Talks in Financial and Insurance Mathematics, Summer 2001

• Tom Fischer (Institute for International Comparative Economics and Social Statistics, University of Heidelberg)
  Differentiability of coherent risk measures
  
  Abstract: We show that differentiable coherent risk measures are linear and therefore minimal. As a consequence smooth coherent risk preferences can be seen as negative expected payoffs due to subjective probability measures. We further propose differentiability properties for coherent risk measures, which seem to be suitable for unambiguous risk capital allocation by Euler's relation. Examples depending on the mean and the one-sided moments of a risky position are given. Risk capital allocation in portfolios of very general distributions, e.g. discrete ones, is possible.

• Marcello Galeotti (University of Firenze)
  Global dynamics in asymmetric coordination games
  
  Abstract: In a coordination game each player maximizes one's utility by adopting the same strategy as his/her opponent. We consider an evolutionary context, where asymmetry comes from the randomly matched players being members of two distinct populations (with generally different structure of preferences), while coordination consists in the fact that strategies available to each population can be listed in pairs of mutual best replies. In other words, in the two pay-off square matrices the diagonal terms are dominant in the columns. The strategy selection is regulated by a replicator dynamics (although many results continue to hold in the general case of a pay-off monotonic dynamics).
     After an exhaustive local analysis of the equilibria, we turn to global dynamics. As to this problem, in fact, very little is known, except in the "trivial" case of 2 strategies and in the particular case, explored by Hofbauer and Sygmund, of c-partnership games (grossly, this corresponds to the fact that in any match the players of each population get the same proportion of the stake).
     We consider the case of 3 strategies, giving place to a 4-dimensional compact phase space. Assuming an interior equilibrium exists (otherwise the interior orbits all tend to the boundary of the phase space), we show that in any neighborhood of such a point (generically a hyperbolic saddle) trajectories start converging to each one of the attractors of the system (3 pure strategy Nash equilibria). In other words, near the interior saddle the uncertainty as to the eventual outcome of the strategy selection is maximum ("the more so" if the eigenvalues of the Jacobian matrix are complex).
     In particular, if the two pay-off matrices are equal (although the two populations remain distinct), a more precise description of the global phase portrait is possible: we conjecture that in this case all the omega- (and alpha-) limit sets are pointwise. (Joint work with Angelo Antoci, Univ. of Sassari)

• Prof. Murad S. Taqqu (Boston University and IHP, Paris)
  Structure of some self-similar stable processes
  
  Abstract: Fractional Brownian motion is the only Gaussian self-similar process with stationary increments. But when one replaces the Gaussian distribution by an infinite variance stable one, this process has many extensions. We will study an important class of such extensions, those involving "mixed self-similar processes with stationary increments". By associating a flow to the process, one can decompose it in three independent parts. One part corresponds to dissipative flow, the second to the conservative flow "identity" and a third part associated with other conservative flows, for example, cyclic flows. Such a decomposition is useful because one can use it to distinguish between the different processes and understand their structure. We will give examples of processes associated with the different flows. (This is joint work with Vladas Pipiras.)

• Prof. Dr. Hans Bühlmann (Departement Mathematik, ETH Zürich)
  About the prudence of the actuary and the courage of the gambler
  
  Abstract: We presuppose the usual risk process of the free reserves. When is such a process acceptable for a company? We compare the stability criterium of the
  • controlled ruin probability
  with the one for the economical value, i.e.,
  • the sum of the expected discounted dividends.
  We obtain concrete decision rules concerning financing, premium determination and risk acceptance and retrocession, which differ substantially depending of the used stability criterium. (If the audience should wish so, the lecture will be given in English.)

• Prof. Dr. Thomas Braun (Universität Bielefeld, Germany)
  Benchmark oriented portfolio strategies
  
  Abstract: Performance is often measured relative to a benchmark. Managers whose payment depends on whether they beat the benchmark or not should invest according to a strategy that maximizes the probability of beating the benchmark. If a manager is judged by the extent to which a pre-specified target is missed, then he should either minimize the expected shortfall or the variance of the shortfall. In this paper, an analytical solution for all three goals is developed within a continuous dynamic framework. A method from statistical test theory is adapted and explained using linear programming. It is shown that the optimal policy for all three goals is to invest a fraction of wealth in the benchmark and a fraction or multiple in the growth optimum portfolio. The latter portfolio has the same structure as the mean-variance efficient tangent portfolio. The impact of different benchmarks on the investment policy is presented. The more volatile the benchmark, the more likely managers beat the benchmark. If the correlation between the benchmark and the mean-variance efficient tangent portfolio is low then the probability of beating the benchmark is high and also the expected shortfall and shortfall variance is low.

• Dr. Peter Bank (Humboldt University, Berlin)
  On intertemporal utilities, the Gittins index and a stochastic representation problem
  
  Abstract: The problem of intertemporal consumption choice is a corner stone of the modern theory of decision making under uncertainty. However, the classical way to model preferences over consumption streams as proposed in the seminal papers by Merton (1969, 1971) suffers from a number of severe drawbacks from an economic point of view. In an attempt to overcome these deficiencies, Hindy, Huang, and Kreps (1992) proposed a fundamentally new approach to model intertemporal preferences. We analyze the corresponding Hindy-Huang-Kreps utility maximization problem and show in a general semimartingale setting how this singular control problem can be reduced to a new type of stochastic representation problem. It turns out that this representation problem is of independent analytical and probabilitsitc interest. In the case of certainty, its solution can be constructed in terms of certain envelopes which exhibit a generalized form of convexity. The proof of existence in the general stochastic case reveals a close connection between this problem and the theory of Gittins indices. Such indices have been introduced by Gittins (1979) in order to solve dynamic allocation problems (e.g. the multiarmed bandit problem), and our representation problem yields a new characterization of these indices.

Organizer: Dr. Uwe Schmock
Workshop secretary: Eveline Fritsch
Previous workshop: February 28, 2001

• Dr. Vicky Henderson (FORC, University of Warwick, UK)
  Valuation of claims on untraded assets using utility maximisation
  
  Abstract: A recent topical problem is how to best handle claims on 'untraded' assets. Often one can choose a traded asset which is correlated to the untraded asset, to use as a proxy for hedging purposes. This paper addresses this problem in a utility maximisation framework and extends the results in Henderson and Hobson (2000) to cover claims which are bounded below. To model this situation, we introduce a second (untraded) log Brownian asset into the well-known Merton investment model with power-law utility. The investor has an option on the untraded asset and the question is how to price and hedge this random payoff. The presence of the second Brownian motion means that we are in the situation of incomplete markets and there is no unique preference independent price.
     Employing utility maximisation and duality methods, we obtain an approximation to the optimal hedge and reservation price. As expected, the marginal price is the expectation of the claim under the minimal martingale measure of Föllmer and Schweizer and is independent of the utility function. Importantly, unlike in the complete markets case, the marginal price depends on the real world measure and the drifts of the two assets. The price and hedge are computed for some example options and the results compared to those obtained using exponential utility.
     Applications of these results to executive stock options and basket options will be discussed.

• Daniel Straumann (Institute for Mathematical Sciences, University of Copenhagen)
  Whittle estimation in a heavy-tailed GARCH(1,1) model
  
  Abstract: The squares of a GARCH(p,q) process satisfy an ARMA equation with white noise innovations and parameters which are derived from the GARCH model. Moreover, the noise sequence of this ARMA process constitutes a strongly mixing stationary process with geometric rate. These properties suggest to apply classical estimation theory for stationary ARMA processes. We focus on the Whittle estimator for the parameters of the resulting ARMA model. Giratis and Robinson (2000) show in this context that the Whittle estimator is strongly consistent and asymptoticallly normal provided the process has finite 8th moment marginal. We focus on the GARCH case when the 8th moment is infinite. This case corresponds to various real-life log-return series of financial data. We show that the Whittle estimator is consistent as long as the 4th moment is finite and inconsistent when the 4th moment is infinite. Moreover, in the finite 4th moment case rates of convergence of the Whittle estimator to the true parameter are the slower the fatter the tail of the unconditional distribution.
     These findings are in contrast to ARMA processes with i.i.d. innovations. Indeed, in the latter case it was shown in Mikosch et al. (1995) that the rate of convergence of the Whittle estimator to the true parameter is the faster the fatter the tails of the innovations. Thus the analogy between a squared GARCH process and an ARMA process is misleading insofar that one of the classical estimation techniques, Whittle estimation, does not yield the expected analogy of the asymptotic behavior of the estimators.

• Dr. Nicole Bäuerle (Universität Ulm)
  Optimal control of stochastic processes in insurance
  
  Abstract: Two classes of stochastic processes play an important role in insurance: diffusion processes and piecewise deterministic Markov processes. These processes are used to model for example time-dependent risk reserve, capital and stock prices. In our talk we present the general concept of optimal control of diffusion processes and piecewise deterministic Markov processes. The procedure is explained by means of the problem of finding optimal dividend pay-out and reinsurence policies. Since the original problem is hard to solve, we obtain at least asymptotically optimal policies by diffusion approximations.

• Prof. Thorsten Hens (Institute for Empirical Economic Research, University of Zürich)
  Evolution of trading strategies in incomplete markets
  
  Abstract: The purpose of this talk is to apply evolutionary reasoning to financial markets. We consider simple trading strategies (portfolio-rules) being in competition for market capital in a stochastic environment. This defines a market selection process which is subject to mutations. Using methods from random dynamical systems we identify a trading strategy, called lambda, to be the unique evolutionary stable strategy of the process of wealth shares. Hence the population of trading strategies in which all trades use strategy lambda is robust to any innovation of mutations and no other population has this property.

• Dr. Larbi Alili (Department of Mathematics, ETH Zürich)
  A survey on the distribution of the first hitting time of a moving boundary by a Brownian motion

• Dr. Ludger Overbeck (Deutsche Bank, Frankfurt)
  Statistische Aspekte bei der Modellierung von Kredit-Portfolien
  
  Abstract: Im Vortrag wird zunächst ein grundlegendes Modell für Kreditrisiken in großen Portfolien vorgestellt. Dann werden die Problematik des zugrundeliegenden Faktormodells und die Schätzung der Korrelationsparameter in homogenen Portfolien angesprochen. Im letzten Teil des Vortrages wird auf die Modellvalidierung eingegangen.

• Alexander Lipton PhD (Deutsche Bank, New York)
  How to price exotic options in the presence of a smile
  
  Abstract: In this talk we discuss several complementary methods for pricing exotic options in the presence of the smile. We consider both static and stochastic volatility models. Our approach combines reductions of the pricing equation and subsequent eigenfunction expansions in order to obtain semi-analytical expressions for the prices and risk profiles of several representative classes of exotic options. We also briefly discuss practical implications of our findings.

• Harri Nyrhinen (Helsinki)
  Ruin probabilities in a stochastic economic environment
  
  Abstract: Let U₀ be the initial capital of an insurance company and U_n the capital at the end of year n. Consider the transition rule U_n=(1+r_n)(U_n-1-B_n) for n=1,2,... The random variable r_n describes the rate of return on investment in year n. The variable B_n is interpreted as the net payout of the company in year n (i.e. claims less premiums). Our interest is in the case where both positive and negative values are possible for the rates r₁, r₂, ... This means that we allow risky investments in the model. Inflation is also allowed to affect the net payouts. Define the time of ruin by T=inf{n|U_n<0}. We show that under suitable conditions, the ruin probability P(T finite) behaves like U₀^-w in the limit when U₀ tends to infinity. The parameter w can be specified and it only depends on the rates r₁, r₂,... and on the inflation.

• Dr. Patrick Cheridito (Department of Mathematics, ETH Zürich)
  Sensitivity of the Black-Scholes option price to the local path behaviour of the underlying asset
  
  Abstract: We show that a change of the local path behaviour of the stock price in the Black-Scholes model can have a dramatic effect on option prices.

• Dr. Alexander McNeil (Department of Mathematics, ETH Zürich)
  Modellieren von abhängigen Kreditverlusten
  
  Abstract: Für die Kreditportefeuilles der großen Banken kann nicht angenommen werden, daß die einzelnen Kreditrisiken (zum Beispiel Darlehen an Firmen) voneinander unabhängig sind. Es gibt gemeinsame ökonomische Faktoren, die zum Kreditausfall ("Default") mehrerer Firmen in bestimmten Perioden führen können. In realistischen Modellen müssen die dadurch einstehenden Abhängigkeiten berücksichtigt werden, da sie einen bedeutenden Einfluß auf den "Tail" der Gesamtverlustverteilung haben können.
     Bei den existierenden Industriemodellen können zwei allgemeine Ansätze zur Modellierung der Abhängigkeiten betrachtet werden: Modelle mit latenten Variablen (CreditMetrics und KMV) und Mischmodelle (CreditRisk+ und Credit Portfolio View). Bei den Modellen mit latenten Variablen ist es die "Copula" der latenten Variablen, die die Abhängigkeiten zwischen Kreditausfällen bestimmt. (Zum Verständnis dieser Behauptung wird das Thema "Copulas" kurz eingeführt/wiederholt.) Bei den Mischmodellen bestimmt die Mischverteilung die Abhängigkeiten.
     In diesem Vortrag werden die beiden Modellklassen verglichen und ihre Beziehungen zueinander untersucht. Insbesondere betrachten wir die Frage, wie man solche Modelle kalibriert oder an Daten statistisch anpaßt.

• Dr. Fabio Trojani (University of Southern Switzerland, Lugano)
  Robust choice of models for the short term rate: the robust generalized method of moments approach
  
  Abstract: We apply a new statistical methodology based on robust statistics, the Robust Generalized Method of Moments (RGMM), and discuss how it can be used in the econometric analysis of financial models. As an illustration we re-examine the empirical evidence concerning a well-known class of one factor CKLS models for the short rate process, cf. Chan et al. (1992). Standard GMM model selection procedures are highly unstable in this application. When testing the CKLS models with the RGMM on the CKLS data set we find that they are all clearly misspecified and we identify a clustering of influencial observations in the 1979-1982 subperiod. When introducing a temporary parameter shift also the unrestricted CKLS model is rejected. However, even a model with a temporary parameter shift is unable to take into account the particular 1979-1982 subperiod which exibits a cluster of influential points similar to that obtained for the constant parameter model. On the other hand, a Cox-Ingersoll-Ross model (selected by the RGMM) could offer a satisfactory data description for the period after 1982 since there only a few isolated outliers are found. Comparable results are obtained for the Euro-mark case. (Joint work with R. Dell'Aquila and E. Ronchetti)

• Oliver Lintner (Erste Bank, Vienna)
  System trading: building and trading profitable systems
  
  Abstract: The presentation will cover:
  • Benefits of systematic trading
  • Components of a trading system
  • Monte Carlo for risk estimation
  • Evaluation of successful systems
  • Moving into realtime trading
  • Pitfalls in a realtime world
  • Learnings and next steps
  Erste Bank has proprietary investments in alternative investments of U$ 500 mn and trades a portfolio of U$ 40 mn employing systematic trading. The portfolio covers over 30 markets including spot-fx-positions and futures on indices, interest rates and commodities. The team of Oliver Lintner started developing trading systems in 1994 and manages a SICAV fund for institutional, European clients.

• Prof. Raymond Brummelhuis (Université de Reims)
  Multi-period VaR prediction in GARCH models
  
  Abstract: The GARCH(1,1) model is often used to describe time series of, for example, daily financial returns. In this talk we will take a look at the k-day returns for a fixed, but arbitrary, k. We will, in particular, give precise asymptotic estimates for the tails of the probability distributions of these k-day returns. These are relevant for multi-day VaR estimation in such a model.

• Dr. Gilles Zumbach (Olsen & Associates, Zürich)
  Application of statistics to high-frequency finance
  
  Abstract: An introduction to financial time series and their statistical analysis is presented. First, the interbank foreign exchange (FX) market is described. Then, some salient empirical properties of financial data are presented, in particular seasonalities (and intrinsic time), lagged correlation, fat-tail distributions, aggregation and scaling, incoherent prices formation, volatility and heteroskedasticity, volatility derivative and the markets heterogeneity. Two basic time series models are introduced to describe these financial data: the random walk, and the Garch(1,1) model.

Page created and supported by Uwe Schmock until September 2003. Last update: October 10, 2003