# Abstracts

**On the regularity and representation of convex risk measures on Orlicz spaces**

*Keita Owari (University of Tokyo, Japan)*

Wednesday June 4, 16:30-17:00 | session 6.3 | Risk Measures | room EF

Convex risk measures on $L^\infty$ have been studied in various aspects. Among many other fine properties, it is known that a risk measure on $L^\infty$ has the so-called robust representation by probability measures if and only if it has the Fatou property (order lower semicontinuity), and for such a risk measure, there is equivalence between (1) the so-called Lebesgue property (continuity w.r.t. the dominated a.s. convergence),(2) the weak compactness of all the sublevel sets of the conjugate, and (3) the attainment of the supremum in the robust representation (Jouini-Schachermayer-Touzi's theorem). Each of equivalent properties has importance in application, and the implication (3) $\Rightarrow$ (2) may be viewed as a partial generalization of perturbed James' theorem. Recently, Orihuela and Ruiz Galán obtained a similar equivalence for risk measures on certain class of Orlicz spaces. In this talk, we provide this type of equivalence for monotone convex functions on solid subspaces of $L^0$, which improve the one by Orihuela and Ruiz Galán, with a much simpler proof, and unifies several other related results. We then discuss applications and implications in financial mathematics.

**Fundamental theorem of asset pricing without reference measure**

*Ludovic Tangpi (University of Konstanz, Germany)*

Wednesday June 4, 17:00-17:30 | session 6.3 | Risk Measures | room EF

When a financial market is governed by a single probability measure, the absence of arbitrage opportunities is characterized by the existence of equivalent martingale or local martingale measures. In this talk, we focus on the fundamental theorem of asset pricing in the case where the market is governed by a non-dominated set of probability measures. We introduce the concept of free lunch with controlled risk. Our main result shows that, in a continuous time model, if the agent is allowed to trade only with strategies that are simple integrands, then the absence of free lunches with controlled risk is equivalent to the existence of a set of local martingale measures equivalent to the set of possible models. Talk based on a join work with Michael Kupper and Patrick Cheridito.

**A simplified risk representation for structured products to disclose key information about probability distributions to investors**

*Riccardo Cesari (University of Bologna, Italy)*

Wednesday June 4, 17:30-18:00 | session 6.3 | Risk Measures | room EF

The probability distribution of a financial product contains a vast amount of information that is exploited by traders and managers in different ways by building numerous risk measures that highlight or exploit particular pieces of this information set. The average investor, who is the final user of the product and fully bears the associated risks without protection, is instead often precluded to benefit of this relevant information, considered too complex and abstract to be understood properly. In the perspective to fill this asymmetric gap, an innovative statistical method to reconstruct the shape of the probability distribution from a minimum number of moments is introduced. This new method is applied to a sample of four different financial products (in fact real world cases, in spite of their complexity). One important result of the work is that, in many, relevant cases, a huge amount of moments are required to sufficiently reconstruct the probability distributions of the products, apparently confirming that the idea to transfer to the investor the information embedded in the probability distribution is unmanageable. Moving forward, a new simple methodology is then proposed to solve the impasse that allows to partition the probability distribution in a set of few events of clear financial meaning for the investor. In order to identify quantiles that are dynamically updated to the changing market’s conditions, the distribution of an elementary product like the cash account is used. In this way the full information set is reduced to a table of probabilities attached to events that the average investor can easily understand and it is supported by additional indicators of risk for each scenario. This representation however well captures the main statistical features of the probability distribution as dispersion, asymmetry, kurtosis by enhancing the investor awareness of the risks embedded in the product. It is generally accepted that model risk is relevant for pricing / hedging activities and a great effort is made by risk managers and traders in the choice of the correct model: the proposed methodology produces results that minimize the unavoidable differences between models and succeed in conveying to the investor the relevant piece of information (the “core”) embedded in the probability distribution not dependent on the choice of a particular quantitative model.