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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**

We have to deal with … Market risk Market impact Price appreciation, Timing risk Market impact trade fast trade slowly We want to balance market risk and market impact.

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**Benchmark: Arrival Price**

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

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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 (xi)i=0..N+1 where xk = #units hold at t=k (i.e. we sell nk=xk-xk+1 at price Sk) Boundary conditions: x0 = X and xN+1 = 0 Price dynamics: Exogenous: Arithmetic Random Walk Sk = Sk-1 + (k+), k=1..N with k ~ N(0,1) 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 ~ N(0,1) i.i.d Temporary market impact Simplest case: Linear impact functions („Quadratic cost model“)

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**Shortfall of a Trading Strategy**

In fact, permanent impact is fairly easy tractable. Hence, we will focus on temporary impact. The capture of a trading strategy (xi)i=1..N is with nk=xk-1-xk. Assuming linear impact, the implementation shortfall is

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**Mean-Variance Optimization**

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

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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, 2000. It was extended in a series of publications, e.g. Konishi, Makimoto: “Optimal slice of a block trade”, Journal of Risk, 2001. Almgren, Chriss: "Bidding principles“, Risk, 2003. Almgren: "Optimal execution with nonlinear impact functions and trading-enhanced risk", Applied Mathematical Finance, 2003. Huberman, Stanzl: “Optimal Liquidity Trading”, Review of Finance, 2005.

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**Response of Finance Industry**

The model has been remarkably influential in the 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

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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 xk are static variables. Then

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**Optimal Static Trading Strategies (II)**

Then becomes a straightforward convex minimization problem in x1,…,xN with solution But is xk really path-independent?

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**Binomial Model (I) Consider the following arithmetic binomial model:**

sell x2+ (S0+2, x2+) (S0+, x1) 2 sell (x1 - x2+) (S0, x2+) sell (X-x1) sell x2+ 1 sell x2- (S0, X) sell (x1 - x2-) (S0, x2-) 2 (S0 - , x1) (S0 – 2, x2-) sell x2- Then we have the shortfall

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**Binomial Model (II) A trading strategy is defined by (x1,x2+,x2-)**

For the variance we have to deal with path dependent stock holdings x2 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:**

Less than anticipated cost (investment gain) Sell faster and allow to burn off some of the profit Increase in cost anticorrelated with investment gain 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.

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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 doesn‘t 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**

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. Ongoing work: 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 …

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