What really causes large price changes ?

Fabrizio LilloUniversity of Palermo (Italy)

in collaboration with: Doyne Farmer, L.Gillemot, S.Mike and A.Sen (Santa Fe Institute)

Quantitative Finance (in press, 2004)

Large price changesThe most known stylized fact of financial time series is the leptokurtosis (fat tails) of the distribution of price returns

over the entire length of the time series. Figure 4!a" showsthe cumulative distribution of returns for #t!1 min. Forboth positive and negative tails, we find a power-law

asymptotic behavior

P!g"x "$1

x%, !4"

similar to what was found for individual stocks &34'. For theregion 3(g(50, regression fits yield

%!! 2.95#0.07 !positive tail"

2.75#0.13 !negative tail",!5"

well outside the Levy stable range, 0(%$2 . Consistentvalues for % are also obtained from the density function. For

a more accurate estimation of the asymptotic behavior, we

use the modified Hill estimator &Figs. 5!a" and 5!b"; see alsoAppendix B'. We obtain estimates for the asymptotic slopein the region 3(g(50:

%!! 3.45#0.07 !positive tail"

3.29#0.07 !negative tail".!6"

For the region g(3, regression fits yield smaller esti-mates of % , consistent with the possibility of a Levy distri-bution in the central region. The values of % obtained in this

range are quite sensitive to the bounds of the region used for

fitting. Our estimates range from %)1.35 up to %)1.8 fordifferent fitting regions in the interval 0.1(g(6. For ex-ample, in the region 0.5(g(3, we obtain

%)! 1.6 !positive tail"

1.7 !negative tail",!7"

which are consistent with the result %)1.4 found for smallvalues of g in Ref. &10'. Note that in Ref. &10' the estimatesof % were calculated using the scaling form of the return

probability to the origin P(0). It is possible that for the

financial data analyzed here, P(0) is not the optimal statistic,

because of the discreteness of the individual-company distri-

butions that comprise it &64'. It is also possible that our val-ues of % for small values of g could be due to the discrete-

ness in the returns of the individual companies comprising

the S&P 500.

B. Scaling of the distribution of returns for !t up to 1 d

Next, we study the distribution of normalized returns for

longer time scales. Figure 6!a" shows the cumulative distri-bution of normalized S&P 500 returns for time scales up to

512 min !approximately 1.5 d". The distribution appears toretain its power-law functional form for these time scales.

We verify this scaling behavior by analyzing the moments of

the distribution of normalized returns g,

FIG. 5. Inverse local slopes of the cumulative distributions of

normalized returns for #t!1 min for the !a" positive and !b" nega-tive tails. Each point is an average over 100 different inverse local

slopes. Extrapolation of the regression lines provides estimates for

the asymptotic slopes %!3.45#0.07 !positive tail" and %!3.29#0.07 !negative tail".

FIG. 6. !a" Log-log plot of the cumulative distribution of nor-malized returns of the positive tails for #t!16, 32, 128, and 512min. Power-law regression fits yield estimates of the asymptotic

power-law exponents %!2.69#0.04, %!2.53#0.06, %!2.83#0.18, and %!3.39#0.03 for #t!16, 32, 128, and 512 min, re-spectively. !b" The moments of the distribution for #t!1, 32, 128,and 512 min. The change in the behavior of the moments from the

1-min scale is probably the effect of the gradual disappearance of

the Levy slope for small values of g. For #t"30 min there is noregion with slopes in the Levy range, and we observe good agree-

ment between all time scales.

PRE 60 5309SCALING OF THE DISTRIBUTION OF FLUCTUATIONS . . .

Mandelbrot (1963)Fama (1965)Mantegna and Stanley (1995)Longin (1996)Lux (1996)

Stanley et al (1999)

P (r > x) ∼1

xα

Investigated Market

We investigate 16 high-cap stocks traded at the London Stock Exchange (LSE) in the period 1999-2002.

The number of investigated events is approximately 50 million

We study only the electronic exchange (SETS) of LSE, which contains approximately 60% of the total order flow

Large price changesSeveral theories for this stylized facts have been proposed. For example:

Subordinated processes (Clark, 1973)

Volume fluctuations (Gabaix et al., 2003)

We present an alternative explanation for fat tails of return based on liquidity fluctuations

Subordinated stochastic processes (Clark, 1973)

Clark’s idea is that price shift due to individual transactions are Gaussian, but when many trades are aggregated in a time interval, the return distribution can be fat tailed.

This is due to the fluctuation of number of trades (i.e. of market activity) in the time interval

The asymptotic properties of the pdf do not depend on the type of aggregation (time vs. event)

The fat tail property is present also for individual transactions !

Volume fluctuations (Gabaix et al. 2003)

P (V > x) ∼ x−γ

r = kVβ

P (r > x) ∼ x−α

α = γ/β

The trade volume distribution is fat tailed

The price impact scales as

Hence the distribution of returns decays as

γ ≈ 1.5

β ≈ 0.5

α ≈ 3

Price impact

102

103

104

105

106

volume

10-4

10-3

pri

ce s

hif

t

E(r|V ) =sign(V )|V |!

"

NYSE (Lillo, Farmer, Mantegna, Nature 2003)

LSE(Lillo and Farmer, QF 2004)

β ≈ 0.2 β ≈ 0.3

What is the role of volume?

p(r|ω) = (1 − g(ω))δ(r) + g(ω)f(r|ω)

Let us decompose the conditional probability of a return conditioned to an order of volume as ωr

and we investigate the cumulative probability

F (r > X|ω) =

∫∞

X

f(r|ω)dr

for several different value of ω

The role of the transaction volume is marginal

The fluctuations in market impact could be important

E(r|ω) = g(ω)

∫f(r|ω)r dr ∝ g(ω)

The fat tails of returns seem NOT to be an effect of temporal aggregation, because even for individual transactions a fat tailed return distribution is observed

The role of volume fluctuations is marginal because approximately the same is observed for the entire range of volumes.

The impact function is NOT deterministic and the fluctuations of price impact are very large.

How can the impact fluctuate so wildly ?

F (r > X|ω)

Limit Order BookMany stock exchanges (LSE, NYSE, Paris Bourse) works

through a double auction mechanism

Large price changes are due to the granularity of supply and demand

The granularity is quantified by the size of gaps in the Limit Order Book

9

FIG. 6: A typical configuration of the limit order book forAZN before and after a large price fluctuation. The two panelsplot the volume (in shares) of limit orders at each price level;sell limit orders are shown as positive, and buy limit ordersas negative. In panel (a) we see that there is a gap of 43 ticksbetween the best ask price and the next highest occupiedlevel. The arrival of a crossing limit order to buy removes allthe volume at the best ask, giving the new limit order bookconfiguration shown in panel (b), which has a much higherbest ask price than previously. The only thing atypical aboutthis event is that in most cases the volume at the best priceis removed by a market order of exactly the right size.

turn distribution can be generated by a constant order ofmedian size – this correlation is interesting, but not es-sential). Note that the trader initiating the change doesnot pay a large spread – that would only happen to thenext trader, if she were to immediately place a marketorder11.

This is by far the most common scenario that gen-erates large price changes. In Figure 7(a) we comparethe distribution of the first gaps to the distribution ofprice returns for the stock AZN. We see that the dis-tributions are very similar. For the first gap size thetail index α = 2.52 ± 0.07, and for the return distribu-tion α = 2.57± 0.08, showing that the scaling behaviorsare similar. However, the similarity is not just evidentin the scaling behavior – the match is good throughout

11 In general after a large shift in the bid or ask price, the nextorders tend to be limit orders, but we have not yet been ableto study the statistical properties of the sequence of subsequentevents in detail. See [4, 9, 24, 28], and [51] for an empirical studyof the role of quote size after a market order.

FIG. 7: The cumulative distribution P (g > x) of the size offirst gaps g (red continuous line), compared to the cumulativedistribution of returns generated by market orders P (r > x)(black dashed line). Panel (a) refers to buy market ordersfor AZN, in double logarithmic scale to highlight the tail be-havior. The two distributions are very similar. The result iseven more impressive when we consider the average over the16 stocks described in Table I (Panel (b))

the entire range, illustrating that most of the large pricechanges are caused by events of this type. Panel (b) ofFigure 7 shows the same comparison for the pool of 16stocks. The agreement in this case is even more strik-ing12.

To demonstrate that the correspondence in the abovefigure is not just a coincidence for AZN, we have com-puted the tail exponents for returns and first gap size for

12 We have also studied the distribution of higher order gaps. Mov-ing away from the best price π0, the nth order gap for n = 1, 2, . . .can be recursively defined as gn = | log πn−1− log πn|, where πn

is the nth occupied price level. Interestingly, we find that thetail behavior of higher order gaps is the same as that of g1.

first gap

7

on how much it moves. From Eq. (1) one easily obtains

E(r|ω) = g(ω)∫

f(r|ω)r dr ∝ g(ω) (2)

where for the last proportionality relationship we haveused the result from Figure 2 that f(r|ω) is almost in-dependent of ω. Thus, the expected price change scaleslike the probability of a price change, a relationship thatwe have verified for both the LSE and the NYSE. How-ever this variation is still small in comparison with theintrinsic variation of returns; the mean market impactof a very large order is less than the average size of thespread, but the largest market impacts are often morethan ten times this large.

C. Correlations between order size and liquidity

One possible explanation of the independence of priceresponse and order size is that there is a strong correla-tion between order size and liquidity. There is an obviousstrategic reason for this: Agents who are trying to trans-act large amounts split their orders and execute them alittle at a time, watching the book, and taking whateverliquidity is available as it enters. Thus, when there is alot of volume in the book they submit large orders, andwhen there is less volume, they submit small orders. Thiseffect tends to even out the price response of large andsmall orders. We will see that this effect indeed exists,but it is only part of the story, and is not the primarydeterminant of the behavior we observe here.

In fact, the unconditional correlation between marketorder size and volume at the best is rather small. ForAZN, for example, it is about 1%. However, if we re-strict the sample to orders that change the midprice, thecorrelation soars to 86%. The reason for this is that fororders that do not change the price, there is essentiallyno correlation between order size and volume at the best.For the rarer case of orders that do change the price, incontrast, most market orders exactly match the volumeat the best. As shown in Table II, for the stocks in oursample, 86% of the buy orders that change the price ex-actly match the volume at the best price. (This is 85%for sell orders.)

The relationship between the volume of market ordersand the best price becomes more evident with a nonlinearanalysis. Figure 4 shows E(ω|Vbest), where ω is the mar-ket order size and Vbest is the volume at the correspond-ing best price. We see in this figure that the expectedorder size is nonzero even for the smallest values of Vbest.It grows monotonically with Vbest, but with slope that issubstantially less than one, and a roughly concave shape.This makes the nonlinear correlation between order sizeand liquidity clear. However, in the following section wewill see that the dependence of order size on liquidityis not strong enough to substantially supress large pricefluctuations.

% of nonzero return % of nonzero returnequal to first gap with ω = Vbest

tick sell buy sell buyAZN 94.3 99.6 83.7 90.1BAA 95.9 99.1 86.2 87.0BLT 95.0 99.2 85.8 85.6BOOT 95.9 99.2 85.7 84.7BSY 94.3 99.7 85.0 88.0DGE 96.3 99.7 86.5 87.4GUS 95.8 99.5 85.0 83.5HG. 95.9 99.5 85.8 83.0LLOY 97.3 99.8 88.4 88.6PRU 95.9 99.5 85.5 78.1PSON 93.1 99.6 81.2 86.2RIO 95.7 99.7 84.6 86.4RTO 96.1 99.5 84.4 85.3RTR 93.0 99.7 83.5 85.4SBRY 95.6 99.4 85.3 83.2SHEL 98.7 99.9 93.0 92.8average 95.6 99.5 85.6 86.0

TABLE II: Summary table of the percentage of the time thatnonzero changes in the best prices are equal to the first gap(left) and that the market order volume ω exactly matchesthe volume at the corresponding best price Vbest (right). As-suming a Bernoulli process the sample errors are the order of0.1− 0.2%.

D. Liquidity fluctuations drive price fluctuations

In this section we demonstrate in concrete terms thatprice fluctuations are driven almost entirely by liquidityfluctuations. To do this we study the virtual market im-pact, which is a useful tool for probing the supply anddemand curves defined by the limit order book. Whereasthe true market impact p(r|ω) tells us about the distri-bution of impact of actual market orders, as discussedin Section III A, the virtual market impact is the pricechange that would occur at any given time if a market or-der of a given size were to be submitted. More formally,at any given time t the limit orders stored in the orderbook define a revealed supply function S(π, t) and re-vealed demand function D(π, t). Let V (π, t) be the totalvolume of orders stored at price π. The revealed supplyfunction is

S(π, t) =∑

π

V (π, t) (3)

The revealed supply function is non-decreasing, and sofor any fixed t has a well defined inverse π(S, t). Thevirtual market impact is the price shift caused by ahypothetical order of size S, e.g. for buy orders it isπ(S, t)− a(t). The virtual market impact for sell marketorders can be defined in terms of the revealed demandin a similar manner. By sampling at different values oft, for any fixed hypothetical order size we can create asample distribution of virtual market impacts. This nat-urally depends on the sampling times, but these can bechosen to match any given set of price returns.

Origin of large price returnsFirst gap distribution (red) and return distribution (black)

Large price returns are caused by the presence of large gaps in the order book

Tail exponents

ConclusionsLarge price fluctuations are present also at the level of individual transaction.

The volume of the trade is not so important in determining the size of price shift

Large price shifts occur when fluctuations of liquidity create large gaps in the limit order book. A market order matching the volume at the best can create a big price shift

What creates these large gaps in the limit order book?