abstracts

• Arun Bagchi, University of Twente

Parameter Estimation in Continuous-Time Financial Data: Application to Exponential-Affine Term Structures
The exponential-affine term structure model is a class of models in which the yields to maturity are affine functions of some state vector x(t). Since the interest rate factors x(t) are not directly observed, unknown parameters in these models need to be estimated on the basis of observing the bond prices of different maturities. Although the state space model is set-up in continuous time, all existing parameter estimation techniques discretize the observation equation in time in order to use known statistical/filtering methods. We resolve this incongruity in the present paper by working throughout with the original continuous-time formulation. We explain the maximum likelihood parameter estimation methodology in this framework and discuss modifications needed when the observation noise covariance is unknown. Finally we illustrate the methods by means of extensive simulation studies. Application to real treasury data will also be discussed.

• Hans-Peter Deutsch, Andersen

Second Order Approximations for Fast Value-at-Risk Computations
As soon as a portfolio contains significant optionality standard (delta normal) variance-covariance Value-at-Risk calculations lead to very inaccurate results while full fledge Monte Carlo Simulations with full re-pricing often involve unacceptably long computing time. A natural compromise is to extend the delta normal variance-covariance methods to include 2nd order terms of the portfolio value's taylor expansion. This talk highlights the algebraic, analytical and statistical tasks involved when doing such a delta-gamma approximation and presents several different alternative ways for calculating a Value at Risk in such a framework.
[Presentation]

• Jürgen Hakala, Commerzbank

Stochastic volatility models: A Finite Difference Approach
Many exotic options are very sensitiv to changes in implied volatility. In such cases it is essential to have a model of the underlying which reflects the change of volatility with time quite well. One approach is Heston's stochastic volatility model which assumes that the volatility of an asset is a stochastic process itself. However for exotic options no closed form solutuions are known. Instead a PDE derived from the stochastic model is solved. We present some ideas and first results on how to improve the speed to calculate quite accurate prices using the Finite Difference Method.
This is a joint presentation of Jürgen Hakala and Tino Kluge.
[Presentation]

• Wolfgang Härdle, Humboldt University of Berlin

The Dynamics of Implied Volatilities: A Common Principle Component Approach
It is common practice to identify the number and sources of shocks that move implied volatilities across space and time by applying Principal Components Analysis PCA) to pooled covariance matrices of changes in implied volatilities. This approach, however, is likely to result in a loss of information, since the surface structure of implied volatilities in the maturities and moneyness dimension is neglected. In this paper we propose to estimate the implied volatility surface at each point in time nonparametrically and to analyze the implied volatility surface slice by slice with a common principal components analysis (CPCA). As opposed to traditional PCA, the basic assumption of CPCA is that the space spanned by the eigenvectors is identical across groups, whereas variances associated with the components are allowed to vary. This allows us to study a p variate random vector of k groups, say the ''volatility smile'' at p different grid points of moneyness for k maturities, simultaneously. Our evidence suggests that surface dynamics can indeed be traced back to a common eigenstructure between covariance matrices of the surface ''slices'', which allow for the usual shift, slope, and twist interpretation of shocks to implied volatilities. This insight is a suitable starting point for VaR Monte Carlo Simulations of delta-gamma neutral, vega sensitive option portfolios.
[Presentation] [Paper]

• Norbert Hofmann, Goethe-University

Approximating the square root process
We consider numerical methods for the simulation of paths of the square root process. We introduce a special implicit method that reflects the positivity of the exact dynamics. Moreover we show that the new method is suited to overcome stability problems. By means of simulation studies we compare the new method with the Euler scheme. It turns out that modifications of the Euler scheme fail.
This is joint work with Eckhard Platen (University of Technology Sydney, Australia).
[Presentation]

• Tino Kluge, Chemnitz University of Technology

Stochastic volatility models: A Finite Difference Approach
Many exotic options are very sensitiv to changes in implied volatility. In such cases it is essential to have a model of the underlying which reflects the change of volatility with time quite well. One approach is Heston's stochastic volatility model which assumes that the volatility of an asset is a stochastic process itself. However for exotic options no closed form solutuions are known. Instead a PDE derived from the stochastic model is solved. We present some ideas and first results on how to improve the speed to calculate quite accurate prices using the Finite Difference Method.
This is a joint presentation of Jürgen Hakala and Tino Kluge.
[Presentation]

• Lutz Molgedey, Andersen

Libor market model with stochastic time homogenous mean reverting volatility
We present a variant of the Libor market model with stochastic volatility. As for the usual deterministic volatility Libor market model we require the parametrization of the model to be as time homogenous as possible. Here, this is achieved by using time homogenous mean reversion levels and speeds for the stochastic volatilities of the respective forward rates. Correct (perfect) pricing of the (at-the-money) caplets corresponds then to non-stationary initialvalues of the forward rate volatilities. However, demanding a time homogenous model restricts possible caplet smile surfaces. Those restrictions (and advantages) will be discussed in the talk.
[Presentation]

• Jörn Rank, Andersen

Improving VaR Calculatuions by Using Copulas and Non-Gaussian Margins
Apart from historical simulation, most Value-at-Risk (VaR) methods assume a multivariate normal distribution of the risk factors. In this work we present the application of copulas for the calculation of the VaR. This enables us to use arbitrary distribution functions for the risk factros. The risk factors themselves are linked together by a copula function that describes the dependence structure between them. We discuss the modification of the Monte-Carlo (MC) method of the VaR calculation under this generalization. Usinga financial portfolio based on historical FX rates over a period of ten years, we compare the backtesting results obtained from the "traditional" MC method with the one from the "copula" MC method, using various copulas and various distribution functions for the margins.
[Presentation]

• L. C. G. Rogers, University of Bath

Monte Carlo valuation of American options
This paper introduces a `dual' way to price American options, based on simulating the path of the option payoff, and of a judiciously-chosen Lagrangian martingale. Taking the pathwise maximum of the payoff less the martingale provides an upper bound for the price of the option, and this bound is sharp for the optimal choice of Lagrangian martingale. As a first exploration of this method, three examples are investigated numerically; the accuracy achieved with even very simple-minded choices of Lagrangian martingale is surprising. The method also leads naturally to candidate hedging policies for the option, and estimates of the risk involved in using them.
[Paper]

• Uwe Schmock, ETH and University of Zürich

Term structure models for credit risks
This talk gives an overview on the available theoretical methods for pricing defaultable bonds. We review popular models for the term structure of risk-free interest rates, introduce hazard rates and loss fractions and thereby motivate default-adjusted interest rates. Several modelling assumptions for recovery at default are mentioned. We proceed by discussing the term structure of credit spreads and their dependence on risk-free interest rates. We conclude with remarks on the modelling of dependent defaults.
[Presentation]

• Ingo Schneider, BHF-Bank

Using Finite Differences for Pricing Options
This is a pratical session which emphasizes the details implementing a numerical method. We will show a step by step implementation of various finite difference schemes in order to solve a Partial Differential Equation. Based on a a paper about "Pricing Arithmetic Average Asian Options (Vecer 2001) we apply the finite difference methodologie and compare the results with Monte Carlo. The interested participant is asked to bring his laptop with EXCEL and VBA installed.
[Presentation]

• Peter Schwendner, Sal. Oppenheim jr. & Cie

Quantitative Aspects of Equity Derivatives Trading
Academics usually analyse the pricing of very complicated derivatives, regardless if they are actively traded or not. But also plain vanilla option books can be a rich playground for quantitative concepts. In this talk, I present some quant aspects of the risk management of a large number of simple equity derivatives. I will refer to some joint work with colleagues and friends.
Contents:

• What is special about equity derivatives in comparison to other asset classes?
• Retail derivatives business model
• Hedging of equity derivatives books using EUREX options
• Fast pricing of a large number of options for market-making using price caching
• Implementation of Monte Carlo simulation for value-at-risk estimation
• Tino Senge, Commerzbank

Jump-Diffusion Models in Foreign Exchange Markets
Jump-diffusion models to price FX options will be considered. Different distributions for the jump size will be stated and analysed. Features and limitations of the obtained models will be presented. Besides theoretical results and notes on implementation, the question will be discussed whether this models can be used to price options taking the volatility smile in todays FX markets into account.
[Presentation] [Charts]

• Steven E. Shreve, Carnegie Mellon University

A Unified Model for Credit Derivatives
This is joint work with Alain Belanger of Scotia Capital Markets and Dennis Wong of Bank of America. A framework is provided for pricing derivatives on defaultable bonds and other credit-risky contingent claims. The framework includes structural models (those in which the time of default is determined by the value of the issuing firm), general reduced-form models (those in which default is exogenous), and reduced-form models in which default can occur only at specific times, such as coupon payment dates. Within the general framework, multiple recovery conventions for contingent claims are considered: recovery of a fraction of par, recovery of a fraction of a no-default version of the same claim, and recovery of a fraction of the pre-default value of the claim. These recovery conventions are matched to appropriate default protection contracts. A stochastic-integral representation for credit-risky contingent claims is provided, and the integrand for the credit exposure part of this representation is identified. In the case of intensity-based reduced-form models, credit spread and credit-risky term structure are studied.
[Paper]

• Gerhard Stahl, Federal Banking Supervisory Office

Modelling Event Risk
The talk will review the on-going joint work with E. Platen, Sydney, on modelling specific market risk for equities. The talk will cover the following topics: benchmarked prices as basic inputs of risk models, modelling event risk based on t-distributions and regulatory implications. Finally, an empirical study for the relevant equity marktes provides insights on the validity of the proposed models.

• Hermann Stahl, Commerzbank

Documentation of OTC Derivatives and other Financial Instruments
The markets for OTC derivatives and securities repurchase and lending have created their own documentation standards. Transactions are documented with trade confirmations which refer to a master agreement. The master agreement provides for the legal and credit terms and integrates all transactions into the master agreement by forming one single agreement. Organizations on national and international level have produced master agreements for various types of business and published sets of definitions that simplify the task of documenting individual trades.
A positive side effect of master agreements is that the credit exposure that both parties have under the various transactions may be netted for capital adequacy purposes.
[Presentation]

• Felix Streichert, University Tübingen

Evolutionary Alogrithms and Finanical Applications
Evolutionary Algorithms (EA) constist of serveral heuristics which are able to solve optimization tasks by imitating some aspects of natural evolution. They may use different levels of abstraction, but they are always working on whole populations of possible solutions for a given task. EAs are an approved set of heurisitcs which are flexible to use and postulate only neglectible requirements on the optimization task.
As a practical application technical trading rules found by the use of EA will be presented.
[Paper]

• Josef Teichmann, Technical University Vienna

On finite dimensional Term structure models
We provide the characterization of all finite-dimensional Heath--Jarrow--Morton models that admit arbitrary initial yield curves. It is well known that affine term structure models with time-dependent coefficients (such as the Hull--White extension of the Vasicek short rate model) perfectly fit any initial term structure. We find that such affine models are in fact the only finite-factor term structure models with this property. We also show that there is usually an invariant singular set of initial yield curves where the affine term structure model becomes time-homogeneous. We also argue that other than functional dependent volatility structures -- such as local state dependent volatility structures -- cannot lead to finite-dimensional realizations. Finally, our geometric point of view is illustrated by several examples.
[Paper]

• Robert Tompkins, Technical University Vienna

The relation between implied and realised probability density functions
A number of financial regulators [see Neuhaus (1995), Bahra (1996, 1997), McManus (1999) and Shiratsuka (2001)] have suggested that risk neutral densities (RND) associated with options markets could provide useful indicators of future market turbulence. Critical to this assumption is that such RNDs should provide an unbiased forecast of realised probability density functions. To date, this assumption has not been fully examined.
In this research, we test the ability of RNDs for options on the S&P 500 and the British Pound / US Dollar to predict future probability densities. We consider three approaches to estimate the RNDs, which are consistent with approaches proposed and used by financial regulators. We also provide a number of new testing procedures to assess the efficiency and unbiasness of the forecasts. These tests provide more power than the usual Komolgorov/Smirnov tests.
Using non-overlapping quarterly data from the mid 1980s to 2000, we find that we can reject the hypothesis that the RNDs for both the S&P 500 and British Pounds are unbiased forecasts. Even with a limited number of observations, the tests are powerful enough to allow rejection. These results are consistent with Weinberg (2001) and are more robust as this work relied upon the use of overlapping data.
These results tend to support the conclusions of Shiratsuka (2001), that RNDs should not be used by financial regulators as financial indicators, and that such use could prove counterproductive; actually increasing future market turbulence rather than alleviating it.
[Presentation]

• Jürgen Topper, Andersen

Applying Generalized Passport Options
Except for special cases, generalized passport options do not have closed-form solutions. Here we show how to derive approximate solutions using finite element methods. We also show that finite elements offer advantages in computing the hedge parameters. These techniques are applied to special cases of the generalized passport option which include Asian options and diverse passport options with caps and/or barriers.
[Paper] [Presentation]

sponsors

Andersen Financial and Commodity Risk Consulting	Commerzbank AG	Deutsche Bank AG	Landeszentralbank Hessen