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Fourier inversion formulas for multiple-asset option pricing

Bruno Feunou and Ernest Tafolong


Plain vanilla options have a single underlying asset and a single condition on the payoff at the expiration date. For this class of options, a well known result of Duffie, Pan, and Singleton (Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68: 1343–1376. shows how to invert the characteristic function to obtain a closed-form formula for their prices. However, multiple-asset and multiple-condition derivatives such as rainbow options cannot be priced within this framework. This paper provides an analytical solution for options whose payoffs depends on two or more conditions. We take the advantage of the inversion of the Fourier transform, resorting to neither Black and Scholes’s framework, nor the affine models’s settings. Numerical experiments based on the aforementioned class of derivatives are provided to illustrate the usefulness of the proposed approach.

JEL classifications:: G12

Corresponding author: Bruno Feunou, Bank of Canada, 234, Wellington Street, Ottawa, ON, K1A 0G9, Canada, Phone: +(613) 782-8302, Fax: +(613) 782-7713, e-mail:


We thank an anonymous referee and Bruce Mizrach (the editor) for comments and suggestions that improved the article substantially. We also thank Guillaume Nolin and Maren Hansen, Glen Keenleyside and Foutse Khomh for comments and suggestions. This research was initiated while Tafolong was working at Desjardins Group. A previous version circulated under the title: “Pricing Multiple Triggers Contingent Claims.” The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada or the National Bank of Canada. Remaining errors are ours.


A Proof of Proposition 2

For 0<τ1, τ2<+∞ and y1, y2∈ℝ, let us define the expression 𝕀 by the following relation:

(19)Iτ1τ1τ2τ2eiv2y2iv2[eiv1y1ψ(aiv1b1iv2b2,x,t,T)eiv1y1ψ(a+iv1b1iv2b2,x,t,T)iv1]eiv2y2iv2[eiv1y1ψ(aiv1b1+iv2b2,x,t,T)eiv1y1ψ(a+iv1b1+iv2b2,x,t,T)iv1]dv1dv2. (19)

Because ψ(a+iv1b1+iv2b2,x,t,T)=2eiv1z1+iv2z2Ga,b1,b2(dz1,dz2;x,T), the relation (19) can be expressed as


Since we disregard the exact order in which the integral has been composed, integrand permutations conserve the value of the integral, and 𝕀 can be expressed as follows:

(20)I=2τ1τ1τ2τ2eiv1y1+iv2y2iv1z1iv2z2eiv1y1+iv2y2+iv1z1iv2z2(iv1)(iv2)eiv1y1iv2y2iv1z1+iv2z2+eiv1y1iv2y2+iv1z1+iv2z2(iv1)(iv2)dv1dv2Ga,b1,b2(dz1,dz2;x,T). (20)

To proceed with the determination of 𝕀, we make use of the following relation. For τ1>0, τ2>0,

(21)τ1τ1τ2τ2eiv1y1+iv2y2iv1z1iv2z2eiv1y1+iv2y2+iv1z1iv2z2eiv1y1iv2y2iv1z1+iv2z2+eiv1y1iv2y2+iv1z1+iv2z2(iv1)(iv2)dv1dv2=τ1τ1τ2τ2eiv1y1+iv2y2iv1z1iv2z2+eiv1y1iv2y2+iv1z1+iv2z2iv1iv2eiv1y1+iv2y2+iv1z1iv2z2+eiv1y1iv2y2iv1z1+iv2z2iv1iv2dv1dv2. (21)

Using the usual trigonometric identity, (21) becomes (22) as follows

(22)τ1τ1τ2τ2eiv1y1+iv2y2iv1z1iv2z2eiv1y1+iv2y2+iv1z1iv2z2eiv1y1iv2y2iv1z1+iv2z2+eiv1y1iv2y2+iv1z1+iv2z2(iv1)(iv2)dv1dv2=τ1τ1τ2τ22cos(v1(y1z1)+v2(y2z2))2cos(v1(y1z1)v2(y2z2))v1v2dv1dv2=τ1τ1τ2τ22[cos(v1(y1z1)v2(y2z2))cos(v1(y1z1)+v2(y2z2))v1v2]dv1dv2=τ1τ1τ2τ22[cos(v1(y1z1))cos(v2(y2z2))+sin(v1(y1z1))sin(v2(y2z2))v1v2]2[cos(v1(y1z1))cos(v2(y2z2))sin(v1(y1z1))sin(v2(y2z2))v1v2]dv1dv2=τ1τ1τ2τ24sin(v1(y1z1))sin(v2(y2z2))v1v2dv1dv2=4τ1τ1sin(v1(y1z1))v1dv1τ2τ2sin(v2(y2z2))v2dv2. (22)

Furthermore, the following result holds for every τ>0:

(23)ττeiv(zy)eiv(zy)ivdv=ττ2sin(v(zy))vdv=2sgn(zy)ττsin(v|zy|)vdv. (23)


Pooling together the relation (23) with the bounded convergence theorem and using the fact that lim(y1,y2)(,)Ga,b1,b2(y1,y2;x,T)=ψ(a,x,0,T), 𝕀 as defined in (19), when letting τ1, τ2, →∞, this brings about

(24)lim(τ1,τ2)(,)I4π2=2sgn(z1y1)sgn(z2y2)Ga,b1,b2(dz1,dz2;x,T)=y1+y2+Ga,b1,b2(dz1;dz2)Iy1+y2Ga,b1,b2(dz1;dz2)IIy2+y1Ga,b1,b2(dz1;dz2)III+y1y2Ga,b1,b2(dz1;dz2)IV. (24)

We determine each of the expressions above in order to compute 𝕀. Firstly, we proceed by computing the expression I


We know that, whenever one of y1 or y2 is +∞, we recover Duffie, Pan, and Singleton (2000) one-condition framework; then, from their Proposition 2, we have


Plugging these relations into I yields

(25)I=Ga,b1,b2(y1;y2)14π+eivy1ψ(aivb1,x,t,T)eivy1ψ(a+ivb1,x,t,T)ivdv14π+eivy2ψ(aivb2,x,t,T)eivy2ψ(a+ivb2,x,t,T)ivdv. (25)

The second expression, II, is derived as follows


where Ga,b1,b2(y1,y2,x,T)=limz1y1,z1<y1;z2y2,z2>y2Ga,b1,b2(z1,z2,x,T) and Ga,b1,b2(y1,y2,x,T)=limz1y1,z1<y1;z2y2,z2<y2Ga,b1,b2(z1,z2,x,T).

From Proposition 2 of Duffie, Pan, and Singleton (2000) we also have

Ga,b1,b2(+;y2)=ψ(a,x,t,T)2+14π+eivy2ψ(aivb2,x,t,T)eivy2ψ(a+ivb2,x,t,T)ivdv and

Ga,b1,b2(y1;y2)=Ga,b1,b2(y1;y2). The expression II then becomes

(26)II=ψ(a,x,t,T)2+14π+eivy2ψ(aivb2,x,t,T)eivy2ψ(a+ivb2,x,t,T)ivdvGa,b1,b2(y1;y2). (26)

Likewise, III is computed as

(27)III=y2+y1Ga,b1,b2(dz1;dz2)=y2+dGa,b1,b2(y1;z2)=Ga,b1,b2(y1;+)Ga,b1,b2(y1;y2)=ψ(a,x,t,T)2+14π+eivy1ψ(aivb1,x,t,T)eivy1ψ(a+ivb1,x,t,T)ivdvGa,b1,b2(y1;y2). (27)

Finally, the last term, IV, is expressed as

(28)IV=y1y2Ga,b1,b2(dz1;dz2)=y1dGa,b1,b2(z1;y2)=Ga,b1,b2(y1;y2)=Ga,b1,b2(y1;y2). (28)

In conclusion, we can express the following limit:

(29)14π2lim(τ1,τ2)(,)I=4Ga,b1,b2(y1;y2)ψ(a,x,t,T)12π+eivy2ψ(aivb2,x,t,T)eivy2ψ(a+ivb2,x,t,T)ivdv12π+eivy1ψ(aivb1,x,t,T)eivy1ψ(a+ivb1,x,t,T)ivdv. (29)

B Proof of Lemma 1

The proof of Lemma 1 relies on the following combinatory identity:

(30)jA(rj+sj)=BA(jBrj)(jA\Bsj). (30)

Substituting rjeivj(yjzj)2i,sjeivj(yjzj)2i and using the fact that sin(vj(yjzj))=eivj(yjzj)eivj(yjzj)2i, we have

2sin(vj(yjzj))vj=eivj(yjzj)eivj(yjzj)ivj and (30) becomes

(31)(2)|A|jAsin(vj(yjzj))vj=BA(1)|B|eiv(B)(y(B)z(B))iv(A\B)(y(A\B)z(A\B))jA(ivj). (31)

We therefore show equation (11) of Lemma 1 by combinatory identity.

Substituting QA by its expression (11) in 𝕀A leads to


C Proof of Lemma 2

Henceforth, to simplify our notations, we drop the multiple occurrences of Xt, T, χ from the expression of Ga,b(z, Xt, T). By definition, we have


Using the fact that


we have




Replacing QA by its expression in 𝕀A leads to


We are now ready to demonstrate the lemma. The goal is to show that


In other words, we endeavor to prove that

(32)JA|A|jAsgn(zjyj)Ga,b(dz(A))=k=0|A|(2)kAkAGa,b(y(Ak)). (32)

To do so, we proceed by induction. Suppose that the relation holds for each subset of E whose member’s cardinality is less than or equal to |A|. Let us show that the relation is also fulfilled for the subset with |A|+1 elements. Denoting A+1≡A∪{yA∣+1}, we have


By induction, we can apply (32) to |A|jAsgn(zjyj)Ga,b(dz(A);z|A|+1). This implies that

(33)|A|jAsgn(zjyj)Ga,b(dz(A);z|A|+1)=k=0|A|(2)kAkAGa,b(y(Ak);z|A|+1), (33)





By the definition of Ga,b(·), we have


and therefore


Replacing AkAGa,b(y(Ak);y|A|+1) by


implies that




By noting that


we can rewrite 𝕁A+1 as the following


Rearranging the summation leads to


which implies


Using the induction assumption up to 𝕁A, we have




In other words


This ends the proof of Lemma 2.

D Proof of Proposition 3

The proof of Proposition 3 relies on Lemma 2 and an application of the Möbius inversion for the Boolean algebra of subsets of a finite set [see Hazewinkel (2002) and Rota (1964)].

Define AE,

(34)G(A)=2|A|Ga,b(y(A)),F(A)=(2π)|A|IA, (34)

where 𝕀A, AE is given by (10).

Lemma 2 implies that

(35)F(A)=BA(1)|A||B|G(B). (35)

The Möbius inversion says that (35) is equivalent to

(36)G(A)=BAF(B)BA, (36)

and the proof of Proposition 3 is completed with A=E in (36).

E Some ARV model features

Below we provide some ARV model properties.

E.1 The density

The joint conditional density of returns (Rt) and realized variance-covariance matrix (RVt) is









E.2 Model characteristics

It follows from (14) that the conditional variance can be expressed as


Given that


the model is well defined (in the sense that the support of distribution of Σt is the symmetric positive definite real matrix) whenever the following condition is fulfilled:


and we can express ω as


Further, given that


the variance matrix is covariance-stationary if all the eigenvalues of ββ+αα′ are <1.

E.3 Proof of Proposition 4, the conditional expectation of log returns






E.4 Proof of the moment-generating functions

E.4.1 Proof of Proposition 5





In the following, we provide more details on the proof of that result:



E.4.2 Proof of Proposition 6



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Supplemental Material

The online version of this article (DOI: 10.1515/snde-2014-0034) offers supplementary material, available to authorized users.

Published Online: 2015-3-7
Published in Print: 2015-12-1

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