Extreme Measures in Continuous Time Conic Finace

TL;DR

This paper derives explicit formulas for extreme measures in continuous-time conic finance, enabling estimation of valuation bounds using market data and historical prices, with implications for risk assessment.

Contribution

It provides explicit Radon-Nykodim derivative formulas for extreme measures and applies them to market-calibrated and historical data scenarios.

Findings

Market determines upper bounds through scenarios with significantly lower losses.

Lower bounds are influenced by scenarios with slightly lower gains.

Estimates align with observed bid-ask spreads and historical high-low prices.

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.