  In this paper we modify the model of Itkin, Shcherbakov and Veygman, (2019)
(ISV2019), proposed for pricing Quanto Credit Default Swaps (CDS) and risky
bonds, in several ways. First, it is known since the Lehman Brothers bankruptcy
that the recovery rate could significantly vary right before or at default,
therefore, in this paper we consider it to be stochastic. Second, to reduce
complexity of the model, we treat the domestic interest rate as deterministic,
because, as shown in ISV2019, volatility of the domestic interest rate does not
contribute much to the value of the Quanto CDS spread. Finally, to solve the
corresponding systems of 4D partial differential equations we use a different
flavor of the Radial Basis Function (RBF) method which is a combination of
localized RBF and finite-difference methods, and is known in the literature as
RBF-FD. Results of our numerical experiments presented in the paper demonstrate
that the influence of volatility of the recovery rate is significant if the
correlation between the recovery rate and the log-intensity of the default is
non-zero. Also, the impact of the recovery mean-reversion rate on the Quanto
CDS spread could be comparable with the impact due to jump-at-default in the FX
rate.
