Nonlinear limits to arbitrage

• Published date: 01 June 2022
• Author: Jingzhi Chen,Charlie X. Cai,Robert Faff,Yongcheol Shin
• Date: 01 June 2022
• DOI: http://doi.org/10.1002/fut.22320

Received: 21 July 2020

|

Accepted: 3 February 2022

DOI: 10.1002/fut.22320

RESEARCH ARTICLE

Nonlinear limits to arbitrage

Jingzhi Chen

1

|Charlie X. Cai

2

|Robert Faff

3

|Yongcheol Shin

4

1

Sun Yat‐Sen Business School, Sun Yat‐Sen University, Guangzhou, China

2

University of Liverpool Management School, University of Liverpool, Liverpool, UK

3

Bond Business School, Bond University, Gold Coast, Australia

4

Department of Economics and Related Studies, University of York, York, UK

Correspondence

Charlie X. Cai, University of Liverpool

Management School, University of

Liverpool, Liverpool L69 7ZH, UK.

Email: x.cai7@liverpool.ac.uk

Funding information

Economic and Social Research Council,

Grant/Award Number: ES/S010238/1

Abstract

We study the nonlinear limits to arbitrage in a model. When mispricing is small,

arbitrage activity increases with mispricing because of the higher cost‐adjusted

return. However, at high levels of mispricing, arbitrageurs are deterred by larger

mispricing as funding constraints become more binding. Testing the model pre-

dictions on the index spot‐futures arbitrage with a Markov‐switching model, we

document an inverse U‐shaped relationship between mispricing and arbitrage

activity. The extreme regime is with the largest mispricing but least arbitrage

activity, and coincides with the market turmoil, suggesting that funding con-

straints become the main driver behind the limit to arbitrage.

KEYWORDS

index arbitrage, limits to arbitrage, Markov‐switching GECM, mispricing correction,

noise momentum

1|INTRODUCTION

Arbitrageurs aggressively search for mispricing opportunities, which ensures that mispricing is short lived. However,

arbitrage is far from a free lunch in practise. Extent finance literature has long documented that arbitrage activity is

impeded by the market frictions, leading to mispricing and resource misallocations (Gromb & Vayanos, 2010).

Meanwhile, larger mispricing may affect the perception on arbitrage frictions inversely, and, in turn, trigger arbitrage

trades. The latter idea draws little attention in the literature, but is of great importance in understanding the complex

joint determination between mispricing, arbitrage friction, and arbitrage activity.

There are two distinct and countervailing views of what limits arbitrage: arbitrage costs and funding constraints. On the

one hand, previous studies (e.g., Bai & Collin‐Dufresne, 2019; Gyntelberg et al., 2017;Rolletal.,2007) suggest that

conducting arbitrage trade is costly and risky (e.g., market illiquidity, transaction cost, and compensation for risk). In this

case, arbitrageurs are willing to exploit the mispricing only when it exceeds a certain threshold that reflects the cost of

conducting the arbitrage trade. By allowing for heterogeneous arbitrage costs, a wider mispricing will trigger more ag-

gressive arbitrage activity since it provides a higher cost/risk‐adjusted return. We call it the positive capital allocation effect.

On the other hand, various studies build on the idea that conducting arbitrage trade requires funding, and

document the importance of funding constraints in limiting arbitrage activity. The slow‐moving capital hypothesis

J Futures Markets. 2022;42:1084–1113.1084

|

wileyonlinelibrary.com/journal/fut

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided

the original work is properly cited.

© 2022 The Authors. The Journal of Futures Markets published by Wiley Periodicals LLC

posits that severe and prolonged mispricing especially during times of market turmoil is mainly due to the tightening of

funding constraints (Acharya et al., 2010; Akbas et al., 2015,2016; Duffie, 2010; Garleanu & Pedersen, 2011; Gromb &

Vayanos, 2002; Karnaukh et al., 2015; Mitchell et al., 2007; Mitchell & Pulvino, 2012; Shleifer & Vishny, 1997).

Furthermore, Brunnermeier and Pedersen (2009) suggest that larger mispricing can exaggerate expectations on future

volatility, which tightens the funding constraint. In this case arbitrage activity is rather deterred in the presence of

larger mispricing. We call this the negative funding constraint effect.

The two sources of arbitrage frictions drive opposite predictions of how arbitrageurs will respond to mispricing. The

former view has been examined empirically through threshold regression models (Dwyer et al., 1996; Martens et al., 1998;

Tse, 2001), while the effect of funding constraints has been studied mainly through arbitrage activity (Cielinska et al., 2017),

arbitrage capital flow (Akbas et al., 2015,2016), and violations from no‐arbitrage relations (Fontaine & Garcia, 2011b;

Garleanu & Pedersen, 2011). Up to our knowledge, however, there is no single study in the literature to analyze the

combined impact of these two frictions on how arbitrage activity responds to mispricing, theoretically or empirically. In this

paper,weaddressthislong‐standing but important knowledge deficit in the limits to arbitrage literature.

At its foundation, our empirical setup follows the standard multiperiod model of Shleifer and Vishny (1997,

henceforth SV). To explicitly analyze arbitrage activities in the multiperiod setting, we also follow Cai et al. (2018,

henceforth CFS) and introduce two important arbitrage parameters: the first is the initial mispricing correction

parameter (

κ

), which measures the proportion of immediate mispricing correction achieved by arbitrageurs, and the

second is the subsequent noise momentum parameter (

λ

), which captures the persistence of the unarbitraged pricing

errors into the next period.

1

Specifically, we investigate the interaction between arbitrage costs and funding constraints

through the interplay between these two parameters

κ

and

λ

with respect to the size of mispricing. When the

mispricing error is small and funding is relatively ample, arbitrageurs strategically limit their investment due to

concerns of arbitrage risk. In this case the arbitrage activity will intensify with the size of mispricing such that the

capital allocation effect prevails. On the other hand, extremely large mispricing is likely to make funding constraints

more binding. Beyond some thresholds, arbitrage activity declines with the size of mispricing because of the associated

funding liquidity scarcity, suggesting that the funding constraint effect becomes the dominant driver. Accordingly,

combining the two countervailing effects, the model predicts that the overall arbitrage activity displays an inverse

U‐shape against the size of mispricing error due to the exchange of dominance between arbitrage costs and funding

constraints in limiting arbitrage.

In our empirical analysis, we apply the GECM with the Markov‐switching extension (MS‐GECM) to the S&P 500

index spot and futures markets over the period 1986–2015.

2

The construction of GECM, advanced by CFS, captures

how arbitrage activity (both mispricing correction and noise momentum) respond to past observable mispricing, which

provides a great tool to test our model predictions.

3

We find strong evidence in favor of regime‐dependent nonlinear

limits to arbitrage. In particular, we can identify three distinct regimes: a normal market state with a small mispricing

error and low mispricing volatility, a transition market state with both medium mispricing error and volatility, and an

extreme market state with a large mispricing error and high mispricing volatility. We observe a relatively low mis-

pricing correction in the normal state, but a dramatic increase during the transition state. This suggests that arbitrage

activity tends to intensify with the size of mispricing error when the mispricing level increases from low to medium. In

contrast, the mispricing correction is the lowest during the extreme state. This suggests that when mispricing increases

from a medium to a high level, arbitrageurs are less capable to raise external funds due to the tightening funding

constraints even when the arbitrage opportunity is at its best. These extreme periods coincide with the market turmoils

in the years 1987, 1998, 2001, and 2008, which provides empirical supports to the existing studies (e.g., Brunnermeier &

Pedersen, 2009) documenting that the amplification effect attributed to funding illiquidity significantly jeopardizes

market resiliency. Overall, arbitrage activity displays an inverse U‐shape against the magnitude of mispricing errors.

To verify whether our estimation results meaningfully capture variations in the tightness of funding constraints, we

examine the potential linkages between the three hidden market states and various observable measures of the funding

1

To capture such multiperiod arbitrage activities, CFS develop a generalized error correction model (GECM) and estimate both parameters. Applying

the model to a wide range of international spot–futures market pairs, CFS document pervasive evidence of noise momentum around the world.

2

As a robustness check, we provide the estimation results over a shorter sample (1990–2015) and results employing other S&P 500 futures contracts

(e.g., 6and 9 monthsto maturity),which areelaborated inthe appendix.

3

In the extent literature with empirical applications using the error correction model (e.g., Balke & Fomby, 1997; Dwyer et al., 1996; Gyntelberg et al.,

2017; Martens et al., 1998; Tao & Green, 2013; Theissen, 2012; Tse, 2001), past mispricing is often treated as an exogenous state variable that

determines the arbitrage activity.

CHEN ET AL.

|

1085