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Risk management is an indispensable component of the financial system. In this context, capital requirements are built by financial institutions to avoid future bankruptcy. Their calculation is based on a specific kind of maps, so-called risk measures. There exist several forms and definitions of them. Multi-asset risk measures are the starting point of this dissertation. They determine the capital requirements as the minimal amount of money invested into multiple eligible assets to secure future payoffs. The dissertation consists of three main contributions: First, multi-asset risk measures are used to calculate pricing bounds for European type options. Second, multi-asset risk measures are combined with recently proposed intrinsic risk measures to obtain a new kind of a risk measure which we call a multi-asset intrinsic (MAI) risk measure. Third, the preferences of an agent are included in the calculation of the capital requirements. This leads to another new risk measure which we call a scalarized utility-based multi-asset (SUBMA) risk measure.
In the introductory chapter, we recall the definition and properties of multi-asset risk
measures. Then, each of the aforementioned contributions covers a separate chapter. In
the following, the content of these three chapters is explained in more detail:
Risk measures can be used to calculate pricing bounds for financial derivatives. In
Chapter 2, we deal with the pricing of European options in an incomplete financial market
model. We use the common risk measures Value-at-Risk and Expected Shortfall to define
good deals on a financial market with log-normally distributed rates of return. We show that the pricing bounds obtained from Value-at-Risk may have a non-smooth behavior under parameter changes. Additionally, we find situations in which the seller's bound for a call option is smaller than the buyer's bound. We identify the missing convexity of the Value-at-Risk as main reason for this behavior. Due to the strong connection between the obtained pricing bounds and the theory of risk measures, we further obtain new insights in the finiteness and the continuity of multi-asset risk measures.
In Chapter 3, we construct the MAI risk measure. Therefore, recall that a multi-asset risk measure describes the minimal external capital that has to be raised into multiple eligible assets to make a future financial position acceptable, i.e., that it passes a capital adequacy test. Recently, the alternative methodology of intrinsic risk measures
was introduced in the literature. These ask for the minimal proportion of the financial position that has to be reallocated to pass the capital adequacy test, i.e., only internal capital is used. We combine these two concepts and call this new type of risk measure an MAI risk measure. It allows to secure the financial position by external capital as well as reallocating parts of the portfolio as an internal rebooking. We investigate several properties to demonstrate similarities and differences to the two
aforementioned classical types of risk measures. We find out that diversification reduces
the capital requirement only in special situations depending on the financial positions. With the help of Sion's minimax theorem we also prove a dual representation for MAI risk measures. Finally, we determine capital requirements in a model motivated by the Solvency II methodology.
In the final Chapter 4, we construct the SUBMA risk measure. In doing so, we consider the situation in which a financial institution has to satisfy a capital adequacy test, e.g., by the Basel Accords for banks or by Solvency II for insurers. If the financial situation of this institution is tight, then it can happen that no reallocation of the initial
endowment would pass the capital adequacy test. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the SUBMA risk measure which optimizes the hedging costs and the expected utility of the institution simultaneously subject to the capital adequacy test. We find out that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark capital adequacy tests.