Detection of Arbitrage in a Market with Multi‐Asset Derivatives and Known Risk‐Neutral Marginals

Bertrand Tavin

(Université Paris 1 ‐ Panthéon Sorbonne)

Paper presented at the

10th International Paris Finance Meeting

December 20, 2012

www.eurofidai.org/december2012.html

Organization: Eurofidai & AFFI

Detection of arbitrage in a market with multi-asset derivatives

and known risk-neutral marginals

Bertrand TAVIN ∗

Universite Paris 1 - Pantheon Sorbonne

btavin@gmail.com

November 6, 2012

Abstract

In this paper we study the existence of arbitrage opportunities in a multi-asset market when risk-

neutral marginal distributions of asset prices are known. We first propose an intuitive characterization

of the absence of arbitrage opportunities in terms of copula functions. We then address the problem

of detecting the presence of arbitrage by formalizing its resolution in two distinct ways that are both

suitable for the use of optimization algorithms. The first method is valid in the general multivariate

case and is based on Bernstein copulas that are dense in the set of all copula functions. The second

one is easier to work with but is only valid in the bivariate case. It relies on recent results about

improved Frechet-Hoeffding bounds in presence of additional information. For both methods, details

of implementation steps and empirical applications are provided.

Keywords : Arbitrage, Multi-Asset Derivative, Incomplete Market, Risk-Neutral Measure, MultivariateDistribution, Copula Function.

JEL Classification : G10, C52, D81.

AMS2010 Classification : 91G20, 62H20, 91G60.

∗First version : January 10, 2012. Acknowledgments : I am grateful to my PhD supervisor Prof. Patrice PONCET(ESSEC Business School) for his comments and helpful discussions. This paper has been presented at the MathematicalFinance Days, held in Montreal (May 2012), at the 7th Congress of the Bachelier Finance Society (BFS2012), held in Sydney(June 2012), at the 16th Congress on Insurance Mathematics and Economics (IME2012), held in Hong Kong (June 2012) andat the Copulae in Mathematical and Quantitative Finance workshop, held in Krakow (July 2012). I thank the participantsof these events for their questions and remarks. I also thank Jan Dhaene and Steven Vanduffel for their comments onearlier versions. All remaining errors are mine. Contact: Laboratoire PRISM - EA4101, 17 rue de la Sorbonne 75005 ParisFrance. Tel: +33140463170 Fax: +33140463011

1

1 Introduction

The notion of arbitrage is fundamental in economics and finance, it underpins the setup in which aca-demics and practitioners solve questions about equilibrium, portfolio allocation and contingent claimvaluation. In these disciplines, many theoretical developments are thus built on the absence of arbitrageopportunity as central assumption. For institutions involved in the financial industry, it is a strategicissue to ensure that their systems do not produce such opportunities. Hence, the availability of methodsto detect arbitrage is of great interest.

In a market with a single underlying asset and a given set of vanilla options, the assessment of theabsence of arbitrage is addressed in [6] (Carr & Madan, 2005), [15] (Davis & Hobson, 2007) and [10](Cousot, 2007). Essentially, the set of option prices is free of arbitrage as soon as butterfly spreads,call spreads and calendar spreads have positive prices. Assessing the absence of arbitrage among a set ofderivative prices becomes a much more involved task when the set under scrutiny has some exotic optionsin addition to vanillas, or in the case of a market with multiple underlying assets. The concern of ourpaper is to address the latter case in a general way that does not rely on the structure of a particularpayoff and that is valid beyond the two-dimensional case. To the best of our knowledge, it has not yetbeen done in the existing literature. Our setup corresponds to a one period multi-asset market withknown risk-neutral marginals, in which we obtain a characterization of the absence of arbitrage amonga set of derivative prices in terms of copula functions. This characterization allows us to derive twonecessary conditions of no-arbitrage, one of which is also sufficient, that both naturally lead to detectionmethods in the sense that if a condition is not verified then the market is not free of arbitrage. Henceour contribution is twofold. First, from a theoretical standpoint it allows a better understanding of theabsence of arbitrage in our market model. And second, with practical perspectives, we detail the detectionmethods that are deduced from the theoretical part and we apply them to real market situations.

For a single risky asset, it is possible to build risk-neutral diffusions that are compatible with a givenset of vanilla options. Early references on that topic are [24] (Dupire, 1993) and [41] (Laurent &Leisen, 2000), the former considers a local volatility diffusion coefficient and the latter considers theconstruction of a risk-neutral Markov chain consistent with observed call option prices. It is possible togo further and to obtain no-arbitrage bounds for an additional derivative when some are already available.This question has already been partially addressed. It is closely related but yet different from our concern.An approach to obtain the desired bounds for Asian options is based on the concept of comonotonicity,see [7] (Chen et al., 2008) and references therein. Another approach to obtain the desired bounds is via aSkorokhod embedding problem formulation, see [32] (Hobson, 2010) and references therein. Yet another,more recent, approach is to apply optimal mass transportation theory to obtain the desired bounds, see[3] (Beiglbock, Henry-Labordere & Penkner, 2011) and [29] (Galichon, Henry-Labordere &Touzi, 2011).

In the multi-asset case, bounds on prices of options written on several underlyings are available.When marginals are known, upper and lower bounds for two-asset basket options are initially obtainedin [23] (Dhaene & Goovaerts, 1996), in a context of actuarial analysis of portfolios of dependentrisks. The same upper and lower bounds are obtained for other two-asset option payoffs in [52] (Rapuch

& Roncalli, 2001). [59] (Tankov, 2011) derives improved bounds when some two-asset options arealready quoted. For basket options, when single underlying vanillas are quoted, upper and lower boundsare available and the associated replicating strategies are explicit. The lower bound result is only validin the two-asset case. As for Asian options, the above mentioned comonotonicity approach can be used,see [21] (Dhaene et al., 2002a) and [22] (Dhaene et al., 2002b). Key results for basket options are in[33] (Hobson at al., 2005a) and [7] (Chen et al., 2008). See also [38] (Laurence & Wang, 2005) and[34] (Hobson, et al., 2005b). [2] (d’Aspremont & El Ghaoui, 2006) work with a linear programmingapproach and obtain upper and lower bounds on basket option price when other basket options, withdifferent weights, are already available. Upper and lower bounds are also available for spread options.The respective bounds are obtained in [39] (Laurence & Wang, 2008) and [40] (Laurence & Wang,2009). The case of spread options is particular because arbitrage opportunities did exist during year 2009among such options written on Constant Maturity Swap rates. This occurrence is documented in [46](McCloud, 2011).

2

The pricing of European options written on several underlying assets has been widely studied. Thisbody of research is linked to our question but, as it is, does not answer it. The classical approachis to postulate a joint distribution for the underlying asset price returns and to calibrate the distri-bution parameters to available data in order to obtain prices and hedge ratios. For example, with thisapproach [45] (Margrabe, 1978) and [58] (Stulz, 1982) both work in a two-asset extension of the Black-Scholes-Merton model and obtain valuation formulas, respectively, for exchange and rainbow options. [1](Alexander & Scourse, 2004) propose a bivariate distribution built as a mixture for the pricing andhedging of spread options. [20] (Dempster et al., 2008) also study spread options and directly model thespread process in a cointegrated two-commodity framework. Nevertheless, in many cases, it is preferableto proceed in two steps by first specifying the marginals and then choosing the dependence structure.This alternative approach relies on the power of copula functions for the modeling of dependence andit allows an easier identification and understanding of the potential sources of risk. See among others[52] (Rapuch & Roncalli, 2001), [11] (Coutant & Durrleman, 2001), [8] (Cherubini & Luciano,2002) and [53] (Rosenberg, 2003).

The remainder of the paper is organized as follows. In section 2 we explain our financial frameworkand detail the implications of our assumptions. In section 3 we propose a characterization of the absenceof arbitrage in terms of copula functions. In section 4 we develop a first methodology based on the familyof Bernstein copulas. In section 5 we propose, for the two-asset case, another methodology based onimproved Frechet-Hoeffding bounds. Section 6 concludes.

3

2 Model and Assumptions

The presentation of our model and its structure is in the spirit of Chap. 1 in [28] (Follmer & Schied,2002) and Chap. 1 and 2 in [13] (Dana & Jeanblanc, 1998). We consider a fundamental probabilityspace (Ω,F ,P) with P the historical probability measure. Our financial market has one period and n+1primary assets (n ≥ 2), t = 0 is the initial time and t = T < +∞ is the final time. The primary assets aredenoted

(

B,S1, . . . , Sn)

. Their initial prices(

B0, S10 , . . . , S

n0

)

∈ ]0,+∞[nare known (non-random) and

their final prices are positive random variables on (Ω,F ,P) and are denoted by(

BT , S1T , . . . , S

nT

)

. The0th asset, B, is a risk-free asset. It earns the risk-free rate r ≥ 0 and its final value is non-random andknown at initial time. This asset can be seen as a money-market account or a zero-coupon bond as thosesecurities are equivalent in a one-period setting. We adopt the zero-coupon convention corresponding toB0 = 1

1+rand BT = 1, other equivalent choices are possible here.

We suppose vanilla call options of all positive strikes to be available for the n risky assets of ourmarket. For i = 1, . . . , n, we denote Ci(Ki) the call option written on Si and struck at Ki ∈ [0,+∞[

with the special case Ci(0) = Si. Its final payoff is written CiT (Ki) =

(

SiT −Ki

)+and Ci

0(Ki) denotesits initial price.

Our financial market model departs from reality on two notable characteristics. First, we considera one-period market (T = 1) where, in reality, trading can almost be done in continuous time. Thisassumption corresponds to a restriction of trading strategies to only static strategies. By static strategieswe mean buy at initial time and hold until final time. Second, we assume the availability of vanilla callprices for a continuum of positive strikes. In reality, vanilla options are traded only at a finite numberof strikes hence leaving space for ambiguity in empirical applications. This remaining ambiguity is welldocumented and can be kept acceptable for liquidly traded underlyings such as equity indices or foreignexchange rates. See among others [36] (Jackwerth & Rubinstein, 1996).

We now introduce the notion of Risk-Neutral Measure for our financial market. The set of suchmeasures is the cornerstone of the results presented in this paper because it is linked to the existence ofarbitrage.

Definition 2.1. Risk-Neutral Measure

A probability measure Q on (Ω,F), equivalent to P, is a Risk-Neutral Measure (RNM) if, for i =1, . . . , n

Ci0(K

i) = B0EQ[

CiT (K

i)]

for all Ki ∈ [0,+∞[ (2.1)

We define Q as the set of Risk-Neutral Measures for our financial market.

The First Fundamental Theorem of Asset Pricing establishes the link between the set of risk-neutralmeasures and the absence of arbitrage opportunity. It has been first obtained in discrete time in [30](Harrisson & Kreps, 1979) and in continuous time in [31] (Harrisson & Pliska, 1981). For proofs,details, further references and extensions see [28] (Follmer & Schied, 2002) and [18] (Delbaen &Schachermayer, 2006).

Theorem 2.2. (Harrisson & Kreps, 1979) (Harrisson & Pliska, 1981)There is no arbitrage opportunity in the financial market if and only if Q is non-empty.

In the sequel we suppose the above described basic market to be free of arbitrage so that Q is non-empty. Let Q ∈ Q. We denote by HQ the joint distribution of (S1

T , . . . , SnT ) under Q and FQ

1 , . . . , FQn its

marginals that are the univariate distributions of S1T , . . . , S

nT under Q. For definitions and details about

univariate and multivariate distributions see [57] (Sklar, 1973), Chap. 1 in [37] (Joe, 1997) and §2.10in [48] (Nelsen, 2006).

4

With the knowledge of initial call prices for all positive strikes, univariate distributions can be re-covered independently of Q via the Breeden-Litzenberger formula. See [5] (Breeden & Litzenberger,1978). Under any risk-neutral measure in Q, univariate distributions of S1

T , . . . , SnT are identical, written

F1, . . . , Fn and the Breeden-Litzenberger formula writes, for i = 1, . . . , n

Fi(x) = 1−1

B0

∂Ci0

∂Ki

(x) x ≥ 0 (2.2)

We also suppose that the available call prices are such that the Fi : [0,+∞[−→ [0, 1] are continuousfunctions. This assumption is essentially technical and corresponds to the realistic and fairly general casein which no particular final value taken by a risky asset is charged with a strictly positive probabilisticmass.

In our setup there are n sub-markets composed of the risk-free asset, one risky asset and all thecorresponding call options. All such sub-markets are complete in the sense that any derivative written onthe considered risky asset admits a unique no-arbitrage price. The only relevant information to computethis no-arbitrage price is the univariate distribution of its underlying asset price. And this univariatedistribution is shared by all measures in Q. On the contrary our multi-asset financial market staysincomplete as a derivative written on two or more risky assets will, in general, have multiple no-arbitrageprices.

Multi-asset derivatives are now introduced to complement the basic market. A multi-asset derivativeZ is a derivative written on up to n risky assets. Its final payoff is a positive random variable on (Ω,F ,P)written ZT = z(S1

T , . . . , SnT ) for some positive function z on [0,+∞[n. The notion of no-arbitrage price

corresponds to a price at which the considered multi-asset derivative can be added to the basic marketwithout creating arbitrage opportunity. The definition is formally stated below.

Definition 2.3. No-arbitrage Price for a multi-asset derivative

Let Z be a multi-asset derivative with payoff function z. We say that π ≥ 0 is a no-arbitrage pricefor Z if it is compatible with, at least, one risk-neutral measure. That is if there exists Q ∈ Q such thatπ = B0E

Q [ZT ]. Π(Z) denotes the set of no-arbitrage prices for Z and is written

Π(Z) =

B0EQ [ZT ] | Q ∈ Q such that EQ [ZT ] < ∞

(2.3)

The notion of no-arbitrage price vector extends the one of no-arbitrage price. It corresponds to avector of prices at which a finite set of multi-asset derivatives

(

Z1, . . . , Zq)

can be sequentially addedto the basic market without creating arbitrage opportunity at any step and irrespective of the additionorder.

Definition 2.4. No-arbitrage Price Vector

Let(

Z1, . . . , Zq)

be a finite set of multi-asset derivatives (q ≥ 1) written on up to n risky assets (and

at least two) and with final payoffs(

Z1T , . . . , Z

qT

)

. π = (π1, . . . , πq) ∈ [0,+∞[q is a no-arbitrage price

vector for(

Z1, . . . , Zq)

if it is compatible with, at least, one risk-neutral measure in Q, that is if there

exists Q ∈ Q such that πk = B0EQ[

ZkT

]

for k = 1, . . . , q. Π(Z1, . . . , Zq) denotes the set of no-arbitrage

price vectors for(

Z1, . . . , Zq)

, it is a subset of [0,+∞[q and is written

Π(Z1, . . . , Zq) =(

B0EQ[

Z1T

]

, . . . , B0EQ [Zq

T ])

| Q ∈ Q such that EQ[

ZkT

]

< ∞, k = 1, . . . , q

(2.4)

To conclude this section, recall that our goal is to detect the existence of arbitrage in a multi-assetmarket. In our setup, the check is straightforward for single underlying derivatives because the associatedsub-markets are complete. Hence our focus in the sequel of the paper will be on the q multi-assetderivatives rather than on the n risky assets.

5

3 Arbitrage and copula functions in a multi-asset market

We begin this section with useful elements about copula functions. We then build a characterization ofthe absence of arbitrage in our market model in terms of such functions.

3.1 Basic properties of copula functions

Definition 3.1. Copula Function

A n-dimensional copula function (or copula) is a n-dimensional joint distribution function on [0, 1]n

with uniform marginals. We denote C(n) the set of all n-dimensional copula functions.

Sklar’s Theorem is the fundamental result for the use of copulas in multivariate probabilistic modeling.For the original statement and proof, see [56] (Sklar, 1959) and [57] (Sklar, 1973).

Theorem 3.2. (Sklar, 1959) (Sklar, 1973)If H is a n-dimensional joint distribution function on [0,+∞[n with continuous marginals denoted

F1, . . . , Fn, then there exists a unique copula CH ∈ C(n) such that, ∀x ∈ [0,+∞[n

H(x) = CH (F1 (x1) , . . . , Fn (xn)) (3.1)

And, ∀u ∈ [0, 1]n

CH(u) = H(

F−11 (u1) , . . . , F

−1n (un)

)

(3.2)

Conversely, if F1, . . . , Fn are continuous one-dimensional distribution functions on [0,+∞[, and ifC ∈ C(n), then HC : [0,+∞[n−→ [0, 1] such that HC(x) = C (F1 (x1) , . . . , Fn (xn)) is a n-dimensionaljoint distribution function with marginals F1, . . . , Fn.

Thanks to the second part of the Theorem it is possible to build a joint distribution in two stepsby separately specifying the marginals and the dependence structure. In finance this two-step approachallows a proper understanding and analysis of risk factors in a multi-asset framework. Let C ∈ C(n). Cis n-increasing and it is a Lipschitz function with constant 1, that is, for all a, b ∈ [0, 1]n

|C(a) − C(b)| ≤n∑

i=1

|ai − bi|

It is well known that C(n) admits upper and lower bounds with respect to the concordance order. Thesebounds are named the Frechet-Hoeffding bounds and respectively denoted C+ and C− [see AppendixA.1]. The set C(n) is a convex and compact set under the topology of uniform convergence. In C(n),pointwise and uniform convergences are equivalent. See [16] (Deheuvels, 1978) and [14] (Darsow etal., 1992).

Further details and properties of multivariate copulas can be found §2.10 in [48] (Nelsen, 2006) andin [25] (Durante & Sempi, 2010). The latter reference also has a recent and wide literature review onthe topic.

Two classical representatives of multivariate copula functions are the Gaussian and Student’s t copulas,both corresponding to the dependence structure of the associated multivariate distributions. We use thesecopulas in the sequel of the paper for illustrative purposes and their expressions are given below The n-dimensional Gaussian copula is written, for u ∈ [0, 1]n

CGR (u) =

∫ Φ−1(u1)

−∞

. . .

∫ Φ−1(un)

−∞

1√

(2π)n| detR|exp

(

−1

2x′R−1x

)

dx (3.3)

where R is a n × n correlation matrix and Φ−1 is the inverse cumulative distribution function of thestandard Gaussian distribution. The n-dimensional Student’s t copula is written, for u ∈ [0, 1]n

Ctν,R(u) =

∫ t−1ν (u1)

−∞

. . .

∫ t−1ν (un)

−∞

Γ(

ν+n2

)

Γ(

ν2

)√

(νπ)n| detR|

(

1 +1

νx′R−1x

)−( ν+n2 )

dx (3.4)

6

where x′ is the transposed of x, Γ denotes the usual gamma function, R is a n × n correlation matrix,ν ≥ 1 and t−1

ν is the inverse cumulative distribution function of the standard Student’s t distributionwith ν degrees of freedom. For more details on the Student’s t copula, see [19] (Demarta & McNeil,2005).

3.2 Characterization in the general case

We consider that a finite set of multi-asset derivatives(

Z1, . . . , Zq)

is available in the market at initialprice vector π = (π1, . . . , πq). Our arbitrage detection problem is solved by assessing whether or not πbelongs to Π

(

Z1, . . . , Zq)

, the associated set of no-arbitrage price vectors. Given the structure of Π in(2.4), we need to explore the set of risk-neutral models Q to check whether π is compatible with, at least,one of them. As we are working with fixed marginals, copula functions are ineluctably going to appear.We begin the section by a Proposition on the link between Q, the set of risk-neutral measures, and theset C(n) of n-dimensional copula functions. Following this remark we will work with copula functionsbecause it is easier than working with multivariate distributions, as argued in [57] (Sklar, 1973).

Proposition 3.3. In the multi-asset market model detailed in Section 2, where univariate risk-neutraldistributions of asset prices are known and continuous, the set Q is isomorphic to the set C(n) of n-dimensional copula functions.

Proof. The isomorphism between Q and C(n) is built thanks to Sklar’s theorem and its corollary. Thecorrespondence is written as follows.

(from Q to C(n)) Let Q ∈ Q and denote HQ the joint distribution of(

S1T , . . . , S

nT

)

under Q. Thecorresponding copula CQ is defined for u ∈ [0, 1]n as

CQ(u) = HQ(

F−11 (u1), . . . , F

−1n (un)

)

(3.5)

(from C(n) to Q) Let C ∈ C(n) and denote HC the corresponding distribution, defined for x ∈ [0,+∞[n

asHC(x) = C (F1(x1), . . . , Fn(xn)) (3.6)

Let QC be the probability measure on [0,+∞[n associated with HC . By construction, the jointdistribution of

(

S1T , . . . , S

nT

)

under QC has marginals F1, . . . , Fn, given by the Breeden-Litzenbergerformula (2.2) so that all vanilla calls are properly repriced. Hence QC ∈ Q.

For C ∈ C(n) we introduce the notation EC denoting the expectation under QC the probability measureassociated with HC given by (3.6). Under QC ,

(

S1T , . . . , S

nT

)

has a dependence structure described by

the copula C and marginals F1, . . . , Fn. For C ∈ C(n) and k = 1, . . . , q the expectation EC[

ZkT

]

can bewritten as a multi-dimensional integral on [0, 1]n as

EC[

ZkT

]

= EC[

zk(

S1T , . . . , S

nT

)]

=

∫

[0,+∞[nzk(s1, . . . , sn)dC(F1(s1), . . . , Fn(sn))

=

∫

[0,1]nzk(F−1

1 (u1), . . . , F−1n (un))dC(u1, . . . , un) (3.7)

If C is absolutely continuous, it has density c = ∂nC∂u1...∂un

and the integral becomes

EC[

ZkT

]

=

∫

[0,1]nzk(F−1

1 (u1), . . . , F−1n (un))c(u1, . . . , un)du1, . . . , dun (3.8)

Proposition 3.4. Under the same conditions as in Definition (2.4), the set of no-arbitrage vectors forZ1, . . . , Zq can be straigthforwardly rewritten

Π(Z1, . . . , Zq) =

(

B0EC[

Z1T

]

, . . . , B0EC [Zq

T ])

| C ∈ C(n)

(3.9)

7

We denote by ρ the pricing rule between C(n) and Π(Z1, . . . , Zq) so that elements of Π are writtenρ(C) =

(

ρ1(C), . . . , ρq(C))

.

ρ : C ∈ C(n) 7−→ ρ(C) ∈ Π(Z1, . . . , Zq) ⊂ [0,+∞[q (3.10)

It is important to note that Π(Z1, . . . , Zq) is a strict subset of the Cartesian product of individualsets of no-arbitrage prices Π(Z1)×· · ·×Π(Zq). In most cases it has a complex structure. As an example,consider a market with three risky assets on which three multi-asset derivatives are available. Thesederivatives are calls with identical strikes. The first one is written on the best of the three assets, thesecond is written on the worst and the third is written on the second to best. The price of any of thethree derivatives is fully determined by the prices of the two others and of the three corresponding singleasset vanilla calls with identical strikes.

Intuitively, when a price is fixed for a first multi-asset derivative, the freedom to choose a copula toobtain a price for a second one is reduced. For the two prices to be free of arbitrage the pricing copulahas to be chosen among those compatible with the first price.

Our concern is to determine whether a given π belongs or not to Π(Z1, . . . , Zq). With (3.9) the newexpression of Π(Z1, . . . , Zq), it can be done by exploring the set C(n). If the search fails to find a copulacompatible with π we will deduce that π is out of the set and hence not free of arbitrage.

We define the q subsets of C(n) composed of n-dimensional copulas compatible with single componentsof π. These subsets are written, for k = 1, . . . , q

Ck,(n) =

C ∈ C(n) | B0EC[

ZkT

]

= πk

=

C ∈ C(n) | ρk (C) = πk

and Ck,(n) is empty if πk /∈ Π(Zk). The twofold characterization of no arbitrage in terms of copulascan now be stated. It is directly obtained by examining definitions of Π, ρ and Ck,(n).

Proposition 3.5. Let π ∈ [0,+∞[q, the following two characterizations hold

π ∈ Π(Z1, . . . , Zq) ⇐⇒ ∃C ∈ C(n) such that ρ(C) = π (3.11)

π ∈ Π(Z1, . . . , Zq) ⇐⇒

q⋂

k=1

Ck,(n) 6= ∅ (3.12)

Each characterization lead to a method to detect arbitrage. The first one allows us to express theassociated problem as a calibration problem. The second one is useful when n = 2, because in thatparticular case best-possible bounds of the Ck,(n) sets can be computed in a quasi-explicit form. Beforedeveloping the details of these arbitrage detection methods we discuss the particular case of a singlemulti-asset derivative.

3.3 The case of a single multi-asset derivative

In this subsection we make some comments about the particular case of a market in which a singlemulti-asset derivative Z, written on exactly n risky assets, is available at initial price π ≥ 0. Recall thatΠ(Z) =

ρ(C) = B0EC [ZT ] | C ∈ C(n)

with ZT = z(

S1T , . . . , S

nT

)

for some positive payoff function zand we are interested in deciding whether π belongs to Π(Z) or not. If n = 2 and z is either 2-increasingor 2-decreasing, the Frechet-Hoeffding bounds directly lead to the bounds of Π(Z).

The definitions of, and details about, Frechet-Hoeffding bounds and n-increasing functions are gath-ered in Appendix A.1.

Proposition 3.6. (Rapuch & Roncalli, 2001)

If z is 2-increasing then ∀C ∈ C(2) we have EC−

[ZT ] ≤ EC [ZT ] ≤ EC+

[ZT ] and

Π(Z) =[

ρ(

C−)

, ρ(

C+)]

(3.13)

If z is 2-decreasing then ∀C ∈ C(2) we have EC+

[ZT ] ≤ EC [ZT ] ≤ EC−

[ZT ] and

Π(Z) =[

ρ(

C+)

, ρ(

C−)]

(3.14)

8

This result is from [52] (Rapuch & Roncalli, 2001) and is based on results in [60] (Tchen, 1980)and [47] (Muller & Scarsini, 2000). An early version of this result for basket options is obtained in [23](Dhaene & Goovaerts, 1996). The expectations involving the Frechet-Hoeffding bounds are computedas one dimensional integrals because C+ and C− respectively have support u2 = u1 and u2 = 1− u1 on[0, 1]2. See Example 2.11 in [48] (Nelsen, 2006).

EC+

[ZT ] =

∫ 1

0

z(

F−11 (u), F−1

2 (u))

du (3.15)

EC−

[ZT ] =

∫ 1

0

z(

F−11 (u), F−1

2 (1− u))

du (3.16)

As it is remarked in [52] (Rapuch & Roncalli, 2001) and seconded in [59] (Tankov, 2011), allcommon two-asset options have a 2-increasing or 2-decreasing payoff function. Hence, in most encounteredcases, explicit bounds of Π(Z) are fast to compute and the check for no-arbitrage is straightforward toperform. On the contrary, for n ≥ 3 the key property is no longer valid so that the Frechet-Hoeffdingbounds do not necessarily lead to bounds on the multi-asset derivative price. See [52] (Rapuch &Roncalli, 2001) and [47] (Muller & Scarsini, 2000). Even though there exists some partial results,such as the upper bound for the basket option price obtained in [33] (Hobson, Laurence & Wang,2005a), bounds of Π(Z) cannot usually be directly computed and the check for no-arbitrage must beaddressed the same way as in the general case.

To conclude the section we present two empirical applications in which we obtain individual no-arbitrage bounds for some two-asset options, prices obtained with a Gaussian copula are also providedfor comparison. In the first application, the underlying assets are the French and German equity marketindices CAC40 and DAX30, denominated in EUR and with initial forward values normalized at 100. It isperformed with market data of May 20th 2008. The considered two-asset options have one-year maturityand are a call on the spread, a call on the equally weighted basket, a put on the maximum and a callon the minimum. In the second application, the underlyings are the foreign exchange rates EURUSDand USDJPY that correspond, respectively, to the exchange rates between the euro and the U.S dollarand between the U.S. dollar and the Japanese yen. The considered two-asset options have one-monthmaturity and are vanilla options on the cross-rate EURJPY which is the exchange rate between the euroand the Japanese yen. The application is performed with market data of January 13th 2006. Spot andforward values of EURJPY and USDJPY are divided by 100 for better readability of figures.

The marginal distribution of log-returns of each underlying asset price is a Normal Inverse Gaussian(NIG) satisfying the forward no-arbitrage condition and calibrated to vanilla option prices. The NIGdistribution parameters and the market data details are gathered in Appendix A.3. The computationsof individual no-arbitrage bounds are performed on a personal computer using the Matlab quadratureroutine quadgk.

Table 1 presents upper and lower bounds of Π (Z) for various two-asset options written on CAC40and DAX30 indices, respectively denoted by S1 and S2. This table also presents prices obtained witha Gaussian copula with correlation matrices R1 and R2 defined below. The computation time for thistable is 5 seconds.

R1 =

(

1.00 -0.50-0.50 1.00

)

R2 =

(

1.00 0.500.50 1.00

)

Table 2 presents upper and lower bounds of Π (Z) for vanilla put and call options on EURJPY, denotedX , and seen as two-asset options written on EURUSD and USDJPY exchange rates, respectively denotedS1 and S2. The relation between the two primary exchange rates and the cross is XT = S1

TS2T . Obtained

bounds on the option price are expressed according to the convention in use on FX options markets thatis JPY pips per EUR notional. It corresponds to the amount in JPY to be paid to acquire an option onEURJPY with 10000 EUR notional. Prices obtained with a Gaussian copula with correlation matricesR1 and R2 are also presented. The computation time for this table is 5 seconds.

9

ZT K ρ (C−) ρ (C+) Π (Z) ρ(

CGR1

)

ρ(

CGR2

)

(

S2T − S1

T −K)+

0 15.6 0.77 [0.77, 15.6] 13.6 7.99(

12 (S

1T + S2

T )−K)+

100 0.75 7.83 [0.75, 7.83] 4.03 6.82(

K −max (S1T , S

2T ))+

110 1.36 12.93 [1.36, 12.93] 3.18 7.53(

min (S1T , S

2T )−K

)+80 7.82 20.49 [7.82, 20.49] 9.81 14.79

Table 1: No-Arbitrage bounds for single two-asset derivatives written on CAC40 (S1) and DAX30 (S2)indices. Marginal distributions of log-returns are NIG, maturity is T = 1 year. See Appendix A.3 formore details on market data.

ZT K ρ (C−) ρ (C+) Π (Z) ρ(

CGR1

)

ρ(

CGR2

)

(K −XT )+

1.3307 101.98 0.00 [0.00, 101.98] 71.77 16.43

(K −XT )+

1.3570 173.06 0.09 [0.09, 173.06] 129.45 54.26

(K −XT )+ 1.3827 279.07 9.74 [9.74, 279.07] 244.52 144.55

(XT −K)+

1.3827 284.84 15.51 [15.51, 284.84] 215.61 150.32

(XT −K)+

1.4081 176.53 0.00 [0.00, 176.53] 135.76 41.63

(XT −K)+

1.4320 107.96 0.00 [0.00, 107.96] 78.17 13.93

Table 2: No-Arbitrage bounds for single vanilla options on EURJPY seen as two-asset options written onEURUSD and USDJPY rates. Marginal distributions of log-returns are NIG, maturity is T = 1 month.See Appendix A.3 for more details on market data.

As expected the prices obtained with a Gaussian copula lie within the computed no-arbitrage bounds.The no-arbitrage intervals obtained in Table 1 and Table 2 are wide if compared with typical option prices.This is because the derivatives under scrutiny in both tables are strongly linked to the dependence risk,making their prices very sensitive to the chosen copula function.

10

4 A method to detect arbitrage with Bernstein copulas

In this section we develop a method to detect arbitrage in the general multivariate case. This methodcorresponds to characterization (3.11) and relies on particular properties of Bernstein copulas within theset of all copulas. The possibility to use such an approach in bivariate foreign exchange markets is brieflysketched out in the conclusions of [54] (Salmon & Schleicher, 2006) and [35] (Hurd & et al., 2007).To the best of our knowledge this approach has not yet been developed further in the literature. Webegin the section by considering Bernstein copulas and their properties.

4.1 Bernstein Copulas and their properties

Bernstein copulas and their properties were introduced in [42] (Li et al., 1997) and [43] (Li et al., 1998).This family is usually used in the context of the approximation of copulas or non-parametric estimation ofdependence structure. The bivariate case is studied in [26] (Durrleman et al., 2000) and the multivariatecase is studied in [55] (Sancetta & Satchell, 2004). In [54] (Salmon & Schleicher, 2006) and [35](Hurd & et al., 2007), a bivariate Bernstein copula is used for the pricing of bivariate currency derivativesand its parameters are calibrated to vanilla options on the cross exchange rate via a reconstruction of itsprobability density.

Bernstein polynomials were introduced by S. Bernstein in the early twentieth century in his proofof the Weierstrass theorem, and are now widely used in the theory of approximation and smoothing.See [44] (Lorentz, 1986) and Chap. 7 in [51] (Phillips, 2003). Bernstein polynomials constitute thebuilding blocks of Bernstein copulas.

Definition 4.1. Bernstein polynomial

For a fixed degree m ∈ N, there are m+ 1 Bernstein polynomials (Bi,m)m

i=0 defined on [0, 1] as

Bi,m(x) =

(

mi

)

xi(1− x)m−i (4.1)

where

(

mi

)

= m!i!(m−i)! is the binomial coefficient.

Figure 1 plots the values of the Bernstein polynomials of degree two (left graph) and five (right graph)as a functions of x ∈ [0, 1].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Bi,m

(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

Bi,m

(x)

Figure 1: Values taken on [0, 1] by Bernstein polynomials of degree m = 2 (left) and m = 5 (right)

The derivative of a Bernstein polynomial is the sum of two Bernstein polynomials of lower degree andwe use the convention Bi,m = 0 if either i < 0 or i > m.

∂Bi,m

∂x(x) = m (Bi−1,m−1(x)−Bi,m−1(x)) (4.2)

11

Before stating the definition of a Bernstein copula, it is necessary to introduce the notion of dis-cretization of the n-dimensional unit hypercube [0, 1]n. As there is no reason for discretizing dimensionsdifferently and for simplicity, we consider m ∈ N a common number of steps for all dimensions. Wedenote Ln,m such a discretization, formally defined as

Ln,m =(α1

m, . . . ,

αn

m

)∣

∣

∣αj ∈ N and 0 ≤ αj ≤ m for j = 1, . . . , n

(4.3)

There are (m + 1)n points in Ln,m and Ln,m ⊂ [0, 1]n. For better readability, elements of Ln,m aredenoted by v = (v1, . . . , vn) and elements of [0, 1]n are denoted by u = (u1, . . . , un).

Definition 4.2. Bernstein Copula

Let ξ be a real-valued function on Ln,m and define CmB : [0, 1]n −→ [0, 1] as

CmB (u) =

∑

v∈Ln,m

ξ(v)

n∏

i=1

Bαi,m(ui) =

m∑

α1=0

· · ·m∑

αn=0

(

ξ(α1

m, . . . ,

αn

m

)

n∏

i=1

Bαi,m(ui)

)

(4.4)

If ξ is n-increasing and verifies the Frechet-Hoeffding bounds, then CmB is a copula, named Bernstein

Copula. It corresponds to the fulfillment of the two conditions below.

1. for 0 ≤ αj ≤ m− 1 (j = 1, . . . , n) and with δn = 0 or 1 whether n is even or odd, respectively.

1∑

l1=0

· · ·1∑

ln=0

(−1)(δn+∑n

j=1lj)ξ

(

α1 + l1m

, . . . ,αn + ln

m

)

≥ 0

2. for v ∈ Ln,m

max

n∑

j=1

vj − n+ 1, 0

≤ ξ (v1, . . . , vn) ≤ minj=1,...,n

(vj)

The summation in (4.4) involves (m+1)n terms as it is done over the elements of the grid Ln,m and ineach term there is a product involving n terms. For Cm

B to be fully determined, it is necessary to know thevalues taken by ξ on the (m+1)n points of the grid. The function ξ is called parameter function becauseits values are the parameters of the Bernstein copula. The family of multivariate Bernstein copulas isdefined and studied in [55] (Sancetta & Satchell, 2004).

Before stating the expression of the density of Bernstein copulas, we introduce ∆nξ the n-dimensionalvolume operator applied to the parameter function ξ and involving a multiple summation with 2n terms.It is defined, for 0 ≤ αj ≤ m− 1 (j = 1, . . . , n), as

∆nξ(α1

m, . . . ,

αn

m

)

=

1∑

l1=0

· · ·1∑

ln=0

(−1)(δn+∑n

j=1lj)ξ

(

α1 + l1m

, . . . ,αn + ln

m

)

(4.5)

Bernstein copulas are absolutely continuous and, as such, always have a proper density. It is a firstimportant property.

Definition 4.3. Bernstein Copula Density

Let CmB be a n-dimensional Bernstein copula and ξ its associated parameter function. We denote cmB

the density of CmB defined as

cmB (u) =∂nCm

B

∂u1 . . . ∂un

(u) =

m−1∑

α1=0

· · ·m−1∑

αn=0

(

∆nξ(α1

m, . . . ,

αn

m

)

n∏

i=1

Bαi,m−1(ui)

)

(4.6)

12

Within the set of copulas, the family of Bernstein copulas have two additional and closely relatedproperties. The first one is an approximation property and the second one is a denseness property inC(n), more relevant in our context. We recall both below. For details and proofs, see [42] (Li et al., 1997),[26] (Durrleman et al., 2000) and [55] (Sancetta & Satchell, 2004). Let C be a copula. The orderm Bernstein copula approximation of C is denoted Cm

B (C) and defined, for u ∈ [0, 1]n, as

CmB (C)(u) =

∑

v∈Ln,m

C(v)

n∏

i=1

Bαi,m(ui) =

m∑

α1=0

· · ·m∑

αn=0

(

C(α1

m, . . . ,

αn

m

)

n∏

i=1

Bαi,m(ui)

)

(4.7)

We denote ξ(C) the associated parameter function, written ξ(C)(v) = C (v1, . . . , vn), for v ∈ Ln,m.ξ(C) satisfies the two conditions in Definition 4.2 so that Cm

B (C) is always a proper copula. The Bernsteincopula approximation of C uniformly converges to C as the approximation order m grows to infinity

CmB (C) −→ C

The other property of the family of Bernstein copulas is their denseness within C(n). This propertymeans we can always find a Bernstein copula as close as desired to a given copula in C(n). It is the keyto their application to our arbitrage detection problem. In particular, we have

∀C ∈ C(n), ∃ (CmB )m∈N such that Cm

B −→ C

where the convergence is the uniform convergence on [0, 1]n. We denote (ξm)m∈N the correspondingsequence of parameter functions. Obviously, one of the converging sequences is the Bernstein copulaapproximation of C. We denote by (ξm(C))m∈N the associated series of parameter functions.

In Figure 2 below we plot the density of the Bernstein copula approximation with degree m = 10 ofa Student’s t copula with parameters ν = 4 and correlation matrix R3 defined below. The parameter ofthe plotted copula is ξ(Ct

4,R3).

R3 =

(

1.00 -0.35-0.35 1.00

)

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

u2

u1

cm B(u

1,u2)

u1

u 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.8

1

1.2

1.4

1.6

1.8

2

Figure 2: Density of a Bernstein copula with degree m = 10 and parameter ξ(Ct4,R3

).

In the next subsection we express the arbitrage detection problem as a calibration problem and wetake advantage of the properties of Bernstein copulas reviewed in the current subsection.

13

4.2 Arbitrage detection as a calibration problem

The setup in this subsection is the same as in Section 3.2. For a given price vector π ∈ [0,+∞[q, werecall the characterization (3.11)

π ∈ Π(Z1, . . . , Zq) ⇐⇒ ∃C ∈ C(n) such that ρ(C) = π

Hence assessing whether or not π belongs to Π can be done by searching a copula in C(n) that perfectlyreproduces π. If this search fails then π is not an arbitrage free price vector for the considered set ofderivatives. Unfortunately the search over C(n) cannot realistically be performed because there exists toomany copulas to explore them all.

As Bernstein copulas are dense in C(n), if there exists a copula reproducing π, then there is also asequence of Bernstein copulas converging to that particular copula. This property allows us to obtain anecessary and sufficient condition of no-arbitrage based on the above characterization. That is a necessaryand sufficient condition for π to belong to Π.

Proposition 4.4. Necessary and sufficient condition of no-arbitrage with Bernstein copulas

π ∈ Π ⇐⇒ ∃ (CmB )m∈N such that ‖ρ (Cm

B )− π‖2 −→ 0 (4.8)

where ‖.‖ is the Euclidean norm on Rq and measures the pricing error as the discrepancy between

ρ (CmB ) and the observed price vector π. That is ‖ρ(C)− π‖2 =

∑qk=1

(

ρk(C) − πk)2.

Here the Euclidean norm is a common choice. Note that other choices are also possible.

Proof. (=⇒) From characterization (3.11), if π ∈ Π then there exists C ∈ C(n) such that ρ(C) = π.Where ρ(C) =

(

ρ1(C), . . . , ρq(C))

=(

B0EC[

Z1T

]

, . . . , B0EC [Zq

T ])

. From their denseness property, thereexists a sequence of Bernstein copulas (Cm

B )m∈Nsuch that Cm

B uniformly converges to C.

The convergence of (CmB )m∈N

to C implies the weak convergence of the associated probability measures

on [0, 1]n [see [17] (Deheuvels, 1981) for a proof]. Thus, for non-pathological payoff functions zk,k =1, . . . , q, we have

∫

[0,1]nzk(F−1

1 (u1), . . . , F−1n (un))dC

mB (u) −→

∫

[0,1]nzk(F−1

1 (u1), . . . , F−1n (un))dC(u)

That is |ρk(CmB ) − ρk(C)| = |ρk(Cm

B ) − πk| −→ 0. Hence, the pricing error converges to zero as mgrows to infinity. That is

‖ρ(CmB )− π‖2 −→ 0

Again, one of the converging series is (CmB (C))m∈N

, the Bernstein copula approximation of C withgrowing order. Series converging faster may exist here.

(⇐=) Suppose there exists (CmB )m∈N

such that ‖ρ(CmB ) − π‖2 −→ 0. From the compactness of C(n),

there exists a sequence extracted from (CmB )m∈N

that converges within C(n). Denote(

Cϕ(m)B

)

m∈Nthe

extracted sequence and C its limit. Hence we have the uniform convergence

Cϕ(m)B −→ C ∈ C(n)

and‖ρ(C

ϕ(m)B )− π‖2 −→ 0

So that, by continuity of the pricing rule ρ from C(n) to [0,+∞[n, we obtain ρ(C) = π.

14

From the necessary and sufficient condition (4.8) we deduce an arbitrage detection method that hasthe form of a calibration task for a Bernstein copula. For that purpose we need to fix a finite value forthe Bernstein copula order m. This value has to be chosen large enough to have an acceptable precisionwhile keeping the computational load manageable. Consider the following minimization problem

P1 :

min ‖ρ(CmB (ξ)) − π‖2

for ξ ∈ R(m+1)n and under constraintsξ(C−) ≤ ξ ≤ ξ(C+)∆nξ ≥ 0

(4.9)

where the inequality constraints ensure that CmB (ξ) is a proper copula and are taken element wise in

R(m+1)n for ξ and in Rmn

for ∆nξ. P1 has the typical form of a calibration task where parameters of amodel, here ξ, are fitted to a set of observed prices, here π. The first constraint, corresponding to theFrechet-Hoeffding bounds, imposes the values of the first and last elements of ξ in each dimension. Henceit reduces the dimensionality of the searched solution to (m− 1)n.

The particular structure of P1 makes it difficult to solve. To overcome this difficulty, a possibility is toregularize the problem by adding a penalty term to the objective function. The idea is to solve a problemthat is close to the initial one but that has a better behavior. For details and a formal approach to theregularization of the model calibration problem, see [12] (Crepey, 2003), [9] (Cont & Tankov, 2004)and references therein. We choose the penalty term to be the squared Euclidean distance to a prior setof parameters. This is a common choice in the spirit of Tikhonov regularization. Note that other choicesare also possible. The regularized calibration task that is, in fact, solved writes

P′

1 :

min(

‖ρ(CmB (ξ)) − π‖2 + α‖ξ − ξ0‖2

)

for ξ ∈ R(m+1)n and under constraintsξ(C−) ≤ ξ ≤ ξ(C+)∆nξ ≥ 0

(4.10)

where α > 0 is the regularization parameter and ξ0 ∈ R(m+1)n is an a priori guess. α controlsthe magnitude of the regularization term when compared to the pricing error term in the objectivefunction. α has to be chosen large enough so that the regularization is useful while keeping the lossof precision acceptable. We denote by ξ∗ a solution of the regularized calibration problem P

′

1 and byβ = ‖ρ(Cm

B (ξ∗))− π‖2 the associated final value of the pricing error. The a priori guess can be obtainedby solving an auxiliary calibration task for a standard copula such as the Gaussian copula. For a Gaussiancopula, the auxiliary calibration problem writes

P0 :

min ‖ρ(CGR )− π‖2

for R a n× n correlation matrix(4.11)

We denote R∗ a solution of P0.

The intuition behind this approach is that if π ∈ Π then the final value of the pricing error in P′

1

should be small. It may not be zero because the degree of the Bernstein copula is fixed and finite so asto obtain a solvable calibration problem. Hence it is necessary to introduce a threshold ǫ > 0, dependingon the chosen degree m and so that a value of β found above this level will lead to the conclusion π /∈ Π.The steps of the arbitrage detection heuristic are as follows.

1. fix a degree m ∈ N and a threshold ǫ > 0

2. solve P0 and compute ξ0 = ξm(

CGR∗

)

3. solve P′

1 and compute β the final pricing error value

4. if β < ǫ then conclude π ∈ Π, otherwise conclude π /∈ Π

15

Besides being based on a necessary and sufficient condition, the main strength of this method isto remain valid when the number of risky assets becomes greater than 2. And, even if it is not theinitial purpose, the calibrated Bernstein copula can be used to price and hedge an additional multi-assetinstrument, coherently with all the available prices.

We now provide two empirical applications to illustrate how the proposed method operates on practicalcases. The two considered basic markets are identical to those of empirical applications in Section 3.3,both with two primary assets. We refer to it for details. In the first application we consider a set of threetwo-asset options, written on the CAC40 and DAX30 indices, namely two calls on the spread and a puton the maximum. In the second application we consider a set of two out-of-the-money vanilla optionson EURJPY, namely a call and a put, which are considered as two-asset options on the EURUSD andUSDJPY exchange rates. EURJPY stands for the exchange rate between the euro and the Japaneseyen. EURUSD and USDJPY correspond, respectively, to the exchange rates between the euro and theU.S. dollar and between the U.S. dollar and the Japanese yen. The observed premiums of the EURJPYvanilla options are expressed in JPY pips per EUR notional. The minimization problems, P0 and P ′

1

are respectively solved with Matlab routines lsqnonlin and fmincon.

Table 3 presents observed and calibrated prices for two-asset options written on the CAC40 andDAX30 indices, respectively denoted by S1 and S2. Individual no-arbitrage bounds are also given as areference. The calibrated Bernstein copula with degree m = 5 is obtained by successively solving theminimization problems P0 and P ′

1. The latter is solved with a regularization parameter α = 0.8 and wedenote ξ∗1 the calibrated Bernstein parameters. It leads to a final pricing error β1 = ‖ρ(C5

B(ξ∗1 ))− π‖2 =

0.0040 that is small enough to conclude that there is no arbitrage among the observed prices. Thenumerical value of ξ∗1 is reproduced in Appendix A.2.

Table 4 presents observed and calibrated prices for EURJPY vanilla options considered as two-assetoptions written on EURUSD and USDJPY FX rates. Individual no-arbitrage bounds are also givenas a reference. The calibrated Bernstein copula with degree m = 8 is obtained by successively solvingminimization problems P0 and P ′

1. The latter is solved with a regularization parameter α = 200 and wedenote ξ∗2 the calibrated Bernstein parameters. It leads to a final pricing error β2 = ‖ρ(C8

B(ξ∗2 ))− π‖2 =

0.1312 that is small enough to conclude that there is no arbitrage among the observed prices. Thenumerical value of ξ∗2 is reproduced in Appendix A.2.

The numerical values of the regularization parameters and the final pricing errors are of different mag-nitudes in the considered applications. This is because the calibration problems have different numericalscales. It is, however, possible to rely on an average relative error to compare the obtained final pricing

errors. This relative measure of error is written β′ = 1q

∑qk=1

|ρk(CmB (ξ∗))−πk|πk . In the first application this

average relative error is β′1 = 0.0061 and, in the second, it is β′

2 = 0.0078. These values have similar orderof magnitude, indicating that comparable levels of pricing error have been reached in both applications.In Figures 3 and 4 we plot the densities of the calibrated Bernstein copulas obtained in the empiricalapplications. The first one has degree m = 5 and the second one has degree m = 8, both are smooth andslightly asymmetric.

k ZkT K πk ρk

(

C5B

)

Π(

Zk)

1(

S2T − S1

T −K)+

0 8.35 8.40 [0.77, 15.60]

2(

S2T − S1

T −K)+

20 2.43 2.45 [0.06, 8.25]

3(

K −max (S1T , S

2T ))+

110 7.21 7.25 [1.36, 12.93]

Table 3: Observed and calibrated prices for the set of two-asset options on CAC40 and DAX30 consideredin the first empirical application of Section 4.2. Marginal distributions of log-returns are NIG, maturityis T = 1 year. See Appendix A.3 for more details on market data.

16

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.810

0.5

1

1.5

2

2.5

3

u1u

2

cm B(u

1,u2)

u1

u 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

1

1.5

2

2.5

Figure 3: Density of the calibrated Bernstein copula. Degree is m = 5 and parameter is ξ∗1 .

k ZkT K πk ρk

(

C8B

)

Π(

Zk)

1 (K −XT )+ 1.3307 28.21 28.56 [0.09, 173.06]

2 (XT −K)+

1.4081 31.01 31.11 [0.00, 176.53]

Table 4: Observed and calibrated prices for the set of EURJPY vanilla options considered in the secondempirical application of Section 4.2 and seen as two-asset options on EURUSD and USDJPY. Marginaldistributions of log-returns are NIG, maturity is T = 1 month. See Appendix A.3 for more details onmarket data.

0

0.2

0.4

0.6

0.8

1

00.2

0.40.6

0.810

0.5

1

1.5

2

2.5

3

3.5

u1u

2

cm B(u

1,u2)

u1

u 2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

1

1.5

2

2.5

3

Figure 4: Density of the calibrated Bernstein copula. Degree is m = 8 and parameter is ξ∗2 .

17

5 The appeal of improved Frechet-Hoeffding bounds

The two-asset case is particular because the partial order on copulas leads, for derivatives with either2-increasing or 2-decreasing payoff functions, to monotonicity in the price and to the associated upper andlower bounds, as seen in Section 3.3. From a calculation standpoint, the two-asset case is considerablyeasier to handle than the general multivariate case. In this section we work within this particular case,we obtain a condition of no-arbitrage and we detail the associated detection method that improveson the method relying on Bernstein copulas. To do so we take advantage of recent results in twodimensions obtained in [59] (Tankov, 2011) and [4] (Bernard et al., 2012) about improved Frechet-Hoeffding bounds when additional information is available. These results are applied to the case of optionswritten on two underlyings in [59] (Tankov, 2011) and to the case of optimal investment strategies in[4] (Bernard et al., 2012).

We consider a restricted financial market with only two risky assets, S1 and S2. We also considera finite set

(

Z1, . . . , Zq)

of derivatives written on these assets and with payoff functions that are either2-increasing or 2-decreasing. As already remarked, in the two-asset case, all common payoff functionsverify this condition. The dimensionality being fixed at n = 2, we drop, for readability, the explicitdependence on n from our notations. Our approach is to start with characterization (3.12), rewrittenbelow for a given price vector π ∈ [0,+∞[q.

π ∈ Π(

Z1, . . . , Zq)

⇐⇒

q⋂

k=1

Ck 6= ∅

For each Ck set, we introduce pointwise best possible upper and lower bounds, respectively denotedby Ak and Bk, and defined for (u1, u2) ∈ [0, 1]2 and k = 1, . . . , q

Ak(u1, u2) = max

C(u1, u2) | C ∈ Ck

Bk(u1, u2) = min

C(u1, u2) | C ∈ Ck

(5.1)

The bounds Ak and Bk are said pointwise best possible in the sense that for all points (u1, u2) ∈ [0, 1]2

there exists a copula in Ck that reaches the bound value.

Extending results in [49] (Nelsen et al., 2001) and [50] (Nelsen et al., 2004), a quasi-explicit ex-pression for the improved bounds is derived in [59] (Tankov, 2011). This key result makes easy thecomputation of values taken by Ak and Bk. These improved bounds are proper copulas under sufficientconditions obtained in [59] (Tankov, 2011) and weakened in [4] (Bernard et al., 2012). We refer tothese articles for details and proofs. The corresponding formulas for the bounds in our framework aregiven below.

Proposition 5.1. (Tankov, 2011)The quasi-explicit expressions of the upper and lower bounds of Ck are written, if Zk has a 2-increasing

payoff function

Ak(u1, u2) =

ρk,−1− (u1, u2, π

k) if πk ∈ [ρk (C−) , ρk−(u1, u2, C+(u1, u2))]

C+(u1, u2) otherwise(5.2)

Bk(u1, u2) =

ρk,−1+ (u1, u2, π

k) if πk ∈ [ρk+(u1, u2, C−(u1, u2)), ρ

k (C+)]C−(u1, u2) otherwise

(5.3)

and, if Zk has a 2-decreasing payoff function

Ak(u1, u2) =

ρk,−1− (u1, u2, π

k) if πk ∈ [ρk−(u1, u2, C+(u1, u2)), ρ

k (C−)]C+(u1, u2) otherwise

(5.4)

Bk(u1, u2) =

ρk,−1+ (u1, u2, π

k) if πk ∈ [ρk (C+) , ρk+(u1, u2, C−(u1, u2))]

C−(u1, u2) otherwise(5.5)

18

where

ρk,−1− (u1, u2, π

k) =max

θ ∈ [C−(u1, u2), C+(u1, u2)] | ρ

k−(u1, u2, θ) = πk

ρk,−1+ (u1, u2, π

k) =min

θ ∈ [C−(u1, u2), C+(u1, u2)] | ρ

k+(u1, u2, θ) = πk

and, for (a, b) ∈ [0, 1]2 and θ ∈ [C−(a, b), C+(a, b)]

ρk−(a, b, θ) =

∫ a−θ

0

zk(

F−11 (u), F−1

2 (1− u))

du+

∫ a

a−θ

zk(

F−11 (u), F−1

2 (a+ b− θ − u))

du

+

∫ 1−b+θ

a

zk(

F−11 (u), F−1

2 (1 + θ − u))

du+

∫ 1

1−b+θ

zk(

F−11 (u), F−1

2 (1 + θ − u))

du

ρk+(a, b, θ) =

∫ a−θ

0

zk(

F−11 (u), F−1

2 (1− u))

du+

∫ a

a−θ

zk(

F−11 (u), F−1

2 (a+ b− θ − u))

du

+

∫ 1−b+θ

a

zk(

F−11 (u), F−1

2 (1 + θ − u))

du+

∫ 1

1−b+θ

zk(

F−11 (u), F−1

2 (1 + θ − u))

du

ρ−1+ (u1, u2, π

k) and ρ−1− (u1, u2, π

k) are values taken by copulas. ρ+(a, b, θ) and ρ−(a, b, θ) are derivativeprices obtained when pricing, respectively, with the upper and lower bounds of the set of copulas takingspecified value θ at point (a, b) in [0, 1]2 and are monotonic functions of θ. With formulas (5.2), (5.3) and(5.4), (5.5) the computation of values taken by Ak and Bk is easy to implement because it involves onlyone dimensional quadratures and a root search.

We can now go back to our initial problem. For⋂q

k=1 Ck to be non-empty the lowest upper bound

should always stay greater than, or equal to, the largest lower bound. The proposed necessary conditionof no-arbitrage is based on this idea. The involved ”super” bounds are denoted by A and B, both takevalues in [0, 1] and are defined pointwise on [0, 1]2 as

A(u1, u2) =min

Ak(u1, u2) | k = 1, . . . , q

B(u1, u2) =max

Bk(u1, u2) | k = 1, . . . , q

Proposition 5.2. Necessary condition of no-arbitrage in the two-asset case

For π to be a no-arbitrage price vector for(

Z1, . . . , Zq)

, A should be greater than, or equal to, Beverywhere in the unit square. This necessary condition is formally written

π ∈ Π =⇒ ∀(u1, u2) ∈ [0, 1]2 A(u1, u2) ≥ B(u1, u2) (5.6)

Proof. Suppose there exists (u∗1, u

∗2) ∈ [0, 1]2 such that A (u∗

1, u∗2) < B (u∗

1, u∗2). By construction of A

and B the minimum and the maximum are always reached. So that, for A, the minimum is attained fork = kA ∈ 1, . . . , q and for B, the maximum is attained for k = kB ∈ 1, . . . , q with kB 6= kA. Hence

AkA (u∗1, u

∗2) < BkB (u∗

1, u∗2)

that ismax

C (u∗1, u

∗2) | C ∈ CkA

< min

C (u∗1, u

∗2) | C ∈ CkB

so that a copula cannot be at the same time in CkA and CkB . Thus CkA⋂

CkB = ∅

19

And because⋂q

k=1 Ck ⊂

(

CkA⋂

CkB)

, we finally have

q⋂

k=1

Ck = ∅.

That is equivalent to π /∈ Π.

The intuition behind the result is that if the inequality is invalidated somewhere in [0, 1]2 there existsno model reproducing the observed set of prices and as a consequence this set is not jointly arbitragefree. It allows us to express our arbitrage detection problem as a minimization task with the conclusiondepending on the final value of its objective function. Consider A−B as a function of (u1, u2) and denoteby P2 the following minimization problem.

P2 : min(u1,u2)∈[0,1]2

A(u1, u2)−B(u1, u2) (5.7)

The objective function in P2 takes values in [−1,+1] and the minimization is done over [0, 1]2, acompact set. Denote (u∗

1, u∗2) a solution of P2 and γ the corresponding value of the objective function.

The value of γ is straightforwardly linked to the detection of arbitrage because if it is negative then theintersection of the Ck sets is empty and the price vector π is not free of arbitrage.

This second arbitrage detection heuristic only has two steps, and writes

1. solve P2 and keep γ the final objective function value

2. if γ ≥ 0 then conclude π ∈ Π, otherwise conclude π /∈ Π

A suitable approach to solve P2 is to use an optimization algorithm that uses multiple start pointsand keeps the lowest local minimum found. This global optimization approach has good chances to reacha global minimum because the objective function has a reasonable number of peaks and basins. SeeFigures 5 and 6 below.

In comparison to the first method that works in the space of prices, the method proposed in thissection works in the space of models (copulas). It has two main strengths, first it is easy to implementand second it flags true arbitrage situations because the algorithm is not truncated as it is the case withBernstein copulas. The output information of this arbitrage detection method cannot be used for pricingadditional derivatives because, by construction, it does not include a calibration step. For this specificpurpose a calibration has to be performed separately once market prices have been checked to be jointlyarbitrage free.

To conclude the section we present two numerical applications to illustrate how the method operates.In both applications we consider the CAC40 and DAX30 indices as primary assets. The first applicationis performed with the same market conditions and multi-asset option quotes as in 4.2, only the detectionmethod differs. In the second application we consider a set made of 2 two-asset derivatives, a call onthe spread and a call on the equally weighted basket, the prices of which are artificially built in order toobtain a case with arbitrage. In both applications, the minimization task P2 is solved with a multi-startversion of the Matlab routine fmincon and the chosen number of start-points is 20.

In the first application, the minimization task P2 leads to a minimum value of the objective functionγ = 0 and this value is reached for any (u∗

1, u∗2) such that u∗

1 or u∗2 is either 1 or 0. And we conclude

that π is free of arbitrage. Figure 5 plots the surface of values and the associated contour graph of theobjective function A−B as a function of (u1, u2) ∈ [0, 1]2.

20

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

u2

u1

A−

B

u1

u 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Figure 5: Values and contour plot of the objective function in P2

Table 5 presents the observed prices for two-asset options considered in the second application. Assaid, these prices are built to obtain a case with arbitrage. Options under scrutiny are written on CAC40and DAX30 indices, respectively denoted S1 and S2. Individual no-arbitrage bounds are also given as areference. The minimization task P2 leads to a minimum value of the objective function γ = −0.1079and this value is reached near the center, for (u∗

1, u∗2) = (0.4671, 0.4885). In this case we conclude that

π is not free of arbitrage. Figure 6 plots the surface of values and the associated contour graph of theobjective function A−B as a function of (u1, u2) ∈ [0, 1]2. A look at the plots confirms that the objectivefunction goes below zero.

k ZkT K πk Π

(

Zk)

1(

S2T − S1

T −K)+

0 15.00 [0.77, 15.60]

2(

12 (S

1T + S2

T )−K)+

100 7.20 [0.75, 7.83]

Table 5: Observed prices for the set of two-asset options on CAC40 and DAX30 considered in the secondnumerical application of Section 5. Marginal distributions of log-returns are NIG, maturity is T = 1 year.See Appendix A.3 for more details on market data.

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

u1

u2

A−

B

u1

u 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.05

0

0.05

0.1

0.15

Figure 6: Values and contour plot of the objective function in P2

21

6 Conclusion

In a multi-asset financial market with known risk-neutral marginal distributions, we have proposed atwofold characterization of the absence of arbitrage opportunities in terms of copula functions. In thesecharacterizations a set of multi-asset option prices is free of arbitrage if there exists, at least, one copulafunction that is compatible with it. It allows us to better understand, for the considered market, themathematical structure underpinning the set of risk-neutral measures, as it is then isomorphic to the setof copula functions. Two no-arbitrage necessary conditions have then been deduced and the associateddetection methods have been detailed. The first relies on special properties of Bernstein copulas, inparticular their denseness within the set of all copulas, which is also a sufficient condition and it remainsvalid when dimensionality increases. The second is made appealing by the availability of quasi-explicitformulas for improved Frechet-Hoeffding bounds. Which is, however, valid only in the bivariate case.Both detection methods have been formalized in such a way optimization algorithms can solve it. Thestructure of our market model makes them particularly suitable for applications to markets with severalequity indices or exchange rates, which correspond to the practical applications we have provided.

The proposed methods can easily be used, within a financial institution, by a risk management ormodel validation department to control for the overall coherence of the pricing tools used by front officeoperators to quote, trade and value multi-asset options, and by doing so, avoiding their institution to bearbitraged by other players. Thinking the other way round, hedge funds or proprietary trading teamscan, of course, implement the method to detect and eventually take advantage of inconsistencies betweenquoted prices of multi-asset options available in the market.

22

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26

A Appendix

A.1 Additional definitions

The definitions of n-increasing function, concordance ordering and Frechet-Hoeffding bounds are gatheredin this Appendix.

Definition A.1. n-increasing function

A n-box in Rn is a set B = [a, b] = [a1, b1]× . . .× [an, bn] ⊂ Rn with ai < bi ∀i = 1, . . . , n.

A n-dimensional function H : Rn −→ R is n-increasing if ∀B n-box

1∑

k1=0

. . .

1∑

kn=0

(−1)∑n

i=1kiH (x1k1

, . . . , xnkn) ≥ 0 (A.1)

with xi1 = ai and xi0 = bi for i = 1, . . . , n.

The sum is taken over all the vertexes of B and involves 2n terms.

In the case n = 2, H is 2-increasing if ∀B = [a1, b1]× [a2, b2] with a1 < b1 and a2 < b2

H(b1, b2)−H(b1, a2)−H(a1, b2) +H(a1, a2) ≥ 0

and H is 2-decreasing when

H(b1, b2)−H(b1, a2)−H(a1, b2) +H(a1, a2) ≤ 0

In the case n = 3, the sum involves 8 terms and H is 3-increasing if ∀B = [a1, b1]× [a2, b2]× [a3, b3]with ai < bi for i = 1, 2, 3

H(b1, b2, b3)−H(a1, b2, b3)−H(b1, a2, b3)−H(b1, b2, a3)

+H(a1, a2, b3) +H(b1, a2, a3) +H(a1, b2, a3)−H(a1, a2, a3) ≥ 0 (A.2)

Definition A.2. Concordance ordering

For C1, C2 ∈ C(n), we denote C1 ≺ C2 if ∀u ∈ [0, 1]n C1(u) ≤ C2(u)

≺ is called concordance order, and is a partial order on C(n).

Proposition A.3. Frechet-Hoeffding bounds

The Frechet-Hoeffding upper and lower bounds C+ and C− are as follows

C+(u1, . . . , un) = mini

(ui) (A.3)

C−(u1, . . . , un) = max

(

∑

i

ui − n+ 1, 0

)

(A.4)

For all copula C ∈ C(n), the following inequality holds for any u ∈ [0, 1]n

C−(u) ≤ C(u) ≤ C+(u)

That isC− ≺ C ≺ C+

Hence the set of copulas admits upper and lower bounds with respect to the concordance ordering ≺even if it is a partial order. C+ is always a copula. C− is a copula only for n = 2 and for n ≥ 3 theassociated bound is best possible.

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A.2 Bernstein copulas calibration outputs

The output of calibration tasks performed in the numerical applications of Section 4 are reproducedbelow.

ξ∗1 =

0 0 0 0 0 00 0.15 0.16 0.17 0.18 0.20 0.15 0.31 0.34 0.37 0.40 0.16 0.33 0.51 0.56 0.60 0.17 0.36 0.55 0.75 0.80 0.2 0.4 0.6 0.8 1

ξ∗2 =

0 0 0 0 0 0 0 0 00 0.09 0.09 0.1 0.11 0.11 0.12 0.12 0.130 0.11 0.14 0.17 0.2 0.22 0.23 0.24 0.250 0.11 0.18 0.23 0.27 0.31 0.34 0.36 0.380 0.11 0.21 0.28 0.34 0.4 0.45 0.48 0.50 0.12 0.23 0.32 0.4 0.48 0.54 0.6 0.630 0.12 0.24 0.35 0.46 0.55 0.64 0.72 0.750 0.12 0.24 0.37 0.49 0.61 0.73 0.85 0.880 0.13 0.25 0.38 0.5 0.63 0.75 0.88 1

A.3 Market data and parameters of marginal distributions

We gather in the first part of this appendix the relevant data used for the underlying assets throughout thepaper. The considered spot, forward and option prices are synchronous quotes observed in the interbankmarket. For equity market indices, it corresponds to May 20th 2008 and considered options and forwardshave one-year maturity. For foreign exchange rates, it corresponds to January 13th 2006 and consideredoptions and forwards have one-month maturity. In the second part of this appendix we reproduce thevalues of Normal Inverse Gaussian (NIG) distribution parameters as used in numerical applications tomodel risk-neutral marginals.

Date Currency Maturity Price

05/20/2008 EUR 1 year 0.952001/13/2006 EUR 1 month 0.997901/13/2006 JPY 1 month 1.000001/13/2006 USD 1 month 0.9962

Table 6: Zero-coupon bond prices for EUR, USD and JPY currencies for the considered dates andmaturities.

Date Name Currency Spot Price Forward Price

01/13/2006 EURUSD USD 1.2136 1.215801/13/2006 USDJPY JPY 114.21 113.7801/13/2006 EURJPY JPY 138.61 138.3205/20/2008 CAC40 EUR 5055.0 5200.005/20/2008 DAX30 EUR 7740.0 7970.0

Table 7: Spot and forward prices for underlying assets of FX and equity index types.

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ATM 25∆RR 25∆BFLY 50∆RR 50∆BFLY

EURUSD 8.95 −0.18 0.15 −0.28 0.40USDJPY 9.15 −1.05 0.20 −1.75 0.80EURJPY 9.30 −0.70 0.20 −1.20 0.65

Table 8: Implied volatility smile quotes for FX rates EURUSD, USDJPY and EURJPY. Values areexpressed in volatility points and the ATM column corresponds to the zero delta straddle strategy.Maturity is 1 month and date is January 13th 2006.

CAC40

K 2528 3033 3539 4044 4550 5055 5561 6066 6572 7077 7583σimp(K) 32.32 30.36 28.12 25.76 23.34 20.92 18.64 16.66 15.04 14.51 14.51

DAX30

K 3870 4644 5418 6192 6966 7740 8514 9288 10062 10836 11610σimp(K) 35.91 31.90 28.55 25.80 23.50 21.53 20.09 19.29 18.98 19.05 19.32

Table 9: Implied volatility smile quotes for CAC40 and DAX30 indices. Values are expressed in volatilitypoints. Maturity is 1 year and date is May 20th 2008.

We model each underlying asset log-returns with a NIG distribution. Its parameters are fitted to theabove market data and reproduced in the below table.

α β µ δ

EURUSD 47.964 −1.912 0.001 0.035JPYUSD 45.345 7.683 −0.006 0.033CAC40 19.175 −15.594 0.269 0.210DAX30 6.163 −3.182 0.114 0.235

Table 10: Values of NIG parameters used in numerical applications as risk-neutral marginal distributionof log-returns.

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