Good deals and benchmarks in robust portfolio selection

•The paper computes the market price of risk for investors with multiple priors.•The standard deviation may be replaced by coherent risk measures.•The most important formulas of the Capital Asset Pricing Model are extended.•The presence of good deal is deeply studied and solved in the general setting above. This paper deals with portfolio selection problems under risk and ambiguity. The investor may be ambiguous with respect to the set of states of nature and their probabilities. Both static and discrete or continuous time dynamic pricing models are included in the analysis. Risk and ambiguity are measured in general settings. The considered risk measures contain, as particular cases, the usual deviations and the coherent and expectation bounded measures of risk. Four contributions seem to be reached. Firstly, necessary and sufficient optimality conditions are given. Secondly, the portfolio selection problem may be frequently solved by linear programming linked methods, despite the fact that risk and ambiguity cannot be given by linear expressions. Thirdly, if there is a market price of risk then there exists a benchmark that creates a robust capital market line when combined with the riskless asset. The global risk of every portfolio may be divided into systemic and specific. Moreover, if there is no ambiguity with respect to the states of nature (only their probabilities are uncertain), then classical CAPM-formulae may be found. Fourthly, some recent pathological findings for ambiguity-free analyses also apply in ambiguous frameworks. In particular, there may exist arbitrage free markets such that the ambiguous agent can guarantee every expected return with a maximum risk bounded from above by zero, i.e. , the capital market line (risk, return) becomes vertical. For instance, in the (non-ambiguous) Black and Scholes model this property holds for important risk measures such as the absolute deviation or the CVaR. Nevertheless, in ambiguous settings, adequate increments of the ambiguity level will allow us to recover capital market lines consistent with the empirical evidence. The introduction of ambiguity may overcome several caveats of many important pricing models.