## UE Stochastic calculus and applications to finance

Diplômes intégrant cet élément pédagogique :

### Descriptif

This MSc course aims at presenting the fundamental concepts of Stochastic Calculus, and the way these 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 ».

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.

Evaluation :

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.

### Pré-requis

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.

### 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

### Informations complémentaires

Méthode d'enseignement : En présence
Langue(s) : Anglais