Talks in Financial and Insurance Mathematics, Winter 2000/01

• Aydin Akgün (RiskLab and Swiss Banking Institute, University of Zürich)
  Risk management, capital budgeting and capital structure policy for financial institutions: An integrated approach
  
  Abstract of the paper by K. Froot and J. Stein:
  We develop a framework for analyzing the capital allocation and capital structure decisions facing financial institutions such as banks. Our model incorporates two key features: i) value-maximizing banks have a well-founded concern with risk management; and ii) not all the risks they face can be frictionlessly hedged in the capital market. This approach allows us to show how bank-level risk management considerations should factor into the pricing of those risks that cannot be easily hedged. We examine several applications, including: the evaluation of proprietary trading operations; and the pricing of unhedgeable derivatives positions. We also compare our approach to the RAROC methodology that has been adopted by a number of banks. (Reference: Kenneth A. Froot and Jeremy C. Stein, Journal of Financial Economics, 47, January 1998, 55-82)

Organizer: Dr. Uwe Schmock
Workshop secretary: Gerda Schacher
Previous events: Risk Day 2000, Workshop June 22, 2000

• Sebastian Vu (Swiss Re New Markets)
  The good, the bad, and the ugly in run-off reinsurance - an actuarial perspective
  
  Abstract:
  The run-off market is large, growing, and offers profitable opportunities. One estimate puts the size of this market to be at USD 230 billion. This talk will comprise of 3 parts: the development of the run-off market, the role of an actuary in a run-off transaction, and three examples of real live run-off deals.

• Henrik Hult (KTH Stockholm, currently visiting RiskLab)
  On approximating some Volterra-type stochastic integrals and applications to parameter estimation.
  
  Abstract:
  By using a general representation theorem for continuous Gaussian processes as the limit of a sequence in the associated Cameron-Martin space, we obtain a representation for a class of Volterra-type stochastic integrals with respect to Brownian motion. This class includes the fractional Brownian motion. As special cases of this representation we can for instance derive the Karhunen-Loeve decomposition for the standard Brownian motion and a wavelet representation for the fractional Brownian motion. We also discuss how the representation can be used to estimate parameters. In particular we can derive an estimator for the mean-reverting parameter in a fractional Ornstein-Uhlenbeck process.

• Marco Avellaneda (New York University and Lamb Analytics Consulting)
  Arbitrage pricing of equity basket options
  
  Abstract:
  With the current strong institutional and public interest in equity products, such as Exchange Traded Funds (ETFs), equity indices such as the NASDAQ 100 Trust (QQQ) and HLDR Shares from the American Stock Exchange, there has been an increased attention given to equity volatility trading and the pricing of options on baskets. An important conceptual theme in this area is: can we price the implied volatility skew of an equity basket based on the implied volatility skews of the component stocks and 'correlation information' available from market data? This talk presents a new methodology for doing this, based upon sophisticated calibration methods for options on different underlying assets in a large-scale Monte Carlo simulation. Calibration for baskets of up to 35 underlying assets with 25-30 options per asset and up to one year can be achieved within reasonable time on commonly available platforms. The talk will present the methodology and some examples of how this method is able to capture the volatility structure of the basket and compare it with contemporaneous market quotes on index options.

• Marco Avellaneda (New York University and Lamb Analytics Consulting)
  Monte Carlo simulation in quantitative finance: Calibration and hedging using the formalism of Gibbs measures

• Conferment of the Walter Saxer Insurance Prize and a talk by
  
  Daniel Greber (CEO Providentia Nyon)
  Appraisal Valuation eines Lebensversicherers: Bedeutung und Herausforderung in der Praxis
  
  Abstract:
  The market's deregulation, the shareholders' request for transparency and profit, as well as the participation in joint venture forced life insurers to estimate "their value". An approach, often used in practice, consists in defining this value (Appraisal Value) as the sum of
  • the net asset value (in some sense the value of the past)
  • the current value of the expected future profit arising from the business in force (present)
  • the current value of the expected future profit of the new business (future).
  Revenue items are projected from the in-force data using economic and demographic assumptions. A practical consequence of this evaluation is that the sensitivity of the results with respect to the changes of some significant parameters is a useful mean of decision for the management. In this presentation we make a turn of relevant aspects related to calculation of the Appraisal Value in practice and concrete examples will be shown.

• Nizar Touzi (Université Paris 1 Panthéon-Sorbonne)
  Hedging, stochastic target problem, and geometric flows
  
  Abstract:
  The stochastic target problem is a natural generalization of the super-replication problem in finance, and is defined as follows. Given a controlled process Z^v, find the set V(t) of all initial conditions starting from which one can place the terminal value Z^v_T in a given target \Gamma, for some control v. This problem is strongly connected to forward-backward SDE's.
  We introduce a geometric dynamic programming principle for the stochastic target problem, which allows to obtain a characterization of the family of sets {V(t), t\in [0,T]} as viscosity solution the associated HJB equation. Examination of the latter PDE shows that the stochactic target problem provides a stochastic representation of geometric flows viewed in the reverse time.

• Moncef Meddeb (Université Paris 1 Panthéon-Sorbonne)
  On generically dynamic complete markets
  
  Abstract:
  In this talk we generalize Magill and Shafer's (1990) paper: characterize the financial market structures leading to (generically) dynamic complete markets. Doing this we are faced with difficulties specific to the presence of infinitely many periods. Until now these difficulties did not allow any satisfactory generalization of the genericity argument in Duffie and Shafer (1985, 1986). These difficulties are threefold: how to get differentiability of an equilibrium equation; which transversality theorem can be applied (and what is the corresponding notion of genericity); and how to 'count' dimensions? (Joint work with Jean-Marc Bottazzi)

• Mete Soner (Koc University, Istanbul)
  A stochastic representation for mean curvature type flows: smooth case

• Enno Mammen (StatLab, University of Heidelberg)
  Nonparametric estimation of yield curves
  
  Abstract:
  We discuss kernel smoothing estimates for some nonstandard nonparametric regression problems. The estimate is based on minimizing a smoothed least squares criterion. The main example in this talk is estimation of discount functions, yield curves and forward curves from government issued coupon bonds. Other examples are additive models, panels of time series, semiparametric GARCH models and regression models with correlated errors. The definition of the estimates is not explicit and our estimation procedure is iterative, rather like the backfitting method of estimating additive nonparametric models. We establish the asymptotic normality of our methods using the asymptotic representation of our estimator as an infinite series with declining coefficients. The rate of convergence is standard for one-dimensional nonparametric regression.

• Alexander Schied (Humboldt Universität zu Berlin)
  Risk measures and trading constraints
  
  Abstract:
  For an extension of the notion of a coherent risk measures that allows for a non-linear growth in the size of the underlying position, one gets a representation theorem which involves a certain penalty function defined on the set of absolutely continuous probabilities. Two case studies for such risk measures are considered: In the first case, the risk measure is defined in terms of a robust notion of bounded shortfall risk. Then we consider risk measures in a financial market model with trading constraints. This is joint work with Hans Föllmer.

• Alexander McNeil (ETH Zürich)
  Modelling dependence with applications to credit risk modelling
  
  Abstract:
  We review the use of copulas for modelling dependent risks. We then consider the modelling of dependent defaults using latent variable models - the approach that underlies KMV and CreditMetrics - and Bernoulli mixture models - the approach underlying CreditRisk+. We show that these seemingly different classes of models can often by easily mapped into each other. In particular we explore the role of copulas in the latent variable framework and present results from a simulation study showing that assumptions concerning the dependence of the latent variables can have a large effect on the distribution of credit losses. Our contribution can be viewed as an analysis of the model risk associated with the modelling of dependence between credit losses.

• Sergei Novak (Eurandom)
  Extreme quantile estimation from dependent heavy-tailed data
  
  Abstract:
  Statistics of extremes has been well developed for the case of i.i.d. observations. In a growing number of financial and insurance applications, however, data appears dependent and heavy-tailed. We deal with problems of tail index and extreme quantile (Value at Risk) estimation from a sample of dependent heavy-tailed variables. Consistency and asymptotic normality of estimators is established under mild mixing conditions. The feature of our results is that accuracy of estimation is shown to be of the same order as if the data were independent. Besides limit theorems, we present a procedure of practical estimation.

• Wolfgang Breymann (RiskEye)
  Prediction of market volatility with a cascade model
  
  Abstract:
  Volatility is central to the characterization of financial time series for both theoretical considerations and applications, and forecasting its future values is important for risk management and option pricing. Unfortunately, the time dependence of volatility is rather complicated. It displays the following important characteristics: (i) anomalous scaling behavior of the returns, (ii) slowly decaying autocorrelation functions, and (iii) asymmetric correlations of volatility defined with respect to different time horizons. These features are caused by the heterogeneous structure of financial markets. Models designed to forecast volatility should take into account these facts. The model presented here has originally been inspired by an analogy with hydrodynamical turbulence (S. Ghashghaie et al., Nature 381, 767-770 (1996)) and takes into account the heterogeneous structure of the market. It is based on a multiplicative stochastic cascade for the volatility, which is composed of a range of different time horizons, as in the case of multifractals or the Kolmogorov cascade. Accordingly, the volatility is composed of a product of log-normally distributed random factors which model the parts of the volatility corresponding to the different time horizons. The volatility dynamics results from the dynamics of the set of factors. This model reproduces well the above-mentioned empirical characteristics of FX rates. The present approach differs from traditional approaches in two important aspects:
  • The volatility dynamics is modeled by assuming exogenous shocks (independent of the return dynamics).
  • The multivariate distribution of the factors is inferred from the observed volatility, thus giving information about the volatilities corresponding to the different market components.
  Inference of the distribution of the factors from observed data is possible because the value of the volatility observed at a given time imposes a condition on this distribution. Forecasting the volatility and its uncertainty is done in the next step, taking into account the relaxation of the modified distribution to there equilibrium distribution. The results of this approach are compared with the results of other models.
  
  Slides (49 pages) are available: PDF (908kB), PS (3016kB), Compressed PS (455kB)

• Caterina Savi (Winterthur)
  Franchisen in der obligatorischen Krankenpflegeversicherung: Konstruktion und Konstruktionsfehler

• Heiner Schwarte (Partner Re)
  Tarifierung und Risikomanagement von hybriden Deckungen

• Thorsten Hens (Institute for Empirical Research in Economics, University of Zurich)
  Der individuelle Investor: Von der klassischen Investitionstheorie zum evolutionären Finance
  
  Abstract:
  
  • Frage 1 (deskriptiv): Wie legen Investoren an der Börse Geld an?
  • Frage 2 (präskriptiv): Wie sollten Investoren an der Börse Geld anlegen?
  Die Vorlesung wird aufzeigen, welche Anhaltspunkte die Portfoliotheorie zur Beantwortung dieser Fragen liefert. In der historischen Reihenfolge betrachten ich klassisches Finance, Behavioral Finance und Evolutionary Finance.
  
  Klassisches Finance: Hier stelle ich die Erwartungsnutzenhypothese und die Mean-Variance-Optimierung vor und belege durch Zitate von Arrow und Markowitz, daß die Klassiker behaupten, jeder Investor solle das Geld gemäß dieser Hypothesen anlegen und, daß jeder Investor (zumindest nach ein wenig Unterrichtung) auch tatsächlich nach diesen Hypothesen anlegt. Bemerkenswert ist, daß die Begründung der Klassiker nur auf der Plausibiltät der Axiome, aus denen man das klassische Entscheidungskalkül ableiten kann, basierte.
  
  Behavioral Finance: Behavioral Finance zeigt durch eine Reihe von statistisch signifikanten Erhebungen (experimentell und in Felddaten), daß der typische Investor sein Geld NICHT nach der klassichen Theorie anlegt, auch dann nicht, wenn ihm diese Theorie erläutert wird. Effekte wie Verlustaversion, Sicherheitseffekt, Habit Formation und Herding sind die Grundbausteine von behavioral Theorien des Anlageverhaltens, wie zum Beispiel der Prospekt Theorie.
  
  Evolutionary Finance: Evolutionary Finance wendet sich der präskriptiven Frage zu: Wie sollte ein Investor sein Geld anlegen? Die Modelle des Evolutionary Finance basieren auf Analogien zur biologischen Evolution. Es wird ein Marktselektionsprozeß modelliert, in welchem Anlagestrategien um Geld/Vermögen konkurrieren. Wie in der biologischen Evolution ist ein wesentliches Grundprinzip die vielfache Wiederholung elementarer Situationen. Aus meiner neuesten Arbeit stelle ich dann ein Beispiel hierfür vor: In einem einfachen Modell (i.i.d. Unsicherheitsstruktur) charakterisiere ich die Anlageregel, welche auf lange Sicht alle anderen Anlageregeln (Mean-variance, Prospect theory) schlägt. Anhand von Daten des SMI (1999) zeige ich dann, daß die evolutionäre Anlageregel auch bei realistischer Unsicherheitsstruktur in wenigen Wochen das Gesamtvermögen des Marktes erobert.
  
  Resumee: Während Behavioral Finance klassisches Finance als descriptive Theorie erstetzt hat, wird Evolutionary Finance klassisches Finance als präskriptive Theorie ablösen.

• Jesper Lund Pedersen (RiskLab, ETH Zürich)
  Some optimal stopping problems for the maximum process
  
  Abstract:
  In this talk, I will illustrate by two examples a method for solving optimal stopping problems for the maximum process. The method is based upon the maximality principle, i.e. the optimal stopping boundary is the maximal solution to a first-order nonlinear differential equation which stays below the diagonal in the plane. The first example is to derive a (sharp) maximal inequality for Bessel processes. The second example is to calculate the fair price of perpetual lookback options in the Black-Scholes model.

Announcement:

Extreme Value Theory and Risk Management - Summer School,
Geneva 2000 ICMB/FAME Executive Courses in Finance,
International Center for Monetary and Banking studies),
October 2-6, 2000

Page created and supported by Uwe Schmock until September 2003. Last update: September 13, 2003