A convex duality method for optimal liquidation with participation constraints

In spite of the growing consideration for optimal execution in the financial mathematics literature, numerical approximations of optimal trading curves are almost never discussed. In this article, we present a numerical method to approximate the optimal strategy of a trader willing to unwind a large portfolio. The method we propose is very general as it can be applied to multi-asset portfolios with any form of execution costs, including a bid-ask spread component, even when participation constraints are imposed. Our method, based on convex duality, only requires Hamiltonian functions to have $C^{1,1}$ regularity while classical methods require additional regularity and cannot be applied to all cases found in practice

Optimal Real-Time Bidding Strategies

The ad-trading desks of media-buying agencies are increasingly relying on complex algorithms for purchasing advertising inventory. In particular, Real-Time Bidding (RTB) algorithms respond to many auctions -- usually Vickrey auctions -- throughout the day for buying ad-inventory with the aim of maximizing one or several key performance indicators (KPI). The optimization problems faced by companies building bidding strategies are new and interesting for the community of applied mathematicians. In this article, we introduce a stochastic optimal control model that addresses the question of the optimal bidding strategy in various realistic contexts: the maximization of the inventory bought with a given amount of cash in the framework of audience strategies, the maximization of the number of conversions/acquisitions with a given amount of cash, etc. In our model, the sequence of auctions is modeled by a Poisson process and the \textit{price to beat} for each auction is modeled by a random variable following almost any probability distribution. We show that the optimal bids are characterized by a Hamilton-Jacobi-Bellman equation, and that almost-closed form solutions can be found by using a fluid limit. Numerical examples are also carried out

Ecological intuition versus economic "reason"

This article discusses the discount rate to be used in projects that aimed at improving the environment. The model has two different goods, one is the usual consumption good whose production may increase exponentially, the other is an environmental good whose quality remains limited. The stylized world we describe is fully determined by four parameters, reflecting basic preferences "ecological" and intergenerational concerns and feasibility constraints. We define an ecological discount rate and examine its connections with the usual interest rate and the optimized growth rate. We discuss, in this simple world, a variety of forms of the precautionary principle.discount rate ; ecological discount rate ; environmental goods ; relative prices ; irreversible damage ; precautionnary principle

Mean Field Games and Applications.

This text is inspired from a “Cours Bachelier” held in January 2009 and taught by Jean-Michel Lasry. This course was based upon the articles of the three authors and upon unpublished materials they developed. Proofs were not presented during the conferences and are now available. So are some issues that were only rapidly tackled during class.Mean Field Games;

Large investor trading impacts on volatility

International audienc