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**Optimal Adaptive Execution of Portfolio Transactions**

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

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**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?

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**Market Model … Discrete times Stock price follows random walk**

Sell program for initial position of X shares Execution strategy: s.t. , = shares hold at time i.e. sell shares between t0 and t1 t1 and t2 … Pure sell program:

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**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 Benchmark: Pre-Trade Book Value Cost C() = Pre-Trade Book Value – Capture of Trade C() is independent of S0 x X=x0=100 N=10 x

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**Œ ð Trader‘s Dilemma Random variable! Minimal Risk**

Obviously by immediate liquidation No risk, but high market impact cost Minimal Risk Œ t T x(t) X Linear strategy Minimal Expected Cost But: High exposure to price volatility ð High risk t T x(t) X Optimal trade schedules seek risk-reward balance

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**Admissible Strategies**

Efficient Strategies Risk-Reward Tradeoff: Mean-Variance Variance as risk measure E-V Plane Minimal variance Œ Admissible Strategies Minimal expected cost Linear Strategy Immediate Sale Efficient Strategies Œ

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**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)

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**Almgren/Chriss Deterministic Trading (2/2)**

Trajectories for some E-V Plane t T X x(t) T=1, =10 Dynamic strategies: xi = xi(1,…,i-1) Almgren/Chriss Trajectories: xi deterministic C() normally distributed ð Straightforward QP ð By dynamic programming Urgency controls curvature Dynamic strategies improve (w.r.t. mean-variance) ! We show:

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

adapted strategies for X shares in N periods with expected cost Admissible trading strategies for expected cost Efficient trading strategies „no other admissible strategy offers lower variance for same level of expected cost“ i.e.

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**Tail of Efficient Strategies**

Suppose is efficient, Note: deterministic, but may depend on 1 Conditional on 1, define the “tail“ of Lemma: For all outcomes a of 1, the tail is efficient ð Dynamic programming!

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**Dynamic Programming (1/4)**

Define value function i.e. minimal variance to sell x shares in k periods with and optimal strategies for k-1 periods Optimal Markovian one-step control + and optimal strategies for k periods …ultimately interested in For type “ “ DP is straightforward. Here: in value function & terminal constraint … ?

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**Dynamic Programming (2/4)**

We want to determine k periods and x shares left Limit for expected cost is c Current stock price S Next price innovation is x ~ N(0,2) Situation: Construct optimal strategy for k periods In current period sell shares at Œ Use efficient strategy for remaining k-1 periods Specify by its expected cost z() ð Note: must be deterministic, but when we begin , outcome of is known, i.e. we may choose depending on

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**Dynamic Programming (3/4)**

ð 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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**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 ð Solve recursively!

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**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}.

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**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

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Numerical Example Respond only to up/down Discretize state space of

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**Sample Trajectories of Adaptive Strategy**

Aggressive in the money …

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**Family of New Efficient Frontiers**

Frontiers are parametrized by Sample cost PDFs: Adaptive strategies Almgren/Chriss deterministic strategy (“market power“) Larger improvement for large portfolios ( i.e ) Almgren/Chriss frontier Distribution plots obtained by Monte Carlo simulation Improved frontiers

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**Family of New Efficient Frontiers**

Family of frontiers parametrized by size of trade X Sample cost PDFs: Adaptive strategies Almgren/Chriss deterministic strategy Larger improvement for large portfolios (i.e ) Almgren/Chriss frontier Distribution plots obtained by Monte Carlo simulation Improved frontiers

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**More Cost Distributions**

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**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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**Thank you very much for your attention! Questions?**

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