Extreme measures in continuous time Conic Finance

Abstract

Dynamic spectral risk measures define a claim's valuation bounds as supremum and infimum of expectations of the claim's payoff over a dominated set of measures. The measures at which such extrema are attained are called extreme measures. We determine explicit expressions for their Radon-Nykodim derivatives with respect to the common dominating measure. Based on the formulas found, we estimate the extreme measures in two cases. First, the dominating measure is calibrated to mid prices of options and valuation bounds are given by options bid and ask prices. Second, the dominating measure is estimated from historical mid equity prices and valuation bounds are given by historical 5-day high and low prices. In both experiments, we find that the market determines upper bounds by testing scenarios in which losses are significantly lower than expected under the dominating measure, while lower bounds by ones in which gains are only slightly lower than in the base case.