key: cord-0814117-ckzhisvv
authors: Ouzan, Samuel
title: Loss aversion and market crashes
date: 2020-07-07
journal: Econ Model
DOI: 10.1016/j.econmod.2020.06.015
sha: 19831935d59ed4afb940fe492b952a0a531a2254
doc_id: 814117
cord_uid: ckzhisvv

This study proposes a rational expectation equilibrium model of stock market crashes with information asymmetry and loss averse speculators. We obtain a state-dependent linear optimal trading strategy, which makes the equilibrium price tractable. The model predicts nonlinear market depth and the result that small shocks to fundamentals (e.g., supply or informational shocks) can cause abrupt price movements. We demonstrate that short-sale constraints intensify asset price collapses relative to upward movements. The model also generates contagion between uncorrelated assets. These results are consistent with the main puzzling features observed during market crashes, namely abrupt and asymmetric price movements that are not driven by major news events but coupled with a spillover effect between unrelated markets.

Economists find it difficult to rationalize the growing number of local stock market crashes leading to widespread turbulence in the global capital market. It is now widely agreed that macroeconomic shocks There is some consensus in the literature 1 about the definition of "crash". A crash should include the following three striking features, most generally observed during times of market disturbance:

1. Large movements in stock price without a correspondingly large public news event 2. More severe average downward price movements than equivalent upward price movements 3. Financial contagion (shock spillover between fundamentally unrelated securities or markets) While interpretation of the above features remains puzzling, constrained information asymmetry frameworks can provide them at least partial support. This encouraged us to develop a tractable equilibrium with asymmetric information to investigate the relationship between loss aversion and market crashes. The theory we propose predicts the above phenomena. We find that speculators' loss aversion might be a major cause of market turmoil, accompanied by financial contagion. Despite the tremendous effort devoted to applying loss aversion to finance problems, 2 no previous theoretical work has investigated the intuitive relationship that might exist between a market meltdown and loss aversion. We believe it is important to fill this gap.

Until recently, behavioral models of information asymmetry in financial markets focused mostly on economies where traders are overconfident. 3 Only Pasquariello (2014) proposes one with prospect theory preferences. Specifically, he highlights the impact of speculators' loss aversion on market quality. 4 In our setting, we assume that speculator preferences are not dictated to by prospect theory (Kahneman and Tversky 1979; Tversky and Kahneman 1992) but rather by the theory of von Gaudecker et al. (2011) . In our model, speculator preferences augment the standard exponential utility function with the loss aversion parameter, and assume the same type of curvature on gains and losses, which makes them much less 2 Benartzi and Thaler (1995) apply loss aversion in order to provide a theoretical explanation of the equity premium puzzle; Odean (1998b) finds that individual investors are reluctant to realize losses; Barberis et al. (2001) find that stock returns have a high mean, are excessively volatile and are significantly predictable, and in line with observations of historical data; Genesove and Mayer (2001) show that house sellers are unwilling to sell below the buying price; Gomes (2005) , and Berkelaar et al. (2004) find that investors abstain from holding stocks unless they expect the equity premium to be high; Dittmann et al. (2010) explain observed compensation practices, given certain assumptions. Pasquariello (2014) explains patterns related to liquidity and price efficiency. More recently, Easley and Yang (2015) document the impact on investor's survival prospect. complex to include in the model than prospect theory preferences. Besides their ability to capture loss aversion, it is mainly due to the properties they share with the standard exponential utility function, and the difficulty of avoiding approximation when deriving market equilibrium with asymmetric information under non-standard (or exponential) CARA preferences. 5 Our rational expectation equilibrium model of crashes departs also obviously from Pasquariello (2014) economy since we introduce short-sale constraints and multiple risky assets to study asymmetries in the dynamic of returns, and contagion between a priori unrelated markets, respectively.

The model we develop is a noisy rational expectation equilibrium model à la Grossman and Stiglitz (1980) in which competitive price-taking, loss-averse speculators endowed with private information maximize their expected utility, trading with liquidity traders and risk-neutral uninformed market makers (MMs) . We obtain a closed-form characterization of the non-linear equilibrium, and a state-dependant linear optimal trading strategy, where loss aversion restrains speculators from trading for weak signals.

Trading intensity within the trading-region only remains unchanged in comparison to risk averse speculators, which makes MMs' inference problem tractable for the first time in an economy with informed loss averse traders. It permits to highlight MMs' confusion phenomenon and demonstrate how small trigger shocks from noise traders or insignificant news events can create large price movements in the intermediate price region. 6 It is also consistent with evidence reported by Cutler et al. (1989) , according to which both crashes and market bubbles appear without any preceding public news. 7 They appear when the absolute value of the aggregate order flow is low. In that state, market makers feel extremely confused about the trading regime of informed traders, and so the market depth becomes nonlinear. Typically, asset collapses occur when the market depth is low.

In a second stage, we extend the model to incorporate short-sale constraints. The unique interaction between loss aversion and short-sales produces asymmetric price movements. Short-sale restriction makes high prices reveal less information than low prices, and produces asymmetric price movements, as reported in the literature (see, e.g., Pindyk 1984; French et al. 1987; Bekaert and Wu 2000; Yuan 2005) .

Specifically, prices are more likely to fall than to rise. In other words, we show that, unlike under certain restrictive conditions, in which both risk aversion and the number of traders that cannot sell short are very low, markets melt down more often than they melt up. 8 When we analyze historical stock return data, we observe that the net change of eight of the ten largest one-day movements of the S&P 500 since 1947 were declines. Thus, from a regulatory perspective, the presence of short-sale constrained, betterinformed, loss-averse traders mitigates the effectiveness of bans on short sales defended by regulators, which are very often imposed in times of market stress. This is indeed consistent with Beber and Pagano's (2013) evidence that short-selling bans were detrimental to liquidity, slowed price discovery, and failed to support prices.

Finally, we extend the proposed model to multiple assets, showing a contagion effect between uncorrelated assets. In other words, we highlight how idiosyncratic shocks unique to one market affect asset prices in other markets. We demonstrate this formally for the optimal demand of informed traders.

Following a news event concerning one asset, loss-averse informed traders might refrain from trading assets in a portfolio consisting only of uncorrelated assets. Unfortunately, the inference problem for the equilibrium price in a multiple-risky-assets economy becomes intractable. We provide numerical approximation in a two-risky-asset economy and demonstrate that contagion occurs. Namely, that small news event in one market might create substantial price movement in another uncorrelated market. This spillover mechanism is the third acknowledged feature of market crashes. The fact that crashes appear to 8 We check for negative skewness in economies where the risk aversion is much greater than other technology parameters, or which have a short-selling ban or a high level of short sales. be contagious is supported by the empirical observation that the correlation between assets increases sharply in falling markets and cannot often be explained by fundamentals. 9

The literature on stock market crashes involving trading constraints often attributes such crashes to either increased or reduced uncertainty. The models developed by Romer (1993) , Cao et al. (2002) , ascribe crashes to constrained traders for whom uncertainty is suddenly resolved, bringing prices closer to fundamental values. However, the models of Barlevy and Veronesi (2003) , Yuan (2005) , and Marin and Olivier (2008) do not accredit crashes to any release of information, but on the contrary, to sharp increase in uninformed traders' uncertainty with regard to fundamentals. Our model blurs this distinction. Market meltdown arises principally because speculators' optimal demand is partially non-revealing when uninformed traders' uncertainty is high, but when the release of trivial information might suddenly solve the market makers' inference problem, causing prices to collapse. Our setting is closer to the rational expectation equilibrium models of Barlevy and Veronesi (2003) , and Yuan (2005) , than to that of Marin and Olivier (2008) . Unlike these authors, we do not make informed-investor trading observable by all market participants. Rather, uninformed traders constantly speculate on the constraint status of insiders. However, our model departs also from the models of Barlevy and Veronesi (2003) and Yuan (2005) in that high prices are not necessarily more informative than low prices.

The analysis presented in this paper is also related to the work of Ozsoylev and Werner (2011) , Condie and Ganguli (2011) and, Mele and Sangiorgi (2015) . Using a rational expectation equilibrium framework, they study how ambiguity (Knightian uncertainty (Knight, 1921) ) over fundamentals affects asset prices. Typically, these papers, based on the standard Grossman and Stiglitz (1980) model, assume that markets are subject to ambiguity and that it is impossible to quantify the risky payoff probabilistically. Like loss aversion, ambiguity aversion is a robust, well-documented behavioral bias. As 9 In this paper, we do not attempt to develop an exhaustive theory of contagion between financial assets, since many channels might explain this phenomenon (Kaminsky et al., 2003) . Instead, we include multiple assets to attempt to produce a unifying theory of crash from a behavioral perspective that robustly explains its three striking features of market crashes. As stated, these are strong price variations without major news events, asymmetrical upward and downward movements, and contagion between apparently uncorrelated assets. One limitation of our model is however that unfortunately, the contagion effect is symmetrical and practical limitations of our model prevent us to introduce short-sale constraints in a two risky asset economy. in our study, Ozsoylev and Werner (2011) , and Mele and Sangiorgi (2015) incorporate a noisy supply in their model and induce partial revelation of information. Whereas in our model, substantial price movements follow insignificant supply or informational shocks, Ozsoylev and Werner (2011), Condie and Ganguli (2011) , and Mele and Sangiorgi (2015) show how large price swings occur after a small change in the uncertainty parameters.

The remainder of this paper is organized as follows. In section 2, we present a theoretical model of trading between loss averse speculators, market makers, and noise traders. We derive optimal demand, equilibrium price and market depth for our economy. We then perform comparative statics. Section 3 introduces short-sale constraints, and we obtain the asymmetry between downward and upward price movement. In Section 4, we extend the model to multiple risky assets and derive the contagion between two uncorrelated assets. Finally, Section 5 concludes and proposes avenues for further research.

In this section, we describe a noisy Rational Expectation Model (REE) of sequential trading in the presence of better informed, loss averse speculators. Like Grossman and Stiglitz (1980) , Diamond and Verrecchia (1981) and Vives (1995a) , we assume that speculators are competitive, submit limit orders instead of market orders and that all random variables are normally distributed. Our model departs moderately from the Constant-Absolute-Risk-Aversion (CARA) Normal model. It incorporates a loss aversion parameter in the utility function while assuming the same type of curvature in gains and losses.

The economy is populated with informed traders, liquidity ("noise") traders whose demand is exogenous and who trade for idiosyncratic lifecycle or liquidity reasons, and risk-neutral competitive market makers (MMs). 10 Informed traders are competitive and form a continuum with measure one. The model includes two dates, time 0 and time 1. At time 0, investors trade competitively in the market based on their private information. At time 1, they realize payoffs from the assets and consumption occurs.

There is one risk-free asset and one risky asset. The risk-free asset is a claim to one unit of terminalperiod wealth, and the risky asset pays v units of the single consumption good. While taking the riskfree asset to be the numeraire, we let P be the price for the risky asset. Prior to trading, informed investors receive private information related to the payoff of the risky asset. The signal s is a noisy signal of the asset final payoff v , given s v ε = + . We assume that all the informed investors receive the same private signal . s 11 We assume the random variables v and ε to be mutually independent and normally distributed with mean zero 12 and variance 2 v σ and 2 ε σ . Liquidity ("noise") traders produce a random, normally distributed demand z with mean zero and variance 2 . z σ Moving first, liquidity traders submit market orders and speculators submit demand schedules or generalized limit orders to the MMs, 13 contingent on their information before the equilibrium price P is set. When speculators optimize their demand, they consider the relation between the functional equilibrium price and the random variables in the economy. Then, competitive, risk neutral MMs set the price efficiently, given the observed aggregate order flow. We know that in large markets, competitive noisy rational equilibriums are implementable, thus allowing agents to use demand schedules as strategies. As in Vives (1995a) and Pasquariello (2014) we denote a speculator demand schedule by ( ,.) i x s . Thus, when the price is P, the desired position of the informed trader is ( , ). i x s P We assume that the speculator perceives the investment of all her initial wealth ,0 i W in the risk-free asset as the reference point, and any other outcomes as changes or profits with respect to this reference point. Since we assume that the risk-free rate is zero, speculator i perceives the reference point as ,0 i W . So, the profits at time 1, from speculator i, are given by ( ).

i i

x v P π = − 11 We assume that speculators observe identical signals and have identical preferences. A model with diverse signals and/or diverse preferences is much more complicated and cannot derive a tractable equilibrium. 12 We assume, to save on notation, that the mean of v is zero. However, for the general value of [ ] E v the derivation remains the same for Vives (1995b) suggests an equivalent yet more intuitive way to describe the market clearing process by referring to the notion of simultaneous placement of orders to a centralized auctioneer (CA) (see also Yuan (2005) , and Ozsoylev and Werner (2011)).

Since the introduction of the Allais paradox (Allais, 1953) several authors have documented violations of the basic expected utility theory. Camerer's (2000) review of the literature finds that one particularly persistent empirical finding is a greater sensitivity to losses than to similar size gains. This idea that people are loss averse with respect to changes in wealth is a central feature of prospect theory (Kahneman and Tversky 1979 

where ?? > 1 represents the degree of loss aversion and ?? > 0 is the coefficient of absolute risk aversion.

In our model, all speculators, have the same utility function ( , , ) U π γ λ , so we drop the subscripts i.

The "CARA Normal" model is very popular. In various settings, it admits linear equilibria derived in a closed form. 14 The objective of our preference choice is to maintain as much as we can of its modeling features, while successfully introducing loss aversion. The kink at the origin of ( , , ) U π γ λ , with a steeper slope for losses than for gains, represents our only departure from CARA. This functional form, plotted in Fig. 1 , disentangles preference parameters from utility curvature (risk aversion) and loss aversion.

14 See, e.g., Hellwig (1980) , Grossman and Stiglitz (1980) , Diamond and Verrecchia (1981) , Admati (1985) and Vives (1995a) .

Note: In line with loss aversion, this function exhibits a kink at its origin. The starred line represents one with a loss aversion parameter of 2.5 λ = and the crossed line represents the case when 1 λ = , (CARA preferences). The risk aversion parameter is 1 γ = .

Although prospect theory considers the utility function as concave over gains and convex over losses, 15

Equation (1) assumes concavity over gains as well as losses. 16 In their web appendix, von Gaudecker et al. (2011) compare the proposed specification with prospect theory specification and report slightly larger values of λ for prospect theory preferences. They consider this as a mechanical consequence of different assumptions with regard to the shape of the utility function in the negative domain. For high-incentive treatment, they report a median estimate parameter of loss aversion of 2.38, in line with previous estimates. 17

Let ( ,.) x s represent the demand schedule for the risky asset of an informed trader given private signal s.

When the price realization is P, the demand function is then ( , ) x s P . The only information available to the informed trader at time 0 is the noisy signal s. Speculators neither learn from market price nor strategically anticipate how they demand will affect equilibrium price. Thus, the demand of the informed trader submitted at time 0, is given by the maximization of the expected utility x is the sign function. 18 We present the derivation of Equation (2) and of the conditional mean and variance in the appendix. Equation

(2) admits for each region (either positive demand or negative demand) only one bounded maximum value, since, as we will see in the ensuing analysis, the first-order condition of Equation (2) is solved for at most one value in each region. For unbounded values of x in each region, the objective function is equal to minus infinity.

Taking the first order condition of Equation (2) with respect to x, yields that the impact on loss aversion is more pronounced for anticipated outcomes, when evaluating experienced returns, the effect diminishes by more than half.

Whether each speculator would engage in short or long position, the term in the bracket of Equation (3) should be equal to zero since the exponential function 

For any degree of loss aversion ( 1) λ ≥ , one can solve Equation (4) λ Λ = Λ PROPOSITION 1. In the economy described above, the optimal demand for the loss averse informed trader is given by ( 1)

We notice that for 1 λ = , ( ) 0 λ Λ = , and the optimal demand falls to the optimal generalized limit order under the regular CARA-Normal model with negative exponential utility (see, e.g., Vives 1995a; Grossman and Stiglitz,1980) .

Optimal demand is therefore a state-dependant linear function of the private signal and the equilibrium price. As for the standard CARA-Normal setting, the proposed model predicts that informed traders submit cautious limit orders. Loss aversion however prevents speculators from trading at all for a sufficiently weak signal in terms of absolute value.

Trading intensity (Vives, 1995a) is defined as the sensitivity of speculators' demand function to

Increasing loss aversion or increasing risk aversion increases the cautiousness of the trade. The losses induced by trading, which obviously increase in proportion to the speculator's loss aversion, are reflected in a reduction of optimal trading activity compared with risk averse speculators only. Outside the no-trade interval, the measure of loss averse trading aggressiveness is the same as for the standard CARA-normal model, depending solely on the precision of the private signal and on risk tolerance. In our model, since we disentangle risk aversion and loss aversion in speculators' preferences, loss aversion does not affect the trading intensity for sufficiently large signals. Intuitively, an infinitesimal informational shock does not increase the likelihood of expected losses but affects the distribution of gains (change in final wealth).

Thus, if speculators already trade on their private information, the degree of loss aversion should not impact trading intensity while risk aversion should.

We now describe equilibrium prices and trading behavior in the model. MMs set the market clearing price from their expectation of the asset payoff v conditional on the aggregate order flow realization , x z ω = + according to semi-strong market efficiency.

( )

Risk neutrality and dealership competition imply the semi-strong market efficiency rule expressed by

Equation (8). 19 The MMs earn zero expected profit, conditional on the order flow. According to Vives (1995b) , this condition can be justified by Bertrand competition among risk neutral MMs who observe the limit order book and have symmetric information. It can also be explained by a situation where liquidity traders submit market orders z jointly with competitive, price taking privately informed speculators who submit optimal demand schedules x and a continuum of risk neutral MMs who submit demand schedules based on prices to a central mechanism according to the semi-strong efficiency rule of Equation (8).

Equilibrium prices are thereby set by a Walsarian centralized auctioneer (CA) (see also Yuan (2005) , Ozsoylev and Werner (2011) , and Pasquariello (2014)) to equate the aggregate excess demand from all the model's market participants to zero. In this case, in equilibrium, Equation (8) is necessarily verified, since otherwise MMs would likely take unbounded positions.

From Equation (5), this implies that the optimal demand schedule * LA x depends on risk aversion, loss aversion, market clearing price and the magnitude of the private signal. For a given private signal at a given equilibrium price, speculators optimal demand falls either within the no-trade interval or the trading interval. Thus, MMs have to guess speculators' trading status. Following Pasquariello (2014) and in the spirit of Yuan (2005) 

are the probability of the order flow being informative, while

is the probability of the order flow giving no information about the risky payoff v.

Since the optimal demand schedule * LA x of Equation (5) 

Equation (11) is identical to the mean variance preferences, equilibrium price found by Pasquariello (2014) and, in particular, is a special case of the linear equilibrium in (Vives 2010, Proposition 1.11) when a continuum of risk averse speculators receive identical noisy signals of the asset payoff.

In equilibrium, informed agent i, buys or sells according to whether s, the private estimate of v is larger than 2 P ρ + ∆ or smaller than 2 P ρ − ∆ , and does not trade otherwise. In their trading region, Trade occurs in this type of model because of the presence of noise traders and because of the information advantage held by informed agents over the MMs. The asymmetric information between speculators and MMs creates typically two opposite effects, selection and information (efficiency). While higher-quality private signals encourage informed agents to trade more aggressively, and thus to exploit their information premium more efficiently, they also typically reveal more private information to the MMs. The insider's information advantage still holds, but it could diminish or increase depending on risk aversion, the quality of private signal and the noise. In that sense, the camouflage concealing informed agents' trading from MMs varies with the economy's parameters. In our model, a third effect, related to the MMs' uncertainty regarding the informed investors' trading region, influences the information discovery process. Because of loss aversion, when the magnitude of the aggregate order flow is very low (high), MMs can be relatively sure (unsure) that the informed traders did not submit any limit order, inferring that the information advantage held by insiders is not (is) exploited (Equation (5)). In case of informative aggregate order flow, the trading intensity ζ outside the no-trade region is similar to that of the CARA model, and therefore both the information content of the equilibrium price and the price itself should be close to that given by the CARA model. 20

To highlight the effect of loss aversion on information sharing and on the equilibrium price formation process, we numerically analysed an economy with a typical market-specific calibration, whose parameters make the expected return on the risky asset to 6% and the standard deviation 20%. We follow Hirshleifer et al.'s (1994) setting 2.5 The value of the risk aversion coefficient is consistent with historical estimates of the market risk premium. We retrieve similar inferences from other market specification calibrations proposed in the literature (see, e.g., Gennotte and Leland 1990; Leland 1992; and Yuan 2005 The unique interaction between speculators' loss aversion and MMs' adverse selection can generate steep price movements, up or down. Fig. 4 graphs the sensitivity of the equilibrium price to signal and noise trading shocks for an economy populated by loss-averse informed traders endowed with private information.

Note: The lines in the left-hand graph represent equilibrium price as a function of the signal shock when the supply shock is -20, -8, 20, and 8. The lines in the right-hand graph represent equilibrium price as a function of supply shock when the signal shock is -50, 0, 80, and 400.

The equilibrium price becomes sensitive to shocks in the intermediate price region when it is more difficult for the MMs to infer the quality of the private signal and to conjecture the trading status of informed traders. The magnitude of such sensitivity decreases with the degree of precision of the private signal and increases with the level of speculators' loss aversion. In this model, large market downturns or upturns may occur following insignificant supply or informational shocks, regardless of the value of the underlying asset. Our model is consistent with empirical findings, reported by Cutler et al. (1989) , that important prices movement can occur without any particular news event.

The market liquidity measure 1 LA λ − can be expressed as the inverse of the price impact LA LA P z λ ∂ = ∂

. By implicit function theorem, the equilibrium price impact is For the intermediate price region, the market depth is highly nonlinear in noise trading demand. As for the CARA normal case, the price impact is nonnegative, since the market maker attempts to offset losses due to the presumably adverse selection of the speculator with profits from noise trading. Fig. 5 illustrates a numerical example of price impact for a given signal shock, with the specific calibration of the technology parameters discussed above, and for different degrees of loss aversion.

Note: The graph represents the price impact (inverse measure of liquidity) of noise traders' shocks where the private signal shock is 0, and for risk averse speculators and loss averse speculators with loss-aversion coefficients of 2.5 and 4 respectively.

We can separate the price impact into three distinct states corresponding to three different levels of inferred likelihood of informed trading status for MMs. A price impact close to zero minimizes MMs' adverse selection problem. However, when the price impact is constant, the problem is merely that of mean variance. Finally, in between these extremes, the market depth emphasizes how difficult it is for MMs to infer the trading status, and thus a small supply shock can have a great effect on the equilibrium price initially unjustified while MMs misinterpret the trading status of the informed trader. Indeed, MMs cannot distinguish between a shock in the private signal and a shock in the noisy demand. Our model supports recent empirical evidence suggesting that the relationship between orders and price adjustment may be nonlinear where large price fluctuations occur when the market depth is low, in line with the comparative static analysis we presented. This is consistent with the empirical study by Pastor and Stambaugh (2003) , which used a related measure of price sensitivities as a measure of market liquidity.

The authors found several episodes of extremely low aggregate liquidity, including the October 1987 crash and the LTCM crisis of September 1998.

In section 2, we demonstrated how asymmetric information in the presence of loss averse informed traders creates large price movements in financial markets. In addition, we showed that large price changes might occur without any particularly dramatic news events. Yet this section fails to support the other main empirical evidence in the literature on market crashes; namely, the fact that large price movements are more likely to be decreases than increases.

Several models of asymmetric information introduce short-sale constraints while providing partial support for episodes where stock prices fall substantially. When crashes appear because of an increase (rather than a sudden fall) in confusion and uncertainty, 23 the models of Barlevy and Veronesi (2003) , Bai et al. (2005) and Marin and Olivier (2008) all require at least a fraction of traders to be unable to sell assets short, to explain the markets' pervasive tendency to melt down. In an entirely different setting, Hong and Stein (2003) develop a disagreement (differences of opinion) model with no asymmetric information, which successfully predicts the three distinct features of market crashes. Interestingly, in their competing behavioral theory of crashes, they also introduce short-sale constraints in their model. 24 We follow the same line of thought and extend our model to study the impact of short-sale constraints on optimal demand and equilibrium price. In this section, we derive the necessary conditions for the technology parameters. The unique interaction between asymmetric information, loss aversion and short-23 In practice, however, crashes appear to involve an increase in uncertainty. The empirical part of Marin and Olivier's (2008) work indeed strongly supports this fact, showing that informed traders tend to leave the market before crashes. 24 Differences of opinion are associated with one of the two principal aspects of overconfidence, the better than average effect, since in their model each group of traders believes his signal is of better quality than that of the other group.

sale constraints supports the empirical evidence for large price movements tending to be downward rather than upward.

We consider a market identical to that in section 2. Furthermore, we assume that informed traders might be subject to short-sale constraints. Short-sale constraints mean that investor i's position is bounded below by a non-positive number:

We assume that 0 1 κ ≤ < of informed traders are subject to short-sale constraints, 25 and index them by

, and that the rest, with mass (1 ) κ − , are unconstrained. For convenience, the short-sale constraint is assumed to be the same for all constrained speculators, so we drop the subscript i associated with constraint . b One can extend the development for the optimal demand of unconstrained speculators to show that the optimal demand schedules for constrained informed traders are

where uc LA x refers to the optimal demand of Equation (5).

Following our previous development, this result is straightforward, and given by the following.

PROPOSITION 3. In the economy described above, the optimal demand for the loss averse and shortsale constrained informed trader is given by 

where ( ) ( ) b For b = ∞ , Equation (14) is reduced to Equation (5).

The equilibrium price results from the presence of both unconstrained and constrained informed traders. We denote the aggregate order flow by (1 ) , c uc x x z ω κ κ = + − + which refers to the noisy limitorder book schedule observed by market makers, where uc x is given by Equation (5) and c x refers to Equation (14). MMs earn zero expected profit, conditional on the order flow. The market-clearing price P satisfies the semi-strong market efficiency rule expressed by Equation (8) constrained optimal demand schedules of Equation (5) and (14) make ω a linear function of the private signal s, and since the boundaries are not functions of the received private signal (e.g., ∆ and Γdo not depend s). Since the inference problem uses the same mathematical properties and follows very similar steps to those described in appendix, we skip the intermediary steps. As the equilibrium price is quite cumbersome and lengthy, we report it in the appendix.

Once we derive the equilibrium price function in an economy where some speculators have short-sale constraints, we can find the conditions for asymmetric price movements. Similarly to Barlevy and Verosini (2003) and Yuan (2005) , but unlike Bai et al. (2005) 

Besides the fact that high prices are less informative than low prices, since in the high-price regime, short-sale constraints bind, speculators trade less aggressively overall, revealing less information to the market makers, the selection effect and efficiency effect still play an important role in a constrained economy, particularly in determining the sign of the skewness of the risky asset. For a sufficiently large 26 It is straightforward to demonstrate that the equilibrium price with short-sale constraints (Equation (A.16)) is reduced to the equilibrium price without short-sale constraints (Equation (10) κ coefficient of risk aversion compared with other technology parameters of the economy, or for an economy with a short-selling ban, or at least an economy with a high level of short-sale restrictions (Equation (16)), we observe an asymmetry of large price movements supporting the empirical evidence of negative market skewness. This indeed confirms the economic intuition; i.e. a positive shock moves the price above the fundamentals, and uninformed investor are more likely to be short-sale constrained.

Hence, market makers are less willing to accommodate noise selling and thus dampen the upward price movements. However, our model might also generate positive skewness. Although under more restrictive conditions, this arises when the efficiency effect dominates the selection effect

, and when the slight information disadvantage (asymmetry) between the high-price and low-price regimes that uninformed traders face cannot be compensated because of the insufficient number of constrained

 that fail to exploit their private information. 28 Therefore, in that situation, when constraints bind, the efficiency effect still prevails over the selection effect, but less than for a low equilibrium price. Therefore, lim lim

This result indeed departs from the out-of-sample predictions of Yuan's (2005) model. Her result indeed, unambiguously supports large asymmetric price falls. Markets are differently organized;

uniformed traders and informed traders share the same preferences, and information is revealed through the self-fulfilling equilibrium price, rather than through the aggregate order flow, as in the present model.

efficiency effect and does not always lead to negative skewness. If we allow in Yuan's (2005) setting uninformed traders to have different risk aversion, and particularly to be risk neutral, (unbiased efficient pricing rule) as in our model, MMs would like to take unbounded positions. This is also the reason why to justify efficient pricing rule we need to assume sequential trading coupled with competition among riskneutral market makers with symmetric information, observing the limit order book (Vives (1995b) ). 29

Although, as in Barlevy and Veronesi (2003) and Yuan (2005) , crashes can occur even when the fundamental are strong, and the magnitude of the crashes depends on the number of unsophisticated passive investors present in the market, crashes also depend on uncertainty with regard to asset payoff and risk aversion. In the above conditions, the less uncertain the asset payoff, the more pronounced negative skewness, suggesting that skewness and price volatility are positively correlated. Fig. 6 graphs the sensitivity of equilibrium price to signal and noise trading shocks.

Note: This graph represents equilibrium price as a function of the signal shock when the noise traders' intensity is zero. 29 Since in Yuan (2005) fictitious economy without uniformed traders is used to solve the equilibrium (Blackwell Theorem) (DeGroot, 1986), we might assume that it isn't the difference in risk aversion that leads to the asymmetry of price movements. Moreover, unreported development shows that if we replace in the proposed model, loss aversion coupled with short sale constraints with borrowing constraint (when assuming that 0 1 κ ≤ < of informed traders are subject to borrowing constraints) we would get asymptotically opposite conditions for the asymmetry between upward and downward movements.

The multiplier effect of short-sale constraints, information asymmetry, and loss aversion is clearly visible in Fig. 6 . In our model, the unique interaction between loss aversion of the speculator, adverse selection between the informed traders and the MMs, and the presence of short-sale constraints among a fraction of speculators can produce a market crisis. 30

In Fig. 7 , a kernel estimation of the price distribution for a short-sale constrained economy with lossaverse speculators has fatter tails than the standard model with unconstrained speculators and CARApreferences predicts and is negatively skewed. This indeed highlights the asymmetry between market meltdown and upward market price movement. Large price drops are more severe than upward movements. Equilibrium price becomes sensitive to shocks in the intermediate price region, when it is more difficult for the market maker to infer the quality of the private signal and to make conjectures about the trading status of informed traders.

Note: The dashed line represents the kernel distribution of the equilibrium price distribution based on 200 draws of ( ) , s z in an economy with loss aversion, asymmetric information, and short-sale constraints. The solid line represents the kernel estimation and the graph represents equilibrium distribution for an economy with asymmetric information only.

The magnitude of such sensitivity decreases with the private signal's degree of precision and increases with the level of speculators' loss aversion. Notice that asymmetry increases with the number of constrained informed traders and the level of constraint b. Short-sale constraints interact with loss aversion at a very fundamental level. One speculator may not trade because of either loss aversion, shortsale constraint, or these two factors combined. The more speculators are constrained, the more difficult it is for MMs to infer a positive signal from the insider.

The pattern of co-movement across apparently uncorrelated assets is not easy to explain. Financial crises often spill over to unrelated markets or to markets with little economic linkage. We extend our model to multiple, independent, and uncorrelated risky assets, and test whether it sheds light on the third important stylized fact about market crashes, namely, the fact that an idiosyncratic shock unique to one market affects asset prices in uncorrelated markets. In this section, we attempt to show that in a multiple-riskyassets economy, price movements can occur even without any shock to its fundamentals. A small informational or supply shock in a single uncorrelated asset of one speculator's portfolio might suddenly induce MMs to update their opinion about the trading status of informed traders and therefore to revise the reservation price of more than one risky asset. Therefore, the price of one asset can be affected by a shock to another independent and uncorrelated asset through this specific uninformed traders' uncertainty channel, induced by speculators' loss aversion. We refer to this spillover mechanism as the contagion effect.

To illustrate this effect, we consider the same two-date economy as defined in section 2 but with N risky assets instead of one. While taking the risk-free asset to be the numeraire, we let be the price for the risky asset .

j Prior to trading, each informed investor receives the same N private signals 1 2 ( , ,..., ), 2 for one risky asset to multiple risky assets, we find that there is a one-to-one correspondence between each pair of assets' optimal demand

and we obtain the following proposition 31 PROPOSITION 4. In an uncorrelated multiple-risky-asset economy, loss averse informed traders either trade for all the risky assets or do not trade at all, and the optimal demand for the risky asset j is given by

The implicit Equation Interestingly, from Equation (17) we see that speculators do not trade for a subset of risky assets only.

The trading region applies uniformly to all assets. Either informed traders submit their demand schedules for all assets simultaneously, or they remain silent and do not seek to trade in any risky asset. This might seem counterintuitive at first, but one can observe that if a private signal on a single asset is much stronger than those on the remaining assets, speculators would certainly trade aggressively in that asset (Equation (18)), independently of the low signal quality of the remaining assets, traded at an inversely proportional intensity (Equation (17)). From a geometric viewpoint, one strong 1-dimension signal (coordinate) that departs from the origin 

Similar to the one-risky-asset economy, MMs infer the private information from the aggregate order flow for each security 1,..., j N =

. Following the semi-strong market efficiency rule, the equilibrium price

j j x z ω = + represents the aggregate order flow for asset market .

j To illustrate the spillover effect, let restrict our economy to 2. N = 32 MMs inference problem in a two-riskyasset is therefore 

where ( ) 

 make the conditional moments in Equation (19) analytically intractable. 33 All the probabilities of Equation (19) can be found analytically. It is easy to

show that in a two-risky-asset economy r follows a rice distribution ( ,1)

r Rice ν with non-centrality parameter 2 2 2 1 .

(1 )

We employ a numerical approach to express ( ) 

We decide to illustrate the mechanism with two uncorrelated markets for the sake of clarity. 33 Note however that recently Arismendi and Broda (2017) derive analytical solution for the unconditional elliptical truncated moments. For conditional truncation, closed formed expression has not been developed yet. 34 While the Marcum-Q function. can be used to represent the cumulative distribution function 

, to be able to solve numerically the implicit functions of the equilibrium prices, we use non-central Wilson-Hilferty approximation (Abdel-Aty, 1954) to approximate the probability of the order flow being informative or not.

1, 2; j = estimated via ordinary least Square (OLS). 35 We describe the approach in the appendix, which yields to the following proposition PROPOSITION 5. In an uncorrelated two-risky-asset economy with loss averse informed traders equilibrium price functions 1 P , and 2 P are the solutions of the following implicit functions Note that it is clear from Equations (19) and (20) Hayashi, 2000) for each market in the same spirit of Pasquariello (2014) for a single-risky-asset economy. 36 MMs' inference problem for 1 2 ( ) P P is a function of 2 1 ( ). P P status of informed traders, forcing them to adjust equilibrium prices abruptly. This time however, the cause of an abrupt adjustment on one market might be a change in the fundamentals of an uncorrelated market. Because MMs know the speculators' decision process, they might suddenly realize that the aggregate order flow has just become uninformative because of the homogeneous conditions of the statedependent optimal demand schedule (Equation (18)) for every risky asset. The source of multiple equilibria in our model is due to the confusion of the MMs on trading status because of speculators' loss aversion. In two-risky asset risky economy this confusion becomes even more pronounced because of the possibility for the traders to be confused from multiple sources (two signal spaces). Multiple equilibria are becoming more trivial. 37 37 Extensive simulations show also that the elliptical region affects substantially the stability of the equilibrium. While the main implication of our model on contagion is robust to alternative calibrations of the technology Finally, it is important to note that one limitation to our model is that the contagion effect is symmetrical. To address the empirical evidence that the spillover effect is often larger during market downturn, than during market upturn, 38 we should introduce short-sale constraints in our multiple-riskyasset setting. However, when introducing short sale constraints the elliptical trading region does not apply uniformly to all assets, making the derivation of the equilibrium unpractical. 39

The study of market crashes has traditionally attracted a great deal of attention both in academia and among practitioners. The proposed theory illustrates the consequences of speculators' loss aversion on the formation of market crashes.

The equilibrium clarifies the role loss aversion plays on market makers' inference problem. Loss aversion restrains speculators from trading for weak signals. Therefore, when the aggregate order flow is low, market makers are very confused, and have difficulty to infer properly the speculators' trading status, so both the equilibrium price and the market depth become highly non-linear. In that situation, a small adverse shock to the fundamentals can trigger a large movement in asset value.

When introducing short-sale constraints, the model provides asymmetry between upward and downward price movements, in line with empirical evidence. For a large set of technology parameters, downward price movements are indeed likely to be more severe than equivalent upward price movements. The proposed model also provides new testable implications for the nature of crashes. Asset returns are likely to exhibit more positive skewness in markets where short-sale constraints are less binding. Finally, we demonstrate that in a multiple-risky-asset economy, contagion can occur even between uncorrelated risky assets. Our model's predictions match historical evidence of market crashes. parameters, the stability of the equilibrium varies nonetheless with the technology parameters. We allow therefore the private signals and supply shocks to be more precise and report the comparative statics for 2 2 1, 1, z ε σ σ = = for both markets. 38 See e.g., Ang and Chen, (2002) and Boyer et al., (2006) . Recently, the result of Ahmed and Huo (2019) mitigates this evidence. They show that during the Chinese stock market crash of 2015 price spillovers from China to other regional markets were more significant during a bullish period. 39 When introducing short-sale constraints in multiple asset setting, we are not able to derive analytically the optimal demand, which makes the derivation of the equilibrium unpractical.

Policymakers can always benefit from studying the impact of cognitive psychology on the price formation process. From a regulatory perspective, the presence of short-sale constrained, better-informed, loss-averse traders mitigates the effectiveness of bans or restrictions on short sales, often defended by regulators during market stress.

We also believe that the tractability of the equilibrium presents a starting point for further research to investigate the role of asymmetric information with more realistic preferences. For example, examining the impact of private information on insurance and hedging under loss aversion might explain the welfare consequences of improvements in private information release from a behavioral economics perspective. It would also be interesting to isolate the role of insiders, because insider trades might have some power to predict market crashes. 40 Making informed trades observable by all market participants would make it possible to distinguish between the impact of insiders and more general forms of informed traders. Our intuition is that such a model would stress even further the role of insiders in predicting market crashes.

The utility function can be written as

The conditional expectation is therefore given by

Note also that depending on positive or negative demand schedule ( ) (2).

Define (4) , the solution * Λ of (4) exists and it is unique, for any degree of loss aversion ( 1) λ ≥ . □

Each conditional expectation and probability of Equation (9) is tractable in our setting. We can express the conditional moments of the truncated normal variables in closed-form (Maddala 1986 

where * ρ refers to the correlation coefficient of the conditional bivariate normal variable , . v s ω The conditional expectation and standard deviation of normally distributed variables (Greene 2003, pp. 90 

The equilibrium price of Equation (10) MMs would do analytically in case of CARA normal setting for uncorrelated assets (see Equation (11)).

We specify first two prices grids: 1 ( ) P n made of N points between 1L P and 1H P , and 2 ( ) P n made of N points between 2 L P and 2 H P , 41 where for 1, 2; j = 

We then assign vectors ( ) ( ) 

Approximate formulae for the percentage points and the probability integral of the non-central χ 2 distribution

A noisy rational expectations equilibrium for multi-asset securities markets

Impacts of China's crash on Asia-Pacific financial integration: Volatility interdependence, information transmission and market co-movement

Le comportement de l'homme rationnel devant le Risque: critique des postulats et axiomes de l'école americaine

Asymmetric correlations of equity portfolios

Multivariate elliptical truncated moments

Asset prices under short-sale constraints

Violations of cumulative prospect theory in mixed gambles with moderate probabilities

Prospect theory and asset prices

Rational panics and stock market crashes

Short-selling bans around the world: Evidence from the 2007-09 Crisis

Asymmetric volatility and risk in equity markets

Myopic loss aversion and the equity premium puzzle

Aggressiveness and survival of overconfident traders

Optimal portfolio choice under loss aversion

How do crises spread? Evidence from accessible and inaccessible stock indices

Prospect theory in the wild: evidence from the field

Sidelined Investors, Trading-Generated News, and Security Returns

Ambiguity and rational expectations equilibria

Do behavioral biases affect prices?

What moves stock prices?

Investor psychology and security market under-and overreactions

Probability and Statistics Addison

Information aggregation in a noisy rational expectations economy

Sticks or carrots? Optimal CEO compensation when managers are loss averse

Loss aversion, survival and asset prices

Stock return variances: The arrival of information and the reaction of traders

Expected stock returns and volatility

Overconfidence and market efficiency with heterogeneous agents

Loss aversion and seller behavior: evidence from the housing market

Market liquidity, hedging, and crashes

Portfolio choice and trading volume with loss-averse investors

Econometric analysis

On the impossibility of informationally efficient markets

Econometrics

On the aggregation of information in competitive markets

Security analysis and trading Patterns when some investors receive information before others

Differences of opinion, short-sales constraints, and market crashes

Prospect Theory: an analysis of decision under risk

The unholy trinity of financial contagion

Risk, uncertainty and profit

Arrogance can be a virtue: Overconfidence, information acquisition, and market efficiency

An index of loss aversion

Continuous auctions and insider trading

Speculation duopoly with agreement to disagree: Can overconfidence survive the market test?

Insider trading: should it be prohibited?

Limited-dependent and qualitative variables in econometrics

The dog that did not bark: insider trading and crashes

Uncertainty, information acquisition, and price swings in asset markets

Financial loss aversion illusion

Are investors reluctant to realize their Losses?

Volume, volatility, price, and profit when all traders are above average

Liquidity and asset prices in rational expectations equilibrium with ambiguous information

Prospect Theory and market quality

Wealth, information acquisition, and portfolio choice

Risk, inflation, and the stock market

Methods of multivariate analysis

Orange juice and weather

Rational asset-price movements without news

Advances in prospect theory: Cumulative representation of uncertainty

Information and learning in markets: the impact of market microstructure

Short-term investment and the informational efficiency of the market

The speed of information revelation in a financial market mechanism

Heterogeneity in risky choice behavior in a broad population

Asymmetric price movements and borrowing constraints: A rational expectations equilibrium model of crises, contagion, and confusion

(A.10) 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 * 2 4 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1where * Σ is the same for 1 ω as well as for 2 ω . From * Σ , we find thatThe probabilities of Equation (9) are given byThe correlation coefficient of the bivariate normal variable , v s ω of the first and third term of Equation (15) is given in (A.12). For the second term, it is simply , ρ and therefore, for the fourth term,Having all the parameters of the conditional truncated normal distribution of each of the four terms of Equation (15), the price function for the equilibrium price can be expressed as the following fixed-point problem ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 1 1 The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.Sincerely, Samuel Ouzan, PhD Assistant Professor of Finance, NEOMA Business School