PRisMa 2013

Prof. Dr. Uwe Schmock
(FAM @ TU Wien)
Welcome

Prof. Dr. Andreas Kyprianou (Department of Mathematical Sciences @ The University of Bath)

Censored Stable Processes

Abstract: We look at a general two-sided jumping strictly alpha-stable process where alpha is in (0,2). By censoring its path each time it enters the negative half line we show that the resulting process is a positive self-similar Markov process. Using Lamperti's transformation we uncover an underlying driving Lévy process and, moreover, we are able to describe in surprisingly explicit detail the Wiener-Hopf factorization of the latter. Using this Wiener-Hopf factorization together with a series of spatial path transformations, it is now possible to produce an explicit formula for the law of the original stable processes as it first enters a finite interval, thereby generalizing a result of Blumenthal, Getoor and Ray for symmetric stable processes from 1961. This is joint work with Juan Carlos Pardo and Alex Watson.

Coffee Break

Chairperson: Prof. Dr. Thorsten Rheinländer

Dr. Christa Cuchiero (FAM @ TU Wien)

An HJM Approach to Multiple-Curve Modeling

Abstract: We propose a general framework for interest rate modeling in the multiple-curve setup which has occurred in the course of the recent financial crisis. More precisely, we provide an HJM approach for simultaneous modeling of the riskfree forward rates deduced from OIS rates and the multiplicative spreads between the simply compounded riskfree forward rates and the forward rates implied by forward rate agreements for some future time interval [T; T + δ]. We specify in particular the HJM drift condition in this setting and establish conditions which ensure that the multiplicative spreads are greater than 1. This general framework allows to unify and extend several approaches which have been proposed in literature in the context of multiple curve modeling. For instance, we can obtain a log-normal LIBOR market model (similarly as in the seminal paper by Brace, Gatarek and Musiela) or recover the Lévy driven HJM model studied by Crépey et al. [1, 2] but also short rate models as for example considered by Kenyon [3]. When the driving process of both the riskfree forward rates and the spreads is specified to be affine, we obtain a Markovian structure which allows for simple pricing formulas of LIBOR interest rate derivatives by exploiting the affine property of the driving process. The talk is based on joint work with Claudio Fontana and Alessandro Gnoatto.
 
[1] S. Crépey, Z. Grbac and H. N. Nguyen. A multiple-curve HJM model of interbank risk. Mathematics and Financial Economics, 6(3): 155-190, 2012.
[2] S. Crépey, Z. Grbac, H. N. Nguyen and D. Skovmand. A multiple-curve CVA interest rate model. Working paper
[3] C. Kenyon. Short-Rate Pricing after the Liquidity and Credit Shocks: Including the Basis. SSRN-eLibrary, 2010.

Sühan Altay, MSc (FAM @ TU Wien)

Yield Curve Scenario Generation with Independent Component Analysis

Abstract: We propose a yield curve scenario generator capable of extracting the state variables (factors) driving the dynamics of the default-free and/or defaultable bonds. Our objective is to extend the methodology proposed by Jamshidian and Zhu (1997) in a way that the original state variables are transformed to a new set of state variables via Independent Component Analysis (ICA) instead of Principal Component Analysis (PCA). Main advantage of ICA is to extract components that are statistically independent, which makes the estimation of parameters and generation (simulation) of future yield curves more efficient. One nice feature of this method is its ability to extract non-Gaussian factors due to the fact that the estimation requires non-Gaussian independent components. Under this proposed model, the yield curve is described by a linear mixture of independent components, which are assumed to follow non-Gaussian dynamics such as non-Gaussian Lévy driven Ornstein-Uhlenbeck processes or Sato processes in general.

Lunch Break

Chairperson: Ao.Prof. Dr. Friedrich Hubalek

PD Dr. Stefan Gerhold (FAM @ TU Wien)

Local Volatility Models: Approximation and Regularization

Abstract: We aim at understanding the typical shape of a local volatility surface, by focusing on its extremes (w.r.t. strike and maturity). The asymptotic behavior is governed by a saddle-point based formula, akin to Lee's moment formula for implied volatility. Applications include local vol parametrization design and assessing volatility model risk. Secondly, it is well known that local vol models cannot deal with jumps in the underlying. We propose a simple regularization procedure as a remedy. Its validity is related to recent work of Yor et al. on Kellerer's theorem from the theory of peacocks (processus croissants pour l'ordre convexe). The talk is based on joint work with S. De Marco, P. Friz, and M. Yor.

Jonas Hirz, MSc (FAM @ TU Wien)

Risk Measures: From the Unconditional to the Conditional Case

Abstract: We start with an introduction to risk measures and why they are important in the insurance industry. Then we define the rich class of distortion risk measures and analyse their properties. Value-at-risk, expected shortfall and beta-value-at-risk arise as special cases. Thereafter, we switch to a conditional setting which allows dynamic analysis of risk measures. Based on the upper envelope and conditional lower quantiles we define conditional distortion risk measures, via a stochastic integral representation, and give a collection of different properties. Conditional expected shortfall arises as a special case of conditional distortion risk measures. We also give a definition via an explicit density on a modelling setup with stochastic levels, involving generalised conditional expectations based on sigma-integrability. Then we prove several properties and give several alternative representations of conditional expected shortfall. Further, we point out the link to dynamic risk measures and show a supermartingale property. In the next step we introduce contributions to conditional expected shortfall and prove several properties. In particular, it is possible to derive the contribution of a subportfolio to the whole portfolio in order to be able to identify main risks. We end with some motivating examples including a time series application. Joint work with Karin Hirhager and Uwe Schmock.

Coffee Break

Chairperson: Ao.Prof. Dr. Peter Grandits

Dr. Julia Eisenberg (FAM @ TU Wien)

Optimal Consumption Under Deterministic Income

Abstract: We consider an individual or household endowed with an initial wealth, having an income and consuming goods and services. The wealth development rate is assumed to be a deterministic continuous function of time. The objective is to maximize the discounted consumption over a finite time horizon. Via the Hamilton-Jacobi-Bellman approach, we prove the existence and the uniqueness of the solution to the considered problem in the viscosity sense. Furthermore, we derive an algorithm for explicit calculation of the value function and optimal strategy. It turns out that the value function is in general not continuous. The method is illustrated by two examples.

DI I. Cetin Gülüm (FAM @ TU Wien)

On the Existence of an Equivalent Martingale Measure in the Dalang-Morton-Willinger Theorem, which Preserves the Dependence Structure

Abstract: We study financial market models where the discounted asset price processes are Markovian. We show the existence of a martingale measure with bounded density which preserves this dependency structure, given that the model is arbitrage-free. Moreover we study financial markets models where the additive respectively multiplicative increments of the securities are independent. We show that in an arbitrage-free market this structure can be preserved: it is possible to find an equivalent martingale measure with bounded density under which the additive respectively multiplicative increments are independent. The talk is based on joint work with Uwe Schmock.

Bread and Wine