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

Optimal Adaptive Execution of Portfolio Transactions

Julian Lorenz

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


Execution of portfolio transactions

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

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:


Market impact and cost of a strategy

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


Trader s dilemma

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!


Efficient strategies

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

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 deterministic trading 2 2

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


Definitions

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


Dynamic programming 1 4

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


Dynamic programming 2 4

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


Dynamic programming 3 4

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

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

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

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

Numerical Example

  • Respond only to up/down

  • Discretize state space of


Sample trajectories of adaptive strategy

Sample Trajectories of Adaptive Strategy

Aggressive in the money …


Family of new efficient frontiers

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

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

Thank you very much for your attention!Questions?


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