Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter March 7, 2015

Fourier inversion formulas for multiple-asset option pricing

Bruno Feunou and Ernest Tafolong

Abstract

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. http://dx.doi.org/10.1111/1468-0262.00164.) 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:

Acknowledgments

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.

Appendix

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

I=τ1τ1τ2τ22eiv1y1+iv2y2iv1z1iv2z2eiv1y1+iv2y2+iv1z1iv2z2(iv1)(iv2)eiv1y1iv2y2iv1z1+iv2z2+eiv1y1iv2y2+iv1z1+iv2z2(iv1)(iv2)Ga,b1,b2(dz1,dz2;x,T)dv1dv2.

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)

Andlimτττsin(v|zy|)vdv=π.

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

I=y1+y2+Ga,b1,b2(dz1;dz2)=y1[Ga,b1,b2(dz1;+)Ga,b1,b2(dz1;y2)]=y1Ga,b1,b2(dz1;+)y1Ga,b1,b2(dz1;y2)=Ga,b1,b2(+;+)Ga,b1,b2(y1;+)Ga,b1,b2(+;y2)+Ga,b1,b2(y1;y2).

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

Ga,b1,b2(y1;+)=ψ(a,x,t,T)2+14π+eivy1ψχ(aivb1,x,t,T)eivy1ψ(a+ivb1,x,t,T)ivdvGa,b1,b2(+;y2)=ψ(a,x,t,T)2+14π+eivy2ψχ(aivb2,x,t,T)eivy2ψ(a+ivb2,x,t,T)ivdv.

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

II=y1+y22Ga,b1,b2(z1;z2)=y1+dGa,b1,b2(z1;y2)=Ga,b1,b2(+;y2)Ga,b1,b2(y1;y2),

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

IA=|A||A|(2)|A|j=1|A|sin(vj(yjzj))vjjAdvjGa,b(dz(A),x,T)=(2)|A||A|jA[sin(vj(yjzj))vjdvj]Ga,b(dz(A),x,T)=(2π)|A||A|jAsgn(zjyj)Ga,b(dz(A),x,T).

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

IA=|A|k=0|A|(1)kAkAeiv(Ak)y(Ak)iv(A\Ak)y(A\Ak)ψ(aiv(Ak)b(Ak)+iv(A\Ak)b(A\Ak))jA(ivj)jAdvj.

Using the fact that

ψ(aiv(Ak)b(Ak)+iv(A\Ak)b(A\Ak))=|A|eiv(Ak)z(Ak)+iv(A\Ak)z(A\Ak)Ga,b(dz(A),Xt,T),

we have

IA=|A|k=0|A|(1)kAkAeiv(Ak)y(Ak)iv(A\Ak)y(A\Ak)|A|eiv(Ak)z(Ak)+iv(A\Ak)z(A\Ak)Ga,b(dz(A))jA(ivj)jAdvj=|A||A|k=0|A|(1)kAkAeiv(Ak)(y(Ak)z(Ak))iv(A\Ak)(y(A\Ak)z(A\Ak))jA(ivj)jAdvjGa,b(dz(A))=|A||A|QAjAdvjGa,b(dz(A),Xt,T),

where

QAk=0|A|(1)kAkAeiv(Ak)(y(Ak)z(Ak))iv(A\Ak)(y(A\Ak)z(A\Ak))jA(ivj).

Replacing QA by its expression in 𝕀A leads to

IA=|A||A|(2)|A|j=1|A|sin(vj(yjzj))vjjAdvjGa,b(dz(A),Xt,T)=(2)|A||A|jA[sin(vj(yjzj))vjdvj]Ga,b(dz(A),Xt,T)=(2π)|A||A|jAsgn(zjyj)Ga,b(dz(A),Xt,T).

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

IA=(2π)|A|k=0|A|(2)kAkAGa,b(y(Ak)).

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

JA+1=|A|+1jA+1sgn(zjyj)Ga,b(dz(A+1))=sgn(z|A|+1y|A|+1)d[|A|jAsgn(zjyj)Ga,b(dz(A);z|A|+1)].

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)

thus

JA+1=sgn(z|A|+1y|A|+1)[k=0|A|(2)kAkAGa,b(y(Ak);dz|A|+1)],

hence

JA+1=k=0|A|(2)kAkAsgn(z|A|+1y|A|+1)Ga,b(y(Ak);dz|A|+1)=k=0|A|(2)kAkA[y|A|+1Ga,b(y(Ak);dz|A|+1)+y|A|+1Ga,b(y(Ak);dz|A|+1)]=k=0|A|(2)kAkA[2Ga,b(y(Ak);y|A|+1)+Ga,b(y(Ak);)+Ga,b(y(Ak);)].

By the definition of Ga,b(·), we have

Ga,b(y(Ak);)=0,Ga,b(y(Ak);)=Ga,b(y(Ak)),

and therefore

JA+1=k=0|A|(2)kAkA[2Ga,b(y(Ak);y|A|+1)+Ga,b(y(Ak))]=k=0|A|(2)k+1AkAGa,b(y(Ak);y|A|+1)+JA=(2)|A|+1Ga,b(y(A+1))+k=0|A|1(2)k+1AkAGa,b(y(Ak);y|A|+1)+JA.

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

AkAGa,b(y(Ak);y|A|+1)=[AkAGa,b(y(Ak);y|A|+1)+Ak+1AGa,b(y(Ak+1))Ak+1AGa,b(y(Ak+1))]

implies that

JA+1=(2)|A|+1Ga,b(y(A+1))+k=0|A|1(2)k+1[AkAGa,b(y(Ak);y|A|+1)+Ak+1AGa,b(y(Ak+1))Ak+1AGa,b(y(Ak+1))]+JA.

Hence

JA+1=(2)|A|+1Ga,b(y(A+1))+k=0|A|1(2)k+1[AkAGa,b(y(Ak);y|A|+1)+AkA+1Ga,b(y(Ak))]k=0|A|1(2)k+1[Ak+1AGa,b(y(Ak+1))]+JA.

By noting that

AkAGa,b(y(Ak);y|A|+1)+AkA+1Ga,b(y(Ak))=AjA+1Ga,b(y(Aj)),

we can rewrite 𝕁A+1 as the following

JA+1=(2)|A|+1Ga,b(y(A+1))+k=0|A|1(2)k+1Ak+1A+1Ga,b(y(Ak+1))k=0|A|1(2)k+1[Ak+1AGa,b(y(Ak+1))]+JA.

Rearranging the summation leads to

JA+1=(2)|A|+1Ga,b(y(A+1))+j=1|A|(2)jAjA+1Ga,b(y(Aj))k=0|A|1(2)k+1[Ak+1AGa,b(y(Ak+1))]+JA,

which implies

JA+1=j=1|A|+1(2)jAjA+1Ga,b(y(Aj))+JAj=1|A|(2)j[AjAGa,b(y(Aj))].

Using the induction assumption up to 𝕁A, we have

JAj=1|A|(2)j[AjAGa,b(y(Aj))]=Ga,b(y())=ψ(a,x,t,T).

Thus

JA+1=j=1|A|+1(2)jAjA+1Ga,b(y(Aj))+ψ(a,x,t,T).

In other words

JA+1=j=0|A|+1(2)jAjA+1Ga,b(y(Aj)).

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

ft1(Rt,RVt)=ft1(Rt|RVt)ft1(RVt)

ft1(Rt|RVt)=(2π)n2|RVt|12exp(12z1,tz1,t)

where

z1,t=RVt1/2[Rtμt1(RVtΣt1)δ],

and

ft1(RVt)=2pn/2|σVσ|p/2Γn(p2)1|Wt|pn12exp(Tr((σVσ)1Wt)2),

where

Wt=σ1(RVtΣt1)(σ1)+pV.

E.2 Model characteristics

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

Σt=ω+βΣt1β+αRVtα=ω+βΣt1β+α{Σt1+σ[WtpV]σ}α=ω+βΣt1β+αΣt1α+ασWtσαpασVσα=ωpασVσα+βΣt1β+αΣt1α+ασWtσα.

Given that

Σt=ωpασVσα+βΣt1β+αΣt1α+ασWtσα,

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:

ωpασVσα0,

and we can express ω as

ω=pασVσα+γγ.

Further, given that

Et1[Σt]=ω+βΣt1β+αΣt1α,

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

Et1[exp(eiRt)]=exp(rf+λieiΣt1ei)

Et1[exp(eiRt)]=Et1[exp(eiμt1+ei(RVtΣt1)δ+eiRVt1/2z1,t)]=Et1[exp(eiμt1+eiσ[WtpV]σδ+eiRVt1/2z1,t)]=Et1[exp(eiμt1+eiσ[WtpV]σδ+12eiRVtei)]=Et1[exp(eiμt1+eiσ[WtpV]σδ+12ei(Σt1+σ[WtpV]σ)ei)]=Et1[exp(eiμt1+eiσ[WtpV]σδ+12(eiΣt1ei+eiσ[WtpV]σei))]

=Et1[exp(eiμt1+eiσWtσδpeiσVσδ+12eiΣt1ei+12eiσWtσeip2eiσVσei)]=Et1[exp(eiμt1+12eiΣt1eipeiσVσδp2eiσVσei+eiσWtσδ+12eiσWtσei)]=Et1[exp(eiμt1+12eiΣt1eipeiσVσδp2eiσVσei+Tr((σδeiσ+12σeieiσ)Wt))]=Et1[exp(eiμt1+12eiΣt1eipeiσVσδp2eiσVσeip2ln[det(In2(σδeiσ+12σeieiσ)V)])].

Hence

eiμt1=rf+(λi12)eiΣt1ei+peiσVσδ+p2eiσVσei+p2ln[det(In2(σδeiσ+12σeieiσ)V)].

E.4 Proof of the moment-generating functions

E.4.1 Proof of Proposition 5

Et1[exp(uRt+Tr(θΣt))]=exp(A(u,θ)+Tr(B(u,θ)Σt1))

with

A(u,θ)=i=1nui(rf+peiσVσδ+p2eiσVσei+p2ln[det(In2(σδeiσ+12σeieiσ)V)])p2ln[det(In2(12σuuσ+σδuσ+σαθασ)V)]puσVσδp2uσVσu+Tr(θ(ωpασVσα))

B(u,θ)=βθβ+αθα+12uu+i=1neiui(λi12)ei.

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

Et1[exp(uRt+Tr(θΣt))]=Et1[exp(u(μt1+(RVtΣt1)δ+RVt1/2z1,t)+Tr[θ(ωpασVσα+βΣt1β+αΣt1α+ασWtσα)])]=Et1[exp(u(μt1+σWtσδpσVσδ)+12uRVtu+Tr[θ(ωpασVσα+βΣt1β+αΣt1α+ασWtσα)])]=Et1[exp(u(μt1+σWtσδpσVσδ)+12uΣt1u+12uσWtσup2uσVσu+Tr[θ(ωpασVσα+βΣt1β+αΣt1α+ασWtσα)])]=Et1[exp(uμt1puσVσδp2uσVσu+Tr(θ(ωpασVσα))+Tr((βθβ+αθα+12uu)Σt1)+Tr[(12σuuσ+σδuσ+σαθασ)Wt])]

=exp(uμt1puσVσδp2uσVσu+Tr(θ(ωpασVσα))+Tr((βθβ+αθα+12uu)Σt1)p2ln[det(In2(12σuuσ+σδuσ+σαθασ)V)])=exp(i=1nuieiμt1puσVσδp2uσVσu+Tr(θ(ωpασVσα))+Tr((βθβ+αθα+12uu)Σt1)p2ln[det(In2(12σuuσ+σδuσ+σαθασ)V)])=exp(i=1nui(rf+(λi12)eiΣt1ei+peiσVσδ+p2eiσVσei+p2ln[det(In2(σδeiσ+12σeieiσ)V)])puσVσδp2uσVσu+Tr(θ(ωpασVσα))+Tr((βθβ+αθα+12uu)Σt1)p2ln[det(In2(12σuuσ+σδuσ+σαθασ)V)])=exp(i=1nui(rf+peiσVσδ+p2eiσVσei+p2ln[det(In2(σδeiσ+12σeieiσ)V)])p2ln[det(In2(12σuuσ+σδuσ+σαθασ)V)]puσVσδp2uσVσu+Tr(θ(ωpασVσα))+Tr[(βθβ+αθα+12uu+i=1neiui(λi12)ei)Σt1]).

E.4.2 Proof of Proposition 6

Et[exp(ui=1τ+1Rt+i)]=Et[exp(uRt+1)Et+1[exp(ui=2τ+1Rt+i)]]=Et[exp(uRt+1)Et+1[exp(uj=1τRt+1+i1)]]=Et[exp(uRt+1+C(u;τ)+Tr(D(u;τ)Σt+1))]=exp(C(u;τ))Et[exp(uRt+1+Tr(D(u;τ)Σt+1))]=exp(C(u;τ)+A(u,D(u;τ))+Tr(B(u,D(u;τ))Σt)).

References

Ahn, D., and B. Gao. 1999. “A Parametric Nonlinear Model of Term Structure Dynamics.” Review of Financial Studies 12: 721–762. http://rfs.oxfordjournals.org/content/12/4/721.abstract.10.1093/rfs/12.4.721Search in Google Scholar

Andersen, L., and J. Andreasen. 2000. “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing.” Review of Derivatives Research 4: 231–262. http://dx.doi.org/10.1023/A%3A1011354913068.Search in Google Scholar

Andersen, T. G., T. Bollerslev, F. X. Diebold, and H. Ebens. 2001. “The Distribution of Realized Stock Return Volatility.” Journal of Financial Economics 61: 43–76. http://www.sciencedirect.com/science/article/pii/S0304405X01000551.Search in Google Scholar

Bakshi, G., C. Cao, and Z. Chen. 1997. “Empirical Performance of Alternative Option Pricing Models.” The Journal of Finance 52: 2003–2049. http://dx.doi.org/10.1111/j.1540-6261.1997.tb02749.x.10.1111/j.1540-6261.1997.tb02749.xSearch in Google Scholar

Bates, D. 1996. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” Review of Financial Studies 9: 69–107. http://rfs.oxfordjournals.org/content/9/1/69.abstract.10.1093/rfs/9.1.69Search in Google Scholar

Buraschi, A., P. Porchia, and T. Fabio. 2010. “Correlation Risk and Optimal Portfolio Choice.” The Journal of Finance 65: 393–420. http://dx.doi.org/10.1111/j.1540-6261.2009.01533.x.10.1111/j.1540-6261.2009.01533.xSearch in Google Scholar

Buraschi, A., P. Porchia, and T. Fabio. 2014. “When Uncertainty Blows in the Orchard: Comovement and Equilibrium Volatility Risk Premia.” The Journal of Finance 69: 101–137. http://dx.doi.org/10.1111/jofi.12095.10.1111/jofi.12095Search in Google Scholar

Buss, A., and G. Vilkov. 2012. “Measuring Equity Risk with Option-Implied Correlations.” Review of Financial Studies 25: 3113–3140. http://rfs.oxfordjournals.org/content/25/10/3113.abstract.10.1093/rfs/hhs087Search in Google Scholar

Chang, C.-C., S.-L. Chung, and M.-T. Yu. 2006. “Loan Guarantee Portfolios and Joint Loan Guarantees with Stochastic Interest Rates.” The Quarterly Review of Economics and Finance 46: 16–35. http://www.sciencedirect.com/science/article/pii/S1062976903000917.10.1016/j.qref.2003.07.004Search in Google Scholar

Christoffersen, P., S. Heston, and K. Jacobs. 2006. “Option Valuation with Conditional Skewness.” Journal of Econometrics 131: 253–284. http://www.sciencedirect.com/science/article/pii/S0304407605000126.10.1016/j.jeconom.2005.01.010Search in Google Scholar

Christoffersen, P., B. Feunou, K. Jacobs, and N. Meddahi. 2014. “The Economic Value of Realized Volatility: Using High-Frequency Returns for Option Valuation.” Journal of Financial and Quantitative Analysis 49: 663–697. http://journals.cambridge.org/article_S0022109014000428.10.1017/S0022109014000428Search in Google Scholar

Davies, R. B. 1973. “Numerical Inversion of a Characteristic Function.” Biometrika 60: 231–262.10.1093/biomet/60.2.415Search in Google Scholar

Duffie, D., J. Pan, and K. Singleton. 2000. “Transform Analysis and Asset Pricing for Affine Jump-Diffusions.” Econometrica 68: 1343–1376. http://dx.doi.org/10.1111/1468-0262.00164.10.1111/1468-0262.00164Search in Google Scholar

Duffie, D., D. Filipović, and W. Schachermayer. 2003. “Affine Processes and Applications in Finance.” Annals of Applied Probability 13: 984–1053.10.1214/aoap/1060202833Search in Google Scholar

Duffy, D. J. 2009. “Numerical Analysis of Jump Diffusion Models: A Partial Differential Equation Approach.” Wilmott Magazine.Search in Google Scholar

Dufresne, D., J. Garrido, and M. Morales. 2009. “Fourier Inversion Formulas in Option Pricing and Insurance.” Methodology and Computing in Applied Probability 11: 359–383. http://dx.doi.org/10.1007/s11009-007-9049-z.10.1007/s11009-007-9049-zSearch in Google Scholar

Engle, R. 2000. “Dynamic Conditional Correlation – A Simple Class of Multivariate Garch Models.” Journal of Business and Economic Statistics 20: 339–350.10.1198/073500102288618487Search in Google Scholar

Feunou, B., and J.-S. Fontaine. 2010. “Discrete Choice Term Structure Models: Theory and Applications.” Working Paper, Duke University and Bank of Canada.10.2139/ssrn.1772971Search in Google Scholar

Feunou, B., and N. Meddahi. 2009. “Generalized Affine Model, Provides a General Framework that Characterizes Infinite Order Affine Model.” Working Paper, Duke University.10.2139/ssrn.1367033Search in Google Scholar

Forsberg, L., and T. Bollerslev. 2002. “Bridging the Gap Between the Distribution of Realized (ecu) Volatility and Arch Modelling (of the euro): The Garch-nig Model.” Journal of Applied Econometrics 17: 535–548. http://dx.doi.org/10.1002/jae.685.10.1002/jae.685Search in Google Scholar

Gay, G. D., and S. Manaster. 1984. “The Quality Option Implicit in Futures Contracts.” Journal of Financial Economics 13: 353–370. http://www.sciencedirect.com/science/article/pii/0304405X84900047.10.1016/0304-405X(84)90004-7Search in Google Scholar

Gerber, H. U., and E. S. W. Shiu. 1994. “Option Pricing by Esscher Transforms.” Transactions of the Society of Actuaries 46: 99–191.Search in Google Scholar

Glasserman, P. 2003. Monte Carlo Methods in Financial Engineering. 1st ed. New York, NY, USA: Springer.10.1007/978-0-387-21617-1Search in Google Scholar

Gourieroux, C. 2006. “Continuous Time Wishart Process for Stochastic Risk.” Econometric Reviews 25: 177–217. http://dx.doi.org/10.1080/07474930600713234.10.1080/07474930600713234Search in Google Scholar

Gourieroux, C. and R. Sufana. 2010. “Derivative Pricing with Wishart Multivariate Stochastic Volatility.” Journal of Business and Economic Statistics 28: 438–451. http://dx.doi.org/10.1198/jbes.2009.08105.10.1198/jbes.2009.08105Search in Google Scholar

Hazewinkel, M. 2002. Encyclopaedia of Mathematics. 1st ed. Berlin Heidelberg New York, NY, USA: Springer-Verlag.Search in Google Scholar

Heston, S. 1993. “A closed-Form Solution for Options with Stochastic Volatility with Applications To Bond and Currency Options.” Review of Financial Studies, 6: 327–343. http://rfs.oxfordjournals.org/content/6/2/327.abstract.10.1093/rfs/6.2.327Search in Google Scholar

Johnson, H. 1987. “Options on The Maximum or The Minimum of Several Assets.” Journal of Financial and Quantitative Analysis 22: 277–283. http://journals.cambridge.org/article_S002210900001262X.10.2307/2330963Search in Google Scholar

Kempf, A., O. Korn, and S. Saßning. 2014. “Portfolio Optimization Using Forward-Looking Information.” Review of Finance. DOI: 10.1093/rof/rfu006. http://rof.oxfordjournals.org/content/early/2014/03/07/rof.rfu006.abstract.Search in Google Scholar

Margrabe, W. 1978. “The Value of An Option to Exchange One Asset for Another.” The Journal of Finance 33: 177–186. http://dx.doi.org/10.1111/j.1540-6261.1978.tb03397.x.10.1111/j.1540-6261.1978.tb03397.xSearch in Google Scholar

Martzoukous, S. H. 2001. “The Option on n Assets with Exchange Rate and Exercise Price Risk.” Journal of Multinational Financial Management 11: 1–15.10.1016/S1042-444X(00)00039-6Search in Google Scholar

Navatte, P., and C. Villa. 2000. “The Information Content of Implied Volatility, Skewness and Kurtosis: Empirical Evidence From Long-term cac 40 Options.” European Financial Management, 6: 41–56. http://dx.doi.org/10.1111/1468-036X.00110.10.1111/1468-036X.00110Search in Google Scholar

Qu, D. 2010. “Pricing Basket Options with Skew.” Wilmott Magazine.Search in Google Scholar

Rota, G.-C. 1964. “On the Foundations of Combinatory Theory of Mö,bius Functions.” Wahrscheinlichkeitstheorie und Verw 2: 340–368.10.1007/BF00531932Search in Google Scholar

Shephard, N., and K. Sheppard. 2010. “Realising the Future: Forecasting with High frequency-Based Volatility (heavy) Models.” Journal of Applied Econometrics 25: 197–231. http://dx.doi.org/10.1002/jae.1158.10.1002/jae.1158Search in Google Scholar

Stoll, R. M. 1969. “The Relationship Between Put and Call Option Prices.” Journal of Finance 24: 801–824.10.1111/j.1540-6261.1969.tb01694.xSearch in Google Scholar

Stulz, R. M. 1982. “Options on The Minimum or The Maximum of Two Risky Assets: Analysis and Applications.” Journal of Financial Economics 10: 161–185. http://www.sciencedirect.com/science/article/pii/0304405X82900113.10.1016/0304-405X(82)90011-3Search in Google Scholar

van Binsbergen, J., M. Brandt, and R. Koijen. 2012. “On the Timing and Pricing of Dividends.” American Economic Review 102: 1596–1618. http://www.aeaweb.org/articles.php?doi=10.1257/aer.102.4.1596.10.1257/aer.102.4.1596Search in Google Scholar

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

©2015 by De Gruyter