Numerical methods and models in market risk and financial valuations Area
Numerical methods and models in market risk and financial valuations area This work is organized in two themes : (i) A novel numerical method to price options on many assets, (ii) The liquidity risk, the limit order book modeling and the market microstructure. First theme : Greedy algorithms and applications for solving partial differential equations in high dimension Many problems of interest for various applications (material sciences, finance, etc) involve high- dimensional partial differential equations (PDEs). The typical example in finance is the pricing of a basket option, which can be obtained by solving the Black-Scholes PDE with dimension the number of underlying assets. We propose to investigate an algorithm which has been recently proposed and ana- lyzed in [ACKM06, BLM09] to solve such problems and try to circumvent the curse of dimensionality. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum using a greedy algorithm. The resolution of high dimensional partial differential equations is highly related to the representation of high dimensional functions. In Chapter 1, we describe various linear approaches existing in literature to represent high dimensional functions and we introduce the high dimensional problems in finance that we will address in this work. The method studied in this manuscript is a non-linear approximation method called the Proper Generalized Decomposition. Chapter 2 shows the application of this method to approximate the so- lution of a linear PDE (the Poisson problem) and also to approximate a square integrable function by a sum of tensor products. A numerical study of this last problem is presented in Chapter 3. The Poisson problem and the approximation of a square integrable function will serve as basis in Chapter 4 for solving the Black-Scholes equation using the PGD approach. In numerical experiments, we obtain results for up to 10 underlyings. Besides the approximation of the solution to the Black-Scholes equation, we propose a variance reduction method, which permits an important reduction of the variance of the Monte Carlo method for option pricing. Second theme : Liquidity risk, limit order book modeling and market microstructure Liquidity risk and market microstructure have become in the past years an important topic in mathematical finance. One possible reason is the deregulation of markets and the competition between them to try to attract as many investors as possible. Thus, quotation rules are changing and, in general, more information is available. In particular, it is possible to know at each time the awaiting orders on some stocks and to have a record of all the past transactions. In this work we study how to use this information to optimally execute buy or sell orders, which is linked to the traders' behaviour that want to minimize their trading cost. The structure of Limit Order Books (LOB) is very complex. Orders can only be made on a price grid. At each time, the number of waiting buy (or sell) orders for each price is stored. For a given price, orders are executed according to the First In First Out rule, as soon as two orders match together. Thus, since it is really complex, an exhaustive modeling of the LOB dynamics would not lead, for example, to draw conclusions on an optimal trading strategy. One has therefore to propose models that can grasp important features of the LOB structure but that allow to find analytical results. In [AFS10], Alfonsi, Fruth and Schied have proposed a simple LOB model. In this model, it is possible to explicitly derive the optimal strategy for buying (or selling) a given amount of shares before a given deadline. Basically, one has to split the large buy (or sell) order into smaller ones in order to find the best trade-off between attracting new orders and the price of the orders. Here, we focus on an extension of the Limit Order Book (LOB) model with general shape in- troduced by Alfonsi, Fruth and Schied. The additional feature is a time-varying LOB depth that represents a new feature of the LOB highlighted in [JJ88, GM92, HH95, KW96]. We solve the op- timal execution problem in this framework for both discrete and continuous time strategies. This gives in particular sufficient conditions to exclude Price Manipulations in the sense of Huberman and Stanzl [HS04] or Transaction-Triggered Price Manipulations (see Alfonsi, Schied and Slynko). These conditions give interesting qualitative insights on how market makers may create price manipulations.