  We introduce the general arbitrage-free valuation framework for counterparty
risk adjustments in presence of bilateral default risk, including default of
the investor. We illustrate the symmetry in the valuation and show that the
adjustment involves a long position in a put option plus a short position in a
call option, both with zero strike and written on the residual net value of the
contract at the relevant default times. We allow for correlation between the
default times of the investor, counterparty and underlying portfolio risk
factors. We use arbitrage-free stochastic dynamical models. We then specialize
our analysis to Credit Default Swaps (CDS) as underlying portfolio,
generalizing the work of Brigo and Chourdakis (2008) [5] who deal with
unilateral and asymmetric counterparty risk. We introduce stochastic intensity
models and a trivariate copula function on the default times exponential
variables to model default dependence. Similarly to [5], we find that both
default correlation and credit spread volatilities have a relevant and
structured impact on the adjustment. Differently from [5], the two parties will
now agree on the credit valuation adjustment. We study a case involving British
Airways, Lehman Brothers and Royal Dutch Shell, illustrating the bilateral
adjustments in concrete crisis situations.
