improved fréchet hoeffding bounds, optimal transport and ... · improved fr´echet–hoeﬀding...

Improved Frechet–Hoeffding bounds, optimal transportand model-free finance

Antonis Papapantoleon

Collaborators: Thibaut Lux; Daniel Bartl, Michael Kupper, Stephan Eckstein; Paulo Yanez

Financial & Actuarial Mathematics Seminar · Ann Arbor · 24 April 2019

Introduction Derivation & Properties Integration, Ordering & Applications Duality, Transport & Applications Arbitrage Summary

Outline

ImprovedFrechet–Hoeffding

bounds

Derivation,properties

Integration,Ordering,

Applications

Duality,Transport

Arbitragedetection

Antonis Papapantoleon — Improved Frechet–Hoeffding bounds 2


Outline


bounds



Applications

Duality,Transport

Arbitragedetection



A paradigm shift in mathematical finance

‘Old’ paradigm:You are given the model and your task is to compute option prices,value-at-risk, . . .

‘New’ paradigm:You are not given the model and your task is to say something about optionprices, value-at-risk, . . . compute bounds





‘New’ paradigm:You are not given the model and your task is to say something about optionprices, value-at-risk, . . .

compute bounds





‘New’ paradigm:You are not given the model and your task is to say something about optionprices, value-at-risk, . . . compute bounds



MotivationCoin tossing / Dice rolling

We are rolling two dices D1,D2 and are interested in the distribution of the sum.

Simplest choice: D1 and D2 are independent dicesChoices with dependent dices:

D1, D2 = D1 (comonotonicity)D1, D2 = 7− D1 (countermonotonicity). . .

I Dependence uncertainty: the marginal distributions are known, thedependence structure is not known





Simplest choice: D1 and D2 are independent dices

Choices with dependent dices:D1, D2 = D1 (comonotonicity)D1, D2 = 7− D1 (countermonotonicity). . .






Simplest choice: D1 and D2 are independent dicesChoices with dependent dices:

D1, D2 = D1 (comonotonicity)D1, D2 = 7− D1 (countermonotonicity). . .




MotivationRandom variables

(X1, . . . ,Xd ): random variables with marginal distributions (F1, . . . ,Fd )Dependence structure: determined by joint distribution F or copula CSklar’s Theorem: given F ,F1, . . . ,Fd , there exists C s.t.

F (x1, ..., xd ) = C(F1(x1), ...,Fd (xd )) for all x ∈ Rd

Dependence uncertainty: the marginal distributions are known, thedependence structure is not known

I Main question: f ‘nice’ function, compute

inf{EC [f ] : C copula} and sup{EC [f ] : C copula}



History

This problem was inspired and formulated by KolmogorovSolved independently by Makarov and Ruschendorf in the early 1980’s ford = 2 and f supermodular

(∂f

∂x1∂x2≥ 0)

or f submodular(

∂f∂x1∂x2

≤ 0)

Schweizer and Nelsen provided an alternative proof in the late 80’s, stillfor d = 2Recently, the problem was reformulated under additional constraints byTankov

inf / sup {EC [f ] : C copula + partial information on C}

Very few results for d > 2 (Optimal Transportation, Optimization)



Roadmap

Define Cd /Qd , partial order �

Find upper/lower boundsQ/Q on Qd

? w.r.t. �

Check whether Q/Q lie in Cd

Define EQ[f ] for Q ∈ Qd Q � Q′ =⇒ EQ[f ] ≤ EQ′ [f ]

inf{EQ[f ]

∣∣Q ∈ Qd?

}= EQ[f ]



DefinitionsCopulas and Quasi-Copulas

Definition

A function Q : Id = [0, 1]d → I is a d-quasi-copula ifQ(u1, ..., ui = 0, ..., ud ) = 0 and Q(1, ...1, ui , 1, ..., 1) = ui , ∀i ∈ {1, ..., d}.Q is non-decreasing in each argument.Q is Lipschitz continuous, i.e.|Q(u1, ..., ud )− Q(v1, ..., vd )| ≤

∑di=1 |ui − vi | for all u, v ∈ Id .

Moreover Q is a d-copula ifQ is d-increasing.

Cd : the set of all d-copulasQd : the set of all d-quasi-copulas (Cd ⊂ Qd )



Definitions2-increasing vs. increasing in 2 variables

Definition

A function f : Rd → R is called d-increasing if for all rectangular subsetsH = (a1, b1]× · · · × (ad , bd ] ⊂ Rd it holds that

Vf (H) := ∆dad ,bd ◦ · · · ◦∆1

a1,b1 f ≥ 0. (2.1)

We call Vf (H) the f -volume of H.



A partial order

Definition (Orthant Order)

Let Q,Q′ be quasi-copulas. The lower orthant order is defined by

Q � Q′ :⇐⇒ Q(u) ≤ Q′(u) for all u ∈ Id .

� defines a partial order on Qd .

RemarkIn case d = 2, this is the concordance order.



Frechet–Hoeffding boundsNo-information case

Theorem (Frechet, Hoeffding)

Let Q be a d-quasi-copula. Then for all u ∈ Id it holds

Wd (u) := max{

0,d∑

i=1

ui − d + 1}≤ Q(u) ≤ min{u1, ..., ud} =: Md (u).

Moreover, Md ∈ Cd for all d ≥ 2, Wd ∈ Cd only if d = 2. However, Wd ispointwise sharp:

Wd (u) = infC∈Cd{C(u)}.



Improved Frechet–Hoeffding boundsPrescription on a subset

Theorem

Let S ⊆ Id be a compact set and Q∗ be a d-quasi-copula. Consider the set

QS,Q∗ :={

Q ∈ Qd : Q(x) = Q∗(x) for all x ∈ S}.

Then it holds that

QS,Q∗ (u) ≤ Q(u) ≤ QS,Q∗ (u) for all u ∈ Id

and QS,Q∗ (u) = Q(u) = QS,Q∗ (u) for all u ∈ S(2.2)

for all Q ∈ QS,Q∗ , where the bounds QS,Q∗ and QS,Q∗ are provided by

QS,Q∗ (u) = max(

0,d∑

i=1

ui − d + 1,maxx∈S

{Q∗(x)−

d∑i=1

(xi − ui )+})

QS,Q∗ (u) = min(

u1, ..., ud ,minx∈S

{Q∗(x) +

d∑i=1

(ui − xi )+})

.

(2.3)

Furthermore, the bounds QS,Q∗ ,QS,Q∗ are d-quasi-copulas.



Improved Frechet–Hoeffding boundsMeasures of association (or option prices!)

Theorem

Let ρ : Qd → R be non-decreasing with respect to the lower orthant order onQd and continuous w.r.t. the pointwise convergence of quasi-copulas. Define

Qθ := {Q ∈ Qd : ρ(Q) = θ}

then

Qθ(u) := min{

Q(u) : Q ∈ Qθ}

=

{ρ−1

+ (u, θ), θ ∈[ρ(

Q{u},Wd (u)), ρ(Md )

]Wd (u), else

and

Qθ(u) := max{

Q(u) : Q ∈ Qθ}

={ρ−1− (u, θ), θ ∈

[ρ(Wd ), ρ

(Q{u},Md (u))]

Md (u), else

where

ρ−1− (u, θ) = max

{r : ρ(

Q{u},r)

= θ}

ρ−1+ (u, θ) = min

{r : ρ(

Q{u},r)

= θ}.

[38]



Improved Frechet–Hoeffding boundsPresciption on lower-dimensional copulas / Aggregation

TheoremLet I1, ..., Ik be subsets of {1, ..., d} with |Ij | ≥ 2 for j = 1, ..., k and |Ii ∩ Ij | ≤ 1for i , j ∈ {1, ..., k}, i 6= j . Consider the set

QI ={

Q ∈ Qd : Qj � QIj � Qj , j = 1, ..., k}

where QIj is the Ij -margin of Q and Qj ,Qj are lj -quasi-copulas for j = 1, ..., k.Then

QI (u) := min{

Q(u) : Q ∈ QI}= max

(max

j=1,...,k

{Qj (uIj ) +

∑m∈{1,...,d}\Ij

(um − 1)},Wd (u)

)QI (u) := max

{Q(u) : Q ∈ QI} = min

(min

j=1,...,k

{Qj (uIj )

},Md (u)

).

Moreover QI ,QI ∈ QI , hence the bounds are sharp.



Improved Frechet–Hoeffding boundsDistance to a reference copula

Definition

We call D : Qd ×Qd → R+ a statistical distance if for Q,Q′ ∈ Qd

D(Q,Q′) = 0 ⇐⇒ Q(x) = Q′(x) for all x ∈ Id .

D is monotonic with respect to the lower orthant order, if for Q,Q′,Q′′ ∈ Qd

Q � Q′ � Q′′ =⇒ D(Q′,Q′′) ≤ min{D(Q,Q′′),D(Q′′,Q)}.

and D is min-stable if

D(Q,Q′) ≥ max{D(Q ∧ Q′,Q),D(Q,Q ∧ Q′)},

where Q ∧ Q′ denotes the minimal convolution x 7→ Q(x) ∧ Q′(x).

ExampleStatistical distances monotononic wrt the lower orthant order and min-stable:Kolmogorov-Smirnov ‖A−B‖∞, Cramer-von-Mises ‖A−B‖2

2 and LP -distances.



Improved Frechet–Hoeffding boundsDistance to a reference copula

Theorem

Let C∗ be a d-copula and D be a statistical distance on Qd which iscontinuous wrt the pointwise convergence of quasi-copulas, monotonic wrt thelower orthant order and min-stable. Consider the set

QD,δ := {Q ∈ Qd : D(Q,C∗) ≤ δ}

for δ ∈ R+, then

QD,δ(u) := min{

Q(u) : Q ∈ QD,δ}

= min{α ∈ [Wd (u),Md (u)] : D

(Q{u},α ∧ C∗,C∗

)≤ δ}

and the bound is a quasi-copula.

Example

Take the Kolmogorov–Smirnov distance DKS(Q,Q′) := supu∈Id |Q(u)− Q′(u)|, then

QDKS ,δ(u) = max{

C∗(u)− δ,Wd (u)}

and QDKS ,δ(u) = min{

C∗(u) + δ,Md (u)}.



Are the improved bounds copulas?Dichotomy for d > 2

Tankov showed that QS,Q∗ ,QS,Q∗ ∈ C2 if the set S is increasing, resp.decreasing. Bernard et al. weakened this result.

Example

Let S ={

(u, u, u) : u ∈[0, 1

2

]∪[ 3

5 , 1]}⊂ I3 and Π be the independence

copula. Then

VQS,Π

([ 56100 ,

35

]3)

= −0.029 < 0,

hence QS,Π is not a copula.



The improved bounds are not copulasStatement

Theorem

Consider the compact subset S of I3

S =(

[0, 1] \ (s1, s1 + ε1))×(

[0, 1] \ (s2, s2 + ε2))×(

[0, 1] \ (s3, s3 + ε3)),

for εi > 0, i = 1, 2, 3 and let C∗ be a 3-copula (or a 3-quasi-copula) such that

3∑i=1

εi > C∗(s + εεε)− C∗(s) > 0,

C∗(s) ≥W3(s + εεε),

where s = (s1, s2, s3), εεε = (ε1, ε2, ε3). Then QS,C∗ is a proper quasi-copula.



Roadmap

Define Cd /Qd , partial order �

Find upper/lower boundsQ/Q on Qd

? w.r.t. �

Check whether Q/Q lie in Cd

Define EQ[f ] for Q ∈ Qd Q � Q′ =⇒ EQ[f ] ≤ EQ′ [f ]

inf{EQ[f ]

∣∣Q ∈ Qd?

}= EQ[f ]



Expectations wrt Quasi-CopulasDefinitions

Aim: define expectation wrt quasi-copulas

Idea: use a partial integration formula (cf. Ruschendorf, Tankov)

Definition (Finite difference operator, Volume)

Let f : Rd → R, i ∈ {1, ..., d}, yi > xi

∆yi f (x1, ..., xd ) = f (x1, ..., xi−1, yi , xi+1, ..., xd )− f (x1, ..., xd )

and for a subset H := [a1, b1]× · · · × [ad , bd ] ⊂ Rd

Vf (H) = ∆bi ◦ · · · ◦∆bd f (a1, ..., ad )

DefinitionA function f is measure inducing if Vf (H) induces a (signed) measure on theBorel-σ-algebra over Rd



Expectations wrt Quasi-CopulasIntegration by parts

Definition (Special case)

Let f : Rd → R be measure inducing and Q : Id → I be a quasi-copula. Thenwe define

πf (Q) :=∫Rd

Q(F (x1), ...,F (xd )) dµf (x1, ..., xd ),

where Q is the survival function of Q and µf the measure induced by f .

Lemma (General, Alternative representation, P.-Yanez)

Let f : Rd → R be measure inducing and Q : Id → I be a quasi-copula. Then

πf (Q) = f (0, . . . , 0) +d∑

n=1

∑J⊆{1,...,d}J={i1,...,in}

∫Rn

QJ (F (xi1 ), . . . ,F (xin )) dµfJ (xi1 , . . . , xin ).

[38]



Expectations wrt Quasi-CopulasOrder preserving maps

Definition

f : Rd → R is called ∆-antitonic if for all subsets {i1, ..., ik} ⊂ {1, ..., d}

(−1)n−1∆i1a1,b1◦ · · · ◦∆in

an,bn f ≥ 0

Each antitonic function induces a signed measure.

Theorem (Muller & Stoyan)

Let f be ∆-antitonic and C ,C ′ copulas then

C � C ′ ⇐⇒ EC [f ] ≤ EC′ [f ].

TheoremLet f be ∆-antitonic and Q,Q′ quasi-copulas then

Q � Q′ =⇒ πf (Q) ≤ πf (Q′).

Moreover, if F1, . . . ,Fd are continuous then the converse is also true.



Model-free bounds on option pricesResults

LetEQ,C [f (S1, . . . , Sd )] =: πf (C).

and consider the sets of arbitrage-free prices (with and without additionalinformation)

Π := {πf (C) : C ∈ Cd} and Π∗ := {πf (C) : C ∈ C∗}

Proposition

Let f be ∆-antitonic and Q∗,Q∗ be the lower and upper bound on theconstrained set of copulas C∗. Then

πf (Wd ) ≤ πf (Q∗) ≤ inf Π∗ ≤ πf (C) ≤ sup Π∗ ≤ πf (Q∗) ≤ πf (Md ) = sup Π

for all C ∈ C∗, while inf Π = πf (Wd ) if d = 2.

Sharpness, i.e. sup Π∗ = πf (Q∗), under strong conditions on f .



Model-free bounds on option pricesExamples of payoff functions

Payoff f (x1, . . . , xd ) ∆-tonicity∫

g(xi1 , . . . , xin ) dµfI

Digital call on minimum1min{x1,...,xd}≥K f monotonic

{g(K , . . . ,K), I = {1, . . . , d}0, else

Call on minimum(min{x1, . . . , xd} − K)+ f monotonic

{∫∞K

g(x , . . . , x)dx , I = {1, . . . , d}0, else

Call on maximum(max{x1, . . . , xd} − K)+ −f antitonic

{−∫∞

Kg(x , . . . , x)dx , |I| even∫∞

Kg(x , . . . , x)dx , |I| odd

Table: Examples of payoff functions for multi-asset options and the respectiverepresentation of the integral with respect to the measure µf .



Numerical illustrationModel

Let (S1, S2,S3) be asset prices that follow the Black–Scholes model, withS i

0 = 10, r = 0 and σi = 1.Observe market prices of single asset options known marginalsObserve market prices of bivariate options H(S i , S j ) = 1max{S i ,S j}<K forK = 2, 4, . . . , 16 and i , j = 1, 2, 3Observed market prices are expectations under a risk-neutral measure Q:

EQ[H(S j , S j )] = Q(S i < K , S j < K) =⇒ Prescription on compact set

Reference model: multivariate log-normal (Gaussian copula) with

ρi,j = Corr(S i ,S j )

I Arbitrage bounds for f (S1, S2, S3) = 1max{S1,S2,S3}<K



Numerical illustrationExample: ρi,j = −0.3 (left) and ρ1,2 = −0.5, ρ1,3 = 0.5, ρ2,3 = 0 (right)

Figure: Arbitrage bounds as functions of K



Outline


bounds



Applications

Duality,Transport

Arbitragedetection



Questions — open problems

1 The ‘nice’ functions are ∆-tonic — basket options are excluded . . .

2 The improved Frechet–Hoeffding bounds are not sharp for d > 2, although. . .

Tankov showed that they are copulas for d = 2,Bernard et al. weakened this result (d = 2).

Are they pointwise sharp, e.g. Q(u) = supQ∈Q?Q(u)?

3 The marginals are known. Is that realistic?



Transport and relaxed transport dualitySetting

Aim: upper bound – superhedging strategy for f (X) E[f (X)]

Classical ingredients:f1, . . . , fd : R→ R bounded, measurable functions (‘call options’)ν1, . . . , νd marginal distributions, µ joint distribution

Then

supµ∈...

∫f dµ = inf

{∫f1dν1 + · · ·+

∫fd dνd : f1 + · · ·+ fd ≥ f

}

New ingredients:πi price of multi-asset digital 1Ai , Ai =×d

j=1(−∞,Aij ], i ∈ I

ai amount invested in 1Ai



Transport and relaxed transport dualityDuality under additional information

Define

Θ(f ) :={

(f1, . . . , fd , a) : f1(x1)+· · ·+fd (xd )+∑i∈I

ai 1Ai (x) ≥ f (x), for all x ∈ Rd},

π(f1, . . . , fd , a) :=∫R

f1 dν1 + · · ·+∫R

fd dνd +∑i∈I

(ai+πi − ai−πi).

Theorem

Let f : Rd → R be an upper semicontinuous and bounded function, then

maxµ∈Q

∫Rd

f dµ = inf{π(f1, . . . , fd , a) : (f1, . . . , fd , a) ∈ Θ(f )},

where

Q :={µ ∈ ca+

1 (Rd ) : µ1 = ν1, . . . , µd = νd and πi ≤ µ(Ai ) ≤ πi , for all i ∈ I}.



Transport and relaxed transport dualityDuality under additional information, relaxed version

Shortselling costraints:

Θ+(f ) :={

(f1, . . . , fd , a) ∈ Θ(f ) : f1, . . . , fd ≥ 0 and ai ≥ 0, for all i ∈ I},

π(f1, . . . , fd , a) :=∫R

f1 dν1 + · · ·+∫R

fd dνd +∑i∈I

ai+πi .

Theorem

Let f : Rd → R be an upper semicontinuous and bounded function, then

maxµ∈Q+

∫Rd

f dµ = inf{π(f1, . . . , fd , a) : (f1, . . . , fd , a) ∈ Θ+(f )}, (4.1)

where

Q+ :={µ ∈ ca+

≤1(Rd ) : µ1 ≤ ν1, . . . , µd ≤ νd and µ(Ai ) ≤ πi , for all i ∈ I}.

Uncertainty in the dependence structure and the marginals!



An explicit solution for f = 1Bunder shortselling constraints

Let d = 2, fix a box B = (−∞,B1]× (−∞,B2], and assume that I is finite.

TheoremThe following holds:

maxµ∈Q+

∫1B dµ = max

µ∈Q+µ(B)

= min{ν1((−∞,B1]), ν2((−∞,B2]),min

i∈I

(πi + ν1((Ai

1,B1])) + ν2((Ai2,B2])

)}.

B B B

A



Sharpness of improved Frechet–Hoeffding boundsA positive and a negative result

A positive result (shortselling constraints / uncertain marginals)

Let S ⊂ [0, 1]d , C∗ be a reference copula, and consider the set

CS,C∗um =

{C(F

′1 (·), . . . ,F

′d (·)) : F

′i (x) ≤ Fi (x) ∀x , i ,C(u) ≤ C∗(u) ∀u ∈ S

}.

The upper improved Frechet–Hoeffding bound on CS,C∗um is provided by

QS,C∗um (F1(y1), . . . ,Fd (yd ))

= min{

F1(y1), . . . ,Fd (yd ),minx∈S

{C∗(x) +

d∑i=1

(Fi (yi )− Fi (xi ))+}}

.

Then, the previous Theorem yields that the QS,C∗um is pointwise sharp, i.e.

QS,C∗um (u) = sup

C∈CS,C∗um

C(u).

A counterexample by S. EcksteinThe improved Frechet–Hoeffding is not pointwise sharp for the original problem, evenin d = 2!



Outline


bounds



Applications

Duality,Transport

Arbitragedetection



Detection of arbitrageSetting, results of Tavin

Assets S1, . . . , Sd with known distributions F1, . . . ,Fd .Derivatives Z 1, . . . ,Z q with payoff Z i = z i (S1, . . . , Sd )Price vector π = (π1, . . . , πq) with πi = EQ[Z i ] (Q EMM)

Theorem (Tavin, JBF 2015)The set Q of martingale measures with fixed marginals F1, . . . ,Fd is isomorphicto the set Cd of d-copulas.

No-arbitrage price vector:

Π(Z 1, . . . ,Z q) ={

(EQ[Z 1], . . . ,EQ[Z q])|Q ∈ Q}

={

(EC [Z 1], . . . ,EC [Z q])|C ∈ Cd}⊆ Π(Z 1)× · · · × Π(Z q).

Question

When does π ∈ Π(Z 1, . . . ,Z q)? Does πi ∈ Π(Z i ) for all i suffice?



Detection of arbitrageRelation to improved FH bounds

Pricing rules:ρ : Cd 3 C 7→ ρ(C) ∈ Π(Z 1, . . . ,Z q) and ρk : Cd 3 C 7→ ρk (C) ∈ Π(Z k )

Define: subsets of Cd compatible with elements of Π

Cd,k ={

C ∈ Cd |ρk (C) = πk ∈ Π(Z k )}

Theorem (Tavin, JBF 2015)The following holds:

π ∈ Π(Z 1, . . . ,Z q)⇐⇒ ∃C ∈ Cd : ρ(C) = π

⇐⇒q⋂

k=1

Cd,k 6= ∅

QuestionHow can we make this operational?

improved FH bounds




Pricing rules:ρ : Cd 3 C 7→ ρ(C) ∈ Π(Z 1, . . . ,Z q) and ρk : Cd 3 C 7→ ρk (C) ∈ Π(Z k )

Define: subsets of Cd compatible with elements of Π

Cd,k ={

C ∈ Cd |ρk (C) = πk ∈ Π(Z k )}

Theorem (Tavin, JBF 2015)The following holds:

π ∈ Π(Z 1, . . . ,Z q)⇐⇒ ∃C ∈ Cd : ρ(C) = π

⇐⇒q⋂

k=1

Cd,k 6= ∅

QuestionHow can we make this operational? improved FH bounds




Introduce bounds on Cd,k ={

C ∈ Cd |ρk (C) = πk ∈ Π(Z k )}

C d,k (u) = min{C(u)|C ∈ Cd,k} and C d,k (u) = max{C(u)|C ∈ Cd,k}then

explicit expressions for C d,k (u),C d,k (u) already available [14]explicit expressions for ρk (C) already available [22]

Proposition (Necessary condition for no-arbitrage)q⋂

k=1

Cd,k 6= ∅ =⇒ mink

C d,k (u) ≥ maxk

C d,k (u) ∀u ∈ Id .

Optimization formulation

minu∈Id

{min

kC d,k (u) ≥ max

kC d,k (u)

}={≥ 0, no decision< 0, π /∈ Π.



Conclusion

Open problems – Future work

Model-free bounds for basket options, with market information (underuncertainty in the marginals)Comparison with option price bounds stemming from neural networks

References

D. Bartl, M. Kupper, T. Lux, A. Papapantoleon, S. Eckstein:Marginal and dependence uncertainty: bounds, optimal transport, and sharpnessarXiv:1709.00641

T. Lux, A. Papapantoleon:Improved Frechet-Hoeffding bounds on d-copulas and applications in model-freefinanceAnnals of Applied Probability 27, 3633–3671, 2017

T. Lux, A. Papapantoleon:Model-free bounds on Value-at-Risk using extreme value information andstatistical distancesInsurance: Mathematics and Economics 86, 73–83, 2019


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