Risk Measures beyond Frictionless Markets

We develop a general theory of risk measures to determine the optimal amount of capital to raise and invest in a portfolio of reference traded securities in order to meet a prespecified regulatory requirement. The distinguishing feature of our approach is that we embed portfolio constraints and transaction costs into the securities market. As a consequence, the property of translation invariance, which plays a key role in the classical theory, ceases to hold. We provide a comprehensive analysis of relevant properties, such as star shapedness, positive homogeneity, convexity, quasiconvexity, subadditivity, and lower semicontinuity. In addition, we establish dual representations for convex and quasiconvex risk measures. In the convex case, the absence of a special kind of arbitrage opportunity allows one to obtain dual representations in terms of pricing rules that respect market bid -ask spreads and assign a strictly positive price to each nonzero position in the regulator's acceptance set.