This document discusses optimal execution strategies for minimizing trading costs. It begins by outlining the key components of execution algorithms, including the macrotrader, microtrader, and order router layers. It then reviews several empirical studies showing that market impact follows a square-root law and that prices revert after a trade is completed. Several models for optimal scheduling are examined, including constant rate trading and trading in bursts. The document also discusses factors influencing the choice between market and limit orders, like order book signals and order flow predictability.

1. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Minimizing execution costs
Jim Gatheral
Advances in Financial Mathematics
Order book modeling and microstructure mini-symposium
January 8, 2014

2. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Overview of this talk
What is optimal execution?
Components of an algorithmic execution
How much does it cost to trade?
How to reduce cost
Optimal scheduling
Choice of market or limit order
Optimal order routing
How much can trading costs be reduced?

3. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Overview of the execution puzzle
Typically, an execution algorithm has three layers:
The macrotrader
This highest level layer decides how to slice the meta-order:
when the algorithm should trade, in what size and for roughly
how long.
The microtrader
Given a slice of the meta-order to trade (a child order), this
layer of the algorithm decides whether to place market or limit
orders and at what price level(s).
The smart order router
Given a limit or market order, to which venue should the order
be sent?

4. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Market impact
The market impact function relates expected price change to
the volume of a transaction.
However, there is no reason a priori to expect that market
impact should be a function of volume only.
Market impact could be a function of:
Market capitalization
Bid-ask spread
The timescale over which the trade is executed.
Note that it is not only market orders that impact the market
price; limit orders and cancelations should also have market
impact.

5. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The square-root formula for market impact
For many years, traders have used the simple
sigma-root-liquidity model described for example by Grinold
and Kahn in 1994.
Software incorporating this model includes:
Salomon Brothers, StockFacts Pro since around 1991
Barra, Market Impact Model since around 1998
Bloomberg, TCA function since 2005
The model is always of the rough form
∆P = Spread cost + α σ
r
Q
V
where σ is daily volatility, V is daily volume, Q is the number
of shares to be traded and α is a constant pre-factor of order
one.

6. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Empirical question
Does the square-root formula work in practice?

7. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Impact of proprietary metaorders (from Tóth et al.)
Figure 1: Log-log plot of the volatility-adjusted price impact vs the ratio
Q/V

8. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Notes on Figure 1
In Figure 1 which is taken from [Tóth et al.], we see the
impact of metaorders for CFM1 proprietary trades on futures
markets, in the period June 2007 to December 2010.
Impact is measured as the average execution shortfall of a
meta-order of size Q.
The sample studied contained nearly 500,000 trades.
We see that the square-root market impact formula is verified
empirically for meta-orders with a range of sizes spanning two
to three orders of magnitude!
1
Capital Fund Management (CFM) is a large Paris-based hedge fund.

9. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Some implications of the square-root formula
The square-root formula refers only to the size of the trade
relative to daily volume.
It does not refer to for example:
The rate of trading
How the trade is executed
The capitalization of the stock
Surely impact must be higher if trading is very aggressive?
The database of trades only contains sensible trades with
reasonable volume fractions.
Were we to look at very aggressive trades, we would indeed
find that the square-root formula breaks down.

10. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Empirical question
What happens on average to the stock price while a metaorder is
being executed? And what happens to the stock price after
completion?

11. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Path of the stock price during execution (from Moro et al.)
Figure 2: Average path of the stock price during execution of a
metaorder on two exchanges

12. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Empirically observed stock price path
From Figure 2, we see that
There is reversion of the stock price after completion of the
meta-order.
Some component of the market impact of the meta-order
appears to be permanent.
The path of the price prior to completion looks like a power
law.
From [Moro et al.]
mt − m0 ≈ (4.28 ± 0.21)
t
T
0.71±0.03
(BME)
mt − m0 ≈ (2.13 ± 0.05)
t
T
0.62±0.02
(LSE)
where T is the duration of the meta-order.

13. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Summary of empirical observations
The square-root formula gives an amazingly accurate rough
estimate of the cost of executing an order.
During execution of a meta-order, the price moves on average
roughly according to (t/T)2/3.
Immediately after completion of a meta-order, the price begins
to revert.

14. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The FGLW market impact model
[Farmer, Gerig, Lillo, and Waelbroeck] (FGLW from now on) show
that the typical impact profile associated with the execution of a
metaorder may be recovered starting from two assumptions:
The Martingale Condition: The price process is a
martingale.
The market maker does not know how long a given metaorder
will continue.
The Fair Pricing Condition: On average, the price reverts
after completion of the metaorder to a level equal to the
average price paid by the informed trader.
If metaorder sizes are power-law distributed with exponent β,
the price reverts on average to a level which is a factor 1/β of
the peak price reached at completion.

15. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
FGLW impact profile in the power-law case
If the distribution of order sizes is power-law with tail
exponent β, one can show that the impact profile (expected
path of the stock price) is given by
S̃t ∼ tβ−1
for t T
S∞ = ST /β.
The price reverts to a level of 1/β times the price reached at
completion.
If β = 3/2, we get
a square-root market impact profile consistent with the
square-root market impact formula.
that permanent impact is 2/3 of peak impact.

16. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Empirical confirmation of FGLW
In a recent paper, [Bershova and Rakhlin] perform an
empirical study of a proprietary dataset of large
AllianceBernstein equity orders.
They confirm that the distribution of order sizes is power-law
with a tail exponent of 3/2 (at least for not too large orders).
They broadly confirm the predictions of FGLW, including the
power-law impact profile, the reversion level of 2/3 and
roughly square-root permanent impact.
They also study reversion of market impact in detail, finding
initial power-law decay followed by exponential decay.
In an even more recent paper, [Waelbroeck and Gomes] study
a proprietary dataset of orders in the Portware OMS (Order
Management System).
They find the predicted 2/3 reversion level for informed orders;
the price reverts completely when orders are uninformed.

17. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The jury is still out
There are both theoretical and empirical reasons why the
story is not over.
In an even more recent paper with another proprietary dataset
of futures trades, [Mastromatteo, Tóth and Bouchaud] report
that they cannot reproduce the decay of market impact
predicted by FGLW.
FGLW implies nonlinear permanent market impact which
would seem to imply the possibility of price manipulation
(though [Waelbroeck and Gomes] argue that such
manipulation is not possible).
For the purposes of this talk, the main point is that the
square-root law for cost of execution is verified in all of these
papers. Moreover, all empirical studies agree that the market
price reverts strongly (on average) after completion of a
metaorder.

18. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Implications for optimal execution
According to a literal reading of the square-root formula, the
cost of trading doesn’t depend on trading strategy.
Does this mean that there is nothing that we can do to reduce
trading costs?
No! We can reduce the size of the prefactor α by breaking down
the execution puzzle into its components and attacking each one
in turn.
We now turn our attention to optimal scheduling.

19. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Statement of the optimal scheduling problem
Given a model for the evolution of the stock price, we would
like to find an optimal strategy for trading stock, the strategy
that minimizes some cost function over all permissible
strategies.
A static strategy is one determined in advance of trading.
A dynamic strategy is one that depends on the state of the
market during execution of the order, i.e. on the stock price.
Delta-hedging is an example of a dynamic strategy. VWAP is
an example of a static strategy.
It turns out, surprisingly, that in many models, a statically
optimal strategy is also dynamically optimal.
For all the models we will describe, a static strategy set in
advance of trading is optimal.

20. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Almgren and Chriss
The seminal paper of [Almgren and Chriss] treats the
execution of a hidden order as a tradeoff between risk and
execution cost.
According to their formulation:
The faster an order is executed, the higher the execution cost
The faster an order is executed, the lower the risk (which is
increasing in position size).
The Almgren-Chriss model can be considered the conventional
market standard.
If there is no risk penalty, the optimal strategy is a boring
VWAP.
Constant rate of trading in volume time.

21. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Models of order book resilience
In the model of [Obizhaeva and Wang] and its subsequent
major generalization by [Alfonsi, Fruth and Schied],
metaorders are executed using market orders only.
Individual executions eat into the order book creating a hole
(referred to as volume impact) or equivalently a wider bid-ask
spread which is then filled up (presumably by limit orders).
The filling-up process is referred to as order book resilience.
It was shown by [Predoiu, Shaikhet and Shreve] that if
resiliency is a function of the volume impact process (or
equivalently the spread) only, the optimal strategy has block
trades at inception and completion and continuous trading at
a constant rate in-between.
These conditions may appear quite general but in fact, there
are many other models that do not satisfy them.

22. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
A transient market impact model
The price process assumed in [Gatheral] is
St = S0 +
Z t
0
f (vs) G(t − s) ds + noise (1)
The instantaneous impact of a trade at time s is given by
f (vs) – some function of the rate of trading.
A proportion G(t − s) of this initial impact is still felt at time
t s.

23. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The square-root model
Consider the following special case of (1) with f (v) = 3
4σ
p
v/V
and G(τ) = 1/
√
τ:
St = S0 +
3
4
σ
Z t
0
r
vs
V
ds
√
t − s
+ noise (2)
which we will call the square-root process.
It turns out that the square-root process is consistent with the
square-root formula for market impact:
C
X
= σ
r
X
V
(3)
Of course, that doesn’t mean that the square-root process is
the true underlying process!

24. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The optimal strategy under the square-root process
Because f (·) is concave, an optimal strategy does not exist in
this case.
It is possible to drive the expected cost of trading to zero by
increasing the number of slices and decreasing the duration of
each slice.
To be more realistic, f (v) must be convex for large v and in
this case, an optimal strategy does exist that involves trading
in bursts, usually more than two.

25. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Optimal scheduling summary
The optimal strategy depends on modeling assumptions.
In the Almgren-Chriss model where there is no price reversion
after completion of the meta-order, the optimal strategy is a
simple VWAP.
In other models where there is reversion, the optimal strategy
is to make big trades separated in time, perhaps with some
small component of continuous trading.
The intuition is easy to see:
The price reversion idea
If the price is expected to revert after completion, stop trading
early and start again later after the price has reverted!

26. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The market or limit order decision
Having decided how to slice the meta-order, should we send
market or limit orders?
Many market participants believe that market orders should
only be sent when absolutely necessary – for example when
time has run out.
Conventional wisdom has it that the more aggressive an
algorithm is, the more costly it should be.
This cannot be true on average. Traders are continuously
monitoring whether to send market or limit orders so in
equilibrium, market and limit orders must have the same
expected cost.
Market orders incur an immediate cost of the half-spread but
limit orders suffer from adverse selection.

27. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The order book signal
If we know the price is going against us, we should send a
market order. Otherwise we should send a limit order.
In practice, we cannot predict the future; we can compute
relative probabilities of future events.
One simple idea is to look at the shape of the order book. If
there are more bids than offers, the price is more likely to
increase than decrease.
Even in the very simple zero-intelligence model of
[Smith, Farmer, et al.], the shape of the order book allows
prediction of price movements.
Traders really would need to have zero intelligence not to
condition on book shape!

28. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Microprice
The relationship between the imbalance in the order book and
future price movements is sometimes described in terms of the
microprice.
This can be thought of as a fair price, usually between the bid
and the ask.
As an example, in the context of the zero-intelligence model,
[Cont and de Larrard] derived the following asymptotic
expression:
Proposition 2 of Cont de Larrard
The probability φ(n, p) that the next price is an increase,
conditioned on having n orders at the bid and p orders at the ask
is:
φ(n, p) =
1
π
Z π
0
dt
2 − cos t −
q
(2 − cos t)2 − 1
p
sin n t cos t
2
sin t
2

29. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Order splitting and the order flow signal
The typical size of a meta-order is a large multiple of the
quantity typically available at the best quote.
Consequently, meta-orders need to be split into smaller child
orders.
This gives rise to a long-memory autocorrelation function
which is significant at all lags.
Order sign (or order flow) is thus highly predictable.

30. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Autocorrelation of order signs
Figure 3: Autocorrelation function of the time series of signs of Vodafone
market orders in the period May 2000 – December 2002, a total of
580,000 events (from [Bouchaud, Farmer, Lillo])

31. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Market impact relates to unexpected order flow
Figure 4: Average impact of AZN market orders vs expected order sign.
Buy orders in green; sell orders in red (from [Bouchaud, Farmer, Lillo])

32. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Queue length, fill probability and rebates
Whether (and where) to send a limit order must depend in
particular on the length of the queue and your position in it
and by any rebates offered by the exchange.
This particularly significant for large tick stocks where the
queue is long.
In the analysis of [Cont and Kukanov], the tradeoff is between
the cost of a limit order and execution probability.
On a given exchange, the lower the price of the limit order, the
lower the probability of execution
On multiple exchanges, the higher the rebate, the longer the
queue length, the lower the probability of execution.
With a risk penalty, it can be optimal to send a market order.

33. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
The market/ limit order decision
We have two (related) signals:
From the current state of the order book, we can predict the
sign of the next price change.
From recent order flow history, we can judge whether active
meta-orders are more on the buy side or on the sell side.
Also, we can estimate the probability of a limit order fill.
The practical recipe is therefore to:
Send market orders when the market is going against us, limit
orders otherwise.
However place limit orders only if the probability of execution
is sufficiently high.
Trade more when others want to trade with us, less when there
are fewer counterparties.

34. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Order routing
Having optimally scheduled child orders and for each such
child order, having decided whether to send a market or a
limit order, where should the order be sent?
In the US, there are currently approximately 13 lit venues and
over 40 dark venues.
On the one hand, it would be prohibitively complicated and
expensive to route to all of them.
On the other hand, by not routing to a particular venue, the
trader misses out on potential liquidity, and all things being
equal, will cause incur greater market impact.
Most traders have to use a smart order routing (SOR)
algorithm provided by a dealer.

35. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Smart order routing (SOR)
The goal of an SOR algorithm is to buy (or sell) as many
shares as possible in the shortest time by optimally allocating
orders across both lit and dark venues.
In the case of lit venues, there are hidden orders so there is
typically more liquidity available than is displayed.
In dark venues, by definition, all liquidity is hidden.

36. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
A stochastic Lagrangian algorithm
[Laruelle, Lehalle and Pagès] propose a clever dark pool
allocation algorithm where
More quantity is sent to dark pools with higher liquidity, as
estimated from the history of executions.
More quantity is sent to the dark pools offering lower-cost
execution.
Relative execution cost may be proxied for example by the
realized spread.
Under various technical assumptions,
[Laruelle, Lehalle and Pagès] prove the convergence of their
algorithm.

37. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Is fragmentation a good thing?
In equilibrium, it is easy to see that queue lengths should be
synchronized across venues with relative lengths depending on
the fee and rebate structure of each exchange.
One answer comes from [Cont and Kukanov] again. They
show how to minimize execution cost by taking advantage of
deviations from the equilibrium configuration.
So, it’s hard to say whether market fragmentation is good or
bad for society in general.
It is clear that market fragmentation is better for you if order
flow is not perfectly correlated between exchanges and you
have a good smart order router.

38. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Another philosophical question
Question
Why should it be so complicated to trade stock?
One answer goes something like this:
Changes in market structure together with technological
innovation have massively reduced trading costs.
Nevertheless, some market participants achieve significantly
lower costs.
This requires either substantial investment in technology and
trading expertise or
careful selection of broker algorithms.

39. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Potential cost savings from optimal scheduling
With reasonable assumptions, assuming the square-root model (1),
savings relative to a simple VWAP are roughly of the following
order:
Around 15% for the Alfonsi-Fruth-Schied bucket-like strategy
Around 25% for a suitable bursty strategy (intermittent
interval VWAPS) with a capped participation rate.

40. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Potential cost savings from microtrader improvements
Were we just to blindly send market orders, we would estimate
that the cost of each child order would be around a
half-spread.
Practical experience shows that it is not possible to reduce
this cost much below one third of a spread.
We conclude that we could potentially save up to one sixth of
the spread.

41. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Potential cost savings from smart order routing
A naı̈ve estimate would be to use the square-root market
impact formula, changing the denominator V to reflect the
potential increase in liquidity from routing to extra venues.
If the potential liquidity accessed is doubled, costs should be
decreased by 1 − 1/
√
2 ≈ 30% according to this simple
computation!
This could be one explanation for the multiplicity of trading
venues in the US.
We expect actual savings to be much less than this because
the different venues are all connected and information leaks
from one venue to the other.

42. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
Final conclusion
Recent empirical and theoretical work on market
microstructure has led to a much improved understanding of
how to trade optimally.
Potential savings from careful order execution relative to
VWAP using a simple first-generation algorithm are
substantial.
Relative to a naı̈ve strategy, cost savings of 25% for reasonably
sized orders are not unreasonable.

43. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
References
Aurélien Alfonsi, Antje Fruth and Alexander Schied, Optimal execution strategies in limit order books with
general shape functions, Quantitative Finance 10(2) 143–157 (2010).
Robert Almgren and Neil Chriss, Optimal execution of portfolio transactions, Journal of Risk 3 5–40 (2001).
Nataliya Bershova and Dmitry Rakhlin, The Non-Linear Market Impact of Large Trades: Evidence from
Buy-Side Order Flow, Quantitative Finance 13(11) 1759–1778 (2013).
Jean-Philippe Bouchaud, J. Doyne Farmer, and Fabrizio Lillo, How Markets Slowly Digest Changes in
Supply and Demand, in Handbook of Financial Markets: Dynamics and Evolution 57-156. (2009).
Rama Cont and Adrien de Larrard, Price dynamics in a Markovian limit order market, SIAM Journal on
Financial Mathematics 4(1) 1–25 (2013).
Rama Cont and Arseniy Kukanov, Optimal order placement in limit order markets, available at
http://ssrn.com/abstract=2155218 (2013).
J Doyne Farmer, Austin Gerig, Fabrizio Lillo, and Henri Waelbroeck, How efficiency shapes market impact,
Quantitative Finance 13(11) 1743–1758 (2013).
Jim Gatheral, No-dynamic-arbitrage and market impact, Quantitative Finance 10(7) 749–759 (2010).

44. Introduction Stylized facts Optimal scheduling Microtrader Order routing Conclusion
References
Sophie Laruelle, Charles-Albert Lehalle, and Gilles Pagès, Optimal split of orders across liquidity pools: a
stochastic algorithm approach, SIAM J. Finan. Math. 2(1) 1042–1076 (2011).
I Mastromatteo, Bence Tóth, and Jean-Philippe Bouchaud, Agent-based models for latent liquidity and
concave price impact, arXiv (2013).
Esteban Moro, Javier Vicente, Luis G Moyano, Austin Gerig, J. Doyne Farmer, Gabriella Vaglica, Fabrizio
Lillo, and Rosario N Mantegna, Market impact and trading prole of hidden orders in stock markets, Physical
Review E 80(6) 066102 (2009).
Anna Obizhaeva and Jiang Wang, Optimal trading strategy and supply/demand dynamics, Journal of
Financial Markets 16(1) 1–32 (2013).
Silviu Predoiu, Gennady Shaikhet and Steven Shreve, Optimal execution in a general one-sided limit-order
book, SIAM Journal on Finance Mathematics 2 183–212 (2011).
Eric Smith, J Doyne Farmer, Laszlo Gillemot, Supriya Krishnamurthy, Statistical theory of the continuous
double auction, Quantitative Finance 3 481–514, (2003).
Bence Tóth, Yves Lempérière, Cyril Deremble, Joachim de Lataillade, Julien Kockelkoren, and
Jean-Philippe Bouchaud, Anomalous price impact and the critical nature of liquidity in financial markets,
Physical Review X 021006, 1-11(2011).
Henri Waelbroeck and Carla Gomes, Is Market Impact a Measure of the Information Value of Trades?
Market Response to Liquidity vs. Informed Trades, available at
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2291720 (2013).