Fast and stable second-order credit sensitivities of credit valuation adjustment

Credit Valuation Adjustment is a balance sheet item which is nowadays subject to active risk management by specialized traders. However, the most important risk factors, which are the default intensities of the counterparties, affect in a nondifferentiable way the most general Monte Carlo estimator of the adjustment, through simulation of default times. Thus the computation of first- and second-order (pure and mixed) sensitivities with respect to inputs affecting these risk factors cannot rely on direct path-wise differentiation, while any approach involving finite differences is slow and shows very high statistical noise. We present ad hoc estimators which have empirically a much smaller variance while offering very low runtime overheads over the baseline computation of the price adjustment, regardless of the number of sensitivities of interest, by leveraging adjoint (i.e., backward) algorithmic differentiation in their implementation. These estimators allow for a generic copula-based dependence structure among the default events, and can be applied beyond our main application, to payoffs depending on more than two of them. We also discuss the conversion of the so-obtained sensitivities to model parameters (e.g., default intensities) into sensitivities to the market quotes used in calibration (e.g., Credit Default Swap spreads) by generalizations of an existing implicit-function based first-order algorithm.