abstracts

• Dr Jörg Behrens, Ernst & Young, Switzerland

Modelling dependencies: some practical examples
We discuss the impact of modelling dependencies for a few selected examples of practical relevance.

• Dr Stefan Benvegnu, Deutsche Bank

Credit Risk for CDO Portfolios
A model that assesses the credit risk for a CDO portfolio is presented. Starting with the introduction of different types of CDOs, it is shown how a Merton type model can be used to obtain risk characteristics of different tranches of CDOs. A short comparison to other existing models that evaluate CDO structures is given.

• Dr Nicole Branger, Goethe University

Is Volatility Risk Priced? Properties of Tests Based on Option Hedging Errors
This paper provides an in-depth analysis of the properties of popular tests for the existence and the sign of the market price of volatility risk. These tests are frequently based on the fact that for some option pricing models under continuous hedging the sign of the market price of volatility risk coincides with the sign of the mean hedging error. Empirically, however, these tests suffer from both discretization error and model risk. We show that these two problems may cause the test to be either no longer able to detect additional priced risk factors or to be unable to identify the sign of their market prices of risk correctly. Our analysis is performed for the model of Black Scholes and the stochastic volatility (SV) model of Heston. In the model of BS, the expected error for a discrete hedge is positive, leading to the wrong conclusion that the stock is not the only priced risk factor. In the model of Heston, the expected hedging error for a hedge in discrete time is positive when the true market price of volatility risk is zero, leading to the wrong conclusion that the market price of volatility risk is positive. If we further introduce model risk by using the BS delta in a Heston world we find that the mean hedging error also depends on the slope of the implied volatility curve and on the equity risk premium. Under parameter scenarios which are similar to those reported in many empirical studies the test statistics tend to be biased upwards. This means that sometimes the test does not detect negative volatility risk premia, or it signals a positive risk premium when it is truly zero. The results of this test furthermore strongly depend on the location of current volatility relative to its long-term mean, and the degree of moneyness of the option. As a consequence the empirical tests may suffer from the problem that the researcher cannot draw the hedging errors from the same distribution repeatedly. This implies that there is no guarantee that the empirically computed t-statistic has the assumed distribution.

• Prof Pierre Collin-Dufresne, Carnegie Mellon University

Identification and estimation of maximal affine term structure models: an application to stochastic volatilty
We propose a canonical representation for affine term structure models where the state vector is comprised of the first few Taylor-series components of the yield curve and their quadratic (co-)variations. With this representation: (i) the state variables have simple physical interpretations such as level, slope and curvature, (ii) their dynamics remain affine and tractable, (iii) the model is by construction `maximal' (i.e., it is the most general model that is econometrically identifiable), and (iv) modelinsensitive estimates of the state vector process implied from the term structure are readily available. We find that the `unrestricted' A1 (3) model of Dai and Singleton (2000) estimated by `inverting' the yield curve for the state variables generates volatility estimates that are negatively correlated with the time series of volatility estimated using a standard GARCH approach. This occurs because the variance of the short rate is at the same time a linear combination of yields (i.e., it impacts the cross-section of yields), and the quadratic variation of the spot rate process (i.e., it impacts the time-series of yields). We then investigate the A1 (3) model which exhibits `unspanned stochastic volatility' (USV). This model predicts that the cross section of bond prices is independent of the volatility state variable, and hence breaks the tension between the time-series and cross-sectional features of the term structure inherent in the unrestricted model. We find that explicitly imposing the USV constraint on affine models significantly improves the volatility estimates, while maintaining a good fit cross-sectionally.
[Paper]

• Dr Bernd Engelmann, Bundesbank

The Basel II IRB Approach - How Could a Validation Procedure Look Like?
Under the proposed new capital adequacy framework, Basel II, capital charges for credit risk in the IRB approach are based on several risk parameters of an individual exposure estimated by financial institutions, the probability of default, the loss given default, and the exposure at default. In this talk we will focus on rating systems and analyse several quality measures we have found in the literature. We will discuss their usefulness for validation purposes and draw conclusions on how a validation procedure could look like.

• Dr Robert Fiedler, Algorithmics Frankfurt

Quantification of Funding Liquidity Risk in a Common Framework with ALM
The quantification of Funding Liquidity Risk is developed and discussed. The concepts of Expected Liquidity (Forward Cash Exposure), Expected Liquidity-at-Risk, Counterbalancing Capacity and Day-Count-to-Default are introduced and it is discussed how they fit into an overall ALM strategy as well as into more general regulatory / supervisory requirements.
The transfer of methodologies from trading risk to ALM is examined; difficulties and alterations are discussed and it is considered how they fit into recent regulatory initiatives (BIS2).

• Prof Rüdiger Frey, University of Leipzig

On Dynamic Models for Portfolio Credit Risk and Credit Contagion
It is by now well known that the performance of models for portfolio credit risk is very sensitive to the modelling of dependence between defaults of different obligors. In this talk we will be concerned with dynamic models for portfolios of dependent defaults. After a brief survey of existing approaches, we concentrate on models for credit contagion, i.e. models where the default of one company has a direct impact on the default intensity of other firms. We introduce a Markovian model and discuss the various types of interaction. Finally we present limit results for large portfolios in a homogeneous model with mean-field interaction and analyze the impact of credit contagion on the portfolio loss distribution.

• Dr Jürgen Hakala, Commerzbank

A time-discrete model for forward volatility
Jürgen Hakala, Ulrike Polte, Dimitri Topaj
A model for forward volatilities in a time­discrete volatility setting is applied to first generation exotic options. To get a tractable simplified model of the real world the assumption of observable market volatilities is relaxed to the point that market volatilities are evolving continuously through time, but can be observed at few points in time only. The natural question arising within that setting is a suitable hedge strategy and the implied costs arising from the chosen strategy. The strength of this approach is the intuitive meaning of the leading terms in a Taylor­Expansion of the pricing equation.

• Dr Vicky Henderson, University of Oxford

A Comparison of q-optimal option prices in a Stochastic Volatility Model
This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result that convex option prices are decreasing in the market price of volatility risk.
We investigate the q-optimal class of pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance optimal pricing measures. If the mean-variance tradeoff is deterministic, this collapses to the well known result that option prices computed under these three pricing measures are the same.
Specialising to the Heston model with mean-variance tradeoff increasing in volatility, enables us to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Choice of q is shown to influence the shape of the implied volatility smile for varying maturity options.

• Dr David Hobson, University of Bath

Real options, non-traded assets and utility indifference prices
We consider a financial model with both traded and non-traded assets, and show that the utility indifference (bid) price for a contingent claim on a non-traded asset is bounded above by the expectation under the minimal martingale measure. This bound also represents the marginal price for the claim. The bound and the marginal bid price are independent of both the utility function and initial wealth of the agent.

• Prof Claudia Klüppelberg, Technical University Munich

Optimal Portfolios with Bounded Capital-at-Risk
We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. For the mean-CaR problem we make use of an approximation of the Lévy process as a sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are in vestigated.
[Paper] to appear in Finance and Stochastics

• Prof Christoph Kühn, Frankfurt MathFinance Institute

Game Options: Pricing, Hedging, and Optimal Exercise
A game option (also referred to as Israeli option or recall option) is a generalization of an American option which also enables the seller to terminate it before maturity, but at the expense of a penalty.
In the first part of the talk we introduce a general approach how to price derivatives in incomplete markets which can also be applied to game options.
Then, we describe the optimal exercise and hedging strategy for the Israeli put option and present a Monte Carlo valuation approach similar to Rogers (2002) (joint work with Jan Kallsen and Andreas Kyprianou).

• Dr Jürgen Linde, Dresdner Bank

Efficient Numerical Techniques for Pricing Multivariate Options
The solution of multidimensional PDEs can only be achieved by interplay of several sophisticated numerical techniques. For high-dimensional problems it is essential to identify a few principal components that accurately approximate the full system. These equations can be dealt with by sparse grid techniques, e.g. the combination technique that extrapolates the solution from substantially less degrees of freedom and additionally allows parallelisation by simple data distribution. For the solution of lower dimensional problems (or thus reduced systems) efficient numerical techniques - combining the building blocks discretisation, grid generation and solver - are at hand. These methods ensure a stable and high order discretisation. Non-smooth regions as they typically appear in option pricing problems can be handled by a priori (e.g. refinement by mesh grading) or more advanced a posteriori (e.g. adapted to finite element methods) adaptivity strategies. Large discretised systems can be solved efficiently by multigrid techniques that can handle unrestricted problems (resulting from European options) as well as obstacle problems (stemming from American style options). In several cases high-order schemes directly yield estimates for sensitivities.
This is a joint presentation of Dr Jürgen Linde and Christoph Reisinger.

• Dr Fehmi Özkan, University of Freiburg

Levy processes in credit risk
Mathematical credit risk models in the literature are mainly models based on Brownian motion although it is known that real-life financial data provides a different statistical behaviour than that implied by these models. Lévy processes are an appropriate tool to increase accuracy of models in finance. They have been used to model stock prices, and term structures of interest rates, thus allowing more accurate derivative pricing and risk management. This presentation shows how Lévy processes can be applied to credit risk models.

• Christoph Reisinger, University of Heidelberg

Efficient Numerical Techniques for Pricing Multivariate Options
The solution of multidimensional PDEs can only be achieved by interplay of several sophisticated numerical techniques. For high-dimensional problems it is essential to identify a few principal components that accurately approximate the full system. These equations can be dealt with by sparse grid techniques, e.g. the combination technique that extrapolates the solution from substantially less degrees of freedom and additionally allows parallelisation by simple data distribution. For the solution of lower dimensional problems (or thus reduced systems) efficient numerical techniques - combining the building blocks discretisation, grid generation and solver - are at hand. These methods ensure a stable and high order discretisation. Non-smooth regions as they typically appear in option pricing problems can be handled by a priori (e.g. refinement by mesh grading) or more advanced a posteriori (e.g. adapted to finite element methods) adaptivity strategies. Large discretised systems can be solved efficiently by multigrid techniques that can handle unrestricted problems (resulting from European options) as well as obstacle problems (stemming from American style options). In several cases high-order schemes directly yield estimates for sensitivities.
This is a joint presentation of Dr Jürgen Linde and Christoph Reisinger.

• Dr Richard Rossmanith, d-fine

Data Quality Measures and Completion of Market Data
Data Quality Measures and Completion of Market Data "Outliers" and "incomplete data" are very common problems that financial institutions face when they collect financial time series in IT data bases from commercial data vendors for regulatory, accounting, and benchmarking purposes. We argue that outlier detection and data completion are closely (almost interchangeably) related. Then we present several data completion techniques, some of which are productively used in practice, and others which are less established. Finally, optimal completion methods are recommended, based on an empirical study for the completion of swap and forward rate curves in the currencies Deutsche Mark (respectively Euro), Pound Sterling, and US Dollar.
This talk is based on the speaker's thesis for the Mathematical Finance postgraduate program at Oxford University (the thesis is available upon request, or can be downloaded from www.d-fine.de). It represents a snapshot of ongoing market data and time series research at d-fine, which continues to concern the speaker and several of his colleagues.

• Prof Wolfgang Schmidt, Hochschule für Bankwirtschaft, Frankfurt

Modeling Default Dependence with Threshold Models
We investigate the problem of modeling defaults of dependent credits. In the framework of the class of structural default models we study threshold models where for each credit the underling ability-to-pay process is a transformation of a Wiener processes. We propose a model for dependent defaults based on correlated Wiener processes whose time scales are suitably transformed in order to calibrate the model to given marginal default distributions for each underlying credit. At the same time the model allows for a straightforward analytic calibration to dependency information in the form of joint default probabilities. We illustrate the application of the model providing some examples of the pricing of basket default swaps.
This is a joint paper with Ludger Overbeck.
[slides]

• Prof Ronald Smith, Loughborough University

Optimal compact finite-difference scheme for the Black-Scholes Equation
Compact numerical schemes use the minimum number of successive mesh points and of time steps. For the Black-Scholes equation this is 3 (arbitrarily-spaced) share price mesh points by 2 time steps. A general compact numerical scheme would be a linear implicit equation involving the computed option values at 6 grid points. Thus, 5 degrees of freedom are available to make the discretization errors be small. The optimal scheme uses all that adjustability to achieve errors at the grid points of fifth order in the step length and to all orders in time. There are stability restrictions upon the permissible time step. Unconditional stability can be achieved at the sacrifice of one order in the accuracy, or with the simplifications of fixed interest rate, fixed volatility and specified time-dependence (related to interest rate and volatility) of logarithmically-spaced grid points. Simple test cases suggest that for typical levels of accuracy, the computational resources can be 0.001 of conventional explicit schemes. It would seem feasible to perform the profusion of calculations needed to span scenarios of changing market rates, asset volatilities and strike prices.
[Paper] published in Proc. Roy. Soc. Lond.

• Dr Mikhail Soloveitchik, d-fine

Growth Optimal Portfolios in a semimartingale context
I am a senior-consultant at d-fine (former Financial and Commodity Risk Consulting practice of Arthur Andersen ). I studied Mathematics at the Moscow State University (PhD 1989 in probabilty theory and stochastic processes) and hold a research positionat at the University of Heidelberg (Habilitation 1997). Last years I work as a consultant for the financial industry. My interest include quantative mathematical analysis of financial markets and it's application to practical problems of financial risk management. I have been working on several projects in German and Eropean financial institutes.

• Dr Robert Tompkins, Hochschule für Bankwirtschaft, Frankfurt

Flexible Complete Models with Stochastic Volatility: Generalising Hobson & Rogers (1998)
Friedrich Hubalek, Josef Teichmann and Robert G. Tompkins
Hobson and Rogers (1998) propose an option pricing model where the volatility is a deterministic function of the moving average of past (logarithm of) underlying prices. They show that such a model can also generate implied volatilities that vary across striking price and term to expiration. In this research, this model is tested on actual option markets.
While the Hobson and Rogers (1998) model produces divergences from Black Scholes (1973) prices on a microscopic scale, we have not been able to replicate actual option prices with this model. To determine prices from this model we develop a robust analytic approximation.
To better fit observed options prices, we generalise the model Hobson and Rogers (1998) by the addition of two additional parameters. This model is able to match option prices on the British Pound/US Dollar across both the striking price dimension (smiles) and across different maturities (the term structure of implied volatility). By use of Mavillian calculus, we are able to determine partial derivatives of the generalised model and compute hedging ratios.

• Dr Torsten Wegner, d-fine

Valuation of Swing Options Considering Seasonality of Power Prices
A log-normal mean-reverting diffusion model with time-dependent parameters is used to describe the stochastic process followed by prices at the electricity spot-market. The time-dependence is utilized to account for effects of seasonality which are an immediate consequence of the fact that electricity is hard to store.
A Black-Scholes-like derivation is used to give a valuation model for pricing derivative securities. This model will be applied to swing contracts. The latter represent a very flexible kind of options that can even be endowed with contractual penalties. Different penalty functions are studied within a finite-difference approximation scheme. The model parameters are calibrated to market data from the European Electricity Exchange in Leipzig (Germany).