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Center for Mathematical Economics (IMW)

Frank Riedel Patrick Beissner

Center for Mathematical EconomicsBielefeld University

Université Paris–DauphineCeremade

Séminaire Décision, Interaction et MarchésNovember 18, 2014

Center for Mathematical Economics (Ceremade 2014)

The Non–Implementability of Arrow-DebreuEquilibria under Knightian Uncertainty



1. Informal Presentation of Results

2. Equivalence under Risk

3. Equilibria under Volatility Uncertainty


Outline



Financial Market with ambiguity–averse agents

Individuals face risk and ambiguity at the individual level

no aggregate ambiguity

existence of static Arrow–Debreu equilibrium

Characterization of Equivalence to Radner Equilibrium

generically, no Radner implementation


Storyboard



Static versus Dynamic Equilibrium

Arrow–Debreu equilibrium: agents trade efficiently all contingentplans at time 0

somewhat unrealistic market institution

perfect insurance in Arrow-Debreu equilibrium

can these efficient equilibria be implemented by dynamic trading insuitably chosen assets?


Equilibrium in Uncertain Models



Equivalence Results

The answer is yes in the classic setting at different levels if the assetmarket is dynamically complete:

real assets, endogenous asset prices, “most demanding case”Anderson–Zame 2005, Riedel–Herzberg 2014, Malamud et al. 2013

“intermediate” case: no dividends, bondDuffie–Zame 1989, Dana, Pontier 1992, Karatzas et al. 1990

asset prices nominal, can be freely chosen by the market, “easiest case”Duffie–Huang

Our claim: even in the easiest case, it will be difficult to get equivalenceof static and dynamic equilibria under Knightian uncertainty


Equilibrium in Uncertain Models







Outline



Let (Ω,F , P) be a probability space endowed with a d–dimensionalBrownian motion W = (Wt) and canonical filtration (Ft)The commodity space is X = L∞(Ω,FT , P) for consumption at terminaltime T > 0 (Lp–spaces possible if relevant objects are square–integrable)I Agents with endowments ei ∈ L∞+ and expected utility functionsUi(c) = EPui(c) for some standard Bernoulli utility function ui


Model



All trade takes place on a contingent market at time 0

An AD equilibrium consists of a price functional Ψ : X → R andan allocation (ci) such that markets clear,

∑(ci − ei) = 0 and agents

maximize utility subject to their budget constraint: if Ui(d) > Ui(ci), thenΨ(d − ei) > 0

In general,Ψ ∈ ba(Ω,F , P). With suitable assumptions,Ψ(x) = EPψx for some ψ ∈ L1(Ω,F , P). With e bounded away from zero,

even ψ bounded.


Arrow–Debreu Model



Agents trade dynamically in financial markets and buy consumptiongoods on spot markets in the moment of consumption

A nominal asset market consists of a bond with price S0t = 1(numéraire) and d risky assets with price processes Sjt > 0, given bypositive semimartingales

A feasible trading strategy is a predictable, S–integrable process θ withvalues in Rd with gains from trade

∫ T0 θudSu

Agent i finances the consumption plan ci for a spot consumption priceψ with a feasible trading strategy θi such that (ci − ei)ψ =

∫ T0 θ

iudSu


Radner’s Dynamic Equilibrium



A Radner equilibrium consists of trading strategies (θi) financingconsumption plans (ci), and a spot consumption price ψ such that marketsclear and agents maximize utility subject to their budget constraint


Radner’s Dynamic Equilibrium



Let ((ci),Ψ) be an Arrow–Debreu equilibrium.Ψ can be identified with a positive, suitably bounded random variable ψ

Can we find a Radner equilibrium with the same (efficient) allocation?

Theorem (Duffie, Huang 1985)

Let Md, d = 1, ... , D be a martingale generator (e.g., one can take S0t =1 (numéraire) and Sd = Wd, d = 1, ... , D the Brownian motion itself(Bachelier model of finance))Then there exist trading strategies θi and a spot price ψ such that ((θi, ci),ψ)form a Radner equilibrium.

Intuition: In diffusion models, finitely many assets span the market.


Duffie–Huang Theorem







Outline



Now we assume that the volatility of the Brownian motion W is uncertain.This can be described by a family of probability measures Pσ where σ isan adapted process taking values in some convex, compact subset of Rd

Construction: P0 Wiener measure on the canonical space with Brownianmotion W

Pσ = law(∫ ·

0σudWu

)Important: the measures are not dominated by one common measureQuasi–sure Analysis necessaryAn event is negligible for agents if it is null simultaneously under all Pσ


Uncertain Volatility



I agents with endowment ei bounded

Aggregate endowment e =∑

ei is ambiguity–free:for all P, Q ∈ P we have P[e ∈ ·] = Q[e ∈ ·]Utility functions of the Gilboa–Schmeidler expected utility form

Ui(c) = Eui(c) = infP∈P

EPui(c)

for smooth, strictly increasing, strictly concave Bernoulli utilityfunctions ui that satisfy an Inada condition

Ambiguity washes out in the aggregate - option for insurance


The Economy



Arrow–Debreu Model

An allocation (ci) isfeasible if we have

∑i ci = e quasi–surely

efficient if there is no other feasible allocation (di) with Ui(di) > Ui(ci)for all agents i

A price is a positive linear functionalΨ : X → RAn equilibrium consists of an allocation (ci) and a priceΨ such that

1.∑

ci =∑

ei

2. ci maximizes Ui subject to the budget constraintΨ(c) ≤ Ψ(ei)


Static Equilibrium Notion



Agents trade dynamically in a financial market with asset pricesS =

(Sdt

), , d = 0, ... , D, t ≥ 0; the spot price of consumption at time T is

ψ.

1. agents finance net demand ci − ei, i.e. there are S-integrable portfolioprocesses θi such that

ψ(ci − ei) =∫ T

0θidSd

2. asset markets clear :∑I

i=1 θi = 0

3. ci maximizes utility Ui over all c that can be financed with tradingdynamically S


Dynamic (Radner) Equilibrium



Let ((ci),Ψ) be an Arrow–Debreu equilibrium.Ψ can be identified with a positive, suitably bounded random variable ψ

Can we find a Radner equilibrium with the same (efficient) allocation?Under risk, in diffusion settings, the answer is yes!

If the filtration has a martingale generator Md, d = 1, ... , D, then we can setS0t = 1 (numéraire) and S

d = Md, d = 1, ... , D

In Brownian settings, one can thus take the Brownian motion itselfBachelier model of finance

Our claim: “usually” this result breaks down under Knightian (volatility)uncertainty.


Duffie–Huang Theorem (Repetition)



TheoremEvery efficient allocation (ci) is ambiguity–free.It satisfies the probability–free characterization of identical marginalrates of substitution among agents: for some weights αi > 0 we have

αiui′(ci) = αjuj′(cj)

As a consequence, ci = f i(e) for some monotone, continuous function f i.

Proof different from Dana, 2002: no comonotonicity, no dominating measure.


Analysis of the Market: Efficient Allocations



We denote EP the expected utility economy with homogenous priors P.

TheoremLet (ci),ψ) be an AD equilibrium in the expected utility economy EP.Then ((ci),Ψ) with

Ψ(X) = EP(Xψ)

is an AD equilibrium in the economy E.

RemarkThe market chooses P and state-price ψ.Ψ is not unique in general.Indeterminacy


Static Equilibrium



e = 1, no aggregate uncertaintyWe use two financial assets, a riskless one with price 1, and theG–Brownian motion W as the “uncertain” assetUnder risk, these assets suffice to span a complete market

TheoremImplementation of an Arrow–Debreu equilibrium ((ci),Ψ) is possible ifand only if the net trade values (ci − ei)ψ are mean–ambiguity–free.In particular, if some individuals face proper Knightian uncertainty inthe mean, implementation will not be possible.

Intuition: It is possible to hedge perfectly under each Pσ, but impossibleto do so under all Pσ simultaneously


Implementation under no Aggregate Uncertainty



Prevalence (Hunt, Sauer, Yorke, Anderson, Zame): a measure–theoreticnotion of “large sets” for infinite–dimensional spacesA ⊂ X is (finitely) prevalent if there is a finite–dimensional subspace V of X such that for all x ∈ X the complement of A has

Lebesgue measure zero in x + V .

TheoremThe set of economies for which no Arrow–Debreu equilibrium can beimplemented is (finitely) prevalent.


“Usually” Impementation Fails



Knightian Uncertainty is important

Asset markets work well when we are faced with risk and diffusions

risk = well–defined probabilities

diffusion = no jumps, trembling paths

asset markets are inefficient when there are jumps (known)

new: when there is Knightian uncertainty about volatility, even the“nice” asset markets can break down


The Message



Knightian Uncertainty is important

Asset markets work well when we are faced with risk and diffusions

risk = well–defined probabilities

diffusion = no jumps, trembling paths

asset markets are inefficient when there are jumps (known)

new: when there is Knightian uncertainty about volatility, even the“nice” asset markets can break down


The Message

Informal Presentation of ResultsEquivalence under RiskEquilibria under Volatility Uncertainty

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