16 research outputs found

    Numerical methods for LĂ©vy processes

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    We survey the use and limitations of some numerical methods for pricing derivative contracts in multidimensional geometric LĂ©vy model

    Essays in Quantitative Finance

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    This thesis contributes to the quantitative finance literature and consists of four research papers.Paper 1. This paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function that allows the model to simultaneously fit the implied volatility of commodity and interest rate options. Because liquid market prices are only available for options on commodity futures (not forwards), a convexity correction formula is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given cross-correlation structure between forward interest rates and commodity prices. When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), whereas in the case of futures, a (rapidly converging) iterative fitting procedure is presented. The cross-correlation fitting is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the “orthonormal Procrustes” problem. The calibration approach is demonstrated on market data for oil futures.Paper 2. This paper describes an efficient American Monte Carlo approach for pricing Bermudan swaptions in the LIBOR market model using the Stochastic Grid Bundling Method (SGBM) which is a regression-based Monte Carlo method in which the continuation value is projected onto a space in which the distribution is known. We demonstrate an algorithm to obtain accurate and tight lower–upper bound values without the need for the nested Monte Carlo simulations that are generally required for regression-based methods.Paper 3. The credit valuation adjustment (CVA) for over-the-counter derivatives are computed using the portfolio’s exposure over its lifetime. Usually, future exposure is approximated by Monte Carlo simulations. For derivatives that lack an analytical approximation for their mark-to-market (MtM) value, such as Bermudan swaptions, the standard practice is to use the regression functions from the least squares Monte Carlo method to approximate their simulated MtMs. However, such approximations have significant bias and noise, resulting in an inaccurate CVA charge. This paper extend the SGBM to efficiently compute expected exposure, potential future exposure, and CVA for Bermudan swaptions. A novel contribution of the paper is that it demonstrates how different measures, such as spot and terminal measures, can simultaneously be employed in the SGBM framework to significantly reduce the variance and bias.Paper 4. This paper presents an algorithm for simulation of options on Lévy driven assets. The simulation is performed on the inverse transition matrix of a discretised partial differential equation. We demonstrate how one can obtain accurate option prices and deltas on the variance gamma (VG) and CGMY model through finite element-based Monte Carlo simulations

    Approximation d’espérances conditionnelles guidée par le problème en optimisation stochastique multi-étapes

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    RÉSUMÉ: Dans cette thèse, nous considérons d’une façon générale la résolution de problèmes d’optimisation stochastique multi-étapes. Ces derniers apparaissent dans de nombreux domaines d’application tels que la finance, l’énergie, la logistique, le transport, la santé, etc. Ils sont généralement insolubles de façon exacte car ils contiennent des espérances mathématiques qui ne peuvent pas être calculées analytiquement. Il est donc nécessaire de considérer pour cela des méthodes numériques. Nous nous intéressons particulièrement aux méthodes de génération d’arbres de scénarios. Ceux-ci remplacent le processus stochastique sous-jacent au problème afin de ramener ce dernier à une taille raisonnable permettant sa résolution pratique. Numériquement, cela permet de remplacer les opérateurs d’espérance qui apparaissent dans la formulation originale du problème (et qui tiennent compte de toutes les scénarios possibles en les pondérant avec une certaine densité de probabilité), par des sommes finies qui, pour leur part, ne prennent en compte qu’un sous-ensemble de scénarios seulement. Cette approximation permet ensuite à un ordinateur de résoudre le problème discrétisé à l’aide de solveurs classiques d’optimisation. L’arbre de scénarios doit satisfaire un compromis entre la qualité d’approximation, qui voudrait que l’arbre soit le plus grand possible, et la complexité de résolution du problème discrétisé qui, à l’inverse, voudrait qu’il soit le plus petit possible. Alors que ce compromis est relativement facile à satisfaire pour les problèmes à deux étapes, il l’est beaucoup moins pour les problèmes multi-étapes (c.-à-d. à partir de trois étapes). Ceci est dû à la nécessité de considérer des structures d’arbres dont la taille (le nombre de noeuds) croît exponentiellement avec le nombre d’étapes. Dans ce contexte multi-étapes, la recherche d’un compromis satisfaisant entre qualité et complexité a mené la communauté d’optimisation stochastique à développer de nombreuses approches de génération d’arbres de scénarios basées sur des justifications théoriques ou pratiques différentes. Ces justifications portent essentiellement sur la qualité d’approximation du processus stochastique par l’arbre de scénarios. Pour cette raison, ces approches sont dites guidées par la distribution, étant donné qu’elles souhaitent reproduire le mieux possible –suivant leur propre critère de qualité– la distribution du processus stochastique (ou certaines propriétés de celle-ci). Prendre en compte la distribution permet sous certaines conditions assez faibles d’assurer la consistance de la méthode de résolution. Pour cette raison, ces méthodes sont utilisées avec succès dans de nombreux problèmes. Cependant, cette stratégie ne permet pas de tirer profit de la structure même du problème d’optimisation, par exemple la variabilité de sa fonction objectif ou l’influence de ses contraintes, qui joue aussi un rôle important dans la qualité d’approximation. La prise en compte de ces caractéristiques permettrait de construire des arbres de scénarios plus adaptés aux problèmes et ainsi de satisfaire un meilleur compromis entre qualité et complexité. En pratique, cela permettrait de pouvoir résoudre des problèmes avec un plus grand nombre d’étapes.----------ABSTRACT: In this thesis, we consider solution methods for general multistage stochastic optimization problems. Such problems arise in many fields of application, including finance, energy, logistic, transportation, health care, etc. They generally do not have closed-form solutions since they feature mathematical expectations, which cannot be computed exactly in most applications. For this reason, it is necessary to consider solutions through numerical methods. One of them, which is the focus of this thesis, is the scenario-tree generation approach. Its aim is to substitute the underlying stochastic process with a finite subset of scenarios so as to replace the conditional expectations with their finite sum estimators. This reduces the size of the problem, which is then solved using some generic optimization solvers. The generation of scenario trees is subject to a trade-off between the approximation accuracy and the complexity of the resulting discretized problem. The former tends to increase the number of scenarios, whereas the latter tends to decrease it. This trade-off turns out to be fairly easy to satisfy when dealing with two-stage problems. However, it becomes much more difficult when problems are multistage, that is, when they have 3 stages of more. This stems from the fact that multistage problems require specific tree structures whose size (the number of nodes) grow exponentially as the number of stages increases. For this reason, a lot of attention has been drawn on generating scenario trees in the multistage setting. Many methods have been developed based on different theoretical or practical grounds. Most of them can be described as distribution-driven, as they aim at approximating the distribution of the stochastic process (or some features of it), according to their own idea of what a good approximation is. The distribution-driven strategy allows to have consistent scenario-tree estimators under some weak conditions. For this reason, these methods have been successfully applied to many problems. However, it does not allow to capitalize on some specific features of the multistage problem (e.g., the variability of its revenue function or the influence of its constraints), although they play an important role in the scenario-tree approximation quality as well. Taking them into account would lead to more suitable scenario trees that may satisfy a better trade-off between accuracy and complexity. This, in turn, may allow to consider problems with more stages. In this thesis, we introduce a new problem-driven scenario-tree generation approach. It takes into account the whole structure of the optimization problem through its stochastic process, revenue (or cost) function and sets of constraints. This approach is developed in a general setting of multistage problems, hence it is not tied to a particular application or field of applications. The conditions that are introduced along the lines of this thesis about the revenue function, constraints, or probability distribution essentially aims at making sure that the problems is mathematically well-defined

    Numerical schemes and Monte Carlo techniques for Greeks in stochastic volatility models

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    The main objective of this thesis is to propose approximations to option sensitivities in stochastic volatility models. The first part explores sequential Monte Carlo techniques for approximating the latent state in a Hidden Markov Model. These techniques are applied to the computation of Greeks by adapting the likelihood ratio method. Convergence of the Greek estimates is proved and tracking of option prices is performed in a stochastic volatility model. The second part defines a class of approximate Greek weights and provides high-order approximations and justification for extrapolation techniques. Under certain regularity assumptions on the value function of the problem, Greek approximations are proved for a fully implementable Monte Carlo framework, using weak Taylor discretisation schemes. The variance and bias are studied for the Delta and Gamma, when using such discrete-time approximations. The final part of the thesis introduces a modified explicit Euler scheme for stochastic differential equations with non-Lipschitz continuous drift or diffusion; a strong rate of convergence is proved. The literature on discretisation techniques for stochastic differential equations has been motivational for the development of techniques preserving the explicitness of the algorithm. Stochastic differential equations in the mathematical finance literature, including the Cox-Ingersoll-Ross, the 3/2 and the Ait-Sahalia models can be discretised, with a strong rate of convergence proved, which is a requirement for multilevel Monte Carlo techniques.Open Acces
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