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

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?

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.

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

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

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

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“)

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

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.

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

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.

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

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

Optimal Static Trading Strategies (II) Then becomes a straightforward convex minimization problem in x1,…,xN with solution But is xk really path-independent?

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

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.

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.

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.

Mean-Variance and Expected Utility Theorem: Corollary:

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.

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:

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 …