# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Dr. Gunther
Leobacher**, Institut für Finanzmathematik, JKU Linz,
will give a public talk (followed by a discussion) on **Wed, March 21, 2012**
at **16:15 o'clock** at **HS 13**
on the topic of
"Path Construction, Monte Carlo Pricing and Calibration, Utility Indifference Pricing"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Path Construction, Monte Carlo Pricing and Calibration, Utility Indifference Pricing

I will present a selection of topics in the field of computational finance. First we will concern ourselves with the (fast) generation of discrete Brownian paths, by which we mean generation methods which need at most $O(n\ log(n))$ floating point operations for a discrete path with $n$ nodes. I will present some classical and some new constructions, highlight some of their relations and show how they can be used in fast generations of discrete Levy paths. We discuss how our method can be applied to more general simulation problems.

Next we consider the problem of (quasi-)Monte Carlo valuation and (quasi-)Monte Carlo calibration of credit risk models. We will see theoretical reasons why quasi-Monte Carlo simulation is more suited to the calibration problem than Monte Carlo. This is also discussed for a stochastic volatility model.

In the final part of the talk we will see applications of the so-called utility indifference pricing method for rainfall derivatives and catastrophe bonds. One intriguing aspect of the first problem is that a large portion of the simulation part disappears, thus rendering it more tractable. In the second problem it emerges that the problem has a semi-exact solution, i.e. the price of the derivative can be written as a solution to an integro partial differential equation.