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15. 05. 2007 Optimal Adaptive Execution of Portfolio Transactions Julian Lorenz Joint work with Robert Almgren (Banc of America Securities, NY)

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 2 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 impact Trading Large Volumes Moves the Price

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 3 Market Model Stock price follows random walk Execution strategy: sell shares between t 0 and t 1 Sell program t 1 and t 2 … s.t. for initial position of X shares Pure sell program:, Discrete times = shares hold at time i.e.

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 4 Benchmark: Pre-Trade Book Value Cost C( ) = Pre-Trade Book Value – Capture of Trade C( ) is independent of S 0 Market Impact and Cost of a Strategy Linear Temporary Market Impact Selling x k-1 – x k shares in [t k-1, t k ] at discount to S k-1 with x x X=x 0 =100 N=10

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 5 Traders Dilemma Random variable! Optimal trade schedules seek risk-reward balance Obviously by immediate liquidation No risk, but high market impact cost Minimal Risk t T x(t) X Linear strategy Minimal Expected Cost But: High exposure to price volatility High risk t T x(t) X

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 6 Efficient Strategies Minimal variance Admissible Strategies Efficient Strategies Linear Strategy Immediate Sale E-V Plane Minimal expected cost Risk-Reward Tradeoff: Mean-Variance Variance as risk measure

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 7 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 (x 1,…,x N )

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 8 Almgren/Chriss Deterministic Trading (2/2) Deterministic Trajectories for some Dynamic strategies: x i = x i ( 1,…, i-1 ) Almgren/Chriss Trajectories: x i deterministic Dynamic strategies improve (w.r.t. mean-variance) ! We show: C( ) normally distributed Straightforward QP E-V Plane t T X x(t) t T X T=1, =10 x(t) Urgency controls curvature By dynamic programming

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 9 Definitions Adapted trading strategy: x i 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

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 10 Tail of Efficient Strategies Lemma: For all outcomes a of 1, the tail is efficient Suppose is efficient, Conditional on 1, define the tail of Dynamic programming! Note: deterministic, but may depend on 1

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 11 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

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 12 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 ~ N(0, 2 ) Construct optimal strategy for k periods In current period sell shares at Use efficient strategy for remaining k-1 periods Specify by its expected cost z( ) Note: must be deterministic, but when we begin, outcome of is known, i.e. we may choose depending on

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 13 Dynamic Programming (3/4) 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

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 14 Dynamic Programming (4/4) Theorem: where Control variable new stock holding (i.e. sell x – x in this period) Control function targeted cost as function of next price change Solve recursively!

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 15 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, z 1, …, z k }. Nice convexity property For fixed determine

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 16 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 costs to reduce total variance

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 17 Numerical Example Respond only to up/down Discretize state space of

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 18 Sample Trajectories of Adaptive Strategy Aggressive in the money …

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 19 Family of New Efficient Frontiers Frontiers are parametrized by Almgren/Chriss deterministic strategy Adaptive strategies Sample cost PDFs: Distribution plots obtained by Monte Carlo simulation (market power) Almgren/Chriss frontier Improved frontiers Larger improvement for large portfolios ( i.e. )

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 20 Family of New Efficient Frontiers Family of frontiers parametrized by size of trade X Almgren/Chriss deterministic strategy Adaptive strategies Sample cost PDFs: Distribution plots obtained by Monte Carlo simulation Almgren/Chriss frontier Improved frontiers Larger improvement for large portfolios (i.e. )

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 21 More Cost Distributions

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 22 Extensions Non-linear impact functions Multiple securities (basket trading) Dynamic Programming approach also applicable for other mean-variance problems, e.g. multiperiod portfolio optimization

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2007 Julian Lorenz, jlorenz@inf.ethz.ch 23 Thank you very much for your attention! Questions?

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