Detection of arbitrage opportunities in multi-asset derivatives markets
DDetection of arbitrage opportunities in multi-assetderivatives markets
Antonis Papapantoleon a, b,1, ∗ , Paulo Yanez Sarmiento c,2, ∗ A BSTRACT
We are interested in the existence of equivalent martingale measuresand the detection of arbitrage opportunities in markets where severalmulti-asset derivatives are traded simultaneously. More specifically,we consider a financial market with multiple traded assets whosemarginal risk-neutral distributions are known, and assume that sev-eral derivatives written on these assets are traded simultaneously. Inthis setting, there is a bijection between the existence of an equiva-lent martingale measure and the existence of a copula that couplesthese marginals. Using this bijection and recent results on improvedFréchet–Hoeffding bounds in the presence of additional information,we derive sufficient conditions for the absence of arbitrage and formu-late an optimization problem for the detection of a possible arbitrageopportunity. This problem can be solved efficiently using numericaloptimization routines. The most interesting practical outcome is thefollowing: we can construct a financial market where each multi-assetderivative is traded within its own no-arbitrage interval, and yet whenconsidered together an arbitrage opportunity may arise.K
EYWORDS : Arbitrage, equivalent martingale measures, detection ofarbitrage opportunities, multiple assets, multi-asset derivatives, copu-las, improved Fréchet–Hoeffding bounds. A
UTHORS I NFO a Department of Mathematics, NTUA, ZografouCampus, 15780 Athens, Greece b Institute of Applied and Computational Mathe-matics, FORTH, Vassilika Vouton 70013 Herak-lion, Greece c Institute of Mathematics, TU Berlin, Straße des17. Juni 136, 10623 Berlin, Germany [email protected] [email protected] ∗ We thank Thibaut Lux for fruitful discussionsduring the work on these topics. P APER I NFO
AMS C
LASSIFICATION : 91G20, 62H05, 60E15.
1. Introduction
We consider a financial market where multiple assets and several derivatives written on single ormultiple assets are traded simultaneously. Assuming we are given a set of traded prices for thesemulti-asset derivatives, we are interested in whether there exists an arbitrage-free model that isconsistent with these prices or not. A consistent arbitrage-free model will exist if we can find anequivalent martingale measure such that we can describe these prices as discounted expected payoffsunder this measure. We assume that the marginal risk-neutral distributions of the assets are known, e.g. they have been estimated from single-asset options prices using Breeden and Litzenberger [3].Then, there exists a bijection between the existence of an equivalent martingale measure and theexistence of a copula that couples these marginal distributions. Using recent results about improvedFréchet–Hoeffding bounds on copulas in the presence of additional information, we can formulatea sufficient condition for the existence of a copula and thus for the absence of arbitrage in thisfinancial market. Moreover, the formulation of this condition as an optimization problem allows forthe detection of an arbitrage opportunity via numerical optimization routines.Arbitrage is a fundamental concept in economics and finance, because the modern theory of optionvaluation is rooted on the assumption of the absence of arbitrage, while it is also closely related with1 a r X i v : . [ q -f i n . P R ] F e b otions of equilibrium in financial markets. Arbitrage is also a concept of practical importance, asfinancial institutions are interested in ensuring that their systems for option valuation, simulation,scenario generation, etc , are free of arbitrage, in order to be useful and relevant. Therefore, topicsrelated to the existence of arbitrage and the consistency of arbitrage-free models with given tradedprices are of significant theoretical and practical interest.There is a sufficiently rich literature by now devoted to the case where a single asset and options onthis asset are traded in a financial market. Laurent and Leisen [11] in their pioneering work providea procedure to check for the absence of arbitrage in a discrete set of market data. Carr and Madan[6] provide a sufficient condition for the absence of arbitrage in a market where countably-infinitemany European options with discrete strikes can be traded. These results where later generalized andextended by Cousot [7], by Buehler [4], and in particular by Davis and Hobson [8] who providednecessary and sufficient conditions for the existence of an arbitrage-free model consistent with a setof market prices. More recently, Gerhold and Gülüm [10] considered the same problem in case theonly observables are the bid and ask prices of the underlying asset.The literature is not that developed when one turns to multiple underlying assets and multi-assetderivatives. Actually, to the best of our knowledge, the only work treating this problem is Tavin [19].The setting in [19] is exactly the same as here, i.e. the author considers multiple underlying assetswith known risk-neutral marginals and several traded derivatives on multiple assets, and providestwo methods for detecting arbitrage opportunities, one based on Bernstein copulas and another basedon improved Fréchet–Hoeffding bounds, which is however restricted to the two-asset case. In ourwork, we extend the results of [19] to the general multi-asset case using the recent results of Luxand Papapantoleon [12] on improved Fréchet–Hoeffding bounds for d -copulas, with d ≥ .The remainder of this article is structured as follows: In Section 2 we review some necessary resultsabout copulas, quasi-copulas and improved Fréchet–Hoeffding bounds. In Section 3 we present re-sults on integration and stochastic dominance for quasi-copulas; these include also a new represen-tation of the integral with respect to a quasi-copula that could be of independent interest. In Section4 we revisit the bijection between the existence of an equivalent martingale measure and a copulathat couples the marginals of the underlying assets already present in Tavin [19], and derive neces-sary conditions for the absence of arbitrage in the presence of several multi-asset derivatives tradedsimultaneously. In Section 5 we apply our results in a model with three underlying assets. In partic-ular, we show that we can construct a financial market where each multi-asset derivative is tradedwithin its own no-arbitrage interval, and yet when considered together an arbitrage opportunity mayarise. Finally, the appendices collect some additional results and proofs.
2. Copulas, quasi-copulas and improved Fréchet–Hoeffding bounds
This section serves as an introduction to the notation that will be used throughout this work, as wellas to some basic results about copulas, quasi-copulas and improved Fréchet–Hoeffding bounds. Let d ≥ be an integer, and set I = [0 , and = (1 , . . . , ∈ R d . In the sequel, boldface letters, suchas u or v , denote vectors in I d or R d with entries u , . . . , u d or v , . . . , v d , while we distinguishstrictly between ⊂ and ⊆ , i.e. if J ⊂ I then J (cid:54) = I . Moreover, for a univariate distribution function F we define its inverse as F − ( u ) := inf { x ∈ R | F ( x ) ≥ u } , while we call a function f : R d → R right-continuous if it is right-continuous in each component.2he finite difference operator ∆ for a function f : R d → R and a, b ∈ R with a ≤ b is defined as ∆ ia,b f ( x , . . . , x d ) := f ( x , . . . , x i − , b, x i +1 , . . . , x d ) − f ( x , . . . , x i − , a, x i +1 , . . . , x d ) , while the f -volume V f for a hyperrectangle R = × di =1 ( a i , b i , ] ⊂ R d is defined as V f ( R ) := ∆ da d ,b d ◦ · · · ◦ ∆ a ,b f. The f -volume of R admits also the following representations, which are more suitable for most ofour purposes, V f ( R ) = (cid:88) v ∈V ( − N ( v ) f ( v )= f ( b , . . . , b d ) − d (cid:88) i =1 f ( b , . . . , b i − , a i , b i +1 , . . . , b d ) (2.1) + d (cid:88) j =2 i Let f : R d + → R be right-continuous and d -increasing. Define for every hyper-rectangle R := × di =1 ( a i , b i ] ⊂ R d + the function µ f (cid:0) R (cid:1) := V f (cid:0) R (cid:1) , (2.2) and set µ f ( ∅ ) := 0 . Then µ f is a measure on R d + . Definition 2.2. A function Q : I d → I is a d -quasi-copula if it satisfies the following properties:(C1) boundary condition: Q ( u , . . . , u i = 0 , . . . , u d ) = 0 , for all ≤ i ≤ d .(C2) uniform marginals: Q (1 , . . . , , u i , , . . . , 1) = u i , for all ≤ i ≤ d .(C3) Q is non-decreasing in each component.(C4) Q is Lipschitz continuous, i.e. for all u , v ∈ I d | Q ( u ) − Q ( v ) | ≤ d (cid:88) i =1 | u i − v i | . Q is a d - copula if it satisfies in addition:(C5) Q is d -increasing.The set of all d -quasi-copulas is denoted by Q d and the set of all d -copulas by C d . Obviously, C d ⊂ Q d . Moreover, we call Q ∈ Q d \ C d a proper quasi-copula. In case the dimension d is clear,we refer to a d -(quasi-)copula as a (quasi-)copula.There exists a clear link between copulas and probability distributions. In fact, for C ∈ C d andunivariate distribution functions F , . . . , F d , F ( x ) := C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) (2.3)defines a d -dimensional distribution function with marginals F , . . . , F d . The celebrated theorem ofSklar [17] tells us that the converse is also true, i.e. given a d -dimensional distribution function F with univariate marginals F , . . . , F d , there exists a copula C such that (2.3) holds true. We will call C the copula corresponding to F .Let Q ∈ Q d . We define its survival function (cid:98) Q : I d → I as follows: (cid:98) Q ( u ) := V Q (cid:16) d × i =1 ( u i , (cid:17) , (2.4)and denote by (cid:98) C d := { (cid:98) C | C ∈ C d }. A well-known result states that if C ∈ C d , then u (cid:55)→ (cid:98) C ( − u ) isagain a copula, namely the survival copula of C , while there exists also a version of Sklar’s theoremfor survival copulas. In case Q is a proper quasi-copula, then u (cid:55)→ (cid:98) Q ( − u ) is not a quasi-copulain general; see e.g. Example 2.5 in Lux and Papapantoleon [12]. Moreover, note that (cid:98)(cid:98) C (cid:54) = C , ingeneral. However, we present below another inverse transformation that is injective and, to the bestof our knowledge, has not appeared in the literature. The proof is again relegated to Appendix C. Proposition 2.3. Let C ∈ C d with survival function (cid:98) C . Then C ( u ) = ( − d V (cid:98) C (cid:16) d × i =1 (0 , u i ] (cid:17) . Hence, the map C (cid:55)→ (cid:98) C is injective. When dealing with random vectors X = ( X , . . . , X d ) , we are often interested in the distribution ofa lower-dimensional vector thereof, i.e. the law of ( X i , . . . , X i n ) with { i , . . . , i n } ⊆ { , . . . , d } .If we know the multi-variate distribution, then we can deduce the lower-dimensional marginals. Thesame applies to copulas. Proposition 2.4. Let Q ∈ Q d and I = { i , . . . , i n } ⊆ { , . . . , d } . We call Q I : I n → I with ( u i , . . . , u i n ) (cid:55)→ Q ( u , . . . , u d ) with u k = 1 if k / ∈ I the I - margin of Q . Then, Q I is an n -quasi-copula. Moreover, if Q ∈ C d then Q I is an n -copula. roof. The properties (C1) to (C4) carry over to Q I immediately. Therefore, consider a d -copula C and let | I | = d − . Without loss of generality we can assume I = { , . . . , d − } . Then we havefor all R = × d − i =1 ( a i , b i ] ⊆ (0 , d − that ≤ V C ( R × (0 , C ( b , . . . , b d − , − C ( b , . . . , b d − , − d − (cid:88) i =1 C ( b , . . . , b i − , a i , b i +1 , . . . , b d − , ± · · · + ( − d − C ( a , . . . , a d − , 1) + ( − d − d − (cid:88) i =1 C ( a , . . . , a i − , b i , a i +1 , . . . , a d − , − d C ( a , . . . , a d − , V C I ( R ) . Hence, C I ∈ C d − . The claim follows inductively for all I ⊆ { , . . . , d } . (cid:3) Let us now define a partial order on Q d , and thus also on C d . Definition 2.5. Let Q , Q ∈ Q d .(i) If Q ( u ) ≤ Q ( u ) for all u ∈ I d , then Q is smaller than Q in the lower orthant order ,denoted by Q (cid:22) LO Q .(ii) If (cid:98) Q ( u ) ≤ (cid:98) Q ( u ) for all u ∈ I d , then Q is smaller than Q in the upper orthant order ,denoted by Q (cid:22) UO Q .The celebrated Fréchet–Hoeffding bounds provide upper and lower bounds for all quasi-copulaswith respect to the lower orthant order. Indeed, for Q ∈ Q d , we have that W d ( u ) := max (cid:26) d (cid:88) i =1 u i − d + 1 , (cid:27) ≤ Q ( u ) ≤ min { u , . . . , u d } =: M d ( u ) , for all u ∈ I d , which readily implies that W d (cid:22) LO C (cid:22) LO M d . W d and M d are respectively calledthe lower and upper Fréchet–Hoeffding bounds. Analogous results hold true for the upper orthantorder and the survival functions, i.e. we have that W d ( − u ) ≤ (cid:98) C ( u ) ≤ M d ( − u ) , for all u ∈ I d , while an easy computation shows that M d ( − · ) = (cid:99) M d ( · ) for all d ≥ , while W d ( − · ) = (cid:99) W d ( · ) only for d = 2 .The Fréchet–Hoeffding bounds are derived under the assumption that the marginal distributionsare fully known and the copula is fully unknown. However, in several applications such as financeand insurance, partial information on the copula is available from market data. Therefore, there hasbeen intensive research in the last decade on improving the Fréchet–Hoeffding bounds by addingpartial information on the copula, see e.g. Lux and Papapantoleon [12, 13], Nelsen [14], Puccetti,Rüschendorf, and Manko [15] and Tankov [18]. The following results from [12, Sec. 3] describe5 mproved Fréchet–Hoeffding bounds under the assumption that the copula is known in a subset ofits domain, or that a functional of the copula is known. Analogous statements for survival copulasare relegated to Appendix A.Let S ⊆ [0 , d be compact and Q ∗ ∈ Q d . Define the set Q S ,Q ∗ := (cid:8) Q ∈ Q d | Q ( x ) = Q ∗ ( x ) for all x ∈ S (cid:9) . Then, for all Q ∈ Q S ,Q ∗ Q S ,Q ∗ L (cid:22) LO Q (cid:22) LO Q S ,Q ∗ U , where the improved Fréchet–Hoeffding bounds Q S ,Q ∗ L , Q S ,Q ∗ U ∈ Q d and are provided by Q S ,Q ∗ L ( u ) = max (cid:110) , d (cid:88) i =1 u i − d + 1 , max x ∈S (cid:8) Q ∗ ( x ) − d (cid:88) i =1 ( x i − u i ) + (cid:9)(cid:111) ,Q S ,Q ∗ U ( u ) = min (cid:110) u , . . . , u d , min x ∈S (cid:8) Q ∗ ( x ) + d (cid:88) i =1 ( u i − x i ) + (cid:9)(cid:111) . Remark 2.6. A natural question is whether the bounds Q S ,Q ∗ L and Q S ,Q ∗ U are copulas or properquasi-copulas. Nelsen [14] showed that in the case of S being a singleton and for d = 2 the lowerand upper improved Fréchet-Hoeffding bounds are copulas using the concept of shuffles of M . Thisstatement was generalized by Tankov [18] and Bernard, Jiang, and Vanduffel [2], still for d = 2 ,under certain ‘monotonicity’ conditions. On the contrary, Lux and Papapantoleon [12] showed thatfor d > , the improved Fréchet–Hoeffding bounds are copulas only in trivial cases and properquasi-copulas otherwise. Moreover, Bartl, Kupper, Lux, Papapantoleon, and Eckstein [1] showedthat the improved Fréchet–Hoeffding bounds are not pointwise sharp (or best-possible), even in d = 2 , if the aforementioned ‘monotonicity’ conditions are violated. (cid:7) The next result provides improved Fréchet–Hoeffding bounds in case the value of a functional of thecopula is known. Examples of functionals could be the correlation or another measure of dependence( e.g. Kendall’s τ or Spearman’s ρ ), but also prices of multi-asset options in a mathematical financecontext. Let ρ : Q d → R be non-decreasing with respect to the lower orthant order and continuouswith respect to the pointwise convergence of quasi-copulas, and consider the set of quasi-copulas Q ρ,θ := (cid:8) Q ∈ Q d | ρ ( Q ) = θ (cid:9) , (2.5)for θ ∈ [ ρ ( W d ) , ρ ( M d )] . Then, for all Q ∈ Q ρ,θ , holds Q ρ,θL (cid:22) LO Q (cid:22) LO Q ρ,θU , where the improved Fréchet–Hoeffding bounds Q ρ,θL , Q ρ,θU ∈ Q d are provided by Q ρ,θL ( u ) := (cid:40) ρ − ( u , θ ) , if θ ∈ [ ρ + ( u , W d ( u )) , ρ ( M d )] ,W d ( u ) , otherwise , (2.6) Q ρ,θU ( u ) := (cid:40) ρ − − ( u , θ ) , if θ ∈ [ ρ ( W d ) , ρ − ( u , M d ( u ))] ,M d ( u ) , otherwise . (2.7)6ere we use the following notation: for u ∈ [0 , d , let r ∈ I u = [ W d ( u ) , M d ( u )] and Q ∗ ∈ Q d with Q ∗ ( u ) = r , and define Q { u } ,rL := Q { u } ,Q ∗ L , Q { u } ,rU := Q { u } ,Q ∗ U and ρ − ( u , r ) := ρ (cid:0) Q { u } ,rL (cid:1) and ρ + ( u , r ) := ρ (cid:0) Q { u } ,rU (cid:1) . Then, for fixed u , the maps r (cid:55)→ ρ − ( u , r ) and r (cid:55)→ ρ + ( u , r ) are non-decreasing and continuous.Hence, we can define their inverse mappings θ (cid:55)→ ρ − − ( u , θ ) := max { r ∈ I u : ρ − ( u , r ) = θ } ,θ (cid:55)→ ρ − ( u , θ ) := min { r ∈ I u : ρ + ( u , r ) = θ } , for all θ such that the sets are non-empty. Analogous statements for non-increasing functionals arerelegated to Appendix B. 3. Integration and stochastic dominance for quasi-copulas This section provides results on the definition of integrals with respect to quasi-copulas and onstochastic dominance for quasi-copulas. These results are largely taken from Lux and Papapantoleon[12, Sec. 5], however we also provide a new representation of the integral with respect to a quasi-copula, as well as some useful results on stochastic dominance for quasi-copulas.Let (Ω , F , P ) be a probability space. Consider an R d + -valued random vector X = ( X , . . . , X d ) with distribution function F and marginals F , . . . , F d . Then, from Sklar’s Theorem, we know thereexists a copula C ∈ C d such that P ( X < x , . . . , X d < x d ) = C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) and P ( X > x , . . . , X d > x d ) = (cid:98) C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) . Hence, there exists an induced measure d C (cid:0) F ( x ) , . . . , F ( x d ) (cid:1) on R d + . Consider a function f : R d + → R . In this section we focus on calculating E [ f ( X )] and its properties with respect to C .Assuming the marginals are given, we define the expectation operator π f as follows π f ( C ) := E [ f ( X )] = (cid:90) R d + f ( x , . . . , x d ) d C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) = (cid:90) [0 , d f (cid:0) F − ( u ) , . . . , F − d ( u d ) (cid:1) d C ( u , . . . , u d ) . (3.1)However, if Q is a proper quasi-copula then d Q (cid:0) F ( x ) , . . . , F ( x d ) (cid:1) does not induce a measureanymore, because the Q -volume V Q is not necessarily positive. The idea is now to switch the func-tion we integrate against, i.e. to perform a Fubini transformation. In order to do so, the function f has to induce a measure. Therefore, we consider functions of the following type.7 efinition 3.1. (i) A function f : R d + → R is called ∆ -antitonic if for every subset I = { i , . . . , i n } ⊆ { , . . . , d } with | I | ≥ and every hypercube × nj =1 ( a j , b j ] ⊂ R n + ( − n ∆ i a ,b ◦ · · · ◦ ∆ i n a n ,b n f ≥ . (ii) A function f : R d + → R is called ∆ -monotonic if for every subset I = { i , . . . , i n } ⊆{ , . . . , d } with | I | ≥ and every hypercube × nj =1 ( a j , b j ] ⊂ R n + ∆ i a ,b ◦ · · · ◦ ∆ i n a n ,b n f ≥ . We will frequently deal with marginals of functions f and quasi-copulas Q , therefore the followingdefinition is useful. We have already proved in Proposition 2.4 that marginals of (quasi)-copulasremain (quasi)-copulas. Definition 3.2. (i) Let f : R d + → R . Then, for I = { i , . . . , i n } ⊆ { , . . . , d } , we define the I -margin of f as f I : R n + → R , ( x i , . . . , x i n ) (cid:55)→ f ( x , . . . , x d ) , with x k = 0 for k / ∈ I . (ii) Let Q ∈ Q d . Then, for I = { i , . . . , i n } ⊆ { , . . . , d } , we define the I -margin of Q as Q I : [0 , n → [0 , , ( u i , . . . , u i n ) (cid:55)→ Q ( u , . . . , u d ) , with u k = 1 for k / ∈ I . According to Proposition 2.1, we can associate a measure to every right-continuous and ∆ -monotonicor ∆ -antitonic function f : R d + → R via µ f I ( ∅ ) := 0 and µ f I (cid:0) R (cid:1) := V f I (cid:0) R (cid:1) , (3.2)for every hyperrectangle R ⊆ R | I | . Then, we get that µ f I is a positive measure on R | I | + if f is ∆ -monotonic, and that ( − n µ f I is a positive measure on R | I | + if f is ∆ -antitonic. If I = { , . . . , d } ,then we write µ f instead of µ f I . In addition, we define µ f ∅ := δ , where δ denotes the Diracmeasure. Remark 3.3. Let f : R d + → R be a right-continuous function, such that − f is either ∆ -antitonic or ∆ -monotonic. Then, we have for I = { i , . . . , i n } ⊆ { , . . . , d } with | I | ≥ and every hypercube × nj =1 ( a j , b j ] ⊂ R n + that ( − n ∆ i a ,b ◦ · · · ◦ ∆ i n a n ,b n f ≤ , if − f is ∆ -antitonic, and ∆ i a ,b ◦ · · · ◦ ∆ i n a n ,b n f ≤ , if − f is ∆ -monotonic. Hence, − µ f I is a positive measure on R | I | + if − f is ∆ -monotonic and ( − n +1 µ f I is a positivemeasure on R | I | + if − f is ∆ -antitonic. (cid:7) I -marginals of functions in con-junction with the I -marginals of copulas, can be used to define an integration operation. We defineiteratively: for | I | = 0 : ϕ If ( C ) := f (0 , . . . , , for | I | = 1 : ϕ If ( C ) := (cid:90) R + f i ( x i ) d F i ( x i ) , for | I | = n ≥ ϕ If ( C ) := (cid:90) R | I | + (cid:98) C I (cid:0) F i ( x i ) , . . . , F i n ( x i n ) (cid:1) d µ f I ( x i , . . . , x i n )+ (cid:88) J ⊂ I ( − n +1 −| J | ϕ Jf ( C ) , (3.3)where (cid:98) C I denotes the survival function of the I -margin of C . Lux and Papapantoleon [12, Prop.5.3] proved that the operator ϕ { ,...,d } f ( C ) defined above coincides with the expectation operator π f ( C ) in (3.1) in case f : R d → R is right-continuous, ∆ -antitonic or ∆ -monotonic and C ∈ C d .However, the operator ϕ { ,...,d } f ( C ) does not depend on C being a copula, and can be also defined forquasi-copulas. This motivates the following definition, which generalizes the expectation operatorto quasi-copulas. Definition 3.4. Let f : R d → R be right-continuous, ∆ -antitonic or ∆ -monotonic and d ≥ . Then,the expectation operator is defined as follows π f : Q d → R , Q (cid:55)→ π f ( Q ) , with π f ( Q ) := ϕ { ,...,d } f ( Q ) . Remark 3.5. Let Q ∈ Q d and consider its survival function (cid:98) Q . We define the dual to the operations ϕ If and π f as follows: (cid:98) ϕ If (cid:0) (cid:98) Q (cid:1) := ϕ If (cid:0) Q (cid:1) and (cid:98) π f (cid:0) (cid:98) Q (cid:1) := π f (cid:0) Q (cid:1) , since both operations actually only depend on the knowledge of (cid:98) Q and not of Q itself. (cid:7) Remark 3.6. Using that V f i (cid:0) (0 , x ] (cid:1) = f i ( x ) − f i (0) , we can rewrite the case | I | = { i } from (3.3)as follows (cid:90) R + f i ( x i ) d F i ( x i ) = (cid:90) R + (cid:0) − F i ( x i ) (cid:1) d µ f i ( x i ) + f i (0) . (3.4)Depending on the way the integrals are computed, this representation might be more useful. If wecompute the one-dimensional integrals as in (3.3) instead of (3.4), then we do not need f { i } to inducea measure. Therefore, in [12] the authors define ∆ -antitonic and ∆ -monotonic in the sense that only f I , | I | ≥ , has to induce a measure. (cid:7) The following result provides an alternative, simpler representation for the expectation operator π f ( Q ) . 9 heorem 3.7. Let f : R d → R be right-continuous, ∆ -antitonic or ∆ -monotonic and Q ∈ Q d .Then, the following representation holds π f ( Q ) = (cid:90) R d + (cid:98) Q (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) d µ f ( x , . . . , x d )+ (cid:88) J ⊂ I | J | = d − (cid:90) R d − (cid:98) Q J (cid:0) F ( x i ) , . . . , F i d − ( x i d − ) (cid:1) d µ f J ( x i , . . . , x i d − )+ · · · + d (cid:88) i =1 (cid:90) R + f { i } ( x i ) d F i ( x i ) − ( d − f (0 , . . . , f (0 , . . . , 0) + d (cid:88) n =1 (cid:88) J ⊆ IJ = { i ,...,i n } (cid:90) R n + (cid:98) Q J (cid:0) F ( x i ) , . . . , F i n ( x i n ) (cid:1) d µ f J ( x i , . . . , x i n ) . (3.5) Proof. Without loss of generality we assume I = { , . . . , d } . For | I | = 1 the claim is given by(3.4). Now assume it holds all n < d for some d ∈ N . Define α J := (cid:90) R | J | + (cid:98) Q J (cid:0) F ( x i ) , . . . , F i n ( x i n ) (cid:1) d µ f J ( x i , . . . , x i n ) , | J | ≥ ,α ∅ := f (0 , . . . , . Then we deduce by (3.3) and the induction hypothesis ϕ If ( Q ) = α I + (cid:88) J ⊂ I ( − d +1 −| J | ϕ Jf ( Q )= α I + (cid:88) J ⊂ I ( − d +1 −| J | (cid:88) J (cid:48) ⊆ J α J (cid:48) . (3.6)Hence, we have to show that for every J (cid:48) ⊂ I the term α J (cid:48) appears exactly once in (3.6) withpositive sign. Consider J (cid:48) = { j , . . . , j k } ⊆ J = { i , . . . , i n } . There are (cid:0) d − kn − k (cid:1) many J ⊂ I with J (cid:48) ⊆ J because for J \ J (cid:48) we can choose n − k elements out of I \ J (cid:48) . We have (cid:88) J ⊂ I ( − d +1 −| J | (cid:88) J (cid:48) ⊆ J α J (cid:48) = n (cid:88) k =0 | J (cid:48) | = k d − (cid:88) n = k ( − d +1 − n (cid:18) d − kn − k (cid:19) α J (cid:48) . Further, d − (cid:88) n = k ( − d +1 − n (cid:18) d − kn − k (cid:19) = d − k − (cid:80) n =0 ( − n +1 (cid:0) d − kn (cid:1) , if d − k is even, d − k − (cid:80) n =0 ( − n (cid:0) d − kn (cid:1) , if d − k is odd.Since (cid:80) ml =0 ( − l (cid:0) ml (cid:1) = 0 , m ∈ N , we have (cid:80) d − n = k ( − d +1 − n (cid:0) d − kn − k (cid:1) = 1 for both cases. Thisproves (3.5). The other representation of ϕ If ( Q ) follows by (3.4). (cid:3) π f is increasing or decreasing with respect to thelower and upper orthant order, depending on the properties of the function f . Proposition 3.8. Let Q , Q ∈ Q d and f : R d + → R . Then(i) for all f ∆ -antitonic s.t. the integrals exist Q (cid:22) LO Q = ⇒ π f ( Q ) ≤ π f ( Q ) , (ii) for all f ∆ -monotonic s.t. the integrals exist Q (cid:22) UO Q = ⇒ π f ( Q ) ≤ π f ( Q ) . (iii) for all − f ∆ -antitonic s.t. the integrals exist Q (cid:22) LO Q = ⇒ π f ( Q ) ≥ π f ( Q ) , (iv) for all − f ∆ -monotonic s.t. the integrals exist Q (cid:22) UO Q = ⇒ π f ( Q ) ≥ π f ( Q ) . Proof. The first two statements are Lux and Papapantoleon [12, Theorem 5.5], while the next twoare a direct consequence of them and Remark 3.3. (cid:3) 4. Copulas and arbitrage In this section, we apply the results on improved Fréchet–Hoeffding bounds and on stochastic dom-inance for quasi-copulas to mathematical finance. We will first derive bounds for the arbitrage-freeprices of certain classes of multi-asset derivatives. Then, we will formulate a necessary condition forthe absence of arbitrage in markets where several multi-asset derivatives are traded simultaneously. Let (Ω , F , P ) be a probability space. We consider the following financial market model: There existsone time period with initial time t = 0 and final time t = T < ∞ . Let d ≥ . There exist d + 1 non-redundant primary assets denoted by B, S , . . . , S d . We assume that their initial prices are known, i.e. ( B , S , . . . , S d ) ∈ R d +1+ . B denotes the risk-free asset that earns the interest rate r ≥ and,for the sake of simplicity, we set Q T = 1 , while S T , . . . , S dT are R + -valued random variables on thegiven probability space.A probability measure Q on (Ω , F ) , equivalent to P , that satisfies S i = B E Q (cid:2) S iT (cid:3) , i = 1 , . . . , d , is called an equivalent martingale measure (EMM) for our financial market. Let P denote the setof all EMMs for our financial market model, i.e. P = { Q | Q ∼ P , Q EMM } . This definition has a11ell-known implication for the pricing of derivatives of S iT . Consider a derivative of S iT with payoff H ( S iT ) at time T , where H is a function such that E Q [ H ( S iT )] exists. Then, the arbitrage-free priceis provided by H i = B E Q (cid:2) H ( S iT ) (cid:3) , i = 1 , . . . , d. We assume that the risk-neutral marginal distributions of each S iT are known and unique for all i = 1 , . . . , d , i.e. the univariate marginal distribution of S iT under Q is equal for all Q ∈ P . Wefurther assume that these distributions are continuous, and denote them by F i . Hence, Q ∈ P if Q (cid:0) S T ∈ R + , . . . , S i − T ∈ R + , S iT ≤ x, S i +1 T ∈ R + , . . . , S dT ∈ R + (cid:1) = F i ( x ) , (4.1)for all i = 1 , . . . , d . The assumption that the marginal distributions are known is not unrealistic,because their dynamics can be derived from market data; see e.g. Breeden and Litzenberger [3]. Thisproperty implies, by the second Fundamental Theorem of Asset Pricing, that the prices of single-asset options are unique, and is referred to in the literature as static-completeness of a financialmarket, see e.g. Carr and Madan [5]. Let us stress that this does not imply | P | = 1 , because thedependence structure of S , . . . , S d might not be uniquely determined.The financial market, beside options on the single assets S , . . . , S d , consists also of a finite numberof multi-asset derivatives, denoted by Z , . . . , Z q , for q ∈ N . Their final payoffs at time T are givenby Z iT = z i (cid:0) S T , . . . , S dT (cid:1) , i = 1 , . . . , q, where the payoff functions z i : R d + → R + (resp. their negation, i.e. − z i ) are either ∆ -antitonic or ∆ -monotonic. We assume that Z , . . . , Z q are “truly” multi-asset derivatives, i.e. they are writtenon at least two and up to d of the risky assets. Definition 4.1 (Arbitrage-free price vector). Let ( Z , . . . , Z q ) be a set of multi-asset derivativesas described above, for q ∈ N . We call p = ( p , . . . , p q ) ∈ R q + an arbitrage-free price vector for ( Z , . . . , Z q ) if there exists a measure Q ∈ P such that p k = B E Q (cid:2) Z kT (cid:3) , for all k = 1 , . . . , q. We denote the set of all arbitrage-free price vectors for ( Z , . . . , Z q ) by Π( Z , . . . , Z q ) . This set isdescribed by Π( Z , . . . , Z q ) = (cid:110)(cid:16) B E Q (cid:2) Z T (cid:3) , . . . , B E Q (cid:2) Z qT (cid:3)(cid:17) (cid:12)(cid:12)(cid:12) Q ∈ P and E Q (cid:2) Z kT (cid:3) < ∞ , k = 1 , . . . , q (cid:111) . In this sub-section, we study the relation between copulas and the set of arbitrage-free price vectors Π( Z , . . . , Z q ) . The first result is essentially Tavin [19, Corollary 3] and we provide a short prooffor the sake of completeness. Proposition 4.2. In the multi-asset financial market model described above, there is a bijectionbetween P and C d . roof. The proof uses essentially Sklar’s Theorem and the association between copulas and proba-bility measures. Let Q ∈ P and denote by F Q the joint distribution of ( S T , . . . , S dT ) under Q . Then,define the function C Q via C Q ( u ) := F Q (cid:0) F − ( u ) , . . . , F − d ( u d ) (cid:1) , u ∈ [0 , d . By Sklar’s Theorem, C Q is indeed a copula.On the other hand, let C ∈ C d and denote by F C the corresponding distribution function defined as F C ( x ) := C (cid:0) F ( x ) , . . . , F d ( x d ) (cid:1) , x ∈ [0 , ∞ ) d . Then F C has marginals F i , i = 1 , . . . , d , and therefore Q ∈ P by (4.1). (cid:3) This bijection allows us to express the arbitrage-free price of a derivative Z iT , and therefore alsoexpectations of the form E Q [ Z iT ] for Q ∈ P , in terms of the associated copula C Q . That is, E Q (cid:2) Z iT (cid:3) = E Q (cid:2) z i ( S T , . . . , S dT ) (cid:3) = (cid:90) R d + z i ( x , . . . , x d ) d F Q ( x )= (cid:90) [0 , d z i (cid:0) F − ( u ) , . . . , F − d ( u d ) (cid:1) d C Q ( u ) . (4.2)We denote the expectation under the measure associated with a copula C by E C . The bijectionbetween the set of equivalent martingale measures and the set of copulas in Proposition 4.2 allowsnow to describe the set of arbitrage-free price vectors in terms of copulas, i.e. Π( Z , . . . , Z q ) = (cid:110)(cid:16) B E C (cid:2) Z T (cid:3) , . . . , B E C (cid:2) Z T (cid:3)(cid:17) (cid:12)(cid:12)(cid:12) C ∈ C d and E C (cid:2) Z kT (cid:3) < ∞ , k = 1 , . . . , q (cid:111) . (4.3)Finally, recall the definition of the expectation operator π f from the previous section. Using (3.1)and (4.2) we get that π z k ( C ) = E C [ Z kT ] for k = 1 , . . . , q . Hence, for the multi-asset derivatives Z , . . . , Z q we define the following pricing rule between the set of copulas and the set of arbitrage-free price vectors Π( Z , . . . , Z q ) , (cid:37) : C d → R q + , C (cid:55)→ (cid:37) ( C ) := (cid:0) B π z ( C ) , . . . , B π z q ( C ) (cid:1) . Consequently, we can prove the following equivalence result. Proposition 4.3. Let p ∈ R q + . Then p ∈ Π( Z , . . . , Z q ) ⇐⇒ ∃ C ∈ C d such that (cid:37) ( C ) = p . Proof. The equivalence follows immediately from the definition of the pricing rule together with(3.1), (4.2) and (4.3). (cid:3) emark 4.4. Using the definition of the dual operator (cid:98) π f , see Remark 3.5, the previous result carriesover analogously to the set of survival copulas (cid:98) C d , i.e. (cid:98) (cid:37) : (cid:98) C d → R q + , (cid:98) C (cid:55)→ (cid:98) (cid:37) ( (cid:98) C ) := (cid:0) B (cid:98) π z ( (cid:98) C ) , . . . , B (cid:98) π z q ( (cid:98) C ) (cid:1) and p ∈ Π( Z , . . . , Z q ) ⇐⇒ ∃ (cid:98) C ∈ (cid:98) C d such that (cid:98) (cid:37) ( (cid:98) C ) = p . (cid:7) We have assumed that the payoff functions z i : R d + → R + , resp. their negations − z i , are either ∆ -antitonic or ∆ -monotonic. Therefore, we get from Proposition 3.8 that π z i is non-decreasing, resp.non-increasing, with respect to the lower or upper orthant order. Hence, we can use the Fréchet–Hoeffding bounds and the parametrization of arbitrage-free price vectors in terms of copulas inorder to derive arbitrage-free bounds for the set Π( Z i ) for each multi-asset derivative in the market.Moreover, assume there exists additional information about the copulas, i.e. consider a constrainedset C ∗ ⊆ C d such as C S ,C ∗ or C ρ,θ . Then, we also have a constrained set of arbitrage-free prices, i.e. Π ∗ ( Z i ) = (cid:8) B π z i ( C ) | C ∈ C ∗ (cid:9) ⊆ Π( Z i ) . In other words, the improved Fréchet–Hoeffding bounds allow us to tighten the range of arbitrage-free prices for the derivative Z i . This concept works analogously for the set of survival functions, i.e. for (cid:98) C ∗ ⊂ (cid:98) C d . Corollary 4.5. Let Z be a multi-asset derivative in the financial market described above with payofffunction z .(i) Let z be ∆ -antitonic and Q ∗ L , Q ∗ U be the lower and upper bound for some constrained set C ∗ ⊆ C d . Then, for all C ∈ C ∗ holds π z ( W d ) ≤ π z ( Q ∗ L ) ≤ π z ( C ) ≤ π z ( Q ∗ U ) ≤ π z ( M d ) . (ii) Let z be ∆ -monotonic and (cid:98) Q ∗ L , (cid:98) Q ∗ U be the lower and upper bound for some constrained set (cid:98) C ∗ ⊆ (cid:98) C d . Then, for all (cid:98) C ∈ (cid:98) C ∗ holds (cid:98) π z ( W d ) ≤ (cid:98) π z ( (cid:98) Q ∗ L ) ≤ (cid:98) π z ( (cid:98) C ) ≤ (cid:98) π z ( (cid:98) Q ∗ U ) ≤ (cid:98) π z ( M d ) = π z ( M d ) , where W d ( u ) = W d ( − u ) and M d ( u ) = M d ( − u ) .Proof. These claims follow directly from the ordering of the bounds, the monotonicity results inProposition 3.8, and their analogues for survival functions. (cid:3) Remark 4.6. The inequalities above change direction if − z is either ∆ -antitonic or ∆ -monotonic. (cid:7) Remark 4.7. Lux and Papapantoleon [12, Section 6] provide conditions such that the improved op-tion price bounds are sharp, in the sense that inf Π ∗ ( Z ) = π z ( Q ∗ L ) and sup Π ∗ ( Z ) = π z ( Q ∗ U ) re-spectively. Depending on the payoff function z the computation of the improved option price boundscan be quite complicated. Rapuch and Roncalli [16], Tankov [18] and Lux and Papapantoleon [12]present several derivatives for which the integrals can be enormously simplified. (cid:7) .4. A necessary condition for the absence of arbitrage in the presence of severalmulti-asset derivatives In this subsection, we assume there exist several multi-asset derivatives Z , . . . , Z q in the financialmarket, and consider a price vector p = ( p , . . . , p q ) ∈ R q + for them. Our goal is to check whether p is an arbitrage-free price vector or not, i.e. whether p ∈ Π( Z , . . . , Z q ) . In fact, we will derive anecessary condition for p to be an arbitrage-free price vector.Consider the following constrained sets of copulas C π k ,p k := (cid:8) C ∈ C d | B π z k ( C ) = p k (cid:9) , k = 1 , . . . , q , which are sets of the form (2.5). Clearly, C π k ,p k (cid:54) = ∅ if and only if p k ∈ Π( Z k ) by Proposition4.3. Hence, C π k ,p k contains all copulas compatible with the price p k for the derivative Z k , for each k = 1 , . . . , q . Analogously we define the set of survival functions (cid:98) C π k ,p k := (cid:8) (cid:98) C ∈ (cid:98) C d | B (cid:98) π z k ( (cid:98) C ) = p k (cid:9) , k = 1 , . . . , q . The next result shows that p is an arbitrage-free price vector for ( Z , . . . , Z q ) if and only if itcontains an arbitrage-free price for each derivative. Proposition 4.8. Let p ∈ R q + . Then we have the following equivalences: p ∈ Π( Z , . . . , Z q ) ⇐⇒ q (cid:92) k =1 C π k ,p k (cid:54) = ∅ or p ∈ Π( Z , . . . , Z q ) ⇐⇒ q (cid:92) k =1 (cid:98) C π k ,p k (cid:54) = ∅ . Proof. Let p ∈ Π( Z , . . . , Z q ) , then there exists a d -copula C ∈ C d such that (cid:37) ( C ) = p hence, forevery k = 1 , . . . , q , there exists a C ∈ C π k ,p k such that (cid:37) k ( C ) = p k . This readily implies that q (cid:92) k =1 C π k ,p k (cid:54) = ∅ . Using the same arguments in the opposite direction allows to prove the equivalence. The case forsurvival copulas is completely analogous. (cid:3) Remark 4.9. The previous result implies that the set of arbitrage-free price vectors for Z , . . . , Z q is a subset of the Cartesian product of the sets of arbitrage-free price vectors for each Z i , i.e. Π( Z , . . . , Z q ) ⊆ Π( Z ) × · · · × Π( Z q ) . In other words, we can have derivatives that are priced within their own no-arbitrage bounds, how-ever when they are considered together an arbitrage opportunity may arise. An example in thisdirection will be presented in the following section. (cid:7) C π k ,p k and (cid:98) C π k ,p k , k = 1 , . . . , q , and here the improved Fréchet–Hoeffding bounds play a crucial role. Letus define Q k p ( u ) = (cid:40) Q π k ,p k U ( u ) , if z k is ∆ -antitonic , (cid:98) Q π k ,p k U ( u ) , if z k is ∆ -monotonic , (4.4) Q k p ( u ) = (cid:40) Q π k ,p k L ( u ) , if z k is ∆ -antitonic , (cid:98) Q π k ,p k L ( u ) , if z k is ∆ -monotonic , (4.5)where Q π k ,p k L , Q π k ,p k U , (cid:98) Q π k ,p k L , (cid:98) Q π k ,p k U are defined as in (2.6)–(2.7) and (A.1)–(A.2) respectively.Moreover, we define Q p ( u ) := min (cid:8) Q k p ( u ) | k = 1 , . . . , q (cid:9) and Q p ( u ) := max (cid:8) Q k p ( u ) | k = 1 , . . . , q (cid:9) . (4.6)Now we can state the main result of this section, which provides a necessary condition for theabsence of arbitrage in a financial market in the presence of several multi-asset derivatives. Thisgeneralizes Tavin [19, Proposition 9] to the d -dimensional case. Theorem 4.10. Let p ∈ R q + . In the financial market described above, with several multi-assetderivatives Z , . . . , Z q traded simultaneously, we have p ∈ Π( Z , . . . , Z q ) = ⇒ Q p ( u ) ≤ Q p ( u ) for all u ∈ [0 , d . (4.7) Proof. Let f be ∆ -antitonic. Assume there exists a u ∗ ∈ [0 , d such that Q p ( u ∗ ) > Q p ( u ∗ ) . Byconstruction of Q p and Q p , the minimum and maximum are always attained. Denote by k A , k B ∈{ , . . . , q } the indices for which the minimum and maximum are attained in (4.6). Then we havethat k A (cid:54) = k B , because otherwise inf (cid:8) C ( u ∗ ) | C ∈ C π kA ,p kA (cid:9) = Q k A p ( u ∗ ) = Q p ( u ∗ ) > Q p ( u ∗ ) = Q k A p ( u ∗ ) = sup (cid:8) C ( u ∗ ) | C ∈ C π kA ,p kA (cid:9) . Hence, we get that Q p ( u ∗ ) = inf (cid:8) C ( u ∗ ) | C ∈ C π kB ,p kB (cid:9) > sup { C ( u ∗ ) | C ∈ C π kA ,p kA } = Q p ( u ∗ ) , which readily implies that C π kA ,p kA ∩ C π kB ,p kB = ∅ . Therefore, we also get that q (cid:92) k =1 C π k ,p k ⊆ (cid:16) C π kA ,p kA ∩ C π kB ,p kB (cid:17) = ∅ , which is equivalent to p / ∈ Π( Z , . . . , Z q ) by Proposition 4.8. The proof for ∆ -monotonic functions f and (cid:98) C π k ,p k works completely analogously. (cid:3) We have assumed so far that there exist S , . . . , S d underlying assets in the financial market andthat all multi-asset derivatives Z , . . . , Z q depend on all d assets. This is however not very realistic,16s there might well exist derivatives that depend on some, but not all, of the underlying assets. Thenext result treats exactly that scenario, making use of the results on I -margins of copulas.Assume there exist Z , . . . , Z q multi-asset derivatives in the financial market, and that each deriva-tive Z k depends on d k of the underlying assets with ≤ d k ≤ d . That is, each Z k depends on ( S i , . . . , S i dk ) with I k = { i , . . . , i d k } ⊆ { , . . . , d } and k = 1 , . . . , q . Let us define I ∗ := (cid:84) qk =1 I k and d ∗ := | I ∗ | . Moreover, we assume that d ∗ ≥ , i.e. all multi-asset derivatives share atleast two common underlying assets.Let us now update the definition of the constrained set of copulas C π k ,p k as follows: C π k ,p k := (cid:8) C ∈ C d | B π z k ( C I k ) = p k (cid:9) , k = 1 , . . . , q ; this coincides with the previous definition in case all derivatives depend on all d assets. Moreover,let us also define the following constrained set of copulas, that projects everything in the space ofthe common underlying assets: C π k ,p k I ∗ := (cid:8) C I ∗ ∈ C d ∗ | C ∈ C π k ,p k (cid:9) , k = 1 , . . . , q . We define now the upper and lower improved Fréchet–Hoeffding bounds for the set C π k ,p k I ∗ , denotedby Q k, ∗ p and Q , ∗ p completely analogously to (4.4) and (4.5), and also define Q ∗ p ( u ) := min (cid:8) Q , ∗ p ( u ) | k = 1 , . . . , q (cid:9) and Q ∗ p ( u ) := max (cid:8) Q k, ∗ p ( u ) | k = 1 , . . . , q (cid:9) , (4.8)as in (4.6). Then, we have the following necessary condition for the absence of arbitrage in thisfinancial market. Theorem 4.11. Let p ∈ R q + . In the financial market described above, with several multi-assetderivatives Z , . . . , Z q traded simultaneously, we have p ∈ Π( Z , . . . , Z q ) = ⇒ Q ∗ p ( u ) ≤ Q ∗ p ( u ) for all u ∈ [0 , d ∗ . (4.9) Proof. The idea is again that for p ∈ Π( Z , . . . , Z q ) there must exist a d ∗ -copula C with C ∈ (cid:84) qk =1 C π k ,p k I ∗ . The proof is then completely analogous to the proof of Theorem 4.10, and thus omittedfor the sake of brevity. (cid:3) The intuition behind the last two results is that whenever the inequalities in (4.7) and (4.9) areviolated for some u ∈ [0 , d , then there does not exist a copula that can describe the prices of allderivatives Z , . . . , Z q . Hence, this set of prices is not jointly arbitrage-free. Therefore, followingTavin [19], we can also express the arbitrage detection problem as a minimization problem. Indeed,let us consider, O : min u ∈ [0 , d (cid:8) Q p ( u ) − Q p ( u ) (cid:9) . The objective function u (cid:55)→ Q p ( u ) − Q p ( u ) takes values in [ − , and the minimization is realizedover a compact set. Hence, there exists a (possibly not unique) minimum, say u ∗ ∈ [0 , d . The ideanow is that if Q p ( u ∗ ) − Q p ( u ∗ ) < , then p is not free of arbitrage. Note that the opposite result17ould not necessarily imply p being arbitrage-free, since Theorems 4.10 and 4.11 provide onlya necessary condition. Nevertheless O might detect an arbitrage which is not obvious in the firstplace. In fact, it is possible that p = ( p , . . . , p q ) is not free of arbitrage although all p i ’s lie withinthe arbitrage-free bounds computed from the Fréchet-Hoeffding bounds. In summary, we have thefollowing result: min u ∈ I d (cid:8) Q p ( u ) − Q p ( u ) (cid:9) = (cid:40) ≥ , no decision ,< , π / ∈ Π . 5. Applications In this section, we present some applications of the previous results in the computation of bounds forarbitrage-free prices and in the detection of arbitrage opportunities. We are particularly interestedin the case where the prices of each multi-asset derivative lie within their respective no-arbitragebounds, yet an arbitrage arises when they are considered jointly.The framework for the applications and the numerical examples presented below is summarized inthe following bullet points:• We consider a financial market as described above with final time T = 1 .• We assume, for simplicity, that the interest rate is zero, i.e. B t = 1 , t ∈ [0 , .• There exist three risky assets S , S , S ( d = 3 ) with known marginals distributions F , F , F at t = 1 but unknown dependence structure.• The marginals of ( S , S , S ) are log-normally distributed, i.e. S i = S i exp (cid:16) σ i W ( i )1 − σ i (cid:17) , i = 1 , , where W ( i ) are standard Brownian motions, while the initial values and parameters are i S i σ i Z , Z ( q = 2 ), with payoff functions z , z such that z and − z are ∆ -monotonic.• The payoff functions of Z and Z are provided by z ( x ) = (cid:0) min { x , x , x } − K (cid:1) + ,z ( x ) = (cid:0) K − min { x , x , x } (cid:1) + , K , K ∈ R + , i.e. a call and a put option on the minimum of three assets.18 .1. Bounds for arbitrage-free prices within the two sub-markets We first consider the two sub-markets that consist of the three assets and each multi-asset derivativeseparately, i.e. ( S , S , S , Z ) and ( S , S , S , Z ) , and we are interested in deriving bounds forthe arbitrage-free prices of Z and Z . The functions z and − z are ∆ -monotonic, hence a lowerand upper bound for Π( Z ) and Π( Z ) can be derived by the Fréchet–Hoeffding bounds; indeed,we have (cid:98) π z ( W ) ≤ p ≤ (cid:98) π z ( M ) , for every p ∈ Π( Z ) , (cid:98) π z ( M ) ≤ p ≤ (cid:98) π z ( W ) , for every p ∈ Π( Z ) . The support of the measures induced by z and z is one-dimensional and lies equally distributedalong the diagonal, i.e. supp( µ z ) = (cid:8) x ∈ [ K , ∞ ) | x = x = x (cid:9) , supp( µ z ) = (cid:8) x ∈ [0 , K ] | x = x = x (cid:9) . Moreover, since z ,I ≡ and z ,I ≡ K for all I with | I | = 1 , , we get that µ z ,I = µ z ,I = 0 .This also implies (cid:90) R + | z i,I ( x, x, x ) | d F Q ( x ) = 0 < ∞ , i = 1 , , while for I = { , , } we have (cid:90) R + | z i ( x, x, x ) | d F Q ( x ) = E (cid:2) | z i ( S , S , S ) | (cid:3) < ∞ , i = 1 , . Hence, the expectation operator (cid:98) π z i is well-defined for i = 1 , by Lux and Papapantoleon [12,Proposition 5.8]. Let us also mention that µ z and − µ z are positive measures. Now, noting that z (0 , , 0) = 0 and z (0 , , 0) = K , we deduce the followings bounds for Π( Z ) and Π( Z ) : (cid:98) π z (cid:0) W (cid:1) = (cid:90) [ K , ∞ ) W (cid:0) F ( x ) , F ( x ) , F ( x ) (cid:1) d x , (cid:98) π z (cid:0) M (cid:1) = (cid:90) [ K , ∞ ) M (cid:0) F ( x ) , F ( x ) , F ( x ) (cid:1) d x , (cid:98) π z (cid:0) W (cid:1) = K − (cid:90) [0 ,K ] W (cid:0) F ( x ) , F ( x ) , F ( x ) (cid:1) d x , (cid:98) π z (cid:0) M (cid:1) = K − (cid:90) [0 ,K ] M (cid:0) F ( x ) , F ( x ) , F ( x ) (cid:1) d x . A numerical illustration of these bounds is depicted in Figure 1.19igure 1: Bounds for Π( Z ) (left) and Π( Z ) (right) derived by the Fréchet–Hoeffding bounds as afunction of the strike. Remark 5.1. There exists an x such that W d (cid:0) F ( x ) , . . . , F d ( x ) (cid:1) = 0 for x > x . This x de-pends on the marginal distributions. In general this fact might be unimportant, but in the case ofa call or put on the minimum it has an interesting implication. It means that (cid:98) π z ( W ) = 0 while (cid:82) [0 ,K ] W (cid:0) F ( x ) , F ( x ) , F ( x ) (cid:1) d x is constant for K ≥ x . There is no equivalent statement forthe upper bound because in general F i ( x ) < for x < ∞ . (cid:7) Finally, we present an application of the main result of this work, i.e. Theorem 4.10. More specif-ically, we detect an arbitrage in the market ( S , S , S , Z , Z ) that contains three assets andtwo three-asset derivatives, even though the prices of Z and Z lie inside their respective no-arbitrage bounds. Tavin [19] searches for the global minimum of the objective function f obj ( u ) = Q p ( u ) − Q p ( u ) over the unit square. However, it suffices to find a u ∗ such that f obj ( u ∗ ) < and notnecessarily the global minimum. Since we consider an additional dimension, we restrict ourselvesto checking whether f obj becomes negative or not.Consider the call and put option on the minimum of three assets Z and Z with strikes K = 3 and K = 10 respectively. Then we have approximately the following no-arbitrage bounds: Π( Z ) = [0 . , . , Π( Z ) = [4 . , . . Assume that the traded price for the call equals . and the traded price for the put equals , i.e. p = (3 . , . Obviously both prices lie within their respective no-arbitrage bounds, hence the twosub-markets where either Z or Z is the only multi-asset derivative are free of arbitrage. However,we numerically compute that f obj (0 . , . , . ≈ − . < , therefore Theorem 4.10 yields that the market with both multi-asset derivatives is not free of arbi-trage, i.e. p / ∈ Π( Z , Z ) . Figure shows a plot of the objective function f obj . One can see clearly20ow f obj drops below zero around u = (0 . , . , . .An intuitive explanation behind the appearance of arbitrage for the price vector p = (3 . , couldbe as follows: The prices for Z and Z are taken from the upper part of the intervals Π( Z ) and Π( Z ) ; however, the payoff function π z is non-increasing with respect to the upper orthant order,which diminishes the chance of finding a copula C such that both (cid:98) π z ( (cid:98) C ) = p and (cid:98) π z ( (cid:98) C ) = p .A similar result appears if we choose both prices close to the lower bounds, i.e. for p = (0 . , . we get that f obj (0 . , . , . ≈ − . < . On the other hand, if we select a price away from the upper bound for Z , e.g. p = (3 . , . ,then the objective function does not become negative any longer. Indeed, we find that the globalminimum of the objective function f obj is zero, and is attained for u i = 0 or u i = 1 for some i = 1 , , , i.e. on the boundaries of the unit cube [0 , . Let us point out again that this does notnecessarily imply that the market is free of arbitrage, since Theorem 4.10 only provides a necessarycondition. A. Improved Fréchet–Hoeffding bounds for survival copulas Here we describe improved Fréchet–Hoeffding bounds for survival copulas. We start with the casewhen the value of the survival copula is known on a subset of its domain.Let S ⊆ [0 , d be compact and C ∗ ∈ C d . Define the set (cid:98) C S ,C ∗ := (cid:8) (cid:98) C ∈ (cid:98) C d | (cid:98) C ( x ) = (cid:98) C ∗ ( x ) for all x ∈ S (cid:9) . Then, for all (cid:98) C ∈ (cid:98) C S ,C ∗ , holds (cid:98) Q S ,C ∗ L ( u ) ≤ (cid:98) C ( u ) ≤ (cid:98) Q S ,C ∗ U ( u ) for all u ∈ [0 , d , where the improved Fréchet–Hoeffding bounds are provided by (cid:98) Q S ,C ∗ L ( u ) := Q (cid:98) S , (cid:98) C ∗ (1 −· ) L ( − u ) and (cid:98) Q S ,C ∗ U ( u ) := Q (cid:98) S , (cid:98) C ∗ (1 −· ) U ( − u ) , with (cid:98) S := { ( − x ) | x ∈ S} .Moreover, we are interested in improved Fréchet–Hoeffding bounds in case the value of a functionalof the survival copula is known. Consider a functional ρ : C d → R as in Section 2, and assume it isnon-decreasing with respect to the lower orthant order and continuous with respect to the pointwiseconvergence of quasi-copulas. Define the dual of ρ as follows (cid:98) ρ : (cid:98) C d → R , (cid:98) C (cid:55)→ (cid:98) ρ ( (cid:98) C ) := ρ ( C ) . The property of ρ being non-decreasing with respect to the upper orthant order implies that (cid:98) ρ isnon-decreasing, on the set of survival functions, with respect to the lower orthant order, i.e. (cid:98) C (cid:22) LO (cid:98) C ⇔ C (cid:22) UO C = ⇒ ρ ( C ) ≤ ρ ( C ) ⇔ (cid:98) ρ ( (cid:98) C ) ≤ (cid:98) ρ ( (cid:98) C ) . f obj for p = (3 . , and for themarginals restricted on (a) u = 0 . , (b) u = 0 . , (c) u = 0 . The continuity of ρ with respect to the pointwise convergence of copulas carries over to (cid:98) ρ and theset of survival functions. We define analogously (cid:98) ρ − ( u , r ) := (cid:98) ρ ( (cid:98) Q { u } ,rL ) and (cid:98) ρ + ( u , r ) := (cid:98) ρ ( (cid:98) Q { u } ,rU ) , and, for (cid:98) I u := (cid:2) W d ( u ) , M d ( u ) (cid:3) , (cid:98) ρ − − ( u , θ ) := max (cid:8) r ∈ (cid:98) I u : (cid:98) ρ − ( u , r ) = θ (cid:9) and (cid:98) ρ − ( u , θ ) := min (cid:8) r ∈ (cid:98) I u : (cid:98) ρ + ( u , r ) = θ (cid:9) . θ ∈ (cid:2)(cid:98) ρ ( W d ) , (cid:98) ρ ( M d ) (cid:3) , and consider the set of survival copulas (cid:98) C ρ,θ := (cid:8) (cid:98) C ∈ (cid:98) C d | (cid:98) ρ ( (cid:98) C ) = θ (cid:9) . Then, for all (cid:98) C ∈ (cid:98) C ρ,θ , holds (cid:98) Q ρ,θL ( u ) ≤ (cid:98) C ( u ) ≤ (cid:98) Q ρ,θU ( u ) for all u ∈ [0 , d , where (cid:98) Q ρ,θL ( u ) := (cid:40)(cid:98) ρ − ( u , θ ) , if θ ∈ [ (cid:98) ρ + ( u , W d ( u )) , (cid:98) ρ ( M d )] ,W d ( u ) , otherwise , (A.1) (cid:98) Q ρ,θU ( u ) := (cid:40)(cid:98) ρ − − ( u , θ ) , if θ ∈ [ (cid:98) ρ ( W d ) , (cid:98) ρ − ( u , M d ( u ))] ,M d ( u ) , otherwise. (A.2) B. Improved Fréchet–Hoeffding bounds for non-increasingfunctionals The following two theorems cover the case when the map ρ is non-increasing with respect to theorthant orders. This appears in our work when the negation of the payoff function, say − ρ , is either ∆ -monotonic or ∆ -antitonic. In that case, we get that ρ ( M d ) ≤ ρ ( W d ) . The proofs of these resultsare omitted for the sake of brevity, as they are completely analogous to the proofs of Theorems 3.3and A.2 in Lux and Papapantoleon [12]. Theorem B.1. Let ρ : Q d → R be non-increasing with respect to the lower orthant order andcontinuous with respect to the pointwise convergence of quasi-copulas. Let θ ∈ [ ρ ( M d ) , ρ ( W d )] and define Q ρ,θ := (cid:8) Q ∈ Q d | ρ ( Q ) = θ (cid:9) . Then, for all Q ∈ Q ρ,θ , holds Q ρ,θL ( u ) ≤ Q ( u ) ≤ Q ρ,θU ( u ) for all u ∈ [0 , d , with Q ρ,θL ( u ) := (cid:40) ρ − ( u , θ ) , if θ ∈ [ ρ ( M d ) , ρ + ( u , W d ( u ))] ,M d ( u ) , otherwise ,Q ρ,θU ( u ) := (cid:40) ρ − − ( u , θ ) , if θ ∈ [ ρ − ( u , M d ( u )) , ρ ( W d )] ,W d ( u ) , otherwise . Theorem B.2. Let ρ : C d → R be non-increasing with respect to the upper orthant order andcontinuous with respect to the pointwise convergence of copulas. Let θ ∈ [ (cid:98) ρ ( M d ) , (cid:98) ρ ( W d )] anddefine (cid:98) C ρ,θ := (cid:8) C ∈ C d | (cid:98) ρ ( (cid:98) C ) = θ (cid:9) . hen, for all (cid:98) C ∈ (cid:98) C ρ,θ , holds (cid:98) Q ρ,θL ( u ) ≤ (cid:98) C ( u ) ≤ (cid:98) Q ρ,θU ( u ) for all u ∈ [0 , d , with (cid:98) Q ρ,θL ( u ) := (cid:40)(cid:98) ρ − ( u , θ ) , if θ ∈ [ (cid:98) ρ ( M d ) , (cid:98) ρ + ( u , W d ( u ))] ,M d ( u ) , otherwise , (cid:98) Q ρ,θU ( u ) := (cid:40)(cid:98) ρ − − ( u , θ ) , if θ ∈ [ (cid:98) ρ − ( u , M d ( u )) , (cid:98) ρ ( W d )] ,W d ( u ) , otherwise. C. Proofs Proof (Proof of Proposition 2.1). The function µ f is non-negative, since f is d -increasing, and sat-isfies µ f ( ∅ ) = 0 by definition. Let R = × di =1 ( a i , c i ] ⊂ R d + and cut R along some b with a i < b < c i for some i ∈ { , . . . , d } into two hyperrectangles R and R , i.e. R = ( a , c ] × · · · × ( a i − , c i − ] × ( a i , b ] × ( a i +1 , c i +1 ] × · · · × ( a d , c d ] R = ( a , c ] × · · · × ( a i − , c i − ] × ( b, c i ] × ( a i +1 , c i +1 ] × · · · × ( a d , c d ] . Denote by V i the set of vertices v of R i and by s i ( v ) the sign of the term f ( v ) in V f ( R i ) , i = 1 , , .Clearly, V = ( V ∪ V ) \ ( V ∩ V ) . Hence, for all v ∈ V , s ( v ) = (cid:40) s ( v ) , if v ∈ V \ ( V ∩ V ) s ( v ) , if v ∈ V \ ( V ∩ V ) . Moreover s ( v ) = − s ( v ) for all v ∈ V ∩ V . Therefore, V f ( R ) + V f ( R ) = (cid:88) v ∈V s ( v ) f ( v ) + (cid:88) v ∈V s ( v ) f ( v )= (cid:88) v ∈ ( V ∪V ) \ ( V ∩V ) s ( v ) f ( v ) + (cid:88) v ∈V ∩V (cid:0) s ( v ) + s ( v ) (cid:1) f ( v )= V f ( R ) . It follows inductively that the volume of a set does not depend on its decomposition. Since f isright-continuous so is V f . Hence, µ f in (2.2) defines a measure. (cid:3) Proof (Proof of Proposition 2.3). Using property (C1), (2.1) and (2.4) we get that C ( u ) = V C (cid:16) d × i =1 (0 , u i ] (cid:17) = V C (cid:0) (0 , d (cid:1) − d (cid:88) i =1 V C (cid:0) (0 , × · · · × (0 , × (0 , u i ] × (0 , × · · · × (0 , (cid:1) ± · · · + ( − d V C (cid:16) d × i =1 ( u i , (cid:17) = (cid:98) C (0 , . . . , − d (cid:88) i =1 (cid:98) C (0 , . . . , , u i , , . . . , ± · · · + ( − d (cid:98) C ( u ) = ( − d V (cid:98) C (cid:16) d × i =1 (0 , u i ] (cid:17) . (cid:98) C (cid:55)→ V (cid:98) C (cid:16) × di =1 (0 , · ] (cid:17) is the left inverse of C (cid:55)→ (cid:98) C . This implies that the transformationis injective. (cid:3) References [1] D. Bartl, M. Kupper, T. Lux, A. Papapantoleon, and S. Eckstein. Marginal and dependence uncertainty:bounds, optimal transport, and sharpness. Preprint, arXiv:1709.00641, 2017.[2] C. Bernard, X. Jiang, and S. Vanduffel. 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