Talks in Financial and Insurance Mathematics, Winter 1999/2000

• Pierre Patie (RiskLab, ETH Zürich)
  Estimation of value at risk using extreme value theory
  
  Abstract:
  Value at Risk is becoming somewhat of a revolution. Around the globe, organizations are racing to implement new technology. Most of the actual techniques rely on normal assumption about the marginal distribution of financial returns implying a lack of estimating the known fat-tail properties of financial returns. Our methodology involves modelling the profit&loss portfolio distribution tail explicitly with a tail estimator based on extreme value theory, and modelling the central part using historical simulation. This technique allows us to calculate the unconditional high quantile with accuracy because of the modelling of extremes events. We discuss about the reliability of this approach comparing with the normal and historical simulation, achieved by backtesting. We also investigate the long standing problem of estimating the sample threshold of where the tails of the distribution starts. Finally, in order to bridge the gap between the theory and the practice we carry out several tests on real financial data.

• Filip Lindskog (RiskLab, ETH Zürich)
  Modelling dependence with copulas and applications to risk management
  
  Abstract:
  The dependence between random variables is completely described by their joint distribution. However, dependence and marginal behaviour can be separated. The copula of a multivariate distribution can be considered to be the part describing the dependence structure. Furthermore, strictly increasing transformations of the underlying random variables result in the transformed variables having the same copula. Hence copulas are invariant under strictly increasing transformations of the margins. This provides a way of studying scale-invariant measures of associations and also a starting point for construction of multivariate distributions. Scale-invariant measures of association such as Kendall's tau and Spearman's rho only depend on the copula and are thus invariant under strictly increasing transformations of the margins, which means that we can apply arbitrary continuous margins to our chosen copula leaving among other things the measures of association unchanged.

  Tail dependence and Kendall's tau and Spearman's rho are presented and evaluated for a large number of copula families. Among these copula families are families suitable for modelling extreme events, which are highly relevant as a basis for risk models in insurance and finance.

  The multivariate normal distribution and linear correlation are the basis of most models used to model dependence. Even though this distribution has a wide range of dependence it is quite seldom suitable for modelling real world situations in insurance and finance. We will show that using a model based on the multivariate normal distribution without knowledge of its limitations can prove very dangerous. Linear correlation is a natural measure of dependence in the context of the normal distribution. However, it should be noted that it is not invariant under strictly increasing transformations of the marginals and can be misleading as a measure of dependence.

  The problem of simulating dependent data arises naturally in Monte Carlo approaches to risk management. One main aim of this talk is to show that when addressing this problem knowledge of copulas and copula based dependence concepts is important, and also the usefulness of copula ideas in this approach to risk management. Another main aim of this talk is the construction of multivariate extensions of bivariate copula families. In particular we focus on multivariate extensions with a flexible and wide range of dependence for which efficient algorithms for random variate generation are presented.

  Slides are available in the following formats:
  Postscript (ps, 812k), Portable Document Format (pdf, 312k), Compressed Postscript (gzip, 392k).

• Zheng Ziyu (INRIA, France)
  Model risk management
  
  Abstract:
  In this talk we are interested in the management of model risk. Our objective is to propose a new strategy for the trader which, in a sense, guarantees good performances whatever is the unknown model. Our construction corresponds to a `worst case' worry and, in this sense, can be viewed as a continuous-time extension of discrete-time strategies based upon prescriptions issued from VaR analyses at the beginning of each period. The trader chooses trading strategies to decrease the risk and therefore acts as a minimizer; the market systematically acts against the interest of the trader, so that we consider it acts as a maximizer. Thus we consider the model risk control problem as a two players (Trader versus Market) zero-sum stochastic differential game problem. We give a proper mathematical statement for such a game problem. We proved that the value function to this game problem is the unique viscosity solution to a corresponding Hamilton-Jacobi-Bellman-Isaacs Equation and satisfies the Dynamic Programming Principle. We simulate the value function by solving the HJBI equation numerically.

• Ludger Overbeck (Deutsche Bank AG, Frankfurt)
  Credit portfolio modeling
  
  Abstract:
  In the talk we first present the basic probabilistic concepts in modeling the credit risk in large portfolios. Then a capital allocation scheme based on the notion of expected shortfall risk is constructed and compared with classical approaches based on variance/covariance analysis. In the last part of the talk the issue of model validation will be raised and first possible steps will be discussed.

  Slides are available in the following formats:
  DVI (19k), Portable Document Format (PDF, 98k), Postscript (PS, 290k)

• Francesco Moresino (The Judge Institute of Management Studies, University of Cambridge.)
  Valuation of corporate debt contracts via stochastic game theory.

• Roger Kaufmann (RiskLab, ETH Zürich)
  Strategic long-term financial risks
  
  Abstract:
  The assets managed by Switzerland's large institutional investors are huge. It is hence very important to have reliable measures of risk for these portfolios. The development of a methodology that could be used for the measurement of strategic long-term financial risks is therefore an important task. Existing modelling instruments such as Risk Metrics allow a relatively good measurement of market risks of trading books. These models, however, have some severe deficiencies if they are applied to longer time periods (typically one year), as it is needed in the case of strategic investments of institutional investors.

  The long-term goal of this RiskLab project is the development of a theoretically well-understood and empirically founded conceptual framework for the measurement of long-term financial risk of strategic investment portfolios.

  In this talk I will primarily give an overview on models that are proposed to be used for measuring long-term financial risks.

• Mary Hardy (University of Waterloo, Canada)
  Maturity guarantees for segregated funds: models, valuation and hedging
  
  Abstract:
  Segregated fund insurance comprises a mutual fund investment with a guaranteed benefit on death or maturity. Typically in Canada these embedded put options are very long term and complex in nature.

  Traditional models for stock returns, including the original Black-Scholes approach, assume that returns follow a geometric Brownian motion. This model is simple and tractable, and may provide a reasonable approximation over shorter time intervals, but is less appropriate for longer term problems. Empirical studies indicate in particular that this model fails to capture more extreme price movements or stochastic variability in the volatility parameter.

  We introduce a regime switching model of stock returns and compare the model with others. Using a Markov Chain Monte Carlo approach to parameter estimation gives a method of quantifying the effect of parameter uncertainty.

  Two approaches to reserving for the embedded options are in common use; a quantile reserve was first suggested by the Maturity Guarantees Working Party of the Faculty and Institute of Actuaries (JIA 107, 1980). The financial engineering approach is to hedge the option. We will compare these two approaches, using the regime switching lognormal model for stock returns.

• Neil Shephard (Nuffield College, University of Oxford)
  BIN models for trade-by-trade data. Modelling the number of trades in a fixed interval of time
  
  Abstract:
  In this talk we propose a simple time series model of the number of transactions made in intervals of length 'Delta' seconds. We call this model the BIN model. The properties of the BIN model are evaluated while we explore connections between this model and Cox processes - that is Poisson processes with random intensities. We apply the modelling framework to data on trades in IBM shares. (Joint work with Tina Hviid Rydberg)

• Nicole El Karoui (Ecole Polytechnique, Palaiseau, France)
  Optimal portfolio management with American capital guarantee
  
  Abstract:
  Let us consider a fund manager who insures customers against losses of their capital. More precisely, at any time a customer may leave the fund with a guaranteed percentile a of his initial endowment. For the manager, the problem is to define an investment strategy in such a way that at any time the value V of the portfolio dominates a given floor K. The simplest version of this problem is to define a self-financing trading strategy for the risky asset S such that at any time V_t > K. When the constraint holds only at maturity, this portfolio insurance problem admits a well-known solution by using a European put on a fixed part y of underlying by the budget constraint y S_o + Put(y S_o,K) = x. With an American guarantee, the American put is useful but in a more complex way, because the strategy associated with an American put is not self-financing. We have to dynamically adjust the amount Y_t invested in the risky asset, by constraining Y_t S_t to stay in the continuation region of the American put in a minimal way. The resulting strategy Y_t S_t + AmerPut(Y_t S_t,K) is self-financing, and dominates K. Because the manager is risk-adverse, he is faced with the following optimization problem: to define the optimal portfolio, dominating K, which maximizes the expected utility of the optimal wealth. We show a separation theorem, that is, it is optimal to select the optimal risk-aversion portfolio S_t, without constraint and then to use the previous strategy on this underlying.
  (Joint work with Monique Jeanblanc and Vincent Lacoste)

• Dilip B. Madan (University of Maryland, USA)
  The fine structure of asset returns: an empirical investigation
  
  Abstract:
  Asset returns are modeled, in the short term, as Lévy processes with both a continuous martingale component and a jump component. The Lévy density for the jump component is completely monotone and parametrically differentiates processes of finite and infinite activity, and finite and infinite variation. The resulting model is termed the CGMY model. Empirical investigations of time series indicate that the dynamics of indices is essentially devoid of a continuous martingale component that may be present in the dynamics of individual stocks. This leads to the conjecture that the risk neutral process should be free of a diffusion component. Empirical investigations of options data tend to confirm this conjecture. The statistical and risk neutral process for indices and stocks tends to be a pure jump process of infinte activity and finite variation. (Joint work with Peter Carr, Hélyette Geman and Marc Yor.)

• Paolo Guasoni (Scuola Normale Superiore, Pisa)
  Risk minimization under transaction costs
  
  Abstract:
  We study the general problem of an agent wishing to hedge some liability at a fixed date in a market with a risky asset, where proportional transaction costs are present. Other frictions, such as incomplete information and constraints on strategies, may also be present. We obtain a minimization problem on a space of predictable processes with finite variation. Borrowing a technique from Calculus of Variation, on this space we look for a convergence where minimizing sequences are compact, and risk is lower semicontinuous. This leads to the class of convex decreasing risk functionals, containing coherent risk measures. If the risky asset is a martingale, we show the existence of an optimal strategy for a class of problems including shortfall minimization and utility maximization. Finally, we make a partial extension of these results to the semimartingale case.

• Martin Baumann (Departement Mathematik, ETH Zürich)
  Valuation of electricity derivatives
  
  Abstract:
  When valuing electricity derivatives, one faces three major challenges: First, electricity cannot be effectively stored, and hence no hedging portfolios with the commodity can be built. Second, electricity shows a very special price behaviour, which clearly distinguishes it from other energy commodities. And finally, some of the most wide spread derivatives in the power markets have an American-style and path dependent payoff structure.

  I will present a spot-price model that accounts for mean reversion and the dramatic price "spikes" one may occasionally observe on the market. From this model, a model for the term structure of futures prices can be deduced, and contingent claims can then be hedged by the use of futures contracts. Furthermore, I will introduce some of the above mentioned contracts, like the popular "swing options".

• Marcel Rüegg (Departement Mathematik, ETH Zürich, and Swiss Re)
  Default risk valuation in incomplete markets
  
  Abstract:
  Credit risk related businesses are nice examples of incomplete markets: illiquid, high transaction costs, few and big transactions, few and big market players, intransparent premiums, prices are not just risk based but also influenced by some regulatory arbitrage, ... The assumptions of a perfect market are therefore far too strong. We need weaker assumptions like no-arbitrage, for modeling our prices of default prone instruments. We identify three important issues in valuing default risk:
  1. find an equivalent risk neutral probability measure under which prices are calculated as risk neutral expectations of discounted cashflows;
  2. modeling under the risk neutral measure a consistent multi-company default model, which allows for default caused dependencies; and finally
  3. to model the recoveries in case of a default event.
  So far we have studied the first two issues, first because they are of more quantitative interest and second because recovery rates are more instrument dependent than from the company's credit quality and last because one does not know statistically much about this quantity. Assuming the existence of an equivalent risk neutral measure, we construct first multi-company default time models. Some preliminary models, where default-caused dependencies exist, are presented. These dependencies between default events is given by a natural system of probability equations. These default-dependency models differ from what was presented in the literature, which - as one can show - do not satisfy the natural dependency equations. Assuming further a specific structure for recoveries given default, we can calculate prices of default prone instruments. We started also some investigations about transformations of equivalent risk neutral measures in this multi-default time frame.

• Freddy Delbaen (Departement Mathematik, ETH Zürich)
  Some technical problems regarding risk measures and the sigma-core of a game
  
  Abstract:
  The definition of a risk measure on the space of all random variables poses difficulties. This leads to the characterisation of convex closed sets P of probability measures with the property that every random variable is integrable for at least one element of P. There are also connections between game theory and the theory of coherent risk measures. If time permits I will show the connection with the so called sigma-core of a game. Some known results from this theory will be extended.

• David Svensson (Chalmers University of Technology, Göteborg, Sweden)
  On a class of renewal processes in a random environment
  
  Abstract:
  We consider a certain generalization of the class of renewal processes with absolutely continuous life length distribution, obtained by introducing a random environment. An underlying birth and death process modulates the stochastic intensity process of the renewal process by using a set of underlying failure rate functions, each of which corresponds to one environment state. A certain Poisson embedding technique turns out to be useful and enables us to establish some couplings, from which we derive results concerning asymptotics, monotonicity properties and domination. In particular, some properties of the renewal function will be discussed, and a generalization of Lorden's inequality is obtained.

• Eckhard Platen (University of Technology, Sydney)
  A minimal share market model with stochastic volatility
  
  Abstract:
  The lecture will discuss a continuous time share market model with a minimal number of factors. These factors are powers of Bessel processes. The asset prices are formed by ratios of the factors and have consequently leptocurtic return distributions. In this framework stochastic volatility with properties that are similar to those actually observed arises naturally. The model generates for the market index the well-known leverage effect due to negative correlation between the index and its volatility. Options on the index show the typically observed implied volatility skew and options on share prices are characterised by the well-known implied volatility smile.

• Thorsten Rheinländer (Technische Universität, Berlin)
  Hedging claims - An exponential utility approach
  
  Abstract:
  We study the problem of hedging claims in incomplete markets. Hedging quality is valued according to an exponential criterion. By a suitable measure transform, it is possible to reduce the hedging problem to the investment problem of maximizing expected exponential utility from terminal wealth. We discuss some suitable choices of spaces of admissible strategies and relate our problem by a duality result to the problem of finding the minimal entropy martingale measure. Moreover, by introducing an additional parameter, which models how risk-averse the investor is, it turns out that the exponential utility approach is asymptotically related to mean-variance hedging as well as superhedging. This is joint work together with Delbaen, Grandits, Samperi, Schweizer and Stricker.

• Henrik Hult (Departement Mathematik, ETH Zürich)
  Quadratic hedging
  
  Abstract:
  We study the problem of finding an optimal hedging strategy and the pricing of options in an incomplete market using quadratic criteria. In a general, continous-time model where the stock price is modelled by a semimartingale X, we provide assumptions on X in order to guarantee existence of an optimal strategy and we present results on the structure of this strategy.

• Uwe Schmock (RiskLab, ETH Zürich)
  Valuation of exotic options under shortselling constraints
  
  Abstract:
  Options with discontinuous payoffs are generally traded above their theoretical Black-Scholes prices because of the hedging difficulties created by their large delta and gamma values. A theoretical method for pricing these options is to constrain the hedging portfolio and incorporate this constraint into the pricing by computing the smallest initial capital which permits super-replication of the option. We develop this idea for exotic options, in which case the pricing problem becomes one of stochastic control. Our motivating example is a call which knocks out in the money, and explicit formulas for this and other instruments are provided.

• Dirk Tasche (Technische Universität, München)
  Risk contributions and performance measurement
  
  Abstract:
  Risk adjusted performance measurement for a portfolio involves calculating the risk contribution of each single asset. We show that there is only one definition for the risk contributions which is suitable for performance measurement, namely as derivative of the underlying risk measure in direction of the considered asset. We also compute the derivatives for some popular risk measures including the quantile-based value at risk (VaR) in a rather general context. As a consequence we obtain a mean-quantile CAPM.

• Thomas Mikosch (University of Groningen)
  Ruin probabilities for random walks with heavy-tailed dependent step sizes

• Aydin Akgün (Ecole des HEC, University of Lausanne)
  Model risk with jump-diffusions
  
  Abstract:
  The main goal of the study is to quantify the effects of model risk within the context of the risk management for interest rate derivatives. Assuming that the model risk emanates from the omission of the finite number of jump terms in the instantaneous forward rate specification, the profit and loss (P&L) function of a short position in a zero-coupon bond European option is derived. A closed-form solution for the bond option value is provided for the case where jumps are driven by a finite state-space compound Poisson process. Numerical methods are used to obtain the empirical distribution of the P&L function. The magnitude of model risk is determined in different contexts and a sensitivity analysis is conducted with respect to parameters such as moneyness, time to maturity, and the subjective volatility estimate of the trader.

• Hanspeter Schmidli (University of Aarhus)
  Optimal proportional reinsurance policies in a dynamic setting

• Nicole Bäuerle (Abteilung Mathematik VII, Universität Ulm)
  Comparing dependencies in risk models
  
  Abstract:
  In the first part we consider multivariate risk portfolios, where the risks are dependent. Using the supermodular and the symmetric supermodular ordering we are able to compare portfolios w.r.t. their dependency. For example, it often occurs that the risks can be grouped such that there is a strong dependence between members of one group but much less dependence between members of different groups. It can be shown now that whenever the group structure majorizes another, the portfolio is greater w.r.t. the symmetric supermodular ordering and thus contains more dependency.

  In the second part we investigate a Markov-modulated ruin model. We vary the dependency with which claims arrive by multiplying the generator of the environment process by a constant. For the finite and also for the infinite horizon ruin probability it can be shown that higher dependence leads to greater ruin functions w.r.t. the stop-loss ordering.

  The talk is based on joint papers with A. Müller and T. Rolski.

• Karsten Prause (Institut für Mathematische Stochastik, Universität Freiburg)
  The generalized hyperbolic model: risk measurement and volatility models
  
  Abstract:
  Generalized hyperbolic (GH) distributions have recently been considered as a starting points for models in Finance. First we will review some properties of these distributions which are useful for the construction of stochastic models.

  Univariate GH Lévy processes have been considered as models for asset price processes. We will review briefly estimation results for univariate return distributions and discuss the performance of derivative pricing models for these processes. See www.fdm.uni-freiburg.de/UK/ for a short introduction.

  Multivariate GH distributions provide the possibility to model tails of asset return distributions better than with normal distributions. In particular, NIG distributions are computationally feasible and lead to substantially better results in backtesting studies than standard value-at-risk approaches.

  Finally we will give an outlook to volatility models of the Ornstein-Uhlenbeck type. These models have been proposed by Barndorff-Nielsen (1998) and are a quite natural refinement of GH Lévy processes. In a joint study with Elisa Nicolato (Aarhus) we will derive option prices and discuss numerical results concerning the computation of these prices.

• Lishang Jiang (Institute of Mathematics, Tongji University, Shanghai, China)
  On path-dependent options
  
  Abstract:
  In this talk we will consider the convergence of binomial tree methods for European/American path-dependent options. The numerical analysis and the notion of viscosity solutions are used to show uniform convergence of binomial tree methods for European/American path-dependent options, including arithmetic average options, geometric average options and lookback options. We also show uniform convergence of the forward shooting grid method for arithmetic average options presented by Barraquand et al. (1996).

• Ulrich Müller (Olsen & Associates)
  High frequency data in finance: Properties, methods and models
  
  Abstract:
  Most financial markets produce high-frequency tick-by-tick data: irregularly spaced time series that contain every single price or quote from a market. This often means thousands of observations per day. Traditional research, by contrast, is based on data sampled at much lower frequencies: yearly, monthly, weekly or daily data.

  The young field of finance based on high-frequency data has quite some advantages:

  1. no information is thrown away;
  2. huge numbers of observations allow for new types of statistical analysis and lead to narrow confidence and significance limits;
  3. insight into intra-daily mechanisms and properties of markets is gained.

  On the other hand, researchers of intra-day data have to pay an entry price: outliers in the raw data have to be identified and removed; the irregular spacing of data in time requires new methods such as time series operators; new effects such as the intra-daily seasonality of volatility have to be studied first.

  High-frequency finance is the special field of Olsen & Associates since 14 years. In the framework of this seminar, only few selected topics can be presented: the most important empirical facts, the time series operator method and finally the EMA-HARCH model.

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