Workshop on

**"Multivariate Risk
Management"**

**December 10-11, 2007**

**EURANDOM, Eindhoven, The
Netherlands**

**ABSTRACTS**

Hansjörg Albrecher, Radon Institute for Computational and Applied Mathematics, and Graz University of Technology, Austria (albrecher@tugraz.at )

Tax payments in risk theory

In this talk it is shown that the inclusion of tax payments according to a loss-carried forward system modifies the survival probability of a classical risk process in a surprising and very transparent way. This effect is then also studied for arbitrary spectrally negative Levy processes and a dual risk model. The distribution of discounted tax payments until ruin is investigated and the optimal surplus level to start taxation is identified.

Jake Ansell, University of Edinburgh, UK (ansell@staffmail.ed.ac.uk )

Title: t.b.a.

Abstract: t.b.a.

Svetlana Borovkova, Vrije Universiteit Amsterdam (sborovkova@feweb.vu.nl )

Basket options and implied correlations

We introduce an analytical approach to pricing and hedging of basket options, which we call the Generalized Lognormal approach (GLN). This approach is suited to valuing options on general portfolios (baskets), e.g. those with arbitrary number of assets and containing both long and short positions, such as 3:2:1 crack spread.

The core of the GLN approach is the approximation of the basket distribution by a generalized family of lognormal distributions, which incorporates negative basket values as well as negative skewness of the basket distribution. Moreover, using generalized lognormal distributions allows us to stay within Black-Scholes framework and derive analytical formulae for the basket option value and the greeks, which can be evaluated quickly and accurately: something of a great value for practitioners.

The approach naturally extends to Asian-style basket options, which are much more common in commodity (and particularly energy) markets, than the European-style options. We demonstrate the high accuracy of the GLN approach on the basis of simulated and real oil market data, by comparing GLN option prices to those obtained by an extensive Monte Carlo simulation, and by calculating the costs of hedging the option.

The GLN approach provides an approximate analytic formula for the basket option price, which can be inverted, to reveal the implied correlations between assets in the basket. We apply this to the NYMEX crack spread option prices, to reveal implied correlations between crude oil and unleaded gasoline, and show how the structure of the implied correlation resembles well-known features of implied volatility, such as volatility smile, smirk or a skew.

Hans Buehler, Deutsche Bank AG, Germany (hans.buehler@db.com )

Measuring hedging performance

We focus on a model's ability to actually reduce the variance of a position when its hedging ratios are used. To this end, we compare various models, calibrated to historic marketdata. We argue that this kind of analysis is more important for the choice of a model than other statistical properties of a model.

Nikolaus Hautsch, Humboldt-Universität zu Berlin, Germany (nikolaus.hautsch@wiwi.hu-berlin.de

**Point Process Models for Financial High-Frequency Data**

In this seminar we discuss different types of multivariate continuous-time intensity models to model the activity of financial markets on the lowest possible level of aggregation. We interpret the trading process as a multivariate (marked) point process which is modelled in terms of an intensity specification. In this context, we focus particularly on Hawkes type processes and (generalized) autoregressive conditional intensity (ACI) models.

Furthermore, applying ACI-type models we analyze the contemporaneous buy and sell intensity as well as the implied net buy pressure in a limit order book market. Utilizing detailed information from the trading process of the electronic market at the Australian Stock Exchange we analyze how the trading intensities as well as the net buy pressure depend on the state of the market. Confirming predictions from market microstructure theory traders submit market orders by inferring from the recent order flow and the book with respect to upper and lower tail expectations as well as trading directions. However, we also find evidence that traders tend to take liquidity when the liquidity supply is high.

Inge Koch, University of Antwerpen, Belgium (inge.koch@ua.ac.be )

The comonotonicity coefficient: a multivariate measure for positive dependence

In actuarial and financial mathematics we often deal with dependent random variables. It is of a great importance to have information on this dependence structure. Unfortunately this information is often scarce. In the last decades we observe an increasing interest from researchers and practitioners in the dependence structure of such a random vector.

Many research is already performed in this field e.g. through the theory of comonotonicity. In fact the comonotonic dependence structure in particular has proved to be a very useful tool when approximating an unknown but strongly positive dependence structure. As a consequence of this evolution, there is a need for a measure reflecting how close a given dependence structure approaches the comonotonic one.

In this contribution we design the comonotonicty coefficient. This coefficient can be calculated for multivariate vectors and examines to what extent the considered vector is comonotonic, compared to the independent situation in the same Fr´echet space. This measure takes values in the range [0, 1]. As we want to quantify the degree of comonotonicity, this measure is defined in such a way that it equals 1 in case (X1, X2, · · · ,Xn) is comonotonic and equals 0 in case (X1,X2, · · · ,Xn) is independent.

We show how such a measure can be designed analytically, by making use of copulas for the modeling of the dependency structure. We discuss the properties of this comonotonicity coefficient and in the particular bivariate case, we compare our measure with the standard bivariate dependence measures. In a second stage we focus on the definition and the use of an empirical comonotoncity coefficient, needed when only data is at one’s disposal.

Keywords

comonotonicity, copula, dependence structure, measure of association

Dimitrios Konstantinides, Aegean University, Karlovassi, Samos, Greece (konstant@aegean.gr )

A two-fluid actuarial model with an alternating payoff policy (Section 2)

In this paper we consider the model for an actuarial problem dealing with two types of claims and payoffs subject to seasonal switching. Claims are assumed to occur in a fluid fashion whereas payoffs are made at a unit rate so long as claims remain to be paid. The distribution properties of the accumulated claim sizes {Z1(t), Z2(t)} are derived at finite time as well as in stationary regime. We first investigate this process embedded at the successive switching points. This process is Markovian with independent components. In continuous time the components {Z1(t), Z2(t)} are also independent for each finite t, but are dependent in stationary regime.

Joint work with N.U. Prabhu

Henrik Jönsson, EURANDOM, NL (jonsson@eurandom.tue.nl )

Single name credit derivatives under Lévy models

Credit risk modeling is about modeling losses. These losses are typically coming unexpectedly and triggered by shocks. So any process modeling the stochastic nature of losses should reasonable include jumps. The presence of jumps is even of greater importance if one deals with derivatives on Credit Default Swaps (CDS) because of the leveraging effects. Jump processes have proven already their modeling abilities in other settings like equity and fixed income and have recently found their way into credit risk modeling. In this talk we review a few jump driven models for the valuation of CDSs and show how under these dynamic models also pricing of (exotic) derivatives on single name CDSs is possible. More precisely, we set up fundamental firm value models that allow for fast pricing of the 'vanillas' of the CDS derivative markets: payer and receiver swaptions. Moreover, we detail how a CDS spread simulator can be set up under this framework and illustrate its use for the pricing of exotic derivatives on single name CDSs as underlyers. We finally show how the methodology can be applied to pricing Constant Maturity Credit Default Swaps.

Viktoriya Masol, Katholieke Universiteit Leuven, Belgium, and EURANDOM, The Netherlands (masol@eurandom.tue.nl )

Levy Base Correlation

We consider a collateralized debt obligation (CDO) with standard credit default swap (CDS) indices as the reference portfolio. We investigate and compare one factor Levy models that extend the classical Gaussian copula model for pricing CDOs. The proposed models are very tractable and perform significantly better than the classical Gaussian copula model.

Furthermore, we introduce the concept of Levy base correlation. The obtained Levy base correlation curve is much flatter than the corresponding Gaussian one. This indicates that the models do fit the observed data much better. Additionally, flat base correlation curves are also much more reliable for pricing of bespoke tranches. We illustrate the application of Levy base correlation to price non-standardized tranches of a synthetic CDO and compare the prices obtained under Gaussian and Levy models.

Finally, we compare delta hedge parameters under the different models. We focus on two common approaches to delta-hedge a CDO tranche: first, hedging a tranche with the index, and, second, hedging the equity tranche with the mezzanine tranche.

Philipp Mayer, Graz University of Technology, Austria (mayer@opt.math.tu-graz.ac.at )

Robust calibration methods for financial market models

In this talk we present a non-parametric stable calibration method based on Tikhonov regularization for generalized L\'evy market models. While the original calibration problem is more ill-posed than the Dupire calibration problem, we are able to prove stability and convergence of the regularized problem and in some cases convergence rates can be derived under the common assumption of an abstract source condition. Finally we underpin the theoretical results by numerical illustrations.

Alexander McNeil, Heriot-Watt University, Edinburgh, UK (A.J.McNeil@hw.ac.uk )

**A New Perspective on Archimedean Copulas**

The Archimedean copula family is used in a number of actuarial applications, ranging from the construction of multivariate loss distributions to frailty models for dependent lifetimes. We present some new results that contribute to a greater understanding of this family and point the way to improved simulation and estimation procedures. We derive necessary and sufficient conditions for an Archimedean generator function (a continuous, decreasing mapping of the positive half-line to the unit interval) to generate a copula in a given dimension d. We also show how the Archimedean family coincides with the class of survival copulas of L1-norm symmetric distributions. These results allow us to construct a rich variety of new Archimedean copulas in different dimensions and to solve in principle the problem of generating samples from any Archimedean copula. The practical consequences include new models for negatively dependent risks, simple formulas for rank correlation coefficients and diagnostic tests for Archimedean dependence.

**Ludger Rüschendorf,** **University of Freiburg,
Freiburg, Germany** **
(ruschen@stochastik.uni-freiburg.de )**

**Stochastic ordering and risk measures for portfolio vector (Section 1)**

The main aim to study risk measures for portfolio vectors $X=(X_1,\dots,X_d)$ is to measure not only the risk of the marginals $X_i$ separately but also to quantify the risk caused by the variation of the components and by possible dependence between the components.

Thus an important property of risk measures is consistency with respect to various classes of convex orderings but also with respect to dependence orderings. In the first part we give an introduction to risk measures and in particular derive in the portfolio case a characterization of law invariant risk measures. Then we review some classical and recent results on stochastic and dependence orderings. For some concrete classes of risk measures for portfolio vectors we obtain as consequence consistency results of the type that higher positive dependence leads to higher values indicated by the risk measures.

**Hanspeter Schmidli, University of Cologne, Germany
(**schmidli@math.uni-koeln.de
)

**Optimal Dividend Strategies in a Cramér-Lundberg Model with Reinvestments
**

We consider a classical risk model with dividend payments and reinvestments. Thereby, the surplus has to stay positive. Like in the classical de Finetti problem, we want to maximise the discounted dividend payments minus the penalised discounted reinvestments. We prove the Hamilton-Jacobi-Bellman equation for the problem and show that the optimal strategy is a barrier strategy. We explicitly characterise when the optimal barrier is at 0 and find the solution for exponentially distributed claim sizes.

**Rafael Schmidt, University of Cologne, Cologne,
Germany
(rafael.schmidt@uni-koeln.de )**

Modelling dynamic portfolio risk using risk drivers of elliptical processes

The situation of a limited availability of historical data is frequently
encountered in portfolio risk estimation, especially in credit risk estimation.
This makes it, for example, difficult to find temporal structures with
statistical significance in the data on the single asset level. By contrast,
there is often a broader availability of cross-sectional data, i.e., a large
number of assets in the portfolio.

This paper proposes a stochastic dynamic model which takes this situation into
account. The modelling framework is based on multivariate elliptical processes
which model portfolio risk via sub-portfolio specific volatility indices called
portfolio risk drivers. The dynamics of the risk drivers are modelled by
multiplicative error models (MEM) - as introduced by Engle (2002) - or by
traditional ARMA models. The model is calibrated to Moody's KMV Credit Monitor
asset returns (also known as firm-value returns) given on a monthly basis for
756 listed European companies at 115 time points from 1996 to 2005. This
database is used by financial institutions to assess the credit quality of
firms. The proposed risk drivers capture the volatility structure of asset
returns in different industry sectors. A characteristic temporal structure of
the risk drivers, cyclical as well as a seasonal, is found across all industry
sectors. In addition, each risk driver exhibits idiosyncratic developments. We
also identify correlations between the risk drivers and selected macroeconomic
variables. These findings may improve the estimation of risk measures such as
the (portfolio) Value at Risk. The proposed methods are general and can be
applied to any series of multivariate asset or equity returns in finance and
insurance.

**Wim Schoutens, University of Leuven,
Belgium (wim@schoutens.be )**

**Tham Wing Wah, University of Warwick**

**Time Deformation and the Yield Curve **

This paper considers how trading activity at one maturity of the yield curve affects and is affected by trading at other maturities. We approach the modelling of bond prices from a stochastic volatility perspective based on time deformation. We put forward a new, continuous time, multivariate time deformation model which is coherent with the market microstructure theory of price discovery and captures information flow in the market. We model the stochastic volatility process by estimating the instantaneous volatility as a function of the price intensity in the spirit of Cho and Frees (1988) and Gerhard and Hautsch (2002). The point process model, a Hawkes model, that we use to model the price intensity allows for both the self and cross excitation effects of trading at different maturities. Univariate and multivariate models are estimated using transaction level data from BrokerTec, a highly liquid and widely traded electronic platform for US securities.

We find that the integrated price intensity is statistically supported as an appropriate directing process in the US bond market suggesting that private information is revealed indirectly through trades in the presence of information asymmetry and heterogeneous agents. We also find that the individual yields on US treasury notes and bonds appear to be driven by different ‘market clocks’ as suggested perhaps by the market segmentation theory of the term structure. These separate market time scales are then related to each other through a multivariate Hawkes model which effectively coordinates activity along the yield curve. We also show that bond returns standardized by the instantaneous volatility estimated from our Hawkes model are Gaussian which is consistent with the theory of time deformation for security prices quite generally.

*This page is maintained by **
Lucienne Coolen*