**Lectures on Malliavin calculus and its applications to finance**

**June 23-July 2,
2009**

**University of
Wisconsin-Madison**

**Room Change: The lectures will be held in B321 Van Vleck**

**Lecturer: Eulalia
Nualart, University of Paris 13, France**

There will also be a series of student seminars in the
afternoons during the course.

Eulalia Nualart, University of Paris 13, will present eight lectures on the Malliavin calculus and its applications to finance. The lectures will be given in B321 Van Vleck Hall on the University of Wisconsin-Madison campus starting at 10:00 AM each weekday morning. The first lecture will be on Tuesday, June 23, and the last on Thursday, July 2.

This course is offered on the basis of need or interest to graduate/ Ph.D. students, postdocs and other researchers. The recommended prior knowledge is an advance probability course. Some familiarity with Ito stochastic calculus is also recommended.

Participants from outside Madison are welcome. A small amount of NSF funding is available to help defray the local expenses for these participants. Contact Tom Kurtz, kurtz@math.wisc.edu.

The aim of these lectures is to give an introduction to the stochastic calculus of variations, known as Malliavin calculus, and to give one of its applications in Mathematical Finance to the computations of Greeks, sensitivity parameters of option prices. The Malliavin calculus is an infinite-dimensional differential calculus on the Wiener space that was first introduced by Paul Malliavin in the 70's, with the aim of giving a probabilistic proof of Hormander's theorem. Since that time, the theory has developed further and many new applications of this calculus have appeared.

We will start these lectures by defining in an abstract setting the concepts of Wiener space, derivative operator, and Sobolev spaces associated with differentiable random variables. We will then study the particular case of stochastic integrals with respect to Brownian motion. Secondly, we will define the dual operator, known as the Skorohod integral, that will take us to one of the main tools of the Malliavin calculus which is the integration by parts formula. We will recall some of its applications to the study of probability laws of random variables on the Wiener space.

The second part of this course will discuss one of the applications of this calculus to Mathematical Finance. This application consists in using the integration by parts formula to give a probabilistic method for numerical computations of price sensitivities known as Greeks. In particular, we will introduce the Black-Scholes model and study the corresponding option price.

**References**

** **

Lamberton, D. and Lapeyre,
B. (1996), *Introduction to Stochastic Calculus Applied to Finance*,
Chapman and Hall.

Malliavin, P. (1997), *Stochastic
Analysis*, Springer-Verlag.

Nualart,
D. (1998), Analysis on Wiener space and anticipating stochastic calculus, *Ecole** d'Ete de Probabilites de Saint-Flour XXV, Lect. Notes in Math.* 1690,
Springer-Verlag, 123-227.

Nualart,
D. (2006), *The Malliavin calculus and related
topics*, Second Edition, Springer-Verlag.

**Note:** Specific notes for these
lectures will be distributed at the beginning of the course.