---
abstract: 'In this study we examine the evolution of price, volume, and the bid-ask spread after extreme 15 minute intraday price changes on the NYSE and the NASDAQ. We find that due to strong behavioral trading there is an overreaction. Furthermore we find that volatility which increases sharply at the event decays according to a power law with an exponent of $\approx 0.4$, i.e., much faster than the autocorrelation function of volatility.'
address:
- '$^1$Department of Industrial Management and Business Economics, Budapest University of Technology and Economics, Műegyetem rkp. 9, H-1111, Budapest, Hungary'
- '$^2$Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, H-1111, Budapest, Hungary'
- '$^3$Laboratory of Computational Engineering, Helsinki University of Technology, P.O.Box 9400, FIN-02015 HUT, Finland'
author:
- 'A. G. Zawadowski$^{1,2}$, J. Kertész$^{2,3}$, and G. Andor$^{1}$'
title: Large price changes on small scales
---

Introduction
============

Research in the past years has revealed that extreme price changes are not outliers, they are significantly frequent. Analyzing how markets react to such extreme events is crucial in order to understand the price formation process on the market. Economists have analyzed how markets react to large daily price changes and significant overreaction was found in the past, although markets seem to be getting more and more efficient over time [@cox]. These extreme price changes may at least partially be due to news arriving on the market, although not all events can be easily attributed to major events [@cutler]. Previous studies have revealed that the autocorrelation of returns on the stock markets is significant for only 15-20 minutes [@liu] thus it seems reasonable that major events take place within the trading day and daily data is not appropriate to fully understand market reaction. We will thus investigate the intraday market evolution of prices, volume and the bid-ask spread after extreme price changes of 15 minutes.

Furthermore the effect of external price shocks has been analyzed on multi-agent model simulation of stock markets [@lux] and these studies show that in case there is behavioral trading on the stock market there is overreaction to external price shocks and market volatility increases sharply at the event [@sajat]. Thus if we are able to localize extreme intraday price changes we will probably get an insight into the process in which the agents of different behavior form the new equilibrium market price.

Defining large intraday events
==============================

The dataset used is the TAQ database of the NYSE for the years 2000-2002. The TAQ (Trades and Quotes) Database is that supplied by the NYSE: it includes all transactions and the best bid and ask price for all stocks traded on the NYSE and on the NASDAQ. We include all stocks in our sample traded on the first trading day of 2000 both on NASDAQ and on the NYSE. A minute-to-minute dataset is generated using the last transaction, and the last bid and ask prices during every minute. Since we examine intraday price formation we will only include liquid stocks in our sample. We define liquid stocks as those for which at least one transaction was filed for at least 90% of the trading minutes of the stocks included in the DJIA during the 20 pre-event trading days.

We are studying the intraday reaction to large price shocks but we have not yet defined what we mean under the term “large price changes on small scales”. Defining intraday 15 minute events is not an easy task because volatility is higher on average at the beginning and at the end of the day. Two trivial methods are at hand:

[*1. Absolute filter*]{}: using this first method we look for intraday price changes bigger than a certain level of let us say 4% within 15 minutes. In this case we have to face several problems. Most of the events we find will occur during the first or last couple of minutes of the trading day because of the U-shape intraday volatility distribution of prices. These events represent the intraday trading pattern instead of extreme events. Another problem is that a 4% price jump may be an everyday event for a volatile stock while an even smaller price move may indicate a major event in case of a low volatility stock.

[*2. Relative filter*]{}: in case of this second method we measure the average intraday volatility as a function of trading time during the day (this means measuring the U-shape 15-minute volatility curve for each stock prior to the event) and define an event as a price move exceeding e.g. 8 times the normal volatility during that time of the day. The problem in case of this method is the following: since price moves are very small during the noon hours, the average volatility for the 24 pre-event days in these hours may be close to zero, i.e. a small price movement (a mere shift from the bid price to the ask price for example) may be denoted as an event. In this case the events cluster around the noon hours and no events are found around the beginning and the end of the trading day.

The best solution for localizing events is an intermediary one. Let us use the relative filter and absolute filter combined: this way we can eliminate the negative effect of both filters and combine their advantages. Thus an event is taken into account if and only if it passes both the relative and the absolute filter. We adjust the absolute and relative filter so as to achieve that events are found approximately evenly distributed within the trading day. In addition we omit the first 5 minutes of trading because we do not want opening effects in our average (for market reaction at the opening minutes see [@zak]).

In order to be able to observe the exact price evolution after the intraday event it is crucial to localize the events as precisely as possible. Since some price changes may be faster than others we will allow shorter price changes than 15 minutes as well. If the price change in e.g. 8 minutes already surpasses the filter level than we will assume the price change has taken place in 8 minutes. The end of the time-window in which the event takes place is regarded as the end of the price change and this is the point to which the beginning of the post-event time scale is set: thus minute 0 is exactly the end of the earliest (and of those the shortest) time window for which the price change passes the filter. This method is constructed so as to ensure that one can definitely decide by minute 0 whether an event has taken place beforehand (in the preceding 15 minutes) or not.

We will not include events in our sample for such stocks where the price tick is high compared to nominal price: thus only stocks with a nominal price over 10 USD will be studied. Another problem arises when using NASDAQ data: there are often singular transactions filed at a price outside the bid-ask spread (sometimes even 4-8% from the mean bid-ask price). Since these do not indicate an event, they are merely an outlier they are to be excluded from the sample. Thus in all cases the trigger level specified above has to be surpassed by the change in the mean of the bid and ask prices as well (not only the change in transaction price). Choosing the sample of events included in the average is a crucial step. A major event may effect the whole market, thus a dozens of stocks may experience a price shock at the same time. Including all of the events in the average would yield a sample biased to a given major event. For convenience we will only include the events in the average which are the first during a given day for a given stock. Furthermore no overlapping events will be taken into account, thus events which happen within 15 minutes (the maximum length of an extreme price change) of the previous event taken into account in the average will be omitted.

When studying intraday price changes 1 minute volatility, 1 minute trading volume (measured in USD), and the bid-ask spread are averaged besides the cumulative abnormal returns, bid and ask prices. Volatility, trading volume, and the bid-ask spread are measured in comparison to the average minute-volatility and minute trading volume of the individual stock during the same period of the trading day (intraday volume and volatility distributions are calculated using the average of the 24 pre-event trading days). This step is crucial in order to remove the intraday U-shape pattern of the above quantities from the average since their intraday variation is in the same order of magnitude as the effect itself we are looking for [@liu].

The decay of intraday shocks
============================

Let us turn to the results: in case of the NYSE we find 197 price increases and 182 price decreases. The number of events included in the sample are 623 and 500, respectively, on the NASDAQ. We find overreaction of around 1% both during the first 10-30 minutes after price increases and decreases (see Fig. \[elso\]), which is robust and approximately of the same size on both markets. Investigating the economic significance of these price reversals is not the scope of this paper and is addressed in a separate paper by the same authors [@zak]. In case we look backwards as well, i.e. at the time before the event, significant asymmetry can be observed in the pre-event behavior before price increases and drops, both seem to occur much more frequently when the price of the stock is declining. The above asymmetry holds for NYSE data as well.

![[Cumulative return computed from the last minute of large intraday events on the NASDAQ. Average of 623 price increases and 500 price decreases]{}[]{data-label="elso"}](zkafig1.eps){width="14.0truecm"}

Price itself is not the only important data which gives us information on market behavior after the event. Both volatility and dollar volume increase significantly (500-800%) in comparison to their pre-event daytime-adjusted value (see Fig. \[masodik\]). Volatility decreases according to a power-law on both markets (on the other hand volume does not) in the case of both increases and decreases for at least 3 decades (minute 1 to minute 1000 after the event). The exponent on the NYSE is $-0.39\pm0.01$ for decreases and $-0.34\pm0.01$ for increases, and $-0.43\pm0.01$ and $-0.44\pm0.01$, respectively, on the NASDAQ. These values indicate much faster decay of volatility than measured on autocorrelation functions of daytime-adjusted volatility for individual stocks (using the exact methodology used by Liu [@liu] for the S&P 500 stock index). Autocorrelation function of volatility exhibits power law decay as well but with a power exponent usually above $-0.2$. Exact values on our sample period of 2000-2002 are e.g. $-0.14\pm0.02$ for Citigroup and $-0.20\pm0.02$ for Nokia. This indicates that price shocks decay much faster than usual fluctuations. A probable explanation for this phenomenon is that, as mentioned already in the introduction, price shocks are likely to be caused by external events (news), thus they can be regarded as exogenous while fluctuations in general are predominantly endogenous (and this is what is measured by the autocorrelation function). We thus propose that volatility shocks due to exogenous events decay much faster than endogenous fluctuations. Similar findings have been reported on long-term volatility behaviour after several major exogenous and endogenous daily stock price crashes [@sornette]. We mention here that in a recent study the bid-ask spread autocorrelations on the London stock exchange have also a power law decay [@Farmer]. It would be interesting to compare these results with our findings too, though the effect in this quantity seems market dependent.

![[Daytime-adjusted volatility: excess daytime volatility exhibits power-law decay. Average of 500 intraday price decreases on the NASDAQ.]{}[]{data-label="masodik"}](zkafig2.eps){width="14.0truecm"}

While the increase in volatility and volume is a robust property observable on both markets, the post-event evolution of the bid-ask spread is very different: on the NYSE the bid-ask spread widens at the event exactly like volatility and exhibits power law decay with the same exponent. On the other hand the bid-ask spread on the NASDAQ reacts to the event by a widening of a mere 20% (which is negligible in comparison to the 600% jump in the bid-ask spread on the NYSE). The cause of this is the same as which causes that there is virtually no intraday variation in the bid-ask spread on the NASDAQ [@chan]: the NASDAQ is a competitive dealership market, while only one market-maker operates on the NYSE for a given stock.

Conclusions
===========

We may thus conclude that one gets an interesting insight into the price formation process when examining large price changes on small scales. The overreaction in case of both extreme price increases and decreases on both markets refer to the presence of trading on behavioral motives within the trading day. We find that not only the volatility jumps at the event (which itself is the event) but volume, and on the NYSE the bid-ask spread increase as well. We find furthermore that the volatility decays according to a power law much faster than the autocorrelation of volatility itself. This points to the fact that extreme price shocks (which are quite probable to be exogenous) decay much faster than usual (endogenous) fluctuations of volatility.

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