Memorable Events in Financial Markets
Andrés Carvajal∗ and Keisuke Teeple†
May 14, 2024

Abstract
We assume that financial traders remember certain days—for example, thosewhen the trader was actively trading—and that these days need not coincide acrosstraders. This disagreement leads to trade because those who recall bull markets
buy and those who recall bear markets sell. In our equilibrium, which is plagued bypurely non-fundamental volatility, a volatile price history generates belief dispersion and subsequent trade ensures continued price volatility. We then characterize
the steady-state cross-sectional distribution when traders remember active tradingdays. In the cross-section, young (old) traders, who have not (have) experienced
many trading days, have dispersed (concentrated) beliefs. Equilibrium prices display excess kurtosis due to the small chance that everyone trading on a particular
day is young.
JEL classification: G41, G14, D84, D91
Keywords: Non-Bayesian models, non-fundamental volatility, heavy tails, experience effects∗

acarvajal@ucdavis.edu, University of California Davis, One Shields Ave Davis, California 95616
k2teeple@uwaterloo.ca, University of Waterloo, 200 University Ave West Waterloo, Ontario N2L3G1.
Teeple received financial support for this research from SSHRC Explore and Insight Development Grants.
†

1

Introduction

Why are asset prices so volatile? Shiller (2015) has famously shown that the majority ofthis volatility cannot be attributed to news alone. Here we introduce a theory of asset
price volatility that relies critically on the past personal experiences of traders. Many
empirical studies support this theory: Malmendier and Nagel (2011) show that individuals who experienced low stock market returns are more pessimistic about future stock
returns. Kuchler and Zafar (2019) show that individuals use personal experiences to form
expectations about aggregate economic outcomes like house prices and unemployment.
Kaustia and Knupfer (2008) show that personally experienced returns from past IPOs,
not passively observed returns, affect future IPO subscriptions. Malmendier and Nagel(2015) show that individuals overweight inflation experienced during their lifetimes whenforming expectations about future inflation.
An important theoretical benchmark model has been developed by Nagel and Xu(2022), in which a representative agent has a fading memory. That is, a data point’s
influence on beliefs gradually fades over time. While their setup is appropriate for modeling aggregate shocks like a recession, here we focus on precisely the residual effects.
What we have in mind are memorable events which are idiosyncratic to the trader, likeactive trading days. These are days (which may differ across traders) when the traderholds a nonzero position in a particular asset and personally experiences the gain or
loss. Andersen et al. (2019) show that experiences on such days, as opposed to secondhand experiences, affect an individual’s risk-taking behavior. Similarly, Strahilevitz et al.
(2011) show that investors are reluctant to repurchase stocks previously sold for a loss.
To summarize, feeling the gain or loss seems to impact individuals more profoundly thanpassively observing returns.
Aside from traders failing to recall certain days, we assume that they are Bayesian.1
They gather what they can remember, and use this data to form expectations about
future returns. Idiosyncratic recall generates expectation disagreement amongst traders,which is captured in the cross-sectional distribution over beliefs. And this disagreement
is the motive for trade: pessimists who recall bear markets sell, and optimists who recallbull markets buy. We assume that a random subset of traders are active in financial
markets during any given period; this effectively “draws” traders independently from1

Our model also resembles one of under- and overreaction that can be traced back to Grether (1980).
Mullainathan (2002) also makes this connection: imperfect recall explains under- and overreactions withrespect to Bayes’ Rule.

2the cross-section and is, in fact, the only exogenous random variable in the model. This
captures the empirically-relevant constraint that traders are not all simultaneously active
in financial markets.2
The final participant in our economy is a simple linear market maker like the one
from Teeple (2022) and Teeple (2023). This market maker raises (lowers) prices when
there is excess demand (supply), while providing the required liquidity throughout to ensure that markets clear.3 The following quote from Bagehot (1971) provides an excellent
summary of our model’s microstructure:
It is well known that market makers of all kinds make surprisingly little useof fundamental information. Instead they observe the relative pressure of buy
and sell orders and attempt to find a price that equilibrates these pressures.
The resulting market price at any point in time is not merely a consensus of
the transactors in the marketplace, it is also a consensus of their mistakes.
Under the heading of mistakes we may include errors in computation, errorsof judgment, factual oversights and errors in the logic of analysis.
An equilibrium is established in the following sense. We begin with a history of prices
drawn from a discrete time stochastic process, which forms the basis for traders’ beliefs.
Based on what they can recall, traders apply Bayes rule. Based on these posteriors,a subset of traders trade; and based on trades, the market maker adjusts the price.
Equilibrium prices must have the property that the distribution of current prices matches
that of the history of past prices. Intuitively, it is because prices have always been volatilethat traders have dispersed beliefs; this, in itself, drives future price volatility. Given that
there is no news or other shocks in our model, imperfect memory gives rise to purely nonfundamental volatility.
We show that the discrete time Brownian motion is the unique equilibrium priceprocess when traders have mean-variance utility and remember a fixed, homogeneous
number of periods. Importantly, this utility assumption gives our model a linear structure: demands are linear in past price observations due to the mean-variance assumption,and the sum of demands determines future prices due to the linear market maker. Thenthe equilibrium requirement that past and future prices share a common distribution
becomes equivalent to stability of the equilibrium price distribution.4 Within the stable
2

See Graham et al. (2009) and Richards and Willows (2018).
The classical Walrasian auctioneer appears as a limiting case of our market maker.
4
A distribution is stable when the sum of i.i.d. random variables have that same distribution.
3

3

family, the Normal distribution is the only distribution with a finite variance.
Up to this point, we took the days which a particular trader remembered as given.
In our main application, we assume that traders remember the days on which they
were actively trading. This gives rise to learning, and the cross-sectional distribution
over subjective returns becomes more concentrated over time as traders receive more
information. In order to make the model stationary, we assume that traders die at aconstant rate and are consequently replaced with newborns who have yet to trade. The
steady-state cross-section consists of a mixture of the youth (traders who have traded
less, leading to dispersed beliefs) and the elderly (traders who have traded more, leading
to concentrated beliefs). This is consistent with survey evidence from Giglio et al. (2021)
that older individuals have tighter subjective distributions over future stock returns.
Unlike in the baseline setting, where belief dispersion is fixed across the cross-section andequilibrium prices are Normal, in this extension, there is a mixture of belief dispersion
across demographics which leads to excess kurtosis in equilibrium prices. The intuitionis as follows: extreme prices result from the small chance that all traders trading on agiven day are young. To the knowledge of the authors, this mechanism for generating
heavy-tailed prices—which we term youthful volatility—is novel. We then provide several
testable implications of our model: an older population of traders leads to lower pricevolatility, and a more dispersed population of traders (in terms of age) leads to larger
excess kurtosis in prices.
We consider three policies to reduce price variance, and immediately rule one out:
a capital gains tax. While it does dampen capital gains, a gains tax also dampens wealthvolatility, and the net effect goes in the wrong direction: it increases trade volume. We
then compare the distributional consequences of the two remaining effective policies: a
tax on trade volume and tightened borrowing limits. While transaction taxes truncate
trade for the center of the cross-sectional distribution (i.e. older traders), borrowing limits truncate trade for the tails of the distribution (i.e. younger traders).
Related Literature. Recent papers by Malmandier et al. (2020) and Schraeder (2016),
which are the closest studies to ours, augment the fading memory assumption with trader
heterogeneity using an overlapping generations setup. In Schraeder (2016), the young
generation is assumed to be rational while the adult generation is assumed to overweightthe previous period’s observation. In Malmandier et al. (2020), the young generation
reacts more strongly than the old to recent observations, as these make up a larger part4

of their lifetimes. To compare models, we assume idiosyncratic instead of fading memory;and to compare results, our paper is the only one that connects memory to excess pricekurtosis. Also closely related are papers by Collin-Dufresne et al. (2017) and Ehrling et al.
(2018). In Collin-Dufresne et al. (2017), younger generations update beliefs more so thanolder generations with respect to macroeconomic shocks. In Ehrling et al. (2018), theyoung generation, who act as trend chasers, disregard the price history and lose money
to the old generation, who act as contrarians.
The difference between the fading memory setup and ours is the over-weightingof the previous period observation versus an idiosyncratic observation. First, our assumption is well-supported by studies of human memory. According to the textbook
treatment of strength theory from Kahana (2012), each memory has a fixed numericalvalue representing the degree to which that memory evokes a sense of familiarity. Kahana(2012) comments that it is “reasonable to suppose that items vary in their strength, withsome items being stronger than others.” In fact, strength theory models typically assume
that strength values are drawn from a Gaussian distribution. In our model, we simplify
matters by assuming that observations are either remembered or not (we also extend tohigh versus low memory strength). Second, our idiosyncratic assumption is reinforced byrecent economic survey evidence. For example, Dominitz and Manski (2011) find that
traders’ return expectations are interpersonally variable but intrapersonally stable. Similarly, Giglio et al. (2021) find that beliefs are mostly characterized by large and persistentindividual heterogeneity.
The remainder of the paper is organized as follows. Section 2 provides a motivating
example demonstrating the Brownian equilibrium. Section 3 models the cross-section
when traders remember active trading days, and Section 4 presents the main theoreticalresults in a generalized setting. Section 5 addresses policy implications, and Section 6
concludes.

2

Motivating Example

Traders. Time is discrete, infinite, and indexed by t ∈ Z (not N), and there is acountable number of traders of a single long-lived financial asset that pays no dividends.
These individuals should be thought of as retail investors who use technical analysis

5before placing market orders.5 Suppose that there is a history of past price realizations− even at the beginning of the model (t = 0) − and that these past realizations weredrawn from a discrete time Brownian motion with zero drift. That is, (pt+1−s − pt−s )s≥1
were independently drawn from a N (0, Σ2 ) distribution. In equilibrium, we will requirethat this price history be consistent with prices generated within the model (for t ≥ 0).
Traders, indexed by i, maximize mean-variance utilityρui (xt ) = (Ei [pt+1 |pt ] − pt )xt − Var(pt+1 − pt |pt )x2t2

(1)

where ρ denotes the risk aversion parameter and xt denotes the trader’s position.6 Demand is chosen to maximize utility,xi (pt ) =Ei [pt+1 |pt ] − ptρΣ2What makes each trader unique is that she believes that price increments (pt+1 − pt )have mean µi , because certain past price realizations are more memorable. In this simple example, we let each trader remember only one period, and we assume that eachtrader has a unique memorable time period. Following Nagel and Xu (2022), we assume
that traders apply geometric weights to observations (pt+1−s − pt−s )s≥1 when applying amemory-constrained version of Bayes’ Rule:f (µ|pt − pt−1 , , ..., pt−T +1 − pt−T ) ∝TYif (pt+1−s − pt−s |µ)αs

(2)

s=1

where T denotes the length of the price history and µ denotes the true price drift. Note
that, when αsi = 1 for all s, this reduces to Bayesian updating with a flat prior. Insteadof the equal weighting case à la Bayes or αsi fading over time à la Nagel and Xu (2022),
we assume that traders distinctly remember one period i by setting weights
αii = γT, and αsi =(1 − γ)Tfor s ̸= i
T −1

where 0 < γ ≤ 1 denotes idiosyncratic memory strength. Because there is a one-to-one
mapping between traders and memorable periods in this example, we let the trader’sindex i also denote the period she remembers. Hence αii denotes the weight trader i puts
5

Empirical evidence suggests that the scarring effects of personal experiences affect not only retailinvestors, but also highly specialized individuals. See Malmandier and Wachter (2022).
6
Upcoming assumptions about the timing of trades justify the trader’s myopic objective.

6

on her memorable period. Now consider the total weight put on non-memorable periods,Piis̸=i αs = (1 − γ)T . Adding this to the memorable weight, αi , we see that the weights
add up to the number of observations, T , just like in the Bayesian case. However, here
we distribute γ proportion of that weight to the memorable period, and the rest on allother periods. Because this remaining weight is distributed evenly across periods, in somesense, traders are Bayesian over non-memorable periods. When a memory-constrained
Bayesian after T periods forms a posterior mean, it becomes a weighted average between
what she remembers and what she does not7
µi = γ(pi − pi−1 ) +1−γ X(ps − ps−1 )T − 1 s̸=i

As we send the number of observations in the history, T , to infinity, by the strong law oflarge numbers we have thatµi → γ(pi − pi−1 ) a.s.
so that non-memorable events effectively wash out.8 If γ = 0 and there are no memorable
periods, the expression above shows that traders eventually learn that the price drift istruly zero. Since each trader recalls a different time period by assumption, the crosssectional distribution of beliefs µi approaches the N (0, γ 2 Σ2 ) distribution as the numberof traders, which we denote M , tends to infinity. More formally, by the strong law oflarge numbers, the empirical distributionM

FM (t) =1 X
1µi ≤t
M i=1

approaches the Normal distribution function for each t almost surely as M → ∞.
Market Maker. If a randomly selected subset of traders of size n trades each period,
then a linear market makernX
pt+1 = pt + c
xi (pt )(3)
i=1
7This formula relies on the Normality of price observations.
Unlike in Nagel and Xu (2022), here there is no residual subjective uncertainty. That is, the varianceof the posterior tends to zero. Hence, volatility in this model comes from objective uncertainty overprices, not subjective uncertainty over µ.
87ensures that prices follow a discrete time Brownian motion.9 Normality follows from thefact that the sum of i.i.d. Normal random variables in (3) is also Normal. Furthermore,√
Σ2 = cγ n/ρ is the unique value that makes the variance of historical prices match thatof current prices. It is an equilibrium object. When Σ2 is too large (small), past pricevolatility makes traders trade too little (much), leading to prices characterized by too
little (much) volatility according to (3). Note that the indexes of traders who trade in any
given period are exogenous random variables (in fact, the only ones), and are assumed
to be independently drawn across periods. This modeling assumption is motivated byrecent survey findings that beliefs do not predict when investors trade, only the direction
and magnitude of trades (see Giglio et al. (2021)). Furthermore, in a dataset of UK
investors, Richards and Willows (2018) find that while the top decile of traders trades 69
times per year, the second decile trades 23 times per year and the remaining 80% trade
6 times per year. Similarly, Graham et al. (2009) find that the average number of days
between trading is 89 using data from a UBS/Gallup general investor survey. These data
support a model where not all traders are simultaneously active in financial markets.10
The market maker (3) has the following microfoundation also used in Teeple (2022).
We set up the market maker objective function according to the three published objectivesof the NYSE’s designated market maker: prioritize price discovery, lower volatility, andprovide liquidity.
X1
max pt+1
xi (pt ) − (pt+1 − pt )2
pt+1
2c
i
Note that the first term above corresponds to price discovery, and is precisely the objectiveof the classical Walrasian auctioneer.11 The incentive to raise prices on days with excess
demand and lower prices on days with excess supply is evident in this term. In theclassical general equilibrium setting, the fixed point of such a maximization problem isstudied; here we do not abstract away from preceding dynamics. The second term above
corresponds to the second objective of the designated market maker (lowering volatility).
The parameter c controls the relative weight allocated between these first two objectives.
Such an objective function implies that the market maker must provide liquidity (itsthird goal); she injects extra liquidity into the market by taking the opposite side to9

To achieve a geometric Brownian motion, we redefine prices in logs.
That all traders are memory-constrained and trade on random days may seem extreme. We could
extend the model to also include Bayesians with access to news who trade on all days. In the spirit ofsimplicity, we have refrained from this exercise.
11
This market maker does not rely on limit orders to infer the shape of trader demand functions.
Instead, she only observes the value of aggregate demand.
10

8

excess demand, effectively clearing markets each period.
This objective function leads to the intuitive rule (3): the market maker maps
excess demand into higher prices and excess supply into lower prices at elasticity c. Thisis consistent with empirical evidence from Chordia et al. (2002), who confirm that excess
buy (sell) orders drive up (down) returns, even at lagged time periods. For familiarity,
let us map this unconventional microstructure into a more standard one, maintaining
the same population of memory-constrained traders. Assume that the same group of n
traders trades each period (instead of n randomly drawn), that the asset pays random
dividends dt+1 (instead of zero dividends), and that the interest rate is R > 0 (instead ofzero interest rates). Demand would then be given byxi (pt ) =Ei [pt+1 + dt+1 ] − (1 + R)pt
γdi + (1 − γ)E[dt+1 ] − Rpt
=
2
ρΣ
ρΣ2

where Σ2 now denotes volatility of exogenous dividends, and the second equality holds
when prices are at a steady state. The fixed point of the market maker’s rule, whichPclears markets, is given by p = R1 nγ i di + (1 − γ)E[dt+1 ] . The point of this exercise
is to emphasize that the market maker is a sensible one, which simply generates offequilibrium, or tâtonnement, dynamics in a standard model. However, in our setup,
we do away with dividends so that our equilibrium takes on a self-fulfilling flavor andvolatility in our setting becomes purely non-fundamental.12 The random variable thattakes the place of exogenous dividends is the identities of active traders.
It is important to point out that the timing assumptions differ significantly fromstandard market order models like that of Kyle (1985). There, traders face uncertainty
about the price at which their orders are executed. Here, traders are able to trade
unlimited positions at the fixed price pt . By ignoring price impact, we are effectively
ignoring the bid-ask spread, which simplifies our model and is a reasonable assumption
when spreads are small. Teeple (2023) introduces the spread in a similar setting but withno memory constraints. There, a revenue-maximizing market maker is shown to havethe same qualitative properties as the one described here.
Because traders trade on random days, it is assumed that they buy (sell) on suchdays at price pt then sell (buy) on the very next day at price pt+1 .13 Importantly, they do
12

21% of the S&P500 does not pay dividends. However, this number grows to over 50% when considering all US publicly traded stocks.
13
The second half of this round trip is not included in the excess demand for period (t + 1). In Teeple(2023), where the market maker’s problem is explicitly modeled, market makers have an incentive to9not hold the asset between two randomly chosen trading days. This has implications forboth earnings and inventories of the market maker. In terms of earnings, prices according
to (3) rise (fall) when the market maker takes a short (long) position. In other words,
the market maker loses money each period. This observation is consistent with empiricalevidence from Sofianos (1995) that market makers incur positioning losses on their inventory, which are compensated by revenues from spreads (not modeled here). In termsof inventories, when the market maker takes a long (short) position, she buys (sells) at
price pt then sells (buys) at price pt+1 . Hence her inventories in any given period areP− i xi (pt ), but importantly, she does not accumulate inventories across periods. Thisis observationally consistent with mean-reversion theories of market maker inventories.
For example, see Hasbrouck and Sofianos (1993).
Equilibrium Definition. Here we define our equilibrium concept. The price history is
initially drawn from some distribution. Based on their imperfect memory, traders disagree and trade; and based on the market maker, prices adjust. We require that the
distribution of prices generated by the model be precisely the one from which the pricehistory was initially drawn. Formally, an equilibrium is a price history Ht , trader beliefsµi and demands xi (pt ), and a price distribution Ft satisfying:(a) Given the recollection Hti ⊂ Ht , traders form beliefs µi according to (2).
(b) Given beliefs µi and conjecture Σ2t , xi (pt ) maximizes (1) if they are in the active
group, i ∈ At . Otherwise, xi (pt ) = 0.
(c) The market maker (3) generates prices drawn from Ft , where Ft is consistent withhistory Ht and VarFt (pt+1 − pt ) = Σ2t .
Furthermore, we focus our attention on steady-state equilibria where Ft = F (Σ2t = Σ2 )for all t. While markets do not clear in the traditional sense (they do clear when themarket maker’s position is accounted for), we maintain the standard assumption thatmarkets are competitive. With few traders, each has a non-negligible own price impact.
Optimizing over this price impact, even with memory constraints, could allow traders to
do better than the demand functions described here.
move prices based on initial trades to elicit trader participation. However, this incentive disappears
when traders unwind positions.

10Uniqueness. Our next question is whether the Brownian motion is the unique equilibrium price process in this baseline setting. Interestingly, the drift of the Brownian
motion (or lack thereof) is unique. To understand why, consider the following extensionof the model. Instead of assuming that the asset is in net zero supply, say that the asset
has an exogenous, deterministic supply S > 0. The natural extension of the market
maker’s objective is!
max pt+1
pt+1

X

xi (pt ) − Si−1(pt+1 − pt )22cwhich yields the following generalization of (3)!
pt+1 = pt + cXxi (pt ) − S(4)iso that, just like before, prices rise (fall) when excess demand is positive (negative).
Assuming that the price history has a drift of µ, beliefs converge almost surely toµi = γ(pi − pi−1 ) + (1 − γ)µand the cross-section of beliefs µi across the population then approaches N (µ, γ 2 Σ2 ). The
equilibrium drift µ can then be found by taking the expectation of the market maker’s
rule (4). With some algebra, this condition reduces toµ=

cScn/(ρΣ2 ) − 1(5)

The intuition is that positive drift generates excess demand; this, in turn, requires apositive asset supply.14 Importantly, µ = 0 when S = 0, establishing the uniqueness ofzero drift in the baseline setting.
To summarize so far, both variance and drift are unique. However, we can say
more by making use of the following well-known statistical fact from Durrett (2017): if alinear combination of two independent random variables with some distribution has thatsame distribution, it is said to be stable. Let us consider the implications for our model,
where demand is linear in past prices due to the mean-variance assumption, and prices√
S > 0 does not change the variance calculation, hence Σ2 = cγ n/ρ like before. Then n > 1 ensuresthat the denominator of (5) is positive.
14

11

are linear in demand due to the nature of the market maker objective. Altogether,
Linear in prices

pt+1 − pt = cz }| {X γ (pi − pi−1 )
i

|

ρΣ2
{z

Linear in demand}

Equilibrium requires that the past price distribution match the current one, so the only
candidate for the equilibrium distribution is the stable-α distribution. Furthermore, there
is only one distribution in the stable-α family that has a finite variance: the Normal
distribution. Hence the Brownian motion (with drift and variance pinned down) is notjust an equilibrium distribution. It is the unique equilibrium distribution.

3

Active Trading Days

Up until this point, we have taken the days which each trader remembers as given. The
setting we consider now is that of Section 2 except for the following change. When the
random subset of size n trade, they remember that period’s return (after they trade andthe return is realized). Empirically, this is consistent with evidence that traders disproportionately use active trading days (more than inactive ones) to form beliefs about
future returns (see Andersen et al. (2019) and Strahilevitz et al. (2011)). Theoretically,
this is consistent with the context-dependent memory model of Bordalo et al. (2020),
where the context here would be whether or not the trader was actively trading.15
Expanded Memory. The first difficulty with this setup is characterizing the Bayesianposterior mean when a trader has multiple memorable periods. Despite the well-known
difficulties in deriving a closed-form solution for the posterior mean of arbitrary distributions (equilibrium prices may not be Normal), our first lemma addresses this issue. All
proofs are in Appendix A.
Lemma 1. (Posterior Consistency) Say that traders calculate their posterior according to memory-constrained Bayes’ Rule (2), with weights given byγT /|Ki | for t ∈ Ki
αti =(1 − γ)T /(T − |Ki |) otherwise
15

See also Wachter and Kahana (2024).

12

(6)

As the history T → ∞, their posterior mean approaches
µi =

γ X
(pt − pt−1 ) + (1 − γ)µ
|Ki |
i
t∈K

irrespective of the price distribution.
Note that µ denotes the mean of the price distribution, γ denotes idiosyncratic
memory strength, and Ki denotes the memorable set for trader i. When this set consistsof only one period, the weights in (6) collapse precisely to those described in Section 2.
Now consider |Ki | > 1. Just like in Section 2, the total weight adds up to T , with γ
proportion of that weight distributed evenly across memorable periods. The way that
we circumvent the usual algebraic difficulties in dealing with non-Normal posteriors is bysending T → ∞. Asymptotically, there are results from probability theory that guarantee
consistency of the Bayesian posterior mean. Weights in (6) are as-if the trader not only
observes an infinite number of independent non-memorable prices, but also an infinite
number of repeated observations of (pt − pt−1 ) for each t ∈ Ki . As T → ∞, they become
sure that their posterior mean equals the µi defined in Lemma 1 (i.e. their posteriorvariance tends to zero).
To get a sense of an economy with expanded memory, Proposition 1 extends the
economy from Section 2 with multiple memorable periods. What we do not allow for(yet) is overlapping memory; each trader still remembers a subset of the price history
disjoint from any other trader. What we do allow for here is heterogeneity in memory
length; one trader may remember two periods, while another may remember three. With
heterogeneity in memory length, equilibrium prices need not be Normal. However, in
one special case we recover the Brownian equilibrium.16
Proposition 1. (Expanded Memory) Say that each trader remembers k i disjoint
periods for k i ∈ {1, ..., K}. Equilibrium prices converge in distribution to a discrete time
Brownian motion as n → ∞ and T → ∞.17
First consider the homogeneous case, where all traders remember the same number,k, of disjoint periods. If multiple events are memorable, good days and bad days begin
to cancel each other out within the memorable set. Hence, traders with a larger set
16

Note that we exclude the case where indexes of memorable prices are measurable with respect tothe realization. This, for example, rules out extreme prices being remembered by more traders.
17
Note that sending the history T → ∞ implies that the number of traders M → ∞ at the same rate.

13

of memorable events hold less dispersed, or tighter, beliefs. This, in itself, does not
destroy Normality and simply lowers the equilibrium variance, Σ2 . See Figure 1 for the
cross-sectional distribution over µi for traders who remember 1, 2, and 3 disjoint periods(memory parameter γ is set to one).

Figure 1: Cross-Section with Expanded Memory
Now consider adding heterogeneity in memory length. The cross-sectional distribution of beliefs is no longer Normal, because the mixture of Normal distributions need not
be Normal. That is, when drawing a trader from the cross-section, there is some chance
she is inexperienced (few memorable periods, with dispersed beliefs) and some chance
she is experienced (many memorable periods, with concentrated beliefs). So while thecross-section need not be Normal, the Lindeberg-Lévy theorem applies when n is large
and the market maker ensures that prices continue to follow a Brownian motion. In
contrast to Section 2, sending n → ∞ is with significant loss of generality.18
Compared to the baseline case of Section 2, this last discussion suggests that there
will be two notable changes in the active trading days setup. First, the equilibrium variance will be reduced because the older generations have less dispersed beliefs. Second,
equilibrium prices need not be Normally distributed due to the mixing of younger andolder generations (when n is finite).
18

As n grows, so does the equilibrium variance. To deal with this fact, we normalize the equilibrium
variance to one and find a market maker constant, c, consistent with that normalization.

14

Cross-Section. When traders remember active trading days, the cross-section of traders
would continuously learn. To regain stationarity in our model, we assume that D > 0
traders die each period and are consequently replaced by D new traders.19 The steadystate cross-section of traders is characterized by a sequence (Pj )∞j=0 , which represents the
proportion of the population that remembers j days. We first analyze the case where
the number of traders, M , is finite, then take limits. When M is finite but large (inparticular, M > n + D), the law of motion for Pj is given byP t 1 − D − n  + D , if j = 0
j
M
M
M
t+1
Pj =
P t 1 − D − n  + P t n , otherwise
j−1 M
j
M
M
D
When j = 0, the proportion decreases by a factor M
due to death, and by another factorn
due to active trading, learning, and traders consequently leaving group P0t for group
M
D
P1t+1 . The proportion increases by the amount M
due to the introduction of new traders.
When j > 0, the proportion decreases for the exact same reasons. The proportionntincreases by the amount Pj−1
due to active trading, learning, and traders consequently
M
t+1t. The steady state of such a system is geometric:entering group Pj from group Pj−1


P=


D
Dn
Dn2
,
,
, ...
D + n (D + n)2 (D + n)3(7)

and, importantly, is independent of M . As an example, consider the special case when
D = n so that traders die at the same rate that they learn. Then the steady state is
1 1 1, , , ... .
2 4 8
The cross section (7) forces us to take a stance on trader beliefs in the absence ofany memorable period. We have yet to encounter this issue because each trader remembered one day in Section 2 and one or more days in Proposition 1. Our approach here is
to extend the weights in (6) so that, when |Ki | = 0, αti = 1 for all t. In other words, they
are Bayesian before their memorable period arrives. Note that this does not preclude
newborns from observing the price history; it simply means that no period from that
history stands out, or is memorable, to them.
Overlapping Memory. The active trading days setup introduces an additional complication in the form of overlapping memory. That is, when n > 1 traders remember an
19

Due to an absence of wealth effects in the model, we conjecture that our results would not change
qualitatively under an alternate survival story where traders die off when wealth hits zero.

15

active trading day, their beliefs are no longer mutually independent.20 In general, thisproblem of overlapping beliefs makes the model intractable. However, in the special casewhen traders remember past active trading days, the degree of overlap is “limited” and wehave the following lemma. It states that, with enough traders, this limited overlap does
not preclude the law of large numbers from holding and, consequently, the cross-sectionof trader beliefs converges in distribution much like in Section 2.
Lemma 2. (Limited Dependence) Assume that the maximum number of periods remembered in the economy is K. Then, as the number of traders M → ∞, the empirical
distribution over beliefs FM (t) converges in probability to its expected value for each t.
A natural next question is: What is the expected value of FM (t)? However, we holdoff on this investigation until Section 4. Instead, the point of Lemma 2 is to emphasizethat the cross-section converges to some distribution despite the presence of traders whose
beliefs overlap (albeit in a limited way).
For intuition, consider whether the weak law of large numbers holds (instead ofwhether the empirical distribution converges). By Chebyshev’s Inequality,
P r(|µM − µ| ≥ ε) ≤Var(µM )ε2(8)where µM denotes the average belief in the cross-section of size M , and µ denotes theactual price drift. In the standard i.i.d. case, the variance term above collapses into
something that is O(1/M ) and so the weak law holds. However, here, posterior meansµi are neither independent nor identically distributed. Although the younger (older)
generation has more dispersed (concentrated) beliefs, this is not problematic because
there exists a most-dispersed belief, which serves as an upper bound for the variance ofany belief in (8). And although there is some overlap in beliefs, each trader’s belief canonly overlap with K(n−1) others’ because they trade on K days where there were (n−1)
other traders on each day. Hence the variance of the average belief in (8) decomposes
into some (bounded) variances and (a finite number of) covariances, which all disappear
as M → ∞. This same idea is employed in the actual proof, although the empiricalPM
distribution, M1
i=1 1µi ≤t , is considered.
The key to the lemma is that we truncate the number of remembered periods at alarge number K. That is, we ignore traders who remember an arbitrarily large number of
20

Note that any overlap in beliefs was explicitly assumed away in Section 2 and Proposition 1, but isunavoidable in the active trading days setup.

16

past periods. This is justified for two reasons. First, such traders do not contribute to our
economy in the sense that they trade close to zero. Second, we focus on cross-sectional
distributions (7) where such traders have a small mass. Hence, the economy consideredin Lemma 2 will become arbitrarily close to the true cross-section when K is large.

4

Equilibrium Prices

Section 2 was the simplest possible example in our setting, and Section 3 suggested a
viable microfoundation for heterogeneous memory. To nest both cases, we now consider
a general steady-state sequence (Pj )K
j=0 , representing proportions of the population remembering j days. If this sequence were geometric, we recover the special case of Section
3; if it were degenerate, we recover the special case of Section 2. We emphasize that,while this setup may seem very general, we can only guarantee that upcoming results
apply to the following two settings:
1. An extension of Section 2 where each trader remembers k i disjoint periods fork i ∈ {1, ..., K}.
2. The active trading days setting from Section 3, so that Lemma 2 continues toapply.21
As discussed in Section 3, it is the problematic overlap in beliefs which necessitates these
conditions above. These two requirements are sufficient but not necessary for upcoming
results.22
Next, we recast the failure of Normality from Proposition 1 (when n < ∞) not asa negative one, but as a positive one (i.e. interesting properties of equilibrium prices). In
particular, we show that mixing distributions with different variances in the cross-section
results in excess kurtosis in equilibrium prices.
Theorem 1. (Heavy Tails) Let µ4 denote the fourth (central) moment of prices. Then
1. E[pt+1 − pt |pt ] = 0.
21

In Section 3 the cross-section (Pj )K
j=0 depended on n, while here it does not. This is not critical forour results.
22
For example, consider Section 2 but with heterogeneity in memory strength γ i or risk aversion ρi .
Upcoming results apply in these settings as well.

172. Assume that the cross-section (Pj )K
j=0 is nondegenerate and that n > N (P). Then42 2µ /(Σ ) ≥ 3, and the inequality holds with equality if and only if n → ∞.
Analogous to the no drift discussion in Section 2, equilibrium prices generally area martingale. Importantly, prices are heavy-tailed and the effect disappears only whenn → ∞ and central limit theorems apply. Figure 2 shows how mixing leads to heavy
tails, using two Normal distributions with mean zero and differing variances (σ 2 = 1 andσ 2 = 2). For visual clarity, the mixture fractions here are set to 12 and 12 . In words,
heavy tails result from a small chance of the trader being drawn from the high variancedistribution. However, this argument only guarantees that the cross-section has heavy
tails, not prices. Prices arise from the sum of n traders’ demands, and summing randomvariables reduces excess kurtosis due to the central limit theorem. Theorem 1 Condition
2 can be understood as characterizing the net effect between these two competing forces(mixing and summing).

Figure 2: Heavy-Tailed Mixture Distributions
To get a sense of how the proof of Theorem 1 works, consider the simpler case whenn → ∞. Calculating the fourth moment of the market maker’s rule (3) and dividing by(Σ2 )2 ,µ4
c4 (nE[xi (pt )4 ] + 3n(n − 1)E[xi (pt )2 ]2 )=
(9)(Σ2 )2(Σ2 )2
which holds true when demands are independently drawn from the cross-section and havemean zero. Note the factor of three in the numerator of (9), which arises mathematically18

Pwhen expanding the expression E[{ i xi (pt )}4 ]. By taking the variance of the market
maker’s rule (3), we have that Σ2 = c2 nE[xi (pt )2 ]. Plugging this into the denominatorof (9), only O(n2 ) terms in the numerator of (9) survive, and hence the price kurtosis
tends to three as n tends to infinity.
Interestingly, the heavy tail result can fail for small values of n below N (P). This
notation is meant to emphasize that the cutoff value of n depends only on the crosssectional distribution P ≡ (Pj )K
j=0 , not any other parameters. While so far we haveemphasized the latter, there are two reasons why equilibrium prices might have heavy
tails. First, there are historical effects in the sense that a heavy-tailed history tends togenerate heavy-tailed prices (or vice versa). Second, due to the mixing of different generations in the cross-section, there is a small chance of a youthful draw. This breakdown
can be explicitly seen after manipulating condition (9):Excess kurtosis =N (P) · (Excess kurtosis) + M (P) · (Variance of cross-section)nwhere N (P) > 0 and M (P) > 0, whose explicit expressions are relegated to Appendix A.
For excess kurtosis to inherit the sign of the variance of the cross-section (positive when
the cross-section is nondegenerate), we require that n > N (P). In Figure 3, we plot thiscutoff N but not as a function of the distribution P. Instead, we take all distributions
with support of size K, then find a sufficient cutoff for that family of distributions.
Mathematically, N (K) = maxsupp(P)≤K N (P). We emphasize that this cutoff is a small
number (hence n > N (P) is not restrictive) even when traders are allowed to recall
K = 50 memorable periods.
The explanation for heavy tails proposed here is distinct from those previously
proposed in the literature. For example, Gabaix et al. (2006) cite heavy tails in wealth,Cont and Bouchaud (2000) cite imitation among traders, and Thurner et al. (2012)
cite leverage effects. Here, heavy tails come from the mixing of random variables withdifferent variances. Extreme price events occur when young traders, understood as those
who have experienced only a few past trades, all trade in a given period. This is consistent
with survey evidence in Giglio et al. (2021), who find that older individuals’ subjective
distributions over future stock returns have lower standard deviations (than those ofyoung individuals) and, furthermore, they assign smaller probabilities to extreme events
such as large stock market declines.
Our next proposition considers comparative statics of equilibrium price moments

19

Figure 3: Cutoff Value N (K)with respect to the cross-sectional distribution (Pj )K
j=0 . As Proposition 1 suggested,
increasing the “age” of the population decreases the equilibrium variance. And as Theorem 1 alluded to, increasing the dispersion of age within the population increases excess
kurtosis.
Proposition 2. (Comparative Statics) Let Σ2 and µ4 denote the variance and fourth(central) moment of prices, respectively. Then
1. Σ2 is increasing in EP [1/X].
2. µ4 /(Σ2 )2 is increasing in VarP (1/X) when n > N (P).
First let us understand the random variable, 1/X. X here denotes age, or more
specifically, when X = k, the proportion Pk of the population has k memorable periods.
Hence 1/X is the inverse of age, so when this object is larger (smaller), the trader trades
more (less). Let us refer to 1/X as the trader’s youthfulness. Condition 1 of Proposition
2 then says that more youthful economies are characterized by more volatility. And
as intuition suggests, excess kurtosis in Condition 2 is increasing in the variance of thecross-section. Proposition 2 presents empirically testable implications of our theory.

20

5

PolicyWe consider three policies to reduce price variance in the general model from Section 4:
a transaction tax, a capital gains tax, and tightened borrowing constraints.
Transaction Tax. The trader’s wealth wt+1 with a transaction tax r > 0 is
wt+1 = (pt+1 − pt )xt − r|xt |where r = 0 corresponds to the baseline setting. In terms of implications for trade, thetax creates an inaction zone, where traders for whom Ei [pt+1 |pt ] ≈ pt choose not to trade.
This can be seen formally in the new demand function:
Ei [pt+1 |pt ]−pt −r


,

ρΣ2

i
|pt ]−pt +r
xi (pt ) = E [pt+1ρΣ
,
2



0,

if Ei [pt+1 |pt ] − pt > r
if Ei [pt+1 |pt ] − pt < −rotherwise

Traders who remain active in financial markets trade smaller quantities. While it may
seem immediate that the equilibrium price variance is reduced, we must be mindfulof equilibrium effects. Lowered price volatility can make financial markets look more
attractive, thereby increasing trade. Note that the upcoming Proposition 3 describes a
new equilibrium, presumably a sufficient amount of time after the policy change has beenenacted. This qualifier is needed, because the price history must also reflect the effectsof the policy.23< 0 for small values of r.
Proposition 3. (Transaction Tax) ∂Σ∂rConsider the cross-section of demands. The tails have been shifted inwards toward
the origin, because potential market gains − in either direction − are reduced by theamount of the tax. The middle range of the distribution has been removed altogether,because the cost of trading entirely outweighs the benefits for such traders. See Figure
4 for an illustration of this truncated cross-section of demands xi (pt ). With this lowered
dispersion of demands, we find that equilibrium price variance is reduced when transaction taxes are reasonably small.
23

The same comment applies to Proposition 4.

21

Densityxi (pt )Figure 4: Truncated Cross-Section of Demands

Capital Gains Tax. We model the capital gains tax with an adjusted budget set:
wt+1 = (1 − τ )(pt+1 − pt )xt
where the tax is in the range 0 < τ < 1. Again we must consider the equilibrium effectsof the tax. Price variance must solve a fixed point condition
K
X

P

(1
−
τ
)γ
t∈Ki (pt − pt−1 )/k2
2
Σ =c nPk Varρ(1 − τ )2 Σ2
k=1
where the term inside theqbrackets is the capital gains-adjusted demand. This condition
Pcγ
24
is solved by Σ2 = ρ(1−τn K
k=1 Pk /k, so that taxes are actually counterproductive.
)
To understand why, consider two competing forces: dampened capital gains versus dampened wealth variance. The calculation above shows that the latter force outweighs theformer, leading to more trade in equilibrium and (unintentionally) higher price volatility.
We conclude that a capital gains tax is an ineffective policy tool in this setting.
Borrowing Constraints. The final policy that we consider is a borrowing constraint.
While mean-variance utility was previously used without formal derivation, here we begin
24

From the proof of Theorem 1, without the tax, Σ2 = cγ
ρ

22

q P
K
n k=1 Pk /k.

with the trader’s budget set today:
xt pt + at = 0
where at denotes the quantity saved (or borrowed) of a riskless asset. Notice that wehave normalized today’s wealth (RHS of equation above) to zero, which would be withloss of generality if we were to consider an infinite horizon objective. The budget set
tomorrow is:
wt+1 = xt pt+1 + at
where wt+1 denotes wealth. Plugging the first budget set into the second, we recover theobjective of the trader from Section 2. In addition to this standard budget constraint,
we introduce two additional constraints
at ≥ −pt b,

xt ≥ −b(10)

where b > 0 denotes the real borrowing constraint. When traders take a long positionon the risky asset, xt > 0, the first constraint in (10) applies and represents a borrowing
constraint on the riskless asset. When traders take a short position on the risky asset,
xt < 0, the second constraint in (10) applies and limits the quantity borrowed of therisky asset. Substituting in today’s budget constraint, the two new constraints reduceto:|xt | ≤ b
This effectively truncates demand tails. Demand functions become



b,


xi (pt ) = −b, Ei [pt+1 |pt ]−pt ,ρΣ2i|pt ]−pt
if E [pt+1
>b
ρΣ2
i

|pt ]−pt
if E [pt+1
< −b
ρΣ2

otherwise

Proposition 4. (Borrowing Constraints) ∂Σ> 0 for large values of b.
∂b
As we tighten the constraint (decrease b), price variance is reduced as desired. And
the result holds when b is large (i.e. the borrowing constraint is relatively loose). Lastly,
we discuss differences between the two effective policies in terms of their distributional
effects. In the case of transaction taxes, traders with moderate beliefs trade zero, and23

those with tail beliefs reduce, or shift, their amount of trade by the quantity ρΣr 2 . In thecase of borrowing limits, demand is truncated for traders with extreme beliefs. Hencetransaction taxes effectively target the center of the distribution, while borrowing limits
effectively target the tails of the distribution. In the context of the active trading days
setting, this means that transaction taxes (borrowing limits) are more likely to target
older (younger) individuals.
While both transaction taxes and borrowing limits are effective policy tools, capitalgains taxes are not. The two effective policies truncate expected gains, without directly
affecting the wealth variance (there are indirect effects that work through prices, whichwere the focus of Propositions 3 and 4). In contrast, capital gains taxes are multiplicative.
This multiplicative property not only leads to dampened capital gains, but also directly
dampens wealth volatility; the net effect is (inadvertent) increased trade.

6

Conclusion

We have proposed a simple mechanism − idiosyncratic memory − to explain nonfundamental volatility and heavy tails in financial markets. When some traders recall
bear markets and others recall bull markets, this creates belief dispersion in the crosssection and, hence, a motive for trade. The first key takeaway from the paper is that welive, in some sense, in a bad equilibrium: past volatility is what guarantees enough belief
dispersion to generate current volatility. The Brownian motion is the unique equilibrium
price process, and resulting volatility can be categorized as purely non-fundamental.
We then formally modeled one reason why traders might have idiosyncratic memory:
remembering active trading days. The cross-section of traders consists of the young (those
who remember few trading days and have dispersed beliefs) and the old (those who
remember many trading days and have concentrated beliefs). The mixture distribution
over such demographics is not Normal, which leads to equilibrium prices that are not
Normal. This is the second key takeaway from the paper: we have identified a novel
mechanism for generating prices with excess kurtosis. Extreme prices result from thesmall chance that all traders trading on a given day are young.

24Appendix A
Proof of Lemma 1. The posterior according to memory-constrained Bayes’ Rule (2)and weights (6) is given byf (µ|pt −pt−1 , , ..., pt−T +1 −pt−T ) ∝Yif (pt − pt−1 |µ)γT /|K |

t∈Ki

Y

i

f (pt − pt−1 |µ)(1−γ)T /(T −|K |)

t∈K
/ i

|

{z

Memorable}|{zNon-memorable}

As T → ∞, this posterior approaches the one that would have been generated from i.i.d.
draws from a distribution − call it G − where, with probability γ/|Ki |, (pt − pt−1 ) is
drawn for each t ∈ Ki ; and with probability (1 − γ), prices are drawn independently fromthe price distribution. To see why, first consider the log of the non-memorable terms,(1 − γ)T X
ln[f (pt − pt−1 |µ)]T − |Ki |it∈K/

and divide by (the log of) the product of (1 − γ)T densities evaluated at independent
prices.25 We claim that, as T → ∞, this ratio satisfiesP(1−γ)T P
1
ln[f
(p
−
p
|µ)]
i
t
t−1
i
i
t∈K
/
t∈K
/ i ln[f (pt − pt−1 |µ)]T −|K |
T −|K |
= 1 P(1−γ)T→ 1 a.s.
P(1−γ)T
ln[f
(p
−
p
|µ)]
ln[f
(p
−
p
|µ)]
t
t−1
t
t−1
t=1
t=1
(1−γ)T
where the equality follows from algebra and the convergence follows from the strong law
of large numbers. This line of reasoning justifies replacing non-memorable terms in theposterior with the product of (1 − γ)T densities evaluated at independent prices. Next
consider the log of each memorable term,γTln[f (pt − pt−1 |µ)], for t ∈ Ki
i
|K |
γT
which trivially equals (the log of) the product of |Ki | densities evaluated at the samei
memorable price, (pt − pt−1 ) for t ∈ K . Putting arguments altogether, the frequencies(1−γ) and γ/|Ki | observed in the posterior precisely match those of the proposed mixture
G. Mixture distributions have an expected value that is equal to the mixture of the25

If (1−γ)T is not an integer, the same argument applies to floor[(1−γ)T ]. This comment also applies
γT
to the number of densities associated with memorable prices, |K
i| .

25

expected values:
EG [µ] =γ X(pt − pt−1 ) + (1 − γ)µ
|Ki |
i
t∈K

By the theorem of Doob (1949), the posterior mean µi converges to EG [µ] almost surely.

Proof of Proposition 1. As n grows, so does the equilibrium variance. As mentioned
in the main text, we normalize the equilibrium variance to one, and find a market maker
constant, c, consistent with that normalization. Let Ki denote the set of memorable
periods for trader i, so |Ki | = k i . By Lemma 1, as T → ∞, we have that
µi →

γ X
(pt − pt−1 ) a.s.
ki
i
t∈K

In equilibrium we will confirm that, indeed, the price drift µ = 0. Let Mk denote the setPof traders who remember k periods (so k |Mk | = M ), and let Pk denote their proportionk|in the population (so |M= Pk ). The empirical distribution of beliefs can be written
M
M
K
X
1 X
Pk X
FM (t) =1µi ≤t =1µi ≤t
M i=1|Mk | i∈M
k=1k

For each k, the empirical distribution approaches N (0, γ 2 /k) almost surely. The entire empirical distribution is a mixture of these Normal distributions, weighted by Pk ,which need not be Normal. However, a direct calculation yields a mean of zero andP
γ2variance K
k=1 Pk k . If a randomly selected subset of size n trades each period, then the
Lindeberg-Lévy theorem can be applied to the market maker (3) as n → ∞, ensuringthat prices follow a discrete time Brownian motion with mean zero and variance one
when c = √ Pρ.
γnk Pk /k

Proof of Lemma 2. By Chebyshev’s Inequality, for each t
M

Pr

1 X
1µi ≤t − F (t) ≥ ε
M i=1

26

!
≤

Var(

PM

i=1 1µi ≤t )M 2 ε2(11)

where F (t) denotes the expected value of the empirical distribution. But now consider
the variance of an arbitrary 1µi ≤t in the cross section. It is
E[12µi ≤t ] − E[1µi ≤t ]2 = P r(µi ≤ t) − P r(µi ≤ t)2 ≤ 1
where the inequality follows from probabilities being between zero and one. Repeat theexercise but for an arbitrary covariance.
E[1µi ≤t 1µj ≤t ] − E[1µi ≤t ]E[1µj ≤t ] = P r(µi ≤ t and µj ≤ t) − P r(µi ≤ t)P r(µj ≤ t) ≤ 1
With these two bounds, we continue the line of reasoning in (11)
Var(

PM

i=1 1µi ≤t )=M 2 ε2PMP

i̸=j Cov(1µi ≤t , 1µj ≤t )M 2 ε2i=1 Var(1µi ≤t ) +M + M (n − 1)KM 2 ε2→0≤

as M → ∞. The inequality above firstly uses the previously derived variance and covariance bounds. Secondly, it uses an upper bound for the number of covariance terms,
which is attained when all traders remember K periods. When traders remember K
periods, their beliefs overlap with up to (n − 1)K other traders.
Proof of Theorem 1. To prove the first claim, the mean of demand (assuming thetrader remembers k periods) is
γ P
E

k

t∈Ki (pt − pt−1 ) + (1 − γ)µ
ρΣ2


=

µ
ρΣ2

where µ denotes the equilibrium price drift. Note that this value does not differ acrossgenerations, and hence proportions Pj do not enter into the calculation. Taking the
expectation of the market maker (3):26
cnµ= µ ⇐⇒ µ = 0
ρΣ2

(12)

q P
K
The equilibrium variance is given by Σ2 = cγn k=1 Pk /k. Then the constant that multiplies µρcn
in expression (12), ρΣ
2 , is always greater than one when n > 1.
26

27

Next we consider excess kurtosis. If the trader drawn from the cross section remembers
k periods, the fourth moment of demand would be
" P
γ
k

E

(pt − pt−1 )
ρΣ2

t∈Ki

4 #

= γ4

kµ4 + 3k(k − 1)(Σ2 )2
k 4 ρ4 (Σ2 )4

because the price history is independently drawn and prices have mean zero. Since thecross section is distributed according to (Pj )K
j=0 , each demand has fourth momenti4E[x (pt ) ] =KXPk γ 4

k=1

=γ

4µ

kµ4 + 3k(k − 1)(Σ2 )2
k 4 ρ4 (Σ2 )4KK

3γ 4 X Pk− 3(Σ2 )2 X Pk+
ρ4 (Σ2 )4 k=1 k 3
ρ4 (Σ2 )2 k=1 k 2

4First we confirm equality in the case n → ∞. The fourth price moment is given by theexpressionµ4 = c4 (nE[xi (pt )4 ] + 3n(n − 1)E[xi (pt )2 ]2 )(13)
because demands are independently drawn from the cross-section and have mean zero.
We divide (13) by (Σ2 )2 and plug in two terms: first, the fourth momentof demand from
γP2Pt∈Ki (pt −pt−1 )kPEabove and, second, the demand variance, K
:
k
2
k=1
ρΣ

c

4



3nγ 4 PKPk
k=1 k2 + 3n(n − 1)
ρ4 (Σ2 )2

3 = limγ2
k=1 Pk kρ2 Σ2

2 
(14)

(Σ2 )2n→∞
2 2Using the fact that (Σ2 )2 = c ργ2 n
c4



PK

Pk
k=1 k ,

3nγ 4 PKPk
k=1 k2 + 3n(n − 1)ρ4

3 = limn→∞PK

c4 γ 4 n 2
ρ4

P

P

K
k=1 Pk /kK
γ2
k=1 Pk kρ2

2 

2

Because the denominator is O(n2 ), from the numerator, only O(n2 ) terms remain which
proves the desired equality. Next we show that price kurtosis, µ4 /(Σ2 )2 , is always greaterthan 3 when n is finite. We follow similar steps as in the infinite case, recalculating the28

RHS of (14)µ4=(Σ2 )2c4

3nγ 4 PK
γ 4 nε PKPk
Pk
k=1 k3 + ρ4 (Σ2 )2
k=1 k2 + 3n(n − 1)
ρ4 (Σ2 )2PKγ2
k=1 Pk kρ2 Σ22 (Σ2 )2

whereµ4 − 3(Σ2 )2(Σ2 )2ε≡2 2Again plugging in (Σ2 )2 = c ργ2 nµ4=(Σ2 )2

c

4



PK

Pk
k=1 k

and simplifying,γ 4 nε PK
3nγ 4 PKPk
Pk
k=1 k3 + ρ4
k=1 k2 + 3n(n − 1)
ρ4

 2 2 P
K
c γ n

Pk
k=1 k

ρ2

nε
=3 +

PK Pk
Pk
k=1 k2 − 3nk=1 k3 + 3n

PK

P

K
γ2
k=1 Pk kρ2

2 

2

P

K
1
k=1 Pk k

2

2
 P
Pk
n K
k=1 k

Plugging in the value of ε and solving for the excess kurtosis,µ4=3+(Σ2 )2PKPk
k=1 k2 −3nP

and we call

K
Pk
k=1 k

hP
K

2

Pk
k=1 k

i2 
(15)

−

PK

Pk
k=1 k3PK
N (P) ≡ PPk
k=1 k3

K
Pk
k=1 k

2

so that n > N (P) guarantees that the denominator on the RHS of (15) is positive. Thenfor the desired result, it suffices to show that
K
X
Pk
k=1

k2

−

K
X
Pk
k=1

k

!2
≥0

with inequality strict when (Pj )K
j=0 is nondegenerate. But this inequality can be rewrit2
2
ten E[X ] − E[X] ≥ 0, where probabilities are given by Pk and the random variabletakes on values k1 . The variance is, indeed, strictly positive when the distribution is
nondegenerate.

29

Proof of Proposition 2. We begin with the first claim, which follows from the closedform solution for the equilibrium variance from the proof of Theorem 1:v
u K
u X Pkcγ
Σ2 = tnρk
k=1
The second claim follows immediately from the closed form solution (15) from Theorem
1.
Proof of Proposition 3. The price variance Σ2 must solve the following fixed pointcondition

Z 0
Z ∞
K
c2 γ 2 n X Pk22
Σ = 2 4x fk (x − kr/γ)dx +x fk (x + kr/γ)dx ,ρ Σ k=1 k 2 −∞0
2where f (x) denotes the price density and fk (x) denotes its k-fold convolution. The RHS of
the condition above is the variance of a distribution where the middle range [−kr/γ, kr/γ]of beliefs has been removed, and the tails (−∞, −kr/γ) ∪ (kr/γ, ∞) have been shifted
inwards toward the origin. Note that the term kr/γ follows from the following rewritingof demand (assuming that the trader remembers k periods)
 P
γ t∈Ki (pt −pt−1 )−kr/γ


,

ρΣ2
k P
t−1 )+kr/γ
xi (pt ) = γk t∈Ki (pt −p
,
ρΣ2



0,

if

P

if

P

t∈Ki (pt − pt−1 ) > kr/γt∈Ki (pt − pt−1 ) < −kr/γotherwise

Denote the function g(r, Σ) asK

Z 0

c2 γ 2 n X Pkg(r, Σ) ≡ρ2 k=1 k 2

Z ∞2

x fk (x − kr/γ)dx +
−∞

2



x fk (x + kr/γ)dx − Σ60First we make the change of variable, u = x−kr/γ (for the second integral, u = x+kr/γ)
K

c2 γ 2 n X Pkg(r, Σ) =ρ2 k=1 k 2

"Z

−kr/γ

(u + kr/γ)2 fk (u)du +

−∞

Z ∞
kr/γ

30

#
(u − kr/γ)2 fk (u)du − Σ6

and, applying the Leibniz integral rule, we have that
"Z
#
Z ∞
K
−kr/γ
∂g
c2 γn X Pk
= 2
2(u + kr/γ)fk (u)du −2(u − kr/γ)fk (u)du < 0∂rρ k=1 k−∞kr/γwhere the inequality follows from the fact that, in the first (second) integral, u is always
evaluated below −kr/γ (above kr/γ). When r is near zero, we also have thatZ ∞K
∂g c2 γ 2 n X Pk ∂2
u fk (u)du − 6Σ5≈ 2∂Σρ k=1 k 2 ∂Σ −∞Z ∞
Kc2 γ 2 n X Pk ∂2
= 2
u f (u)du − 6Σ5ρ k=1 k ∂Σ −∞K

c2 γ 2 n X Pk
= 2
2Σ − 6Σ5
ρ k=1 kwhere the second line follows from the fact that the variance of the k-fold convolution is
k times the price variance. But
the derivative is evaluated near r = 0, we know
q becausePKcγ
that the equilibrium Σ2 ≈ ρ n k=1 Pk /k. Squaring this term then plugging in,∂g≈ 2Σ5 − 6Σ5 < 0∂Σ∂g/∂r
Altogether, by the implicit function theorem, we have that ∂Σ= − ∂g/∂Σ< 0.
∂r

Proof of Proposition 4. The price variance solves the following fixed point condition"#
Z kbρΣ2 /γ
Z ∞Z −kbρΣ2 /γ
K
2
X
γ
2
2
Σ2 = c2 n
Pk 2 2 4x2 fk (x)dx +b fk (x)dx +b fk (x)dx
k
ρ
Σ
2
2 /γ
kbρΣ
−kbρΣ
/γ
−∞
k=1
where f (x) denotes the price density and fk (x) denotes its k-fold convolution. The firstterm on the RHS corresponds to traders unconstrained by the borrowing constraint. The
second and third terms on the RHS correspond to all extreme traders whose demands
have been truncated by the borrowing constraint. Note that the limits of integration
follow from the following rewriting of demand (assuming that the trader remembers k31periods)



b,


xi (pt ) = −b,P γ t∈Ki (pt −pt−1 ) ,kρΣ2

2t∈Ki (pt − pt−1 ) > kbρΣ /γ

if

P

if

P

2
t∈Ki (pt − pt−1 ) < −kbρΣ /γotherwise

Define the function g(b, Σ) asg(b, Σ)
2

≡c n

K
X

"
Pk

k=1

γ2
k 2 ρ2

Z kbρΣ2 /γ

2

x fk (x)dx + Σ−kbρΣ2 /γ

4

Z ∞

2

b fk (x)dx + ΣkbρΣ2 /γ

4

Z −kbρΣ2 /γ#
2b fk (x)dx − Σ6−∞and we apply the Leibniz integral rule.
"Z#
Z −kbρΣ2 /γK∞X∂g= c2 nPk Σ42bfk (x)dx > 0
2bfk (x)dx +
2 /γ
∂b
−∞
kbρΣ
k=1
where the inequality follows from direct computation of the integrals. Applying theLeibniz integral rule once again,
∂g
∂Σ
"
#
2
Z ∞
Z −kbρΣ2 /γ
2 Z kbρΣ /γ
∂
γ
2
2
4
4
Pk
=c2 n
x2 fk (x)dx + Σ
b fk (x)dx + Σ
b fk (x)dx
2 ρ2
2
2
∂Σ
k
−kbρΣ /γkbρΣ /γ−∞k=1
KX− 6Σ5 + εK

c2 γ 2 n X Pk ∂≈ 2
ρ k=1 k 2 ∂Σ

Z ∞

K

Z ∞

c2 γ 2 n X Pk ∂= 2
ρ k=1 k ∂Σ=2



x fk (x)dx − 6Σ5 + ε

−∞
2



x f (x)dx − 6Σ5 + ε

−∞

K
X

c2 γ 2 nPk
2Σ − 6Σ5 + ε2
ρ k=1 k2where Σ denotes the equilibrium variance but held constant (with respect to the partial
derivative), and the approximate equality holds when b is large. Note that the second-tolast line follows from the fact that the variance of the k-fold convolution equals k times
the price variance. The variable ε denotes an error term, which we will show disappears
when b is large. Before that, because the expression above is evaluated for a large value
32

cγ
ρ

2

q P
n K
k=1 Pk /k. Plugging in the square ofof b, we know that the equilibrium Σ ≈this expression,∂g≈ 2Σ5 − 6Σ5 + ε < 0
∂Σ

∂g/∂b
when ε is small. By the implicit function theorem, we have that ∂Σ= − ∂g/∂Σ> 0. The∂blast step is to confirm that ε → 0 when b → ∞. Without limits, it isε ≡4c2 nΣ3

K
X

"Z

∞

Pk

k=1

Z −kbρΣ2 /γ

2b fk (x)dx +kbρΣ2 /γ#
2b fk (x)dx

(16)

−∞

The entire term (16) tends to zero because
Z ∞

2

Z ∞

b fk (x)dx ≤kbρΣ2 /γkbρΣ2 /γ



xγ
kρΣ2

2
fk (x)dx → 0, as b → ∞where the inequality holds because x ≥ kbρΣ2 /γ. The same argument applies to thesecond term on the RHS of (16), except now x ≤ −kbρΣ2 /γ.

33

References
S. Andersen, T. Hanspal, and K. Nielsen. Once bitten, twice shy: the power of personal
experiences in risk taking. Journal of Financial Economics, 132:97–117, 2019.
W. Bagehot. The only game in town. Financial Analysts Journal, 27(2):12–14, 1971.
P. Bordalo, N. Gennaioli, and A. Shleifer. Memory, attention, and choice. Quarterly
Journal of Economics, 135(3):1399–1442, 2020.
T. Chordia, R. Roll, and A. Subrahmanyam. Order imbalance, liquidity, and market
returns. Journal of Financial Economics, 65:111–130, 2002.
P. Collin-Dufresne, M. Johannes, and L. Lochstoer. Asset pricing when this time isdifferent. Review of Financial Studies, 30(2):505–538, 2017.
R. Cont and J. Bouchaud. Herd behavior and aggregate fluctuations in financial markets.
Macroeconomic Dynamics, 4(2):170–196, 2000.
J. Dominitz and C. Manski. Measuring and interpreting expectations of equity returns.
Journal of Applied Econometrics, 26:352–370, 2011.
J. Doob. Application of the theory of martingales. Actes du Colloque International Le
Calcul des Probabilites et ses applications, Paris CNRS:23–27, 1949.
R. Durrett. Probability: Theory and Examples. Cambridge University Press, 2017.
P. Ehrling, A. Graniero, and C. Heyerdahl-Larsen. Asset prices and portfolio choice withlearning from experience. Review of Economic Studies, 85(3):1752–1780, 2018.
X. Gabaix, P. Gopikrishnan, V. Plerou, and E. Stanley. Institutional investors and stock
market volatility. Quarterly Journal of Economics, 121(2):461–504, 2006.
S. Giglio, M. Maggiori, J. Stroebel, and S. Utkus. Five facts about beliefs and portfolios.
American Economic Review, 111(5):1481–1522, 2021.
J. Graham, H. Campbell, and H. Huang. Investor competence, trading frequency, andhome bias. Management Science, 55(7):1094–1106, 2009.
D. Grether. Bayes rule as a descriptive model: The representativeness heuristic. Quarterly
Journal of Economics, 95(3):537–557, 1980.
34

J. Hasbrouck and G. Sofianos. The trades of market makers: An empirical analysis ofnyse specialists. Journal of Finance, 48(5):1565–1593, 1993.
M. Kahana. Foundations of Human Memory. Oxford University Press, 2012.
M. Kaustia and S. Knupfer. Do investors overweight personal experience? evidence from
ipo subscriptions. Journal of Finance, 63(6):2679–2702, 2008.
T. Kuchler and B. Zafar. Personal experiences and expectations about aggregate outcomes. Journal of Finance, 74(5):2491–2542, 2019.
A. Kyle. Continuous auctions and insider trading. Econometrica, 53(6):1315–1335, 1985.
U. Malmandier and J. Wachter. Memory of past experiences and economic decisions.
Oxford Handbook of Human Memory, 2022.
U. Malmandier, D. Pouzo, and V. Vanasco. Investor experiences and financial market
dynamics. Journal of Financial Economics, 136(3):597–622, 2020.
U. Malmendier and S. Nagel. Depression babies: Do macroeconomic experiences affectrisk taking? Quarterly Journal of Economics, 126(1):373–416, 2011.
U. Malmendier and S. Nagel. Learning from inflation experiences. Quarterly Journal of
Economics, 131(1):53–87, 2015.
S. Mullainathan. A memory-based model of bounded rationality. Quarterly Journal of
Economics, 117(3):735–774, 2002.
S. Nagel and Z. Xu. Asset pricing with fading memory. Review of Financial Studies, 35(5):2190–2245, 2022.
D. Richards and G. Willows. Who trades profusely? the characteristics of individual
investors who trade frequently. Global Finance Journal, 35:1–11, 2018.
S. Schraeder. Information processing and non-bayesian learning in financial markets.
Review of Finance, 20(2):823–853, 2016.
R. Shiller. Irrational Exuberance. Princeton University Press, 2015.
G. Sofianos. Specialist gross trading revenues at the new york stock exchange. Working
Paper, 1995.

35

M. Strahilevitz, T. Odean, and B. Barber. Once burned, twice shy: how naive learning,
counterfactuals, and regret affect the repurchase of stocks previously sold. Journal ofMarketing Research, 48:102–120, 2011.
K. Teeple. Support, resistance, and technical trading. Working Paper, 2022.
K. Teeple. Optimally stochastic volatility. Working Paper, 2023.
S. Thurner, J. Farmer, and J. Geanakoplos. Leverage causes fat tails and clusteredvolatility. Quantitative Finance, 12(5):695–707, 2012.
J. Wachter and M. Kahana. A retrieved-context theory of financial decisions. Quarterly
Journal of Economics, 139(2):1095–1147, 2024.

36