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Risk-Averse Adaptive Execution of Portfolio Transactions Julian Lorenz Institute of Theoretical Computer Science, ETH Zurich This is joint work with R. Almgren (Bank of America Securities, on leave from University of Toronto)

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Execution of Financial Decisions Portfolio optimization tells a great deal about investments that optimally balance risk and expected returns Markowitz, CAPM, … But how to implement them? How to sell out of a large or illiquid portfolio position within a given time horizon?

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Price Appreciation, Market Impact, Timing Risk Price appreciation, Timing risk Market impact trade fasttrade slowly We want to balance market risk and market impact. Market riskMarket impact We have to deal with …

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Benchmark: Arrival Price Benchmark: Implementation shortfall Other common benchmark: Market VWAP = value of position at time of decision-making - capture of trade Goal: Find optimized execution strategy This benchmark is also known as Arrival Price (i.e. price prevailing at decision-making). Arrival price Average price achieved

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Discrete Trading Model Trading is possible at N discrete times No interest on cash position A trading strategy is given by (x i ) i=0..N+1 where x k = #units hold at t=k (i.e. we sell n k =x k -x k+1 at price S k ) Boundary conditions: x 0 = X and x N+1 = 0 Price dynamics: Exogenous: Arithmetic Random Walk S k = S k-1 + ( k + ), k=1..N with k i.i.d Endogenous: Market Impact -Permanent -Temporary

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Permanent vs. Temporary Market Impact Simplified model of market impact:

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Permanent vs. Temporary Market Impact Permanent market impact with k i.i.d Temporary market impact with k i.i.d (Quadratic cost model) Simplest case: Linear impact functions

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Shortfall of a Trading Strategy The capture of a trading strategy (x i ) i=1..N is with n k =x k-1 -x k. Assuming linear impact, the implementation shortfall is In fact, permanent impact is fairly easy tractable. Hence, we will focus on temporary impact.

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Mean-Variance Optimization ¸ 0 is the Lagrange multiplier or can be seen as a measure of risk aversion by itself. In the spirit of Markowitz portfolio optimization, we want to optimize The Lagrangian for this problem is

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Efficient Trading Frontier Similar to portfolio optimization, this leads to an efficient frontier of trading strategies:

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Bibliography This is the model as first proposed in R. Almgren, N. Chriss: "Optimal execution of portfolio transactions", Journal of Risk 3, It was extended in a series of publications, e.g. Konishi, Makimoto: Optimal slice of a block trade, Journal of Risk, Almgren, Chriss: "Bidding principles, Risk, Almgren: "Optimal execution with nonlinear impact functions and trading-enhanced risk", Applied Mathematical Finance, Huberman, Stanzl: Optimal Liquidity Trading, Review of Finance, 2005.

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Response of Finance Industry Neil Chriss and Robert Almgren pioneered much of the early research in the field... [The] efficient trading frontier will truly revolutionize financial decision-making for years to come. Robert Kissell and Morton Glantz, Optimal Trading Strategies, 2003 Almgren's paper, […] coauthored with Neil Chriss, head of quantitative strategies for giant hedge fund SAC Capital Advisors, helped lay the groundwork for the arrival-price algorithms currently being developed on Wall Street. Justin Shack, The orders of battle", Institutional Investor, 2004 The model has been remarkably influential in the finance industry:

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Optimal Static Trading Strategies (I) Almgren/Chriss brought up arguments, why in this arithmetic Brownian motion setting together with mean-variance utility, an optimal trading strategy would not depend on the stock price process. They therefore considered the model, where x k are static variables. Then

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Optimal Static Trading Strategies (II) Then convex minimization problem in x 1,…,x N with solution becomes a straightforward But is x k really path-independent?

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Binomial Model (I) Consider the following arithmetic binomial model: (S 0, X) sell (X-x 1 ) ( S 0 -, x 1 ) sell (x 1 - x 2 + ) sell (x 1 - x 2 - ) ( S 0 +, x 1 ) ( S 0 – 2, x 2 - ) ( S 0, x 2 - ) ( S 0 +2, x 2 + ) ( S 0, x 2 + ) sell x 2 + sell x 2 - Then we have the shortfall

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Binomial Model (II) A trading strategy is defined by (x 1,x 2 +,x 2 - ) For the variance we have to deal with path dependent stock holdings x 2 and with covariances, e.g.. One calculates (with and ) The path-independent solution forces = 0 with optimum For < 0, first order decrease in variance ( ) and only second-order increase in expectation. ) Path-independent solution is non-optimal. )

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Binomial Model (III) Intuition? Suppose price moves up: How to compute optimal path-dependent strategies? In fact, Optimal Execution can be seen as a multiperiod portfolio optimization problem with quadratic transaction costs and the additional constraint that at the end we are only allowed to hold cash. Less than anticipated cost (investment gain) Sell faster and allow to burn off some of the profit Increase in cost anticorrelated with investment gain

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Continuous Time Continuous-time formulation: Strategy v(t) must be adapted to the filtration of B. s.t. We would like to use dynamic programming, but variance doesnt directly fit into expected utility framework.

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Mean-Variance and Expected Utility Theorem: Corollary:

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Dynamic Programming (I) Hence, mean-variance optimization is essentially equivalent to minimizing expectation of the utility function. Value function at t in state (x,y,s) Terminal utility function { Force complete liquidation There is only terminal utility, no consumption process.

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Dynamic Programming (II) The HJB-Equation for the process leads to with the optimal trade rate. With =T-t we get the final PDE that is to be solved for >0:

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Further Research Directions Find explicit analytic solutions for strategies Multiple-security portfolios (with correlations), basket trading Nonlinear impact functions Other stochastic processes for security e.g. geometric Brownian motion … Ongoing work: Summary: We showed that the path-independent trading strategies given by Almgren/Chriss can be further improved. Using the dynamic programming paradigm, we derived a PDE which characterizes optimal adaptive strategies.

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