Neil Chriss

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

Complex Semisimple Groups

We begin this section by reviewing some basic facts about semisimple groups and Lie algebras which we will need in the rest of this book. For further information the reader is referred to [Bour], [Bo3], [Hum], [Se1], and [Di].

Equivariant K-Theory

This chapter is devoted to the fundamentals of equivariant algebraic K–theory. The reader interested mostly in the applications to representation theory may skip this chapter and use it only as a reference for later chapters. As has been explained in the introduction, most of the results here were proved by Thomason [Th1]–[Th4], sometimes in much greater generality. Our approach is however more elementary and in many places essentially different.

Representations of Convolution Algebras

Our primary goal in this chapter is to obtain a classification of simple modules over the affine Hecke algebra H although the techniques we develop works in much greater generality (we will indicate this on several occasions). In §8.1 we introduce a class of “standard” H-modules. In the same section we define simple H-modules in terms of a certain intersection form on standard module, and formulate the main classification theorem for simple H-modules. After a short overview of derived categories of constructible sheaves and intersection cohomology, given in §§8.3–8.4, we will express in §8.5 the underlying vector space of a simple H-module in terms of the intersection cohomology.

Springer Theory for u (sln)

One might ask whether the work of producing representations of Weyl groups by geometric means, carried out in the previous chapter, was worth-while. Our point is that absolutely the same machinery can be applied to construct representations of sln(C) and perhaps other semisimple Lie algebras, cf. [Na2]. Many of the objects we use for studying the sln(C)-case are analogous to the objects in the Weyl group case.

Hecke Algebras and K–Theory

Let R ⊂ P be a reduced (not necessarily finite) root system as defined, e.g., in 3.1.22. There is a slight difference with 3.1.22, since now we are working with lattices instead of vector spaces. This makes axiom 3.1.22(3) superfluous. Thus it is assumed only that, in addition to the above data, a subset R v ⊂ P v, called the dual root system, and a specified bijection R ↔ R v, α ↔ ᾰ are given such that the following three properties hold.

Flag Varieties, K-Theory, and Harmonic Polynomials

In this chapter we study some further properties of general complex semisimple groups. Most of the results of the result of the chapter play a crucial role in the representation theory of semisimple groups and Lie algebras. We have tried to assemble and give complete proofs for all those results that are, on the one hand, considered “too advanced” to be included in elementry text books on Lie algebras and, on the other hand are regarded as not part of representation theory itself. These latter results are generally assumed as prerequisites–not to be explained–in any advanced book on representation theory.

4 – Optimal portfolios from ordering information

Publisher Summary This chapter illustrates a method for portfolio optimization based on replacing expected returns with sorting criteria. The method is based on information about the order of the expected returns but not their values. In other words, the chapter presenst a framework for portfolio selection when an investor possesses information about the order of expected returns in the cross-section networks, but not the values of the expected returns. It mentions that modern portfolio theory produces an optimal portfolio from estimates of expected returns and a covariance matrix.. It also presents a simple and economically rational definition of optimal portfolios that extends Markowitz’ definition in a natural way; in particular, the construction allows full use of covariance information. Further, it provides efficient numerical algorithms for constructing optimal portfolios. The formulation is stated to be very general and is easily extended to more general cases, where assets are divided into multiple sectors or there are multiple sorting criteria available, and may be combined with transaction cost restrictions. It has three ingredients—ordering information which gives rise to a cone of consistent returns; a probability density within the belief cone that specifies belief about the relative probability of the actual expected return vector being any particular location within the belief cone; and a constraint set in which the portfolio is constrained to lie. Using both real and simulated data, dramatic improvement over simpler strategies is demonstrated.