Combined Risk Measures: Representation Results and Applications

The analysis and interpretation of risk play a crucial role in different areas of modern finance. This includes pricing of financial products, capital allocation and derivation of economic capital. Key to this analysis is the quantification of the risk via risk measures. A promising approach is to define risk measures by a set of desirable properties. This leads to the main topic of this research; the characterization of so called convex risk measures. First, we review the concept of convex risk measures on Lebesgue spaces and provide a structural basis for the following parts of the thesis by stating and proving the different characterization results and adjusting them to our definitions and notation. A key result of this thesis is the characterization of linear combinations and convolutions of convex risk measures. We study different dual correspondences, which are induced by the Fenchel-Legendre transform. In our case we investigated three different duality correspondences, which are sum and inf-convolution, difference and deconvolution and multiplication of scalars and epi-multiplication. These results are used to characterize linear combinations and convolutions of convex risk measures. We investigate when certain combinations of risk measures belong to the class of convex risk measures and investigate the basic properties. Furthermore, two applications based on theoretical results of the first part of the thesisare derived. In the first application we study the pricing and hedging problem for contingent claims in an incomplete market as a trade-off between a trader and a regulator. In our model the regulator allows the trader to take some risk, but insists that the residual risk, which is not hedged away, has to be covered. To achieve this, the regulator introduces an extra bank account, which serves as a capital reserve to cover for eventual losses of the trader, and is dependent on the risk of the trader’s portfolio. The risk attitudes of the trader and the regulator are reflected by different risk measures. We derive risk measure price and the risk indifference price. In both cases, the resulting risk measure is given by a weighted sum of the regulator’s and trader’s risk measures. This new operator is also a convex risk measure as we have proven in the first part of the thesis. In the second application we consider the problem of partial hedging of a contingent claim. Under the assumption of a complete market, it is always possible to replicate the claim. In this case, the claim can be priced using the unique equivalent martingale measure. The question is of a different nature when the initial capital is less than the expectation under the equivalent martingale measure. Under this condition we derive a suitable hedging strategy such that the risk of the difference of the hedging portfolio and the claim is minimized. As risk measure we consider Average Value at Risk and a simple spectral risk measure. We discovered that overhedging may arise. Nevertheless this happens only in special situations, for example, when the level of Average Value at Risk is high or the initial capital is close to the value which is required for a perfect hedging strategy. The results are illustrated by solving the problem for a call and a put option in the Black-Scholes model