Random sets in economics, finance
and insurance
Ilya Molchanov
University of Bern, Switzerland
based on joint works with
I.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
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Early years
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Why do people make particular choices?
How to allocate assets optimally between agents?
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50 years and 3 Nobel Prizes
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Gerard Debreau (1983)
(Debreau expectation of random sets)
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50 years and 3 Nobel Prizes
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Robert Aumann and Thomas Schelling (2005)
(Aumann expectation of random sets)
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50 years and 3 Nobel Prizes
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Alvin Roth and Llloyd Shapley (2012)
(allocations: choice of an element of a random set)
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Overview
Developments over the last 10 years
Lecture 1
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Random sets and selections in economics.
Market imperfections and transaction costs.
Lecture 2
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Sublinearity and risk.
Financial networks.
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A. Beresteanu, I. Molchanov, and F. Molinari. Sharp
identification regions in models with convex moment
predictions. Econometrica, 79:1785–1821, 2011.
I. Molchanov and I. Cascos. Multivariate risk
measures: a constructive approach based on
selections. Math. Finance, 2015.
I. Molchanov and M. Schmutz. Multivariate
extensions of put-call symmetry. SIAM J. Financial
Math., 1:396–426, 2010.
Works in progress ...
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Basics of random sets
Definition
A map X : Ω 7→ F from a probability space (Ω, F, P) to the
family of closed subsets of a Polish space X is said to be
a random closed set if
{ω : X(ω) ∩ G 6= ∅} ∈ F
for all open sets G.
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In X = Rd one can take compact sets K instead of
open G.
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Examples
So far most of examples involve “simple” sets
scattered in space.
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Point process.
Random geometric graphs.
Collection of lines/balls, etc.
Random polytopes.
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Examples: Economics
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Consider a game of d players with random
parameters. Then the set of equilibria (pure or mixed)
is a subset of the unit cube in Rd .
Measurements are often represented as intervals
rather than points. The reason may be not only the
lack of precision or censoring, but also intentional
reporting of intervals.
If the underlying probability measure P is uncertain,
this uncertainty can be described as a family of
random elements.
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Examples: Finance
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Each asset has two prices Sb ≤ Sa : bid and ask
prices.
Adding cash as one axis, this leads to a random
cone.
asset
Sb−1
X
Sa−1
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cash
Multiasset setting leads to more complicated random
cones. Example: currency exchanges with
transaction costs (Kabanov’s model).
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Examples: Finance (link-save)
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Buying carrots and potatoes together brings a
reduction.
Potatoes
Carrots
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Example: Financial networks
Here is a picture
of a large network
connecting major financial
institutions in the world
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Example: Financial network
Holding
Subsidiary I
Subsidiary II
Reinsurance
Life Non−life
Junk
Investment
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Objective: IntraGroup Transfers (in order to help
distressed members of the group)
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Example: Financial network, two agents
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Two agents A and B are assessed by a regulator.
The regulator evaluates their individual exposure to
risks and requests that they set aside (and freeze)
necessary capital reserves.
The agents want to minimise these reserves and
conclude an agreement that at the terminal time (and
under certain conditions) one of them would help to
offset the deficit of another one.
The family of allowed transactions is a random closed
set in the plane.
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Examples: financial network, two agents
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The terminal capital is (C1 , C2 ).
Transfers from a solvent company to another one are
allowed up to the available positive capital.
Disposal of assets is allowed.
(C1 , C2 )
X
(C1 , C2 )
X
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Selections
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A random vector ξ is called a selection of X if both ξ
and X can be realised on the same probability space,
so that ξ ∈ X a.s.
From now on we tacitly assume that all random
elements are realised on the same probability space.
Theorem (Zvi Artstein, 1983)
A probability measure µ is the distribution of a selection of
a random closed set X in Rd if and only if
µ(K ) ≤ P{X ∩ K 6= ∅}
for all compact sets K ⊂ Rd .
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Games and payoffs
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Set K is a coalition of players.
ϕ(K ) is the payoff that K receives.
Payoff functional is not additive, but subadditive.
Allocation is a measure µ such that
µ(K ) ≤ ϕ(K )
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∀ K.
Bondareva–Shapley theorem: existence of an
allocation under some conditions on ϕ (convexity).
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Example: market entry game
(non-coalitional)
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Payoff for the jth player, j = 1, 2,
πj = aj (a−j θj + εj ) ,
where
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aj ∈ {0, 1} is the action (enter or not) of the jth player;
a−j is the action of other player(s),
θj are unknown parameters
εj are random profit shifters with known distribution
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Example: market entry game — equilibria
πj = aj (a−j θj + εj ) ,
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j = 1, 2.
Set Sθ (ε) of (Nash) equilibria is random and depends
on ε.
ε2
{11}
{01}
−θ2
{10, (− θε22 , − θε11 ), 01}
{00}
−θ1
ε1
{10}
Note that θ1 , θ2 < 0.
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Example: market entry game - inference
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Assume pure strategies only
(the method works also for mixed strategies).
The econometrician observes empirical variant of the
distribution
µ = (p00 , p10 , p01 , p11 ) .
These frequencies are sampled from the set of
possible equilibria, i.e.
µ is the distribution of a selection of Sθ (ε) .
Inference
Estimate θ = (θ1 , θ2 ) based on this, i.e. estimate
parameters of a random set by observing its selection:
n
o
θ : µ(K ) ≤ P{Sθ ∩ K 6= ∅} ∀K ⊂ {00, 01, 10, 11} .
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The most serious difficulty
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The family of selections is too rich.
It is numerically difficult to verify inequalities
µ(K ) ≤ P{X ∩ K 6= ∅} for all K .
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Course exercise
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ξ has normal distribution N(µ, σ 2 ).
Observe numbers x1 , . . . , xn chosen (using an
unknown mechanism) such that xi ≥ ξi for i.i.d.
realisations ξ1 , . . . , ξn of ξ.
Estimate µ and σ 2 and utilise the available
information in full!
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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 its
selections.
Definition
Lp (X) denotes the family of p-integrable selections of X.
Expectations of selections
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Assume that X has at least one integrable selection,
i.e. L1 (X) 6= ∅.
The (Aumann) expectation of X
EX = cl{Eξ : ξ ∈ L1 (X)}.
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The expectation is always a convex set if the
probability space is non-atomic (follows from
Lyapunov’s theorem on range of a vector-valued
measure).
Higher moments are not well defined!
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Example: interval least squares I
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Explanatory variable x
Response y ∈ Y = [yL , yU ].
y
yiU
yiL
x
xi
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Example: interval least squares II
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Regression model (for mean values). If y ∈ Y a.s.,
then
E(y) = θ1 + θ2 E(x) .
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Then
(θ1 , θ2 ) = Σ(x)
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−1
E(y)
,
E(xy )
1 Ex
Σ(x) =
.
Ex Ex 2
The expectations E(y) and E(xy) may take various
values depending on the choice of selection
y ∈ Y = [yL , yU ].
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Example: interval least squares III
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y is a selection of Y and xy is a selection of xY
Thus, (y , xy) is a selection of
y
G=
: yL ≤ y ≤ yU ⊂ R2
xy
segment with end-points (yL , xyL ) and (yU , xyU ).
xyU
xy
G
xyL
yL
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y
yU
Identification region θ ∈ Σ(x)−1 EG .
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Course exercises
1. How to amend the setting for the polynomial
regression?
2. How to handle the case of interval-valued
explanatory variable x?
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Options (main idea)
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Call price
EQ (F η − k )+
and put price
EQ (k − F η)+
can be expressed in terms of the expectation of the
random set X.
(1, η)
EX
Eη = 1
X
(0, 0)
1
1
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Option prices I
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Let η be a non-negative random variable (relative
price change).
(1, η)
X
(0, 0)
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The support function of X in direction u is
hX (u) = sup{hu, xi : x ∈ X} = (u1 + u2 η)+
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Option prices II
hX (u) = (u1 + u2 η)+
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If u = (−k, F ), then hX (u) = (F η − k)+ is the payoff
from the call option.
In this case:
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F is the forward price (deterministic carrying costs);
S = F η is the terminal price;
k is the strike price (buying price at the terminal time).
If u = (k, −F ), then hX (u) = (k − F η)+ is the payoff
from the put option.
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Option prices III
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The expected support function EhX (u) is the support
function of the expectation hEX (u).
Then hEX (−k, F ) = EQ (F η − k)+ is the call price if the
expectation is taken with respect to the martingale
measure.
(1, η)
EX
Eη = 1
X
(0, 0)
1
1
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Symmetries
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Point symmetry with respect to ( 12 , 21 ) is equivalent to
European put-call parity
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Line symmetry is equivalent to put-call symmetry.
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Multi-asset symmetry
Asset prices ST 1 = F1 η1 , . . . , STd = Fd ηd
Prices of basket options
EQ (u0 + u1 η1 + · · · + ud ηd )+
(forward prices are included in the weights).
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When is the price invariant for all permutations of the
weights (self-duality)?
If this is the case, then η is exchangeable.
However, the exchangeability property does not
suffice, e.g. η with i.i.d. coordinates.
When is the price invariant for u0 = 0 and
permutations of other weights (swap-invariance)?
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Convex models of transaction costs
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Let (Ω, Ft , t = 0, . . . , T , P) be a stochastic basis.
Let Kt , t = 0, . . . , T , be a sequence of random convex
sets so that Kt is Ft -measurable.
Sets Kt contain the origin and are lower sets.
Assume that they do not contain any line.
Set Kt describes the positions available at price zero
at time t.
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Kabanov’s exchange cone model
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Kt is a random exchange cone (e.g. generated by
bid-ask exchange rates for currencies at time t).
Kt is the family of portfolios available at price zero.
Reflected set −Kt is the family of solvent positions at
time t.
EUR
1
1
USD
Kt
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Self-financing
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A self-financing portfolio process satisfies
Vt − Vt−1 ∈ L0 (Kt , Ft )
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so the increment is available at price zero.
The set
t
X
At =
L0 (Ki , Fi ) ⊂ L0 (Rd )
i=0
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 .
Interpretation
Starting from zero it is not possible to achieve non-zero
solvent position at any time t.
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No arbitrage and martingales (cone models)
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Define the dual cone
K∗t = {u : hu, xi ≤ 0 ∀ x ∈ Kt }
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It describes the family of consistent price systems, if
u are prices, then each portfolio available at price
zero indeed has negative price.
Theorem (Kabanov et al.)
(NAS) condition is equivalent to the existence of an
equivalent 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
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A sequence of random sets Xt , t = 0, . . . , T , is a
martingale if
E(Xt |Fs ) = Xs
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a.s. ∀ s ≤ t.
The conditional expectation is defined as the family of
conditional expectations of all selections.
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Cores and conditional cores
Definition
If X is F-measurable random closed set and A is a
sub-σ-algebra, then the conditional core
Y = m(X|A)
is the largest A-measurable random set Y such that Y ⊂ X
a.s.
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The conditional core exists.
If X = (−∞, ξ], then the conditinal core is the set
Y = (−∞, η], where η is the largest A-measurable
random variable such that η ≤ ξ.
Single asset case
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Recall the sequence of exchange sets Kt and price
intervals Xt = [Sbt , Sat ], t = 0, . . . , T .
(NAS) (no arbitrage) condition.
asset
price −1
Sbt
Kt
−1
Sat
1
cash
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No arbitrage and conditional cores
Theorem
In case of a single asset with bid-ask spread
Xt = [Sbt , Sat ], the (NAS) condition holds if and only if
Xt ⊆ m(Xt+1 |Ft ),
t = 0, . . . , T − 1.
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