1. Optimal Adaptive Execution of Portfolio Transactions
Julian Lorenz
Joint work with Robert Almgren (Banc of America Securities, NY)

2. 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?

3. Market Model … Discrete times Stock price follows random walk
Sell program
for initial position of X shares
Execution strategy:
s.t. ,
= shares hold at time
i.e.
sell shares between t0 and t1
t1 and t2 …
Pure sell program:

4. 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
C() is independent of S0 x
X=x0=100
N=10 x

5. Œ  ð Trader‘s Dilemma Random variable! Minimal Risk
Obviously by immediate liquidation
No risk, but high market impact cost
Minimal Risk Πt T
x(t) X
Linear strategy
Minimal Expected Cost
But: High exposure to price volatility ð
High risk t T
x(t) X
Optimal trade schedules seek risk-reward balance

6. Admissible Strategies
Efficient Strategies
Risk-Reward Tradeoff: Mean-Variance
Variance as risk measure
E-V Plane
Minimal variance Œ
Admissible Strategies
Minimal expected cost
Linear Strategy
Immediate Sale
Efficient Strategies Œ

7. Almgren/Chriss Deterministic Trading (1/2)
R. Almgren, N. Chriss: "Optimal execution of portfolio transactions", Journal of Risk (2000).
Deterministic trading strategy ð
functions of decision variables (x1,…,xN)

8. Almgren/Chriss Deterministic Trading (2/2)
Trajectories
for some
E-V Plane t T X
x(t)
T=1, =10
Dynamic strategies:
xi = xi(1,…,i-1)
Almgren/Chriss Trajectories:
xi deterministic
C() normally distributed ð
Straightforward QP ð
By dynamic programming
Urgency  controls curvature
Dynamic strategies improve (w.r.t. mean-variance) !
We show:

9. Definitions Adapted trading strategy: xi may depend on 1…,i-1
adapted strategies for X shares in N periods with expected cost
Admissible trading strategies for expected cost
Efficient trading strategies
„no other admissible strategy offers lower variance for same level of expected cost“
i.e.

10. Tail of Efficient Strategies
Suppose is efficient,
Note:
deterministic, but
may depend on 1
Conditional on 1, define the “tail“ of 
Lemma:
For all outcomes a of 1, the tail is efficient ð
Dynamic programming!

11. 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
Optimal Markovian one-step control +
and optimal strategies for k periods
…ultimately interested in
For type “ “ DP is straightforward.
Here: in value function & terminal constraint … ?

12. Dynamic Programming (2/4)
We want to determine
k periods and x shares left
Limit for expected cost is c
Current stock price S
Next price innovation is x ~ N(0,2)
Situation:
Construct optimal strategy for k periods
In current period sell shares at Œ
Use efficient strategy for remaining k-1 periods
Specify by its expected cost z() ð
Note: must be deterministic, but when we begin , outcome of is known, i.e. we may choose depending on 

13. Dynamic Programming (3/4)ð
Strategy  defined by control and control function z()
Conditional on :
Using the laws of total expectation and variance
One-step optimization of and by means of and

14. 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  ð
Solve recursively!

15. Solving the Dynamic Program
No closed-form solution
Difficulty for numerical treatment:
Need to determine a control function
Approximation: is piecewise constant ð
For fixed determine
Nice convexity property
Theorem:
In each step, the optimization problem is a convex constrained problem in {x‘, z1, … , zk}.

16. 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) ð
If price goes up ( > 0), sell faster in remainder
Spend part of windfall gains on increased impact costs
to reduce total variance

17. Numerical Example
Respond only to up/down
Discretize state space of

18. Sample Trajectories of Adaptive Strategy
Aggressive in the money …

19. Family of New Efficient Frontiers
Frontiers are
parametrized by
Sample cost PDFs:
Adaptive
strategies
Almgren/Chriss deterministic strategy
(“market power“)
Larger improvement for large portfolios
( i.e )
Almgren/Chriss frontier
Distribution plots obtained by Monte Carlo simulation
Improved frontiers

20. Family of New Efficient Frontiers
Family of frontiers
parametrized by
size of trade X
Sample cost PDFs:
Adaptive
strategies
Almgren/Chriss deterministic strategy
Larger improvement for large portfolios
(i.e )
Almgren/Chriss frontier
Distribution plots obtained by Monte Carlo simulation
Improved frontiers

21. More Cost Distributions

22. Extensions Non-linear impact functions
Multiple securities („basket trading“)
Dynamic Programming approach also applicable for other mean-variance problems,
e.g. multiperiod portfolio optimization

23. Thank you very much for your attention! Questions?