# Finance

Für mich bedeutet der Begriff Finance: mathematische Methoden in den Wirtschaftswissenschaften, oder angewandte Finanzmathematik. Da gibt es eine coole Winterschule, die alljährlich in Lunteren ziemlich in der Mitte Hollands stattfindet: Winter school on mathematical Finance.

Und meine Lieblingskonferenz muss an dieser Stelle natürlich auch beworben werden: CEQURA Conference on Advances in Financial and Insurance Risk Management. Findet immer Ende September in München statt.

• Crises, Crises (ein bisschen was zur Finanzkrise in schönem Seegrün)

## Ein Artikel und Preprints

The Solvency II Standard Formula, Linear Geometry, and Diversification. Journal of Risk and Financial Management 10(2) (2017)
The core of risk aggregation in the Solvency II Standard Formula is the so-called square root formula. We argue that it should be seen as a means for the aggregation of different risks to an overall risk rather than being associated with variance-covariance based risk analysis. Considering the Solvency II Standard Formula from the viewpoint of linear geometry, we immediately find that it defines a norm and therefore provides a homogeneous and sub-additive tool for risk aggregation. Hence, Euler's Principle for the reallocation of risk capital applies and yields explicit formulas for capital allocation in the framework given by the Solvency II Standard Formula. This gives rise to the definition of diversification functions, which we define as monotone, subadditive, and homogeneous functions on a convex cone. Diversification functions constitute a class of models for the study of the aggregation of risk and diversification. The aggregation of risk measures using a diversification function preserves the respective properties of these risk measures. Examples of diversification functions are given by seminorms, which are monotone on the convex cone of non-negative vectors. Each Lp norm has this property, and any scalar product given by a non-negative positive semidefinite matrix does as well. In particular, the Standard Formula is a diversification function and hence a risk measure that preserves homogeneity, subadditivity and convexity.

### Preprints im SSRN (Social Science Research Network)

Expected Shortfall for Discontinuous Random Variables with Application to Accounting Values. 16.03.2012. Verfügbar bei SSRN: http://ssrn.com/abstract=2024962.
We consider the expected shortfall of accounting values, or, mathematically speaking, of random variables that are not continuous (i. e. whose cumulative distribution function is not continuous). Acerbi and Tasche show that one has to abandon the conditional expectation in order to maintain expected shortfall as a coherent risk measure. They also provide a definition of expected shortfall of discontinuous random variables which fulfills the axioms of coherence.
Provided that the market values of the assets are continuous random variables, we show that the expected shortfall of the accounting values can still be interpreted as a conditional expectation, where the condition refers to the market values.

The Impact of Basis Risk on Optimal Hedging and Hedge Efficiency. Advanced Risk & Portfolio Management Research Paper Series, 16.03.2012. Verfügbar bei SSRN: http://ssrn.com/abstract=2024978.
We analyze the dependency of the optimal hedge and its efficiency on the correlation between the asset to be hedged, and the hedge instrument. The optimal portion of the hedge instrument depends on this correlation, and on the ratio of the volatilities. Efficiency is defined as the percental decrease in volatility caused by the hedge. The connection between the efficiency E of the optimal hedge, and the correlation rho is given by
E = 1 - sqrt(1 - rho^2)
This means that basis risk is really substantial whenever the absolute value of the correlation is smaller than 1. The result is independent of any assumptions on distributions. The mathematics of our result are not new. Arguments like this are used in the theory of Monte Carlo simulation.

The Volatility of Interest Rates and Forward Rates in the Hull White Model. Advanced Risk & Portfolio Management Research Paper Series, 17.09.2012. Verfügbar bei SSRN: http://ssrn.com/abstract=2159308.
The interest rate model by Hull and White allows to calculate an explicit formula for the price of zero bonds. From this pricing formula we deduce explicit formulas for the volatility of the instantaneous forward rate, the volatility of the interest rate (both the spot rate and interest rates of any maturity), and the volatility of the forward rate.

Local and Terminal Volatility of Equity in a Hybrid Model with Hull White Interest Rates, 29.08.2016. Verfügbar bei SSRN: http://ssrn.com/abstract=2831438.
We consider a hybrid model for stocks and interest rates as it is proposed by GDV (Gesamtverband der Deutschen Versicherungswirtschaft) to assign market consistent values to the technical provisions of german life insurance companies. In this model, stock prices are modeled with a Black Scholes model with time-varying parameters, and interest rates are modeled with a Hull White short rate model.
We find that local and terminal volatility of equity differ in this model and that the terminal volatility of equity is considerable higher than the local volatility. This raises the question, which of both volatilities is the right reference to calibrate the model. We argue that calibration should be done with respect to local volatility rather than terminal volatility, at least for the application we have in mind.

Zuletzt geändert am 02.08.2018