The Mathematics of Optimal Execution

The goal of the course is to present a generalization of the seminal model of Almgren and Chriss and to show how it can be used in practice in various fields: optimal execution, option pricing and hedging, the pricing of block trades, asset management, etc. Initially built to tackle optimal execution issues, the Almgren-Chriss model can be used (i) in the cash equity area for designing the trading curve of IS orders, but also of Target Close orders and POV orders, and (ii) outside of the cash equity area for hedging and pricing options and for solving optimal portfolio issues. The course will rely on stochastic calculus, stochastic and deterministic optimal control, and will also use ideas coming from Bayesian learning. Part of the course will be based on the book "The Financial Mathematics of Market Liquidity: From Optimal Execution to Market Making".


Lecture 1. THE ALMGREN-CHRISS MODEL REVISITED
(Tuesday October 25, 18:00-20:00, Lecture Theatre 340, Huxley Building)


In this first course, we will present a generalization of the Almgren-Chriss model, and show how the optimal trading curve of an IS order can be characterized by a simple Hamiltonian system. We will also discuss the limitations of the model, the numerical methods that can be used to approximate the trading curves, and how practitioners use this model in practice. Finally, we shall discuss the use of the Almgren-Chriss framework for executing Target Close and POV orders in an optimal way.

 
Lecture 2. PRICING IN THE ALMGREN-CHRISS FRAMEWORK
(Wednesday October 26, 17:00-19:00, Lecture Theatre 308, Huxley Building)


Although the Almgren-Chriss model was built for finding optimal trading curves, it can also be used for pricing. In this second course, we shall address two problems: the pricing of large blocks of shares, and the pricing (and hedging) of vanilla options. Large blocks of shares should not be priced at their MtM value: the Almgren-Chriss framework can be used to define a risk-liquidity premium to be added/subtracted to the MtM price. Equity derivatives with large nominals or equity derivatives written on illiquid underlying assets should not be Delta-hedged (and priced) with a model à la Black and Scholes because execution costs are not taken into account. In this course, we will show how the Almgren-Chriss framework can be used for addressing option pricing and hedging issues.


Lecture 3. ASSET MANAGEMENT WITH EXECUTION COSTS
(Wednesday November 2, 17:00-19:00, Lecture Theatre 308, Huxley Building)

 
Classical asset management models do not account for execution costs. In this third lecture, we will show how the Almgren-Chriss framework can be used to solve portfolio choice and portfolio transition issues. We will also present how Bayesian learning techniques can be used in order to improve the robustness of optimal portfolio choice models, both in the classical framework à la Merton and in the Almgren-Chriss framework.

 

                                              ***The Mathematics of Optimal Execution SLIDES CAN BE DOWNLOADED HERE***