Liquidity Risk and the Dynamics of Arbitrage Capital

• Published date: 01 June 2019
• Author: PÉTER KONDOR,DIMITRI VAYANOS
• Date: 01 June 2019
• DOI: http://doi.org/10.1111/jofi.12757

THE JOURNAL OF FINANCE •VOL. LXXIV, NO. 3 •JUNE 2019

Liquidity Risk and the Dynamics

of Arbitrage Capital

P´

ETER KONDOR and DIMITRI VAYANOS∗

ABSTRACT

We develop a continuous-time model of liquidity provision in which hedgers can trade

multiple risky assets with arbitrageurs. Arbitrageurs have constant relative risk-

aversion (CRRA) utility, while hedgers’ asset demand is independent of wealth. An

increase in hedgers’ risk aversion can make arbitrageurs endogenously more risk-

averse. Because arbitrageurs generate endogenous risk, an increase in their wealth

or a reduction in their CRRA coefficient can raise risk premia despite Sharpe ratios de-

clining. Arbitrageur wealth is a priced risk factor because assets held by arbitrageurs

offer high expected returns but suffer the most when wealth drops. Aggregate illiq-

uidity, which declines in wealth, captures that factor.

LIQUIDITY IN FINANCIAL MARKETS IS OFTEN provided by specialized agents such as

market makers, trading desks in investment banks, and hedge funds. Adverse

shocks to the capital of these agents cause liquidity to decline and risk premia

to increase. Conversely, movements in the prices of assets held by liquidity

providers feed back into these agents’ capital.1

∗P´

eter Kondor is at the London School of Economics and CEPR. Dimitri Vayanosis at the London

School of Economics, CEPR, and NBER. The authors thank Edina Berlinger; Bruno Biais; Itamar

Dreschler; Nicolae Garleanu; Florian Heider; Pete Kyle; Stavros Panageas; Anna Pavlova; Jean-

Charles Rochet; WeiXiong; Hongjun Yan; an anonymous Associate Editor; two anonymous referees;

seminar participants at Atlanta Fed, Bank of Canada, Cass, Columbia, Copenhagen, Duke, HKU,

Imperial, LBS, LSE, Minneapolis Fed, Montreal, NUS, Ohio State, Tinbergen, UIUC, UNC, Wash

U, Wharton, and Zurich; and participants at the AEA, AFA, Arne Ryde, BIS, Cambridge INET,

CRETE, EIEF Market Microstructure, ESEM, ESSFM Gerzensee, EWFC, NBER Asset Pricing,

QMUL, and SED conferences for helpful comments. They are grateful to Mihail Zervos for generous

and valuable guidance with the proofs on the differential equations. P´

eter Kondor acknowledges

financial support from the European Research Council (Starting Grant #336585). Both authors

acknowledge financial support from the Paul Woolley Centre at the LSE. Both authors have read

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

1A growing empirical literature documents the relationships between the capital of liquidity

providers, the liquidity that these agents provide to other participants, and assets’ risk premia.

For example, Comerton-Forde et al. (2010) find that bid-ask spreads quoted by specialists in the

New York Stock Exchange widen when specialists experience losses. Aragon and Strahan (2012)

find that following the collapse of Lehman Brothers in 2008, hedge funds doing business with

Lehman experienced a higher probability of failure and the liquidity of the stocks that they were

trading declined. Jylha and Suominen (2011) find that outflows from hedge funds that perform the

carry trade predict poor performance of that trade, with low interest-rate currencies appreciating

DOI: 10.1111/jofi.12757

1139

1140 The Journal of Finance R

In this paper, we study the dynamic interactions between liquidity providers’

capital, the liquidity that these agents provide to other participants, and as-

sets’ risk premia. We build a framework with minimal frictions, in particular,

no asymmetric information or leverage constraints. The capital of liquidity

providers matters in our model only because of standard wealth effects. We

depart from most frictionless asset-pricing models, however, by fixing the risk-

less rate and by suppressing wealth effects for agents other than the liquidity

providers. These assumptions are sensible when focusing on shocks to the

capital of liquidity providers in an asset class rather than in the entire as-

set universe.

Our combination of assumptions allows us to prove general analytical re-

sults on equilibrium prices and allocations. In particular, we characterize how

liquidity providers’ risk appetite, the endogenous risk that they generate, and

the pricing of that risk depend on both liquidity demanders’ characteristics

and liquidity providers’ capital. We also show that the capital of liquidity

providers is the single priced risk factor, and that liquidity aggregated over

the assets that we consider captures that factor because it increases in capital.

Our results thus suggest that a priced liquidity risk factor may arise even with

minimal frictions.

We assume a continuous-time infinite-horizon economy. There is a riskless

asset with an exogenous constant return and multiple risky assets whose prices

are determined endogenously in equilibrium. There are two sets of competitive

agents: hedgers, who receive a risky income flow and seek to reduce their risk

by participating in financial markets, and arbitrageurs, who take the other

side of the trades that hedgers initiate. Arbitrageurs can be interpreted, for

example, as speculators in futures markets. We consider two specifications

for hedgers’ preferences. Hedgers can be “long-lived” and maximize constant

absolute risk-aversion (CARA) utility over an infinite consumption stream, or

they can be “short-lived” and maximize a mean-variance objective over changes

in wealth in the next instant. Under both specifications, hedgers’ demand for

insurance is independent of their wealth. In contrast, because arbitrageurs

maximize constant relative risk-aversion (CRRA) utility over consumption, the

supply of insurance depends on their wealth.

Arbitrageur wealth impacts equilibrium prices and allocations, and is the

key state variable in our model. Solving for equilibrium amounts to solving

a system of ordinary differential equations (ODEs) in wealth with boundary

conditions at zero and infinity.While these ODEs include nonlinear terms, their

structure makes it possible to prove general analytical results across the entire

parameter space, for example, for all risk-aversion parameters of hedgers and

arbitrageurs. In the case in which hedgers are short-lived, we show that a

solution exists and we characterize how it depends on both wealth and the

model parameters. Moreover, in both the short-lived and long-lived cases, we

characterize the behavior of the solution close to the boundaries.

and high interest-rate ones depreciating. Acharya, Lochstoer, and Ramadorai (2013) find that risk

premia in commodity futures markets are larger when broker-dealer balance sheets are shrinking.

Liquidity Risk and the Dynamics of Arbitrage Capital 1141

Our analysis yields new insights on dynamic risk-sharing and asset pricing.

We show that the risk aversion of arbitrageurs is the sum of their static CRRA

coefficient and a forward-looking component that reflects intertemporal hedg-

ing. The latter component makes the risk aversion of arbitrageurs dependent on

parameters of the economy that affect equilibrium prices. For example, when

hedgers are more risk-averse, arbitrageurs become endogenously more risk-

averse if their CRRA coefficient is less than one. This effect can be sufficiently

strong to imply that more risk-averse hedgers may receive less insurance from

arbitrageurs in equilibrium. Intuitively, when hedgers are more risk-averse,

expected returns rise steeply following a decline in arbitrageur wealth. This

makes arbitrageurs with CRRA coefficient less than one invest more conserva-

tively so as to preserve wealth in bad states and earn the high returns.

On the asset-pricing side, we show that arbitrageurs generate endogenous

risk, in the sense that changes in their wealth affect return variances and co-

variances through amplification and contagion mechanisms. Endogenous risk

is small at both extremes of the wealth distribution: when wealth is close to

zero, this is because arbitrageurs hold small positions and hence have a small

impact on prices, whereas when wealth is close to infinity, this is because

prices are insensitive to changes in wealth. The dependence of endogenous risk

on arbitrageur wealth can give rise to hump-shaped patterns of variances, co-

variances, and correlations. It can also cause risk premia, defined as expected

returns in excess of the riskless asset, to increase with arbitrageur wealth

for small values of wealth even though Sharpe ratios decrease. We show that

risk premia always exhibit this pattern when arbitrageurs’ CRRA coefficient is

small, and can exhibit it for larger values as well provided that hedgers are suf-

ficiently risk-averse. In a similar spirit, we show that risk premia can be larger

if arbitrageurs’ CRRA coefficient is smaller—precisely because endogenous risk

is larger.

Additional asset-pricing results concern liquidity risk and its relationship

with expected returns. A large empirical literature documents that liquidity

varies over time and in a correlated manner across assets within a class. More-

over, aggregate liquidity appears to be a priced risk factor and carry a positive

premium: assets that underperform the most during times of low aggregate liq-

uidity earn higher expected returns than assets with otherwise identical char-

acteristics.2We map our model to this literature by defining liquidity based on

the effect that hedgers have on prices. We show that liquidity is lower for assets

with more volatile cash flows. It also decreases following losses by arbitrageurs,

with this variation common across assets.

2Chordia, Roll, and Subrahmanyam (2000), Hasbrouck and Seppi (2001), and Huberman and

Halka (2001) document the time variation of liquidity in the stock market and its correlation

across stocks. Amihud (2002) and Hameed, Kang, and Viswanathan (2010) link time variation in

aggregate liquidity to the returns on the aggregate stock market. Pastor and Stambaugh (2003)

and Acharya and Pedersen (2005) find that aggregate liquidity is a priced risk factor in the stock

market and carries a positive premium. Sadka (2010) and Franzoni, Nowak, and Phalippou (2012)

find similar results for hedge fund and private equity returns, respectively.For more references, see

Vayanosand Wang (2013), who survey the theoretical and empirical literature on market liquidity.