Compatibility between pricing rules and risk measures: The CCVaR

Abstract

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 measure M (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. Compatibilidad entre reglas de valoración y medidas de riesgo: el CCVaRResumen. Consideraremos una medida de riesgo ρ y un mercado libre de arbitraje (puede ser que incompleto o imperfecto) con regla de valoración Π.Éstos serán compatibles si no hay estrategias alcanzables y tales que Π(y) permanece acotado y ρ(y) se acerca a −∞. Veremos que la falta de compatibilidad conduce a situaciones sin sentido económico en las aplicaciones actuariales o financieras.La compatibilidad será caracterizada mediante propiedades que ligan al Factor de Descuento Estocástico de Π y al sub-gradiente de ρ. Consecuentemente, se podrán dar importantes ejemplos en los que hay falta de compatibilidad. Por ejemplo, el CVaR no es compatible con el modelo de Black-Scholes o el CAPM.Probaremos que para cualquier par incompatible (Π, ρ) se puede construir una medida de riesgo minimal M (Π,ρ) compatible con ρ, y tal que ρ ≤ M (Π,ρ) . Este resultado se particularizará para el CVaR y el CAPM y el modelo de Black-Scholes. Por tanto, construiremos el CVaR Compatible (CCVaR). El CCVaR parece preservar las buenas propiedades del CVaR y superar sus deficiencias.

Good deal indices in asset pricing: actuarial and financial implications

“…The properties of ρ (Assumption ) and Π allow us to establish Proposition . We will omit the proof because it is similar to one that can be found in Balbás and Balbás (). Proposition The following statements are equivalent.…”

“…Let us now recall the notion of “compatibility” of Balbás and Balbás (). Definition The couple is said to be noncompatible if any of ( a ), ( b ), ( c ), ( d ), or ( e ) above hold.…”

“…Problem is concave. Bearing in mind Assumption , , , , and , and proceeding as in Balbás and Balbás () or Balbás et al. (), one can prove the existence of a linear dual problem characterizing the solutions of .…”

“…The existence of GD s is equivalent to the existence of alternative sequences of investment strategies whose pairs ( risk, price ) diverge to . This pathological finding has been analyzed in Balbás and Balbás () and Balbás et al. (), where explicit examples of the sequences above have been constructed and their performance empirically tested.…”

Good Deals and compatible modification of risk and pricing rule: a regulatory treatment

“…Then there existsZ ∈ R such thatZ ∈ C(Z min ) andZ = Z min . Since Z min = Z ∈ C(Z min ), by definition there exists λ ∈ (0, 1] and Z 1 ∈ ρ such that 1 . By convexity of ρ we know that Z 2 ∈ ρ .…”