Institutional Investors and Stock Market Volatility

TLDR: In this paper, the authors present a theory of excess stock market volatility, in which market movements are due to trades by very large institutional investors in relatively illiquid markets, and derive the optimal trading behavior of thse investors, which allows them to provide a unified explanation for apparently disconnected empirical regularities in returns, trading volume and investor size.

Abstract: We present a theory of excess stock market volatility, in which market movements are due to trades by very large institutional investors in relatively illiquid markets. Such trades generate significant spikes in returns and volume, even in the absence of important news about fundamentals. We derive the optimal trading behavior of thse investors, which allows us to provide a unified explanation for apparently disconnected empirical regularities in returns, trading volume and investor size.

Figures

Figure 2: Probability density of the individual trade sizes q for 30 large stocks in the Paris Bourse from January 1995 to October 1999. This represents 35 million observations. OLS fit yields a density ∼ q−(1+ζq), for ζq = 1.5± 0.1. Source: Plerou et al. (2003).

Figure 8: Conditional expectations for (a) E[r | Q0], (b) E[Q0 | r], (c) E[N | Q0], (d) E[N | N 0], and (e) E[N 0/N | Q0]. We form, for each interval ∆t = 15 min, the quantities (i) r, the returns, (ii) QB (resp. QS), the number of shares exchanged in a buyer (resp. seller) initiated trade, (iii) NB (resp. NS), the number of buyer (resp. seller) initiated trades, Q0 ≡ QB −QS , and N 0 ≡ NB −NS . The left panel shows the empirical values for the 116 frequently traded stocks in the Trades and Quotes database for 1994—1995. This represents 1.5 million observations. Variables are normalized to unit variance after setting the mean to zero; for variables such as volume for which the variance is divergent, we have by normalized the first moment instead. The right panel shows the model’s predictions, which agree well with the empirical data.

Figure 5: Cumulative distribution of the normalized number of transactions n∆t = N∆t/ hN∆ti . Each symbol shows the cumulative distribution P (n∆t > x) of the normalized number of transactions n∆t for all stocks in each bin of stocks sorted according to size. An analysis of the exponents obtained by fits to the cumulative distributions of each of the 1000 stocks yields an average value ζN = β = 3.40± 0.05.

Figure 3: Zipf plot for the daily fluctuations in the Nikkei (1984-97), the Hang-Seng (1980-97), and the S&P 500 (1962-96). The apparent power-law behavior in the tails is characterized by the exponents ζr = 3.05 ± 0.16 (NIKKEI), ζr = 3.03 ± 0.16 (Hang-Seng), and ζr = 3.34 ± 0.12 (S&P 500). The fits are performed in the region |r| > 1. Source: Gopikrishnan et al. (1999).

Figure 6: The modular structure of arguments in this paper

Figure 4: Cumulative distribution of the conditional probability P (r > x) of the daily returns of companies in the CRSP database, 1994-5. We consider the starting values of market capitalization K, and define uniformly spaced bins on a logarithmic scale and show the distribution of returns for the bins, K ∈ (105, 106], K ∈ (106, 107], K ∈ (107, 108], K ∈ (108, 109]. (a) Unnormalized returns (b) Returns normalized by the average volatility σK of each bin. The plots collapsed to an identical distribution, with ζr = 2.70 ± .10 for the negative tail, and ζr = 2.96 ± .09 for the positive tail. Source: Plerou et al. (1999).

Full text

A Theory of Large Fluctuations in Stock M ark et Activity
∗
Xavier Gabaix
†
, P aram eswaran Gopikrish nan
‡
, Vasiliki Plerou
‡
, H. E ugen e S tanley
‡
†
MIT, Economics Departm ent and NBER
‡
Boston University, Physics Department, Cen ter for Polym er Studies
August 16, 2003
Abstract
We propose a theory of large movements in stock market activity. Our theory is motivated
by growing empi rical evidenc e on the power-law tailed nature of distributions that characterize
large movements of distinct variables describing stock market activity such as returns, volumes,
number of trad es, and order fl ow. Remarkably, the exponents that characterize these power
laws are similar for different c ou ntries, for d ifferent types and sizes of markets, and for different
market trends, suggesting that a generic theoretical basis may underlie these regularities. Our
theory provides a unified way to understand the power-law tailed distributions of these variables,
their appare ntly universal nature, and the precise va lues of exponents. It links large movements
in market activity to the power-law distribution of the size of large financial institutions. The
trades made by large financia l institutions create large fluctuations in volume and returns. We
show that optimal t rading by su ch large institutions generate power-law tailed distribution s for
market variables with exp onents that agree with those found in empirical data. Furthermore,
our m odel also makes a large number of testable out-of-sample predictions.
Conten ts
1 Introduction 4
1.1 Thepower-lawdistributionofreturns,volume,numberoftrades ........... 4
1.2 Outlineofthetheory.................................... 6
2 The empirical facts that motivate our theory 7
2.1 Methodologytoestimatepowerlaws ........................... 7
2.2 Normalizations used to make differentassetscomparable................ 7
2.3 The cubic law of price fluctuations: ζ
r
' 3 ........................ 8
∗
xgabaix@mit.edu, gop i@cgl.bu.edu, plerou@cgl.bu.edu, hes@ miranda.bu.edu. We thank Tal Fishm an, David
Kang, Alex Weisgerb er, and especially Kirk Do ran for outstanding research assistance. For helpful com m ents, we
tha n k Tobias Ad r ia n , Jon a t h a n Be r k , Oliv ier Bla n cha rd , Je a n -P hillipe Bou cha ud, Jo h n C a mpbe ll, Emanue l De rman ,
Ken French, J o el Hasbrouck, S o eren Hvidjkaer, H arrison Hon g, Ivana K omunjer, David L aibson, Augustin Landier,
An a nth Mad h avan, La s se Pederse n , Th o mas P h ilippon , M a rc Pot te rs , Jon Reu te r, G id e o n S a a r, A n d re i Sh le ife r,
Did ie r S ornette , D imitri Vayan os , Je ssic a Wachter, Jian g Wang, Je ff Wu rg le r , an d s emin a r p ar tic ipants at Berkele y ’s
HaasSchool,Delta,Harvard,JohnHopkinsUniversity,MIT,NBER,NYU’sSternSchool,Princeton,StanfordGSB,
the 2002 E conophysics conferences in Indonesia and Tokyo, and the 2003 W FA meetings. We thank the NSF for
supp ort. X.G . than ks the Ru ssell Sage Foundation and NY U Stern Scho ol for their wondeful hospitality during the
year 2002 -2003.
1

2.4 The half cubic law of volume: ζ
Q
' 3/2 ......................... 9
2.5 The cubic law of number of trades: ζ
N
' 3.3 ...................... 10
2.6 The po wer law distribution of the size of large investors: ζ
S
' 1 ............ 10
3 The theory, assuming a power law price impact of trades 11
3.1 Sketchofthetheory .................................... 11
3.2 Notationsandsomeelementarylinksbetweenexponents................ 12
3.2.1 Notations ...................................... 12
3.2.2 Link bet ween aggregate and trader-based exponents .............. 13
3.2.3 Linkbetweentheexponentsofreturnandvolume ............... 13
3.3 Amainresult........................................ 14
3.4 An optimizing model that illustrates Theorem 3 ..................... 16
4 The square root of price impact of trades: Evidence and a possible explanation 19
4.1 Evidenceonthesquarerootlawofpriceimpact..................... 19
4.2 Sketchofourexplanationforthesquarerootpriceimpact............... 19
4.3 Amicrostructuremodelforthepowerlawimpactofablock.............. 21
4.3.1 Thebehaviorofliquidityproviders........................ 21
4.3.2 PermanentvsTransitorypriceimpact ...................... 21
4.3.3 Thebehaviorofthelargetrader ......................... 21
4.4 The distribution of individual trades q vs distribution of target volumes Q ...... 23
4.5 Anoptimizingintertemporalmodelthatjoinspowerlawsandmicrostructure .... 24
4.6 Somecommentsonthemodel............................... 25
4.6.1 Arbitrage ...................................... 25
4.6.2 Plausibility of very large volumes and price impact .............. 26
5 Assessing some further empirical predictions of the model 26
5.1 Contemporaneousbehaviorofseveralmeasuresoftradingactivity .......... 26
5.1.1 Definitions ..................................... 26
5.1.2 Comparisonbetweentheoryandfacts ...................... 27
5.2 Someuntestedpredictions................................. 28
6 Related literature 28
6.1 Alternativetheories .................................... 28
6.1.1 The public news based (efficientmarkets)model ................ 30
6.1.2 Amechanical“pricereactiontotrades”model ................. 30
6.1.3 Randombilateralmatching ............................ 30
6.2 Related empirical findings................................. 30
6.2.1 Othermodelsforreturns ............................. 30
6.2.2 Buy/Sellasymmetry ............................... 31
6.3 Link with the behavioral and excess volatility literature . ............... 31
6.4 Linkwiththemicrostructureliterature.......................... 31
6.5 Linkwiththe“econophysics”literature ......................... 32
7Conclusion 32
8 Appendix A: Some power la w mathematics 34
9 Appendix B: Generalization of Theorem 3 to non i.i.d. settings 35
2

10 Appendix C: The model with general trading exponents 37
11 Appendix D: A model for the linear supply function (38) 39
12 Appendix E: Algorithm for the simulations 40
13 Appendix F: Confidence intervals and tests when a variable has infinite variance 41
13.1 Construction of the confidenceintervalsforFigure7 .................. 41
13.2 Test of the linear relation E
£
r
2
| Q
¤
= α + βV ..................... 41
3

1Introduction
Even ex post, stock market fluctuations are very hard to explain by movement in fundamentals.
Trading per se seems to move prices. This leads to excess v olatility of stock market prices. Even
crashes seem to happen for no good reason. Trading volume is very high, and its large variations
are also hard to relate to fundamentals. The present paper wishes to present a theory of those large
movements in trading activit y
1
.
For this is useful to have precise empirical facts. Otherwise, theories are too unconstrained. This
is why we use a series of sharp empirical facts, established using dozens of millions of data points.
They are quite precise, hence they constrain any theory of large movements of stock mark et activity.
We first outline some of those key empirical regularities before presenting our theory.
1.1 T he power-la w distribution of returns, volum e, n umber of trades
Our theory is motivated by the following empirical findings on the power law distribution of (i)
returns, (ii) volumes, and (iii) the number of trades, (iv) the power law of price impact, and (v) the
power law distribution of t he size of large investors.
(i) The power law distribution of returns. Let P
t
denote the price of a stock or an index, and
define return over a time interval ∆t as r
t
=ln(P
t
/P
t−∆t
). Empirical studies (Gopikrishnan et al.
(1999), Plerou et al. (1999)) show that the distribution function of returns for 1000 largest U.S.
stocks and several major international indices decays as
P (|r| >x) ∼
1
x
ζ
r
with ζ
r
' 3 (1)
Here, ∼ means asymptotic equality up to numerical constants
2
. This holds for positive and negative
returns separately.
Figure 1 shows that P (|r| >x) on a bi-logarithmic scale displays a linear asymptotic behavior
with slope −ζ
r
= −3.1. Since linear behavior on a log-log plot means a power-law decay, we conclude
that ln P (|r| >x)=−3.1lnx + constant, i.e. (1). There is no tautology that implies that this
graph should be a straigh t line, or that the slope should be -3. A Gaussian would have a concave
parabola, not a straight line. The remarkable nature of the distribution (1) is that it holds for stock
indices and individual stocks different sizes, different time periods (see Section 2 for a systematic
exploration), and we always find ζ
r
' 3. In the following, w e shall refer to Eq. (1) as “the cubic law
of returns”.
(ii) The power law distribution of trading volume. We call volume the number of shares traded
3
.
We find that the distribution is:
P (q>x) ∼
1
x
ζ
q
with ζ
q
' 3/2. (2)
The finding holds both for the volume of individual trades and aggregate volumes:
P (Q>x) ∼
1
x
ζ
Q
with ζ
Q
' 3/2. (3)
1
A condensed version of som e elements of the present paper app eared in Gabaix et al. (2003a).
2
Fo rmally f(x) ∼ g (x) means f (x) /g (x) tend s to a positive constant (not necessarily 1) as x →∞.
3
The dollar value traded yields very similar results. This is expected, as, for a given security, the variations of
number of shares traded and the volume traded are essentially proportional.
4

Figure 1: Empirical cumulative distribution of the absolute values of the normalized 15 minute
returns of the 1000 largest companies in the TAQ database for the 2-yr period 1994—1995. This
represen ts 12 million observations. In the region 2 <x<80 we find an OLS fit ln P (r>x)=
−ζ
r
ln x+b,withζ
r
=3.1±0.1. This means that returns are distributed with a power law P (r>x) ∼
x
−ζ
r
for large x.
We had initially found this in U.S. data (Gopikrishnan et al. 2000), and we hav e recently
confirmed the same on French data (Plerou et al. 2003). Figure 2 illustrates this. It shows the
density satisfies p (q) ∼ q
−2.5
, i.e. (2). The exponent of the distribution of individual trades is close
to 1.5. In the following, we refer to Eq. (2)-(3) as the “half-cubic law of trading volume”.
As before, the exponents describing these power-laws seem to be stable across different types of
stocks, different time periods and t ime horizons etc. (see Section 2).
(iii) The power law distribution of the number of trades. A similar analysis for the number of
trades show s that
P (N>x) ∼
1
x
ζ
N
with ζ
N
' 3.3 (4)
(iii) The power law of pric e impact.
Building on the concave nature of the impact functions in stock markets (Hasbrouck 1991, Has-
brouck and Seppi 2001, Plerou et al. 2002), we will give evidence that the price impact ∆p of a
trade of size V scales as:
∆p ∼ V
γ
with γ ' 1/2.(5)
(v) Power law distribution of the assets of large investors. Our empirical analysis of the size
distribution of mutual funds shows evidence for a power law distribution. As Section 2.6 reports the
nu mber of funds with size (asset under management) greater than x follows:
P (S>x) ∼
1
x
ζ
S
with ζ
S
' 1 (6)
We will present a theory where those fiv e facts are intimately related. It is largely orthogonal to
the existing finance theory. Indeed, much of finance is about the risk premia commanded by various
5