Trading mechanisms, Roll model. Fabrizio Lillo Lecture 2. Market microstructure.
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(from Gopikrishnan et al 1999)
Irregular temporal spacing
at NYSE). Hidden and
a consolidated limit order
(Ponzi, Lillo, Mantegna 2007)
where ut is an i.i.d. noise with variance s2u
and thus the spread is 2c
where qt=+1 (-1) if the customer is buyer (seller).
Moreover qt are assumed serially independent.
and it is zero for lag larger than one
For example (Hasbrouck) the estimated first order covariance for PCO (Oct 2003) is g1=-0.0000294, which implies c=$0.017 and a spread 2c=$0.034. The time weighted spread is $0.032
Autocorrelation function of transaction price returns (solid circles), absolute value of transaction price returns (open circles), returns of midprice sampled before each transaction (filled triangles), and returns of midprice sampled at each market event (filled squares) for the AstraZeneca (AZN) stock traded at LSE in 2002. The lags on the horizontal line are measured in event time.
where et is a zero mean white noise process and kt is a linearly deterministic process (can be predicted arbitrarily well by a linear projection on past observation of xt)
Public information initially consists of common knowledge concerning the probability structure of the economy, in particular the unconditional distribution of terminal security value and the distribution of types of agents
Sequential trade models: randomly selected traders arrive at the market singly, sequentially, and independently. (Copeland and Galai (1983) and Glosten and Milgrom (1985)).
The terminal (liquidation) value of the asset is v, normally distributed with mean p0 and variance S0.
The informed trader wants to trade as much as possible to exploit her informational advantage
and the expected profit is
“Orders do not impact prices. It is more accurate to say that orders forecast prices” (Hasbrouck 2007)
London Stock Exchange
Impact of individual transaction is NOT universal
(Potters et al. 2003)
Individual market impact is a concave function of the volume
(Lillo et al. Nature 2003)
GROUP A -> least capitalized groupGROUP T -> most capitalized group
Let indicate the cumulative number of shares (depth) up to price return r
A market order of size V will produce a return
For example if
then the price impact is
Let us decompose the conditional probability of a return r conditioned to an order of volume V as
and we investigate the cumulative probability
for several different value of V.
This is the cumulative probability of a price return r conditioned to the volume and to the fact that price moves
the volume !!
The impact function is NOT deterministic and the fluctuations of price impact are very large.
Large price returns are caused by the presence of large gaps in the order book
Low liquidity (red),medium liquidity (blue),high liquidity (green)
A similar exponent describes also the probability density of the
DFA of spread (Plerou et al 2005)
Spread is an important determinant in order placement decision process
(see also Mike and Farmer 2006)
Waiting time between two spread variations
We wish to answer the question: how does the spread s(t) return to a normal value after a spread variation?
To this end we introduce the quantity
zero permanent impact
Permanent impact is roughly proportional to immediate impact
(from Zovko and Farmer 2002)
Limit order price is power law distributed with a tail exponent in the range 1.5-2.5
Moreover limit order price is correlated with volatility
For a given limit price D, volatility s, and investment horizon T the investor is faced with the lottery
From first passage time of the price random walk
So the expected utility is where u(D) is the utility function.
Agent optimizes her limit order placement by choosing the lottery (i.e. the value of D) which maximizes the expected utility.
For several choices of u(D) it is possible to solve the problem analytically. For example for power utility function u(x)=C xa
Some variable must be highly fluctuating: market (s) or agent (a or T)?
It is possible to show that heterogeneity in volatility or utility function cannot explain the value of the exponent empirically observed.
The only possibility is a strong heterogeneity in investment horizon T
Heterogeneity in volatility
Heterogeneity in traders preferences
Curiously, the same exponent of the time scale distribution is obtained by considering the time to fill of limit orders, metaorder splitting (see below), and a modified GARCH (Borland and Bouchaud 2005)
- the decisions and strategies of traders
- the exploiting of arbitrage opportunities
Time series of signs of market orders is a long memory process (Lillo and Farmer 2004, Bouchaud et al 2004)
The sign time series of the three types of orders
is a long-memory process
It is not possible to have an impact model where the impact is both fixed and permanent
The model assumes that the price just after the (t-1)-th transaction is
and return is
where the propagator G0(k) is a decreasing function.
The propagator can be chosen such as to make the market exactly efficient. This can be done by imposing that the volatility diffuses normally. The volatility at scale is
where D is a correlation-induced contribution
The model is able to make predictions on the response function defined as
which can be re-expressed in terms of the propagator and of the order sign correlation Cj
We neglect volume fluctuations and we assume that the naïve model is modified as
where W is the information set of the liquidity provider.
Ex post there are two possibilities, either the predictor was right or wrong
Let q+t (q-t) be the probability that the next order has the same (opposite) sign of the predictor and r+t (r-t) are the corresponding price change
-----> MARKET EFFICIENCY
ASYMMETRIC LIQUIDITY MODEL
The history dependent, permanent model is completely defined when one fixes
- the information set W of the liquidity provider
- the model used by the liquidity provider to build her forecast
As an important example we consider the case in which
- the information set is made only by the past order flow
- the liquidity provider uses a finite or infinite order autoregressive
model to forecast order flow
If the order flow is long memory, i.e. the optimal parameters of the autoregressive model are and the number of lags K in the model should be infinite.
If, more realistically, K is finite the optimal parameters of the autoregressive model follows the same scaling behavior with k
Under these assumptions and if K is infinite the linear model becomes mathematically equivalent to the fixed-temporary model (or propagator) model by Bouchaud et al. with
We assume that the distribution of initial hidden order size is a Pareto distribution
We prove that the time series of the signs of the revealed order has an autocorrelation function decaying asymptotically as
The volume of on-book and off-book trades have different statistical properties
Is it possible to identify directly hidden orders?
Credit Agricole trading Santander
We developed a statistical method to identify periods of time when an investor was consistently (buying or selling) at a constant rate -> Hidden orders
Number of transactions
Volume of the order
Circles and squares are data taken from Chan and Lakonishok at NYSE (1995) and Gallagher and Looi at Australian Stock Exchange (2006)
The distributions of large hidden orders sizes are characterized by power law tails.
Power law heterogeneity of investor typical (time or volume) scale
These results are not consistent with the theory of Gabaix et al. Nature 2003)
We measure the relation between the variables characterizing hidden orders by performing a Principal Component Analysis to the logarithm of variables.
BBVA SAN TEF