Forward equations for option prices in semimartingale models

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. This result generalizes Dupire's forward equation to a large class of non-Markovian models with jumps and allows to retrieve various forward equations previously obtained for option prices in a unified framework.

Forward equations for option prices in semimartingale models

We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a -possibly discontinuous- semimartingale. This result generalizes Dupire's forward equation to a large class of non-Markovian models with jumps and allows to retrieve various forward equations previously obtained for option prices in a unified framework.

From Black-Scholes and Dupire formulae to last passage times of local martingales. Part B : The finite time horizon

These notes are the second half of the contents of the course given by the second author at the Bachelier Seminar (8-15-22 February 2008) at IHP. They also correspond to topics studied by the first author for her Ph.D.thesis

Short-time asymptotics for marginal distributions of semimartingales

We study the short-time asymptotics of conditional expectations of smooth and non-smooth functions of a (discontinuous) Ito semimartingale; we compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of \textit{out-of-the-money} options is found to be linear in time, the short time asymptotics of \textit{at-the-money} options is shown to depend on the fine structure of the semimartingale

Projection markovienne de processus stochastiques

This PhD thesis studies various mathematical aspects of problems related to the Markovian projection of stochastic processes, and explores some ap- plications of the results obtained to mathematical finance, in the context of semimartingale models. Given a stochastic process ξ, modeled as a semimartingale, our aim is to build a Markov process X whose marginal laws are the same as ξ. This construction allows us to use analytical tools such as integro-differential equa- tions to explore or compute quantities involving the marginal laws of ξ, even when ξ is not Markovian. We present a systematic study of this problem from probabilistic view- point and from the analytical viewpoint. On the probabilistic side, given a discontinuous semimartingale we give an explicit construction of a Markov process X which mimics the marginal distributions of ξ, as the solution of a martingale problems for a certain integro-differential operator. This con- struction extends the approach of Gy ̈ongy to the discontinuous case and applies to a wide range of examples which arise in applications, in particu- lar in mathematical finance. On the analytical side, we show that the flow of marginal distributions of a discontinuous semimartingale is the solution of an integro-differential equation, which extends the Kolmogorov forward equation to a non-Markovian setting. As an application, we derive a forward equation for option prices in a pricing model described by a discontinuous semimartingale. This forward equation generalizes the Dupire equation, orig- inally derived in the case of diffusion models, to the case of a discontinuous semimartingale. These results give an application to the evaluation of index options allowing to reduce the problem of high dimension.Cette thèse porte sur l'étude mathématique du problème de projection Markovienne d'un processus aléatoire: il s'agit de construire, étant donné un processus aléatoire ξ, un processus de Markov ayant à chaque instant la même distribution que ξ. Cette construction permet ensuite de déployer les outils analytiques disponibles pour l'étude des processus de Markov (équations aux dérivées partielles ou équations integro-différentielles) dans l'étude des lois marginales de ξ, même lorsque ξ n'est pas markovien. D'abord étudié dans un contexte probabiliste, notamment par Gyöngy (1986), ce problème a connu un regain d'intêret motivé par les applications en finance, sous l'impulsion des travaux de B. Dupire. La thèse entreprend une étude systématique des aspects probabilistes (construction d'un processus de Markov mimant les lois marginales de ξ) et analytiques (dérivation d'une équation de Kolmogorov forward) de ce problème, étendant les résultats existants au cas de semimartingales discontinues. Notre approche repose sur l'utilisation de la notion de problème de martingale pour un opérateur integro-différentiel. Nous donnons en particulier un résultat d'unicité pour une équation de Kolmogorov associée à un opérateur integro-différentiel non-dégénéré. Ces résultats ont des applications en finance: nous montrons notamment comment ils peuvent servir à réduire la dimension d'un problème à travers l'exemple de l'évaluation des options sur indice en finance

Projection Markovienne de processus stochastiques

This PhD thesis studies various mathematical aspects of problems related to the Markovian projection of stochastic processes, and explores some applications of the results obtained to mathematical finance, in the context of semimartingale models.Given a stochastic process, modeled as a semimartingale, our aim is to build a Markov process whose marginal laws are the same as the first one. This construction allows us to use analytical tools such as integro-differential equations to explore or compute quantities involving the marginal laws of a general stochastic process, even when it is not Markovian.We present a systematic study of this problem from probabilistic viewpoint and from the analytical viewpoint. On the probabilistic side, given a discontinuous semimartingale we give an explicit construction of a Markov process which mimics the marginal distributions of a general stochastic process, as the solution of amartingale problems for a certain integro-differential operator. This construction extends the approach of Gyöngy to the discontinuous case and applies to a wide range of examples which arise in applications, in particular in mathematical finance. On the analytical side, we show that the flow of marginal distributions of a discontinuous semimartingale is the solution of an integro-differential equation, which extends the Kolmogorov forward equation to a non-Markovian setting.As an application, we derive a forward equation for option prices in a pricing model described by a discontinuous semimartingale. This forward equation generalizes the Dupire equation, originally derived in the case of diffusion models, to the case of a discontinuous semimartingale..Cette thèse porte sur l' étude mathématique du problème de projection Markovienne des processus stochastiques : il s'agit de construire, étant donné un processus aléatoire, un processus de Markov ayant à chaque instant la même distribution que celui-ci. Cette construction permet ensuite de déployer les outils analytiques disponibles pour l'étude des processus de Markov (équations aux dérivées partielles ou équations integro-différentielles) dans l' étude des lois marginales d'un processus aléatoire général, même lorsque ce dernier n'est pas markovien. D'abord étudié dans un contexte probabiliste, notamment par Gyöngy (1986), ce problème a connu un regain d'intérêt motivé par les applications en finance, sous l'impulsion des travaux de B. Dupire.Une étude systématique des aspects probabilistes est entreprise (construction d'un processus de Markov mimant les lois marginales d'un processus aléatoire) ainsi qu'analytiques (dérivation d'une équation integro-différentielle) de ce problème, étendant les résultats existants au cas de semimartingales discontinues et contribue à éclaircir plusieurs questions mathématiques soulevées dans cette littérature. Ces travaux donnent également une application de ces méthodes, montrant comment elles peuvent servir à réduire la dimension d'un problème à travers l'exemple de l' évaluation des options sur indice en finance.PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF