Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$, where $f$ is a smooth function and $V_i,i\in\mathbb{N}$, are random variables with absolutely continuous law $p_i(y) dy$. We assume that $p_i$, $i=1,...,n$, are piecewise differentiable and we develop a differential calculus of Malliavin type based on $\partial\ln p_i$. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$, where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a L\'{e}vy process.Comment: Published at http://dx.doi.org/10.1214/105051606000000592 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$ where $f$ is a smooth function and V_i,i\in\mathbbN are random variables with absolutely continuous law $p_i(y).$ We assume that $p_i,i=1,...,n$ are piecewise differentiable and we develop a differential calculus of Malliavin type based on \partial\lnp_i. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$ where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process

Minoration de densité pour les diffusions à sauts.<br /><br />Calcul de Malliavin pour processus de sauts purs, applications à la finance.

This thesis is concerned withapplications of Malliavin-like calculus for jump processes. In thefirst part, we compute lower bounds for the density of jumpdiffusions with a continuous part driven by a Brownian motion. Weuse a Malliavin conditional integration by parts formula based onBrownian increments only. We then deal with the computation offinancial options, when the asset price follows a pure jump process.In the second part, we develop an abstract calculus of the Malliavin type based on random variables which are not independent and have discontinuous conditional densities. We settle an integration by parts formula that we apply then to the jump times and amplitudes of pure jump processes. In the third part, we use this integration by parts formula for the computation of the Delta of European and Asian options, and we derive representation formulas for conditional expectations and their gradients in order to compute the price and the Delta of American options.Cette thèse donne deux applications du calcul de Malliavin pour les processus de sauts.Dans la première partie, nous traitons la minoration de la densité des diffusions à sauts dont la partie continue est dirigée par un mouvement Brownien. Pour cela, nous utilisons une formule d'intégration par parties conditionnelle basée sur le mouvement Brownien uniquement.Nous traitons ensuite le calcul d'options financières dont le prix du sous-jacent est un processus à sauts pur.Dans la deuxième partie, nous développons un calcul abstrait du type Malliavin basé sur des variables aléatoires non indépendantes, de densité conditionnelle discontinue. Nous établissons une formule d'intégration par parties que nous appliquons aux amplitudes et temps de sauts des processus à sauts considérés. Dans la troisième partie, nous utilisons cette intégration par parties pour calculer le Delta d'options européennes et asiatiques, et pour calculer le prix et le Delta d'options américaines via des formules de représentation pour les espérances conditionnelles et leur gradient

Minoration de densité pour les diffusions à sauts (Calcul de Malliavin pour processus de sauts purs, applications à la finance)

Cette thèse donne deux applications du calcul de Malliavin pour les processus de sauts. Dans la première partie, nous traitons la minoration de la densité des diffusions à sauts dont la partie continue est dirigée par un mouvement Brownien. Pour cela, nous utilisons une formule d'intégration par parties conditionnelle basée sur le mouvement Brownien uniquement. Nous traitons ensuite le calcul d'options financières dont le prix du sous-jacent est un processus à sauts pur. Dans la deuxième partie, nous développons un calcul abstrait du type Malliavin basé sur des variables aléatoires non indépendantes, de densité conditionnelle discontinue. Nous établissons une formule d'intégration par parties que nous appliquons aux amplitudes et temps de sauts des processus à sauts considérés. Dans la troisième partie, nous utilisons cette intégration par parties pour calculer le Delta d'options européennes et asiatiques et le prix et le Delta d'options américaines.This thesis gives applications of Malliavin calculus for jump processes. In the first part, we compute lower bounds for densities of jump diffusions with a continuous part driven by a Brownian motion. For that, we use a Malliavin conditional integration by parts formula based on Brownian increments only. We then deal with the computation of financial options, when the asset price follows a pure jump process. In the second part, we develop an abstract calculus of the Malliavin type based on random variables which are not independent and have discontinuous conditional densities. We settle an integration by parts formula that we apply then to the jump times and amplitudes of pure jump processes. In the third part, we use this integration by parts formula for the computation of the Delta of European and Asian options, and we derive representation formulas for conditional expectations and their gradients in order to compute the price and the Delta of American options.PARIS-DAUPHINE-BU (751162101) / SudocSudocFranceF

Computation of Greeks using Malliavin's calculus in jump type market models

We use the Malliavin calculus for Poisson processes in order to compute sensitivities for European options with underlying following a jump type diffusion. The main point is to settle an integration by parts formula (similar to the one in the Malliavin calculus) for a general multidimensional random variable which has an absolutely continuous law with differentiable density. We give an explicit expression of the differential operators involved in this formula and this permits to simulate them and consequently to run a Monte Carlo algorithm

Integration by parts formula for locally smooth laws and applications to sensitivity computations

We consider random variables of the form $F=f(V_1,...,V_n)$ where $f$ is a smooth function and V_i,i\in\mathbbN are random variables with absolutely continuous law $p_i(y).$ We assume that $p_i,i=1,...,n$ are piecewise differentiable and we develop a differential calculus of Malliavin type based on \partial\lnp_i. This allows us to establish an integration by parts formula $E(\partial_i\phi(F)G)=E(\phi(F)H_i(F,G))$ where $H_i(F,G)$ is a random variable constructed using the differential operators acting on $F$ and $G.$ We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process

Histoire et archéologie du pays de Levroux (Indre). Levroux : Association pour la défense et l’étude du canton de Levroux

ill., cartes, plans, bibliogr. p. 133-136International audienc