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, [email protected]

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


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.


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


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


Efficient Strategies

Minimal varianceŒ Admissible Strategies

Efficient Strategies

Linear Strategy

ImmediateSale

E-V Plane

Minimal expected cost

Risk-Reward Tradeoff: Mean-Variance

Œ

Variance as risk measure


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)


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


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


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



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



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



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!


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


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


Numerical Example

Respond only to up/down

Discretize state space of


Sample Trajectories of Adaptive Strategy

Aggressive in the money …


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. )


Extensions

Non-linear impact functions

Multiple securities („basket trading“)

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


Thank you very much for your attention!

Questions?

Optimal Adaptive Execution of Portfolio Transactions

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