Session 2008-2009
Department of Economics and Related Studies
Financial Market Microstructure
Topic 1 : Overview and Fixed Cost Models of
Spreads
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Introduction
1.1
Some background
• Most of what is taught in the MScs about asset pricing does not
take into account the institutional arrangements that produce the
prices and rates of return that are studied. Part of this is a timeless
perspective - units of time are not very important.
• Prices that are traded are not really continuous, as we (implicitly)
assume - they are often measured in fixed units such as eighths in
US stock markets, ticks in stock markets in general and so on.
• Trading itself is not continuous and evenly spaced - there are actually markets that open and close, have days off for public holidays
and weekends - these are often ignored. In Eqypt, for example, the
stock market opens on a Sunday and not on a Friday, Datastream
does not notice this. Also some assets are thinly traded and may
go for long periods without trades.
• The actual process of trading can affect prices eg order driven
in Euronext, quote driven in NYSE and Taiwan. A number of
potential trading prices can co-exist - bid and ask, buying and
selling prices with a spread between them can affect the serial
correlation in prices and returns.
• Over what period of time between observations of prices or returns
do these issues matter - less for long horizons, essential for short.
But newer arguments suggest that microstructure effects might be
long-lasting eg the effect of order flow on exchange rates; information effects as opposed to pricing errors. The existence of multiple
equilibria might mean that the choice of equilibrium depends on
microstructure parameters.
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Market Efficiency
One issue that will arise frequently in the models that we look at is the
question of whether a financial market is efficient. The classic reference
on this is Fama’s survey in 1970 where he wrote: ”A market in which
prices always ”fully reflect” available information in called ’efficient”’.
One way of assessing this is to ask whether hypothetical trading based
on a specific information set would earn superior returns. A set of information sets and definitions referred to by Fama is:
• Weak-form Efficiency: The information set includes only past
prices themselves.
• Semi-strong Efficiency: The information set includes all information known to all market participants (publicly available information eg macro data series, other prices and returns).
• Strong-form Efficiency: The information set includes all information known to any market participant (private information).
Often in the models that we will look at we will examine the question
of whether the behaviour that is implied by the models concerned implies
market efficiency and of what type. Given the nature of the data that
are examined and the processes of trading that are modelled, there may
be issues of timing or how the accumulation of information is reflected
in prices.
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3.1
The Canonical Model of a Dealer Market with
Fixed Costs
Roll Model Structure
This initial model lays out some of the basic of microstructure theory.
The Roll (1984) model appears in a number of guises throughout the
course. We assume that markets are efficient in the sense that prices
reflect all available information. The efficient price is mt which can be
thourght of as reflecting all available information and be the price that
would hold in the absence of any transactions costs. We assume that is
therefore moves through time as:
mt = mt−1 + ut
(1)
where ut is a random error or innovation term which will reflect the
innovation in expectations in an efficient market. Trading in this market
is assumed to operate through dealers who post bid (bt ) and ask (at )
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prices. Buyers buy at the price at and sellers receive the price bt . The
cost of a transaction is c per transaction. So the bid and ask prices are:
at = mt + c
bt = mt − c
(2)
(3)
So the prices that we observe trading to take place at are pt which is
given by:
pt = mt + cqt
(4)
where qt = +1 for a buyer-initiated trade, ie a buy at the ask and −1
for a seller initiated trade, ie a sell at the bid.
We next assume that the innovation in expectations, ut has a constant variance σ 2u , that buys and sells are equally likely and that the
indicator variable qt is independent of it’s past values and of ut . So what
can we say about the properties of the observed price level pt ? We model
the change in the price, so:
∆pt = pt − pt−1 = −cqt−1 + cqt + ut
(5)
and we can obtain the variance of the change in the price as:
V ar(∆pt ) = 2c2 + σ 2u
(6)
and the covariance of the change of the price with it’s immeadiate
past value:
Cov(∆pt , ∆pt−1 ) = −c2
(7)
The correlation is, therefore:
−c2
Corr(∆pt , ∆pt−1 ) = 2
2c + σ 2u
(8)
The covariance with longer lags of price changes is zero:
Cov(∆pt , ∆pt−k ) = 0, k > 1
(9)
Thus the change in the price is volatile and negatively serially correlated
because of this particular microstructure effect ie because of the bid/ask
bounce. Intuitively this happens because the trading price is either at
the bid or ask (for a fixed mt ). So if the price is currently at the ask,
the change in price to today is either 0 or 2c. The change to the next
period is then either 0 or −2c. The opposite is true if the current price
is at the bid, so the serial correlation of price changes must be negative.
If the transaction cost c rises then the variance of the change in the
price goes up. The extent of the impact on the degree of serial correlation
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depends on the size of the transaction cost relative to the variance of the
innovation in expectations. There is no higher order serial correlation
whatever the level of the transaction cost.
These expressions imply, furthermore, that a measure of c, or the
implied bid-ask spread in this model 2c, can be calculated from the
observed behaviour of prices as:
p
c = 2 −Cov(∆pt , ∆pt−1 )
(10)
and the variance of the innovation in expectations as:
σ 2u = V ar(∆pt ) + 2.Cov(∆pt , ∆pt−1 )
(11)
This model of Roll thus provides us with a way of estimating the
costs of transactions from observed transactions prices in the absence of
bid and ask price data.
3.2
Empirical Evidence
Does the Roll model work? Hasbrouck (2006) presents the following
data: for the firm CBL on July 2 2003 there were 821 NYSE trades.
The estimated first order autocovariance of the price change is -0.0000251
which implies an estimate of c = 0.005($/share) and a spread of 2c =
0.01($/share). In this case the bid and ask prices exist. The average
NYSE spread is 0.022($/share) so the Roll estimate looks too low. What
would make these calculations wrong? There is always potential error
in the assumptions underlying the model and any sampling error in the
data. Also the real costs charged may be different to those quoted. Large
customers may be able to negotiate lower rates and so there may be variation in actutal costs paid. One way of calculating the effective spread
is to observe that trade prices are, on the NYSE, about 0.0079($/share)
away from the midpoint of bid and ask quotes. The effective spread is
therefore 0.0158($/share) which is closer to the Roll estimate.
A problem observed with some of the inital estimates presented by
Roll and by others is that price changes are estimated to be positively serially correlated rather than negatively, as predicted by the model. There
may, of course, be other factors as work which might be the cause of the
serial correlation and we will look at some of them in due course. The
argument presented by Harris (1990) is based on the statistical properties of the estimated first-order covariance. Harris suggests that the
Roll estimates are seriously biased by the degree of measurement error.
Through the use of simulations, Harris establishes that, allowing for this
noise or measurement error, the Roll calculations turn out to not be too
bad. Many generalisations of the Roll model have been developed and
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we will return to them. The first is due to Glosten and Milgrom (1985)
and allows c to be endogenous and related to mt . We will examine this
model when we look at the first set of asymmetric information models
in a couple of weeks.
A simple generalisation of the Roll model is provided by Choi et
al (1988). They allow for serial correlation in transaction type. So
they allow for the conditional probability of of a transaction at the bid
following one at the bid to differ from 0.5. They also make the serial
dependence the same for bids and asks. So if the probability that the
transaction is at bid, given that it was at the bid in the previous period
is δ then the autocovariance between the price change in period t and
t-1 can be shown to be
Cov(∆pt , ∆pt−1 ) = −(2c)2 (1 − δ)2
(12)
where this covariance is the same as that in equation (7) when δ = 0.5.
The implied bid-ask spread is then
p
(13)
2c = [ 2 −Cov(∆pt , ∆pt−1 )]/(1 − δ)
So if there were positive serial correlation in transaction type with δ >
0.5 then the bid-ask spread calculated by equation (10) would be a downward biased estimate of the true spread. Choi et al present two estimates
of δ based on data on options traded on the Chicago Board Options Exchange (CBOE) in the late 1970’s. The serial covariance of returns is
estimated to -0.00262 with a standard error of 0.00424. So therefore the
first order autocovariance of returns is found to be significantly negative.
The estimate of δ is 0.703 with a standard error of 0.068 and therefore
the estimate of the spread from equation ( 13) of 15.79% with a standard
error of 15.57 making it not very well determined. Again, bid and ask
prices can be observed in this dataset and the average spread is 17.89%.
The lower estimate provided by the model may be due to the influence
of adverse selection effects on the observed spread which we will discuss
later in this course.
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