Numerical methods and models applied to market risks and financial valuation.

Summary This thesis addresses two topics: (i) The use of a new numerical method for the valuation of options on a basket of assets, (ii) Liquidity risk, order book modeling and market microstructure. First topic: A greedy algorithm and its applications to solve partial differential equations. The typical example in finance is the valuation of an option on a basket of assets, which can be obtained by solving the Black-Scholes PDE having as dimension the number of assets considered. We propose to study an algorithm that has been proposed and studied recently in [ACKM06, BLM09] to solve high dimensional problems and try to circumvent the curse of dimension. The idea is to represent the solution as a sum of tensor products and to iteratively compute the terms of this sum using a gluttonous algorithm. The solution of PDEs in high dimension is strongly related to the representation of functions in high dimension. In Chapter 1, we describe different approaches to represent high-dimensional functions and introduce the high-dimensional problems in finance that are addressed in this thesis work. The method selected in this manuscript is a nonlinear approximation method called Proper Generalized Decomposition (PGD). Chapter 2 shows the application of this method for the approximation of the solution of a linear PDE (the Poisson problem) and for the approximation of an integrable square function by a sum of tensor products. A numerical study of the latter problem is presented in Chapter 3. The Poisson problem and the approximation of an integrable square function will be used as a basis in Chapter 4 to solve the Black-Scholes equation using the PGD approach. In numerical examples, we have obtained results up to dimension 10. In addition to approximating the solution of the Black-Scholes equation, we propose a variance reduction method of classical Monte Carlo methods for pricing financial options. Second topic: Liquidity risk, order book modeling, market microstructure. Liquidity risk and market microstructure have become very important topics in financial mathematics. The deregulation of financial markets and the competition between them to attract more investors is one of the possible reasons. In this work, we study how to use this information to optimally execute the sale or purchase of orders. Orders can only be placed in a price grid. At each moment, the number of pending buy (or sell) orders for each price is recorded. In [AFS10], Alfonsi, Fruth and Schied proposed a simple model of the order book. In this model, it is possible to explicitly find the optimal strategy to buy (or sell) a given quantity of shares before a maturity. The idea is to split the buy (or sell) order into other smaller orders in order to find the balance between the acquisition of new orders and their price. This thesis work focuses on an extension of the order book model introduced by Alfonsi, Fruth and Schied. Here, the originality is to allow the depth of the order book to depend on time, which is a new feature of the order book that has been illustrated by [JJ88, GM92, HH95, KW96]. In this framework, we solve the optimal execution problem for discrete and continuous strategies. This gives us, in particular, sufficient conditions to exclude price manipulation in the sense of Huberman and Stanzl [HS04] or Transaction-Triggered Price Manipulation (see Alfonsi, Schied and Slynko).