij θ j β i λi, i = 1,, Υ. (3.4) The soluion o (3.4) is, in general, no unique. Neverheless, we assume ha one soluion of he sysem (3.4) is seleced 4 and a pricing measure P is fixed 5. Under he P-measure, { for asse } i-h in he baske, we sill have he compound Poisson processes Q (i), he underlying Poisson process { } N (i) and he jumps Y (i) j bu now he inensiy of he Poisson process { } N (i) is λ i and β i = Ẽ[Y (i) ] = + y f (i) (y)dy. One way o model he 1 size of he jumps is aking, for each asse, jumps iid log-normally disribued 6 (i) (i) such ha Ẽ[log(Y j + 1)] = η i and Ṽ ar[log(y j + 1)] = υi 2. The risk-neural P-dynamics of he asses composing he baske can be described as: ds (i) = (r β i λi )S (i) d + S (i) { } (i) where W measure P. n w j=1 γ ij d W (j) + S (i) dq (i), i = 1,, Υ (3.5) are independen Wiener processes under he maringale 4 There is a large lieraure devoed o he issue of selecing a pricing measure. For a review, see Frielli (2) and references wihin. 5 Henceforh, E and Ẽ are used o indicae he expecaion operaors under he physical measure P and under he risk-neural measure P, respecively. 6 When we impose a log-normal disribuion for Y (i) j + 1, we implicily assume ha he sysem of equaions in (3.4) has a soluion. Furhermore, any oher disribuion f (i) (y) : R + [, 1] could have been chosen, if i leads o a feasible sysem. 7

9 The soluions o (3.5) can be derived in he following convenien closedform: S (i) = S (i) e (r β i λi 1 nw 2 j=1 γij)+ 2 N nw (j) (i) j=1 γ ij W l=1 (Y (i) l + 1), i = 1,, Υ. (3.6) 3.2. Shifed jump-diffusion Model From a modelling poin of view, i would be more appropriae o use models ha are capable of generaing negaive skewness reflecing he empirical evidence in equiy markes. One such flexible model is he generalized GBM process in Borovkova e al. (27). Here, we exend ha model o include jumps, hus obaining a jump-diffusion process for he displaced or shifed asse value: ( d b i S (i) ) δ (i) ( = (α i β i λ i ) b i S (i) + ) ( δ (i) d + b i S (i) δ (i) ) n w j=1 γ ij dw (j) + ( ) b i S (i) δ (i) dq (i), i = 1,, Υ. (3.7) In (3.7), δ (i) is he shif applied o S (i) a ime and b i { 1, 1}. We assume ha b i is negaive when he asse price assumes values in (, δ (i) ) and posiive when he range for he asse price is (δ (i), ). The shif δ (i) is assumed o follow equaion dδ (i) = rδ (i) d, wih δ (i) R and, consequenly, represens he cash posiion a ime. All he oher parameers have he same meaning as described above for equaion (3.1) only ha hey refer now o he shifed asse prices. The soluion of equaion (3.7), under he risk-neural pricing measure P, is clarified in he following proposiion 7. Proposiion 3.1. Consider ha he asses in a baske follow he shifed jumpdiffusion } model wih dynamics given by he SDE (3.7) wih he shifing process { δ (i) saisfying dδ(i) = rδ (i) d. If a soluion (θ, β, λ) of he sysem α i β i λ i r = n w j=1 γ ij θ j β i λi, i = 1,, Υ (3.8) 7 A more general version of Proposiion 3.1 is saed in he Proposiion A.1 for he sake of compleeness, bu i is no used empirically in his paper. 8

10 does exis and is seleced in associaion wih he risk-neural pricing measure P, hen, under his risk-neural measure, ( ) S (i) = S (i) b i δ (i) e (r β i λi 1 nw 2 j=1 γij)+ 2 N nw (j) (i) j=1 γ ij W Proof. See Appendix A.1 l=1 (Y (i) l + 1) + b i δ (i) er. (3.9) In order o simplify he noaion for he empirical work carried ou in Secion 5, we denoe V (i) = n w γ ij (j) j=1 σ i W where σi 2 = { } n w j=1 γ2 ij. Thus V (i) are dependen sandard Brownian moions wih and consequenly ρ l1 l 2 = corr(v (l 1), V (l 2) ) = 1 σ l1 σ l2 ( ) N (i) S (i) = S (i) b i δ (i) e (r β i λi 1 2 σ2 i )+σ i V (i) l=1 n w j=1 (Y (i) l γ l1 j γ l2 j, + 1) + b i δ (i) er (3.1) is used insead of (3.9). Finally, we poin ou ha he shifed jump-diffusion may encompass hree sub-cases: geomeric Brownian moion (GBM) when δ (i) = and λ i = for each asse i; shifed GBM when λ i = for each asse i; sandard jump-diffusion when δ (i) = for each asse i. 4. Pricing and hedging mehodology Our aim is o price European baske opions under he shifed jump-diffusion model. The payoff a mauriy of such opion is (BT K ) +, driven by he underlying variable Υ B = a i S (i), (4.1) i=1 9

11 where K is he srike price, a = (a 1,..., a Υ ) is he vecor of baske weighs, which could be posiive or negaive, and T is he ime o mauriy. Under he majoriy of models applied in pracice, he probabiliy densiy of he baske B canno usually be obained in closed-form. The mehodology proposed here is circumvening his problem using a Hermie approximaion probabiliy densiy ha will replace he risk-neural densiy implied by he model (3.1). In addiion, he approximaion densiy derived in his paper is consruced in such a way o mach exacly up o he firs m momens of he model implied risk-neural densiy. Leccadio e al. (212) proposed he Hermie ree mehod for pricing financial derivaives. In a nushell, he idea is o mach he momens of he log-reurns of he underlying asse wih he momens of a discree random variable. This work elaboraes on some varians of he mehod presened in Leccadio e al. (212) o deal wih baskes ha may ake on negaive values. In paricular, he binomial disribuion has been changed wih he asympoically equivalen Gaussian disribuion (coded as G) and he momen maching is done on wo differen ypes of reurn quaniies (coded as A and B) as specified in Table 1, where B T is defined by equaion (4.2). Henceforh, B is assumed o be differen from. [Table 1 abou here.] 4.1. Momens of he baskes The firs sep in our mehodology is o derive he momens of he baske (4.1) under he specificaion of a model for he underlying asses. For model (3.1), consider he shifed baske B = and he shifed srike price Υ i=1 K = K ( a i S (i) b i δ (i) er) (4.2) Υ i=1 a i b i δ (i) er. (4.3) For pracical purposes we shall calculae he momens of he shifed baske. Proposiion 4.1 shows how o calculae hese momens. Proposiion 4.1. The k-momen of B, under P, is given by µ k = Ẽ[B k ] = Υ i 1 =1 Υ i k =1 ( ) a i1 S (i 1) b i1 δ (i 1) e (r+ω i ) 1 ( ) a ik S (i k) b i1 δ (i k) e (r+ω i ) k mgf(e i e ik ) (4.4) 1

12 where ω j = β j λj 1 2 σ2 j, e j is he vecor having 1 in posiion j and zero elsewhere. Furhermore, he momen generaion funcion of σ i V (i) + N (i) (i) l=1 log (Y l + 1) is given by mgf(u) = exp {u Σu/2} Υ i=1 mgf N (i) ( where Σ denoes he covariance marix of V = Proof. See Appendix A.2. ( ηi u i + υ 2 i u 2 i /2 ) (4.5) V (1) ),,, V (Υ) and mgf (i) N (u) = exp( λ i (e u 1)). (4.6) 4.2. European Baske Call opion pricing and hedging The mechanism of shifing he baske and srike price in equaions (4.2) and (4.3) allows rewriing he European baske call opion price in wo equivalen ways: c = e rt Ẽ[(B T K ) + ] = e rt Ẽ[(B T K ) + ]. (4.7) We are going o use wo Hermie approximaion varians 8 described in Table 1, each varian being associaed wih a paricular arge quaniy for he baske. The nex proposiion provides a formula for he European call baske opion price under he approximaions considered in his paper. Proposiion 4.2. The price of a European call baske opion wih he Hermie expansion varian mga or mgb is given by: where c = B [(ϕ + h 1 )Φ( h 2 z) + h 2 g( z)] K e rt Φ( h 2 z) (4.8) m 2 g( z) = φ( z) ϕ k+1 H k ( z), (4.9) k= K is he shifed srike price, h 1 = for he varian mga and h 1 = 1 for he varians mgb, h 2 = sgn(b ), z is he soluion of [J( z) + h 1 ]Be rt = K, φ( ) is he sandard normal densiy funcion and Φ( ) is he sandard normal cumulaive disribuion funcion. 8 The mehodologies described in his paper are suppored by various compuaional ools ha are described in Appendix B for inernal consisency. 11

13 Proof. See Appendix A.3 The nex proposiion repors he formula for he hedging parameer wih respec o he variable u, which can be any of he quaniies S (i), B, σ i, r, T, a i, λ i, δ (i), β i, η i or υ i. Proposiion 4.3. For c, h 1, h 2, z, g( ), φ( ) and Φ( ) defined in Proposiion 4.2, he hedging parameer of a European call baske opion, wih respec o he variable u, under he Hermie expansion varian mga or mgb, is given by c u where rt e rt = c e u + B + e rt (B e rt ) u Proof. See Appendix A.4 [ h 2 g ( z) + ϕ ] u φ( h 2 z) + ( [h 2 g( z) + ϕ φ( h 2 z) + h 1 h 2 Φ( z) + h m 2 g ( z) = φ( z) k= )] (4.1) ϕ k+1 u H k( z), (4.11) In Secion 5.2, a comparison of our mehod wih oher mehods in he lieraure is carried ou using he Dela-hedging performances as a yardsick. For ha exercise, i is paricularly imporan o apply formula (4.1) for he case when u = B : c u 5. Empirical Comparisons [ = B h 2 g ( z) + ϕ ] u φ( h 2 z) + ( + h 2 g( z) + ϕ φ( h 2 z) + h 1 h 2 Φ( z) + h ) (4.12) Pricing performances The usefulness of a newly proposed mehod can be gauged by comparing i wih oher esablished mehods in he lieraure. To his end, in his secion, he wo mehods mga and mgb of our Hermie approximaion approach are direcly benchmarked wih he mehod in Borovkova e al. (27). In addiion, he Mone Carlo wih conrol variae mehodology oulined in Pellizzari (21) is adaped o deal wih asses having he dynamic specified by equaion (3.1). The model performance is deermined considering hree measures of error: 12

14 C1 number of bes soluions found, defined as number of imes he minimum squared error is reached under he specified mehod 9 : C1 l = { } 1 min SE j i = SE l i (5.1) j {BPW,mGA,mGB} i O where O is he se of opions considered and, for each opion i O, SE j i is he squared error for opion i and mehod j {BP W, mga, mgb}; C2 number of imes a mehod is no able o price an opion, ha is he procedure of momen maching gives poor resuls for he opion. We consider he momen maching o be poor when he relaive error (E r ) is greaer han 5%: C2 l = l 1{E r i > 5%} (5.2) i O By convenion, if C2 is no explicily saed, i is equal o. C3 square roo of MSE, calculaed only relaive o he opions for which he mehod was able o find a numerical soluion Muli-dimensional Model Comparisons This secion is a direc comparison wih he mehod in Borovkova e al. (27). The six baske opions priced in ha paper are summarized in Table 2. The special case λ i =, δ (i) = and b i = 1 combined wih equaion (3.1) falls ono he GBM case for all asses in he baske. Table 3 conains he comparison resuls. The prices obained here for he shifed log-normal model of BP W are differen from he ones in Borovkova e al. (27) because, o be consisen wih he oher models in he paper, we are pricing baske opions where he underlying asses are he sock and no he forward conracs. The empirical resuls indicae ha he mehod 6GA appears o be he bes mehod according o C1. The mehods 4GA and 4GB give, for hese six baske opions, exacly he same prices and under he C3 (RMSE), hey achieve he bes performance. For he baskes analysed here, here is very lile advanage in maching all six momens, he Hermie approximaion mehod working as well when only he firs four momens are mached. [Table 2 abou here.] [Table 3 abou here.] 9 Throughou { his paper 1{ } will denoe he indicaor funcion given, for any se A, by 1, if x A; 1{A}(x) =., oherwise. 13

15 Comparison under a se of simulaed scenarios A general comparison is performed considering a se of 2 randomly generaed opions. In paricular, he parameers of he underlying model (3.1) are drawn as follows: he risk-free rae r is uniformly disribued beween. and.1; he volailiy parameers σ i are uniformly disribued beween.1 and.6; he ime-o-mauriy T is uniformly disribued beween.1 and 1 years; curren spo prices S (i) are uniformly disribued beween 7 and 13; he weighs a i of he asses in he baske are uniformly disribued beween -1 and 1; he raios K over B are uniformly disribued beween.95 and 1.5; he shifs δ (i) ert range uniformly beween -2 and 2; each asse has he same probabiliy o be posiively (b i = 1) or negaively (b i = 1) shifed; he inensiies of he Poisson processes λ i are uniformly disribued beween and.2; For each scenario, he correlaion marix is randomly generaed saisfying he semiposiiveness condiion. Furhermore, he opion prices scenarios are divided ino wo ses of 1 opions each: Se 1 includes 5 opions wih he number of asses uniformly disribued beween 2 and 1, 3 opions wih he number of asses uniformly disribued beween 11 and 15, 1 opions beween 16 and 2 and 1 opions beween 21 and 5. Each asse has jumps wih average size (η) uniformly disribued beween -.3 and, and volailiy (υ) uniformly disribued beween and.3; Se 2 includes 1 opions wih he number of asses uniformly disribued beween 2 and 5, each asse having jumps wih average size (η) uniformly disribued beween -.3 and.3, and volailiy (υ) uniformly disribued beween and.3. For baskes wih less han 1 asses in Se 1, resuls are calculaed maching m = 4 momens and also maching m = 6 momens. As shown in Table 4, he resuls for m = 6 are ouperformed by he resuls wih m = 4. This is consisen wih 14

16 he research of Corrado and Su (1997) who concluded ha considering more han four momens creaes severe collineariy problems since all even... (momens)... are highly correlaed wih each oher... (and)... similarly, all odd-numbered subscriped (momens) are also highly correlaed. Therefore, for baskes wih more han 1 asses in Se 1 and also for all he opions in Se 2, we conduced our empirical analysis only for m = 4. In addiion, for boh ses of baske opions, he Mone Carlo mehod wih conrol variae deailed in Pellizzari (21) is employed as a benchmark. The number of simulaions used are beween 1 5 and 4 1 6, depending on he number of asses considered. The resuls in relaion o Se 1 are summarized in Tables 4, 5, 6 and 7, grouped for scenarios wih he number of asses beween 2-1, 11-15, 16-2 and 21-5 respecively, while Table 8 summarizes he resuls for all he 1 insances in his se. Overall, he mehods 4GA and 4GB give he same resuls in erms of RMSE (C3), wih 4GA slighly beer han 4GB for shor mauriies. Considering he comparison crierion C2, he mehod 4GB is much beer han he ohers. For small mauriies BP W performs slighly beer han our mehod bu he error associaed wih he BP W mehod is a leas double for all he oher comparison crieria. Finally, considering C1 boh mehods 4GA and 4GB perform much beer han BP W. When applying he BP W mehod, increasing he number of asses in he baske has he effec of increasing he RMSE on he mached opions. BP W s performance is almos consan across differen caegories, he only sligh improvemen (1.5 decimal poins on average) can be noiced a small mauriies. Moreover, C2, he percenage of non-mached opions, increases wih he number of asses, performing beer for lower ineres raes, shor mauriies and a-hemoney opions. Considering C1, he number of minimum errors, he bes resuls are obained for a number of asses in he baske beween 11 and 2. The BP W mehod performs well for longer mauriies. For 4GA here does no seem o be an explici relaion beween number of asses and C3. However, our empirical resuls show ha C1 and C2 decreases, increases respecively, wih he number of asses. Overall one can conclude ha boh Hermie approximaion mehods 4GA and 4GB have an excellen performance on large baskes. [Table 4 abou here.] [Table 5 abou here.] [Table 6 abou here.] [Table 7 abou here.] [Table 8 abou here.] 15

17 Table 9 summarizes he resuls for Se 2, reflecing he challenges posed by aking ino consideraion he inensiy of he Poisson processes. For he analysis in his group, we also consider a hybrid mehod spanned by he wo mehods 4GA and 4GB, which will be called 4GAB henceforh. This hybrid mehod 4GAB reurns he soluion of he mehod ha maches correcly he momens if only one of 4GA and 4GB works properly. The comparison is carried considering he error of he mehod ha maches he firs four momens if only one of 4GA and 4GB finds a soluion, or he wors error if boh find a numerical soluion. Even hough he mehod 4GAB considers he wors error beween he A and B varians, i is superior o he oher compared mehods, being able o mach he required baske momens in 96.6% (1-C2) of cases and reaching he minimum error (C1) 84% of he imes. [Table 9 abou here.] 5.2. Dela-hedging performances A comparison of dynamic Dela-hedging performance beween our formula (4.12) and he formula proposed in BP W (see definiion of i in ha paper 1 ) is illusraed in his secion. A sample of 1 simulaed pahs, indexed by s = 1,, 1, wih 1-monh-inerval hedging rolling frequency are generaed for he six baske opions. The baske opions considered are mosly hose in Table 2 wih some modificaions in order o have a more meaningful comparison. In he following, he opions characerisics are deailed: baskes 1 and 2 are exacly he same as 1 and 2 in Table 2; baske 3 is equal o baske 3 bu δ (i) = 1 ie rt, λ i =.3, η i =.3 and υ i =.2 for all i = 1,, Υ; baske 4 is equal o baske 4 bu δ (1) = and δ (2) = 5e rt, b 1 = b 2 = 1, λ i =.3, η 1 =.3, η 2 =.1 and υ i =.2 for all i = 1,, Υ; baskes 5 and 6 are respecively equal o baske 5 and 6 bu δ (i) = 1 ie rt, λ i =. For each pah, he opion price and he opion Dela are calculaed a each ime sep. The evaluaion of he performance for he Dela-hedged porfolios is performed via hree differen measures: 1 Borovkova e al. (27) repor he formula for he sensiiviy of he opion wih respec o individual sock prices in he baske. The sensibiliy wih respec o B can be calculaed by muliplicaing ha formula for Si B = 1 a i. 16

18 C4 average volailiy of he Dela, defined as: C4 l = 1 1 σi l (5.3) 1 where σi l is he volailiy of he Dela calculaed by mehod l along pah i. A pricing mehod implying less volaile Dela is beer because hedging coss do no pu liquidiy pressure on he invesor; C5 Square roo of MSE in he hedged porfolio evaluaed (per monh) as: C5 l = 1 12c n 1 i= i=1 [ c i c i+1 + l i (B i B i+1 )] 2 (5.4) where c i is he Mone Carlo price a ime i = T 12 i, n = 12 and l i is he Dela calculaed a ime i by mehod l. C6-C1 Abiliy of he hedging sraegy o mach he opion value a mauriy. Ouside ransacion coss, we evaluae how far from zero is he value of he hedged porfolio a mauriy T. A ime, he hedged-porfolio conains a shor posiion in a call opion, posiion in he baske and cash in a money accoun ha renders a null value for he porfolio a ime. A each ime sep, he number of posiions in he baske is changed according o and consequenly he money accoun. Five performance measures are used o evaluae he money-performance: he percenage of sub-hedging (C6), he percenage of super-hedging (C7), he average error for he sub-hedged and super-hedged porfolios (C8 and C9 respecively), and he average error among all he simulaions (C1). The resuls for he hedging performance are repored in Table 1. The mehods 4GA and 4GB produce very good resuls ha are very similar wih 4GA only slighly beer bu his may be due o he paricular simulaions used in pricing opions. However, boh Hermie approximaions mehods are superior o he BP W mehod for all measures of performance excep he RMSE (C5). For BP W, C6 and C7 are almos he same, showing ha his mehod may lead o under-hedging bu also over-hedging. A he same ime he mehods 4GA and 4GB seem o be occasionally only under-hedged, bu he hedging error is small as indicaed by C1. [Table 1 abou here.] 17

19 6. Conclusions By inroducing a shif parameer ino he drif of he diffusion process underlying he asses of a baske, one can accoun for he empirical characerisics of hisorical prices of hose asses. In paricular, he modelling is laid on improved foundaions, being able o cover he well-documened negaive skewness. However, recen echniques imposed srong assumpions on he evoluion dynamics of he baske as whole, searching for closed-form soluion and repackaging of log-normal Black-Scholes ype pricing formulae. In his paper, we have shown ha his pah is no necessary and we have highlighed a mehodology ha may work well wih oher fuure models in his area. We focused here on he shifed jump-diffusion model and we demonsraed wih empirical simulaions, ha our Hermie expansion approach may provide pricing resuls ha are as good as compeing mehods, and in many cases superior. In addiion, we followed he hedging performance as a comparison ool and again our echnology provided excellen resuls. In our opinion, he improved resuls emphasized in he paper are no surprising since he echnique is fundamenally based on maching he firs four momens under model specificaion. Thus, we allow granular specificaion of dynamics for each asse and hen only deermine he momens of he baske. While our paper was focused on equiy baskes, i is clear ha he same mehodology can be applied for mixures of asses and models, as long as momens can be calculaed easily. References Alexander, C. and Venkaramanan, A. (212). Analyic approximaions for muliasse opion pricing. Mahemaical Finance, 22(4): Barraquand, J. (1995). Numerical valuaion of high dimensional mulivariae European securiies. Managemen Science, 41(12): Barraquand, J. and Marineau, D. (1995). Numerical valuaion of high dimensional mulivariae American securiies. Journal of Financial and Quaniaive Analysis, 3(3): Bjerksund, P. and Sensland, G. (211). Quaniaive Finance, pages 1 1. Closed form spread opion valuaion. Borovkova, S., Permana, F., and Weide, H. (27). A closed form approach o valuing and hedging baske and spread opions. Journal of Derivaives, 14(4):

20 Borovkova, S., Permana, F., and Weide, H. V. (212). American baske and spread opion pricing by a simple binomial ree. Journal of Derivaives, 19(4): Borovkova, S. and Permana, F. J. (27). Asian baske opions and implied correlaions in oil markes. In Proceedings of he Fourh IASTED Inernaional Conference on Financial Engineering and Applicaions, FEA 7, pages 85 91, Anaheim, CA, USA. ACTA Press. Broadie, M. and Glasserman, P. (24). A sochasic mesh mehod for pricing highdimensional American opions. Journal of Compuaional Finance, 7(4): Carmona, R. and Durrleman, V. (23). SIAM Review, 45(4): Pricing and hedging spread opions. Corrado, C. J. and Su, T. (1997). Implied volailiy skews and sock reurn skewness and kurosis implied by sock opion prices. European Journal of Finance, 3(1): Frielli, M. (2). The minimal enropy maringale measure and he valuaion problem in incomplee markes. Mahemaical Finance, 1(1): Genle, D. (1993). Baske weaving. Risk, pages Hurd, T. and Zhou, Z. (21). A Fourier ransform mehod for spread opion pricing. SIAM Journal on Financial Mahemaics, 1(1): Jarrow, R. and Rudd, A. (1982). Approximae opion valuaion for arbirary sochasic processes. Journal of Financial Economics, 1: Ju, N. (22). Pricing Asian and baske opions via Taylor expansion. Journal of Compuaional Finance, 5(3): Kim, I. J. (199). The analyic valuaion of American opions. Review of Financial Sudies, 3(4): Kirk, E. (1995). Correlaion in he energy markes. Managing Energy Pricing Risk, Risk Publicaions and Euron, pages Korn, R. and Zeyun, S. (213). Efficien baske Mone Carlo opion pricing via a simple analyical approximaion. Journal of Compuaional and Applied Mahemaics, 243: Leccadio, A., Toscano, P., and Tunaru, R. (212). Hermie binomial rees: a novel echnique for derivaive pricing. Inernaional Journal of Theoreical and Applied Finance, 15(8):

21 Levy, E. (1992). Pricing European average rae currency opions. Journal of Inernaional Money and Finance, 11(5): Li, F. (2). Opion pricing: How flexible should he SPD be? The Journal of Derivaives, 7(4): Li, M., Deng, S.-J., and Zhou, J. (28). Closed-form approximaions for spread opion prices and greeks. Technical Repor 3. Li, M., Zhou, J., and Deng, S.-J. (21). Muli-asse spread opion pricing and hedging. Quaniaive Finance, 1(3): Longsaff, F. A. and Schwarz, E. S. (21). Valuing American opions by simulaion: A simple leas-squares approach. Review of Financial Sudies, 14(1): Margrabe, W. (1978). The value of an opion o exchange one asse for anoher. Journal of Finance, 33(1): Milevsky, M. A. and Posner, S. E. (1998). A closed-form approximaion for valuing baske opions. The Journal of Derivaives, 5(4): Pellizzari, P. (21). Efficien Mone Carlo pricing of European opions using mean value conrol variaes. Decisions in Economics and Finance, 24: Posner, S. E. and Milevsky, M. A. (1998). Valuing exoic opions by approximaing he SPD wih higher momens. The Journal of Financial Engineering, 7(2): Rubinsein, M. (1981). Displaced diffusion opion pricing. Journal of Finance, 38(1): Rubinsein, M. (1998). Edgeworh binomial rees. The Journal of Derivaives, 5(3):2 27. Shreve, S. (24). Sochasic Calculus for Finance II: Coninuous-Time Models. Number v. 11 in Springer Finance. Springer. Turnbull, S. M. and Wakeman, L. M. (1991). A quick algorihm for pricing European average opions. Journal of Financial and Quaniaive Analysis, 26(3): Venkaramanan, A. and Alexander, C. (211). Closed form approximaions for spread opions. Applied Mahemaical Finance, 18(5):

22 Zhou, J. and Wang, X. (28). Accurae closed-form approximaion for pricing Asian and baske opions. Applied Sochasic Models in Business and Indusry, 24(4): Table 1: Summary of he varians of he Hermie mehod considered in his work. The firs column conains he names of he varians considered: m sands for he number of momens mached, G highlighs ha a ransformaion of he Gaussian disribuion is considered (as shown by he variable Z in he second column) and A and B idenify he sandardized reurns used as approximaed random variable (las column). In paricular, wo sandardized reurns are considered: varian A is consruced in such a way ha he firs momen of he approximaed random variable is 1 while he reurn in varian B has firs momen equal o. B is defined in (4.2) as shifed baske, he momens of J(Z) and he momens of he quaniies in he las column are in Appendix B.2, H k (x) denoes he kh-order Hermie polynomial H k (x) = ( 1)k φ(x) densiy funcion and ϕ k approximaed random variable (las column). k φ(x) x k where φ( ) is he sandard normal are deermined o exacly mach he firs m momens of he Varian s name Approximaing r.v. Approximaed r.v. mga mgb J(Z) = m 1 k= ϕ kh k (Z) B T B e rt B T B e rt 1 21

23 Table 2: Specificaion of he baske opions under muli-dimensional GBM model. This specificaion follows Borovkova e al. (27). Oher relevan parameers are risk-free rae equal o 3%, 1-year mauriy, λ =, δ (i) = and b i = 1. The firs row indicaes [S (1), S(2), S(3) ], he second [σ 1, σ 2, σ 3 ], he hird [a 1, a 2, a 3 ], he forh he correlaion ρ i,j for each couple (i, j) of asses and he fifh K. The only difference wih he opions in Borovkova e al. (27) is ha hey price opions on baske of forward conracs while we price opions on baske of asses. Baske 1 Baske 2 Baske 3 Baske 4 Baske 5 Baske 6 Sock Prices [1,12] [15,1] [11,9] [2,5] [95,9,15] [1,9,95] Volailiy [.2,.3] [.3,.2] [.3,.2] [.1,.15] [.2,.3,.25] [.25,.3,.2] Weighs [-1,1] [-1,1] [.7,.3] [-1,1] [1,-.8,-.5] [.6,.8,-1] ρ 1,2 =.9, ρ 1,2 =.9, Correlaion ρ 1,2 =.9 ρ 1,2 =.3 ρ 1,2 =.9 ρ 1,2 =.8 ρ 2,3 =.9 ρ 2,3 =.9 ρ 1,3 =.8 ρ 1,3 =.8 Srike price

24 Table 3: Comparison under muli-dimensional GBM model. This able repors he comparison on he six baske opions in Borovkova e al. (27). In he second column, he prices (sandard deviaion in bracke) calculaed by he Mone Carlo mehod wih conrol variae in Pellizzari (21) wih simulaions are repored and hey are considered as benchmark. In he hird column, here are he prices calculaed by he mehod in Borovkova e al. (27). The las four columns conain he prices under he mehods mga and mgb when m = 4 and m = 6. Two of he measures of error considered are repored in he las wo rows: C1 he percenage of imes he minimum squared error is reached under he specified mehod, C3 he square roo of MSE calculaed only relaive o he opions for which he mehod was able o find a numerical soluion. The hird measure of error, C2, ha indicaes he percenage of imes he relaive error is greaer han 5%, is always equal o and i is no repored in he able. Baske 1 Baske 2 Baske 3 Baske 4 Baske 5 Baske 6 MC (SD) (.31) (.52) (.5) (.8) (.27) (.3) BPW 4GA 4GB 6GA 6GB C % 16.67% 16.67% 66.67% 33.33% C

25 Table 4: Comparison I (Se 1): number of asses beween 2 and 1. This able conains he summary of he performances of several mehods for pricing opions in Se 1 wih numbers of asses randomly generaed beween 2 and 1. The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) = [7; 13], a i [ 1; 1], [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3; ] and υ i [;.3]. Three measures of error are K B repored: C1 he percenage of imes he minimum squared error is reached under he specific mehod; C2 he percenage of imes he relaive error is greaer han 5% for he specified mehod; C3 he square roo of MSE calculaed only relaive o he opions for which he mehod was able o find a numerical soluion. The resuls are shown (per column) along hree differen dimensions: risk-free rae, ime o mauriy and srike price. Along he differen rows, he resuls per mehod are showed: in paricular, BP W sands for he mehod in Borovkova e al. (27) and mga and mgb are considered for boh m = 4 and m = C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal 6GA 42.54% 48.71% 39.68% 5.99% 47.18% 44.95% 44.38% 45.4% 6GA 6GB 38.43% 41.81% 32.79% 47.4% 4.14% 42.93% 36.25% 4.% 6GB BPW 15.3% 12.7% 2.24% 7.51% 17.61% 13.13% 11.25% 13.8% BPW 4GA 46.64% 43.97% 51.82% 39.13% 44.37% 42.42% 5.% 45.4% 4GA 4GB 5.% 46.55% 56.68% 4.32% 43.66% 49.49% 51.25% 48.4% 4GB 6GA.%.86%.4%.4%.% 1.1%.%.4% 6GA 6GB 1.12% 1.72% 2.43%.4%.7% 2.53%.63% 1.4% 6GB BPW 11.57% 8.19% 2.43% 17.39% 9.15% 9.9% 11.88% 1.% BPW 4GA 9.33% 1.34% 16.19% 3.56% 8.45% 13.13% 6.88% 9.8% 4GA 4GB 5.6% 3.88% 6.48% 3.16% 3.52% 5.56% 5.% 4.8% 4GB 6GA GA 6GB GB BPW BPW 4GA GA 4GB GB # opions

26 Table 5: Comparison II (Se 1): number of asses beween 11 and 15. This able conains he summary of he performances of several mehods for pricing opions in Se 1 wih numbers of asses randomly generaed beween 11 and 15. The asses follow equaion (3.1). The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) = [7; 13], a i [ 1; 1], K B [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3; ] and υ i [;.3]. 25 C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal BPW 11.89% 15.29% 2.57% 7.55% 13.83% 16.1% 1.23% 13.67% BPW 4GA 81.82% 75.8% 66.67% 89.31% 82.98% 72.88% 81.82% 78.67% 4GA 4GB 86.1% 84.8% 78.1% 91.19% 85.11% 84.75% 85.23% 85.% 4GB BPW 12.59% 15.92% 3.55% 23.9% 14.89% 11.2% 18.18% 14.33% BPW 4GA 11.19% 14.1% 22.7% 3.77% 7.45% 18.64% 1.23% 12.67% 4GA 4GB 5.59% 3.82% 7.8% 1.89% 5.32% 3.39% 5.68% 4.67% 4GB BPW BPW 4GA GA 4GB GB # opions

27 Table 6: Comparison III (Se 1): number of asses beween 16 and 2. This able conains he summary of he performances of several mehods for pricing opions in Se 1 wih numbers of asses randomly generaed beween 16 and 2. The asses follow equaion (3.1). The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) = [7; 13], a i [ 1; 1], K B [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3; ] and υ i [;.3]. 26 C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal BPW 12.5% 15.38% 25.% 5.36% 12.5% 13.79% 15.38% 14.% BPW 4GA 85.42% 82.69% 68.18% 96.43% 87.5% 82.76% 82.5% 84.% 4GA 4GB 83.33% 82.69% 7.45% 92.86% 87.5% 86.21% 76.92% 83.% 4GB BPW 14.58% 23.8% 4.55% 3.36% 31.25% 6.9% 17.95% 19.% BPW 4GA 2.8% 7.69% 11.36%.% 6.25% 3.45% 5.13% 5.% 4GA 4GB 2.8% 7.69% 6.82% 3.57% 3.13%.% 1.26% 5.% 4GB BPW BPW 4GA GA 4GB GB # opions

28 Table 7: Comparison IV (Se 1): number of asses beween 31 and 5. This able conains he summary of he performances of several mehods for pricing opions in Se 1 wih numbers of asses randomly generaed beween 31 and 5. The asses follow equaion (3.1). The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) = [7; 13], a i [ 1; 1], K B [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3; ] and υ i [;.3]. 27 C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal BPW 12.24% 5.88% 15.38% 2.8% 14.81% 2.38% 12.9% 9.% BPW 4GA 85.71% 9.2% 78.85% 97.92% 85.19% 92.86% 83.87% 88.% 4GA 4GB 85.71% 94.12% 82.69% 97.92% 81.48% 97.62% 87.1% 9.% 4GB BPW 16.33% 35.29% 13.46% 39.58% 29.63% 19.5% 32.26% 26.% BPW 4GA 2.4% 3.92% 5.77%.% 3.7% 4.76%.% 3.% 4GA 4GB 2.4%.% 1.92%.% 3.7%.%.% 1.% 4GB BPW BPW 4GA GA 4GB GB # opions

29 Table 8: Comparison V (Se 1): Toal summary. This able conains he summary of he performances of several mehods for pricing opions in Se 1. The asses follow equaion (3.1). The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) = [7; 13], a i [ 1; 1], K B [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3; ] and υ i [;.3]. 28 C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal BPW 13.78% 12.8% 2.25% 6.78% 15.59% 12.92% 11.64% 13.3% BPW 4GA 63.98% 63.1% 6.54% 66.28% 65.8% 6.21% 66.4% 63.5% 4GA 4GB 66.73% 67.28% 66.94% 67.5% 65.8% 68.22% 67.3% 67.% 4GB BPW 12.6% 15.4% 4.13% 22.87% 15.25% 1.59% 16.35% 13.8% BPW 4GA 8.46% 1.57% 16.53% 2.91% 7.46% 13.18% 6.92% 9.5% 4GA 4GB 4.92% 3.86% 6.4% 2.52% 4.7% 3.88% 5.35% 4.4% 4GB BPW BPW 4GA GA 4GB GB # opions

30 Table 9: Comparison (Se 2): Toal summary. This able conains he summary of he performances of several mehods for pricing opions in Se 2. The asses follow equaion (3.1) where he parameers are randomly generaed and uniformly disribued in he following ranges: r [;.1], σ i [.1;.6], T [.1; 1], S (i) K = [7; 13], a i [ 1; 1], B [.95; 1.5], δ (i) ert [ 2; 2], b i [ 1; 1], λ i [;.2], η i [.3;.3] and υ i [;.3]. The resuls are showed (per column) along hree differen dimensions: risk-free rae, ime o mauriy and srike price. Along he differen rows, he resuls per mehod are showed: in paricular, BP W sands for he mehod in Borovkova e al. (27), mga and mgb are considered for m = 4 and 4GAB is a combinaion of 4GA and 4GB. 4GAB reurns he soluion of he mehod ha maches correcly he momens if only one of 4GA and 4GB works properly. The comparison for 4GAB is carried considering he error of he mehod ha maches he momen if only one beween 4GA and 4GB finds a soluion or he wors error if boh find a soluion. 29 C1 C2 C3 r T K K r.5 r >.5 T.5 T >.5 B < K K B 1.2 B > 1.2 Toal BPW 17.95% 17.44% 22.72% 12.37% 17.16% 16.83% 19.45% 17.7% BPW 4GA 78.7% 75.66% 68.74% 86.19% 78.22% 78.22% 74.74% 77.2% 4GA 4GB 76.13% 77.89% 71.46% 82.89% 75.58% 78.22% 76.79% 77.% 4GB 4GAB 83.4% 84.79% 79.2% 89.7% 84.16% 84.65% 82.59% 84.% 4GAB BPW 19.72% 24.14% 7.96% 36.7% 29.4% 17.82% 2.14% 21.9% BPW 4GA 9.27% 13.18% 16.31% 5.77% 11.22% 1.15% 12.63% 11.2% 4GA 4GB 12.3% 11.36% 14.37% 8.87% 13.2% 11.88% 9.9% 11.7% 4GB 4GAB 2.96% 3.85% 3.11% 3.71% 3.96% 2.97% 3.41% 3.4% 4GAB BPW BPW 4GA GA 4GB GB 4GAB GAB # opions

31 Table 1: Comparison: Dela-hedging performances. This able conains he summary of he Dela-hedging performances of hree mehods. BPW sands for he mehod in Borovkova e al. (27) and 4GA and 4GB are he mehods summarized in Table 1. The measures of error considered are: C4 he volailiy of Dela, C5 he MSE on he hedging performance along he life ime of he conrac, C6 and C7 he numbers of sub-hedging and super-hedging respecively, finally C8, C9 and C1 respecively he average error on sub-hedging porfolios, super-hedging porfolios and all porfolios. 3 BPW 4GA 4GB Baske 1 Baske 2 Baske 3 Baske 4 Baske 5 Baske 6 Toal GBM GBM Shifed Jump Shifed Jump Shifed GBM Shifed GBM C C4 C C5 C % C6 C % C7 C C8 C C9 C C1 C C4 C C5 C % C6 C % C7 C C8 C C9 C C1 C C4 C C5 C % C6 C % C7 C C8 C C9 C C1