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Optimal Adaptive Execution of Portfolio Transactions

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Optimal Adaptive Execution of Portfolio Transactions

Julian Lorenz

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

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?

- Discrete times

- Stock price follows random walk

- Sell program

for initial position of X shares

s.t.

,

- Execution strategy:

= shares hold at time

i.e.

sell shares between t0 and t1

t1 and t2

…

- Pure sell program:

X=x0=100

N=10

Benchmark: Pre-Trade Book Value

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

C() is independent of S0

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

with

Linear Temporary Market Impact

x

x

x(t)

x(t)

X

X

t

t

T

T

Œ

Minimal Risk

Obviously by immediate liquidation

No risk, but high market impact cost

Minimal Expected Cost

Linear strategy

ð

But: High exposure to price volatility

High risk

Optimal trade schedules seek risk-reward balance

Random variable!

Risk-Reward Tradeoff: Mean-Variance

Minimal expected cost

Œ

Minimal variance

Variance as risk measure

E-V Plane

Admissible Strategies

Linear Strategy

ImmediateSale

Efficient Strategies

Œ

R. Almgren, N. Chriss: "Optimal execution of portfolio transactions", Journal of Risk (2000).

Deterministic trading strategy

ð

functions of decision variables (x1,…,xN)

Almgren/Chriss Trajectories:

Dynamic strategies:

xi = xi(1,…,i-1)

xi deterministic

E-V Plane

x(t)

X

T=1, =10

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

We show:

ð

C() normally distributed

t

T

ð

Straightforward QP

x(t)

x(t)

X

t

T

Deterministic

Trajectories

for some

ð

By dynamic programming

Urgency controls curvature

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

Admissible trading strategies for expected cost

adapted strategiesfor 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“

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

…ultimately interested in

?

For type “ “ DP is straightforward.

Here: in value function & terminal constraint …

Define value function

Œ

In current period sell shares at

Use efficient strategy for remaining k-1 periods

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

ð

Specify by its expected cost z()

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 x ~ N(0,2)

Construct optimal strategy for k periods

Conditional on :

Using the laws of total expectation and variance

One-step optimization of and by means of and

ð

Strategy defined by control and control functionz()

Theorem:

where

Control variablenew stock holding

(i.e. sell x – x’ in this period)

Control functiontargeted cost as function of next price change

ð

Solve recursively!

- 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}.

„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

- Respond only to up/down

- Discretize state space of

Aggressive in the money …

Family of frontiers

parametrized by

size of trade X

Sample cost PDFs:

Adaptive

strategies

Larger improvement for large portfolios

Almgren/Chriss deterministic strategy

(i.e. )

Almgren/Chriss frontier

Distribution plots obtained by Monte Carlo simulation

Improved frontiers

- Non-linear impact functions

- Multiple securities („basket trading“)

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