# Stochastic Calculus and Applications to Finance

### Credits

3 ECTS, 18h

### Instructor

Pierre Etoré

### Syllabus

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 field of « Mathematical Finance ».

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. 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

Pricing and hedging formulas, illustration of the link between Stochastic Differential Equations and Partial Differential Equations inside Black-Scholes type models.

### Assessment

homework (1/3) + final exam (2/3)