Optimal Adaptive Execution of Portfolio
Transactions
Julian Lorenz
Joint work with Robert Almgren (Banc of America Securities, NY)
15. 05. 2007
Execution of Portfolio Transactions
Sell 100,000
Microsoft shares
today!
Broker/Trader
Fund Manager
Problem: Market impact
Trading Large Volumes Moves the Price
How to optimize the trade schedule over the day?
2007
Julian Lorenz, [email protected]
2
Market Model
 Discrete times
 Stock price
follows random walk
 Sell program for initial position of X shares
s.t.
= shares hold at time
 Execution strategy:
i.e. sell
,
…
shares between t0 and t1
t1 and t2
 Pure sell program:
2007
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3
Market Impact and Cost of a Strategy
Selling xk-1 – xk shares in [tk-1, tk] at discount to Sk-1
with
Linear Temporary
Market Impact
Benchmark: Pre-Trade Book Value
Cost C() = Pre-Trade Book Value – Capture of Trade
X=x0=100
x
x
N=10
C() is independent of S0
2007
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Trader‘s Dilemma
Random
variable!
 Minimal Risk
Obviously by immediate liquidation
X
x(t)
No risk, but high market impact cost
T t
 Minimal Expected Cost
X
Linear strategy
x(t)
T t
But: High exposure to price volatility  High risk
Optimal trade schedules seek risk-reward balance
2007
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Efficient Strategies
Risk-Reward
Tradeoff:
Variance
as risk
measure Mean-Variance
 Minimal variance
 Minimal expected cost

Efficient Strategies
E-V Plane
Admissible Strategies
Linear
Strategy
Immediate
Sale

2007
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Almgren/Chriss Deterministic Trading (1/2)
R. Almgren, N. Chriss: "Optimal execution of portfolio transactions",
Journal of Risk (2000).
Deterministic trading strategy

functions of decision variables (x1,…,xN)
2007
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Almgren/Chriss Deterministic Trading (2/2)
Deterministic
Trajectories
for some
Almgren/Chriss Trajectories:
xi deterministic
T=1, =10
normallydistributed
 C() Urgency
controls
curvature
x(t)
QP
 Straightforward
E-V Plane
Dynamic strategies:
xi = xix(t)
(1,…,i-1)
X
 By dynamic programming
T t
X x(t)
T t
We show: Dynamic strategies improve (w.r.t. mean-variance) !
2007
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Definitions
Adapted trading strategy: xi may depend on 1…,i-1
Admissible trading strategies for expected cost
adapted strategies for X shares in N periods
with expected cost
Efficient trading strategies
i.e.
„no other admissible strategy offers lower
variance for same level of expected cost“
2007
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Dynamic Programming (1/4)
Define value function
i.e. minimal variance to sell x shares in k periods with
and optimal
strategies for k-1 periods
and optimal
strategies for k periods
+
Optimal Markovian
one-step control
For type “
Here:
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…ultimately interested in
“ DP is straightforward.
in value function & terminal
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constraint …
?
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Dynamic Programming (2/4)
We want to determine
Situation:




k periods and x shares left
Limit for expected cost is c
Current stock price S
Next price innovation is  ~ N(0,2)
Construct optimal strategy


In current period sell
Use efficient strategy
for k periods
shares at
for remaining k-1 periods
Note:
must be deterministic, but when we begin
, outcome
of
is known, i.e. we may choose
depending on 
 Specify
2007
by its expected cost z()
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12
Dynamic Programming (3/4)
 Strategy  defined by control
and control function z()
Conditional on :
Using the laws of total expectation and variance
One-step optimization of
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and
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by means of
and
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Dynamic Programming (4/4)
Theorem:
where
Control variable
new stock holding
(i.e. sell x – x’
in this period)
Control function
targeted cost as
function of next
price change 
 Solve recursively!
2007
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Solving the Dynamic Program
 No closed-form solution
 Difficulty for numerical treatment:
Need to determine a control function
 Approximation:
 For fixed
is piecewise constant
determine
 Nice convexity property
Theorem:
In each step, the optimization problem is a convex
constrained problem in {x‘, z1, … , zk}.
2007
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15
Behavior of Adaptive Strategy
„Aggressive in the Money“
Theorem:
At all times, the control function z() is monotone increasing
Recall:
 z() specifies expected cost for remainder
as a function of the next price change 
 High expected cost = sell quickly (low variance)
Low expected cost = sell slowly (high variance)
 If price goes up ( > 0), sell faster in remainder
Spend part of windfall gains on increased impact costs
to reduce total variance
2007
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Numerical Example
 Respond only to up/down
 Discretize state space of
2007
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Sample Trajectories of Adaptive Strategy
Aggressive in
the money …
2007
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Family of New Efficient Frontiers
Sample cost PDFs:

Adaptive
strategies
Family of frontiers
parametrized by
size of trade X

Larger improvement
for large portfolios
Almgren/Chriss
deterministic strategy
(i.e.
)


Improved
frontiers
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
Almgren/Chriss
frontier
Distribution plots
obtained by Monte
Carlo simulation

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20
Extensions
 Non-linear impact functions
 Multiple securities („basket trading“)
 Dynamic Programming approach also applicable for
other mean-variance problems,
e.g. multiperiod portfolio optimization
2007
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Thank you very much
for your attention!
Questions?
2007
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