optimal adaptive execution of portfolio transactions

15. 05. 2007

Optimal Adaptive Execution of Portfolio Transactions

Julian Lorenz

Joint work with Robert Almgren (Banc of America Securities, NY)

22007 Julian Lorenz, jlorenz@inf.ethz.ch

Execution of Portfolio Transactions

Fund Manager Broker/Trader

Sell 100,000 Microsoft shares

today!

How to optimize the trade schedule over the day?

Problem: Market impactTrading Large Volumes Moves the Price

32007 Julian Lorenz, jlorenz@inf.ethz.ch

Market Model

Stock price follows random walk

Execution strategy:

sell shares between t0 and t1

Sell program

t1 and t2 …

s.t.

for initial position of X shares

Pure sell program:

,

Discrete times

= shares hold at time

i.e.

42007 Julian Lorenz, jlorenz@inf.ethz.ch

Benchmark: Pre-Trade Book Value

Cost C() = Pre-Trade Book Value – Capture of Trade

C() is independent of S0

Market Impact and Cost of a Strategy

Linear Temporary Market Impact

Selling xk-1 – xk shares in [tk-1, tk] at discount to Sk-1

with

x

x

X=x0=100

N=10

52007 Julian Lorenz, jlorenz@inf.ethz.ch

Trader‘s Dilemma

Random variable!

Optimal trade schedules seek risk-reward balance

Obviously by immediate liquidation

No risk, but high market impact cost

Minimal RiskŒ

tT

x(t)X

Linear strategy

Minimal Expected Cost

But: High exposure to price volatility

High risk

tT

x(t)X

62007 Julian Lorenz, jlorenz@inf.ethz.ch

Efficient Strategies

Minimal varianceΠAdmissible Strategies

Efficient Strategies

Linear Strategy

ImmediateSale

E-V Plane

Minimal expected cost

Risk-Reward Tradeoff: Mean-Variance

Œ

Variance as risk measure

72007 Julian Lorenz, jlorenz@inf.ethz.ch

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)

82007 Julian Lorenz, jlorenz@inf.ethz.ch

Almgren/Chriss Deterministic Trading (2/2)

DeterministicTrajectories

for some

Dynamic strategies:xi = xi(1,…,i-1)

Almgren/Chriss Trajectories:xi deterministic

Dynamic strategies improve (w.r.t. mean-variance) !We show:

C() normally distributed

Straightforward QP

E-V Plane

tT

X x(t)

tT

x(t)XT=1, =10

x(t)

Urgency controls curvature

By dynamic programming

92007 Julian Lorenz, jlorenz@inf.ethz.ch

Definitions

Adapted trading strategy: xi may depend on 1…,i-1

Efficient trading strategies

„no other admissible strategy offers lower variance for same level of expected cost“

i.e.

adapted strategies for X shares in N periods with expected cost

Admissible trading strategies for expected cost

112007 Julian Lorenz, jlorenz@inf.ethz.ch

Dynamic Programming (1/4)

i.e. minimal variance to sell x shares in k periods with

Define value function

and optimal strategies for k-1 periods

Optimal Markovian one-step control

+ and optimal strategies for k periods

For type “ “ DP is straightforward.

Here: in value function & terminal constraint … ?…ultimately interested in

122007 Julian Lorenz, jlorenz@inf.ethz.ch

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 for k periods

In current period sell shares atŒ

Use efficient strategy for remaining k-1 periods

Specify by its expected cost z()

Note: must be deterministic, but when we begin , outcomeof is known, i.e. we may choose depending on

132007 Julian Lorenz, jlorenz@inf.ethz.ch

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 and by means of and

142007 Julian Lorenz, jlorenz@inf.ethz.ch

Dynamic Programming (4/4)

Theorem:

where

Control variablenew stock holding

(i.e. sell x – x’ in this period)

Control functiontargeted cost as function of next price change

Solve recursively!

152007 Julian Lorenz, jlorenz@inf.ethz.ch

Solving the Dynamic Program

Difficulty for numerical treatment:

No closed-form solution

Need to determine a control function

Approximation: is piecewise constant

Theorem:

In each step, the optimization problem is a convex constrained problem in {x‘, z1, … , zk}.

Nice convexity property

For fixed determine

162007 Julian Lorenz, jlorenz@inf.ethz.ch

Behavior of Adaptive Strategy

Theorem:

„Aggressive in the Money“

At all times, the control function z() is monotone increasing

Recall:

High expected cost = sell quickly (low variance)

z() specifies expected cost for remainder as a function of the next price change

Low expected cost = sell slowly (high variance)

If price goes up ( > 0), sell faster in remainder

Spend part of windfall gains on increased impact coststo reduce total variance

172007 Julian Lorenz, jlorenz@inf.ethz.ch

Numerical Example

Respond only to up/down

Discretize state space of

182007 Julian Lorenz, jlorenz@inf.ethz.ch

Sample Trajectories of Adaptive Strategy

Aggressive in the money …

202007 Julian Lorenz, jlorenz@inf.ethz.ch

Family of New Efficient Frontiers

Family of frontiersparametrized by size of trade X

Almgren/Chriss deterministic

strategy

Adaptivestrategies

Sample cost PDFs:

Distribution plots obtained by Monte Carlo simulation

Almgren/Chriss frontier

Improved frontiers

Œ

‹

‹

Œ

Larger improvement for large portfolios

(i.e. )

222007 Julian Lorenz, jlorenz@inf.ethz.ch

Extensions

Non-linear impact functions

Multiple securities („basket trading“)

Dynamic Programming approach also applicable for other mean-variance problems, e.g. multiperiod portfolio optimization

232007 Julian Lorenz, jlorenz@inf.ethz.ch

Thank you very much for your attention!

Questions?