Random sets in economics, financeand insurance

Ilya Molchanov

University of Bern, Switzerland

based on joint works withI.Cascos (Madrid, Statistics), E.Lepinette (Paris, Finance),

F.Molinari (Cornell, Economics), M.Schmutz (Bern, Probability and Finance),A.Haier (Swiss Financial Market Supervision)

University Austin TX, May 2015

1

Early years

I Why do people make particular choices?I How to allocate assets optimally between agents?

2

50 years and 3 Nobel Prizes

I Gerard Debreau (1983)(Debreau expectation of random sets)

I

3

50 years and 3 Nobel Prizes

I Robert Aumann and Thomas Schelling (2005)(Aumann expectation of random sets)

I

4

50 years and 3 Nobel Prizes

I Alvin Roth and Llloyd Shapley (2012)(allocations: choice of an element of a random set)

I

5

Overview

Developments over the last 10 years

Lecture 1

I Random sets and selections in economics.I Market imperfections and transaction costs.

Lecture 2

I Sublinearity and risk.I Financial networks.

6

I A. Beresteanu, I. Molchanov, and F. Molinari. Sharpidentification regions in models with convex momentpredictions. Econometrica, 79:1785–1821, 2011.

I I. Molchanov and I. Cascos. Multivariate riskmeasures: a constructive approach based onselections. Math. Finance, 2015.

I I. Molchanov and M. Schmutz. Multivariateextensions of put-call symmetry. SIAM J. FinancialMath., 1:396–426, 2010.

I Works in progress ...

7

Basics of random sets

DefinitionA map X : Ω 7→ F from a probability space (Ω,F,P) to thefamily of closed subsets of a Polish space X is said to bea random closed set if

ω : X(ω) ∩G 6= ∅ ∈ F

for all open sets G.

I In X = Rd one can take compact sets K instead ofopen G.

8

Examples

So far most of examples involve “simple” setsscattered in space.

I Point process.I Random geometric graphs.I Collection of lines/balls, etc.I Random polytopes.

9

Examples: Economics

I Consider a game of d players with randomparameters. Then the set of equilibria (pure or mixed)is a subset of the unit cube in Rd .

I Measurements are often represented as intervalsrather than points. The reason may be not only thelack of precision or censoring, but also intentionalreporting of intervals.

I If the underlying probability measure P is uncertain,this uncertainty can be described as a family ofrandom elements.

10

Examples: FinanceI Each asset has two prices Sb ≤ Sa: bid and ask

prices.I Adding cash as one axis, this leads to a random

cone.

1 cash

S−1a

S−1b

X

asset

I Multiasset setting leads to more complicated randomcones. Example: currency exchanges withtransaction costs (Kabanov’s model).

11

Examples: Finance (link-save)

I Buying carrots and potatoes together brings areduction.

Carrots

Potatoes

12

Example: Financial networks

Here is a pictureof a large network

connecting major financialinstitutions in the world

13

Example: Financial network

Reinsurance

Holding

Subsidiary I Subsidiary II

Life Non−life

InvestmentJunk

I Objective: IntraGroup Transfers (in order to helpdistressed members of the group)

14

Example: Financial network, two agents

I Two agents A and B are assessed by a regulator.I The regulator evaluates their individual exposure to

risks and requests that they set aside (and freeze)necessary capital reserves.

I The agents want to minimise these reserves andconclude an agreement that at the terminal time (andunder certain conditions) one of them would help tooffset the deficit of another one.

I The family of allowed transactions is a random closedset in the plane.

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Examples: financial network, two agents

I The terminal capital is (C1,C2).I Transfers from a solvent company to another one are

allowed up to the available positive capital.I Disposal of assets is allowed.

X X

(C1,C2)(C1,C2)

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Selections

I A random vector ξ is called a selection of X if both ξand X can be realised on the same probability space,so that ξ ∈ X a.s.

I From now on we tacitly assume that all randomelements are realised on the same probability space.

Theorem (Zvi Artstein, 1983)A probability measure µ is the distribution of a selection ofa random closed set X in Rd if and only if

µ(K ) ≤ PX ∩ K 6= ∅

for all compact sets K ⊂ Rd .

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Games and payoffs

I Set K is a coalition of players.I ϕ(K ) is the payoff that K receives.I Payoff functional is not additive, but subadditive.I Allocation is a measure µ such that

µ(K ) ≤ ϕ(K ) ∀ K .

I Bondareva–Shapley theorem: existence of anallocation under some conditions on ϕ (convexity).

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Example: market entry game(non-coalitional)

I Payoff for the j th player, j = 1,2,

πj = aj(a−jθj + εj) ,

whereI aj ∈ 0,1 is the action (enter or not) of the j th player;I a−j is the action of other player(s),I θj are unknown parametersI εj are random profit shifters with known distribution

19

Example: market entry game — equilibria

πj = aj(a−jθj + εj) , j = 1,2.

I Set Sθ(ε) of (Nash) equilibria is random and dependson ε.

−θ2

01 11

00 10

10, (− ε2θ2,− ε1

θ1),01

ε1

ε2

−θ1

Note that θ1, θ2 < 0.

20

Example: market entry game - inferenceI Assume pure strategies only

(the method works also for mixed strategies).I The econometrician observes empirical variant of the

distributionµ = (p00,p10,p01,p11) .

I These frequencies are sampled from the set ofpossible equilibria, i.e.

µ is the distribution of a selection of Sθ(ε) .

InferenceEstimate θ = (θ1, θ2) based on this, i.e. estimateparameters of a random set by observing its selection:

θ : µ(K ) ≤ PSθ ∩ K 6= ∅ ∀K ⊂ 00,01,10,11.

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The most serious difficulty

I The family of selections is too rich.I It is numerically difficult to verify inequalitiesµ(K ) ≤ PX ∩ K 6= ∅ for all K .

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Course exercise

I ξ has normal distribution N(µ, σ2).I Observe numbers x1, . . . , xn chosen (using an

unknown mechanism) such that xi ≥ ξi for i.i.d.realisations ξ1, . . . , ξn of ξ.

I Estimate µ and σ2 and utilise the availableinformation in full!

23

Castaing representation

Theorem (Charles Castaing, 1977)X is a random closed set if and only if

X = cl(ξn,n ≥ 1)

meaning that X is the closure of a countable family of itsselections.

DefinitionLp(X) denotes the family of p-integrable selections of X.

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Expectations of selections

I Assume that X has at least one integrable selection,i.e. L1(X) 6= ∅.

I The (Aumann) expectation of X

EX = clEξ : ξ ∈ L1(X).

I The expectation is always a convex set if theprobability space is non-atomic (follows fromLyapunov’s theorem on range of a vector-valuedmeasure).

I Higher moments are not well defined!

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Example: interval least squares I

I Explanatory variable xI Response y ∈ Y = [yL, yU ].

yiU

x

y

xi

yiL

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Example: interval least squares II

I Regression model (for mean values). If y ∈ Y a.s.,then

E(y) = θ1 + θ2E(x) .

I Then

(θ1, θ2) = Σ(x)−1[

E(y)E(xy)

], Σ(x) =

[1 Ex

Ex Ex2

].

I The expectations E(y) and E(xy) may take variousvalues depending on the choice of selectiony ∈ Y = [yL, yU ].

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Example: interval least squares IIII y is a selection of Y and xy is a selection of xYI Thus, (y , xy) is a selection of

G =

(yxy

): yL ≤ y ≤ yU

⊂ R2

segment with end-points (yL, xyL) and (yU , xyU).

y yU

xyL

xyUGxy

yL

I Identification region θ ∈ Σ(x)−1EG .

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Course exercises

1. How to amend the setting for the polynomialregression?

2. How to handle the case of interval-valuedexplanatory variable x?

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Options (main idea)I Call price

EQ(Fη − k)+

and put priceEQ(k − Fη)+

can be expressed in terms of the expectation of therandom set X.

1 1

Eη = 1(1, η)

(0,0)

X

EX

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Option prices I

I Let η be a non-negative random variable (relativeprice change).

1

X

(1, η)

(0,0)

I The support function of X in direction u is

hX(u) = sup〈u, x〉 : x ∈ X = (u1 + u2η)+

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Option prices II

hX(u) = (u1 + u2η)+

I If u = (−k ,F ), then hX(u) = (Fη − k)+ is the payofffrom the call option.

I In this case:I F is the forward price (deterministic carrying costs);I S = Fη is the terminal price;I k is the strike price (buying price at the terminal time).

I If u = (k ,−F ), then hX(u) = (k − Fη)+ is the payofffrom the put option.

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Option prices III

I The expected support function EhX(u) is the supportfunction of the expectation hEX(u).

I Then hEX(−k ,F ) = EQ(Fη − k)+ is the call price if theexpectation is taken with respect to the martingalemeasure.

1 1

Eη = 1(1, η)

(0,0)

X

EX

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Symmetries

I Point symmetry with respect to (12 ,

12) is equivalent to

European put-call parity

I Line symmetry is equivalent to put-call symmetry.

34

Multi-asset symmetryAsset prices ST1 = F1η1, . . . ,STd = Fdηd

Prices of basket options

EQ(u0 + u1η1 + · · ·+ udηd )+

(forward prices are included in the weights).

I When is the price invariant for all permutations of theweights (self-duality)?

I If this is the case, then η is exchangeable.I However, the exchangeability property does not

suffice, e.g. η with i.i.d. coordinates.I When is the price invariant for u0 = 0 and

permutations of other weights (swap-invariance)?

35

Convex models of transaction costs

I Let (Ω,Ft , t = 0, . . . ,T ,P) be a stochastic basis.I Let Kt , t = 0, . . . ,T , be a sequence of random convex

sets so that Kt is Ft -measurable.I Sets Kt contain the origin and are lower sets.

Assume that they do not contain any line.I Set Kt describes the positions available at price zero

at time t .

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Kabanov’s exchange cone modelI Kt is a random exchange cone (e.g. generated by

bid-ask exchange rates for currencies at time t).I Kt is the family of portfolios available at price zero.I Reflected set −Kt is the family of solvent positions at

time t .

USD

EUR

1

1

Kt

37

Self-financing

I A self-financing portfolio process satisfies

Vt − Vt−1 ∈ L0(Kt ,Ft)

so the increment is available at price zero.I The set

At =t∑

i=0

L0(Ki ,Fi) ⊂ L0(Rd )

is the family of attainable claims at time t .

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No arbitrage

Definition(NAS) (strict no-arbitrage) condition holds if

At ∩ L0(−Kt ,Ft) = 0

for all t = 0, . . . ,T .

InterpretationStarting from zero it is not possible to achieve non-zerosolvent position at any time t .

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No arbitrage and martingales (cone models)I Define the dual cone

K∗t = u : 〈u, x〉 ≤ 0 ∀ x ∈ Kt

I It describes the family of consistent price systems, ifu are prices, then each portfolio available at pricezero indeed has negative price.

Theorem (Kabanov et al.)(NAS) condition is equivalent to the existence of anequivalent probability measure Q and a Q-martingale Mt

such that

Mt ∈ relative interior K∗t , t = 0, . . . ,T .

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Conditional expectation and martingales

I A sequence of random sets Xt , t = 0, . . . ,T , is amartingale if

E(Xt |Fs) = Xs a.s. ∀ s ≤ t .

I The conditional expectation is defined as the family ofconditional expectations of all selections.

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Cores and conditional cores

DefinitionIf X is F-measurable random closed set and A is asub-σ-algebra, then the conditional core

Y = m(X|A)

is the largest A-measurable random set Y such that Y ⊂ Xa.s.

I The conditional core exists.I If X = (−∞, ξ], then the conditinal core is the set

Y = (−∞, η], where η is the largest A-measurablerandom variable such that η ≤ ξ.

42

Single asset case

I Recall the sequence of exchange sets Kt and priceintervals Xt = [Sbt ,Sat ], t = 0, . . . ,T .

I (NAS) (no arbitrage) condition.

1 cash

assetprice

Kt

S−1at

S−1bt

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No arbitrage and conditional cores

TheoremIn case of a single asset with bid-ask spreadXt = [Sbt ,Sat ], the (NAS) condition holds if and only if

Xt ⊆ m(Xt+1|Ft), t = 0, . . . ,T − 1.

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