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 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 Larger improvement for large portfolios Almgren/Chriss deterministic strategy (i.e. ) Almgren/Chriss frontier Distribution plots obtained by Monte Carlo simulation 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?

Optimal Adaptive Execution of Portfolio Transactions