Download PDF by Frederick Hoffman: Credit Valuation Adjustment

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However, interestingly as the correlation approaches 1 it began to fall. 99 the UR-CVA dropped. Brigo gives a simple analytic explanation for this. Assume that λ1 and λ2 are constant. Make the natural assumption that λ1 > λ2 the underlying credit reference is risky than the protection seller insurer. 99, the default triggers ξ1 and ξ2 are almost perfectly correlated, so also assume that ξ1 = ξ2 :≈ ξ. This implies that, t Γi (t) = λi ds = λi t 0 36 and since, τi = Γ−1 (ξ) we have, τ1 = ξ/λ1 , τ2 = ξ/λ2 .

Models that neglect this correlation between the credit riskiness of reference credit and protection seller in a CDS would miss this effect and as a result drastically underprice CVA. Likewise uncertainty over the credit worthiness of entities as we have witnessed since the recent crisis lead to highly volatile credit spreads that also impact CVA. Volatility in credit spreads has a particularly pronounced effect when one is pricing the CVA of a CDS. In this case the calculation of CVA involves pricing an option on a CDS and volatility has a large impact on the pricing of options.

However, as discussed we must compute Q(τ1 ≥ u|FTj ). This gives, Q(τ1 ≥ u|FTj ) = E[1{Tj−1 <τ2 ≤Tj } 1{τ1 }>u |FTj ] = E[1{Tj−1 <τ2 ≤Tj } 1{τ1 }>Tj 1{τ1 }>u |FTj ] = E[1{τ1 }>u |FTj , Tj−1 < τ2 ≤ Tj , τ1 > Tj ] = E[1{τ1 }>u |FTj , Tj−1 < τ2 ≤ Tj , τ1 > Tj ] = = = Q(τ1 > u, Tj−1 < τ2 < Tj |FTj ) Q(τ1 > Tj , Tj−1 < τ2 < Tj |FTj ) Q(U1 > 1 − e−Λ1 (u) , 1 − e−Λ2 (Tj−1 ) < U2 < 1 − e−Λ2 (Tj ) |FTj ) Q(U1 > 1 − e−Λ1 (Tj ) , 1 − e−Λ2 (Tj−1 ) < U2 < 1 − e−Λ2 (Tj ) |FTj ) = e−Λ2 (Tj−1 ) − e−Λ2 (Tj ) + E[C(1 − e−Λ1 (u) , e−Λ2 (Tj−1 ) ) − C(1 − e−Λ1 (u) , e−Λ2 (Tj−1 ) )] e−Λ2 (Tj−1 ) − e−Λ2 (Tj ) + C(1 − e−Λ1 (Tj ) , e−Λ2 (Tj−1 ) ) − C(1 − e−Λ1 (Tj ) , e−Λ2 (Tj−1 ) ) It can be checked that if the copula is the independence copula, C(u1 , u2 ) = u1 u2 then this expression reduces to the simpler one prior.

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Credit Valuation Adjustment by Frederick Hoffman


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