Talks in Financial and Insurance Mathematics, Summer 2000

• Vicky Henderson (RiskLab, ETH Zürich)
  A utility maximisation approach to life insurance guarantees
  
  Abstract:
  We examine maturity guarantees in life insurance where the policyholders benefit from a guaranteed interest rate and a percentage of the performance of the company's asset portfolio. In practice, insurers tend to use a 'close' index as a proxy for their fund, and price and hedge with this. The effectiveness of this method for various levels of correlation is the question addressed here. The approach of this work is to convert the problem into an incomplete markets problem and use recent results in utility maximisation to provide an 'optimal' price and hedge using the index. Numerical examples, by solving a non-linear pde, are presented. These illustrate the effect on the price (via the participation rate) of taking the naive 'proxy' approach, for various levels of correlation.

• Larbi Alili (Vienna University of Technology, Austria)
  Further results on some singular linear stochastic differential equations
  
  Abstract:
  Some particular examples of Volterra kernels of Goursat type are considered and associated linear stochastic differential equations are extensively studied. This is applied to extend some results of Jeulin and Yor about a question related to ergodic properties of the corresponding Volterra transform, from kernels with order one to kernels with any order.

• Stanislav Uryasev (Industrial & Systems Engineering, University of Florida)
  Conditional Value-at-Risk: optimization algorithms and applications
  
  Abstract:
  A new approach to optimizing or hedging a portfolio of financial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR). CVaR is defined as the average losses for the worst x% scenarios (e.g., 10%), while VaR answers the question what is the maximum loss with the specified confidence level. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is considered to be a more consistent measure of risk than VaR. Portfolios with low CVaR necessarily have low VaR as well because CVaR is always bigger than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage firms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming.

• Jesper Lund Pedersen (Department of Operations Research, University of Aarhus)
  Optimal prediction of the ultimate maximum of Brownian motion
  
  Abstract:
  

Workshop on Long Term Financial Risks

The workshop is organized by Swiss Re (Dr. Niklaus Bühlmann) in connection with RiskLab, ETH Zürich. Due to the great interest in the workshop, we regret that room constraints don't allow us to accept further registrations.

Program:

Previous RiskLab workshop: May 4, 2001

• Jia-An Yan (Institute of Applied Mathematics, Academica Sinica, Beijing)
  Martingale measure method for utility maximization in an incomplete market
  
  Abstract:
  In this talk I will present a joint work with Dr. Xia Jian-ming about the problem of utility maximization in an incomplete market with a semimartingale model. In this case the "fictitious completion" method is no longer applicable. Instead, for a given utility function, we specify a class of martingale measures Mc. For any x>0 and Q in Mc we construct a contingent claim x_x,Q(T) at time T. If for some Q^*, x_x,Q(T) can be replicated by a self-financing strategy with initial wealth x, then this strategy maximizes the expected utility of the terminal wealth, and the option pricing using the martingale measure Q^* coincides with that using Davis' "marginal rate of substitution" rule and has a transparent financial meaning. For HARA utility function log x (resp. (x^g-1)/g for g<0), the historical probability measure has the minimum relative entropy (resp. maximizes Hellinger integral of order g/(g-1)) w.r.t. Q^* within Mc. and for utility function -(1-gx)^1/g with g<0, Q^*, within Mc, maximizes Hellinger integral of order g/(g-1) w.r.t. the historical probability measure. If the market driven by a Lévy process, the optimal strategy and the related martingale measure are further worked out.

• George Papanicolaou (Stanford University)
  Interest rate models with stochastic volatility and options on interest rates
  
  Paper: Compressed postscript, Portable document format (pdf)

• George Papanicolaou (Stanford University)
  A review of the asymptotic theory of stochastic differential equations

• George Papanicolaou (Stanford University)
  Fast mean reverting stochastic volatility
  
  Paper: Compressed postscript, Portable document format (pdf)

• George Papanicolaou (Stanford University)
  Hedging and portfolio optimization with stochastic volatility
  
  Reference: Derivatives in Financial Markets with Stochastic Volatility
  by J.P. Fouque, G. Papanicolaou and R. Sircar,
  Cambridge University Press, June 2000.

• Olivier Scaillet (Université catholique de Louvain)
  Nonparametric estimation and sensitivity analysis of expected shortfall
  
  Abstract:
  We consider a nonparametric method to estimate the expected shortfall, i.e. the expected loss on a portfolio of financial assets knowing that the loss is larger than a given quantile. We derive the asymptotic properties of the kernel estimators of the expected shortfall and its first oder derivative with respect to portfolio allocation in the context of a stationary process satisfying strong mixing conditions. Monte Carlo experiments with a vector autoregressive process of order one and truncated loss distributions with a generalized Pareto distributed right tail are reported to assess the behavior of the estimators. An empirical illustration is given for a portfolio of French stocks. Another empirical illustration deals with Danish data on fire insurance losses.

• George Papanicolaou (Stanford University)
  Stochastic volatility effects in option pricing
  
  Paper: Compressed postscript, Portable document format (pdf)

• Michael Taksar (Department of Applied Mathematics, State University of New York at Stony Brook)
  Business risk vs. financial risk. Which comes first?
  (Optimal dynamic portfolio selection and dividend distribution policy for a corporation with controllable risk.)
  
  Abstract:
  We consider a problem in which the liquid assets or reserves of a company are modeled by a diffusion process. At each moment of time the management of the company makes a decision of the amount of dividends paid-out to the shareholders. There is also a possibility to reduce risk exposure by conducting a less aggressive business activity, which also results in a smaller potential profit. In the case of an insurance company different business activities correspond to different reinsurance levels, which the insurance company may choose. Mathematically this corresponds to decreasing simultaneously drift and diffusion coefficients of the controlled process. In addition the reserve of the company can be invested in a stock market in which asset prices follow the Black-Scholes model. The objective is to find the policy which maximizes the expected present value of the cumulative dividend distributions.

  Mathematically this problem is a mixed singular/regular stochastic control problem. Its analytical part consists of a solution to a series of related free boundary problems for linear and nonlinear ODEs. The resulting optimal process is a diffusion process whose coefficients are determined via the solutions to those ODEs and which is reflected from one of the free boundaries. The optimal control functional which represents the cumulative dividend distributions is the local time of the optimal process at the reflections boundary. It is singular, i.e., it is continuous but not absolutely continuous w.r.t. time, where the term "singular control" comes from. Despite mathematical sophistication these results have a natural and transparent economic interpretation.

• Thorsten Rheinländer (ETH Zürich)
  Momentum traders and instabilities of financial markets
  
  Abstract:
  There is empirical evidence that price processes of financial assets show stylized facts like heavy tailed returns, volatility clusters and large price movements not accompanied by any dramatic news events. We discuss whether this observed behavior can be explained by the activity of momentum traders. These are agents who take past price movements as a signal for their investment decisions in a trend-chasing fashion. This is joint work with Marcus Steinkamp, TU Berlin.

• Artan Borici (IFOR, ETH Zürich)
  Fast efficient frontier computations for the american style options
  
  Abstract:
  The standard tools of option pricing may be too slow to compute the efficient frontier of an american style option. Alternative computational schemes based on the Black-Scholes continuous dynamics discretized on a homogeneous lattice are presented. They lead to a proper class of the linear complementarity problem (LCP) for Z-type matrices. Combined with the ordered structure of the payoff function, a special algorithm proposed by Dempster et al may be devised. Its complexity scales linearly with the number of the lattice points and it is compared to the complexity of other available methods.

• Hartmut Milbrodt (Mathematisches Institut der Universität zu Köln)
  . und sie bewegt sich doch! Ein-blick in Geschichte und Gegenwart der Personenversicherungsmathematik

• Ernst Eberlein (University of Freiburg i. B., Germany)
  Lévy motions as a basic tool in finance
  
  Abstract:
  In standard mathematical finance Brownian motion plays the dominating role as driving process for modelling price movements. In order to achieve a better fit to real-life data it is, however, preferable to replace Brownian motion by a Lévy process. Generalized hyperbolic Lévy motions are processes which allow an almost perfect fit to financial data. We discuss in detail what the consequences for asset price modelling, derivative pricing and interest rate theory are. We also touch on aspects of multivariate and intraday modelling.

• Wolfgang Stummer (University of Ulm, Germany)
  On exponentials of Markov processes, with applications to mathematical finance and statistical informations
  
  Abstract:
  We present necessary and sufficient conditions in order that exponentials of additive functionals L of Markov processes X have finite expectations. Furthermore, we obtain estimates for these expectations. Special attention will be given to the expectation of exp(L_0,T) := exp(∫ ₀^T f(z, X_z) dz), where f is a positive function, T is a fixed deterministic time, and X_z is a generalized geometric Brownian motion. It will be shown how the above-mentioned results can be used to construct non-lognormally distributed price processes Y. For these, we study the absence of arbitrage opportunities and the valuation of contingent claims, e.g. call options on Y. We also investigate the decision risk which is contained in the following dichotomous Bayesian decision problem: does the price process Y have a certain drift or does it have zero drift? The reduction of the decision risk that can be attained by observing the path of Y, can be regarded as a measure of information. We give some estimates for this risk reduction, as well as connections with some generalized relative entropies.

• Hartmut Milbrodt (Mathematisches Institut der Universität zu Köln)
  On the loss created by a life insurance policy
  
  Abstract:
  Hattendorff's theorem on the decomposition of the variance of the overall loss created by an insurance contract is obtained for policy developments given by an inhomogeneous continuous-time Markov process with a possibly non-smooth transition matrix. Due to the lack of smoothness assumptions, the result covers classical discrete versions of Hattendorff's theorem as well and is also applicable to "mixed" situations in which some transitions have smooth transition probabilities whereas others can only take place at discrete times.

• Guus A. Balkema (Universiteit van Amsterdam)
  On convergence of extremal processes

RiskLab Workshop on Credit Risk

Program:

Organizer: Dr. Uwe Schmock
Workshop secretary: Gerda Schacher
Previous RiskLab event: Risk Day 1999

• Mark H. A. Davis (Financial and Actuarial Mathematics, TU Vienna, Austria)
  Optimal hedging with basis risk
  
  Abstract:
  It often happens that options are written on underlying assets that cannot be traded directly, but where a 'closely related' asset can be traded. Rather than simply using the traded asset as a proxy for the option underlying, one should calculate some 'best' hedging strategy. This is an 'incomplete markets' problem, and we address it using a utility maximization approach. With exponential utility the optimal hedging strategy can be computed in reasonably explicit form using the methods of convex duality. In particular, a perturbation analysis using ideas of Malliavin calculus gives the modification to the exact replication strategy that is appropriate when the option underlying and traded assets are highly, but not perfectly, correlated.

• Deimante Rusaityte (visiting ETH Zürich)
  Continuity estimates for ruin probabilities
  
  Abstract:
  This talk is devoted to the continuity problem of ruin probabilities and the goal is to obtain continuity estimates with respect to the governing parameters (i.e. such as claim size distribution, inter-occurrence times distribution, etc.). In practice these parameters are not known but are estimated from the data and are assumed to be close to the true parameters in some sense. Moreover, in many cases ruin probability cannot be found explicitly and therefore the continuity estimates become crucial.

  We consider the Markov modulated risk model and models with stochastic Lévy processes - driving interest. The continuity estimates are stated in terms of the weighted metrics, allowing us to compare the tail behavior of the corresponding ruin probabilities.

• Patrick Cheridito (ETH Zürich)
  Mixed fractional Brownian motion
  
  Abstract:
  We show that the sum of a Brownian motion and an independent fractional Brownian motion with Hurst parameter H in (0,1] is a semi-martingale if and only if H = 1/2 or H is in (3/4,1].

• Walter Schachermayer (TU Wien)
  Optimal investment in incomplete financial markets
  
  Abstract:
  We study the problem of maximizing the expected utility of terminal wealth in the framework of a general incomplete semimartingale model of a financial market. We show that the necessary and sufficient condition on a utility function for the validity of several key assertions of the theory to hold true is the requirement that the asymptotic elasticity of the utility function is strictly less than one.

• The Financial Stochastics Group at EURANDOM is organizing a workshop on Risk, Insurance and Extremes in Eindhoven (The Netherlands), 24-26 August, 2000. Before the workshop there is a one-day Concentrated Course on Extremes and Financial Risks

  Scientific Committee of the workshop: Thomas Mikosch, Casper de Vries, Claudia Klüppelberg, Paul Embrechts, Holger Rootzén.

• Thomas Møller (Laboratory of Actuarial Mathematics, University of Copenhagen)
  Quadratic hedging approaches and indifference pricing in insurance
  
  Abstract:
  We study the problem of hedging and valuating insurance contracts which combine pure insurance risk and purely financial risk. One example is a unit-linked life insurance contract where premiums and benefits are linked to the development of some stock index or the performance of an investment fund. Other examples are so-called double triggered stop-loss contracts and financial stop-loss contracts. With the last mentioned, the insurer's total losses have both an insurance and a financial component. We first determine risk-minimizing hedging strategies for payment streams generated by unit-linked life insurance contracts and for (non-life) insurance risk processes where claim amounts are affected by a traded asset. We then investigate financial transformations of the actuarial variance and standard deviation principles, which can be obtained via an indifference argument. Existing results are complemented by the derivation of optimal trading strategies for the two financial valuation principles when the discounted price process of the traded assets is a continuous semimartingale. Furthermore, we determine simple upper and lower bounds for the fair premiums under the financial valuation principles. These bounds seem particular relevant for the valuation and hedging of reinsurance contracts with financial risk.

• Rüdiger Frey (Swiss Banking Institut, University of Zurich)
  Rechenkünstler oder akademische Zauberlehrlinge?
  Über Möglichkeiten und Grenzen mathematischer Methoden im finanziellen Risikomanagement

  Slides are available in the following formats:
  Postscript (ps, 2545k), Portable Document Format (pdf, 232k), Compressed Postscript (gzip, 940k).

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