Learning From Liquidation Prices - Harvard University · 2020. 11. 25. · Learning From Liquidation Prices GianlucaRinaldi∗ November25,2020 Clickforthelatestversion Abstract I

Learning From Liquidation Prices

Gianluca Rinaldi∗

November 25, 2020Click for the latest version

Abstract

I develop a model of investor learning driven by mistaken inference from marketprices. Investors have heterogeneous beliefs about the worst case return of a riskyasset and take leverage to buy it. When the worst case becomes more likely, forcedliquidations result in price crashes, which investors mistake for negative informationabout worst case returns. They therefore revise cash flow expectations downwards,henceforth requiring larger returns. The model predicts that crashes lead to persis-tent changes in future average returns and that larger crashes are followed by largerchanges. To link the model to historical crashes, I consider two strategies associatedwith the Black Monday crash in 1987 and the Lehman Brothers bankruptcy in 2008.Hedged put options selling suffered severe losses around Black Monday, while arbitrag-ing the difference in implied credit risk between the corporate bond and CDS marketswas similarly negatively affected after the Lehman bankruptcy. The losses on thesestrategies in those crisis episodes were likely exacerbated by deleveraging, but the in-creased returns after the crashes have been remarkably persistent, consistent with theimplications of my model.

∗Harvard University: [email protected]. I am indebted to Xavier Gabaix, Andrei Shleifer, JeremyStein, Adi Sunderam, and especially John Campbell for their outstanding guidance and support. I am alsograteful for comments and suggestions to Malcolm Baker, Joshua Coval, Tiago Florido, Nicola Gennaioli,Robin Greenwood, Samuel Hanson, Franz Hinzen (discussant), Clémence Idoux, Lukas Kremens, OwenLamont, Andrew Lilley, Ian Martin, Robin Lumsdaine (discussant), Carolin Pflueger, Nicola Rosaia, DavidScharfstein, Erik Stafford, Argyris Tsiaras, and the participants at the Harvard Finance lunch, HarvardMacro lunch, the OFR PhD Symposium, and the TADC conference.

https://scholar.harvard.edu/files/rinaldi/files/rinaldi_jmp.pdf

1 Introduction

The pricing of financial risks often sharply changes after market crashes. For instance, the

returns to providing tail risk protection in equity markets increased substantially after Black

Monday in October 1987 and have not declined since. Likewise, several arbitrage strategies in

fixed income markets have become persistently profitable after the Lehman Brothers collapse

in 2008. Prices after these traumatic episodes seem detached from the risks perceived before.

Why do such stark changes occur, and why do they persist?

I propose an explanation based on the idea that investors learn from prices while neglect-

ing the importance of leverage. In the model, while investors take on leverage, they think

prices are determined as if their individual leverage choices have no impact on equilibrium

prices. Since other investors also take on leverage, this assumption is mistaken and leads

investors to believe that market prices convey more information than they actually do: they

over-learn.

The baseline model has three periods and investors can either hold cash or buy a risky

asset. The asset represents an investment opportunity that most likely delivers a payoff of

one at time 3 but is exposed to losses in an unlikely worst case scenario. At time 1, investors

decide whether to hold the risky asset or cash. At time 2, two states can occur: a good state

and a fragile state. If the good state occurs, the asset will pay off one for sure at time 3.

If instead the fragile state realizes, the asset either pays off one at time 3 or the worst case

scenario realizes, in which case the asset pays off 1− d.

There is a continuum of risk neutral investors who agree to disagree on how to interpret

a public signal about d, which is observed at time 1. Some investors take the signal to

be more positive than others, and are therefore more optimistic. Investors therefore have

heterogeneous beliefs and can be indexed by their optimism about d, the payoff drop in the

worst case scenario.

2

At time 1, more optimistic investors buy the risky asset taking on leverage, subject to

a collateral constraint. At time 2, if the fragile state realizes, levered investors have to pay

back their borrowing, selling some of their holdings to do so. The marginal buyer in the

fragile state is therefore more pessimistic about the worst case payoff 1− d. Investors learn

under a misspecified model: they think that the market clearing price in the fragile state

reflects only new information about d. In particular, they do not understand that the fragile

state price is affected by delevering.

The misspecification in investors’ learning model captures the idea that investors believe

the market knows how to price tail risk. Investors fail to appreciate the extent to which,

had there been less leverage, the price decline in a fragile state would have been less severe.

The mistaken belief that market prices only reflect information about objective risks rather

than technical factors (such as deleveraging) undermines learning.

The main prediction of the three period model is that the average belief across investors

becomes more pessimistic after the fragile state realizes. This is because the fragile state price

decline is exacerbated by deleveraging and disagreement, while investors do not account for

their impact and therefore over-learn. If investors didn’t disagree, could not take on leverage,

or did not learn from fragile state prices, then average beliefs would not change after a fragile

state. Moreover, pessimism increases more for larger price declines in the fragile state.

I extend the model to a dynamic setting in order to analyze the persistence of the effects

of a fragile state realization. Overlapping generations of investors trade multiple vintages of

the risky asset and inherit their beliefs from the previous generation. I define the yield of

the asset as the return from buying it at time 1 and holding until time 3, conditional on the

worst case not realizing.

Because pessimism increases after a fragile state, fragile states are followed by increases

in yields for later vintages of the risky asset. I also show that more disagreement and more

leverage before a fragile state lead to larger crashes in fragile states and therefore to larger

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increases in yields afterwards. Moreover, since investors become less uncertain about the

worst case scenario payoff as time passes, fragile states that realize earlier lead to larger

increases of the risky asset yield.

To map the model to historical crashes, I consider two strategies corresponding to Black

Monday in 1987 and the period of the Lehman Brothers bankruptcy in 2008. Hedged put

options selling suffered severe losses around Black Monday, while arbitraging the difference

in implied credit risk between the corporate bond and CDS markets was similarly negatively

affected after the Lehman Bankruptcy. In the context of my model, undertaking these

strategies corresponds to buying the risky asset and the crashes correspond to fragile states,

which were not followed by worst case scenarios since the terminal payoff to an investor who

did not liquidate during the crash was positive.

The worst case scenario for a delta hedged put selling strategy is a decline in the un-

derlying which is so large and sudden as to result in a default on the hedging leg. For a

CDS-bond convergence trade, instead, a worst case scenario is one in which both the bond

and the CDS counterparty default. While for neither strategy those states realized, they

became more likely in the fragile states in 1987 and 2008.

In both fragile state episodes, leverage likely exacerbated the magnitude of the crash.

Before Black Monday 1987, margin requirements for option market makers were substan-

tially lower than they have been since, and many had to liquidate their positions on Black

Monday. A government report published shortly after explains how their sudden need for

cash contributed to the dramatic option price moves on Black Monday (USGAO, 1988).

Around the Lehman bankruptcy, the winding down of levered trades similarly played an

important role. D.E.Shaw (2009) suggests that dealer positioning was the primary driver of

CDS-bond basis changes around the Lehman bankruptcy and Choi et al. (2018) show that

bonds with larger preexisting basis arbitrage positions had significantly lower returns in this

period, after controlling for other characteristics.

4

Given the importance of leverage in those episodes, my model suggests that the fragile

states realized in 1987 and 2008 led investors to over-learn. Investors updated their beliefs

about delta hedged put option selling strategies and CDS-bond convergence trades, and this

updating led to persistently higher out of the money put option prices and wider CDS-bond

bases.

There are two main alternative explanations for the persistent changes in the returns

of these strategies after crashes: rational learning and slow moving capital. I contrast the

implications of my model with those of these alternatives in the context of the crashes of

1987 and 2008.

The standard explanation for the option prices change in 1987 is a rational learning one:

investors used to rely on the Black and Scholes (1973) model until the crash highlighted its

deficiencies and prompted them to shift to a new model. Differently from my model, this

explanation implies that option prices afterwards should be consistent with their objective

riskiness. However, rationalizing the returns on option strategies has been challenging.1

In particular, I show that risk adjusted average returns to hedged put options selling

strategies were close to zero before Black Monday and have been much larger afterwards, even

though Black Monday is included in the later sub-sample. The large rewards to undertaking

this strategy after 1987 suggest that the options market builds in more crash risk than there

seems to exist. This overshooting is consistent with my model if investors didn’t properly

account for the fact that the large negative returns on Black Monday were caused by the

high leverage of option market makers.

The second alternative, slow moving capital, has been a popular explanation for the

changes after the recent financial crisis. Several cross-market relationships which were con-1Previous research has found one needs to assume that investors are extremely averse to small price

jumps (Pan, 2002), that large equity market crashes are thought to be much more frequent than they havehistorically been (Bates, 2000), or that those crashes coincide with large consumption drops (Barro (2006),Gabaix (2012)).

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sidered arbitrage laws broke down then, and many have not returned to their pre-crisis

state. In the slow moving capital framework, the depletion of specialized capital in 2008

explains the breakdown, while the persistence of the apparent arbitrage violations is due

to increased regulatory constraints on the trading activities of intermediaries, which made

taking advantage of arbitrage opportunities harder.2

Mymodel suggests a different explanation. Taking advantage of those arbitrage violations

in practice exposes an investor to an unlikely risk of incurring large losses: a worst case

scenario in my model. While those technical risks are actually small, investors inferred their

magnitude from the losses on quasi-arbitrage trades around the Lehman bankruptcy and

believe them to be large since then.3 This mistaken belief stems from the failure to adjust

for the impact deleveraging had on arbitrage strategies around the Lehman Bankruptcy.

My model thus complements the regulatory explanation in several ways. First, it provides

a reason for apparent arbitrage to persist even when banks and other regulated intermediaries

are not the only source of funding nor the only participants in these markets. Second, it

explains why those deviations were already large before capital regulation went into effect

and, third, it brings additional cross sectional implications. In particular, the model implies

that, even after controlling for the actual risk of each strategy, those which experienced the

largest losses around the Lehman bankruptcy should also deliver higher returns afterwards.

I test this cross sectional implication for the CDS-bond basis constructed for each US

corporation. In line with the implications of the model, I find a strong relationship between

the post crisis average bases and the losses incurred on the corresponding convergence trades

around the Lehman bankruptcy of 2008, even after controlling for granular characteristics2The implementation of Basel III guidelines, and in particular the adoption of the Supplementary Leverage

Regulation in 2014 increased banks’ cost of entering trades which require holding large exposures on balancesheet, even when those exposures supposedly cancel out. Du et al. (2018) and Boyarchenko et al. (2018)articulate this perspective.

3Previous studies attempt to quantify those risks and find them to be too small to explain post-crisisdeviations. See for instance Bai and Collin-Dufresne (2019) for the CDS-bond basis and Du et al. (2018) forCovered Interest Parity deviations.

6

that should capture the risk in those trades. This cross sectional relationship is not eas-

ily obtained in a slow moving capital model: intermediaries would have to be extremely

specialized and only trade certain CDS-bond pairs.

Relation to previous literature. This paper builds on the literature on information

aggregation and learning from prices, starting with Grossman (1976), Grossman and Stiglitz

(1980) and Hellwig (1980). Investors take on leverage, which can cause steep price declines

in fragile states because levered holders have to sell to more pessimistic investors.4 I use

the heterogeneous belief framework of Geanakoplos (2010) to model fire sales (Shleifer and

Vishny (1992), Shleifer and Vishny (1997)) and, more generally, the impact of non fun-

damental factors on prices (De Long et al. (1990)) in a way that tractably interacts with

learning.5

The only information revealed in a fragile state is that the worst case has become more

likely, but prices decrease by more than this would imply because of leverage. The key

departure from the literature above is that I assume investors do not understand this: they

mistakenly think additional information is being revealed and back it out from prices.6

Therefore, my model is related to a recent strand of literature analyzing the implications of

learning under a misspecified model.7 In particular, Eyster et al. (2019) apply the concept of

cursed equilibrium (Eyster and Rabin, 2005) to financial markets. In their model, investors4I draw from the vast literature on heterogeneous beliefs and asset pricing, beginning with Miller (1977)

and Harrison and Kreps (1978). Specifically, rather than assuming that investors have heterogenous priors, Iassume they interpret public signals differently. This is analogous to the assumptions of Kandel and Pearson(1995) and Banerjee and Kremer (2010). See Hong and Stein (2007) for a more extensive review.

5Several studies employ the Geanakoplos (2010) framework to analyze leverage and its consequences. Sim-sek (2013) highlights how disagreement about good and bad states can asymmetrically influence constraintsand Geerolf (2018) characterizes the heterogeneity in borrowing arrangements when investors disagree onthe recovery value of collateral. Martin and Papadimitriou (2019) model the the dynamics of sentiment butdo not focus on learning.

6Banerjee (2011) proposes a way to determine whether or not investors condition on prices to updatetheir beliefs and provides evidence consistent with investors using prices.

7Gabaix (2014) proposes an explanation for misperceptions in investors’ models, while Schwartzstein(2014) and Gagnon-Bartsch et al. (2018) provide conditions under which mistaken models are likely tosurvive.

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do not fully internalize the fact that prices reflect information. On the other hand, investors

in my setting have a naive model in which prices convey direct cash flows information: they

infer too much from prices, while cursed investors learn too little.

Rare fragile states in my model have an outsized impact on beliefs. Relatedly, Malmendier

and Nagel (2011) and Malmendier et al. (2018) underline the importance of traumatic ex-

periences in beliefs formation.8

While several papers focus on explaining large price moves,9 few explore the impact of

crashes on the subsequent pricing of the affected assets. An exception is Banerjee and Green

(2015), in which investors learn whether others are trading on information or noise. Large

price changes lead investors to think it’s more likely that others are noise traders, increasing

expected returns as compensation for noise trader risk (De Long et al., 1990).

Kozlowski et al. (2015) also focus on understanding the impact of rare events on beliefs,

using a rational learning model. The difference with my model is clear in the context of

the 1987 crash. Their model can be seen as a formalization of the standard explanation

described above: the extreme event is a wake up call to update the working model. On the

other hand, my setup emphasizes the fact that option returns in this episode were affected

by leverage, and therefore should have carried less information about their objective risk

than what investors seem to have inferred.

This paper is also related to the work on the importance of intermediaries for asset

prices. Duffie (2010) is motivated by similar dislocation episodes but takes a different route

to explaining them: capital is slow to move into attractive opportunities (Grossman and

Miller (1988), Mitchell et al. (2007)). Relatedly, Vayanos and Woolley (2013) shows how8Memory of negative experiences and its reinforcement through repeated reminders provides a different

channel through which beliefs can remain persistently biased (Wachter and Kahana, 2019).9For instance, if a fraction of investors follows a rule based strategy leading them to sell at the same

time (Grossman (1988), Gennotte and Leland (1990)), levered traders are forced to liquidate their positions(Geanakoplos, 2010) or funding markets tighten (Brunnermeier and Pedersen, 2009), large price changescan occur even if there is little new information. Moreover, uncertainty about others’ signals or widespreaddispersion of information can cause large price movements without news (Romer, 1992).

8

agency frictions affect the informational content of prices. While this class of models assume

investors understand the market structure, in my setting the misspecification in investors’

models causes the dislocation to induce persistent beliefs shifts, which in turn explains capital

sluggishness.

Finally, I contribute to the empirical literatures on index options pricing and the CDS-

bond basis: I relate my paper to those literatures in more detail in section 3.

2 A model of learning from crashes

The model features a continuum of investors trying to learn the value of an asset. I start by

describing the learning environment in a three period setting. I then extend this framework to

an overlapping generations setting to show how this mistake in the investors’ model generates

persistent changes in asset prices. Finally, I consider a simple closed form example.

2.1 Environment

There are three dates t ∈ {1, 2, 3} and a measure 1 continuum of investors indexed by

i ∈ [0, 1] who consume only at time 3. There is a single consumption good, henceforth

dollars. Each investor i has an endowment of one dollar at time 1 and needs to transfer it

to the last period. Investors can either hold a zero interest rate storage technology (cash) or

a risky asset. The payoff of the risky asset is represented in Figure 1.

9

p1

G 1

FpF

1

1− dh2

h1

t = 3t = 2t = 1

Figure 1: Timing and asset payoffs

At time 2, with probability 1 − h1, the good state G realizes, and the final payoff is 1

for sure. Otherwise, with probability h1, the fragile state F realizes and the time 3 payoff

remains uncertain. From the fragile state F , with probability 1−h2 the asset pays off 1, but

with probability h2 the worst case state realizes and the asset pays off 1− d.10 I denote the

price of the risky asset at time 1 by p1 and in state F by pF . The price in the good state pG

equals 1 since the asset is equivalent to cash in state G.

While cash is supplied inelastically, there is a fixed unit net supply of the risky asset,

which is initially owned by an unmodeled agent who sells it to the continuum of investors

at time 1 and consumes the proceeds. This technical assumption rules out feedback effects

from the initial price of the asset to investors’ wealth, simplifying the analysis.11

Investors can borrow for one period against their holdings of the risky asset. At time 1,

they can raise ` dollars by pledging one unit of the risky asset as collateral. This collateralized10The worst case state is analogous to a rare disaster of Barro (2006). For the US economy, the rare

disasters in Barro (2006) are only the Great Depression of 1929-1933 and the aftermath of World War Two.In both these episodes, GDP declined around 30 percent in the space of four years. From this perspective,it is natural to interpret the Black Monday crash of 1987 and the Lehman Bankruptcy of 2008 as fragilestates, rather than worst case scenarios.

11In the dynamic overlapping generations extension of section 2.6, old investors own the risky asset andsell it to the young when they die. Similar assumptions are made in Simsek (2013) and Geerolf (2018). Theresults are unchanged if instead investors are initially endowed with the asset and can trade it at time 1, butthe market clearing conditions become more cumbersome.

10

loan has to be repaid at time 2, and collateral is seized in the event that its market price is

below ` at that point. Even if collateral is seized, borrowers are still responsible for repaying

their loans in full: they could have negative wealth in the fragile state. I assume that ` is

an exogenous parameter and that unmodeled investors provide the collateralized financing

at time 1 and get paid back at time 2. For simplicity, I rule out borrowing between dates 2

and 3, so that all borrowing has to be repaid at time 2.

Finally, I assume for simplicity that short sales of the risky asset are not possible. While

this is a common assumption in the literature on disagreement and it simplifies the analysis,

the main results of the model do not rest on this assumption.12

2.2 Subjective model, learning, and disagreement

All learning and disagreement is about d: the extent of the payoff drop in the worst case state.

Investors know the probability h1 of transitioning from time 1 to a fragile state at time 2, as

well as the probability h2 of a worst case state realizing after a fragile state. Investors have

a common prior about d: each investor i initially believes d is normally distributed around a

mean d0. Disagreement stems from individual biases in interpreting common public signals.

At time 1, investors observe a noisy public signal s1 = d + �1 about d. Each investor i

updates their beliefs as if the public signal were13

si1 = s1 + ψδi. (1)

The individual bias δi is sampled from a distribution centered around zero with cumu-12For instance, it is employed to obtain equilibrium existence in the baseline models of Miller (1977),

Harrison and Kreps (1978), and Geanakoplos (2010). In Appendix B, I extend the model to allow for shortsales and show that mislearning will still occur in fragile states and that the features of mislearning areanalogous to those arising in the baseline model.

13In the baseline three period model, this assumption is analogous to assuming heterogeneous priors acrossinvestors, but it increases disagreement in later periods in the dynamic version of the model. See Kandel andPearson (1995) and Banerjee and Kremer (2010) for an extensive discussion of how this assumption differsform heterogeneous priors.

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lative distribution function Gδ(δi). The non negative parameter ψ quantifies the extend of

disagreement: if ψ = 0 the model collapses to a representative agent model, and there is no

difference between investors. For analytical tractability, I assume the noise �1 is normally

distributed with mean zero and variance σ2s .

I denote by bit(x) the density function of investor i’s beliefs at time t for a random

variable x, and use φ(x; µ, σ2) to indicate the density function of a normal random variable

with mean µ and standard deviation σ.14 Analogously, I denote the subjective expectation

operator of investor i at time t by Eit. The priors in the subjective model are therefore

described by

bi1(d)

= φ(d; d0, σ20

)(2)

bi1(s1|d

)= φ

(s1 + ψδi; d, σ2s

). (3)

Given this Gaussian structure, the posterior of investor i about the worst case scenario drop

d at time 1, after having observed the public signal s1, is

bi1(d|si1

)= φ

(d; di1, σ21

)(4)

in which, using τs = 1σ2s and τ0 =1σ20

to indicate the precision of signal and prior respectively,

di1 =τs

τs + τ0(s1 + ψδi) +

τ0τs + τ0

d0 (5)

σ21 =1

τs + τ0. (6)

To complete the description of investors’ subjective model, I characterize investors’ beliefs

about the fragile state F at time 2, as well as how they learn if the fragile state does realize.14More formally, for any random variable X, I denote by bit(x) the derivative of Pit[X < x], where the

probability measure is defined by the subjective model of investor i at time t.

12

Here, I depart from the noisy rational expectation benchmark by assuming that investors

do not have rational expectations about the price of the asset in the fragile state F . At

time 1, investors think the fragile state market price will reflect new information, and that

their updated beliefs about d will be consistent with this price. Formally, agent i at time 1

believes

pi,1F = 1− h2Ei2[d] (7)

where the superscripts i, 1 highlight that pi,1F is not the actual market clearing price in state

F , but rather the price that agent i at time 1 thinks will realize in state F . Note that

equation (7) defines pi,1F as a random variable in the mind of agent i: agent i thinks that the

realization of pF if the fragile state F occurs depends on how his own beliefs about d will

change, as reflected by the time 2 subjective expectation on the right hand side.

If state F does realize, investors know they are in the fragile state F and observe the

market clearing pF . I assume that they back out a noisy signal dpF for d from the market

price:

dpF =1− pFh2

. (8)

This is a second misspecification in investors’ model: the market clearing price pF is actually

pre-determined at time 1 and does not convey new information about d, it only reflects the

extent of disagreement and leverage. Yet, investors do not understand this and think that

the market price efficiently reflects available information, so they learn from the price pF . In

particular, each agent i believes that dpF is a noisy normally distributed unbiased signal for

d with standard deviation σp:

bi1(dpF |d

)= φ

(dpF ; d, σ2p

). (9)

The parameter σp captures the extent to which investors learn from fragile state prices. A

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small perceived precision of the signal τp = 1σ2p corresponds to investors putting little weight

on fragile state prices in forming their beliefs about d. Any perceived precision τp > 0 is a

misspecification in the learning model investors employ since fragile state prices actually do

not convey new information about d.

The assumption that investors rely on this mistaken model for learning from fragile state

prices is crucial: it formalizes the idea that investors believe markets are efficient and that

prices reflect only cash flow information when they actually do not. While it is a strong

assumption, it is not unreasonable in the context I am modeling. Firstly, investors rarely

observe fragile states prices, which depend much more strongly on the extent of leverage and

disagreement than the prices at time 1. Secondly, the actual model is complex, with prices

depending on both the exact distribution of disagreement and overall leverage, which are

hard to observe in practice.

At time 2, if the good state G is realized, the market price no longer depends on d and

no learning occurs. If instead the fragile state is realized, investors update using the market

implied signal dpF = 1−pFh2 as described above. The time 2 posterior is therefore

bi2(d|I2

)= φ

(d ; di2, σ22

)(10)

where time 2 information is

I2 =

{s1} if state G realizes

{s1, dpF} if state F realizes(11)

14

and the parameters are

di2 =

di1 if state G realizes

τ1τ1+τpd

i1 +

τpτ1+τpd

pF if state F realizes

(12)

σ22 =

σ21 if state G realizes

1τ1+τp if state F realizes

(13)

In these expressions, τ1 = 1σ21 = τs + τ0 is the precision of investors beliefs at time 1 and

τp = 1σ2p is the perceived precision of the market signal.

2.3 Preferences

Investors are risk neutral and only consume at time 3. At both times 1 and 2, each investor

i maximizes his expected payoff subject to a budget constraint. At time 1, investors choose

how many units of the asset to buy with leverage (ai1) and without (ai1,0), as well as how

much cash to keep (ci1), given their time 1 subjective beliefs on the price of the risky asset

in the fragile state pF and the worst case scenario payoff 1 − d. Their budget constraint is

therefore

ci1 + ai1,0p1 + ai1(p1 − `) ≤ 1. (14)

Investors also need to hold one unit of the risky asset for each ` of cash they borrow: this

collateral constraint is implicit in the way I specify their portfolio choice problem. At time

1, investor i seeks to maximize his expected wealth at time 2. He therefore solves

maxai1≥0,ai1,0≥0,ci1≥0

ci1 + (1− h1)(ai1,0 + ai1(1− `)) + h1Ei1[ai1,0pF + ai1 (pF − `)

]under (14). (15)

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Portfolio weights are constrained to be weakly positive as short sales are not allowed. If

state G realizes at time 2, portfolio choices afterwards are irrelevant to final payoffs since

cash and asset are then equivalent. On the other hand, if state F realizes, investors again

optimize given their updated information set by choosing whether to keep cash (ci2) or buy

the asset without leverage (ai2). Their time 2 budget constraint is given by

ci2 + ai2pF ≤ ci1 + ai1,0pF + ai1(pF − `). (16)

Since in state F there can be no borrowing against the risky asset, investors’ problem is

maxai2≥0,ci2≥0

ci2 + (1− h2)ai2 + h2ai2Ei2 [1− d] under (16). (17)

2.4 Market clearing

Having described the assumptions of the model, I now turn solving it. I begin by characteriz-

ing market clearing. Lemma 1 shows that investors’ portfolio choice problem (15) simplifies

substantially once we take into account the subjective model investors use.

Lemma 1. The solution (ci1, ai1,0, ai1) to problem (15) is such that ai1,0 = 0 for each i: if an

investor wants to buy the risky asset at time 1, he prefers to do so with as much leverage as

possible. Moreover, no investor prefers to keep cash at time 1 to buy the asset later in case

the fragile state F realizes.

Proof. See Appendix A.

At time 1, each investor thinks that there will be no point in buying the asset in the fragile

state since its time 2 price will equal his subjective expected payoff, as in (7). Nevertheless,

when the fragile state F actually realizes, investors might still want to buy the risky asset

given their new posterior beliefs and the actual market clearing pF . This time inconsistency

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is a result of the mistaken subjective model of fragile state prices which investors employ.

Lemma 1 allows us to characterize portfolio choice at time 1: investors either hold cash

or buy the asset with as much leverage as possible. In particular, given risk neutrality, they

buy the asset if and only if their subjective valuation is higher than the market price, which

is equivalent to having a more optimistic view of the worst case scenario than the market

implied one. The market implied d at time 1 is dp1, the value of E[d] which equalizes the

market price p1 and the expected payoff 1− h1h2E[d]. The market implied d at time 1 can

therefore be written as

dp1 ≡1− p1h1h2

(18)

and, recalling the notation di1 = Ei1[d], we have

1− h1h2Ei1[d] > p1 ⇐⇒ di1 < dp1. (19)

The demand function for investor i at time 1 in units of the risky asset is therefore given

by

ai1(p1) =

0 if di1 ≥ d

p1

1p1−` if d

i1 < d

p1

(20)

It is important to keep track of the distribution of mean beliefs across investors. This

distribution endogenously varies over time as investors incorporate information from market

prices and signals. I denote the cumulative distribution function of the distribution of mean

beliefs across investors at time t by Ct(dit), and the mean of this distribution as dt.

Aggregate demand at time 1 is therefore

a1(p1) =ˆi

ai1(p1)C ′1(di1)di (21)

17

and market clearing requires

a1(p1) = 1 ⇐⇒ C1(dp1) = p1 − ` (22)

since the risky asset is in unit net supply at time 1.

The left hand side of the second equation in (22) is the aggregate wealth of optimistic

investors who buy the asset at time 1. For those investors, the market implied d, dp1, is lower

than their subjective mean belief di1: those investors are more optimistic than the marginal

buyer since they expect a lower drop in the worst case scenario. The right hand side is the

cash needed to buy the unit supply of the risky asset. The left hand side of this equation is

a strictly decreasing function of p1: the number of optimistic enough investors decreases as

p1 increases, so that a solution exists and is unique.

Market clearing at time 1 can be seen as finding the marginal buyer: agent i with mean

belief di. This can be visualized as in Figure 2, where I consider a normal distribution of the

individual bias δi. Since investors are risk neutral, in equilibrium it must be the case that

the price is equal to the marginal buyer’s valuation 1−h1h2di. Moreover, di needs to satisfy

(22): C1(di) + ` = p1. The intersection of the two solid blue and black lines in 2 identifies

the marginal buyer and therefore the market price p1.

Investors’ portfolio choice at time 2 is again between buying the risky asset and holding

cash. Investors will want to buy the asset if their subjective expectation of d at time 2

implies a private valuation higher than the market price:

1− h2Ei2[D] > 1− h2dpF = pF ⇐⇒ di2 < d

pF (23)

since all investors update using a common signal, the relative optimism of investors does not

change. Therefore, time 1 buyers would still like to hold the risky asset in state F but are

18

C1(d i) + l 1-h1h2d il

0 1 2 3 di

0.2

0.4

0.6

0.8

1.0

1.2

1.4

p1

d1p

l

-1 0 1 2 3

Density

Figure 2: Example of market clearing at time 1. Gδ is a normal distribution with mean 0and standard deviation 1, its density is reported in the bottom panel. Parameters are set asτ0 = τs = τp = 1, h1 = h2 = .2, �1 = 0 and ` = .49. The horizontal axis indexes investors’ meanbeliefs about d at time 1. The blue solid line is the cash available at time 1 to investors with beliefsequal or more optimistic than di: C1(di) + `. The black line is the valuation of the risky asset attime 1: 1− h1h2di.

forced to liquidate in order to pay back their loans.15

The demand function for agent i at time 2, state F , is therefore given by

ai2(pF ) =

− 1p1−` ·min

(`pF, 1)

if agent i bought at time 1

0 if di2 ≥ dpF and agent i did not buy at time 1

1pF

if di2 < dpF and agent i did not buy at time 1.

(24)

To understand the demand of time 1 buyers, recall they borrowed ` dollars per unit of the

risky asset. As each of them holds 1p1−` units of the risky asset, they borrowed

`p1−` dollars

15They also need to sell some of their asset if state G is realized, but every agent agrees the asset is worth1 in state G so this selling doesn’t have consequences for the price.

19

each. Moreover, they can only sell as much as they have, so if pF < ` they sell everything:1

p1−` units of the risky asset.16 If instead pF > `, they only have to sell 1pF ·

`p1−` units of the

risky asset in order to pay back their debt. 17

Since in state F all transactions are within the continuum of investors, the aggregate

position is unchanged and aggregate demand must be zero:

a2(pF ) =ˆi

ai1(pF )C ′2(di2)di = 0. (25)

Which can be rewritten as

min(`

pF, 1)

1p1 − `

ˆtime 1 buyers

C ′2(di2)di =1pF

ˆstate F buyers

C ′2(di2)di. (26)

The left hand side of equation (26) is the risky asset quantity that time 1 buyers need to

sell to pay back their loans, while the right hand side is the amount state F buyers demand.

Noticing that the mass of time 1 buyers has to equal p1− ` by (22), and that state F buyers

are those who think d is larger than dpF and did not buy earlier,18 market clearing in state

F simplifies as follows

min (`, pF ) = C2 (dpF )− C2(

τ1τ1 + τp

dp1 +τp

τ1 + τpdpF

). (27)

Market clearing in state F is depicted in Figure 3. The distribution of cash available to

investors below di has no mass for di < dp1 as optimists bought as much as they could at16When pF < `, the lenders receive their collateral in state F and sell it on the market, which is equivalent

to the borrowers selling and passing on the proceeds to the lenders.17The assumption that no debt rollover is allowed in the fragile state is certainly stark, but it is not

unrealistic since fragile states capture crisis situations and uncertainty about final payoffs sharply increasesin the fragile state. Assuming that partial rollover is possible only weakens the impact of fire sales, but doesnot eliminate it.

18The least optimistic time 1 buyer iF had diF1 = dp1 and therefore his state F mean belief d

i12 is equal to

τ1τ1+τ d

p1 + ττ1+τ d

pF by (12).

20

-1 0 1 2 3di0.00.2

0.4

0.6

0.8

1.0

1.2

p1

d1p

lpF

dFp

-1 0 1 2 3

Density

τ1τ1 + τp d1p + τpτ1 + τp dFpDensity Before FDensity After F

Figure 3: Example of market clearing in state F . Gδ is a normal distribution with mean 0 andstandard deviation 1, τ0 = τs = τp = 1, h1 = h2 = .2, �1 = 0 and ` = .49. The horizontal axisindexes investors’ mean beliefs about d. The black solid and dashed lines are the valuations of therisky asset in state F and at time 1, respectively 1−h2di and 1−h1h2di. In the top panel, the bluesolid line is C1(di)− C1(dp1): the cash available at time 2 to investors with beliefs more optimisticthan di under the time 1 distribution of beliefs. The red line in the top panel represents the samequantity of cash but under the time 2 distribution of beliefs. The orange line in the top panel is theamount of leverage `. The bottom panel reports the probability density function of mean beliefs diat time 1 in blue and after the state F realization in red.

time 1. Moreover, the distribution of beliefs about d shifts from the blue to the red curve

because of learning from the fragile state price. Notice that the two cumulative distribution

functions intersect at di = dpF : the mass of investors below the marginal buyer stays the

same as the marginal buyer in the fragile state doesn’t update his mean belief about d since

the market price reflects his own belief. Equilibrium obtains when the amount of risky asset

sold by time 1 optimists (worth `) is equal to the cash available to investors in state F .

21

2.5 Equilibrium

I now turn to characterizing the equilibrium. Apart from the prices p1 and pF , equilibrium

also pins down investors’ beliefs and their distribution across investors.

Definition 1 (Equilibrium). An equilibrium is a 4-tuple of distributions of beliefs and

market prices (C1, C2, p1, pF ) such that, given initial prior parameters τ0 and d0, leverage `,

perceived signal precisions τs and τp, as well as the distribution of bias across investors Gδ:

1. Each agent i maximizes subjective expected payoff at each time and state

2. Beliefs update according to (4) and (10)

3. p1, pF , and the distributions of mean beliefs C1 and C2 satisfy the market clearing

conditions (22) and (27).

Interest rates on time 1 collateralized loans are equal to zero, since even if the price

of the risky asset in the fragile state is lower than the amount borrowed, the borrower is

responsible for paying out the difference to the lender. This simplifying assumption allows

avoiding accounting for the option value of default but is not crucial. For the parameter

values I use, investors think the probability that pF will be under ` at time 1 is extremely

small. Moreover, I focus on equilibria in which the actual asset price in the fragile state F

is not below the amount borrowed, pF ≥ `, so that borrowers never end up with negative

payoffs.

Lemma 2 (Conditions for existence and uniqueness). For each combination of prior average

mean d0, prior precision τ0, perceived signals precision τs and τp, initial signal s1, as well

as a differentiable and non atomic cumulative distribution function of mean priors across

investors Gδ, there exists a unique initial price p1(0) which clears the market if no leverage

is available: it solves (22) for ` = 0. If the parameters above are such that p1(0) > 1 − h1

22

then there exists ` > 0 such that for all 0 ≤ ` < ` there exist an equilibrium (C1, C2, p1, pF )

with pF ≥ `. Moreover, this equilibrium is unique.

Proof. See Appendix A.

Lemma 2 shows that such equilibria exist and are unique for small enough values of `.

The restriction p1(0) > 1−h1 rules out parameter values for which the worst case scenario is

believed to be so disastrous as to imply a negative fragile state price.19 Having established

conditions under which equilibrium exists and is unique, I turn to describing the relationship

between price crashes in fragile states, disagreement and leverage.

Proposition 1 (Leverage). If investors disagree (ψ > 0) and ` is such that the equilibrium

exists with pF ≥ ` as in Lemma 2, then the initial price of the risky asset p1 is increasing in

leverage `, and the price in the fragile state pF is decreasing in `. Therefore,

∂

∂`(p1 − pF ) > 0. (28)

Proof. See Appendix A.

Equation (27), the market clearing condition in the fragile state F , implies that if there

is no leverage, i.e. ` = 0, then dp1 = dpF , so that the market implied magnitude of losses in

the worst case does not worsen in the fragile state. Similarly, if there is no disagreement

(for instance if τs = 0), then leverage has no impact on market prices and dp1 = dpF . This

is because, when there is no disagreement, the model collapses to the representative agent

case, and therefore the marginal buyer does not change from time 1 to the fragile state.

Disagreement can be quantified by ψ, the positive scaling parameter magnifying the

individual biases. We can therefore analyze the impact of disagreement by studying the19The worst case scenario payoff is 1−d and investors’ beliefs about d are normally distributed. Therefore,

investors can believe this payoff to be negative, which creates the need for this restriction. While thepossibility of negative prices is unappealing, given investors’ risk neutrality it does not imply theoreticalinconsistencies.

23

comparative static with respect to ψ.

Proposition 2 (Disagreement). Consider an equilibrium as in Lemma 2. Recall that d1 is

the average across agents of the posterior mean belief about d at time 1. If prices p1 and pF

are such that dp1 ≤ d1 < dpF then the magnitude of the price decline in the fragile state is

increasing in ψ:∂

∂ψ(p1 − pF ) > 0. (29)

Proof. See Appendix A.

Proposition 2 states that for any non degenerate distribution of δi and any τs >0, equation

(1) implies that a larger ψ leads to more dispersion in beliefs across investors. More dispersion

in beliefs in turn implies that the impact of leverage on the price crash in the fragile state is

greater. This is because the difference in optimism between initial buyers and fragile state

buyers is larger when there is more disagreement, implying that the decrease in optimism of

the marginal agent is larger.

The assumption that dp1 ≤ d1 < dpF means that the marginal buyer at time 1 is at least

as optimistic as the average investor and that the marginal buyer in the fragile state is more

pessimistic than the average time investor was at time 1. This restriction corresponds to a

lower bound on the amount of leverage `: when there is no leverage, dp1 = dpF , as implied by

the fragile state market clearing equation (27).

The main implication of the model is about learning after a fragile state occurs. In

particular, equation (12) shows how the market clearing price in the fragile state, pF , changes

all investors’ beliefs about d. This is the key to the dynamic implications of the model as

belief shifts can be persistent: Proposition 3 summarizes the beliefs adjustment predictions.

Proposition 3 (Over-learning). Consider an equilibrium as in Lemma 2, and suppose the

fragile state F is realized at time 2. If investors disagree (ψ > 0), initial buyers take on

leverage (` > 0), and the average belief across investors about d at time 1 is not too pessimistic

24

d1 <1

2h1h2 , then the average subjective mean of the magnitude of the worst case drop, d2, is

higher than that before the fragile state realization, d1. Moreover the increase in pessimism

d2 − d1 is:

• Greater for larger values of `

• Greater when there is more disagreement, i.e. when ψ is larger

• Increasing in the perceived informativeness of the price signal τp

• Decreasing in the prior precision τ0

Finally, if there is no leverage ` = 0 or investors do not disagree (ψ = 0) then d2 = d1.

Proof. See Appendix A.

Proposition 3 shows how learning from prices in the fragile states affects investors’ sub-

sequent beliefs. In the fragile state, investors observe dpF , which they interpret as a noisy

signal for d with precision τp. The posterior mean of each individual investor shifts towards

dpF , which we showed is larger and therefore more pessimistic than the prior average d1.

Since all beliefs shift towards a signal which is more pessimistic than the average prior

mean, the average posterior mean belief also becomes more pessimistic. This shift in beliefs

after a fragile state is the central implication of the model since beliefs determine prices in

future periods. Recall that the fragile state price does not actually reflect new information

about d, so this change in beliefs is over-learning.

The proposition also shows that there is more over-learning if dpF is larger or if investors

believe the fragile state price signal to be more informative, which corresponds to a larger

value of τp. As the previous propositions showed, more leverage or more disagreement result

in larger dpF and therefore in more over-learning. Moreover, if agents have more precise prior

beliefs (τ0 is larger), then the increase would be smaller.

25

2.6 Dynamics and time varying leverage

I now turn to an overlapping generations setting in order to analyze the persistence of belief

shifts and the impact of crashes on subsequent leverage. Generation k is born at time 2k+1,

and each agent is endowed with one unit of cash. The old generation owns the whole supply

of the risky asset and sells it to the young at time 2k+1, after consuming the payoff. Beliefs

are inherited through generations: agent i of generation k at time 2k + 1 has the same

beliefs that agent i of generation k − 1 held when they died at time 2k + 1. The timing

of the dynamic model is illustrated in Figure 4. In each odd period, there is a new public

signal sk = d + �k and investors interpret it differently, as in the three period version: they

update their beliefs as if sik = sk + δi were the public signal. Notice that the individual level

disturbance δi doesn’t change from period to period: some agent types are always optimistic

about public signals. Each agent i keeps updating his beliefs about d, based both on the

observed market prices and on the public signals. I denote market prices at time 2k+ 1 and

in the fragile state at time 2k + 2 by p1,k and pF,k, respectively.

Beliefs bi1 , t = 1

F1− dh2

1h1

G 1

Beliefs bi3 , t = 3

F1− dh2

1h1

G 1

· · · · · ·

Figure 4: Timing of the dynamic model

Mean beliefs at each point in time can be characterized as a function of past private signals

26

and market price signals. At time 2k+ 1, the information available to agent i includes the k

past public signals s1, ...sk, as well as any fragile state price observed. In particular, denoting

by NFk the number of fragile state realizations before time 2k+1, which occurred for vintages

f1, ..., fNFkthe information set of agent i is

I2k+1i ={si1, .., s

ik, d

pF,f1 , ..., d

pF,k

NFk

}. (30)

Recalling that the perceived precision of the public and state F price signals are, respectively,

τs and τp, the beliefs of agent i can be written as

bit(d|I it

)= φ

(dit ,

1τt

)(31)

where dit is a linear combination of the signals observed and the agent’s prior, weighted by

their perceived precision:

dit =τs∑kl=1 sl + τp

∑NFtl=1 d

pF,kl

+ τ0di0kτs +NFt τp + τ0

=τ0δi + τs

∑kl=1 sl + τp

∑NFtl=1 d

pF,kl

+ τ0d0kτs +NFt τp + τ0

(32)

since the disturbance δi doesn’t change through time, and the posterior precision is given by

τt = kτs +NFt τp + τ0. (33)

Therefore, the distribution of mean beliefs across investors at time t, with symmetric

cumulative distribution function Ct, is an affine transformation of the initial distribution of

δi across investors, Gδ, keeping the model analytically tractable. I denote the mean of the

distribution of investors’ average beliefs by dt.

In this dynamic setting, I allow the amount of leverage available to vary for each asset

vintage k and denote it by `k. A simple way to link the amount of leverage to investors’

27

beliefs is to assume

`k = p1,k − α (34)

where α is a parameter quantifying the haircut. Intuitively, the lower the price of the risky

asset, the less they can borrow against it. While I assumed leverage is risk free, this reduced

form relationship resembles the one that would obtain in a model in which leverage contracts

are themselves an equilibrium outcome.20 We can now naturally extend the equilibrium

concept of Definition 1.

Definition 2 (Dynamic Equilibrium). Given initial prior parameters τ0 and d0, worst case

scenario drop d, haircut α ∈ [0, 1], perceived signal precisions τs and τp, distribution of bias

across investors Gδ as well as a sequence of signals {sk} and state realizations, a dynamic

equilibrium is a sequence {(C2k+1, C2k+2, p1,k, pF,k)} of equilibria of the 3 period model as

in Definition 1 with leverage given by equation (34), each corresponding to a vintage k of

the risky asset and such that the cumulative distribution functions (C1, C2, ..) are consistent

with equation (32).

I can now characterize price paths given a sequence of state realizations and signals. In

particular, Proposition 4 summarizes the consequences of a fragile state realization.

Proposition 4 (Price and leverage dynamics). Suppose �j = 0 ∀j, that the investors’ prior

is initially centered at the truth, d0 = d, and that state F is realized for the first time at time

2k + 2. Then, as long as α is large enough so that dp1 > d0:

• Initial period risky asset prices are lower for all subsequent vintages: p1,j < p1,k ∀j > k

• The decrease in initial prices from before to after the fragile state realization, |p1,k+1−20The equilibrium determination of collateralized borrowing arrangements is the focus of Simsek (2013),

Geanakoplos and Zame (2014), and Geerolf (2018). Since in my setting investors disagree on the valueof the asset in the fragile state and in the worst case scenario, rather than on the probability of negativestates realizing, the equilibrium borrowing arrangement would resemble the one of Geerolf (2018), in whicha continuum of margin levels exist in equilibrium, one for each level of optimism of the borrowers.

28

p1,k|, is larger when the haircut α is smaller or the perceived precision of the price

signal τp is larger

• After the fragile state realization, prices p1,j are increasing in j as long as state F is not

realized again and the "recovery speed", p1,j+1 − p1,j for j > k is larger if the precision

of the public signal τs is greater.

Proof. See Appendix A.

Proposition 4 shows that fragile state realizations have very persistent consequences:

initial prices will always be lower than they were before a fragile state realizes.21 Initial

period prices of subsequent vintages recover as investors incorporate more public signals into

their beliefs at a rate that is increasing in how informative those signals are.

The restriction that leverage is initially low enough to have dp1 > d0 is not key to those

dynamics, but it simplifies the proof as reduced uncertainty doesn’t mechanically imply a

price increase.

Proposition 5 (Average belief after a fragile state). Consider asset vintage k and suppose

the fragile state is realized at time 2k + 2 but the worst case scenario does not realize and

that the public signal for vintage k + 1 is equal to the true d: �k+1 = 0. For any τs > 0, and

any α ∈ [0, 1), as long as d2k < 12h1h2 :

• The fragile state signal is more pessimistic than the average belief at time 2k, dpf,k > d2k

• dk+1 > dk and dk+1 − dk is decreasing in k.

Proof. See Appendix A.21I set �j = 0 for all j in order to clarify the analysis. Positive random public signals can change the

dynamics of prices and offset the impact of fragile states. Nevertheless, the benchmark in which signals donot contain noise is interesting as it is related to the average path after a fragile state.

29

Proposition 5 demonstrates that as long as there is any leverage and disagreement, the

realization of a fragile state makes investors more pessimistic about the worst case scenario.

While Proposition 4 shows this for the first realization of a fragile state, 5 confirms the same

results for subsequent realizations. Additionally, Proposition 5 shows that later fragile states

have a smaller impact on investors’ beliefs. This is intuitive as investors become less and

less uncertain about the riskiness of the asset as time passes and therefore their beliefs react

less to fragile state signals.

2.7 Uniform bias distribution: closed form solution

If the individual bias under which agents interpret public signals is uniformly distributed

over [−1, 1], the model is analytically tractable. I analyze this case in order to derive closed

form expressions linking observable quantities such as price drops and average returns to

unobservables such as disagreement. Such restrictions are useful to quantitatively assess the

predictions of the model in section 3.3.

Suppose, for simplicity, the initial public signal is s1 = d0, confirming the initial average

belief, then the time 1 mean belief di1 is uniformly distributed across investors over the

interval[d0 − ψ1, d0 + ψ1

], where

ψ1 ≡τs

τ0 + τsψ.

The market clearing price at time 1 solves

p1 − ` = C1(dp1) =dp1 − (d0 − ψ1)

2ψ1(35)

which gives

p1 = 1−h1h2

1 + 2ψ1h1h2

(ψ1 − 2ψ1`+ d0

). (36)

Leverage ` only matters for pricing to the extent that investors disagree, ψ1 > 0, and is

30

more important the more extreme disagreement is. I now turn to the fragile state at time 2.

Given the Gaussian beliefs structure described in the previous section, time 2 learning shifts

and shrinks the distribution of mean beliefs towards the market signal dpF . The distribution

of mean beliefs at time 2 is therefore an affine transformation of the time 1 distribution. In

particular, it is still uniform but over the interval:[d2 − ψ2, d2 + ψ2

]where

d2 =τ1

τ1 + τpd1 +

τpτ1 + τp

dpF (37)

ψ2 =τ1

τ1 + τpψ1. (38)

The state F market clearing condition is given by

` =dpF −

(τ1

τ1+τpdp1 +

τpτ1+τpd

pF

)2 τ1τ1+τpψ1

(39)

which simplifies to

dpF − dp1 = 2`ψ1. (40)

If ` = 0, this equation implies that, no matter the level of disagreement, dpF = dp1: if the

original buyers do not have to sell in the fragile state F , no transaction will occur and the

marginal buyer will have the same beliefs about d. We can rewrite the above as

pF = 1− h2dp1 − 2h2`ψ1, (41)

notice that 1− h2dp1 is the price that would have obtained in the fragile state, had leverage

not influenced prices. In fact, it is the price implied by the belief of the marginal time 1

buyer. This equation also stresses that it is the interaction of disagreement and leverage

that drives price crashes in this model.

Finally, we can obtain comparative statics for the average level of pessimism after state

31

F realizes at time 2 from equations (37) and (40):

d2 =τ1

τ1 + τpd1 +

τpτ1 + τp

(dp1 + 2`ψ1) (42)

investors are more pessimistic after observing pF when `ψ1, the interaction of leverage and

disagreement, is larger and when they put more weight on the market price signal, i.e. when

τp is larger, as long as the initial beliefs are such that 1− h1h2d0 ≥ 12 . This latter restriction

on the average of prior mean beliefs about the worst case scenario payoff 1− d0 is minimal.

Violating the restriction would require an extremely negative 1 − d0, since the probability

h1h2 is small.

3 Historical episodes

I now analyze the change in option prices around the 1987 Black Monday crash and the

change in the CDS-bond basis after the 2008 Lehman Bankruptcy. While this section is not

meant to present conclusive evidence in favor of my model, it illustrates how the changes

around those two distress episodes are consistent with my model. Moreover, in the context

of these episodes, I explain how my model differs from rational learning and slow moving

capital explanations for the same changes.

3.1 Black Monday and option prices

On October 19, 1987, the Dow Jones Industrial Average fell 22.6% in one trading session,

marking the largest one day percentage decline in US equity prices. The options market

radically changed afterwards, as prices deviated from the benchmark Black and Scholes

(1973) formula and the volatility smile appeared, as shown in Figure 5: out of the money

put options became relatively more expensive (Derman and Kani, 1994).

32

The standard explanation for this change is that market participants had been relying

on a misspecified model and the crash served as a wake-up call, forcing them to address the

deficiencies of the existing framework. Prices were "wrong" before but are "correct" after the

crash.

0

10

20

30

1985 1989 1993 1997 2001 2005 2009 2013 2017

OT

M −

AT

M P

uts

IV (

%)

Figure 5: For each put option in the data, I define its moneyness as the ratio of strike priceand underlying spot price. This figure displays, for each date, the difference between the averageimplied volatility of put options with moneyness between .85 and .95 (OTM IV) and the averageimplied volatility of those with moneyness between .98 and 1.02 (ATM IV).

In line with this interpretation, a large literature extending the Black and Scholes (1973)

model developed. Notably, Heston (1993) adds stochastic volatility and Bates (2000) con-

siders state dependent jump risk. Pan (2002) shows that those two factors can empirically

explain option prices after 1987. The main shortcoming of this approach is that it implies

either that investors are extremely averse to small jumps in prices or that large crashes

should be much more frequent than what we actually observe (Bates, 2000). This shortcom-

ing becomes more stark as time goes by, since we still haven’t observed crashes of similar

magnitude.

33

I propose a different and complementary view of the change in the option market following

the 1987 crash. I interpret the risky asset in my model as hedged selling of put options on

the S&P 500. In particular, defining moneyness as the ratio of strike and current underlying

price, I consider a strategy selling all available index put options with moneyness between

0.8 and 1.05 each day and hedging the position by shorting the underlying in proportion to

the Black and Scholes (1973) delta of the option.

Black Monday corresponds to the occurrence of a fragile state for this synthetic asset: a

time in which the probability of the very worst states of the world increases substantially.

To be concrete, a worst case scenario for a delta hedged put selling strategy is one in which

the price decline of the underlying is so large and sudden that short positions hedging gains

are not paid out due to counterparty defaults. Even though Black Monday was exceptional,

futures markets continued working relatively smoothly and no defaults on futures contracts

were recorded (Fenn and Kupiec, 1993). Moreover, if an investor had entered the strategy

the day before Black Monday keeping aside the required margin and had continued following

it, he would had only lost 5% of his initial investment after a month. Nevertheless, the fragile

state resulted in large negative returns as this strategy lost around 30% in two days: the

severity of these losses changed the perception of the riskiness of this strategy as investors

started believing that the worst case scenario could be even worse than they previously

anticipated.

Margin requirements on option positions were much lower before the crash of 1987 than

they have been afterwards: consistently with decreased leverage after the crash, the CBOE

doubled the margin requirement on short put options positions in 1988 (CBOE, 2000).

Importantly, on Black Monday, several option trading firms suffered large losses as option

prices moved in an unprecedented way and had to close their short positions, suggesting

that forced buying of out of the money put options contributed to their price increase in this

episode (USGAO, 1988).

34

In order to compare the prediction of my model to the returns on this strategy, I construct

its historical returns by using data on S&P 100 and 500 index option prices from the Berkeley

Option Data Base (BODB) and the OptionMetrics Ivy database, covering the 1983-2017

period. I describe the data cleaning and strategy construction procedures in Appendix C.

Table 1: This table describes the returns on the strategy before and after October 19 1987. Iconsider three weighting rules to aggregate the individual put option returns into a daily return.The equal weights panel reports statistics for the strategy in which each option is weighted equally,the margin weights panel for that in which each option is weighted by the initial margin requiredto hold the hedged position, and the overweight OTM assigns weight 1Moneyness2 to each put option.Moneyness is defined as the ratio of strike price and underlying spot price, therefore weighting bythe inverse squared overweights heavily out of the money put options with moneyness < 1. Thereare 1149 and 7509 daily observations before and after Black Monday, respectively. Means andvolatilities are in annualized percent. The p-values in the last column correspond to Welch’s t-testsand Levene’s test for means and variances equality, respectively.

Before After p-valueMargin weightsAverage return -0.94 5.05 0.02Standard deviation 4.79 7.10 0.71Sharpe ratio -0.20 0.71Equal weightsAverage return -0.22 6.44 0.01Standard deviation 4.90 7.39 0.59Sharpe ratio -0.04 0.87Overweight OTMAverage returns 0.00 6.68 0.01Standard deviation 4.91 7.43 0.63Sharpe ratio -0.01 0.90

Table 1 shows that average returns on this strategy increased substantially after Black

Monday, even when including the extremely negative returns on Black Monday in the "after"

sample. Since the return on the two days around Black Monday was around -30%, a simple

back of the envelope calculation shows that, in order for average returns to be the same

before and after Black Monday, six episodes with similar losses to Black Monday should

have occurred since then. This suggests that the option market prices in more crash risk

35

than there actually is, and began doing so since the traumatic episode of the crash, consistent

with the over-learning mechanism in my model.22

While there is a large literature on option market anomalies,23 most studies do not focus

on how those arose after the Black Monday crash, as they usually rely on subsequent data.24

I contribute to this literature by demonstrating that at least one of those anomalies (the

expensiveness of out of the money put options) was not present before the Black Monday

crash and appears afterwards.

Santa-Clara and Saretto (2009) show that put option strategies similar to those I analyze

have very desirable properties but that engaging in them requires large amounts of margin

and entails substantial transaction costs. While transaction costs are indeed large for the

strategies I analyze, similar results hold when I consider strategies re-balancing the option

side of the strategy weekly or monthly, while still delta hedging daily. Moreover, if market

makers undertake this strategy, they might be able to pass on some of those costs to their

customers.

3.2 The CDS-bond Basis after Lehman

The value of both bonds and Credit Default Swaps (CDS) depends on the market perception

of the credit worthiness of an entity, so we can infer a credit spread from either. The difference

between those two credit spreads is the CDS-bond basis. As highlighted by Bai and Collin-22The Sharpe ratio metric is lacking when investors do not have mean variance preferences. Nevertheless,

in Figure 7, I show that the yearly returns well approximated by a normal distribution. Moreover, thecorrelation of the option strategies returns with the S&P 500 is .27 for the whole sample, -.12 for the periodbefore Black Monday and .30 after. Together with the low volatility of the option strategy, this makes itdifficult to explain the large average returns after Black Monday by market risk.

23For instance, Coval and Shumway (2001) show that delta neutral straddles realized returns seem outof line with their riskiness. Relatedly, Constantinides et al. (2009) argues that pricing of S&P 500 optionsdoesn’t seem to reflect their riskiness and argue that prices do not seem to have been becoming morerational over time. On the theoretical side, Garleanu et al. (2009) develop an option pricing theory basedon intermediary constraints which rationalizes some of those findings.

24An exception is Jackwerth (2000), who proposes a method to recover risk aversion of investors withdifferent wealth from option prices. While the resulting parameters are intuitively appealing before the 1987crash they become sometimes negative and generally increasing with wealth after.

36

Dufresne (2019), the quasi-arbitrage opportunity implied by a non zero CDS-bond basis is

suited to study limits to arbitrage theories given the large cross section of corporations with

both bonds and CDS.

I construct the basis from corporate bond transaction prices from TRACE and CDS

quotes from the CMA database, as detailed in Appendix D. A negative basis implies that

bond prices are lower than what would be implied by the credit spreads obtained from CDS.

One can engage in so called negative basis trades and earn a positive return by purchasing

the a corporate bond and entering the corresponding CDS to hedge default risk. Figure

6 shows the time series of the basis for five groups of underlying corporations, sorted by

the basis change around the Lehman bankruptcy. While the basis was close to zero before

the financial crisis,25 it became sharply more negative for most reference entities and began

displaying large cross sectional variation after the Lehman bankruptcy.

The negative basis trade for each underlying corporation maps to a different synthetic

risky asset in my model. Negative basis trades initiated before the Lehman bankruptcy

suffered large mark-to-market losses as the basis became more negative: a fragile state in my

model. The worst case scenario is one in which the bond defaults and the CDS contract does

not pay out: the trade is exposed to counterparty risk since the writer of CDS protection

might not honor their obligations. The payoff in this scenario is uncertain since it depends

on the recovery rates of both the CDS and the bond legs of the trade.

It is therefore not surprising that the basis widened as counterparty risk rose after the

Lehman bankruptcy: Lehman had sold CDS protection and its positions had to be unwound

the Sunday before bankruptcy. Since CDS contracts are highly collateralized and the posi-

tions were settled before bankruptcy, Lehman counterparties did not suffer significant direct

losses in this episode, but this might not have been the case if the situation had worsened.26

25Longstaff et al. (2005) and Hull et al. (2004) are early studies documenting the properties of the precrisis CDS-bond basis.

26For instance, the dislocation in the CDS market would have likely been much greater if AIG, which was

37

−400

−300

−200

−100

0

2006 2008 2010 2012 2014

CD

S-B

ond

basi

s

Lehman Jump Percentile

0%−20%20%−40%40%−60%60%−80%80% − 100%

Figure 6: Mean CDS-bond basis for firm groups. Firms are sorted by the difference in theiraverage CDS-bond basis in the month before and after the Lehman bankruptcy. Groups are thendefined by the quantile intervals in the legend. The red vertical line marks September 15 2008.

After the acute crisis period, the basis corresponds to the initial price of subsequent vintages

of the risky asset: a more negative basis is equivalent to a higher expected return if the worst

case does not realize.

Importantly, leverage and market positioning played a key role in the magnitude of the

negative returns in the fragile state. D.E.Shaw (2009) argues that dealer positioning was

the primary driver of basis changes around the Lehman bankruptcy and Choi et al. (2018)

show that bond returns in September 2008 were significantly lower for bonds with larger

preexisting basis arbitrage positions.27

another large provider of CDS protection, were not bailed out by the Federal Government. Some dealersdid incur costs because of the resolution of Lehman’s positions, including the trade replacement costs dueto having to replace lost protection at higher prices after the bankruptcy. Nevertheless, those costs wererelatively small and the market continued functioning smoothly, as described, for instance, in Moodys (2008)

27More generally, Siriwardane (2019) show that capital constraints of intermediaries are priced in the CDSmarket.

38

The cross sectional variation across underlying corporations allows testing the key pre-

diction of the model: that risky assets which experienced the worst fragile state returns will

also have the lowest initial prices for later vintages. This can be seen informally from Figure

6, in which I group firms based on the magnitude of the basis change around the Lehman

bankruptcy: the relative position of the average basis for the 5 groups is mostly unchanged

after Lehman. To confirm this relationship more rigorously, Table 2 reports the results from

cross sectional regressions of the average basis in years following 2008 on the magnitude of

the change around Lehman, controlling for various measures the literature has proposed to

explain the cross section of the CDS-bond basis. The main takeaway from table 2 is that the

decrease in the weeks following the Lehman bankruptcy has a long lasting impact: for each

100 basis points negative change, the basis is 28 basis points more negative in subsequent

years. This is surprising to the extent that the price changes in those weeks were partly

driven by hedge funds deleveraging, as the evidence in Choi et al. (2018) suggests.

In all columns of Table 2 apart from the first, I control for the average level of the basis

of each company in 2008, before the Lehman bankruptcy: one might have thought that after

the temporary dislocation had subsided, the cross section of the basis could be explained

by the previous level for each company. In column 3, I control for the Standard and Poor’s

credit rating of the underlying bond, which is the main determinant of the funding cost of a

negative basis trade (Garleanu and Pedersen, 2011). While those fixed effects substantially

increase the explanatory power of the regression, they do not drive out the Lehman jump

measure. In the model of Oehmke and Zawadowski (2015), the basis arises as a result of

differences in liquidity of bonds and CDS. While it is easy to compute measures of liquidity

for the corporate bonds using TRACE data, obtaining CDS liquidity proxies with only end

of day quoted prices is harder. Nevertheless, some of the differences in liquidity should be

accounted for by the credit rating and industry fixed effects.

To further verify that the Lehman jump measure is not proxying for differences in funding

39

Table 2: Regression of the average CDS-bond basis in each year after 2008 starting from July2009 for each underlying entity on the Lehman jump, defined as the difference between the averagebasis in the month preceding and after the Lehman bankruptcy of September 15 2008, and variousfirm and year level controls. Heteroskedasticity robust standard errors clustered at the firm levelare in parentheses: ∗p

3.3 Back of the envelope calibration

Having shown that the observed changes qualitatively fit with the model predictions, I take

one step further by using the empirical estimates to pin down model parameters and deriving

the magnitude of yield changes implied by the model. I use the uniform distribution of

individual biases version developed in section 2.7 because it delivers closed form expressions

for observable quantities, as detailed in Lemma 3. To further simplify the expressions and

since disagreement and leverage interact in the model and are hard to disentangle without

direct data, I set the leverage parameter ` to equal .5. This implies that the median agent

is the marginal buyer of the risky asset in the initial period and therefore that disagreement

does not impact initial prices.

Lemma 3. Under the simplifying assumptions of section 2.7, consider two consecutive vin-

tages of the risky asset, with initial prices p1,Before and p1,After. Suppose the fragile state

realizes for the first vintage, and that the price in this fragile state is pF,Before. Then, if

the probability of the worst case scenario h1h2 is small and ` = .5, the log expected return

conditional on the worst case not realizing for the first vintage is

log(

1p1,Before

)≈ h1h2dBefore, (43)

where dBefore is the time 1 mean across investors of their belief about D. The price drop in

the fragile state is given by

p1,Before − pF,Before ≈ h2((1− h1)dBefore + ψ1

), (44)

and the expected log return conditional on the worst case not realizing for the subsequent

vintage is

log(

1p1,After

)≈ log

(1

p1,Before

)+ h1h2

(τp

τ1 + τp· ψ1

). (45)

41

Proof. See Appendix A.

Lemma 3 shows that the model pins down returns conditional on the worst case not

realizing both before and after a fragile state realization, as well as the magnitude of the

price drop in a fragile state. Since those three quantities are observable in the data, they

impose restrictions on model parameters. In particular, in the calibration exercises below, I

choose parameters to match the observed returns before the fragile state and the price drop

in the fragile state and then compare the model implied returns after the fragile state to the

empirical ones.

3.3.1 Hedged Puts Selling

In order to match the data to the stylized framework of Lemma 3, suppose that each vintage

of the risky asset corresponds to carrying out the puts selling strategy described above for

a year.28 The left panel of Figure 7 reports the distribution of the yearly returns on this

strategy. While the time series variation in initial prices in the model does not generate these

volatile returns,29 the distribution of yearly returns doesn’t feature fat tails, supporting the

premise that Black Monday was a fragile state rather than a worst case scenario.30

A natural empirical counterpart to a fragile state in the model is a period in which the

strategy suffers an unusually large drawdown, such as the week of Black Monday. The right

panel of Figure 7 shows that there have been two such drawdowns over a period of 35 years

so I set h1 = 0.06.31 Table 1 shows that expected annual returns were close to 0 before

Black Monday: following equation (43), we can set dBefore = 0 to match this. While the data28I use the margin-weighted strategy here but results are very similar with either of the other two.29This is because 1p1 in the model conceptually corresponds to the yield of the quasi-arbitrage. A modifi-

cation which addresses this without changing any of the results or intuition is to assume that the final payoffoutside of the worst case is not 1 with certainty, but rather 1 + � where � is a mean zero random variable.

30Also, the losses on the week of Black Monday were recouped in less than a year.31While it might be more natural to think of h1 and h2 as risk neutral probabilities (the investors in the

model are risk neutral) I take the historical frequency as a conservative estimate: equation (45) shows thatlarger values of h1 imply a larger change in yield after a fragile state.

42

0

2

4

6

−0.2 0.0 0.2Yearly Return

Cou

nt

0

2

4

6

−0.4 −0.3 −0.2 −0.1 0.0Max Drawdown

Figure 7: The left panel reports the demeaned yearly returns frequency for the margin weightedputs selling strategy. Daily returns are demeaned differently before and after Black Monday toreflect the change in average returns documented in Table 1. The right panel shows the maximumdrawdown of the strategy in each year, defined as the most negative return an investor who enteredthis strategy at any point during a year would have experienced during that year. Daily returnsare geometrically compounded to obtain multi-period returns.

doesn’t pin down h2 directly, it does restrict ψ1h2. Equation (44) implies that the maximum

drawdown around Black Monday, which was approximately 35%, is equal to ψ1h2. In the

model, ψ1h2 quantifies the amount of disagreement at time 1: before the fragile state occurs,

the most pessimistic agent values the asset at 1 − h1(h2ψ1) = 1 − h1 · .35 ≈ .98 while the

most optimistic values it at around 1.02.

The size of the maximum drawdown also determines an upper bound for the magnitude

of the yield adjustment after Black Monday. By equation (45), the change in yield equalsτp

τ1+τph1 · h2ψ1 ≈τp

τ1+τp · .15 · 0.35 ≤ 0.02 sinceτp

τ1+τp < 1. The extent of the adjustment

depends on investors’ subjective uncertainty before the fragile state is realized, quantified by

τ1, and on the perceived informativeness of price signals τp. Since this was a relatively new

market in 1987, τ1 is arguably low. On the other hand, the fact that the change has been

extremely persistent suggests that the price signal was thought to be much more informative

than the public signals investors receive in normal times, implying a large value of τp and

43

therefore a value of τpτ1+τp close to 1. Estimating the volatility of payoffs by the volatility of

the demeaned annual returns reported in the left panel of Figure 7, this means the model

can produce an increase in the Sharpe ratio of the options selling strategy from 0 before the

crash to .3 afterwards. While this doesn’t match the .7 in Table 1, it shows the model can

generate changes of comparable magnitude.

3.3.2 CDS-bond basis

I proceed similarly for the CDS-bond basis. An estimate of h1 is the historical frequency

of fragile states, so in this case around one in 10 years: h1 = 0.10. We observe a basis for

each corporation, but to simplify the analysis I group them as in Figure 6, by quintiles of

the Lehman Jump. Figure 6 shows that all five assets had similar mean bases before the

Lehman bankruptcy: around -10 basis points on average. Since the basis represents the yield

from holding the trade to maturity if the worst case scenario doesn’t realize, equation (43)

implies h1h2dBefore = 0.1%, and therefore h2dBefore = 1%.

To focus on the impact of fire sales, I assume all parameter are equal across assets apart

from the disagreement about the worst case scenario, denoted ψj1 for each asset j. Since

` = 0.5 for all assets, a larger ψj1 implies a larger fire sale by Proposition 2.

Mapping the fragile state price drop to the 2008 maximum drawdown, equation (44) gives

MaxDrawdownj = h2dBefore − h1h2dBefore + h2ψj1 = .9% + h2ψj1 (46)

so that we can pin down h2ψj1 for each j and hence obtain the model implied change in yield

for the next vintage of each of the risky assets. In particular, (45) implies that the increase

in yield is

log(

1p1,After

)− log

(1

p1,Before

)≈ τpτ1 + τp

· 10% · (MaxDrawdownj − .9%). (47)

44

In Figure 8, I compare the upper bound of the change implied by the formula above to the

increase in absolute average basis from before the Lehman bankruptcy to 2010 for each asset.

While the upper bound predicted by the model is lower than the actual change in most cases,

the figure shows the model can deliver risk profile changes of comparable magnitude.

●

● ●

●●

0

10

20

30

40

50

0 10 20 30 40 50Model Max Increase

Act

ual I

ncre

ase

●

●

●

●

●

0%−20%

20%−40%

40%−60%

60%−80%

80% − 100%

Figure 8: For each of the 5 assets, the horizontal axis reports the upper bound of the model impliedchange, namely 10% · (MaxDrawdownj − .9%). The vertical axis reports the absolute differencebetween the mean basis before the Lehman bankruptcy and the average basis from the start ofJanuary 2010 to the end of September 2010. The black line is the 45-degree line. The units onboth axes are basis points.

4 Conclusion

I described a learning mechanism through which traumatic episodes can change investors

perception of risk persistently. The mechanism I analyze is likely to be particularly relevant

following crisis situations. Unprecedented times tend to be associated with dislocations in

various corners