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). Execution of Portfolio Transactions. Sell 100,000 Microsoft shares today!. Broker/Trader. Fund Manager. Problem: Market impact.

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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)


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?


Market Model

  • 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

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

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

Trader‘s Dilemma

Random variable!


Risk-Reward Tradeoff: Mean-Variance



Minimal expected cost

Œ

Minimal variance

Efficient Strategies

Variance as risk measure

E-V Plane



Admissible Strategies

Linear Strategy

ImmediateSale

Efficient Strategies

Œ


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)


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

Almgren/Chriss Deterministic Trading (2/2)

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“

Definitions


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 …

Dynamic Programming (1/4)

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()

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 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

Dynamic Programming (3/4)

ð

Strategy  defined by control and control functionz()


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 

ð

Solve recursively!


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}.


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


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 frontiers

parametrized by

size of trade X

Sample cost PDFs:



Adaptive

strategies

Œ

Larger improvement for large portfolios

Almgren/Chriss deterministic strategy

(i.e. )

‹

Almgren/Chriss frontier

‹

Distribution plots obtained by Monte Carlo simulation

Œ



Improved frontiers


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?


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