Optimal investment decisions often rely on assumptions about the models and their associated parameter values. Therefore, it is essential to assess suitability of these assumptions and to understand sensitivity of outcomes when they are altered. More importantly, appropriate approaches should be developed to achieve a robust decision. In this thesis, we carry out a sensitivity analysis on parameter values as well as model speci cation of an important problem in portfolio management, namely the optimal portfolio execution problem. We then propose more robust solution techniques and models to achieve greater reliability on the performance of an optimal execution strategy. The optimal portfolio execution problem yields an execution strategy to liquidate large blocks of assets over a given execution horizon to minimize the mean of the execution cost and risk in execution. For large-volume trades, a major component of the execution cost comes from price impact. The optimal execution strategy then depends on the market price dynamics, the execution price model, the price impact model, as well as the choice of the risk measure. In this study, rst, sensitivity of the optimal execution strategy to estimation errors in the price impact parameters is analyzed, when a deterministic strategy is sought to minimize the mean and variance of the execution cost. An upper bound on the size of change in the solution is provided, which indicates the contributing factors to sensitivity of an optimal execution strategy. Our results show that the optimal execution strategy and the e cient frontier may be quite sensitive to perturbations in the price impact parameters. Motivated by our sensitivity results, a regularized robust optimization approach is devised when the price impact parameters belong to some uncertainty set. We rst illustrate that the classical robust optimization might be unstable to variation in the uncertainty set. To achieve greater stability, the proposed approach imposes a regularization constraint on the uncertainty set before being used in the minimax optimization formulation. Improvement in the stability of the robust solution is discussed and some implications of the regularization on the robust solution are studied. Sensitivity of the optimal execution strategy to market price dynamics is then investigated. We provide arguments that jump di usion models using compound poisson processes naturally model uncertain price impact of other large trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under jump di usion models and compare them with the optimal execution strategies obtained from a di usion process. A jump di usion model for the market price dynamics suggests the use of Conditional Value-at-Risk (CVaR) as the risk measure. Using Monte Carlo simulations, a smoothing technique, and a parametric representation of a stochastic strategy, we investigate an approach to minimize the mean and CVaR of the execution cost. The devised approach can further handle constraints using a smoothed exact penalty function.

In computational finance, finite differences are a widely used tool in the valuation of standard derivative contracts. In a lower-dimensional setting, high accuracy and speed often characterize such methods, which gives them a competitive advantage against Monte Carlo methods. For option contracts with discontinuous payoff functions, however, finite differences encounter problems to maintain the order of convergence of the employed finite difference scheme. Therefore the timesteps are often computed in a conservative manner, which might increase the total execution time of the solver more than necessary.     It can be shown that for American call options written on dividend paying stocks, it may be optimal to exercise the option right before a dividend is paid out. The result is that yet another discontinuity is introduced in the solution and the timestep is often reduced to preserve the intrinsic convergence order. However, it is thought that at least in theory the optimal length of the timestep is an increasing function of the time elapsed since the last discontinuity occured. The objective thus becomes that of finding an explicit method for adjusting the timestep both at the dividend instants and between dividend instants. Keeping the discretization in space constant leads to a time non-equidistant finite difference problem.     The aim of this thesis is to propose a time non-equidistant numerical finite difference algorithm for valuation of American call options on stocks with dividends known in advance. In particular, an explicit formula is proposed for computing timesteps at the dividend instants and between dividend payments given a user-specified error tolerance. A portion of the report is also devoted to numerical stabilization techniques that are applied to maintain the convergence order, including Rannacher time-marching and mollification.

Optimal investment decisions often rely on assumptions about the models and their associated parameter values. Therefore, it is essential to assess suitability of these assumptions and to understand sensitivity of outcomes when they are altered. More importantly, appropriate approaches should be developed to achieve a robust decision. In this thesis, we carry out a sensitivity analysis on parameter values as well as model speci cation of an important problem in portfolio management, namely the optimal portfolio execution problem. We then propose more robust solution techniques and models to achieve greater reliability on the performance of an optimal execution strategy. The optimal portfolio execution problem yields an execution strategy to liquidate large blocks of assets over a given execution horizon to minimize the mean of the execution cost and risk in execution. For large-volume trades, a major component of the execution cost comes from price impact. The optimal execution strategy then depends on the market price dynamics, the execution price model, the price impact model, as well as the choice of the risk measure. In this study, rst, sensitivity of the optimal execution strategy to estimation errors in the price impact parameters is analyzed, when a deterministic strategy is sought to minimize the mean and variance of the execution cost. An upper bound on the size of change in the solution is provided, which indicates the contributing factors to sensitivity of an optimal execution strategy. Our results show that the optimal execution strategy and the e cient frontier may be quite sensitive to perturbations in the price impact parameters. Motivated by our sensitivity results, a regularized robust optimization approach is devised when the price impact parameters belong to some uncertainty set. We rst illustrate that the classical robust optimization might be unstable to variation in the uncertainty set. To achieve greater stability, the proposed approach imposes a regularization constraint on the uncertainty set before being used in the minimax optimization formulation. Improvement in the stability of the robust solution is discussed and some implications of the regularization on the robust solution are studied. Sensitivity of the optimal execution strategy to market price dynamics is then investigated. We provide arguments that jump di usion models using compound poisson processes naturally model uncertain price impact of other large trades. Using stochastic dynamic programming, we derive analytical solutions for minimizing the expected execution cost under jump di usion models and compare them with the optimal execution strategies obtained from a di usion process. A jump di usion model for the market price dynamics suggests the use of Conditional Value-at-Risk (CVaR) as the risk measure. Using Monte Carlo simulations, a smoothing technique, and a parametric representation of a stochastic strategy, we investigate an approach to minimize the mean and CVaR of the execution cost. The devised approach can further handle constraints using a smoothed exact penalty function.

In this article the Finite Difference method is used to solve the Black Scholes equation. A second order and fourth order accurate scheme is implemented in space and evaluated. The scheme is then tried for different initial conditions. First the discontinuous pay off function of a European Call option is used. Due to the nonsmooth charac- teristics of the chosen initial conditions both schemes show an order of two. Next, the analytical solution to the Black Scholes is used when t=T/2. In this case, with a smooth initial condition, the fourth order scheme shows an order of four. Finally, the initial nonsmooth pay off function is modified by smoothing. Also in this case, the fourth order method shows an order of convergence of four.

In this manuscript, we develop a nite liquidity framework for two-asset markets. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity models. We adopt Euler- Maruyama and Milstein scheme in the simulation of asset prices. Exchange and Spread option values are numerically estimated by Monte Carlo with the Margrabe option as a controlled variate. The time complexity of these numerical schemes is included. Finally, we provide some deep learning frameworks to implement these pricing models effectively. / Thesis / Master of Science (MSc)

Cette thèse porte sur deux aspects de la dépendance entre actifs financiers. La première partie concerne la dépendance entre vecteurs aléatoires. Le premier chapitre consiste en une comparaison d'algorithmes calculant l'application de transport optimal pour le coût quadratique entre deux probabilités sur R^n, éventuellement continues. Ces algorithmes permettent de calculer des couplages ayant une propriété de dépendance extrême, dits couplage de corrélation maximale, qui apparaissent naturellement dans la définition de mesures de risque multivariées. Le second chapitre propose une définition de la dépendance extrême entre vecteurs aléatoires s'appuyant sur la notion de covariogramme ; les couplages extrêmes sont caractérisés comme des couplages de corrélation maximale à modification linéaire d'une des marginales multivariées près. Une méthode numérique permettant de calculer ces couplages est fournie, et des applications au stress-test de dépendance pour l'allocation de portefeuille et la valorisation d'options européennes sur plusieurs sous-jacents sont détaillées. La dernière partie décrit la dépendance spatiale entre deux diffusions markoviennes, couplées à l'aide d'une fonction de corrélation dépendant de l'état des deux diffusions. Une EDP de Kolmogorov forward intégrée fait le lien entre la famille de copules spatiales de la diffusion et la fonction de corrélation. On étudie ensuite le problème de la dépendance spatiale atteignable par deux mouvements Browniens, et nos résultats montrent que certaines copules classiques ne permettent pas de décrire la dépendance stationnaire entre des mouvements Browniens couplés.

Nous établissons les liens asymptotique entre deux catégories de modèles à volatilité stochastique décrivant le même marché dérivé: - un modèle générique à volatilité stochastique instantanée (SInsV) , dont le système d'EDS est un chaos de Wiener formel, spécifié sans aucune variable d'état. - une classe à volatilité implicite stochastique glissante (SImpV), qui est un autre modèle de marché, décrivant explicitement la dynamique conjointe du sous-jacent et de la surface d'options Européennes associées. Chacune de ces connexions est atteinte couche par couche, entre un groupe de coefficients SInsV et un ensemble de differentielles SImpV (statiques et dynamiques). L'approche asymptotique conduit à ce que ces différentielles croisees soient prises à l'expiration zéro, au point ATM. Nous progressons d'une configuration simple, bi-dimensionnelle à sous-jacent unique, d'abord vers une configuration multi-dimensionnelle, puis vers un cadre à structure par terme. Nous exposons les contraintes structurelles de modélisation et l'asymétrie entre le problème direct (de SInsV vers SImpV) et inverse. Nous montrons que cette expansion asymptotique en chaos (ACE) est un outil puissant pour la conception et l'analyse de modèles. En se concentrant sur des modèles à volatilité locale et leurs extensions, nous comparons ACE avec la littérature et exhibons un biais systématique dans l'heuristique de Gatheral. Dans le contexte multi-dimensionnel, nous nous concentrons sur des paniers à poids stochastiques, pour lesquels ACE fournit des résultats intuitifs soulignant la recurrence naturelle. Dans l'environnement des taux d'intérêt, nous etablissons la première couche de descripteurs du smile pour les caplets, les swaptions et les options sur obligations, à la fois dans un cadre SV-HJM et un cadre SV-LMM. En outre, nous montrons que ACE peut être automatisé pour des modèles génériques, à n'importe quel ordre, sans calcul formel. L'intérêt de cet algorithme est démontré par le calcul manuel des 2eme et 3eme couches, dans un modèle générique SInsV bi-dimensionnel. Nous présentons le potentiel applicatif d'ACE pour la calibration, l'evaluation, la couverture ou à des fins d'arbitrage, illustré par des tests numériques sur le modèle CEV-SABR.

An assembled algorithm for pricing American options with absolute, discrete dividends using adaptive lattice methods is described. Considerations for hardware-conscious programming on both CPU and GPU platforms are discussed, to provide a foundation for the investigation of several approaches for deploying the program onto GPU architectures. The performance results of the approaches are compared to that of a central processing unit reference implementation, and to each other. In particular, an approach of designating subtrees to be calculated in parallel by allowing multiple calculation of overlapping elements is described. Among the examined methods, this attains the best performance results in a "realistic" region of calculation parameters. A fifteen- to thirty-fold improvement in performance over the CPU reference implementation is observed as the problem size grows sufficiently large.

In Finance, the use of Automated Trading Systems (ATS) on markets is growing every year and the trades generated by an algorithm now account for most of orders that arrive at stock exchanges (Kissell, 2020). Historically, these systems were based on advanced statistical methods and signal processing designed to extract trading signals from financial data. The recent success of Machine Learning has attracted the interest of the financial community. Reinforcement Learning is a subcategory of machine learning and has been broadly applied by investors and researchers in building trading systems (Kissell, 2020). In this thesis, we address the issue that deep reinforcement learning may be susceptible to sampling errors and over-fitting and propose a robust deep reinforcement learning method that integrates techniques from reinforcement learning and robust optimization. We back-test and compare the performance of the developed algorithm, Robust DDPG, with UBAH (Uniform Buy and Hold) benchmark and other RL algorithms and show that the robust algorithm of this research can reduce the downside risk of an investment strategy significantly and can ensure a safer path for the investor’s portfolio value.

Dans le cadre de ma thèse, je me suis intéressée à analyser mathématiquement un indicateur de rupture de volatilité très utilisé par les praticiens en salle de marché. L'indicateur Bandes de Bollinger appartient à la famille des méthodes dites d'analyse technique et donc repose exclusivement sur l'historique récente du cours considéré et un principe déduit des observations passées des marchés, indépendamment de tout modèle mathématique. Mon travail consiste à étudier les performances de cet indicateur dans un univers qui serait gouverné par des équations différentielles stochastiques (Black -Scholes) dont le coefficient de diffusion change sa valeur à un temps aléatoire inconnu et inobservable, pour un praticien désirant maximiser une fonction objectif (par exemple, une certaine utilité espérée de la valeur du portefeuille à une certaine maturité).