**Abstract: **We show that the recent results on the Fundamental Theorem of Asset Pricing and the super-hedging theorem in the context of model uncertainty can be extended to the case in which the options available for static hedging (hedging options) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of robust no-arbitrage which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are non-redundant. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.

**Abstract: **This paper proposes a new method to introduce coherent risk measures for risks with infinite expectation, such as those characterized by some Pareto distributions. Extensions of the conditional value at risk, the weighted conditional value at risk and other examples are given. Actuarial applications are analyzed, such as extensions of the expected value premium principle when expected losses are unbounded.

**Abstract: **We present some new evidence on the tail distribution of commercial property losses based on a recently constructed dataset on large commercial risks. The dataset is based on contributions from Lloyd’s of London syndicates, and provides information on over three thousand claims occurred during the period 2000–2012, including detailed information on exposures. We use occupancy characteristics to compare the tail risk profiles of different commercial property exposures, and find evidence of substantial heterogeneity in tail behavior. The results demonstrate the benefits of aggregating granular information on both claims and exposures from different data sources, and provide warning against the use of reserving and capital modeling approaches that are not robust to heavy tails.

**Abstract: **A spectrum of upper bounds (*Q*_{α}(*X** *; *p) * _{ αε[0,∞]} on the (largest) (1-p)-quantile *Q*(*X*;*p*) of an arbitrary random variable *X *is introduced and shown to be stable and monotonic in *α*, *p*, and *X *, with *Q*_{0}(*X** *;p) = *Q*(*X*;*p*). If *p *is small enough and the distribution of *X *is regular enough, then Q_{α}(X ; p) is rather close to *Q*(*X *; *p*). Moreover, these quantile bounds are coherent measures of risk. Furthermore, *Q*_{α}*(X** *; *p*) is the optimal value in a certain minimization problem, the minimizers in which are described in detail. This allows of a comparatively easy incorporation of these bounds into more specialized optimization problems. In finance, *Q*_{0}(*X*;*p*) and *Q*_{1}(*X** *; *p*) are known as the value at risk (VaR) and the conditional value at risk (CVaR). The bounds *Q*_{α}*(X** *; *p*) can also be used as measures of economic inequality. The spectrum parameter *α *plays the role of an index of sensitivity to risk. The problems of the effective computation of the bounds are considered. Various other related results are obtained.

**Abstract: **We consider a one-period portfolio optimization problem under model uncertainty. For this purpose, we introduce a measure of model risk. We derive analytical results for this measure of model risk in the mean-variance problem assuming we have observations drawn from a normal variance mixture model. This model allows for heavy tails, tail dependence and leptokurtosis of marginals. The results show that mean-variance optimization is seriously compromised by model uncertainty, in particular, for non-Gaussian data and small sample sizes. To mitigate these shortcomings, we propose a method to adjust the sample covariance matrix in order to reduce model risk.

**Abstract: **Assume that claims in a portfolio of insurance contracts are described by independent and identically distributed random variables with regularly varying tails and occur according to a near mixed Poisson process. We provide a collection of results pertaining to the joint asymptotic Laplace transforms of the normalised sums of the smallest and largest claims, when the length of the considered time interval tends to infinity. The results crucially depend on the value of the tail index of the claim distribution, as well as on the number of largest claims under consideration.