The Dynamics of Financially Constrained Arbitrage

• DOI: http://doi.org/10.1111/jofi.12689
• Date: 01 August 2018
• Author: DENIS GROMB,DIMITRI VAYANOS
• Published date: 01 August 2018

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 4 •AUGUST 2018

The Dynamics of Financially Constrained

Arbitrage

DENIS GROMB and DIMITRI VAYANOS∗

ABSTRACT

We develop a model in which financially constrained arbitrageurs exploit price dis-

crepancies across segmented markets. We show that the dynamics of arbitrage cap-

ital are self-correcting: following a shock that depletes capital, returns increase,

which allows capital to be gradually replenished. Spreads increase more for trades

with volatile fundamentals or more time to convergence. Arbitrageurs cut their po-

sitions more in those trades, except when volatility concerns the hedgeable com-

ponent. Financial constraints yield a positive cross-sectional relationship between

spreads/returns and betas with respect to arbitrage capital. Diversification of arbi-

trageurs across markets induces contagion, but generally lowers arbitrageurs’ risk

and price volatility.

THE ASSUMPTION OF FRICTIONLESS ARBITRAGE is central to finance theory and all

of its practical applications. It is hard to reconcile, however, with the large body

of evidence on so-called market anomalies, in particular those concerning price

discrepancies between assets with almost identical payoffs. Such discrepan-

cies arise during both crises and more tranquil times. For example, large and

persistent violations of covered interest parity (CIP) have been documented

for all major currency pairs, during and after the global financial crisis. Price

discrepancies that are hard to reconcile with frictionless arbitrage have also

∗Denis Gromb is at the Department of Finance, HEC Paris. Dimitri Vayanos is at the Depart-

ment of Finance, London School of Economics. We thank Philippe Bacchetta; Bruno Biais (the

Editor); Patrick Bolton; Darrell Duffie; Vito Gala; Jennifer Huang; Dong Lou; Henri Pag`

es; Anna

Pavlova; Christopher Polk; Matti Suominen; an anonymous associate editor; and three referees;

as well as seminar participants in Amsterdam, Bergen, Bordeaux, the Bank of Italy,la Banque de

France, BI Oslo, Bocconi University, Boston University, CEMFI Madrid, Columbia, Copenhagen,

Dartmouth College, Duke, Durham University, ESC Paris, ESC Toulouse, the HEC-INSEAD-PSE

workshop, HEC Lausanne, Helsinki, the ICSTE-Nova seminar in Lisbon, INSEAD, Imperial Col-

lege, Institut Henri Poincar´

e, LSE, McGill, MIT, Naples, NYU, Paris School of Economics, Univer-

sity of Piraeus, Porto, Queen’s University,Stanford, Toulouse, Universit´

e Paris Dauphine, Science

Po–Paris, the joint THEMA-ESSEC seminar, Vienna, and Wharton for comments. Financial sup-

port from the Paul Woolley Centre at the LSE, and grants from the Fondation Banque de France,

from the Institute for New Economic Thinking (INET) and from Labex ECODEC (ANR-11-IDEX-

0003/Labex Ecodec/ANR-11-LABX-0047), are gratefully acknowledged. We have read the Journal

of Finance’s disclosure policy and have no conflicts of interest to disclose.

DOI: 10.1111/jofi.12689

1713

1714 The Journal of Finance R

been documented for stocks, government bonds, corporate bonds, and credit

default swaps (CDS).1

To address these anomalies, one approach is to abandon the assumption

of frictionless arbitrage and study the constraints faced by real-world arbi-

trageurs, for example, hedge funds or trading desks in investment banks. Arbi-

trageurs have limited capital, which can constrain their activity and ultimately

affect market liquidity and asset prices. Empirical studies show that various

measures of arbitrage capital are related to the magnitude of the anomalies.

Since arbitrage capital can be targeted at multiple anomalies, returns to

investing in the anomalies are interdependent and hence so are arbitrageurs’

positions. This interdependence raises a number of questions. How should

arbitrageurs allocate their limited capital across anomalies, and how should

this allocation respond to shocks to capital? Which anomalies’ returns are more

sensitive to changes in arbitrage capital? How do the expected returns offered

by the different anomalies relate to sensitivity to arbitrage capital and other

characteristics? How do the expected returns offered by anomalies evolve over

time, and how do these dynamics relate to those of arbitrage capital? In this

paper, we develop a model to address these questions.

We consider a discrete-time, infinite-horizon economy with a riskless asset

and a number of “arbitrage opportunities” (the anomalies within our model)

that consist of a pair of risky assets with correlated payoffs. Each risky asset is

traded in a different segmented market by risk-averse investors who can trade

only that asset and the riskless asset. Investors experience endowment shocks

that generate a hedging demand for the risky asset in their market. Shocks are

opposites within each pair, so a positive hedging demand for one asset in the

pair is associated with a negative hedging demand of equal magnitude for the

other. This simplifying assumption ensures that arbitrageurs trade only on the

price discrepancy between the two assets. Market segmentation is exogenous

in our model, but could arise because of regulation, agency problems, or a lack

of specialized knowledge.

We make two key assumptions. First, we assume that, unlike other investors,

arbitrageurs can trade all assets. They therefore have better opportunities than

other investors. By exploiting price discrepancies between paired assets, arbi-

trageurs intermediate trade between otherwise segmented investors, providing

them liquidity: they buy cheap assets from investors with negative hedging de-

mand, and sell expensive assets to investors with positive hedging demand. We

refer to the price discrepancies that arbitrageurs seek to exploit as “arbitrage

spreads” and use them as an inverse measure of liquidity.

Second, we assume that arbitrageurs are constrained in their access to ex-

ternal capital. We derive their financial constraint following the logic of market

segmentation and assuming that they can walk away from their liabilities un-

less they are backed by collateral. Consider an arbitrageur who wishes to buy

an asset and short the other asset in its pair. The arbitrageur could borrow the

1References to the empirical literature are provided in Sections I.B.3 and III.C.2.Inthese

sections, we also explain how to map our model and results to the empirical settings.

The Dynamics of Financially Constrained Arbitrage 1715

cash required to buy the former asset, but the loan must be backed by collateral.

Posting the asset as collateral would leave the lender exposed to a decline in its

value. The arbitrageur could post as additional collateral the short position in

the other asset, which can offset declines in the value of the long position. Mar-

ket segmentation, however, prevents investors other than arbitrageurs from

dealing in multiple risky assets, which implies that the additional collateral

must be a riskless asset position. We assume that collateral must be sufficient

to protect the lender fully against default. This implies that positions in as-

sets with more volatile payoffs require endogenously more collateral so that

lenders are protected against larger losses. The need for collateral limits the

positions that an arbitrageur can establish, and that constraint is a function

of his wealth. The positions that arbitrageurs can establish as a group are

constrained by their aggregate wealth, which we refer to as arbitrage capital.2

When assets in each pair have identical payoffs, arbitrage is riskless. We

analyze this case, which is a natural benchmark, first. If spreads are positive,

then the riskless return offered by arbitrage opportunities exceeds the risk-

less rate. Arbitrageurs, however, may not be able to scale up their positions to

exploit that return because of their financial constraint. Their optimal policy

is to invest in the opportunities that offer maximum return per unit of col-

lateral. Equilibrium is characterized by a cutoff return per unit of collateral:

arbitrageurs invest in opportunities above the cutoff, which drives their return

down to the cutoff, and do not invest in opportunities below the cutoff. The

cutoff is inversely related to arbitrage capital. For example, when capital in-

creases, arbitrageurs become less constrained and can hold larger positions.

This drives down the returns of the opportunities they invest in.

The inverse relationship between returns and capital implies self-correcting

dynamics and a deterministic steady state. If arbitrage capital is low,then arbi-

trageurs hold small positions, returns are high, and capital gradually increases.

Conversely, if capital is high, then returns are low and capital decreases be-

cause of arbitrageurs’ consumption. In steady state, arbitrage remains prof-

itable enough to offset the natural depletion of capital due to consumption.

We next analyze the case in which payoffs within each asset pair consist of a

component that is identical across the two assets and hedgeable by arbitrageurs

and a component that differs across the two assets. Because asset payoffs are

not identical, arbitrage is risky.As in the case of riskless arbitrage, arbitrageurs

invest in the opportunities that offer maximum return per unit of collateral.

Unlike in that case, however, the relevant return is the expected return net

of a risk adjustment that depends on arbitrageur risk aversion and position

size. The financial constraint binds when the risk-adjusted return exceeds the

riskless rate.

2In Internet Appendix Section II, we formulate equilibrium in our model under general collat-

eralized contracts. The Internet Appendix may be found in the online version of this article. We

allow, for example, loan repayments to be state-contingent and to extend over several periods. We

show that, under some conditions, contracts can be restricted to the ones that we consider in our

main model. See Section I.C.2.