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Stochastic calculus and applications to finance - WMM9AM13

  • Volumes horaires

    • CM 18.0
    • Projet -
    • TD -
    • Stage -
    • TP -
    • DS -

    Crédits ECTS

    Crédits ECTS 3.0

Objectif(s)

This MSc course aims at presenting the fundamental concepts of Stochastic Calculus, and the way this concepts have been used in order to build models for applications to finance. Stochastic calculus is a theory that uses Brownian motion and Itô’s integral as basic building blocks, and Itô's formula as a multipurpose tool, in order to describe and manipulate a rather large variety of continuous time Stochastic processes , called « continuous semimartingales » (Stochastic calculus for processes with jumps is out of the scope of this course). The theory of Stochastic calculus is largely due to the seminal work by K. Itô, that goes back to the 1940s and 1950s. This work has been rediscovered by economists (among them Myron Scholes) in the 1970s, giving rise to the famous Black-Scholes model. Since the late 1980s the link between Stochastic calculus and economics has been more and more formalized, giving rise to the fleld of « Mathematical Finance ».

Responsable(s)

Pierre ETORE

Contenu(s)

The content is planned to be:

  • Continuous time stochastic processes, Brownian motion (definition and properties)
  • Continuous time martingales
  • Itô’s integral
  • Itô’s formula, Theorem of Lévy, Theorem of Girsanov
  • Black-Scholes model; notion of pricing and hedging
  • Princing and hedging formulas, illustration of the link between Stochastic Differential Equations and Partial Differential Equations inside Black-Scholes type models.

Prérequis

This course requires knowledge of probability and integration theory. Some previous knowledge of Stochastic processes is welcomed. No previous knowledge of Brownian motion or Stochastic Calculus is required.

Contrôle des connaissances

The mark is mainly given by the mark obtained to the final examination (3h). A quite long homework will be given during this course. The mark to the homework is used to modify the mark of the final exam (up to minus or plus 1.5 point). This will give the final mark.

The note is mainly given by the final exam. This note is modified (up to 1.5 or 2 points) according to the work provided by the student in some homework (HM) that will be given during the course.

N1=E1 (modulated by HM)
N2=E2 (modulated by HM)

L'examen existe uniquement en anglais FR

Calendrier

Le cours est programmé dans ces filières :

  • Cursus ingénieur - Master 2 Math. et Applications - Semestre 9 (ce cours est donné uniquement en anglais EN)
cf. l'emploi du temps 2020/2021

Informations complémentaires

Code de l'enseignement : WMM9AM13
Langue(s) d'enseignement : FR

Vous pouvez retrouver ce cours dans la liste de tous les cours.

Bibliographie

"Continuous martingales and brownian motion", D. Revuz, M. Yor
"Brownian motion and stochastic calculus" I.K. Karatzas S.E. Shreve
"Stochastic Calculus for Finance", S.E. Shreve