Robert Almgren

Optimal execution of portfolio transactions

We consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. For a simple linear cost model, we explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty. We may then select optimal strategies either by minimizing a quadratic utility function, or by minimizing Value at Risk. The latter choice leads to the concept of Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best tradeoff between volatility risk and liquidation costs. ∗We thank Andrew Alford, Alix Baudin, Mark Carhart, Ray Iwanowski, and Giorgio De Santis (Goldman Sachs Asset Management), Robert Ferstenberg (ITG), Michael Weber (Merrill Lynch), Andrew Lo (Sloan School, MIT), and George Constaninides (Graduate School of Business, University of Chicago) for helpful conversations. This paper was begun while the first author was at the University of Chicago, and the second author was first at Morgan Stanley Dean Witter and then at Goldman Sachs Asset Management. †University of Toronto, Departments of Mathematics and Computer Science; almgren@math.toronto.edu ‡ICor Brokerage and Courant Institute of Mathematical Sciences; Neil.Chriss@ICorBroker.com

Optimal execution with nonlinear impact functions and trading-enhanced risk

Optimal trading strategies are determined for liquidation of a large single-asset portfolio to minimize a combination of volatility risk and market impact costs. The market impact cost per share is taken to be a power law function of the trading rate, with an arbitrary positive exponent. This includes, for example, the square root law that has been proposed based on market microstructure theory. In analogy to the linear model, a ‘characteristic time’ for optimal trading is defined, which now depends on the initial portfolio size and decreases as execution proceeds. A model is also considered in which uncertainty of the realized price is increased by demanding rapid execution; it is shown that optimal trajectories are described by a ‘critical portfolio size’ above which this effect is dominant and below which it may be neglected.

Optimal Trading with Stochastic Liquidity and Volatility

We consider the problem of mean-variance optimal agency execution strategies, when the market li- quidity and volatility vary randomly in time. Under specific assumptions for the stochastic processes satisfied by these parameters, we construct a Hamilton-Jacobi-Bellman equation for the optimal cost and strategy. We solve this equation numerically and illustrate optimal strategies for varying risk aversion. These strategies adapt optimally to the instantaneous variations of market quality.

Variational algorithms and pattern formation in dendritic solidification

We present a completely new variational algorithm for computing dendritic solidification. This algorithm reproduces the Gibbs-Thomson relation as a balance between bulk and surface energy and is able to operate in the infinite-mobility limit with no unphysical time-step restriction. It may be used with arbitrary non-smooth surface energy functions and may include finite kinetic mobility. We perform computations with isotropic and anisotropic surface energy; from a small irregular initial seed we generate radial tip-splitting structures for isotropic energy and parabolic dendrites with side-branching for anisotropic energy. For anisotropic energy, the final structure is determined by the material and environmental properties; the initial shape is forgotten. For the parabolic dendrite tips, we obtain agreement with the Ivantsov solutions within a few percent and proper dimensional scaling of lengths and velocities with surface energy.

Singularity formation in Hele–Shaw bubbles

We provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two‐dimensional Hele–Shaw cell with no air injection. Constantin and Pugh have proved that no such singularity is possible if the initial shape is close to a circle; thus we show that their result is not true in general. Our evidence takes the form of direct numerical simulation of the full problem, including a careful assessment of the effects of limited spatial resolution, and comparison of the full problem with the lubrication approximation.

Stable and unstable singularities in the unforced Hele‐Shaw cell

We study singularity formation in the lubrication model for the unforced Hele‐Shaw system, describing the breaking in two of a fluid droplet confined between two narrowly spaced glass plates. By varying the initial data, we exhibit four different scenarios: (1) the droplet breaks in finite time, with two pinch points moving toward each other and merging at the singular time; (2) the droplet breaks in finite time, with two asymmetric pinch points propagating away from each other; (3) the droplet breaks in finite time, with a single symmetric pinch point; or (4) the droplet relaxes to a stable equilibrium shape without a finite time breakup. Each of the three singular scenarios has a self‐similar structure with different scaling laws; the first scenario has not been observed before in other Hele‐Shaw studies. We demonstrate instabilities of the second and third scenarios, in which the solution changes its behavior at a thickness that can be arbitrarily small depending on the initial condition. These transit...

Mean–Variance Optimal Adaptive Execution

Abstract Electronic trading of equities and other securities makes heavy use of ‘arrival price’ algorithms that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, mean–variance optimal trading strategies are static: they do not modify the execution speed in response to price motions observed during trading. We show that substantial improvement is possible by using dynamic trading strategies and that the improvement is larger for large initial positions. We develop a technique for computing optimal dynamic strategies to any desired degree of precision. The asset price process is observed on a discrete tree with an arbitrary number of levels. We introduce a novel dynamic programming technique in which the control variables are not only the shares traded at each time step but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. The resulting adaptive strategies are ‘aggressive-in-the-money’: they accelerate the execution when the price moves in the traders favor, spending parts of the trading gains to reduce risk.

CROSSOVER SCALING IN DENDRITIC EVOLUTION AT LOW UNDERCOOLING

We examine scaling in two-dimensional simulations of dendritic growth at low undercooling, as well as in three-dimensional pivalic acid dendrites grown on NASA’s USMP-4 isothermal dendritic growth experiment. We report new results on self-affine evolution in both the experiments and simulations. We find that the time-dependent scaling of our low undercooling simulations displays a crossover scaling from a regime different than that characterizing Laplacian growth to steady-state growth. [S0031-9007(99)09307-2]

Option Hedging with Smooth Market Impact

We consider intraday hedging of an option position, for a large trader who experiences temporary and permanent market impact. We formulate the general model including overnight risk, and solve explicitly in two cases which we believe are representative. The first case is an option with approximately constant gamma: the optimal hedge trades smoothly towards the classical Black–Scholes delta, with trading intensity proportional to instantaneous mishedge and inversely proportional to illiquidity. The second case is an arbitrary non-linear option structure but with no permanent impact: the optimal hedge trades toward a value offset from the Black–Scholes delta. We estimate the effects produced on the public markets if a large collection of traders all hedge similar positions. We construct a stable hedge strategy with discrete time steps.

Financial derivatives and partial differential equations

1. ASSETS AND DERIVATIVES. Assets of all sorts are traded in financial markets: stocks and stock indices, foreign currencies, loan contracts with various interest rates, energy in many forms, agricultural products, precious metals, etc. The prices of these assets fluctuate, sometimes wildly. As an example, Figure 1 shows the price of IBM stock within a single day. The picture would look more or less the same across a month, a year, or a decade, though the axis scales would be different.