Equilibrium Models with Transaction Costs I

Johannes Muhle-Karbe

University of Michigan → Carnegie Mellon University

Dresden, August 28, 2017

IntroductionOutline

Introduction

Partial Equilibrium

General Equilibrium

Summary and Outlook

IntroductionFrictions

I Classical financial theory: based on “frictionless markets”.I Arbitrary amounts can be traded immediately at no cost. Etc.I Substantially simplifies the mathematical analysis.I But neglects many important effects.

I Pedersen (“Overview of Frictional Finance”):I “Frictions affect asset prices, macroeconomics, monetary

policy.”I “Provide unified explanation of a wide variety of phenomena.”I “Empirical evidence is stronger than almost any other influence

on the markets.”I Cochrane ‘11:

I “The problem is that we don’t have enough math. [...]Frictions are just hard with the tools we have right now”.

I Tractable models with frictions? E.g., trading costs?

IntroductionEquilibrium Models

I Classical approach in economics: general equilibrium.I Prices determined endogenously by matching supply and

demand.I Typically intractable beyond simple toy models.

I “Partial” equilibrium models greatly facilitate analysis.I Agents still optimize, but prices are given exogenously by some

estimated or calibrated model. Much more tractable.I Well-suited to analyze individual optimization of small agents.I But cannot address systemic effects or regulatory measures.

I Need for equilibrium models with frictions:I Effects of a financial transaction tax, tighter margin

requirements, etc.?I Impact of market power on asset prices?

IntroductionEquilibrium Models with Frictions?

I Equilibrium models lead to nasty fixed-point problems.I Given prices, agents solve their optimization problems.I Then, need to choose prices to ensure markets clear.I These prices then again feed back into agents’ decision making.

I Tractable models are rare even without frictions.I Problem becomes much worse with frictions.

I Complicated individual decision problems.I Frictionless solutions typically rely on “representative agent”.I Aggregates all market participants.I No trade in equilibrium. Not suitable with frictions.

I Accordingly, literature on equilibrium models with frictions israther limited.

IntroductionGeneral Equilibrium Models with Frictions ct’d

Literature on equilibrium models with frictions:I Numerical solution of discrete-time tree models.

I Heaton/Lucas ‘96. Buss/Dumas ‘15; Buss/Vilkov/Uppal ‘15.I Additional restrictive modeling assumptions.

I No risky assets (Vayanos/Vila ‘99, Weston ‘16).I Constant asset prices (Lo/Mamaysky/Wang ‘04).I Full refund of costs that is not internalized (Davila ‘15).

I Stark contrast to progress on partial equilibrium models.I Starts with Magill/Constantinides ‘76; Dumas/Luciano ‘91;

Davis/Norman ‘90, Shreve/Soner ‘94.I Explicit asymptotic formulas now available for small costs in

general settings. Soner/Touzi ‘14; Kallsen/M-K ‘14,‘15;Cai/Rosenbaum/Tankov ‘15, ‘16.

IntroductionThis Lecture

I Recap intuition behind small-cost asymptotics for partialequilibrium models.

I Different frictions require different mathematical tools.I Regular vs. singular vs. impulse control.I We will therefore focus on the most tractable specification, the

model of Almgren/Chriss ‘01.I However, the same methods apply much more broadly.

I Apply to general equilibrium problems in a second step.I Starting point: sufficiently regular frictionless equilibrium.I Goal: understand leading-order effects of small trading costs on

trading strategies and prices.

IntroductionThe Almgren/Chriss Model

I Unaffected price process St .I Execution price affected linearly by trade size and speed:

St + λ∆ϕt∆t

when trading ∆ϕt shares on [t, t + ∆t].I Trading cost compared to frictionless model is λ( ∆ϕt

∆t )2∆t.I Wealth process in the continuous-time limit:

x +∫ T

0ϕtdSt −

∫ T

0λϕ̇2

t dt

I Only finite variation strategies dϕt = ϕ̇tdt are feasible.

IntroductionThe Almgren/Chriss Model ct’d

I Quadratic costs on turnover rate:I Introduced for optimal execution (Almgren/Chriss ‘01).I Used for portfolio choice in Garleanu/ Pedersen ‘13, ‘16;

Guasoni/Weber ‘17; Almgren/Li ‘16; Moreau/M-K/Soner ‘17.I Interpretation?

I Price impact in a highly-resilient block-shaped order book(Roch/Soner ‘13, Kallsen/M-K ‘14).

I Compensation for risk market makers incur until the locate asuitable counterparty (Garleanu/Pedersen ‘13, ‘16).

I Stylized example of a “progressive” transaction tax(Subrahmanyam ‘98, Schied/Zhang ‘17).

I Most tractable specification among all trading costs.

Partial EquilibriumGoal Functional

I First step: fix exogenous price process S.I Find trading rate ϕ̇λt = dϕλt

dt that maximizes expected utility:

E[U(

x +∫ T

0ϕtdSt −

∫ T

0λϕ̇2

t dt)]→ max

ϕ̇!

I U is increasing, concave utility function.I Here, focus on most tractable example: U(x) = −e−γx .

I Frictionless problem (λ = 0) also requires dynamicprogramming. “Nice” solution ϕ̂ is starting point here.

I For small transaction costs λ ≈ 0, we expect:

ϕλ ≈ ϕ̂ and λ(ϕ̇λ)2 ≈ 0

Partial EquilibriumSmall-Cost Expansion

I Approximate goal functional by (formal) Taylor expansion:

E[

U(

x +∫ T

0ϕtdSt −

∫ T

0λϕ̇2

t dt)]≈ E

[U(

x +∫ T

0ϕ̂tdSt

)]+ E

[U′

(x +∫ T

0ϕ̂tdSt

)(∫ T

0(ϕλt − ϕ̂t )dSt −

∫ T

0λ(ϕ̇λt )2

)dt]

I First-order condition for frictionless optimality:I U ′(x +

∫ T0 ϕ̂tdSt) is density of equivalent martingale measure

Q up to normalizing constant y .I Intuition: optimality implies

E[

U(

x +∫ T

0ϕ̂tdSt

)]≥ E

[U(

x +∫ T

0(ϕ̂t + εϕt )dSt

)]

Partial EquilibriumSmall-Cost Expansion ct’d

I Taylor expansion yields

E[

U(

x +∫ T

0ϕ̂tdSt

)]≥ E

[U(

x +∫ T

0(ϕ̂t + εϕt )dSt

)]≈ E

[U(

x +∫ T

0ϕ̂tdSt

)]+ εyEQ

[∫ T

0ϕtdSt

]I As a consequence, 0 = EQ

[∫ T0 ϕtdSt

]for all integrands ϕ, so

that S has to be a Q-martingale.I In particular:

E[

U′

(x +∫ T

0ϕ̂tdSt

)(∫ T

0(ϕλt − ϕ̂t )dSt

)]= 0

I Need second-order expansion to obtain nontrivialcost-displacement tradeoff.

Partial EquilibriumSmall-Cost Expansion ct’d

I (Formal) second-order Taylor expansion:

E[

U(

x +∫ T

0ϕtdSt −

∫ T

0λϕ̇2

t dt)]

≈ E[

U(

x +∫ T

0ϕ̂tdSt

)]− yEQ

[(∫ T

0λ(ϕ̇λt )2

)dt]

− yEQ

[γ(x +

∫ T0 ϕ̂tdSt )2

(∫ T

0(ϕλt − ϕ̂t )dSt

)2

dt

]

I Here, γ(x) = −U ′′(x)/U ′(x) is the risk-aversion of U.I Constant for exponential utility.I For more general utilities, need extension to “risk-tolerance

process” of Kramkov/Sirbu ‘06, Czichowsky/Kallsen/M-K.

Partial EquilibriumTracking Problem

I For small costs, exponential utility maximization is formallyequivalent to linear-quadratic tracking problem:

EQ

[∫ T

0

(γ̄t(ϕt − ϕ̂t)2 + λϕ̇2

t

)dt]→ min

ϕ̇!

whereγ̄t = γσ2

t2 for σ2

t = d〈S〉tdt

I Variants studied by Rosenbaum/Tankov ‘14, Almgren/Li ‘16,Cai/Rosenbaum/Tankov ‘15, ‘16, Bank/Soner/Voss ‘17.

I Asymptotic equivalence to exponential utility maximization isestablished in Ahrens ‘15; Cayé/Herdegen/M-K ‘17.

Partial EquilibriumRescaled Tracking Problem

I Key simplification: tracking problem for small costs is myopic.I Subproblems on a fine partition of [0,T ] can be solved

independently.I Consider a partition (ti ) with meshwidth O(λa).I Rescale displacement by λb to obtain nontrivial limit.I Rewrite “local” tracking problems via change of variable

s = t/λc as

EQ

[λc∫ ti+1/λc

ti/λc

(γ̄λcsλ

2b(ϕλcs − ϕ̂λcs

λb

)2+ λϕ̇2

λcs

)ds]

I Now: choose a, b, c to obtain nontrivial limit of the targetϕ̂λcs/λ

b, match orders of displacement and costs.

Partial EquilibriumRescaled Tracking Problem ct’d

I Small-cost limit of the rescaled target ϕ̂λcs/λb?

I If frictionless optimizer ϕ̂ is an Itô process:

ϕ̂λcsλb = λc

λb b̂λcsds + λc/2

λb σ̂λcsdWs

I Martingale part dominates.I If c/2 = b: converges to BM Bs with volatility σ̂ti as λ→ 0.I Rescaled tracking strategy:

ψs := ϕλ2bs/λb ⇒ ψ̇s := λbϕ̇λ2bs

I Different orders for pure jump processes (Rosenbaum/Tankov‘14) or fBM (Czichowsky/M-K/Schelling).

Partial EquilibriumRescaled Tracking Problem ct’d

I In summary, the rescaled tracking problem is

EQ

[λ2b

∫ ti+1/λ2b

ti/λ2b

(γ̄λ2bsλ

2b (ψs − Bs)2 + λ1−2bψ̇2s

)ds]

I To match displacement and trading costs: choose b = 1/4.I As γ̄λ2bs = γ̄ti + o(1), this leads to

EQ

[λ

∫ ti+1/λ1/2

ti/λ1/2

(γ̄ti (ψs − Bs)2 + ψ̇2

s

)ds]

I To obtain a stationary infinite horizon problem, choosea < 1/2, so that ti+1−ti

λ1/2 = O(λa−1/2) diverges.

Partial EquilibriumRescaled Tracking Problem ct’d

I Approximate with ergodic tracking for Brownian motion:

Vti := infψ̇

EQ

[lim

∆→∞

1∆

∫ ∆

0

(γ̄ti (ψs − Bs)2 + ψ̇2

s

)ds]

I Local tracking problem approximated by λ ti+1−tiλ1/2 Vti .

I Summing over all local problems in turn gives

λ1/2EQ

[∫ T

0Vtdt

]+ o(λ1/2)

I Relationship between optimal controls:

ϕλλ1/2s − ϕ̂λ1/2s ≈ λ1/4(ψs − Bs), ϕ̇λλ1/2s ≈ λ

−1/4ψ̇s

Partial EquilibriumRescaled Tracking Problem ct’d

I Convergence proof for these approximations:I In probability: Cai/Rosenbaum/Tankov ‘14, ‘15, ‘16.I In Sp: Ahrens ‘15, Cayé/Herdegen/M-K ‘17.

I In summary: remains to solve ergodic tracking problem forBrownian motion.

I Constant “frozen” volatility σ̂ =√

d〈ϕ̂〉tdt .

I Constant “frozen” risk aversion γ̄ = γσ2t

2 = γ2

d〈S〉tdt .

I Can be tackled by different methods.I Dynamic programming.I Calculus of variations as in Bank/Soner/Voss ‘17.

I We will discuss the second approach in the next lecture, so letus sketch the first here.

Partial EquilibriumErgodic Tracking Problem

I The dynamic programming principle suggests

Vt = infϕ̇t

{γ̄(ϕt − Bt)2dt + ϕ̇2

t dt + Et [Vt+dt ]}

I In view of the quadratic structure of the problem, make thefollowing quadratic ansatz for the value function:

Vt = V (ϕt − Bt) = A0(T − t) + A2 (ϕt − Bt)2

I Constant growth as horizon T becomes large, deviation is onlystate variable because target has independent increments.

I Remains true for Lévy processes as in Rosenbaum/Tankov ‘14.I Becomes more involved for fBM (Czichowsky/M-K/Schelling).

Partial EquilibriumErgodic Tracking Problem ct’d

I Recall the dynamic programming principle:

Vt = infϕ̇t

{γ̄(ϕt − Bt)2dt + ϕ̇2

t dt + Et [Vt+dt ]}

I The ansatz Vt = A0(T − t) + A2 (ϕt − Bt)2 and Itô’s formula

yield

Et [Vt+dt ] = Vt +(−A0 + A(ϕt − Bt)ϕ̇t + 1

2Aσ̂2)

dt

I Dividing by dt and sending dt → 0, formally leads to thedynamic programming equation:

0 = infϕ̇t

{γ̄(ϕt − Bt)2 + ϕ̇2

t − A0 + A(ϕt − Bt)ϕ̇t + 12Aσ̂2

}

Partial EquilibriumErgodic Tracking Problem ct’d

I The pointwise minimum is ϕ̇t = A2 (Bt − ϕt).

I To pin down A, plug this back into the dynamic programmingequation:

A = 2√γ̄, A0 = 1

2 σ̂2A =

√γ̄

2 σ̂2

I Therefore, the optimal trading rate is:

ϕ̇t =√γ̄(Bt − ϕt)

I Deviation Bt − ϕλt is Ornstein-Uhlenbeck process withvolatility σ̂ and mean-reversion speed

√γ̄.

I A0 describes long-run growth rate as T →∞.

Partial EquilibriumSolution of Original Tracking Problem

I We have solved the ergodic tracking problem.I Its solution in turn pins down the asymptotical expansion of

the original tracking problem.I Indeed, the corresponding approximate value becomes

EQ

∫ T

0

√λγσ2

t2 d〈ϕ̂〉t

I The corresponding optimal trading rate is the rescaled

feedback control:

ϕ̇λt =

√γσ2

t2λ (ϕ̂t − ϕt)

General EquilibriumTo Do

I Now, we want to apply the asymptotic results to a generalequilibrium model.

I Start from frictionless equilibrium.I Then study the effect of a small trading cost.

I Need to model several agents.I Trading friction not visible through representative agent.

I Crucial: provide motive to trade.I Here: risk sharing.I Agents trade risky asset to hedge correlated endowments.I Alternative: heterogenous beliefs.

I For simplicity: focus on two agents i = 1, 2.I Receive exogenous endowment streams (y i

t )t∈[0,T ].

General Equilibrium: ModelEndowments and Preferences

I Agents i = 1, 2 trade two assets with dynamics to bedetermined in equilibrium:

I A “bank account”, in zero net supply.I A “stock”, in unit net supply, that gives right to an exogenous

dividend process.I Both agents have constant absolute risk aversion γ i and a

deterministic impatience rate (βit)t∈[0,T ].

I Choose (excess) consumption rates (c it)t∈[0,T ] to maximize

expected utility:

E[∫ T

0e−∫ t

0 βiudue−γ i (y i

t +c it )dt

]→ max!

General Equilibrium: ModelTrading Friction

I Exogenous quadratic cost λ for buying and selling the stock.I Transaction tax imposed by the government.I Fees charged by the trading platform.

I Agents bargain how the fee is split between buyers and sellers.I But where does the fee go? Should not disappear in general

equilibrium.I Our proposal:

I Model the entity receiving the fees.I Also receives exogenous endowment (y3

t )t∈[0,T ].I Additionally receives endogenous payments from other agents.I Solves corresponding optimal consumption problem.

Does not trade the stock.

General Equilibrium: ModelRadner Equilibrium

I Quantities to be determined in equilibrium:I Interest rate.I Initial value, drift, and volatility of the stock.I Agents’ optimal strategies and consumption rates.I Bid-ask split λ1 + λ2 = λ of the total transaction cost.

I Equilibrium conditions:I All agents behave optimally.I Markets clear.

I Asymptotic equilibrium:I Only ask for asymptotic optimality for small costs in individual

optimization problems.I Similar to notion of ε-equilibrium in game theory.

General Equilibrium: ModelAsymptotic Perspective

I Start from frictionless equilibrium.I Suppose all corresponding quantities are known and “nice”.I In particular: frictionless interest rate is deterministic.

I Lo/Mamaysky/Wang ‘04: diffusive strategies, but constantstock price.

I Vayanos ‘98, Vayanos/Vila ‘99, Christensen/Larsen ‘14:deterministic, smooth trading strategies.

I Christensen/Larsen ‘14: diffusive stock prices and strategies,but stochastic interest rates.

I Herdegen/M-K ‘16: example with deterministic interest ratesbut diffusive prices and trading strategies.

I What changes with the introduction of the trading cost?

General Equilibrium: ResultsStock Market Clearing

I Recall: asymptotically optimal trading rates

ϕ̇λ,it =

√γ iσ2

t2λi (ϕ̂i

t − ϕλ,it ) := `it(ϕ̂i

t − ϕλ,it )

where ϕ̂i are the frictionless optimal strategies.I Suppose that both agents have the same risk aversion as in

Lo/Mamasky/Wang ‘04 and pay the same transaction cost.I Then, `1t = `2t = `t and the frictionless equilibrium dynamics

still clear the market:I Frictionless stock market clearing implies ϕ̂1

t + ϕ̂2t = 1 and in

turnd(ϕλ,1t + ϕλ,2t ) = (1− (ϕλ,1t + ϕλ,2t ))dt

I If the initial allocation clears the market, this indeed hasthe unique solution ϕλ,1t + ϕλ,2t = 1.

General Equilibrium: ResultsStock Market Clearing ct’d

I For heterogenous risk aversions, the same argument still worksif

λ1/γ1 = λ2/γ2

I Stronger risk-sharing motive makes higher costs acceptable:

λ1 = γ1

γ1 + γ2λ, λ2 = γ2

γ1 + γ2λ

I Ensures stock market clearing holds for the frictionlessequilibrium dynamics.

I But what about the market for the consumption good?

General Equilibrium: ResultsGoods Market Clearing

I Crucial point: transaction costs shift money from Agents 1,2to Agent 3.

I Do asymptotically optimal consumption rates still clearthe goods market?

I Second main ingredient: asymptotic analysis of optimalconsumption problems for small perturbations of theendowment (Herdegen/M-K ‘15).

I Similar asymptotic perspective:I Start from baseline endowment for which solution is well

understood.I Perform sensitivity analysis for general but small perturbation.

I Application here?

General Equilibrium: ResultsGoods Market Clearing

I Asymptotic consumption adjustments (Herdegen/M-K ‘15):I Second-order adjustment for displacement of order O(ε1/3).

Mean zero. Effect of order O(ε2/3).I Total adjustment is zero as aggregate displacement is zero.I First-order optimal adjustment for transaction cost loss of

order O(ε2/3). Effect of order O(ε2/3).I First-order adjustment for Agent 3. Again of order O(ε2/3).I Total consumption change can be chosen to be zero, as total

endowment adjustment is zero.I In summary:

I Frictionless equilibria can still support market clearing evenwith small trading frictions.

I Prices and price dynamics need not change due to introductionof a small transaction tax.

I The more risk averse agent pays most of the costs.

Summary and OutlookScope of Robustness Result

I Robustness of equilibria with respect to small trading costs.I Not an envelope theorem: strategies change, but not prices.

I “Straightforward” extensions:I Other trading costs.I No third agent, transaction costs disappear from model.

I Crucial prerequisites:I Only two agents. Extensions with market maker?I Constant absolute risk aversions. Deterministic impatience

rates. Deterministic frictionless interest rates.I Justified if time horizon is not too long.I Formal alternative: refunds like in the finance literature

(Buss/Dumas ‘15, Davilo ‘15).I Equal risk aversions or costs split accordingly. Will be relaxed

in a simpler model in the next lecture.