Compatibility between pricing rules and risk measures: The CCVaR

Abstract

This paper has considered a risk measure ρ and a (maybe incomplete and/or imperfect) arbitrage-free market with pricing rule Π. They are said to be compatible if there are no reachable strategies y such that Π(y) is bounded and ρ(y) is close to −∞. We show that the lack of compatibility leads to meaningless situations in financial or actuarial applications. The presence of compatibility is characterized by properties connecting the Stochastic Discount Factor of Π and the sub-gradient of ρ. Consequently, several examples pointing out that the lack of compatibility may occur in very important pricing models are yielded. For instance the CVaR is not compatible with the Black and Scholes model or the CAPM. We prove that for a given incompatible couple (Π, ρ) we can construct a minimal risk measureM(Π,ρ) compatible with ρ and such that ρ≤M (Π,ρ). This result is particularized for the CVaR and the CAPM and the Black and Scholes model. Therefore we construct the Compatible Conditional Value at Risk (CCVaR). It seems that the CCVaR preserves the good properties of the CVaR and overcomes its shortcomings.