Equilibrium Models with Small Frictions
Johannes Muhle-Karbe
University of Michigan
Joint work with Martin Herdegen
University of Michigan, January 13, 2016
Introduction
Outline
Introduction
Model
Results
Sketch of the Proofs
Summary
Introduction
Trading Frictions
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Classical financial theory is based on “frictionless markets”.
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Bid-ask spreads. Brokerage commissions. Price impact. Etc.
Heated discussion in the wake of the financial crisis:
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Infinite “liquidity”: trading is instantaneous and costless.
Yet, even small frictions often have drastic effects.
Paul Krugman (‘08 Nobel laureate, “How did economists get it
so wrong?”) vs. John Cochrane (‘10 President of the AFA,
“How did Paul Krugman get it so wrong?”).
About the only agreement: central importance of “flaws and
frictions”.
However: The problem is that we don’t have enough math.
[...] Frictions are just bloody hard with the mathematical tools
we have now. (Cochrane ‘10).
Introduction
Equilibrium Models
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Classical approach in economics: general equilibrium.
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Prices determined endogenously by matching supply and
demand.
Problem: typically intractable beyond simple toy models.
Way out: partial equilibrium models.
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Directly model observables (=prices) rather than endowments
and preferences of all market participants.
Solve decision problems of small agents that take prices as
given.
Widely adopted in Mathematical Finance.
Typically much more tractable.
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But: need general equilibrium to assess policy changes.
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E.g., the introduction of a financial transaction tax.
Introduction
Equilibrium Models ct’d
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General equilibrium models lead to nasty fixed point problems.
Indeed:
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Tractable models are rare even without frictions.
Problem becomes much worse with frictions.
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Given prices, agents solve their optimization problems.
Then, need to choose prices to ensure markets clear.
These again feed back into agents’ decision making.
Frictionless solutions typically rely on “representative agent”.
Aggregates all market participants.
No trade in equilibrium. Not suitable with frictions.
Accordingly, literature on equilibrium models with frictions is
rather limited.
Introduction
Literature
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Two strands of general equilibrium literature with frictions.
Numerical solution of discrete-time tree models.
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Additional restrictive modeling assumptions.
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Buss and Dumas ‘15, Buss, Vilkov, and Uppal ‘15.
No risky asset (Vayanos and Vila ‘99).
Exogenous interest rate (Lo, Mamaysky, and Wang ‘04).
Full refund of costs that is not internalized (Davila ‘15).
Transaction costs disappear from the model (everybody else).
Stark contrast to progress on partial equilibrium models.
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Starts with Magill and Constantinides ‘76. Much recent
progress in the limit for small trading costs.
Explicit formulas for optimal policies in very general settings.
Soner and Touzi (‘14). Kallsen and M-K (‘14,‘15). Martin and
Schöneborn (‘14,‘15). Rosenbaum and Tankov (‘14).
Introduction
Literature ct’d
Transaction costs in partial equilibrium?
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With fluctuating market conditions:
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Relevant case for most hedge fund strategies.
Much bigger effect of transaction costs (Lynch and Tan ‘11).
Asymptotically optimal strategies still known in closed form
(Kallsen and M-K ‘14, ‘15; Martin and Schöneborn ‘14, 15).
Universal relationship (Kallsen and M-K ‘15):
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Large effect on trading volume. But only small welfare effect
(Constantinides ‘86).
2/3 of utility lost due direct transaction costs. Tax payments.
1/3 due to displacement from frictionless target. True friction.
Simple, robust formulas for trading volume.
General equilibrium effects of transaction costs?
Introduction
This Paper
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Study equilibria with small trading costs.
True general equilibrium setting.
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Main result: frictionless equilibrium often need not change.
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Transaction cost paid to extra agent, who consumes them
optimally.
Taxes paid to state. Or fees paid to stock exchange.
Same prices still clear the market.
More risk-averse agents pay most of the trading costs.
Partial equilibrium analysis justified a posteriori.
Main assumptions:
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Only two agents (and state).
All use strategies asymptotically optimal for small costs.
Constant absolute risk aversion. Deterministic impatience
rates. Deterministic interest rates in frictionless equilibrium.
Model
Endowments and Preferences
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Two agents i = 1, 2.
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Receive exogenous endowment streams (yti )t∈[0,T ] .
Trade two assets with dynamics to be determined in
equilibrium:
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A “bank account”, in zero net supply.
A “stock”, in unit net supply, that gives right to an exogenous
dividend process.
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Both agents have constant absolute risk aversion γ i and a
deterministic impatience rate (βti )t∈[0,T ] .
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Choose (excess) consumption rates (cti )t∈[0,T ] to maximize
expected utility:
"Z
T
E
e
0
−
Rt
0
βui du −γ i (yti +cti )
e
#
dt → max!
Model
Trading Friction
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Exogenous proportional cost (εt )t∈[0,T ] for buying and selling
the stock.
Interpretation:
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Transaction tax imposed by the government.
Fees charged by an exchange.
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Agents bargain how fee is split between buyers and sellers.
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But where does the fee go? Should not disappear in general
equilibrium.
Our proposal:
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Also model the entity receiving the fees.
Also receives exogenous endowment (yt3 )t∈[0,T ] .
Additionally receives endogenous payments from other agents.
Solves corresponding optimal consumption problem.
Does not trade the stock.
Model
Radner Equilibrium
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Quantities to be determined in equilibrium:
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Equilibrium conditions:
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Interest rate.
Initial value, drift, and volatility of the stock.
Agents’ optimal strategies and consumption rates.
Bid-ask split ε1t + ε2t = εt of the total transaction cost.
All agents behave in an (asymptotically) optimal manner.
Market clears for bank account and stock.
Challenges?
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Trading strategies of local time type. Reflection off boundaries
of a “no-trade region”.
How to get agents to agree on trading times?
Endowment of state only singular continuous. Intractable.
Model
Asymptotic Perspective
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Start from frictionless equilibrium.
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Difficult problem in its own right.
Limited examples that can be solved explicitly (Christensen,
Munk, and Larsen ‘12, Christensen and Larsen ‘14).
Not the problem we focus on here. Want to study additional
effect of small friction.
Suppose all corresponding quantities are known and “nice”.
In particular: frictionless interest rate assumed to be
deterministic.
These and other assumptions can be checked in examples in
the spirit of Christensen et al. ‘12, ‘14.
So how to tackle the optimization problems with transaction
costs?
Results
Partial Equilibrium
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First ingredient: partial equilibrium analysis with small
transaction costs à la Kallsen and M-K ‘15.
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Optimal policy: minimal amount of trading to keep position in
a “no-trade region” around frictionless target.
Asymptotically optimal trading boundaries given explicitly:
∆NTit
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=
3 i dhϕi it
ε
2γ i t dhSit
1/3
2/3
Optimal performance at the leading asymptotic order O(εt ).
Little activity if trading is expensive, risk aversion is low, or
target strategy is inactive relative to market.
For stock market clearing:
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Trading boundaries have to coincide.
Frictionless market clearing: ϕ1t + ϕ2t = 1.
Results
Stock Market Clearing ct’d
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For any equilibrium stock price S, this implies
ε1t =
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γ1
εt ,
γ1 + γ2
ε2t =
γ2
εt
γ1 + γ2
The more risk averse agent is willing to bear most of the
transaction cost.
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Ensures stock market clearing holds, independent of the
choice of stock dynamics and interest rates.
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So maybe this works even with the original frictionless
equilibrium prices without any adjustment?
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Crucial point: transaction costs shifts money from Agents 1,2
to Agent 3 (state).
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Can asymptotically optimal consumption rates be
adjusted so that the bank account also still clears?
Results
Bank Account Clearing
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Second main ingredient: asymptotic analysis of optimal
consumption problems for small perturbations of the
endowment (Herdegen and M-K ‘15).
Similar asymptotic perspective:
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Start from baseline endowment for which solution is well
understood.
Perform sensitivity analysis for general but small perturbation.
For the setting considered here:
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Leads to explicit formulas, in terms of agents’ risk aversion.
These apply even for the singular continuous transaction cost
payments.
In turn allow to establish market clearing for the bank account
as well.
Results
Robustness of Frictionless Equilibria
In summary:
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Frictionless equilibria can still support market clearing even
with small trading frictions.
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Prices and price dynamics need not change due to
introduction of transaction tax.
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The more risk averse agent pays most of the costs.
Prerequisites:
I Only two agents.
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Mean-field extensions?
Constant absolute risk aversions. Deterministic impatience
rates. Deterministic frictionless interest rates.
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Justified if time horizon is not too long.
Alternative: refunds like in the finance literature
(Buss and Dumas ‘15, Davilo ‘15).
Results
Extensions
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Results do not hinge on particular type of friction.
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Similar results obtain for fixed transaction costs or linear price
impact.
Analysis based on partial equilibrium results of Altarovici, M-K,
and Soner ‘15, Moreau, M-K, Soner ‘15.
Results also remain valid if one drops the third agent and lets
costs disappear form the model.
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Local-time strategies need to be smoothed in this case, to
clear market with absolutely continuous dividends.
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Can one endogenize the decision whether a transaction tax
should be introduced?
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General robustness principle?
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Higher-order effects?
First-order effects with noise traders?
Sketch of the Proofs
Upper and Lower Asymptotic Bounds
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For asymptotic lower bounds:
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For asymptotic upper bounds:
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Derive candidate policy by heuristic arguments. Separation of
time scales.
Analyze performance of the explicitly known candidate.
Ensure integrability by suitable localizations.
Upper bound for all competitors comes from convex duality.
Candidate dual variable again derived heuristically
Rigorous analysis shows upper and lower bound match at the
leading order, completing the verification.
This line of reasoning is used twice:
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Partial equilibrium with small transaction costs.
Sensitivity of optimal consumption with respect to changes in
endowment.
Sketch of the Proofs
Market Clearing
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Two sources of losses due to transaction costs:
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Direct transaction costs of order O(ε2/3 ).
Displacement of order O(ε1/3 ) from frictionless target strategy.
First-order condition: expected loss is of order O(ε2/3 ), too.
State receives transaction costs of order O(ε2/3 ).
Use results form Herdegen and M-K ’15 to derive:
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Second-order adjustment for displacement of order O(ε1/3 ).
Mean zero. Effect of order O(ε2/3 ).
First-order optimal adjustment for transaction cost loss of
order O(ε2/3 ). Effect of order O(ε2/3 ).
First-order adjustment for state. Again of order O(ε2/3 ).
The last two steps allow for enough flexibility to ensure market
clearing.
Summary
General Equilibrium Models with Frictions
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Crucially needed to assess impact of regulatory reforms.
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Typically intractable.
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Difficult fixed point problem already without frictions.
But, asymptotic techniques show:
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Transaction taxes. Margin requirements. Etc.
Frictionless equilibrium still supports market clearing.
Partial equilibrium analysis justified for small trading costs.
For more details:
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Kallsen and M-K ‘15. The general structure of optimal
investment and consumption with small transaction costs.
Math. Finance, to appear.
Herdegen and M-K ‘15. Sensitivity of optimal consumption
streams. Preprint.
Herdegen and M-K ‘15. Equilibrium models with small
frictions. Working paper, (hopefully) available soon.