PRisMa 2010

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

Abstract: Life annuities and pension products usually involve a number of "guarantees", such as, e.g., minimum accumulation rates, minimum annual payments and minimum total payout. Packaging different types of guarantees is the feature of so-called variable annuities. Basically, these products are unit-linked investment policies providing deferred annuity benefits. The guarantees, commonly referred to as GMxBs (namely, Guaranteed Minimum Benefits of type "x"), include minimum benefits both in case of death and in case of survival. Following a risk management oriented approach, this talk first aims at singling out all sources of risk affecting variable annuities ("risk identification phase"). Critical aspects arise from the interaction between financial and demographic issues. In particular, the longevity risk may have a dramatic impact on the technical equilibrium within a portfolio. Then, we deal with risk quantification ("risk assessment phase"), mostly via stochastic simulation of financial and demographic scenarios. Our main contribution is to present an integrated approach to risks in variable annuity products, so providing a unifying and innovative point of view.

Prof. Dr. Walter Farkas
(ETH Zürich, Universität Zürich)
On Modelling and Option Pricing using Lévy Copula Processes

Abstract: We discuss the valuation of derivative contracts on baskets of risky assets whose prices are Lévy-like Feller processes of tempered stable type. The dependence among the marginals’ jump structure is parametrized by a Lévy copula. We analyze the jump measure of the corresponding Lévy copula processes and discuss further analytical and numerical aspects and the efficient numerical solution of the parabolic Kolmogorov equation ut + Au=0 arising in valuation of derivative contracts.

Dr. Zorana Grbac
(Department of Mathematical Stochastics, University of Freiburg)
Conditional Markov Chains and Credit Risk in the Lévy Libor Model

Abstract: In this work we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default-free forward Libor rate to defaultable bonds with credit ratings and develop the rating-based Libor market model. As driving processes for the dynamics of the default-free and pre-default term structure of interest rates a wide class of time-inhomogeneous Lévy processes is used. Credit migration is modeled by a conditional Markov process with finite state space consisting of possible credit ratings. This process is constructed in a canonical way by enlarging the reference filtration carrying the default-free information. We show that the conditional Markov property is preserved under all forward Libor measures. Moreover, we prove that this filtration enlargement satisfies the immersion property under all forward measures. This enables us to derive the conditions for absence of arbitrage in the model. Finally, the valuation formulae for some common credit derivatives in this setup are presented. In particular, we introduce defaultable forward measures to value certain types of defaultable contingent claims. This is joint work with E. Eberlein.

Dr. Stefan Gerhold
(FAM @ TU Wien)
Refined Volatility Expansion in the Heston Model

Abstract: It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes (Roger Lee's moment formula). Motivated by recent tail-wing refinements of this moment formula, we first derive a novel tail expansion for the Heston density, and then show the validity of a refined volatility expansion. Our results may prove useful for calibration of the Heston model to the volatility surface. This is joint work with P. Friz, A. Gulisashvili, and S. Sturm.

Dr. Robert Schöftner
(UBS Zürich)
Market and Credit Risk Aggregation: A Bottom-Up Approach

Abstract: The recent financial crisis has undoubtedly shown that the "thinking in risk types", which is the current practice in the financial industry, neglects essential interactions between risk drivers at product level. In order to circumvent such unpleasant past events, we study market and credit risk aggregation from a bottom-up perspective at product level. In contrast to its counterpart, the top-down approach using the statistical tools of copulas and introducing dependence at risk type level, the bottom-up approach captures dependence at risk driver level. For this purpose we calibrate a scenario generator, which consists of several dependent submodules that are based on both market risk and credit risk related components such as interest rates, foreign exchange rates, equity prices, as well as default intensities. Due to the significant importance to be able to value any kind of financial products at long time horizons in credit risk, we also focus on the modeling of credit exposures using a least squares Monte Carlo algorithm. Finally, formal definitions of (joint) market and credit risk profits and losses from a bottom-up perspective are provided.

Dipl.-Math. Benedikt Blum
(FAM @ TU Wien)
Superreplication and No-Arbitrage in Multiasset Models with Transaction Costs

Abstract: Apart from being an economic reality in financial markets, transaction costs (TC) give us the opportunity to render price processes with interesting modeling properties arbitrage-free, that would have allowed for arbitrage without TC. On the other hand, well-known results from classical financial mathematics more often than not get distorted as soon as TC are introduced to the underlying model. The so-called face-lifting theorem, which deals with super-replication prices in multiasset models with TC, is an especially drastic example. The second important result developed by the PRisMa-Lab's work on TC models is a multiasset version of the famous fundamental theorem of asset pricing, which gives necessary and sufficient conditions for the absence of arbitrage in models with asymptotically small TC. In both cases, the focus lies on the question whether "big" market agents can neglect the presence of TC, i.e. if their situation converges to the frictionless case as TC tend to zero.

Dr. Richard Warnung
(Raiffeisen Capital Management)

Abstract: In this talk we analyze the collective risk model which originally comes from insurance mathematics (non-life insurance). Despite its long history this model has very up-to-date applications in quantitative risk management in banks and other financial institutions (credit and operational risk). In 1981 the classical Panjer recursion was first published as an algorithm to calculate the loss distribution in the collective risk model. With the aim to guarantee the numerical stability of this recursion we first present an improved Panjer recursion and apply it to compound Poisson mixture models which allow for a flexible description of the claim number distribution. A second approach tries to find short and quick recursions. This aim is achieved by considering a special class of claim size distributions. (Joint work with Uwe Schmock and Stefan Gerhold)

Dipl.-Ing. Annemarie Mayer
(Generali Holding Vienna AG)

Abstract: Risiken sind ein Merkmal jeder unternehmerischen Tätigkeit, im Risikomanagement erfolgt die systematische Erfassung und Bewertung von Risiken. Auf erkannte Risiken kann reagiert werden, und mit geeigneten Maßnahmen können diese gesteuert werden. Mittels Zinsstukturmodelle werden Bondoptionen und vor allem Bermuda Optionen bewertet. Mit Hilfe der Term Structure Equation wird das Vasicekmodell eingeführt, das ein Vorläufer des genauer beschriebenen Hull-White-Zinsstrukturmodells ist. Es werden sowohl Formeln für die Bewertung von Bonds als auch von den dazu gehandelten Optionen hergeleitet und die Modellierung eines künstlichen Short-Rate-Prozesses mittels Konstruktion eines Trinominalbaumes vorgenommen.

Christoph Brodowicz, MSc.

Abstract: Throughout this talk we describe a numerical approach to compute CDO tranche prices. Actually we apply the normal approximation of the binomial distribution within the conditional independent framework to obtain the cumulative loss distribution of the CDOs underlying portfolio, which is the essential part to value a CDO tranche. Since the size of the portfolio is fixed for a standard synthetic CDO tranche, the use of correction terms for the approximations is inevitable. To achieve adequate corrections, we combine Stein's method and the zero-bias transformation.