Dynamic Strategic Arbitrage∗

Vincent Fardeau† Frankfurt School of Finance and Managment

This version: June 11, 2014

Abstract

Asset prices do not adjust instantaneously following uninformative demand or supply shocks. This paper oﬀers a theory based on arbitrageurs’ price impact to explain these slow-moving capital dynamics. Arbitrageurs recognizing their price impact break up trades, leading to gradual price adjustments. When shocks are anticipated, prices exhibit a V-shaped pattern, with a peak at the realization of the shocks. The dynamics of market depth and price adjustments depend on the degree of competition between arbitrageurs. When shocks are uncertain, price dynamics depend on the interaction between arbitrageurs’ competition and the total risk-bearing capacity of the market.

JEL codes: G12, G20, L12; Keywords: Strategic arbitrage, liquidity, price impact, limits of arbitrage.

∗I benefited from comments from Miguel Antón,Bruno Biais, Maria-Cecilia Bustamante, Amil Dasgupta, Michael Gordy, Martin Oehmke (discussant), Matt Pritsker, Jean-Charles Rochet, Yuki Sato, Kostas Zachariadis, Jean-Pierre Zigrand and seminar and conference participants at LSE, INSEAD, FRB, the University of Zurich and the 2013 AFA meetings in San Diego. I am grateful to Dimitri Vayanos and Denis Gromb for supervision and insights. Jay Im provided excellent research assistance. This paper is the first part of a paper previously circulated as “Strategic Arbitrage with Entry”. All remaining errors are my own. †Department of Finance, Frankfurt School of Finance and Management, Sonnemannstrasse 9-11, 60314 Frankfurt am Main, Germany. Email: [email protected].

1 1 Introduction

Empirical evidence shows that prices recover slowly from demand or supply shocks that are unrelated to future earnings and that these patterns are due to imperfections in the supply of capital.

Patterns of slow price adjustments due to these sluggish capital responses have been documented, for instance, in the catastrophe reinsurance markets (Froot and O’Connell, 1999), and in numerous popular arbitrage strategies used by hedge funds such as convertible debt and merger arbitrage

(Mitchell, Pedersen and Pulvino, 2007).1 These price patterns present a challenge for models of eﬃcient capital markets based on competitive, rational investors, which predict that the supply of capital is perfectly elastic.

This paper proposes a theory based on arbitrageurs’ price impact to explain why capital appears to be slow-moving. There is widespread evidence that maket depth is limited, even in actively traded markets, and that investors develop strategies to minimize their price impact.2 Arbitrage in particular typically involves setting up positions in relatively illiquid markets that are often dominated by a few highly specialized sophisticated investors.3 I consider a model in which supply shocks and market segmentation generate price discrepancies between two identical assets. I show that when arbitrageurs recognize their price impact, these price discrepancies are not eliminated instantaneously. Strategic arbitrageurs break up their trades to minimize price impact and exploit price discrepancies longer, which results in gradual price convergence. The interesting feature of the model is that arbitrageurs’ price impact and the quantity they trade are both endogenous.

The model generates additional predictions which match well-documented empirical ﬁndings, such as V-shaped price patterns around anticipated (and uninformative) supply shocks, time- varying market depth, and changes in the sign of basis between identical or closely-related assets.

Moreover, the model is extended to analyze arbitrageur’s behaviour when the supply shocks are

1See Duﬃe (2010) for additional examples. 2See Chan and Lakonishok (1993, 1995) and additional references in Section 2. 3Anecdotal evidence reveals that some strategies can be dominated by a one or a handful of arbitrageurs. For instance, LTCM was nicknamed the central bank of volatility (Lowenstein, 2000) as it was selling large amounts of out-of-the-money put options and hedging in the underlying securities. Recently, JP Morgan was put under the spotlight after one of it’s traders in the Chief Investment Oﬃce accumulated large positions in the CDS markets. The trader, who was called the ’London Whale’ by the press and other market participants, was reportedly involved in yield curve arbitrage. See ‘London Whale’ Rattles Debt Market”, Wall Street Journal, April 6, 2012.

2 stochastic, i.e. when the arbitrage is risky, and study whether arbitrageurs “lean against the wind”.

I consider a multiperiod model in which arbitrageurs exploit price differences between two identical assets traded in segmented markets (e.g. Siamese twin stocks, a derivative and the equivalent security based on the underlying, etc.). In each local market, competitive investors receive endowment shocks that are correlated with the asset payoff. Correlations to the asset payoff are

(for simplicity) opposite across local investors, thus market segmentation causes prices to diverge.

Stronger hedging needs from local investors create a larger arbitrage “supply” and thus larger price differences. Arbitrageurs can trade in both markets and act as middlemen until assets mature and pay off the same dividend.4 Arbitrageurs are imperfectly competitive. While intermediating trades, they understand that they face downward-sloping demand curves in each market and take into account the impact of their trades on market-clearing prices. This setting with complete information is suitable to analzye cleanly the effects of uninformative supply shocks on price dynamics.5

If arbitrageurs take prices as given, the spread between the prices of the two assets drops instantaneously to zero. Since arbitrageurs are perfectly hedged across markets A and B, they are effectively risk-neutral, and eliminate any mispricing immediately, resulting in perfect risk-sharing and liquidity. By contrast, imperfectly competitive arbitrageurs intermediate only a limited amount between the two markets to keep the spread open and maximize capital gains. Further, as trades have a permanent effect on prices, arbitrageurs break up their trades to exploit the arbitrage as long as possible. When there is a constant arbitrage supply, arbitrageurs reallocate the asset slowly from one market to the other, giving rise to slow-moving capital. A similar mechanism implies that when unanticipated supply shocks occur at some stage, their effects on prices die out only gradually. In both cases, the equilibrium speed of arbitrage increases as more arbitrageurs compete in the trade.

This predictable price pattern arises in equilibrium even though all investors have perfect

4The assumptions of market segmentation and identical assets emphasize arbitrageurs’ role as intermediaries but are not essential for the results. 5Apart from hedging needs, supply shocks can arise from portfolio rebalancing of passive investors (e.g. index funds or ETFs) around index recompositions, forced liquidations of market participants following shocks to their portfolios (e.g. LTCM unwound its convertible arbitrage activity following losses on other convergence trades), etc. Gromb and Vayanos (2010) provide additional examples.

3 foresight. What induces local investors in, say, market A, to sell today given that in that market the price of the asset will (on average) increase tomorrow? Local investors are willing to sell today because waiting is risky. New information about the fundamental accrues between today and tomorrow and will aﬀect tomorrow’s actual price.6 Nevertheless, the anticipation of a more favorable price reduces local investors’ willingness to sell today. Hence local investors’ perfect foresight erodes arbitrageurs’ market power, as in the classical durable goods monopoly problem.

The slow-moving dynamics rely crucially on the discreteness of trading and the ﬁniteness of the horizon. The discreteness implies that receiving liquidity today is not exactly equivalent to receiving liquidity tomorrow, while the ﬁnite horizon provides arbitrageurs a means to commit to provide only limited liquidity in the future.

The model generates additional predictions, including V-shaped price eﬀects around the realization of anticipated shocks, time-varying market depth, and sign changes of price diﬀerences in case of negative decreasing supply. V-shaped price patterns have been documented around corporate or Treasury bond issuances, and around pre-announced index recompositions, which trigger shifts in uninformed demand for stocks (e.g. from passive investors such as index funds and ETFs), resulting in changes in the net supply.7 These patterns have proven hard to reconcile with rational behavior of market participants because price decreases and reversals are gradual and predictable.

This type of price patterns emerges in the model around the realization of pre-announced shocks, even though all market participants are rational. After the realization of the shocks, prices adjust gradually as in the case of surprise shocks. Indeed, arbitrageurs avoid providing any additional liquidity ahead of the shock, so that much of the shock absorption occurs after realization.

The anticipation of this behavior by local investors generates an increasing spread up to the realization. The striking feature of this result is that, between the announcement and the realization

6Vayanos and Woolley (2008) have a similar effect, which they term a bird-in-the-hand effect. 7Keloharju, Malkmaki, Nyborg and Rydqvist (2002) show that heavy funding needs of the Treasury cause V- shaped price patterns around new issuances in the Finnish market. Prices decline ahead of the issuance and recover afterwards. Lou, Yan and Zhang (2013) find similar secondary market price pressure effects around US Treasury issuances. Newman and Rierson (2003) show that bond issuance of one firm in the European telecom market temporarily raises yields of other firms in the sector. Index recompositions are either announced in advance or can be guessed with a high precision due to mechanical rules. See Lynch and Mendenhall (1997), Madhavan (2000), Greenwood (2005) and Boyer (2011) for evidence about the effects of index recompositions on prices. The V-shaped patterns can be observed for added and deleted stocks.

4 of the supply shock, the spread increases only gradually instead of instantaneously. As with unanticipated shocks, the discreteness of trading and time to maturity of the asset are important for the result. As time passes, two forces determine the spread dynamics. First, arbitrageurs’ trades accumulate, leading to a decreasing spread. Second, the opportunities to retrade vanish, so that local investors anticipate less and less liquidity provision, increasing the impact of future shocks on the spread. In the case of constant supply or surprise shocks, the first effect dominates and the spread decreases gradually in equilibrium. When shocks are pre-announced, the second effect dominates during the interim period between announcement and realization, leading to a peak of the effect at realization of the shock.

Time to maturity is also a key driver of market depth dynamics. Since the opportunities to further share risks become scarcer as time passes, local investors’ willingness to hold the asset decreases over time. This implies that - controlling for conditional dividend volatility, which decreases as more information accrues over time - price impact increases as time passes. This eﬀect is stronger when fewer arbitrageurs compete in the trade, since imperfect competition further limits risk-sharing opportunities. The non-stationarity of endogenous price impact stands in sharp contrast to the assumption of constant exogenous price impact that is usual in the literature on optimal order execution, showing that for large traders/ orders, it is important to consider jointly optimal trading strategies and market depth.8 Time-varying market depth is a feature of equity markets (Chordia, Roll and Subramanyam, 2000, 2001), Treasuries (Fleming, 1999) and corporate bonds (Bao, Pan, Wang, 2008).

In addition to gradual and V-shaped price adjustment, the model also predicts that the spread can change sign when the supply decreases in a predictable way, even if the supply does not change sign. This implies that assets which are undervalued for liquidity reasons, can become overvalued, i.e. prices overshoot the fundamental value. In other words, deterministic decreases in supply, instead of simply narrowing the spread also change its sign. This result may shed light on puzzling empirical evidence of such change in the sign of bases between closely related assets.

For instance, Bergstresser et al. (2010) ﬁnd that insured municipal bonds became cheaper than

8The examples of LTCM and JP Morgan given in footnote 3 are good illustrations of this result.

5 uninsured bonds of the same municipal with similar characteristics in the aftermath of the ﬁnancial crisis, i.e. precisely when liquidity needs were decreasing, which is one interpretation of the decrease in supply in the model.

Finally, I study arbitrageurs’ reactions to arbitrage risk. This analysis speaks to the old question of whether arbitrageurs “lean against the wind” (e.g. Friedman, 1953). When the future supply is stochastic, arbitrageurs’ positions increase in the size of the average future shock. This increase, however, does not reduce the spread because prices also reflect the anticipation of a larger shock, and this effect dominates. Models with financially constrained arbitrageurs (e.g. Shleifer and Vishny,

1997, Gromb and Vayanos, 2002) generate the same price effect, but for a different reason. In these models, the spread increases because arbitrageurs decrease their positions to save on capital for later. I also analyse arbitrageurs and price responses to increases in the volatility of future supply shocks, which is not analysed by the previous models. Since a higher shock volatility makes it more likely to face large price gaps, one could expect that arbitrageurs respond by trading more aggressively. I find that when the volatility of shock increases, arbitrageurs do not necessarily increase their positions, even if they are risk-neutral. This is because a larger shock volatility leads to a thinner market, inducing arbitrageurs to reduce their trades. These two results show that arbitrageurs’ willingness to lean against the wind may not only depend on their risk-aversion or trading capacity (determined, for instance, by inventory considerations, or financial constraints), as emphasized by the previous literature, but more broadly on the interaction between the market structure and the overall risk-bearing capacity of the market.

A number of empirical ﬁndings support or are consistent with the predictions of the model.

Some papers ﬁnd evidence that arbitrageurs’ market power is a driver of slow-moving capital and mispricings. Froot (2001) shows that supply-side stories and market power exerted by intermediaries are the most likely explanations of slow-moving capital in the catastrophe reinsurance markets.

Ruf (2012) studies the diﬀerence between implied and realized skewness in commodities options. He shows that the diﬀerence between the two is larger when traders hold more concentrated positions.

Others document more generally the explanatory role of market thinness (the inverse of market depth) for mispricings. For instance, Pelizzon et al. (2014) show that market thinness predicts the

6 bond-futures basis in the European sovereign bond market.

The paper relates to several strands of the literature. First, the paper belongs to the literature studying models where an oligolopoly of large (Cournot) traders trade with a competitive sector

(DeMarzo and Urosevic, 2006, Pritsker, 2009, Kihlstrom, 2000). My main contribution is to high- light new price effects caused by time variation in the asset supply, and to extend the analysis to risky asset supply.9 Pritsker’s paper is closest to this paper. He studies the effects of distressed sales on price dynamics, but does not consider random changes in asset supply. Further, arbitrageurs have heterogeneous levels of risk-aversion in Pritsker’s model, and price dynamics in the presence of distressed sales are obtained via numerical simulations. In this paper, the arbitrageurs are effectively risk-neutral when the supply is deterministic and risk-averse when the supply is stochastic.

This allows me to characterize the full dynamics of price adjustments in closed-form and to derive new comparative statics. Another related paper is Oehmke (2010), who studies the price eﬀects of large arbitrageurs in a dynamic setting, but models competitive investors as exogenouly given demand curves. Instead, market depth is endogenously time-varying in my model.10

Vayanos (1999), and more closely Rostek and Weretka (2011), study dynamic extensions of Kyle

(1989), where an oligopoly of large traders compete in demand schedules (and there is no competitive sector). Rostek and Weretka study the price effects of unanticipated and pre-announced supply shocks, but do not study random shocks. The main difference between our predictions concerns price dynamics. In the present paper, prices are affected by market illiquidity.11 This implies that unanticipated shocks generate gradually adjusting prices and that pre-announced shocks generate

V-shaped returns around the realization date, which are consistent with empirical evidence. By contrast, equilibrium prices coincide with competitive prices in the absence of shocks in Rostek and Weretka’s model (i.e. the market is illiquid, but there is no liquidity premia). As a result, pre-announced shocks aﬀect prices at announcement and realization but not in-between, as returns

9Kilhstrom studies a three-period model with fixed asset supply and focuses on the analogy between asset pricing with market power and the durable goods monopoly problem. DeMarzo and Urosevic extend Kihlstrom’s model to multiple periods and focus on a moral hazard problem, whereby the arbitrageur is a firm insider whose incentives to exert effort depends on his asset holdings. 10Zigrand (2004) considers a static setting and focuses on strategic arbitrageurs’ entry decisions. 11This feature corresponds to empirical findings. See for instance, Amihud and Mendelson (1986, 2002) and Acharya and Pedersen (2005).

7 coincide with competitive returns in the interim period. It is the interaction between the market illiquidity and the uncertainty about the fundamental value (which decreases as time passes in my model, whereas it is realized in one go in the ﬁnal period in Rostek and Weretka’s setting) that generates liquidity premia in the present model. Thus the comparison of our results shows that liquidity aﬀects prices when trading and the realization of dividend news take place simultaneously.

Previous research has shown that gradual price recovery after shocks can result from ﬁnancial constraints on arbitrageurs or search frictions to contact intermediaries (Duﬃe and Strulovici,

2012).12 Our model oﬀers a complimentary mechanism and delivers diﬀerent predictions. It predicts that these gradual adjustments can be observed not only in bad times (that is, when constraints bind), and that they are slower in more concentrated markets.13

Few models with rational investors generate V-shaped price patterns. That is, most models generate either gradual price declines, or gradual reversals, but not both “at the same time”.

Vayanos and Woolley (2013) and Albuquerque and Miao (2014) are two notable exceptions. Vayanos and Woolley consider a model of delegated investment where gradual ﬂows in and out of active funds generate momentum and reversals. Albuquerque and Miao obtain momentum and reversal in a model where some investors have advance information about future earnings that is unrelated to current earings. However, as noted above, gradual price declines and reversals occur also around uninformative supply shocks. This paper shows that gradual declines and reverals can occur also in the absence of asymmetric information and even if arbitrageurs are not subject to short-term outﬂows (e.g. thanks to lock-up periods).

The paper proceeds as follows. In section 2, I outline the three-period baseline model. Many of the results can be derived in this framework with deterministic supply shocks, as shown in Section

3. I extend the analysis to risky arbitrage in Section 4. Section 5 considers the T -period model and analyse surprise and pre-announced shocks. Section 6 summarizes the empirical predictions and

12See for instance Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009) for an analysis of ﬁnancially- constrained arbitrage. 13Of course, binding constraints or arbitrageurs’ defaults can lead to more market concentration in bad times, and search frictions may be lower in concentrated markets. However, Duﬃe and Strulovici (2012) show that more competition among intermediaries can reduce the speed of price adjustment when search frictions are present.

8 concludes. All proofs of Sections 3 and 4 are in the appendix. The proofs of Section 5 are available in the supplementary appendix.

2 Baseline model

In this section, I introduce the three-period baseline model. I consider an economy in which imperfectly competitive arbitrageurs exploit price diﬀerences between two identical assets traded in segmented markets.

2.1 Set-up

Assets and timeline. The economy has three periods (t = 0, 1, 2), and consists of two identical risky assets (A and B) and a risk-free asset. The risky assets are in zero net supply and pay oﬀ a liquidating dividend at time 2,

D2 = D + 1 + 2,

2 where t are i.i.d. normal variables with mean 0 and variance σ . The innovations t are realized and revealed to all market participants at time t = 1, 2. I denote Dt = Et (D2), the conditional expected value of the dividend. The risk-free asset is in perfectly elastic supply and its return r is normalized to 0. Trading takes places at time 0 and time 1, and consumption at time 2.

Local investors. The economy is made of two types of investors: competitive local investors and imperfectly competitive arbitrageurs. There is continuum of local investors in markets A and B, represented by a competitive agent with CARA utility with absolute risk-aversion coeﬃcient a in

9 each market:14

k k u C2 = − exp −aC2 , k = A, B

Local investors in market k can trade only the k-risky asset and the risk-free asset. Thus, markets A and B are segmented from the point of view of local investors. Market segmentation can stem from institutional constraints or behavioral biases. For instance, mutual funds are usually restricted by their mandates to trade in the cash market (e.g. bonds), while other market participants such as banks or insurance companies might prefer to use derivatives (e.g. CDS). Siamese stocks trade on diﬀerent exchanges and local investors may incur costs in the form of taxes, agency frictions or behavioral costs inducing them to trade in their domestic exchange (Froot and Dabora, 1999).15

Market segmentation limits valuable risk-sharing between local investors, as follows. Local investors in each market receive endowment shocks that are correlated with the payoﬀ of their risky

k asset. Speciﬁcally, investors in market k receive a shock st−1t at time t = 1, 2. st−1 represents the magnitude of local investors’ exposure to dividend risk and will determine local investors’

A hedging demand in period t − 1. For instance, if investors in market A have exposure s0 > 0, they will be willing to sell the asset at time 0 to insure against their endowment risk. Notice that

k although the endowment shock is realized at time t, the exposure st−1 is known (at the latest – A B see next paragraph) at time t − 1. Shocks are opposite across markets: st−1 = −st−1 = st−1, so that local investors could perfectly share risk by trading with each other. However, market segmentation prevents direct trading between local investors and creates a role for arbitrageurs.

Indeed, arbitrageurs can buy from investors with low valuation and sell to investors with high valuation to capture the price diﬀerence (spread).

In the baseline case, all investors know in advance the values of s0 and s1. Thus there is 14The assumption of a representative agent for competitive investors is standard in the finance literature on dynamic markets with large traders (see DeMarzo and Urosevic (2006), Pritsker (2009), Kihlstrom (2000)) but is not without consequences for the analysis. Under this assumption, only unilateral deviations by strategic arbitrageurs affect the outcome of the game. Gul, Sonnenschein and Wilson (1985) consider a game between a monopolist seller and non- atomic buyers but impose regularity conditions on buyers’ strategies that imply that unilateral deviations by buyers do not affect other buyers’ actions nor the monopolist’s actions. (See also Ritzberger (2002)). 15One can also think of local investors as hedgers with opposite hedging needs. See Gromb and Vayanos (2010) for additional examples of supply or demand shocks and market segmentation.

10 no uncertainty about the spread and arbitrageurs face a textbook arbitrage opportunity, which

A B disappears at time 2 when assets A and B pay oﬀ (p2 = p2 = D2). In Section 4, I extend the baseline model to the case where investors know only the distribution of s1 at time 0. In this case, arbitrageurs face the risk that the spread increases or decreases at time 1. This case is called risky arbitrage.

At time 2, local investors consume their entire wealth:

k k k k C2 = W2 = Y2 D2 + E2 , k = A, B

k k where Y2 denote the position in the risky asset and E2 the position in the risk-free asset. I denote k k k k k yt the time t trade in risky asset k and pt its price. The law of motion of positions is: Yt = Yt−1+yt k k k k k 16 for the risky asset and Et = Et−1 − yt pt + st−1t, for the risk-free asset, for k = A, B. Local investors solve the following problem:

h ki for k = A, B, max E u C2 (1) Y k ( t )t=0,1

k k k s.t. C2 = Y2 D2 + E2

k k k Yt = Yt−1 + yt

k k k k k Et = Et−1 − yt pt + st−1t

Arbitrageurs. Arbitrageurs can trade all assets without restriction. Thus they can intermediate trades by buying from investors with low valuation (say, in market A) and selling to investors with high valuation (in market B). Doing so, arbitrageurs will contribute to integrate markets A and B and provide liquidity to local investors. The main assumption is that arbitrageurs recognize the price impact of their trades.

Price impact costs are a ﬁrst-order concern for sophisticated investors seeking to proﬁt from

16In the three-period model, there is no trading at time 2, and no endowment sock at time 0, implying that

k k k k k k k k k E2 = E1 + s12,E1 = E0 − y1 p1 + s01,E0 = E−1 − y0 p0

k Since local investors have CARA preferences, we can set their initial endowment E−1 = 0 without loss of generality.

11 misvaluations such as hedge funds. Given that mispricings are often caused by some unmet liquidity demand, limited liquidity is a natural issue for arbitrageurs. A well-known example is that of LTCM: many of the hedge fund’s trades involved betting on assets with identical or closely related payoﬀs to exploit price discrepancies. However, as Perold (1999) and Lowenstein (2000) report, as the fund size increased, the impact of LTCM’s trades on prices were limiting the proﬁtability of it’s strategies. 17

Price impact costs are likely to be a major issue in trades where only a few arbitrageurs are active or in specialized markets (Attari and Mello, 2002). For instance, convertible arbitrage involves buying illiquid convertible debt to exploit price diﬀerences with the synthetic asset made of the underlying securities. Further, there is a large body evidence showing that even very active markets such as the equity, Treasury and currency markets of developed markets may oﬀer limited depth.18

Price impact is modeled via a Cournot game of complete information among arbitrageurs.

There are n imperfectly competitive arbitrageurs indexed by i (i = 1, . . . , n < ∞), each endowed with CARA utility with absolute risk-aversion coeﬃcient b:

i i U C2 = − exp −bC2 , i = 1, ..., n

k Arbitrageurs choose trades in the risky asset xt , for k = A, B, to maximize their expected utility subject to the price schedules in each market. The price schedules are derived from local investors’ inverted demand schedules, and imposing market-clearing:

n k X i,k Yt + Xt = 0, k = A, B, t = 0, 1 (2) i=1

17These costs became also exacerbated LTCM’s problems when funding tightened after the Russian default. Other examples include Amaranth, who became a dominant player in the gas market, and more recently JP Morgan’s Chief Invesment Office, where one trader was nicknamed the “London whale” because of the sheer size of his positions in the CDS market. 18Chan and Lakonishok (1993, 1995) show how institutional investors respond to limited depth by breaking up trades. Chen, Stanzl and Watanabe (2002) document high price impact costs in value and momentum strategies. Brandt and Kavajecz (2004) show that the order-flow has a permanent impact on US Treasury yields. In FX markets, changes in the order-flow accounts for short term price movements (Evans and Lyons, 2002).

12 The price schedules map the eﬀect of arbitrageurs’ trades into the price in each market. That is, a price schedule represent the market-clearing price at which the competitive fringe of local investors in each market is ready to trade all possible quantities submitted by arbitrageurs. In other words, arbitrageurs observe local investors’ demands and choose their trades accordingly subject to the constraint that markets clear. An important feature of the model is that price impact will be determined as part of the equilibrium.

i Arbitrageurs’ ﬁnal wealth W2 is equal to:

i i X i,k W2 = B2 + X2 D2, i = 1, . . . , n, (3) k=A,B

i,k i where Xt and Bt denote arbitrageur i’s position in the risky asset k and risk-free asset, respectively. i,k i,k i,k i,k The arbitrageur’s position in asset k evolves as follows: Xt = Xt−1 + xt , where xt is the trade i i P i,k k 19 of the period. The position in the risk-free asset evolves as: Bt = Bt−1 − k=A,B xt pt . For simplicity, arbitrageurs have no endowment in the risky asset (and given the zero net supply, so do local investors). Given the symmetry of the model, it will be optimal for arbitrageurs to hold

20 i,A i,B i opposite positions across markets. Without loss of generality, I set xt = −xt = xt, t = 0, 1. Given that assets A and B are both in zero net supply, this implies that arbitrageurs do not bear any aggregate risk.

Remarks. The key ingredient of the model is that arbitrageurs are not price-takers. The assumptions that both assets are exactly identical, that markets are segmented and that local investors incur perfectly negatively correlated endowment shocks are for simplicity only. In fact, the model is equivalent to a model where i) there is only one group of local investors (representing both markets) initially owning the arbitrage asset, ii) arbitrageurs are risk-neutral, and iii) the arbitrage asset supply is 2st at time t = 0, 1. For this reason, I refer to st as the time-t supply. When there are two assets in segmented markets, what matters is that there is some correlation between the payoﬀs of assets A and B to create a role for arbitrageurs. Similary, the symmetry of st−1 is not

19 k k Since there is no trading at time 2, B2 = B1, and X2 = X1 . 20In a model with exogenous prices, Liu and Timmerman (2013) show that the optimality of oﬀsetting positions obtain only when assets are identical and with symmetric divergence from fair value, which is the case here.

13 crucial. There simply needs to be gains from trade between local investors for arbitrageurs to have a role in the model. The current setting with two identical assets highlights the arbitrageurs or

ﬁnancial intermediaries’ role in (slowly) moving capital across markets.

2.2 Competitive benchmark

It is useful to ﬁrst consider the competitive benchmark of the baseline model. Since arbitrageurs can perfectly hedge across markets, they face no fundamental risk, and are thus the natural holders of the asset supply 2st. Hence, in equilibrium, arbitrageurs hold the supply and bring prices in line with each other. That is, arbitrageurs’ positions in market A are (positions in market B are symmetric):

n n n X i X i X i x0 = s0, x1 = s1 − s0, i.e. X1 = s1 i=1 i=1 i=1

∆0 = ∆1 = ∆2 = 0,

At time 0, arbitrageurs absorb the asset excess supply generated by local investors’ endowment shock s0. At time 1, as a new shock induces local investors to adjust their hedge, arbitrageurs absorb the change in supply s1 − s0, eventually holding the asset supply of the ﬁnal date, s1.

3 Risk-free arbitrage with imperfectly competitive arbitrageurs

In this section, I solve for local investors’ and arbitrageurs’ equilibrium strategies when the arbitrage is risk-free and arbitrageurs are imperfectly competitive. Due to imperfect competition, arbitrage is limited, and markets are no longer perfectly liquid. After deriving the equilibrium in the general case with arbitrary s0 and s1, I focus on the price eﬀects in the case of constant and decreasing supply. The case of increasing supply is treated in the extended model of Section 5. Finally, I review the implications of imperfect competition for the dynamics of price impact and optimal order execution.

14 3.1 Price schedules and equilibrium

Price schedules. Here I assume that s0 and s1 are positive shocks and are known in advance by all market participants. I ﬁrst derive the price schedules faced by arbitrageurs at time 1. In the standard CARA-normal framework, local investors’ demand in market A is:

(D ) − pA Y A = E1 2 1 − s (4) 1 aσ2 1

Local investors in market A experience a positive shock s1, which reduces their demand for asset

B A. In market B, local investors have similar demand functions (in p1 ), except that they experience an opposite shock, increasing their demand for asset B. Using the assumption of opposite positions in markets A and B, and imposing market-clearing (2), gives the schedule for the arbitrage spread,

Pn i B Pn i A Pn i ∆1 i=1 X1 = p1 i=1 X1 − p1 i=1 X1 :

" n # " n n # 2 X i 2 X i X i ∆1 (.) = 2aσ s1 − X1 = 2aσ s1 − x0 − x1 (5) i=1 i=1 i=1

i i i where I used the law of motion X1 = X0 + x1 and the assumption that arbitrageurs have no i i preexisting position in any of the risky assets, i.e. x0 = X0. The schedule has an intuitive form. 2 Pn i The ﬁrst component, 2aσ s1 − i=1 x0 is the spread that would prevail between assets A and B in the absence of trading at time 1. The residual excess supply faced by arbitrageurs at time 1

Pn i is s1 − i=1 x0 instead of s1, because their time 0 trades reduce local investors’s need to hedge. This is because shocks are correlated to the asset payoﬀ. Thus the positions set up at time 0 work as an imperfect hedge against the shock of time 1.

This means that there is some substitutability between insurance (liquidity) received from arbitrageurs at time 0 and that received at time 1. The fact that the liquidity received by local investors at time 0 “durably” reduces their hedging demand at time 1 erodes arbitrageurs’ market power, as in the classical durable goods monopoly problem. However, here, Coasian dynamics are limited by the fact that there is discrete trading and a ﬁxed horizon at which assets mature, which

15 provides a commitment for arbitrageurs to limit liquidity provision.21 22

∂∆1 The second component of (5) represents the impact of arbitrageurs’ trades at time 1, i = ∂x1 2aσ2, which depends on local investors’ risk-aversion and the risk of the fundamental.23 When they are more risk-averse, local investors are more reluctant to hold the risky asset, and thus will require larger price concessions at the margin, which results in a larger price impact.

Equilibrium strategies and spreads. To illustrate the strategic choice faced by arbitrageurs, note that because arbitrageurs set up opposite positions, their objective at time 1 boils down to

i maximizing the trading proﬁt, x1∆1 (.):

n ! i 2 i X i X −i i max x1∆1 (.) = max 2aσ x1 s1 − x0 − x1 − x1 (6) xi xi 1 1 i=1 −i

i ∆1 (.) is given by (5) and depends not only on arbitrageur i’s trade, x1, but also all other arbi- P −i P −i i Pn i trageurs’ trades −i x1 , with −i x1 + x1 = i=1 x1, and on the positions established at time P i 0, i x0. Since price impact is permanent, arbitrageurs’ trades have a cumulative eﬀect on prices. Hence at time 0, arbitrageurs take into account the dynamic impact on the spread of their own

21Bulow (1982), Stockey (1981) and Kahn (1986) provide a formal analysis of the Coase (1972) conjecture. Stockey shows that the conjecture obtains when the time between two sale periods shrinks to zero. Bulow and Stockey show that a monopolist able to commit to a specific price path captures monopoly rents and discuss the role of time discreteness as a form of limited commitment. Kahn shows that the conjecture holds only in the case of constant marginal costs. In this paper, marginal costs are constant since arbitrageurs are perfectly hedged across markets. Thus the first difference of this paper with the classic durable goods problem is that the asset matures at a fixed horizon. This limits the durability of the “liquidity good” sold by arbitrageurs. The second difference is that trading occurs discretely. When financial assets are infinitely-lived and trading occurs very frequently, the conjecture holds true. This is shown by DeMarzo and Urosevic (2006) who prove the Coase conjecture in a continuous time model with a Cournot trader selling an infinitely-lived asset with shrinking time between trading periods. Vayanos (1999) also proves the conjecture in an infinite horizon market with shrinking time where trading is organized as a Walrasian auctions between n imperfectly competitive traders. See also Kihlstrom (2000), Pritsker (2009), and Edelstein, Sureda-Gomilla, Urosevic, and Wonder (2010). 22 One might infer from the previous example that the fact that s0 and s1 are both positive, i.e. that local investors’ hedging needs are positively correlated over time, is important. However, this is not the case, and the case of positively correlated shocks over time is just for simplicity of the intuition. For instance, suppose that s0 ≥ 0 and s1 ≤ 0. Then, in market A, local investors would like to buy the risky asset and then sell it. If local investors anticipate a high price at t = 1, this decreases their willingness to hedge at t = 0, since they anticipate that high profits at t = 1 will compensate them for potential losses at time 0. Thus, arbitrageurs compete with themselves over time also in this case, and the crucial element is that they cannot refrain from trading a second time at time 1. In fact, a simple exercise along the lines of Proposition 1 shows that if a monopolistic arbitrageur can commit to specific prices (i.e. s0 s1−s0 choose x0 and x1 simultaneously at t = 0, his optimal trades are x0 = 2 and x1 = 2 . Thus, with constant supply s0 = s1 = s, the monopolist able to commit trades only at t = 0 and earns monopoly profits. 23I take the absolute value of the derivative as it is more intuitive to compare positive numbers.

16 trades as well as of other arbitrageurs’ trades. The next result characterizes equilibrium trades and prices.

Proposition 1 For given s0, s1, there is a unique equilibrium in which arbitrageurs’ trades in market A are:

i 1 n − 1 x0 = x0 = s0 + 2 s1 (7) φn (n + 1) φn i n ¯ x1 = x1 = − s0 + φns1, (8) (n + 1) φn

The equilibrium spread is:

2 ∆0 = 2aσ ψns0 + ψ¯ns1 (9) 2 n ∆1 = 2aσ − s0 + φ¯ns1 (10) (n + 1) φn

n3+4n2+3n+2 ¯ 1 n(n−1) n2+n+2 ¯ 3n+2 with φn = 2 , φn = − 3 , ψn = 3 2 , ψn = 3 2 (n+1) n+1 (n+1) φn n +4n +3n+2 n +4n +3n+2

The proposition shows how imperfect competition aﬀects liquidity provision and equilibrium prices.

The trades and spread are not only functions of st but also of the number of arbitrageurs n, which determines the degree of competition. The equilibrium is symmetric: arbitrageurs have identical preferences and each arbitrageur has the same impact on the price. Thus, all arbitrageurs play a symmetric role, and trade the same quantity in equilibrium. At time 0, arbitrageurs no longer fully absorb the supply s : as long as n is ﬁnite, s0 < s . Arbitrageurs’ time-0 trades now also depend 0 φn 0 on s1 (provided n > 1, see remark below). Arbitrageurs anticipate that the change in supply will generate additional trade and are wary of their price impact if they absorb the change in one shot at time 1. Thus, in contrast to the competitive benchmark, arbitrageurs adjust their portfolio prior to and at the realization of the shock. The limited liquidity provision typically generates a non-zero spread at both dates. The dynamics of the spread are particularly interesting in two special cases:

ﬁrst, the case of constant supply (s0 = s1 = s, with s > 0, to ﬁx ideas), and second, the case of decreasing supply (s0 ≥ s1 ≥ 0).

17 3.2 Gradual arbitrage with constant supply (s0 = s1 = s > 0)

Lemma 1 When s0 = s1 = s > 0, the equilibrium is as follows:

• Each arbitrageur buys a fraction of the supply in each period:

i i x0 = κ0,ns, x1 = κ1,ns, where ∀n ≥ 1, κ0,n ∈ ]0, 1[ and κ1,n ∈ ]0, 1[

• Arbitrageurs increase their individual and aggregate positions over time, but do not fully

absorb the supply in each period:

n n i i X i X i x0 < x1 and X0 < X1 < s i=1 i=1

• The spread is positive at time 0 and 1 and decreases over time:

2 2 ∆0 = 2aσ κ¯0,ns > ∆1 = 2aσ κ¯1,ns > ∆2 = 0

The intuition is straightforward. Arbitrageurs break up their trades to limit price impact. Hence their positions increase gradually, resulting in gradual convergence of prices towards the fundamental. The main driver of the price dynamics is the degree of competition among arbitrageurs:

Corollary 1 Suppose s0 = s1 = s, then

∂∆t • the spread decreases with the number of arbitrageurs at time 0 and time 1: ∂n < 0, t = 0, 1, h i ∂ ∆1−∆0 ∆0 • and it decreases faster as n increases: ∂n < 0.

• When n → ∞, the arbitrageurs absorb the entire supply at time 0 and time 1, and the spread

24 converges to zero: limn→∞ ∆t = 0, t = 0, 1.

24 Note that this result does not depend on the assumption s0 = s1 = s.

18 When competition increases, each arbitrageurs buys (sells) a smaller amount in market A (B).

However the aggregate quantity traded in equilibrium increases, as Figure 1 shows. In the limit, arbitrageurs fully intermediate trades between A- and B-investors and the equilibrium spread converges to zero at all dates, as in the competitive case.

These results extend Oehmke (2010)’s results about “gradual arbitrage” to a setting where all investors are rational. In Ohemke’s model, arbitrageurs trade against an exogenously given price schedule, while the price schedule is endogenous here. Hence, my results show that gradual arbitrage is consistent with rational behavior from both sides of the market, which was not obvious with exogenous price schedules.

Consider for instance market A, where local investors hedge by selling the asset. The price of asset A increases on average between time 0 and time 1, leading to a negative expected return on the hedge. Local investors in market A are willing to sell at time 0 in spite of this because of risk-aversion. Waiting until time 1 is risky since new information on the asset accrues at time

1 and will be reflected in the price. Nevertheless, future prices affect local investors’ demand for liqudity today, i.e. the anticipation of a higher price tomorrow reduces local investors’ willingess to sell today. As arbitrageurs cannot commit not to provide further liquidity at time 1, they are competing with themselves over time as in the classical durable good monopoly problem. The extent of the problem is limited by the discreteness of trading and the fixed horizon. Local investors face price risk between the time elapsing between date 0 and date 1. Further, the fixed horizon allows arbitrageurs to commit to provide limited liquidity at time 1. (i.e. the fixed horizon represents an imperfect commitment device, as arbitrageurs do provide liquidity but in limited quantity).

The latter suggests that slow-moving capital price dynamics should affect more assets with fixed maturity (e.g. fixed income vs stocks), and arbitrage strategies with a known convergence date.25

This mechanism can account for the observed slow reversal of prices towards fundamentals following shocks documented, for instance, by Mitchell, Pulvino and Staﬀord (2002) in the merger

25However, note that the model does not only apply to assets with ﬁxed maturity. Indeed, date 2 could be seen as the date at which the markets becomes perfectly liquid again. DeMarzo and Urosevic (2007) and Pritsker (2009) consider an inﬁnitely-lived asset but restrict investors to trade it during T periods, and simply consume dividends thereafter. Alternatively, the model may describe the periods before dividend payments.

19 and convertible arbitrage markets, and Coval and Stafford (2007) in the equity market. The mechanism differs from and complements Duffie and Strulovici (2012), who consider a related model where intermediaries can search for local investors to move capital across markets. In Duffie and

Strulovici’s paper, capital ﬂows slowly because ﬁnding new investors takes time. Here we can think of date 2 as a shortcut for the arrival of fresh capital. The paper shows that even some investors

(arbitrageurs) are ready to invest before time 2, they have no incentive to do so aggressively as this would exhaust their rents too quickly.

3.3 Sign change of the spread with decreasing supply (s0 ≥ 0, s1 ≥ 0)

A positive st implies that A-investors should value the asset less than B-investors. Thus, in the absence of perfect risk-sharing, ∆t should be positive. However, if the supply decreases suﬃciently over time, the time-1 spread can become negative, even though s0 and s1 are positive.

Corollary 2 Suppose that s0 and s1 are positive.

∂∆0 • At time 0, the spread is always positive and decreases with the number of arbitrageurs: ∂n < 0.

• At time 1, the spread is negative if and only if s1 is small enough relative to s0: ∆1 ≤ 0 ⇔

s1 ≤ αns0, with 0 < αn < 1.

To understand more precisely what the change of sign means, it is useful to think about the evolution of prices within one market. For instance, in market A, where local investors have a low valuation for the asset, the change of sign of the spread means that the price of the risky asset is lower than the fundamental value E0(D2) at time 0, and higher than E1(D2) at time 1. The 1 liquidity discount on asset A is, by construction, 2 ∆t at time t. A large supply st implies a large liquidity discount. As the supply decreases over time, one expects the discount to be reduced.

However, the result shows that the discount becomes a premium. In other words, the change of sign of the spread means that the the price of asset A overshoots its fundamental value. (Figure 2)

The intuition of the mechanism is simple. The negative supply shock implies that arbitrageurs sell the asset in market A at time 1. To make a proﬁt on this sale, it must be that the expected

20 capital gain is negative, i.e. that the price overshoots the fundamental value at time 1. To illustrate the mechanism, consider an example in which s0 > 0 and s1 = 0. In this case, local investors in market A initially short the asset, receiving partial insurance from arbitrageurs against the positive supply shock. At time 1, since there is no reason to hedge anymore (s1 = 0), local investors seek to

A close their hedge by buying back the asset (indeed, y1 > 0). Arbitrageurs take the other side of the A A trade, i.e. they sell the asset (x1 = −y1 < 0). To make a proﬁt on this trade, it thus must be that A A A A A x1 E(p2 ) − p1 > 0, i.e. that the capital gain is negative, that is, E(p2 ) < p1 , which is equivalent to ∆1 < 0. Hence, given the direction of local investors’ hedging trade, arbitrageurs maximize proﬁts by setting a price above the fundamental value. This explains why the price overshoots.

We can verify that local investors’ demand is optimal given these prices. Since liquidity is limited,

A local investors cannot fully close their short position and remain short: Y1 < 0, as one can see by A setting s1 = 0 and substituting p1 for its equilibrium price in equation (4). This is optimal since 26 the price of asset A will (on average) drop at time 2. As Lemma 2 shows, s1 does not have to be zero, but small enough relative to s0. Intuitively, the need to revert the hedge must simply be large enough so that arbitrageurs must sell the asset at time 1, leading to the price overshoot.

A decrease in st represents a reduction in local investors’ liquidity needs. Hence the model predicts that spreads or bases should change sign following periods of low liquidity (or equivalently large price divergence). Interestingly, it is when asset prices should get closer to their fundamental value that arbitrageurs cause a breakdown of the intuitive relationship between A-and B-asset prices. What causes this breakdown is that arbitrageurs limit liquidity both when local investors need to sell and to buy.

The result shows that decrease in supply can generate counterintuively negative “bases” between closely-related assets. It may shed light on some recently observed price patterns. Several standard and intuitive relationships between pairs of assets broke down in the aftermath of the

2007-2009 ﬁnancial crisis. For instance, the 7-and 10-year swap spread turned negative for the ﬁrst time in 2010 (Business Week, 23/03/2010). Uninsured municipal bonds became more expensive

26Arbitrageurs are not subject to this eﬀect since they are not exposed to market risk, taking opposite positions across markets.

21 than similar insured bonds issued by the same city also in 2010 (Bergstresser et al., 2011). Concerns about monoline insurers may reduce the premium attached to insured bonds to zero. However, it is hard to see how it could generate a negative premium. The mechanism of the model relies on imperfect competition and decrease in supply. The timing is consistent with the model prediction that spreads change sign when liquidity needs from constrained investors decrease.27

Monopolistic vs oligopolistic arbitrage. Proposition 1 shows that the time-0 trade (7) depends on both s0 and s1, unless there is a single arbitrageur. In this special case, the arbitrageur’s trade does not depend on s1, although the spread does. x0 does not depend on s1 because s1 affects the marginal cost and the marginal benefit of increasing the time 0 trade by one unit in the same way. One can see this by setting n = 1 in the first order condition of an arbitrageur’s problem at time 0 – See equation (27) in the Appendix. The shock s1 nevertheless affects the time-0 spread since local investors anticipate that this shock will not be fully absorb due to the imperfect liquidity of the market. I discuss this effect further in Section 5.3.

3.4 Liquidity implications

Several papers in the literature focus on arbitrageurs’ trading strategies given an exogenously given price schedule with constant price impact coeﬃcient.(e.g. Carlin, Sousa-Lobo and Viswanathan,

2007, Brunnermeier and Pedersen, 2005).28 When price impact is derived as part of the equilibrium, it is time-varying, which in turn aﬀects the arbitrageurs’ optimal trading strategies.

Time-varying price impact. With endogenous price schedules, the arbitrageurs’ price impact is no longer constant over time. It decreases as time passes and depends on the market structure:29

Corollary 3 Price impact decreases over time, even more so if the market is concentrated (n

27Inflation-protected Treasuries also became cheaper than similar nominal bonds (Pflueger and Viceira, 2011). Pflueger and Viceira show that the negative breakeven inflation in the Treasury market can be attributed to a larger liquidity discount for the TIPS and not to deflation expectations. 28As some of these models are framed in continuous time, there is also a temporary price impact component that helps pin down the equilibrium speed of trading. 29This result is general and does not depend on the assumption that shocks are constant over time, or that shocks are known in advance.

22 small):

∂∆1 2 • At time 1, arbitrageurs’ price impact is i = 2aσ (i = 1, . . . , n) ∂x1

• At time 0, the equilibrium spread schedule is

" # s1 n + 2 X ∆ (Q ) = 2aσ2 s + − xi , (11) 0 0 0 (n + 1) n + 1 0 i

∂∆0 2 n+2 ∂∆1 i.e. arbitrageurs’ price impact is i = 2aσ n+1 > i . ∂x0 ∂x1

Two opposite effects determine the evolution of price impact over time. First, given that new information accrues over time, the conditional variance of the asset payoff is decreasing over time as uncertainty realizes. This implies that local investors in each market are “more risk-averse” at time 0 than at time 1. Since the variance of each innovation t is constant over time, price impact should be twice as large at time 0 than at time 1. This is not the case, however, because a second effect tends to reduce price impact.30 As local investors anticipate that arbitrageurs will provide further liquidity at time 1, they understand that they will have another trading opportunity to share risk, and this reduces their effective level of risk-aversion ex-ante. Said differently, local investors are less desperate to receive liquidity if they anticipate that more liquidity is coming later

2 n+2 on. The more concentrated the market is, however, the more rationed liquidity will be (aσ n+1 is maximal for n = 1), and therefore price impact is higher at time 0 if the market is concentrated.

Note that at time 1, arbitrageurs’ price impact depends only on risk-aversion and not on the market structure. This is because at time 2, the asset pays oﬀ, which is equivalent to restoring perfect liquidity in the market. If the market was perfectly competitive also at time 1, the market structure adjustment of time 0 price impact would disappear, and price impact would be constant: when

2 n+2 2 n → ∞, aσ n+1 → aσ .

Market power has a similar eﬀect on price impact dynamics in Rostek and Weretka (2010), who study a setting with n strategic arbitrageurs and no competitive fringe. Their model, however,

30 2 n+2 2 It is easy to see that, indeed for any n ≥ 1, 2aσ n+1 < 4aσ .

23 predicts that price impact should increase over time, because only the second eﬀect, stemming from the opportunity to retrade and diversify risk further in the future, is present.

Optimal execution with endogenous market depth. An interesting implication of the time- varying price impact is that a monopolistic arbitrageur does not equally split his trade across periods

2 3 (even if s0 = s1 = s): for n = 1, κ0,1 = 5 > κ1,1 = 10 , i.e. x0 > x1. (More generally, for an arbitrary number of arbitrageurs, x0 > x1) This is a key diﬀerence with the literature on optimal execution of large orders (e.g. Bertsimas and Lo, 1998), which shows that with constant price impact, it is optimal for a monopolistic trader to break up orders equally over time. Thus the model highlights that in concentrated markets, optimal order execution and market depth are jointly determined and depend on the deep characteristics of the market, such as investors’ risk-aversion, asset volatility and the market structure.

4 Risky arbitrage

In this section, I assume that s1 is not known at time 0. Investors only know that it is normally dis-

2 tributed with means ¯1 and variance z1. I also make the following assumption about the parameters, which ensures that utility is bounded:

2 2 2 2 (n+1) Assumption 1 a σ z1 < 2n+1

Since the second shock is random from the point of view of time 0, the arbitrage is no longer risk-free. Therefore, even if arbitrageurs can eliminate all fundamental risk by taking opposite positions in assets A and B, they face uncertainty about the future proﬁtability of the arbitrage.

As in standard noise trader risk models, the potential deepening of the mispricing is short-lived, and the prices assets A and B converge at time 2 when the assets pay oﬀ. In the ﬁrst part of this section, I show how the risky arbitrage case generalizes the previous case and delve more deeply into the mechanisms. In the second part (sections 4.2 to 4.4), I discuss the results in relation to the literature on the (de)stabilizing role of arbitrageurs.

24 4.1 Price schedules and equilibrium

At time 1, the problem is not diﬀerent from the risk-free case. However at time 0, all investors face uncertainty about the magnitude of the future liquidity shock. I show in the appendix that at time

0, the spread schedule faced by arbitrageurs is the following:

" # s¯1 n + 2 X ∆ (.) = 2aσ2 s + − (1 + φ ) xi , (12) 0 0 (n + 1) r n + 1 a 0 a i 2 2 2 a σ z1 2 2 2 2n + 1 with φa = 2 and ra = 1 − a σ z1 2 (13) (n + 1) ra (n + 1)

There are two key differences with respect to the risk-free case, in which the schedule is given by h i 2 s1 n+2 P i equation (11), which I reproduce here for convenience: ∆0 (.) = 2aσ s0 + (n+1) − n+1 i x0 . The first part of the schedule, s + s¯1 , represents the price divergence that would prevail in 0 (n+1)ra h i equilibrium in the absence of trade. Given that r < 1, we have: s + s¯1 > s + s1 , a 0 (n+1)ra E0 0 (n+1) which captures the effect of convexity, as in Jensen’s inequality. The second part represents arbitrageurs’ price impact. It increases by a factor 1 + φa > 1 relative to the risk-free case. The

2 2 increase is larger if the volatility of the liquidity shock z1, fundamental volatility σ , or risk aver- n+2 sion a is large. The eﬀect of the market structure, captured by the term n+1 , is ampliﬁed by the uncertainty about future liquidity shocks. Price impact increases because local investors require larger discounts to hold their risky asset when future shocks are random.

Uncertainty about future liquidity shocks also aﬀects arbitrageurs’ strategies at time 0, both through their own risk aversion and through the change in the price schedules. The diﬀerent channels appear clearly in their value function:

Proposition 2 At time 0, the arbitrageurs’ value function is given by

" P i 2 !!# − 1 x s¯ i 2 2 i ˆ i 0 1 X i J0 = max −rb exp −2baσ x0∆0 + (1 − φb) 2 − 2 2 x0 − s¯1 xi 0 (n + 1) (n + 1) rb i 2 2 2 2 ˆ ∆0 (.) 4abσ z1 4abσ z1 where ∆0 = 2 , rb = 1 + 2 and φb = 2 (14) 2aσ (n + 1) (n + 1) rb

25 The arbitrageurs’ value function is made of three components:

i ˆ ˆ 1. Their time-0 trading proﬁt 0, x0∆0, i.e. quantity times (normalized) price gap ∆.

2. The time-1 continuation proﬁt, in which we can distinguish two parts, depending on their

relation to risk aversion:

2 (P xi ) (a) The ﬁrst part, (1 − φ ) i 0 , is decreasing in arbitrageurs’ risk aversion b, and more b (n+1)2 2 generally in z1, σ2, and a. Hence I will refer to it as the precautionary (or hedging) mo-

tive. The coeﬃcient φb measures by how much arbitrageurs reduce their aggressiveness in tackling the arbitrage gap at time 0 for fear of facing too much risk at time 1. Note that

φb depends on the product of a and b because an increase in local investors’ risk-aversion makes them more reluctant to hold the risky asset and thus restricts arbitrageurs’ risk-

sharing opportunities. The hedging motive is, perhaps surprisingly, increasing in the

P i total size of previous trades, i x0. This is because trading aggressiveness at time 0 works as an indirect hedge against large shocks at time 1 by reducing the spread perma-

nently. The strength of the hedging motive also depends on the number of arbitrageurs.

An increase in competition has two conﬂicting eﬀects:

Corollary 4 At time 1, when the number of arbitrageurs increases, there is

1 ∂ 2 • a business-stealing eﬀect: (n+1) < 0, which reduces the coeﬃcient 1−φb , ∂n (n+1)2

∂(1−φb) • a co-insurance effect: ∂n > 0, which increases it. 1−φ ∂ b (n+1)2 The business-stealing effect always dominates the co-insurance effect, i.e. ∂n < 0.

The business-stealing eﬀect is the standard consequence of stronger competition in a

Cournot setting. The co-insurance eﬀect is positive because the total liquidity supply

increases, even though individually arbitrageurs scale down their positions.

(b) The second part of the time-1 continuation payoﬀ represents the “strategic motive”:

! s¯1 X i − 2 2 x0 − s¯1 (n + 1) rb i

26 It is increasing in arbitrageurs’ risk-aversion b and decreasing in previous trades, as

arbitrageurs have an incentive to strategically limit their positions at time 0 to be able

to fully exploit the arbitrage opportunity later. The key driver of the strategic motive

2 is the expected level of arbitrage risks ¯1 instead of the risk of the arbitrage risk z1, as explained in more details below. In sum, arbitrageurs face the following trade-oﬀ when

choosing their positions at time 0: large positions allow them to co-insure, but this may

reduce the proﬁtability of the trade in the next period.

Proposition 3 When arbitrage is risky, there is a unique equilibrium characterized by:

s¯1 2¯s1 s0 + − 2 i (n+1)ra (n+1) rb x0 = (15) φn + (n + 2) φa + 2nφb s − P xi xi = 1 i 0 (16) 1 n + 1

The equilibrium spread between asset B and asset A is

2 s¯1 2¯s1 ∆0 = 2aσ Φa s0 + + (1 − Φa) 2 , with Φa ∈ [0, 1] (17) (n + 1) ra (n + 1) rb 2 2aσ n n (n + 1) rb − 2ra ∆1 = − s0 + s1 − 2 s¯1 (18) n + 1 d (n + 1) drarb with d = φn + (n + 2) φa + 2nφb.

The time 1 subgame is similar to the risk-free case, thus I focus on time 0 where uncertainty about future shocks generates a number of interesting eﬀects. By comparing (7) and (15), one can see that arbitrageurs equilibrium trades generalize in a very intuitive way. Arbitrageurs buy

−1 s¯1 2¯s1 a fraction d of s0 + − 2 . The denominator, d = φn + (n + 2) φa + 2nφb, depends (n+1)ra (n+1) rb not only on the degree of competition but also on local investors’ and arbitrageurs’ risk-aversions.

Arbitrageurs buy a smaller fraction as their risk aversion b increases, due to precautionary concerns

(term in φb), and as local investors’ risk aversion a increases, because arbitrageurs have a larger price impact at time 0, which prompts them to scale back their trade (as captured by the coeﬃcient

φ ). The ﬁrst term of the numerator, s + s¯1 , is the spread that would prevail in the absence a 0 (n+1)ra of liquidity provision (the maximum spread). This maximum spread represents the demand for

27 2¯s1 liquidity addressed to arbitrageurs at time 0. The second term, − 2 , represents the impact of (n+1) rb dynamic considerations. I.e., arbitrageurs avoid buying too aggressively at time 0 to preserve the proﬁtability of the arbitrage at time 1. This eﬀect depends on the “strategic motive” highlighted in the analysis of arbitrageurs’ value function. The strategic motive depends on arbitrageurs’ risk-aversion, the degree of competition, and uncertainty about the future shock. Arbitrageurs

“speculate” more when the market is more concentrated (n small), the expected shocks ¯1 is large, and when they are more risk-tolerant. The strategic motive is highest when arbitrageurs become risk-neutral, since arbitrageurs no longer care about the hedging beneﬁt of the ﬁrst trade.

s¯1 2¯s1 s0 + 2 i (n+1)ra (n+1) when b → 0, x0 → − (19) φn + (n + 2) φa φn + (n + 2) φa

When uncertainty vanishes, we converge to the risk-free arbitrage case (assuming s1 =s ¯1):

s¯1 2¯s1 s0 + 2 i (n+1) (n+1) 1 n − 1 When z1 → 0, x0 → − = s0 + 2 s1 φn φn φn (n + 1) φn

One can see clearly here that when n = 1, the shock s1 influences the time 0 and time 1 profitability of the trade in exactly offsetting ways, as noted above.

Since the precautionary and the strategic motives have opposite dependence on arbitrageurs’ risk aversion, an increase in b has an ambiguous eﬀect.

4.2 Arbitrageurs’ risk aversion and liquidity

According to Friedman (1953), speculators reduce price volatility by smoothing out temporary price

ﬂuctuations. Given that this view implies a contrarian behaviour, it may seem desirable to have risk-loving arbitrageurs for markets to be eﬃcient. This is no longer the case when arbitrageurs have price impact: the spread between assets A and B may increase as arbitrageurs become risk- neutral. On one hand, a decrease in risk aversion decreases the precautionary motive. On the other hand, it increases the strategic motive.

Corollary 5 An increase in arbitrageurs’ risk-aversion may result in them providing more or less

28 liquidity at time 0. There are two opposite eﬀects:

 

i   ∂x0  2 s¯1 2n  = κ −n (n + 1) s0 + +s ¯1 d +  , κ > 0 ∂b  (n + 1) ra rb  | {z } | {z }  precautionary motive < 0 reduction in strategic motive > 0

The reduction in strategic motive dominates iﬀ s1 is large enough relative to s0:

i ∂x0 n ((n + 1) rb − 2ra) 2 ≥ 0 ⇔ s¯1 d − ≥ n (n + 1) s0 ∂b rarb

The following lemma shows a special case in which the strategic motive is so strong that a decrease in risk aversion does lead to a decrease in liquidity provision and higher spreads (and conversely, an increase in risk aversion leads to higher liquidity and lower spreads):

Lemma 2 Suppose that s0 → 0. If n ≤ 2 and local investors’ risk-aversion a is small enough

2 2 (or equivalently, σ or z1 small enough), then, following a small increase in their risk-aversion, arbitrageurs provide more liquidity, which decreases the time 0 spread. This eﬀect is stronger if they are not very risk-averse.

Unsurprisingly, the strategic motive dominates in a very concentrated market, and even more so if arbitrageurs are not too risk-averse. Note that if s0 is very small, on average, the spread will decrease between time 0 and time 1, implying a negative return. As b increases, arbitrageurs increase their trade at time 0, and this reduces the time 0 spread more than the time 1 spread, leading to a less negative return.

4.3 How do arbitrageurs respond to an increase in arbitrage risk?

In the presence of arbitrage risk, it is important to understand whether arbitrageurs’ reactions to changes in risk are stabilizing (i.e. leading to smaller spreads), or destabilizing. In the limits of arbitrage literature, it is common to study how positions and prices respond to an increase in

“noise trader risk” (Shleifer and Vishny, 1997), or demand pressures / supply imbalances (Gromb

29 and Vayanos, 2010, Brunnermeier and Pedersen, 2009). It is shown that arbitrageurs do not necessarily increase their positions ex-ante when they face larger future shocks, and this may push prices further away from their fundamental values. Here, I analyze arbitrageurs’ responses to an increase in the level of the future shock,s ¯1, and in the volatility of the shock z1. Surprisingly, the literature on limits of arbitrage has to the best of my knowledge focused only on the ﬁrst comparative static (dubbed noise trader risk).

Corollary 6 Following an increase in the expected shock s¯1, arbitrageurs increase their positions ∂xi at time 0, but the spread nevertheless increases: 0 ≥ 0 and ∂∆0 ≥ 0. ∂s¯1 ∂s¯1

The two parts of the result may seem contradictory, as one would expect the increase in arbitrageurs’ positions to lead to a smaller spread. It is not the case because an increase ins ¯1 also causes an increase in local investors’ liquidity demand, and arbitrageurs’ response, albeit positive, is not commensurate with local investors’ increased need for liquidity. This is in particular due to the fact that an increase ins ¯1 increases the profitability of the arbitrage but also arbitrageurs’ strategic motive. In models of financially-constrained or performance-based arbitrage, a similar increase in s¯1 leads to a larger spread, but for a different reason. There the larger shock induces arbitrageurs to reduce their positions to have more financial slack in the next period (e.g. Shleifer and Vishny,

1997, Brunnermeier and Pedersen, 2009).

Next, it is interesting to understand how arbitrageurs respond to increased uncertainty about the future profitability of the arbitrage. As one would expect, increased uncertainty reduces arbitrageurs’ strategic motives and increases their precautionary motives. However, uncertainty about future profitability matters even when arbitrageurs are risk-neutral, as it affects local investors’ liquidity demand, as well as arbitrageurs’ price impact.

Corollary 7 Consider the limit case where arbitrageurs are risk neutral, i.e. b → 0. Then arbi-

2 trageurs respond to an increase in arbitrage risk z1 by taking larger positions if and only if volatility is small enough and the expected shock is large enough relative to the current shock. Otherwise,

30 arbitrageurs decrease their positions.

 2 i  2 2 2 (n+1) ∂x  a σ z1 < cn with cn < 0 ≥ 0 ⇔ 2n+1 ∂z2 1  s¯1 ≥ θn,as0

2 ∂∆0 No matter how arbitrageurs respond, the spread always increases following an increase in z1: 2 ≥ ∂z1 0.

The result shows that even if arbitrageurs are risk-neutral, they may scale down their positions when uncertainty about arbitrage profitability increases. There are two effects: first, local investors are demanding more liquidity, as the convexity effect (i.e. the need to insure against shocks) increases with z1 (see equation (12)). This increases the potential price differences, which prompts arbitrageurs to increase their positions, but also increases the spread. Second, an increase in uncertainty steepens local investors’ demand for the asset in each market, as local investors are more reluctant to hold the asset. This results in larger price impact, which pushes arbitrageurs to decrease their positions. When volatility is already high, this effect dominates, and an increase in volatility leads to a reduction in arbitrageurs’ positions. Interestingly, whether arbitrageurs increase their positions or not, the spread always increases, showing that the increase in liquidity demand always outweighs the increase in arbitrageurs’ positions.

4.4 Market power and spread autocorrelation

Corollary 8 The spread has the following properties:

• Comparative statics: the current shock increases the spread.

∂∆ t > 0, t = 0, 1 ∂st

• Serial correlation: suppose s0 is random from the point of view of time -1. Then, when the number of arbitrageurs is ﬁnite, the half spread exhibits negative serial correlation between

31 time 0 and time 1:

∆ ∆ autocov 0 , 1 < 0 −1 2 2

When perfect competition obtains, the serial correlation vanishes:

∆0 ∆1 lim autocov−1 , = 0 n→∞ 2 2

Given that arbitrageurs revert trades in proportion of previous shocks to keep the spread open as long as possible, previous shocks continue to aﬀect the current spread. This generates serial correlation at one lag. As competition increases, arbitrageurs absorb liquidity shocks fully in each period, thus serial correlation disappears, and the price follows a random walk.

5 Surprise and anticipated shocks

As we saw in section 3, in particular in proposition 1, when markets are imperfectly competitive, shocks that are known in advance aﬀect portfolios both ahead of and at realization. Conversely, past shocks aﬀect trades and prices even after they are realized. To better understand how investors trade in response to shocks, it is useful to introduce several periods between the announcement and the realization of the shock. Hence, in this section, I present an extension of the baseline model to

T periods.

5.1 T-period model

The model is the same as in Section 3, except that I now consider a T + 1-period economy (t =

0, 1,...,T ), with T trading dates and a ﬁnal consumption date. The ﬁnal period liquidating dividend is

T +1 X DT +1 = D + t t=1

32 I denote Dt = Et(DT +1) the conditional expected value of the dividend at t. At each date t =

1,...,T , local investors receive a shock st−1t, where st−1 (t = 1,...,T ) is known by all investors from time 0 on. All the assumptions about distribution and preferences remain as in Section 3. The

Pn i only diﬀerence is that I assume that arbitrageurs have an aggregate initial endowment i=1 X−1 in market A, and the opposite in market B.

The model has a simple recursive structure.31

Proposition 4 Let T denote the number of trading dates. For k ∈ {1,...,T },

1. The local investors’ equilibrium certainty equivalent at T − k is

T −1 2 1 X ET −k [Eτ (pτ+1) − pτ ] CE =E + s + Y p + T −k T −k−1 T −k−1 T −k T −k−1 T −k 2 aσ2 τ=T −k T −1 2 X − aσ ET −k (Eτ (pτ+1) − pτ ) sτ (20) τ=T −k

2. The spread in the subgame perfect equilibrium is

 T −1 n  2 X X j ∆T −k = 2aσ  hT −k (τ) sτ − lT −k XT −k−1 (21) τ=T −k j=1

3. The arbitrageur’s value function is

" k T −1 T −1 # i 2 X X X 2 JT −k = 2aσ sT −m · aT −k (u, m) su + bT −k (τ) sτ m=2 u=T −m+1 τ=T −k 2  T −1 n  n   2 X X j X j +2aσ  cT −k (τ) sτ · XT −k−1 + dT −k  XT −k−1  (22) τ=T −k j=1 j=1

The parameters h, l, a, b, c, d are functions of n and k and are deﬁned recursively in the proof.

From Proposition 4, we can express local investors’ certainty equivalent and arbitrageurs’ value function at date 0, and from then obtain the equilibrium as a function of the initial parameters by

31All proofs of this section are relegated to the Supplementary Appendix.

33 forward induction.

Proposition 5 The equilibrium spread at time t ∈ {0,...,T − 1} is32

 t−1 τ T −1 t−1 t−1  2 X X X X Y ∆t = 2aσ −lt ζt (τ, p) sτ + ρt (τ, p) sτ − lt βτ Q−1 τ=0 p=0 τ=t p=0 τ=0

where ζt (τ, p) = γp (τ) Ip, ρt (τ, p) = ht (τ) − ltγp (τ) Ip, Ip = πp1{p≤t−2} + 1{p=t−1}, πp = Qt−1 Pn j q=p+1 βq, and Q−1 = j=1 X−1. γ and β are functions of the parameters l, d, h, c.

This result generalizes Proposition 1 to the multiperiod case. As the market is imperfectly liquid, prices at time t are determined not only by current and future shocks (second summation in the bracket), but also by the entire history of shocks up to time t − 1 (ﬁrst summation). Thus supply shocks have long-lived eﬀects, even if they are short-lived.

The functions ζ and ρ have intuitive interpretations. ζ measures the impact of past shocks through past trades, while ρ represents the impact of future shocks through both past and future

(anticipated) trades. The function ρ indeed has two components: ρt (τ, p) = ht (τ) − ltγp (τ) Ip.

Anticipations (of local investors) at time t are captured by the function ht, while the eﬀect of past trades is given by ltγ. lt represents the time-varying component of price impact, while γp(τ) measures how the future shock sτ aﬀects the time-p trade (p < t ≤ τ). Of course, γ itself depends on anticipations and price impact in a complex way: arbitrageurs determine their current trading taking into account local investors’ anticipations of future prices, etc.

Local investors’ anticipations at time t about how the future shock sτ will aﬀect prices enter the spread through ht(τ). Thus h represents the purely anticipatory component of the spread.

Note that in general h increases with t, i.e. the eﬀect of shock sτ on the spread (through local investors’ anticipations) is larger at time t + 1 than at time t. Why is it the case? Future shocks matter because local investors anticipate limited liquidity provision. As time passes, there are fewer trading opportunities, and thus fewer opportunities to receive liquidity, hence ht(τ) < ht+1(τ), for

32The formula holds for any t ≥ 0 since the functions γ and β are deﬁned only for t ≥ 0 and are nil otherwise, and with the usual convention that sums must exist.

34 t + 1 ≤ τ. The anticipatory component of the price becomes more sensitive to the future shock sτ , as less cumulative liquidity will be provided to absorb sτ in the future. This effect should thus generate an increasing pattern of the spread over time. Of course, arbitrageurs’ liquidity provision through time has the opposite effect and in general offsets the pure anticipatory component.

To sum up, two eﬀects determine the impact of future shocks on the current spread: local investors’ anticipations and the component of past trades that was based on anticipated shocks.

These two forces are important to understand the price eﬀects of anticipated shocks, as discussed below.

As a benchmark, it is useful to start as in the baseline model with the case of constant supply.

As expected, we get back to the gradual arbitrage dynamics:

33 Corollary 9 Suppose s0 = s1 = ··· = sT −1 = s, then

t−1 2 Y ∆t = 2aσ lt βq [s − Q−1] q=0 where β = 1+lq+1 . q (n+1)(1+lq+1)−2ndq+1

With arbitrageurs’ aggregate endowments, the total supply of assets to be transferred from local Qt−1 investors to arbitrageurs is 2(s − Q−1). The factor lt q=0 βq determines the convergence rate of prices, which increases as n increases.

Introducing endowments produces new insights. Speciﬁcally, with non-zero endowments for arbitrageurs, the model shows how the ownership distribution aﬀects liquidity and the convergence of prices. First, the spread is smaller when Q−1 is close to s, i.e., markets are more liquid when most of the supply is already held by arbitrageurs. Second, for a given Q−1, prices converge faster when this aggregate endowment is split across a larger number of arbitrageurs. Hence markets where the asset ownership is initially highly concentrated within a few arbitrageurs are likely to exhibit more sluggish price convergence.

33I use the convention that if a product or a sum is not well deﬁned, e.g. for t = 0, then it is equal to 1.

35 5.2 Eﬀects of surprise shocks

The gradual arbitrage dynamics obtain as well when unanticipated shocks happen. The next proposition characterizes the price adjustment in this case.

Proposition 6 Suppose that endowment shocks are constant over time, i.e. sτ = s, and assume that at t∗ < T − 1, there is a surprise and constant jump from s to su = s + ∆su. Then the

2 u ∗ equilibrium spread jumps by 2aσ lt∗ ∆s at t , and the jump gradually decreases afterwards. The spread is given by:

t−1 ∗ 2 Y At t < t , ∆t = 2aσ lt βτ [s − Q−1] τ=0

∗ 2 u At t = t , ∆t∗ = ∆˜ t∗ + 2aσ lt∗ ∆s t−1 ∗ 2 Y u At t > t , ∆t = ∆˜ t∗ + 2aσ lt βτ ∆s τ=t∗ where ∆˜ t is the spread based on the constant shock s.

The announcement and realization of the shock ∆su happen in the same period t∗. Thus the spread immediately jumps and arbitrageurs start adjusting their portfolios at t∗. The unanticipated shock has a prolonged eﬀect since arbitrageurs adjust their portfolios only progressively as in the gradual arbitrage case. Local investors anticipate the gradual liquidity provision, hence require a larger liquidity discount to hold the asset. Figures 4 and 5 plot the price and portfolio dynamics.

The price dynamics ressemble the patterns of price adjustment observed in many markets with slow-moving capital. For instance, premia for catastrophe risk insurance recover only in a matter of months after insurers suﬀer losses due to natural disasters (Froot and O’Connell, 1999). Froot

(2001) shows empirically that market imperfections on the supply side and arbitrageurs’ (in this context, reinsurers’) market power can explain the gradual price responses, which is in line with the mechanism studied in this paper.34 Gradual price adjustments have also been documented in equity markets. Early evidence by Harris and Gurel (1986) shows that index recompositions of the

34Froot (2001): “the most important explanations [of the slow adjustments] are supply-side stories of capital market imperfections facing reinsurers and the exercise of market power by reinsurers.”

36 S&P500 (which were announced and realized on the same day until 1989) generate temporary price pressure for added and deleted stocks.

5.3 Eﬀects of anticipated shocks

I now consider the eﬀects of pre-announced shocks on prices.

Proposition 7 Suppose that arbitrageurs learn at time t1 > 0 that the endowment shock will jump

u u permanently from s to s = s+∆s at time t2 > t1. The shock aﬀects the spread at announcement, realization, in between announcement and realization, and after the realizaton of the shock. The eﬀect on the spread is as follows:

t−1 2 Y At t < t1, ∆t = 2aσ lt βτ [s − Q−1] = ∆˜ t τ=0 T −1 ˜ 2 X u At t = t1, ∆t1 = ∆t1 + 2aσ ht1 (τ) ∆s τ=t2 T −1 t T −1 ! ˜ 2 X X X u At t ∈ [t1 + 1, t2 − 1], ∆t = ∆t + 2aσ  ht (τ) − lt Ip γp−1 (τ)  ∆s τ=t2 p=t1+1 τ=t2   T −1 t2−1 t2−1 T −1 ! ˜ 2 X X Y X u At t = t2, ∆t2 = ∆t2 + 2aσ lt2 1 − γt2−1 (τ) − βq γp−1 (τ)  ∆s τ=t2 p=t1+1 q=p τ=t2   t−1 t2 t−1 T −1 ! ˜ 2 Y X Y X u At t ∈ [t2 + 1,T − 1], ∆t = ∆t + 2aσ lt  βτ − βq γp−1 (τ)  ∆s τ=t2 p=t1+1 q=p τ=t2

Qt−1 1 1 where Ip = q=p βq {p≤t−1} + {p=t}.

Standard theory based on competitive arbitrageurs would predict that prices adjust immediately and once-for-all at announcement and that portfolios are rebalanced in one go at realization of the shock. In the present setting where arbitrageurs are hedged against fundamental risk, the price would not react if arbitrageurs were competitive, as local investors would anticipate to be able to oﬄoad immediately all the risk onto arbitrageurs at realization of the shock. When arbitrageurs have price impact, instead, prices increase slowly and gradually up to the realization and decrease thereafter (Figure 6). Hence prices have a V-shape pattern and returns are predictable although all

37 investors are rational and forward-looking. As Figure 7 shows, arbitrageurs start providing some extra liquidity as the shock is announced but realize the bulk of the trading in the periods following the realization of the shock. The exact pattern of the price adjustment depends on the degree of competition and the amount of time that elapses between the annoucement and the realization of the shock. When the realization is closer to the announcement (i.e. t2 − t1 is small), the price reacts more strongly at announcement and the slope of the gradual increase is steeper (Figure

8). The V-shape patterns are less pronounced in more competitive markets. When the market is monopolistic, however, the price adjustment is slightly diﬀerent because the monopolist continues to trade as if there was no announcement at t1 (Figure 13).

Corollary 10 Suppose that n = 1. Then the monopolistic arbitrageur behaves as if the shock had not been announced, i.e. the arbitrageur does not change his trading strategy until t2. The spread, however, jumps at t1 as prices reﬂect new information. The jump stays constant until t2, where the arbitrageur starts to gradually absorb the additional endowment. The dynamics of the equilibrium spread are given by:

t−1 2 Y At t < t1, ∆t = 2aσ lt βτ [s − Q−1] = ∆˜ t τ=0 T −2 ˜ 2 X 1 1 u At t = t1, ∆t1 = ∆t1 + 2aσ + ∆s lτ+1 + 2 2 τ=t2 T −2 2 X 1 1 u At t ∈ [t1 + 1, t2 − 1], ∆t = ∆˜ t + 2aσ + ∆s lτ+1 + 2 2 τ=t2 ˜ 2 u At t = t2, ∆t2 = ∆t2 + 2aσ lt2 ∆s t−1 2 Y u At t ∈ [t2 + 1,T − 1], ∆t = ∆˜ t + 2aσ lt βτ ∆s τ=t2

The reason why the monopolistic arbitrageur does not alter his trading strategy before t2 is the same as in the baseline model. After t1, local investors’ willingness to hold the asset decreases as they anticipate that the increase in supply will not be fully absorbed at t2. This makes the arbitrage more proﬁtable, which induces arbitrageurs to increase liquidity supply. Given that price

38 impact is permanent, the arbitrageur’s trades have a cumulative effect on the price. Thus buying more now decreases the spread permanently. The marginal value of increasing liquidity provision now (before t2) is exactly offset by the cost of facing a smaller spread after t2 for the monopolist. Specifically, the partial derivatives of the marginal cost and the marginal benefit with respect to

∆su are equal in that special case. Thus the monopolist does not change the speed at which he provides liquidity until t2. Local investors anticipate that no additional liquidity will be provided between t1 and t2 relative to the baseline case. Hence their anticipations about the cumulative liquidity provision during the interim period are the same at t1 and at, say, t1+s < t2. In other words, the anticipatory component of the spread is constant between t1 and t2. But since there is no interim additional trading relative to the baseline case in that period, the anticipatory component is the sole determinant of the (extra) spread dynamics. Thus there is no extra eﬀect on the spread before the realization of the shock, apart from the one-shot anticipation adjustment at t1. Hence the spread jumps at t1, and follows the same dynamics as in the baseline case up to t2 where the actual liquidity provision occurs (Figure 12).

In the oligopolistic case, the marginal value of accelerating liquidity provision before t2 is higher than the cost of facing a smaller spread at t2 (i.e. the partial derivative of the marginal benefit with respect to ∆su is higher than the partial derivative of the marginal cost). As mentioned above, the acceleration is not pronounced, and the bulk of the trading occurs from t2 onwards. Arbitrageurs have an incentive to limit liquidity as much as possible to exploit the arbitrage longer and more profitably. But since there is now additional interim liquidity provision, the spread dynamics between t1 and t2 depend on the interplay between this interim liquidity provision and local investors’ adjusted anticipations. Because some additional liquidity is provided between t1 and t2, ht1 (t2) is lower than ht1+s (t2). In general, we saw that this does not imply that the spread increases over time. Indeed, past liquidity provision has a cumulative effect and offsets the anticipatory effects. This is what we observe, for instance, with a constant supply. However, in this case, arbitrageurs’ liquidity provision is tenuous, and the anticipatory effect dominates. Hence the spread exhibits a gradually increasing pattern up to t2. After t2, gradual arbitrage dynamics obtain again.

39 The model may contribute to explain the V-shaped price patterns observed in actual equity and bond markets. Gradual secondary market price declines and reversals are observed around

Treasury issuances, which are announced in advance.35 For instance, Keloharju et al. (2002) report this pattern in the Finnish Treasury market during the 1990s. Although the model is very stylized and abstracts from specific institutional details, the key ingredients of our theory can be mapped to the evidence. The Finnish sovereign debt increased during the period, so that issuances were increasing the net supply of Treasuries. The intermediation between the issuer (Treasury) and investors is the privilege of a small group of primary dealers, leaving room for market power. In this respect, Keloharju et al. find that the V-shaped price effects are particularly pronounced during the beginning of the sample, when the market of primary dealers was particularly concentrated, which is consistent with the model predictions. The authors also find that the effect is stronger when issuances are larger, which is also consistent with our predictions. Lou, Zan, and Zhang (2013) find similar patterns in the US Treasury market. Their evidence points to the role of primary dealers’ risk-bearing capacity, a channel which is shut in the model since arbitrageurs are effectively risk- neutral. Hence our theory predicts an explanatory role for measures of market power or of market thinness for V-shaped price effects after controlling for dealers’ limited risk-bearing capacity.36

The model may also shed light on price pressure eﬀects observed around stock index recompositions. Although stocks are inﬁnitely-lived assets, the model may be mapped to this case, as discussed in footnote 25. Index recompositions trigger changes in demand from passive investors tracking the index such as index funds and ETFs but are unlikely to be informative about fundamentals. Since 1989, S&P announces a week in advance the additions/ deletions to the S&P 500.

Lynch and Mendenhall (1997) show that these pre-announced recompositions have price eﬀects not only at announcements but also between the announcement and the recomposition, as well as after

35These events are publicly announced and are not subject to asymmetric information. This sets our theory apart from Albuquerque and Miao (2014) who obtain momentum and reversal in a model where some informed agents obtain advance information about future earnings, which is uncorrelated with their information about current earnings. This makes their theory more relevant for momentum and reversals in stocks, while the model of this paper is suitable to study price effects following uninformed demand or supply shifts. 36The model delivers predictions about one asset only, but supply shocks in one asset may affect secondary market prices of other assets. For instance, Newman and Rierson (2003) document spillover effects from bond issuances in the European telecom sector. They shows that issuances by one large firm depresses secondary bond market prices of other firms.

40 the recomposition. The returns of added stocks follow a inverted V-shaped pattern around the inclusion, while the returns of deleted stocks have opposite (symmetric) patterns. Similar eﬀects are observed around the reconsitution of the Russell 1000, 2000 and 3000 indices where the mechanical rules allow market participants to predict the additions and deletions with a high degree of conﬁdence (Madhavan (2000)).

V-shaped price patterns are diﬃcult to obtain in models with rational agents. Some frameworks generate gradual declines, others gradual reversals, but very few produce both “at the same time”.

Here, the key ingredients are arbitrageurs’ market power and the model’s ﬁnite horizon. The mechanism diﬀers from extant theories. Vayanos and Woolley (2013) propose a theory where

V-shaped patterns are caused by the flow-performance relationship. Uncertain gradual outflows induce agents to buy an asset whose price is expected to decrease further. After a string of bad returns, expected returns are so large that some investors start reinvesting in the asset, causing the price to rebound. Arbitrageurs such as hedge funds, although not completely sheltered from outflows, have some leaway in designing their capital structure, and typically impose gates and lock-up periods (Hombert and Thesmar, 2014). Our mechanism relies on the presence of large strategic liquidity providers and survives the argument that some investors are less sensitive to performance-based outflows.

6 Empirical predictions and concluding remarks

In standard models where all investors are price takers, non-informative supply shocks are quickly absorbed by financial markets and arbitrageurs eliminate any price discrepancies, leading to perfect risk-sharing. In practice, arbitrage is mostly carried out by sophisticated financial institutions which account for their impact on prices. When arbitrageurs have price impact, they break up trades and provide limited liquidity. Consequently, the effects of supply shocks die out only gradually, and risk-sharing is imperfect. Further, the effects of anticipated supply shocks increase up to the realization of the shock and revert only gradually afterwards. When future shocks are not known with certainty, price effects depend on the complex interaction between the total risk-bearing

41 capacity of the market and arbitrageurs’ strategic considerations. Arbitrageurs’ risk-aversion can oﬀset the eﬀects of market power, so that it is sometimes desirable to have risk-averse arbitrageurs.

Further, while uncertainty about future shocks increases liquidity needs and makes the arbitrage more likely to be proﬁtable, it also makes the market thinner, which feeds back into arbitrageurs’ trading strategies and may induce them to provide less liquidity.

The analysis of imperfectly competitive arbitrage delivers a number of empirical predictions, which we can summarize as follows:

1. The degree of competition between arbitrageurs determines price impact, the level of spreads

or liquidity premia and the speed at which they decrease over time.

2. Controlling for volatility and risk-bearing capacity, price impact (e.g. proxied by Kyle’s

lambda) increases over time, even more so in more concentrated markets.

3. A change of sign in bases / spreads is more likely to occur when aggregate liquidity conditions

improve.

4. A change of sign in bases / spreads is likely to occur around a negative supply shock (s0 −s1 > 0 in the model).

5. For a given share of assets owned by arbitrageurs/ ﬁnancial institutions, supply shocks

(e.g. following surprise downgrades, unanticipated losses triggering portfolio reblancing etc.)

should have more impact and generate more sluggish price responses if a small number of

arbitrageurs initially hold these assets.

6. V-shaped patterns around pre-announced supply shocks should be more pronounced in more

concentrated markets, or when announcement and realization dates are closer. For instance,

in Treasury markets, a larger number of primary dealers should be associated with smaller

price eﬀects around Treasury issuances.

This paper shows how imperfect competition aﬀects the equilibrium dynamics of prices and portfolio holdings in ﬁnancial markets. An important question that this paper left aside is that

42 of the origins and dynamics of imperfect competition. A step towards answering this question is made in Zigrand (2004), Oehmke (2010) and Fardeau (2014). However, more research is needed to understand the static and dynamic determinants of the market structure of the ﬁnancial industry.

43 Appendix

A Two lemmas

2 Lemma 3 Let (p, q) ∈ R with p 6= q and consider the (n, n) matrix

  p q ··· q     q p ··· q M =   n . . . . . . .. . . . .   q q ··· p

Mn is invertible and its inverse is given by:

  p + (n − 2) q −q · · · −q     1  −q p + (n − 2) q · · · −q  M −1 =   n  . . . .  (p − q)(p + (n − 1) q)  . . .. .   . . .    −q −q ··· p + (n − 2) q

Proof. Mn being a square matrix with independent lines and columns, it is invertible.

−1 −1 It is straightforward to check that Mn.Mn = Mn .Mn = I.

2 1 Lemma 4 Let X ∼ N µ, σ and (A, B) ∈ − 2σ2 , ∞ × R, then

2 exp (y) B2σ2+2µ(B−Aµ) E exp −AX + BX = √ , with y = 2 2Aσ2 + 1 2(2Aσ +1)

44 Proof. Since X is normally distributed:

2 2 (x−µ) Z ∞ exp −Ax + Bx exp − 2 2 2σ E exp −AX + BX = √ dx −∞ σ 2π 2 µ+σ2B µ ∞ 2Aσ +1 2 Z exp − 2σ2 x + σ2 x − 2σ2 = √ dx −∞ σ 2π

(x−m)2 Rewrite the exponential in the integrand as exp − 2z2 exp (y). This gives, by identiﬁcation of the terms:

1 2Aσ2 + 1 σ2 = ⇒ z2 = 2z2 2σ2 2Aσ2 + 1 m µ + Bσ2 µ + Bσ2 = ⇒ m = z2 σ2 2Aσ2 + 1

m2 µ2 B2σ2+2µ(B−Aµ) This implies that 2z2 − y = 2σ2 , i.e. y = 2(2Aσ2+1) . Thus,

(x−m)2 ∞ exp (y) Z exp − 2z2 exp (y) exp −AX2 + BX = √ √ dx = √ E 2 2 2Aσ + 1 −∞ z 2π 2Aσ + 1

B Risk-free arbitrage

Proposition 1

Proof. At each date, going backward, I ﬁrst solve for the demand of local investors in markets

A and B, and then solve for the arbitrageurs’ optimal trades, given local investors’ demands and market-clearing.

Time 1 - local investors’ problem It is enough to solve for the demand of local investors in market A, as market B is the symmetric case, thus I drop superscripts k = A, B, unless the context requires it. At time 1, W2 = E2 + Y2D2 = E1 + s12 + Y1D2 = E0 + s01 − y1p1 + s12 + Y1D2. At time 1, 1 is revealed to all agents and s1 is known, thus only 2 is random.

45 The local investors’ maximization problem is

2 aσ 2 V1 = max E1 [u (W2)] = max − exp −a E0 + s01 − y1p1 + Y1D1 − (Y1 + s1) y1 y1 2

D − p From the FOC, Y + s = 1 1 (23) 1 1 aσ2 n ! A 2 X i,A Using market-clearing, p1 = D1 − aσ s1 − X1 (24) i=1 n ! B 2 X i,B By analogy, p1 = D1 − aσ −s1 − X1 (25) i=1

Arbitrageurs’ problem at time 1

Starting from arbitrageurs’ wealth at time 2 given by equation (3), and using the law of motions

i i for Bt and Xt and the assumption of opposite positions across makets, we can rewrite wealth as:

i i i B A W2 = B0 + x1 p1 − p1

Arbitrageurs maximize their expected utility of wealth, subject to equations (24) and (25). Sub- stituting (24) and (25) into arbitrageur’s wealth gives the following maximization problem:

" n !!# i i 2 i X i J1 = max E u B0 + 2aσ x1 s1 − X1 xi 1 i=1

(Note that by an abuse of notation, I use i both as a counting variable and to refer to arbitrageur i).

Pn i Pn i Pn −i i Pn i Pn −i i We can write i=1 X1 = i=1 X0 + −i x1 + x1 = i=1 x0 + −i x1 + x1, where −i denote all arbitrageurs but arbitrageur i, and solve for the zero of the ﬁrst-order condition for each arbitrageur i:

n i X −i X i 2x1 + x1 = s1 − x0, i = 1, . . . , n −i i=1

46 Stacking the n equations together and using matrix notation gives:

n ! X i Anx˜1 = s1 − x0 I, i=1 where An is an (n, n) matrix with 2’s on the diagonal and 1’s elsewhere,x ˜1 is a (n, 1) vector of

−1 1 n −1 trades,x ˜1 = x1, . . . , x1 and I is the identity matrix. We can then use Lemma 4 to ﬁnd An , invert the system and get the equilibrium trade at time 1,

s − P xi xi = 1 i 0 , i = 1, . . . , n (26) 1 n + 1

Then, plugging equation (26) into (24) and arbitrageur i’s objective function gives the equilibrium price and value function in the time 1 subgame:

s − Pn xi pA = D − aσ2 1 i=1 0 1 1 n + 1 Pn i 2 ! s1 − x J i = u Bi + 2aσ2 i=1 0 1 0 (n + 1)2

A Similarly, plugging the previous expression for p1 into (23) gives the local investors’ equilibrium certainty equivalent in the subgame:

aσ2 CE = E + s + Y p + Y (D − p ) − (Y + s )2 1 0 0 1 0 1 1 1 1 2 1 1 2 Pn i 2 Pn i aσ s1 − i=1 x0 2 s1 − i=1 x0 = E0 + s01 + Y0p1 + − aσ s1 2 (n + 1)2 n + 1 with E0 = E−1 − y0p0 = −y0p0 (E−1 = 0).

Time 0 - local investors

Going backward and using the expression for their certainty equivalent and for E0, the A investors’ problem is:

2 P i 2 P i !! aσ s1 − i x0 2 s1 − i x0 V0 = max −E0 exp −a −y0p0 + Y0p1 + 2 − aσ s1 + s01 y0 2 (n + 1) n + 1

47 Hence, evaluating the expectation,

P i 2 2 s1 − i x0 aσ 2 V0 = max − exp −a −y0p0 + Y0 D − aσ − (Y0 + s0) y0 n + 1 2 2 P i 2 P i ! aσ s1 − i x0 2 s1 − i x0 . exp −a − aσ s1 2 (n + 1)2 n + 1

From the ﬁrst-order condition,

s − P xi aσ2 (Y + s ) = D − aσ2 1 i 0 − p 0 0 n + 1 0 P i s1 − x X i.e. p = D − aσ2 i 0 − aσ2s + aσ2 xi by market-clearing 0 n + 1 0 0 i s1 n + 2 X i.e. p = D − aσ2 s + + aσ2 xi 0 0 n + 1 n + 1 0 i s1 n + 2 X By analogy, pB = D + aσ2 s + − aσ2 xi 0 0 n + 1 n + 1 0 i

i i,A i,B where I also used the fact that xt = xt = −xt for the last equation. The spread between A and B is thus

! s1 n + 2 X ∆ = pB − pA = 2aσ2 s + − xi 0 0 0 0 n + 1 n + 1 0 i

Time 0 - Arbitrageurs

i i P i,k k i Using this expression for ∆0 and B0 = B−1 − k=A,B x0 p0 = −x0∆0 (B−1 = 0), arbitrageur i’s problem is:

P i 2 ! s1 − x i 2 i ˆ 2 i 0 ˆ ∆0 J = max − 0 exp −b 2aσ x ∆0 + 2aσ , with ∆0 = 2 0 i E 0 2 2aσ x0 (n + 1) ! P i 2 ! 2 i s1 n + 2 X i 2 s1 − i x0 = max − exp −b 2aσ x0 s0 + − x0 + 2aσ 2 xi n + 1 n + 1 0 i (n + 1)

48 From the ﬁrst-order condition,

n ! n ! s1 n + 2 X 2 X s + − xi + xi = s − xi , i = 1, . . . , n (27) 0 n + 1 n + 1 0 0 2 1 0 i=1 (n + 1) i=1 n n + 2 n2 + 3n X n − 1 xi + xi = s + s n + 1 0 2 0 0 2 1 (n + 1) i=1 (n + 1) 2(n2 + 3n + 1) n2 + 3n X n − 1 xi + x−i = s + s (n + 1)2 0 2 0 0 2 1 (n + 1) −i (n + 1)

Stacking the n equations together and solving for the equilibrium using Lemma 4, I get after some simple algebra:

i s0 n − 1 s1 n3+4n2+3n+2 x0 = + 2 , with φn = 2 φn (n + 1) φn (n+1)

i i The equilibrium quantities for x1, ∆0 and ∆1 follow from substituting x0 into (26) and the price schedules.

Lemma 1 and Corollary 1

Proof. If s0 = s1 = s, then after some algebra, we get:

n2 + 3n κ = 0,n n3 + 4n2 + 3n + 2 n + 2 κ =κ ¯ = 1,n 1,n n3 + 4n2 + 3n + 2 (n + 2)2 κ¯ = 0,n n3 + 4n2 + 3n + 2

For any n ≥ 1, κ1,n < κ0,n ⇒ x0 < x1, and κ0,n + κ1,n < 1 ⇒ x0 = X0 < X1, and thus

P 1 P i ∂∆t i x0 < i X1. Further, for any n ≥ 1,κ ¯0,n > κ¯0,n, thus ∆0 > ∆1 > 0 = ∆2, and ∂n < 0 (t=0,1). Further, ∆1 = 1 is decreasing in n and lim ∆ = 0 (t = 0, 1). In the more general ∆0 n+2 n→∞ t case with s0, s1, limn→∞ φn = 0 implies that ∆0 and ∆1 converge to 0 when n becomes large.

Corollary 2

Proof. The ﬁrst comparative statics is straightforward. For the second part of the result, note

49 n3+2n2+n that ∆1 ≤ 0 iﬀ s1 ≤ n3+3n2+4n+2 s0 ≡ αns0. Clearly, 0 < αn < 1, for any n ≥ 1.

Corollary 3

This result is proved in the proof of Proposition 1.

C Risky arbitrage

Proposition 2

Proof. At time 1, the problem is similar to the risk-free arbitrage case. From the proof of

Proposition 1, recall that:

s − Pn xi pA = D − aσ2 1 i=1 0 1 1 n + 1 Pn i 2 ! s1 − x J i = u Bi + 2aσ2 i=1 0 1 0 (n + 1)2 aσ2 CE = E + s + Y p + Y (D − p ) − (Y + s )2 1 0 0 1 0 1 1 1 1 2 1 1 2 Pn i 2 Pn i aσ s1 − i=1 x0 2 s1 − i=1 x0 = E0 + s01 + Y0p1 + − aσ s1 2 (n + 1)2 n + 1

Hence, after rearranging terms,

Pn i Pn i 2 2 i=1 x0 2 i=1 x0 CE1 = E0 + s01 + Y0 D + aσ + aσ n + 1 (n + 1)2 ! Y0 n X 2n + 1 −aσ2s − xi − aσ2 s2 (28) 1 n + 1 n + 1 0 2 1 i 2 (n + 1) with E0 = −y0p0.

Time 0 - local investors

At time 0, the local investors in market A solve the following problem:

s1,1 V0 = max E [− exp (−a (CE1))] y0

50 s1,1 E [− exp (−a (CE1))] = " Pn i Pn i 2 2 !# 2 i=1 x0 2 i=1 x0 aσ 2 − exp −a −y0p0 + Y0 D + aσ + aσ − (Y0 + s0) n + 1 (n + 1)2 2 Y n P xi 2n + 1 s1 2 0 i 0 2 2 × E exp −a −aσ s1 − − aσ s1 n + 1 (n + 1)2 2 (n + 1)2

n P xi By Lemma 4, and setting −A = a2σ2 2n+1 , B = a2σ2 Y0 − i 0 , µ =s ¯ and σ2 = z2, we 2(n+1)2 n+1 (n+1)2 1 x 1 have under Assumption 1 in the text:

1 2 − 2 1 E exp −As1 + Bs1 = ra exp C (29) 2ra !2 ! ! Y0 n X Y0 n X 2n + 1 with C = a4σ4z2 − xi + 2¯s a2σ2 − xi +s ¯ 1 n + 1 2 0 1 n + 1 2 0 1 2 (n + 1) i (n + 1) i 2 (n + 1) 2 2 2 2n + 1 and ra = 1 − a σ z 1 (n + 1)2

Thus investors in market A solve the following problem:

2 ! 1 P i P xi 2 − 2 2 i x0 2 i 0 aσ 2 max −ra exp −a −x0p0 + Y0 D + aσ + aσ 2 − (Y0 + s0) y0 n + 1 (n + 1) 2 3 4 2 P i 2 2 P i ! a σ z1 Y0 n i x0 aσ s¯1 Y0 n i x0 2n + 1 . exp −a − − 2 − − 2 + 2 s¯1 2ra n + 1 (n + 1) ra n + 1 (n + 1) 2 (n + 1)

The FOC yields

P i 3 4 2 P i 2 2 i x0 2 a σ z1 Y0 n i x0 aσ s¯1 D + aσ − aσ (Y0 + s0) − − 2 − = p0 n + 1 (n + 1) ra n + 1 (n + 1) (n + 1) ra

P i By market-clearing: Y0 = − i x0, thus regrouping terms gives:

s¯1 n + 2 X p = D − aσ2 s + + aσ2 (1 + φ ) xi (30) 0 0 (n + 1) r n + 1 a 0 a i 2 2 2 a σ z1 with φa = 2 (n + 1) ra

51 By symmetry, the price schedule faced by arbitrageurs in market B is:

s¯1 n + 2 X pB = D + aσ2 s + − aσ2 (1 + φ ) xi (31) 0 0 (n + 1) r n + 1 a 0 a i

Arbitrageurs’ problem at time 0

Arbitrageurs’ value function:

s − P xi J i = max u Bi + 2aσ2 1 i 0 0 i E 0 2 x0 (n + 1) " 2 P i P i 2 !# i 2 s1 − 2s1 i x0 + i x0 = max − exp −b x ∆0 + 2aσ (32) i E 0 2 x0 (n + 1)

Using Lemma 4, I get

2 2 P i 2abσ 2 4abσ i x0 E exp − s1 + = (n + 1)2 (n + 1)2 2 !! 1 8a2bσ4z2 P xi 2 − 2 1 i 0 s¯1 2 X i 2aσ rb exp −b − 4 − 2 4aσ x0 − 2 s¯1 (33) (n + 1) rb (n + 1) rb i (n + 1)

From equations (30) and (31), I get ∆ = 2aσ2 s + s¯1 − n+2 (1 + φ ) P xi . Denoting 0 0 (n+1)ra n+1 a i 0

ˆ ∆0 ∆0 = 2aσ2 , and using equations (32) and (33), and rearranging terms gives the value function stated in the proposition.

Corollary 4

Proof. Direct from Proposition 2.

Proposition 3

Proof. Using Proposition 2, the ﬁrst-order condition gives, for all i ∈ {1, ..., n}:

! P i s¯1 n + 2 X x s¯1 s + − (1 + φ ) xi + xi + 2 (1 − φ ) i 0 − 2 = 0 0 (n + 1) r n + 1 a 0 0 b 2 2 a i (n + 1) (n + 1) rb

Stacking the n equations together and using Lemma 3 to solve for the equilibrium, gives, after

52 some algebra:

s¯1 2¯s1 s0 + − 2 i (n+1)ra (n+1) rb x0 = (34) φn + (n + 2)φa + 2nφb

n+2 n+2 Note that the facts that 1 − φb < 1 and n+1 (1 + φa) > n+1 > 1 ensures that the maximand is i concave in x0, which guarantees that the optimum is a maximum.

Using equations (30) and (34), one can get the equilibrium price of asset and the spread between assets B and A:

X n s¯1 2¯s1 xi = s + − 0 d 0 (n + 1) r (n + 1)2r i a b s¯1 n + 2 X pA = D − aσ2 s + + aσ2 (1 + φ ) xi 0 0 (n + 1) r n + 1 a 0 a i A 2 2 s¯1 2 s¯1 ⇒ p0 = D − aσ Φas0 − aσ Φa − 2aσ (1 − Φa) 2 (n + 1) ra (n + 1) rb n (n + 2) with Φ = 1 − (1 + φ ) a (n + 1) d a

d = φn + (n + 2) φa + 2nφb

n (n + 2) (1 + φa) Note that Φa = 1 − (n + 1) (φn + (n + 2) φa + 2nφb) n2+n+2 + n (n + 2) φa + 2n (n + 1) φb = n+1 (n + 1) (φn + (n + 2) φa + 2nφb)

The second equation follows from the deﬁnition of φn given in Proposition 2. Since (n + 1) φn > n2+n+2 n+1 ,Φa ∈ [0, 1].

Corollary 5

i Proof. From the expression of x0 given in Proposition 3,

i " s¯1 # " s¯1 # ∂x ∂ s0 + ∂ 2 0 = (n+1)ra − 2 (n+1) rb ∂b ∂b φn + (n + 2) φa + 2nφb ∂b φn + (n + 2) φa + 2nφb

53 I ﬁrst calculate the second term in brackets.

2 2 2 2 First, note that (n + 1) rb = (n + 1) + 4abσ z1

s¯1 s¯1 (n+1)2 Thus 2 = 2 2 , with f (b) = 4abσ2z2 (n + 1) rb 4abσ z1 [1 + f (b)] 1 ∂ 1 1 4aσ2z2 1 ⇒ = − 1 = − (35) ∂b 2 2 22 2 2 2 2 2 (n + 1) rb 4abσ z1 [1 + f (b)] 4ab σ z1 [1 + f (b)]

∂φ ∂ 1 f 0 (b) Second, b = = − ∂b ∂b 1 + f (b) [1 + f (b)]2 2 0 (n + 1) f (b) Noting that f (b) = − 2 2 2 = − b 4aσ z1 b ∂φ f (b) gives b = (36) ∂b b [1 + f (b)]2

Hence, using equations (35) and (36), and the notation d = φn + (n + 2) φa + 2nφb, gives:

s¯1 " # ∂ 2 s¯ d 2nf (b) (n+1) rb = 1 − − ∂b φ + (n + 2) φ + 2nφ d2 2 2 2 2 2 2 n a b 4ab σ z1 [1 + f (b)] b [1 + f (b)] (n + 1) rb s¯ 2n = − 1 d + (37) 2 2 2 2 2 r 4ab σ z1d [1 + f (b)] b

f(b) 1 The second line follows from the fact that 2 = 2 2 2 . I now turn to the ﬁrst term in b(n+1) 4ab σ z1 brackets:

0 " s¯1 # s¯1 s¯1 s + 2nφ s0 + 2nf (b) s0 + ∂ 0 (n+1)ra b (n+1)ra (n+1)ra = − 2 = − 2 (38) ∂b φn + (n + 2) φa + 2nφb d bd2 [1 + f (b)]

f(b) (n+1)2 Combining equations (38) and (37), noting that = 2 2 2 and rearranging terms gives: b 4ab σ z1

i ∂x0 1 2 s¯1 2n = −n (n + 1) s0 + +s ¯1 d + ∂b 2 2 2 2 2 (n + 1) r r 2ab σ z1 [1 + f (b)] d a b

The rest of the corollary follows immediately.

Lemma 2

54 Proof.

I prove a slightly more general result. The results of Lemma 2 obtain by taking s0 → 0 in Lemma 5.

Lemma 5 It holds that:

i ∂x0 i) The sign of ∂b is independent of b.

∂xi ii) If n ≤ 2, d + 2n − n(n+1) > 0 if a2σ2z2 is small enough. Thus, in this case, 0 ≥ 0 is rb ra 1 ∂b equivalent to s¯1 large enough if s > 0, and is always satisﬁed if s ≤ 0. s0 0 0

∂xi iii) If n > 2 or n ≤ 2 and a2σ2z2 is large enough, 0 ≥ 0 is equivalent to s¯1 small enough 1 ∂b |s0| for s0 < 0, and is never satisﬁed if s0 > 0.

i 2 i ∂x0 ∂ x0 iv) ∂b ≥ 0 ⇒ ∂b2 ≤ 0.

Proof. Recall from Corollary 5 that

i ∂x0 2 s¯1 2n = κ −n (n + 1) s0 + +s ¯1 d + (39) ∂b (n + 1) ra rb

1 with κ = 2 2 2 2 2 > 0. Hence the sign of the derivative depends on the expression in 2ab σ z1 d [1+f(b)] parenthesis. Given that d = φn + (n + 2) φa + 2nφb, the terms in b are given by

2 2 2n 4abσ z1 2n 2nφb + = 2n 2 + rb (n + 1) rb rb 2 2 2n 4abσ z1 2n = 1 + 2 = rb = 2n rb (n + 1) rb

This proves i). As a consequence, one can write:

2 2 2 3 2n n (n + 1) n (n + 1) (n + 2) a σ z1 − n (n + 1) d + − = φn + (n + 2) φa − + 2n = φn + 2n + 2 rb ra ra (n + 1) ra

55 Developing and rearranging the terms,

3 2 2 2 2 3 2n n (n + 1) 3n + 8n + 5n + 2 (n + 2) a σ z1 − n (n + 1) d + − = 2 + 2 rb ra (n + 1) (n + 1) ra n(6n3+18n2+14n+4) −n4 + 5n2 + 4n + 2 − a2σ2z2 (n+1)2 1 = 2 (n + 1) ra

42 2 2 2 2 2 2 4 Note that if n = 1, the numerator equals 10 − 4 a σ z1. From Assumption 1, a σ z1 < 3 . Hence, d + 2n − n(n+1) ≥ 0 iﬀ a (or σ2 or z2) is small enough. The same applies if n = 2. If n > 2, rb ra 1 −n4 + 5n2 + 4n + 2 < 0, thus d + 2n − n(n+1) < 0. ii) and iii) follow. rb ra

Finally, I compute the second derivative of xi with respect to b using (39). Since d + 2n is 0 rb independent of b,

2 i ∂ x0 ∂κ 2 s¯1 2n 2 = × −n (n + 1) s0 + +s ¯1 d + ∂b ∂b (n + 1) ra rb

∂κ Given that ∂b < 0, we have the result.

Corollary 6 h i ∂∆0 2 Φa 2(1−Φa) Proof. From the expression of the equilibrium spread (17), = 2aσ + 2 > ∂s¯1 (n+1)ra (n+1) rb

0 since Φa ∈ ]0, 1[. Similarly, from equation (15),

i ∂x0 1 1 2 = − 2 , with d = φn + (n + 2) φa + 2nφb ∂s¯1 d (n + 1) ra (n + 1) rb

Replacing ra and rb by their expressions, and rearranging terms, this simpliﬁes into:

2 2 4abσ z1 2 2 2n+1 ∂xi n − 1 + n+1 + 2aσ z1 2 0 = (n+1) > 0, for any n ≥ 1 ∂s¯1 d

Corollary 7

56 i 2 Proof. Since both the denominator and the numerator of x0 depend on z1, I ﬁrst calculate i the derivative of each part. Starting from the expression of the equilibrium trade x0 given by (15), and the deﬁnitions of ra, φa, rb and φb given by equations (13) and (14), we get:

2 2 2 2 2 2n+1 2 2 2 ∂r 2n + 1 ∂φ a σ (n + 1) r1 + a σ 2 a σ z1 a2σ2 a = −a2σ2 < 0; a = (n+1) = > 0 2 2 2 2 2 2 2 ∂z1 (n + 1) ∂z1 (n + 1) ra (n + 1) ra 2 2 4abσ2 2 2 ∂r 4abσ2 ∂φ 4abσ (n + 1) rb − 2 4abσ z1 4abσ2 b = > 0, b = (n+1) = > 0 ∂z2 2 ∂z2 4 2 2 2 1 (n + 1) 1 (n + 1) rb (n + 1) rb

i This implies that the derivative of the numerator of x0 is:

s¯1 2¯s1 " # − 2 −1 −1 (n+1)ra (n+1) rb s¯1 ∂ra 2 ∂rb ∂ 2 = 2 − 2 ∂z1 n + 1 ∂z1 n + 1 ∂z1 " # s¯ (2n + 1) a2σ2 8abσ2 = 1 + n + 1 2 2 3 2 (n + 1) ra (n + 1) rb

i Similarly, the derivative of the denominator of x0 is:

∂ (φ + (n + 2) φ + 2nφ ) (n + 2) a2σ2 4abσ2 n a b = + 2n ∂z2 2 2 2 2 1 (n + 1) ra (n + 1) rb

Thus combining both derivatives and using the notation d = φn + (n + 2) φa + 2nφb, I get:

h 2 2 2 i 2 2 2 ds¯1 (2n+1)a σ 8abσ s¯1 2¯s1 (n+2)a σ 4abσ i 2 + 3 − s + − 2 2 + 2n 2 n+1 2 2 0 (n+1)ra 2 2 ∂x0 (n+1) ra (n+1) rb (n+1) rb (n+1) ra (n+1) rb 2 = 2 ∂z1 d

i ∂x0 Thus 2 ≥ 0 iﬀ ∂z1

" 2 2 2 # 2 2 2 ! ds¯1 (2n + 1) a σ 8abσ s¯1 ((n + 1) rb − 2ra) (n + 2) a σ 4abσ + ≥ s0 + + 2n n + 1 2 2 3 2 2 2 2 2 2 (n + 1) ra (n + 1) rb (n + 1) rarb (n + 1) ra (n + 1) rb

Now let’s consider the limit case where arbitrageurs become risk-neutral, b → 0. The previous condition becomes:

i b→0 2 2 2 2 ∂x0 d s¯1 (2n + 1) a σ s¯1 (n + 1 − 2ra) (n + 2) a σ ≥ 0 ⇔ ≥ s0 + 2 2 2 2 2 2 ∂z1 n + 1 (n + 1) ra (n + 1) ra (n + 1) ra

57 b→0 Since d = φn + (n + 2) φa, we can rearrange terms and get:

i ∂x0 (2n + 1) (φn + (n + 2) φa) n + 1 − 2ra If b → 0, 2 ≥ 0 ⇔ − 2 s¯1 ≥ s0 ∂z1 (n + 1) (n + 2) (n + 1) ra | {z } Θ

2 2 After a simple calculation, I get Θ > 0 iﬀ a2σ2z2 < c ≡ (n+1) (n+1)(2n+1)φn+n+3−(n+1) . Clearly, 1 n 2n+1 (n+1)(2n+1)φn+n+3 2 2 (n+1)(2n+1)φn+n+3−(n+1) < 1, thus c < (n+1) . Hence there are two cases: (n+1)(2n+1)φn+n+3 n 2n+1

2 i 2 2 2 (n+1) ∂x0 • If cn ≤ a σ z1 < , then Θ < 0 and 2 < 0 2n+1 ∂z1

i 2 2 2 ∂x0 −1 • If 0 < a σ z1 < cn, then Θ > 0 and 2 ≥ 0 iﬀ s1 > Θ s0 ∂z1

−1 In the result, I use the notation θn,a = Θ .

To derive the comparative statics of the spread, I start from the expression of the spread schedule (12), and get:

 s¯1  ∂ s0 + Pn i n ∂∆0 2 (n+1)ra n + 2 ∂ i=1 x0 n + 2 X i ∂φa = 2aσ  − (1 + φa) − x0  ∂z2 ∂z2 n + 1 ∂z2 n + 1 ∂z2 1 1 1 i=1 1

When b → 0, this gives:

2 2 ds¯1 2 2 2n+1 s¯1(n+1−2ra) (n+2)a σ n+1 a σ 2 2 − s0 + 2 2 2 ∂∆0 2n + 1 2 2 n + 2 (n+1) ra (n+1) ra (n+1) ra = a σ s¯1 − (1 + φa) 2 3 2 2 ∂z1 (n + 1) ra n + 1 d s¯1(n+1−2ra) n + 2 a2σ2 n s0 + 2 − (n+1) ra 2 2 n + 1 (n + 1) ra d

After rearranging terms, I get:

∂∆0 2 2 2 2 s¯1 (n + 1 − 2ra) 2 = αa σ s¯1 + βa σ s0 + 2 , with ∂z1 (n + 1) ra

58 2n + 1 (n + 2) (2n + 1) 2n + 1 n3 + 2n2 + n α = − (1 + φa) = + (n + 2) (n − 1) φa > 0 3 2 4 2 4 2 (n + 1) ra (n + 1) rad (n + 1) rad n + 1 (n + 2)2 (1 + φ ) n (n + 2) β = a − 3 2 3 2 (n + 1) ra (n + 1) rad   n + 2 = (n + 2) φn − n +φa (n + 2) (φn + (n + 2) φa + n + 2) 3 3   (n + 1) rad | {z } >0

Hence with α and β strictly positive, the derivative is positive.

Corollary 8

Proof. Follows from Proposition 3.

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65 0.5 1

0.45 0.9

0.4 0.8

0.35 0.7

0.3 0.6

0.25 0.5

0.2 0.4

0.15 0.3

0.1 0.2

0.05 0.1

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

x0^i nx0^i

Figure 1: Individual and aggregate arbitrageurs’ trades at time 0 as a function of the number of arbitrageurs (risk-free case). The parameters are: s0 = s1 = 1.

Figure 2: Spread dynamics with con- Figure 3: Arbitrageurs’ trades with stant and decreasing shocks. The constant and decreasing shocks. parameters are: n = 5, a ∗ σ2 = 1, s = 2 The parameters are: n = 5, a ∗ σ2 = 1, in constant case, and s0 = 2, s1 = 1.3 in s = 2 in constant case, and s0 = 2, decreasing case. s1 = 1.3 in decreasing case.

66 ∆t Qt

7.5 12.5 12 6.5 11.5 5.5 11 4.5 10.5 10 3.5 9.5 2.5 9 1.5 8.5 8 0.5 t 7.5 t ‐0.5 135791113151719212325272931 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Baseline Surprise Baseline Surprise

Figure 4: Eﬀect of a surprise shock Figure 5: Eﬀect of a surprise shock on the equilibrium spread. The pa- on arbitrageurs’ positions. The pa- u u rameters are: s = 10, Q−1 = 5, ∆s = 2, rameters are: s = 10, Q−1 = 5, ∆s = 2, a = 1, σ2 = 1, n = 2, T −1 = 30, t∗ = 15. a = 1, σ2 = 1, n = 2, T −1 = 30, t∗ = 15.

∆t Qt

t t

Figure 6: Eﬀect of an announced Figure 7: Eﬀect of an announced shock on the equilibrium spread. shock on arbitrageurs’ positions. The parameters are: s = 10, Q−1 = 5, The parameters are: s = 10, Q−1 = 5, ∆su = 2, a = 1, σ2 = 1, n = 2, ∆su = 2, a = 1, σ2 = 1, n = 2, T − 1 = 30, t1 = 10, t2 = 20. T − 1 = 30, t1 = 10, t2 = 20.

67 ∆t Qt

t t

Figure 8: Eﬀect of an announced Figure 9: Eﬀect of an announced shock on the equilibrium spread - shock on arbitrageurs’ positions - shorter announcement. The param- shorter announcement. The param- u u eters are: s = 10, Q−1 = 5, ∆s = 2, eters are: s = 10, Q−1 = 5, ∆s = 2, a = 1, σ2 = 1, n = 2, T − 1 = 30, a = 1, σ2 = 1, n = 2, T − 1 = 30, t1 = 10, t2 = 15. t1 = 10, t2 = 15.

∆t Qt

t t

Figure 10: Eﬀect of a surprise shock Figure 11: Eﬀect of a surprise shock on the equilibrium spread with a on arbitrageurs’ positions with a monopolistic arbitrageur. The pa- monopolistic arbitrageur. The pa- u u rameters are: s = 10, Q−1 = 5, ∆s = 2, rameters are: s = 10, Q−1 = 5, ∆s = 2, a = 1, σ2 = 1, n = 1, T −1 = 30, t∗ = 15. a = 1, σ2 = 1, n = 1, T − 1 = 30, t∗15.

68 ∆t Qt

t t

Figure 12: Eﬀect of an announced Figure 13: Eﬀect of an announced shock on the equilibrium spread shock on arbitrageurs’ positions with a monopolistic arbitrageur. with a monopolistic arbitrageur. The parameters are: s = 10, Q−1 = 5, The parameters are: s = 10, Q−1 = 5, ∆su = 2, a = 1, σ2 = 1, n = 1, ∆su = 2, a = 1, σ2 = 1, n = 1, T − 1 = 30, t1 = 10, t2 = 20. T − 1 = 30, t1 = 10, t2 = 20.