Learning From Liquidation Prices
Gianluca Rinaldi
∗November 7, 2020
Click for the latest versionAbstract
I develop a model of investor learning driven by mistaken inference from market prices. Investors have heterogeneous beliefs about the worst case return of a risky asset and take leverage to buy it. When the worst case becomes more likely, forced liquidations result in price crashes, which investors mistake for negative information about 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 larger changes. To link the model to historical crashes, I consider two strategies associated with 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 markets was similarly negatively affected after the Lehman bankruptcy. The losses on these strategies in those crisis episodes were likely exacerbated by deleveraging, but the in-creased returns after the crashes have been remarkably persistent, consistent with the implications of my model.
∗Harvard University: [email protected]. I am indebted to Xavier Gabaix, Andrei Shleifer, Jeremy Stein, Adi Sunderam, and especially John Campbell for their outstanding guidance and support. I am also grateful for comments and suggestions to Malcolm Baker, Joshua Coval, Tiago Florido, Nicola Gennaioli, Robin Greenwood, Samuel Hanson, Franz Hinzen (discussant), Clémence Idoux, Lukas Kremens, Owen Lamont, Andrew Lilley, Ian Martin, Robin Lumsdaine (discussant), Carolin Pflueger, Nicola Rosaia, David Scharfstein, Erik Stafford, Argyris Tsiaras, and the participants at the Harvard Finance lunch, Harvard Macro lunch, the OFR PhD Symposium, and the TADC conference.
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.
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
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.
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 have historically been (Bates,2000), or that those crashes coincide with large consumption drops (Barro(2006), Gabaix(2012)).
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
My model 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 characteristics
2The 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 balance sheet, 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-crisis deviations. See for instanceBai and Collin-Dufresne(2019) for the CDS-bond basis andDu et al.(2018) for Covered Interest Parity deviations.
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 withGrossman (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, investors
4I draw from the vast literature on heterogeneous beliefs and asset pricing, beginning withMiller(1977) andHarrison and Kreps(1978). Specifically, rather than assuming that investors have heterogenous priors, I assume they interpret public signals differently. This is analogous to the assumptions ofKandel and Pearson (1995) andBanerjee and Kremer(2010). SeeHong and Stein(2007) for a more extensive review.
5Several studies employ theGeanakoplos(2010) framework to analyze leverage and its consequences. Sim-sek(2013) highlights how disagreement about good and bad states can asymmetrically influence constraints and Geerolf (2018) characterizes the heterogeneity in borrowing arrangements when investors disagree on the recovery value of collateral. Martin and Papadimitriou(2019) model the the dynamics of sentiment but do not focus on learning.
6Banerjee (2011) proposes a way to determine whether or not investors condition on prices to update their 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 to survive.
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 isBanerjee 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 how
8Memory 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 changes can occur even if there is little new information. Moreover, uncertainty about others’ signals or widespread dispersion of information can cause large price movements without news (Romer,1992).
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.
p1 G pG = 1 1 F pF 1 1 − d h2 h 1 t = 3 t = 2 t = 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 collateralized
10The worst case state is analogous to a rare disaster of Barro (2006). For the US economy, the rare disasters inBarro(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 fragile states, rather than worst case scenarios.
11In the dynamic overlapping generations extension of section2.6, old investors own the risky asset and sell it to the young when they die. Similar assumptions are made inSimsek(2013) andGeerolf(2018). The results are unchanged if instead investors are initially endowed with the asset and can trade it at time 1, but the market clearing conditions become more cumbersome.
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), andGeanakoplos (2010). In AppendixB, I extend the model to allow for short sales and show that mislearning will still occur in fragile states and that the features of mislearning are analogous to those arising in the baseline model.
13In the baseline three period model, this assumption is analogous to assuming heterogeneous priors across investors, but it increases disagreement in later periods in the dynamic version of the model. SeeKandel and Pearson (1995) and Banerjee and Kremer (2010) for an extensive discussion of how this assumption differs form heterogeneous priors.
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 σ2
s.
I denote by bi
t(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 Ei
t. The subjective model is therefore described by
bi1d= φd; d0, σ02 (2) bi1s1|d = φs1+ ψδi; d, σs2 . (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
bi1d|si1= φd; di1, σ12 (4) in which, using τs= σ12 s and τ0 = 1 σ2 0
to indicate the precision of signal and prior respectively,
di1 = τs τs+ τ0 (s1+ ψδi) + τ0 τs+ 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. Here, I depart from the noisy rational expectation benchmark by assuming that investors
14More formally, for any random variable X, I denote by bi
t(x) the derivative of Pit[X < x], where the
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 − pF
h2
. (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:
bi1dpF|d= φdpF; d, σp2. (9)
The parameter σpcaptures the extent to which investors learn from fragile state prices. A
small perceived precision of the signal τp = σ12
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−pF
h2 as described above. The time 2 posterior is therefore
bi2d|I2
= φd ; di2, σ22 (10)
where time 2 information is
I2 = {s1} if state G realizes {s1, dpF} if state F realizes (11)
and the parameters are di2 = di 1 if state G realizes τ1 τ1+τpd i 1+ τp τ1+τpd p F if state F realizes (12) σ22 = σ2 1 if state G realizes 1 τ1+τp if state F realizes (13) In these expressions, τ1 = σ12 1
= τs+ τ0 is the precision of investors beliefs at time 1 and
τp = σ12
p 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
max ai 1≥0,ai1,0≥0,ci1≥0 ci1+ (1 − h1)(ai1,0+ a i 1(1 − `)) + h1Ei1 h ai1,0pF + ai1(pF − `) i under (14). (15)
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 (ci
2) or buy
the asset without leverage (ai
2). Their time 2 budget constraint is given by
ci2+ ai2pF ≤ ci1+ a
i
1,0pF + ai1(pF − `). (16)
Since in state F there can be no borrowing against the risky asset, investors’ problem is
max ai 2≥0,ci2≥0 ci2+ (1 − h2)ai2+ h2ai2E i 2[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 clearcharacteriz-ing. 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
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. Denoting the market implied d as dp1 ≡ 1−p1
h1h2, we have
1 − h1h2Ei1[d] > p1 ⇐⇒ di1 < d
p
1. (18)
The demand function for investor i at time 1 in units of the risky asset is therefore given by ai1(p1) = 0 if di1 ≥ dp1 1 p1−` if d i 1 < d p 1 (19)
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) = ˆ i ai1(p1)C10(d i 1)di (20)
and market clearing requires
a1(p1) = 1 ⇐⇒ C1(d
p
1) = p1− ` (21)
since the risky asset is in unit net supply at time 1.
investors who buy the asset at time 1. For those investors, the market implied d, dp1, is lower
than their subjective mean belief di
1: 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.
C1(di) + l 1-h1h2di l 0 1 2 3di 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 0
and standard deviation 1, its density is reported in the bottom panel. Parameters are set as
τ0 = τs = τp = 1, h1 = h2 = .2, 1 = 0 and ` = .49. The horizontal axis indexes investors’ mean
beliefs about d at time 1. The blue solid line is the cash available at time 1 to investors with beliefs equal or more optimistic than di: C1(di) + `. The black line is the valuation of the risky asset at time 1: 1 − h1h2di.
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 Figure2, 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
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 − h2d
p
F = pF ⇐⇒ di2 < d
p
F (22)
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 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) = − 1 p1−` · min ` pF, 1
if agent i bought at time 1
0 if di
2 ≥ d
p
F and agent i did not buy at time 1
1
pF if d
i
2 < d
p
F and agent i did not buy at time 1.
(23)
To understand the demand of time 1 buyers, recall they borrowed ` dollars per unit of the risky asset. As each of them holds p1
1−` units of the risky asset, they borrowed
`
p1−` dollars
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 p
F > `, they only have to sell p1F · 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
15They also need to sell some of their asset if state G is realized, but every agent agrees the asset is worth 1 in state G so this selling doesn’t have consequences for the price.
16When p
F < `, 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 increases in the fragile state. Assuming that partial rollover is possible only weakens the impact of fire sales, but does not eliminate it.
position is unchanged and aggregate demand must be zero: a2(pF) = ˆ i ai1(pF)C20(d i 2)di = 0. (24)
Which can be rewritten as
min ` pF , 1 ! 1 p1− ` ˆ time 1 buyers C20(di2)di = 1 pF ˆ state F buyers C20(di2)di. (25)
The left hand side of equation (25) 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 (21), 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(d p F) − C2 τ1 τ1+ τp dp1+ τp τ1 + τp dpF ! . (26)
Market clearing in state F is depicted in Figure 3. The distribution of cash available to investors below di has no mass for di < dp
1 as optimists bought as much as they could at
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 .
18The least optimistic time 1 buyer i
F had di1F = d
p
1 and therefore his state F mean belief d
i1 2 is equal to τ1 τ1+τd p 1+ τ τ1+τd p F by (12).
-1 0 1 2 3di 0.0 0.2 0.4 0.6 0.8 1.0 1.2 p1 d1p l pF dFp -1 0 1 2 3 Density τ1 τ1+ τpd1 p+ τp τ1+ τp dF p Density Before F Density After F
Figure 3: Example of market clearing in state F . Gδ is a normal distribution with mean 0 and
standard deviation 1, τ0 = τs = τp = 1, h1 = h2 = .2, 1 = 0 and ` = .49. The horizontal axis
indexes investors’ mean beliefs about d. The black solid and dashed lines are the valuations of the risky asset in state F and at time 1, respectively 1 − h2di and 1 − h1h2di. In the top panel, the blue
solid line is C1(di) − C1(dp1): the cash available at time 2 to investors with beliefs more optimistic
than di under the time 1 distribution of beliefs. The red line in the top panel represents the same quantity of cash but under the time 2 distribution of beliefs. The orange line in the top panel is the amount of leverage `. The bottom panel reports the probability density function of mean beliefs di at time 1 in blue and after the state F realization in red.
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
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 (21) and (26).
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 (21) for ` = 0. If the parameters above are such that p1(0) > 1 − h1
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. (27)
Proof. See Appendix A.
Equation (26), 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 d
p
1 = d
p
F. 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 the 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
19The 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 the possibility of negative prices is unappealing, given investors’ risk neutrality it does not imply theoretical inconsistencies.
increasing in ψ:
∂
∂ψ(p1− pF) > 0. (28)
Proof. See Appendix A.
Proposition2states 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 (26).
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: Proposition3 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 d1 < 2h11h2, 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 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.
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 bi 1 , t = 1 F 1 − d h2 1 h1 G 1 Beliefs bi3 , t = 3 F 1 − d h2 1 h1 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 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
f1, ..., fNF
k the information set of agent i is
Ii2k+1=nsi1, .., sik, dpF,f1, ..., dpF,k
N F k
o
. (29)
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
bitd|Iti= φ dit , 1 τt
!
(30)
where di
t is a linear combination of the signals observed and the agent’s prior, weighted by
their perceived precision:
dit= τs Pk l=1sl+ τpP NtF l=1d p F,kl+ τ0d i 0 kτs+ NtFτp+ τ0 = τ0δi+ τs Pk l=1sl+ τpP NtF l=1d p F,kl+ τ0d0 kτs+ NtFτp + τ0 (31)
since the disturbance δi doesn’t change through time, and the posterior precision is given by
τt = kτs+ NtFτp+ τ0. (32)
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’
beliefs is to assume
`k = p1,k − α (33)
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 (33), each corresponding to a vintage k of the risky asset and such that the cumulative distribution functions (C1, C2, ..) are consistent
with equation (31).
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−
p1,k|, is larger when the haircut α is smaller or the perceived precision of the price
signal τp is larger
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 value of the asset in the fragile state and in the worst case scenario, rather than on the probability of negative states realizing, the equilibrium borrowing arrangement would resemble the one ofGeerolf(2018), in which a continuum of margin levels exist in equilibrium, one for each level of optimism of the borrowers.
• 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 < 2h11h2:
• 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.
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 Proposition4shows this for the first realization of a fragile state, 5confirms the same
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 do not contain noise is interesting as it is related to the average path after a fragile state.
results for subsequent realizations. Additionally, Proposition5shows 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 section3.3.
Suppose, for simplicity, the initial public signal is s1 = d0, confirming the initial average
belief, then the time 1 mean belief di
1 is uniformly distributed across investors over the
intervalhd0− ψ1, d0+ ψ1 i , where ψ1 ≡ τs τ0+ τs ψ.
The market clearing price at time 1 solves
p1− ` = C1(dp1) = dp1− (d0− ψ1) 2ψ1 (34) which gives p1 = 1 − h1h2 1 + 2ψ1h1h2 ψ1 − 2ψ1` + d0 ! . (35)
Leverage ` only matters for pricing to the extent that investors disagree, ψ1 > 0, and is
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: hd2− ψ2, d2+ ψ2
i where d2 = τ1 τ1+ τp d1 + τp τ1+ τp dpF (36) ψ2 = τ1 τ1+ τp ψ1. (37)
The state F market clearing condition is given by
` = d p F − τ1 τ1+τpd p 1+ τp τ1+τpd p F 2 τ1 τ1+τpψ1 (38) which simplifies to dpF − dp1 = 2`ψ1. (39)
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, (40)
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
F realizes at time 2 from equations (36) and (39):
d2 = τ1 τ1+ τp d1+ τp τ1+ τp (dp1 + 2`ψ1) (41)
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).
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
O
TM − A
TM Puts IV (%)
Figure 5: For each put option in the data, I define its moneyness as the ratio of strike price
and underlying spot price. This figure displays, for each date, the difference between the average implied volatility of put options with moneyness between .85 and .95 (OTM IV) and the average implied volatility of those with moneyness between .98 and 1.02 (ATM IV).
In line with this interpretation, a large literature extending theBlack 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.
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).
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. I
consider three weighting rules to aggregate the individual 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 required to hold the hedged position, and the overweight OTM assigns weight moneyness1 2 to each put option. There are 1149 and 7509 daily observations before and after Black Monday, respectively. Means and volatilities are in annualized percent. The p-values in the last column correspond to Welch’s
t-tests and Levene’s test for means and variances equality, respectively.
Before After p-value
Margin weights Average return -0.94 5.05 0.02 Standard deviation 4.79 7.10 0.71 Sharpe ratio -0.20 0.71 Equal weights Average return -0.22 6.44 0.01 Standard deviation 4.90 7.39 0.59 Sharpe ratio -0.04 0.87 Overweight OTM Average returns 0.00 6.68 0.01 Standard deviation 4.91 7.43 0.63 Sharpe 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 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
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, the correlation of the option strategies returns with the S&P 500 is .27 for the whole sample, -.12 for the period before Black Monday and .30 after. Together with the low volatility of the option strategy, this makes it difficult to explain the large average returns after Black Monday by market risk.
