# Courses

## Beyond Value at Risk and Expected Utility: Convex risk measures and robust preferences (Prof. Hans Föllmer)

In these lectures we discuss the problem of quantifying financial risk in monetary terms from a mathematical point of view.

The method of Value at Risk, which is widely used in practice, has serious deficits, and some of them have played a role in the recent financial crisis, as analyzed in "The Turner Review: A regulatory response to the global banking crises" of the British Financial Services Authority (FSA). On the academic level, these deficits have been recognized early on. Beginning with the seminal work of Artzner, Delbaen, Eber and Heath (1999), this has led to a systematic theory of capital requirements specified by coherent and convex risk measures. This is closely related to recent advances in the microeconomic theory of preferences, which go beyond the classical paradigm of expected utility and deal explicitely with the pervasive problem of "Knightian uncertainty", another issue which is raised in the Turner Review. We describe some of these developments and discuss some of the implications, in particular robust versions of the classical problem of optimal portfolio choice.

## Functional Ito calculus and applications in finance (Prof. Rama Cont)

We present a new calculus for functionals of semimartingales, which extends the Ito calculus to path-dependent functionals in a non-anticipative way. The approach builds on H. Föllmer's deterministic proof of the Ito formula and a notion of pathwise functional derivative recently proposed by B. Dupire. In this framework one can derive a functional extension of the Ito formula, which has numerous applications in probability and mathematical finance.

The functional Ito formula is used to derive two key results. First, we obtain a martingale representation formula for square integrable functionals of a semimartingale S. Second, regular functionals S, which have the local martingale property, are characterized as solutions of a functional differential equation, for which a uniqueness result is given.

These results have obvious applications to the pricing and hedging of path-dependent derivatives. We derive universal pricing and hedging equations, which hold for any path-dependent option, written on a financial asset, whose price is modeled as a continuous semimartingale S. Using these results, we derive a general formula for the hedging strategy of a path-dependent contingent claim and present a numerical method for computing this hedging strategy. By contrast with methods based on Malliavin calculus, this representation is based on non-anticipative quantities, which many be computed pathwise, and leads to simple simulation-based estimators of hedge ratios.

These lectures are based on joint work with David FOURNIE (Columbia University).

A complete outline of the course (PDF, 111KB) and the slides of the course (PDF, 550KB) are available now.