# Impact of the First to Default Time on Bilateral CVA

14 Pages Posted: 21 Jun 2011

See all articles by Damiano Brigo

## Damiano Brigo

Imperial College London - Department of Mathematics

## Cristin Buescu

King's College London, Department of Mathematics

## Massimo Morini

Banca IMI; Bocconi University

Date Written: June 17, 2011

### Abstract

We compare two different bilateral counterparty valuation adjustment (BVA) formulas. The first formula is an approximation and is based on subtracting the two unilateral Credit Valuation Adjustment (CVA)'s formulas as seen from the two different parties in the transaction. This formula is only a simplified representation of bilateral risk and ignores that upon the first default closeout proceedings are ignited. As such, it involves double counting. We compare this formula with the fully specified bilateral risk formula, where the first to default time is taken into account. The latter correct formula depends on default dependence between the two parties, whereas the simplified one does not. We also analyze a candidate simplified formula in case the replacement closeout is used upon default, following ISDA's recommendations, and we find the simplified formula to be the same as in the risk free closeout case. We analyze the error that is encountered when using the simplified formula in a couple of simple products: a zero coupon bond, where the exposure is unidirectional, and an equity forward contract where exposure can go both ways. For the latter case we adopt a bivariate exponential distribution due to Gumbel to model the joint default risk of the two parties in the deal. We present a number of realistic cases where the simplified formula differs considerably from the correct one.

Keywords: Credit Valuation Adjustment, Unilateral CVA, Bilateral CVA, Simplified Bilateral CVA, Debit Valuation Adjustment, Closeout, Equity Forward Contract, Zero coupon bond, Bivariate exponential distributions, Gumbel bivariate exponential distributions

JEL Classification: G12, G13

Suggested Citation

Brigo, Damiano and Buescu, Cristin and Morini, Massimo, Impact of the First to Default Time on Bilateral CVA (June 17, 2011). Available at SSRN: https://ssrn.com/abstract=1868853 or http://dx.doi.org/10.2139/ssrn.1868853

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