Thesis (M.Sc. (Applied Mathematics)) -- University of Limpopo, 2020 / In this study,we consider two problems.The first one is the problem of computing hedging portfolios for options that may have discontinuous payoff functions.For this problem we use the Malliavin property called the Clark-Ocone formula and give some examples for diferent types of pay of functions of the options of European type.The second problem is based on the computation of price sensitivities (derivatives of the probabilistic representation of the pay off functions with respect to the underlying parameters of the model) also known as`Greeks' of discontinuous payoff functions and also give some examples.We restrict ourselves to the computation of Delta, Gamma and Vega.For this problem we make use of the properties of the Malliavin calculus like the integration by parts formula and the chain rule.We find the representations of the price sensitivities in terms of the expected value of the random variables that do not involve the actual direct differentiation of the payout function,that is, E[g(XT ) ] where g is a payoff function which depend on the stochasticdic differential equation XT at maturity time T and is the Malliavin weigh tfunction. In general, we study the regularity of the solutions of the stochastic differentia lequations in the sense of Malliavin calculus and explore its applications to Mathematical finance. / Services SETA and National Research Foundation (NRF)

Thesis (M. Sc.(Mathematics and Applied Mathematics))--University of Pretoria, 2009. / Summary in English. Includes bibliographical references.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

Thesis (MSc)--University of Stellenbosch, 2008. / Bibliography. Also available via the Internet.

This dissertation provides a brief theoretical introduction to the Malliavin calculus leading to a particular application in finance. The Malliavin calculus concepts are used to aid in the simulation of the Greeks for financial contingent claims. Particular focus is placed on creating efficiency in the more exotic type option simulations, where no closed solution pricing formulae exist. Copyright / Dissertation (MSc)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

Submitted by Camilla Antunes Silva (camillantunes@gmail.com) on 2018-08-03T17:19:02Z No. of bitstreams: 1 DissertacaoCamilla.pdf: 5244367 bytes, checksum: 9756ab39ef0bc845b78f13156bb9eee7 (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2018-09-03T18:50:34Z (GMT) No. of bitstreams: 1 DissertacaoCamilla.pdf: 5244367 bytes, checksum: 9756ab39ef0bc845b78f13156bb9eee7 (MD5) / Made available in DSpace on 2018-09-11T13:51:44Z (GMT). No. of bitstreams: 1 DissertacaoCamilla.pdf: 5244367 bytes, checksum: 9756ab39ef0bc845b78f13156bb9eee7 (MD5) Previous issue date: 2018-07-06 / Um dos métodos de análise de derivativos financeiros é o estudo de seu comportamento em relação a variações do seu ativo subjacente. Quando o payoff do derivativo não é uma função suave, temos problemas para calcular esse comportamento. Usamos o Cálculo de Malliavin para encontrar um método para calcular a primeira derivada do preço em relação ao valor inicial do ativo subjacente mesmo quando o payoff correspondente não é diferenciável. Para isso, estudamos a derivada de Malliavin e seu adjunto, a integral de Skorohod. / One of the methods of analyzing financial derivatives is the study of their behavior in relation to variations in the underlying asset. When the derivative payoff is not a smooth function, we have trouble calculating this behavior. We use Malliavin calculus to find a method to calculate the first derivative of price in relation to the initial value of the underlying asset even when the corresponding payoff is not differentiable. For this, we study the derivative of Malliavin and its adjoint, the Skorohod integral.

L'ensemble de ce travail est dédié à l'étude de certaines propriétés concernant les processus de sauts d-dimensionnels X = (Xt) dont le générateur est donné par Lψ(x) = 1/2 ∑ aᵤᵥ(x)∂²ψ(x)/∂xᵤ∂xᵥ + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) où µ est de masse totale infinie. Si γ ne dépendait pas de x, nous nous trouverions dans une situation classique où le processus X pourrait être représenté comme une solution d'une équation stochastique comportant une mesure ponctuelle de Poisson de mesure d'intensité γ(z)µ(dz) ; lorsque γ dépend de x, on peut s'en représenter l'heuristique en imaginant le processus comme la trajectoire d'une particule, la loi des sauts pouvant alors dépendre de la position de la particule. Dans la première partie, nous donnons des conditions pour obtenir l'existence et l'unicité de tels processus. Ensuite, nous considérons ce type de processus comme une généralisation des PDMP ; nous montrons qu'ils peuvent être vus comme une limite d'une suite (Xᵣ(t)) de PDMP standards pour lesquels l'intensité des sauts tend vers l'infini quand r tend vers l'infini, suivant deux régimes : un lent et un rapide qui, en supposant que les processus en question sont centrés et normalisés convenablement, produit une composante de diffusion à la limite. Finalement, on prouve la récurrence au sens de Harris de X en utilisant un schéma régénératif entièrement basé sur les sauts du processus. De plus, nous dégageons des conditions explicites par rapport aux coefficients du processus qui nous permettent de contrôler la vitesse de convergence vers l'équilibre en terme d'inégalités de déviation pour des fonctionnelles additives intégrables. Dans la seconde partie, nous considérons à nouveau le même type de processus X = (Xt(x)) partant du point x. Utilisant une approche basé sur un Calcul de Malliavin fini-dimensionnel, nous étudions la régularité jointe de ce processus dans le sens suivant : on fixe b≥1 et p>1, K un ensemble compact de Rᵈ, et nous donnons des conditions suffisantes pour avoir P(Xt(x)∈dy)=pt(x,y)dy avec (x,y)↦pt(x,y) appartenant à Wᵇᵖ(K×Rᵈ) / This work is dedicated to the study of some properties concerning the d-dimensional jump type diffusion X = (Xt) with infinitesimal generator given by Lψ(x) = 1/2 ∑ aᵤᵥ(x)∂²ψ(x)/∂xᵤ∂xᵥ + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. If γ did not depend on x, we would be in a classical situation where the process X could be represented as the solution of a stochastic equation driven by a Poisson point measure with intensity measure γ(z)µ(dz) ; when γ depends on x, we may have the heuristic idea that, if we were to imagine the process as a trajectory of a particle, the law of the jumps may depend on the position of the particle. In the first part, we give some conditions to obtain existence and uniqueness of such processes. Then, we consider this type of processes as a generalization of Piecewise Deterministic Markov Processes (PDMP) ; we show that they can be seen as a limit of a sequence (Xᵣ(t)) of standard PDMP's for which the intensity of the jumps tends to infinity as r tends to infinity, following two regimes: a slow one, which leads to a jump component with finite variation, and a rapid one which, supposing that the processes at hand are centered and renormalized in a convenient way, produces the diffusion component in the limit. Finally, we prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals. In the second part, we consider again the same type of process X = (Xt(x)) starting from x. Using an approach based on a finite dimensional Malliavin Calculus, we study the joint regularity of this process in the following sense : we fix b≥1 and p>1, K a compact set of Rᵈ, and we give sufficient conditions in order to have P(Xt(x)∈dy)=pt(x,y)dy with (x,y)↦pt(x,y) in Wᵇᵖ(K×Rᵈ)

De ce fait, le premier objectif de notre travail consiste à proposer des générateurs de nombres aléatoires appropriés pour des architectures parallèles et massivement parallèles de clusters de CPUs/GPUs. Nous testerons le gain en temps de calcul et l'énergie consommée lors de l'implémentation du cas linéaire du pricing européen. Le deuxième objectif est de reformuler le problème non-linéaire du pricing américain pour que l'on puisse avoir des gains de parallélisation semblables à ceux obtenus pour les problèmes linéaires. La méthode proposée fondée sur le calcul de Malliavin est aussi plus avantageuse du point de vue du praticien au delà même de l'intérêt intrinsèque lié à la possibilité d'une bonne parallélisation. Toujours dans l'objectif de proposer des algorithmes parallèles, le dernier point est l'étude de l'unicité de la solution de certains cas linéaires inverses en finance. Cette unicité aide en effet à avoir des algorithmes simples fondés sur Monte Carlo / Handling multidimensional parabolic linear, nonlinear and linear inverse problems is the main objective of this work. It is the multidimensional word that makes virtually inevitable the use of simulation methods based on Monte Carlo. This word also makes necessary the use of parallel architectures. Indeed, the problems dealing with a large number of assets are major resources consumers, and only parallelization is able to reduce their execution times. Consequently, the first goal of our work is to propose "appropriate" random number generators to parallel and massively parallel architecture implemented on CPUs/GPUs cluster. We quantify the speedup and the energy consumption of the parallel execution of a European pricing. The second objective is to reformulate the nonlinear problem of pricing American options in order to get the same parallelization gains as those obtained for linear problems. In addition to its parallelization suitability, the proposed method based on Malliavin calculus has other practical advantages. Continuing with parallel algorithms, the last point of this work is dedicated to the uniqueness of the solution of some linear inverse problems in finance. This theoretical study enables the use of simple methods based on Monte Carlo

Cette thèse se décompose en six chapitres plus ou moins distincts. Cependant, tous font appel au calcul de Malliavin, aux notions de processus gaussien et processus de Lévy, et à leur utilisation en statistique. Chacune des trois parties a fait l'objet de deux articles. <br />Dans la première partie, nous établissons les théorèmes d'Itô et deTanaka pour le mouvement brownien bifractionnaire multidimensionnel. Ensuite nous étudions l'existence de la densité d'occupation pour certains processus en relation avec le mouvement brownien fractionnaire.<br />Dans la deuxième partie, nous analysons, dans un premier temps, le comportement asymptotique de la variation cubique pour le processus de Rosenblatt. Dans un deuxième temps, nous construisons d'une part des estimateurs efficace pour la dérive de mouvement brownien fractionnaire et d'autre part des estimateurs biaisés de type James-Stein qui dominent, sous le riqsue quadratique usuel, l'estimateur du maximum de vraisemblance.<br />La dernière partie présente deux travaux. Dans le premier, nous utilisons une approche menant à un calcul de Malliavin pour les processus de Lévy, qui a été développée récemment par Solé et al. , et nous étudions des processus anticipés de type intégrale d'Itô-Skorohod sur l'espace de Lévy. Dans le deuxième, nous étudions le lien entre les processus stables et les processus auto-similaires, à travers des processus qui sont infiniment divisibles en temps.

In this thesis we propose a new approach to solve single-agent investment problems with deterministic coefficients. We consider the classical Merton’s portfolio problem framework, which is well-known in the modern theory of financial economics: an investor must allocate his money between one riskless bond and a number of risky stocks. The investor is assumed to be "small" in the sense that his actions do not affect market prices and the market is complete. The objective of the agent is to maximize expected utility of wealth at the end of the planning horizon. The optimal portfolio should be expressed as a ”feedback” function of the current wealth. Under the so-called complete market assumption, the optimization can be split into two stages: first the optimal terminal wealth for a given initial endowment is determined, and then the strategy is computed that leads to this terminal wealth. It is possible to extend this martingale approach and to obtain explicit solution of Merton’s portfolio problem using the Malliavin calculus and the Clark-Ocone formula.