A structural credit model is a framework for modelling default risk in which default is triggered when a firm’s asset value process crosses a pre-specified threshold (typically related to the firm’s debt structure, such as short-term debt plus half of long-term debt). This contrasts with reduced-form models, where default is driven by an exogenous intensity process. In the structural approach, the asset value follows a diffusion-type stochastic differential equation, and the default time is defined as the first hitting time of the barrier — a stopping time of the filtration generated by the asset process.
In the context of xVA computation, structural models have the advantage that default is endogenously derived from firm dynamics, which enables a natural linkage between credit risk and market exposure. This is particularly useful for capturing wrong-way risk (the adverse dependence between exposure and default likelihood), since the same Brownian drivers that move the derivative portfolio also drive the firm value processes. The correlation structure between the d non-defaultable risk factors and the 2 defaultable risk factors (bank and counterparty assets) is captured through a full (d+2) x (d+2) correlation matrix.
A key numerical advantage in the context of deep BSDE methods is that structural models produce BSDEs driven solely by Brownian motions, avoiding the jump processes that arise in reduced-form models. This means the standard Euler-Maruyama discretization applies directly to the forward SDE, and the BSDE involves only continuous stochastic integrals. However, the default barrier introduces a bounded domain for the BSDE, which requires careful error analysis. The convergence rate degrades from O(h^{1/2}) to O(h^{1/4-epsilon}) due to the difficulty of accurately resolving boundary exits in time-discretized schemes.
Key Details
- Default times for the bank and counterparty are given by tau_B = inf{t : X^{d+1}_t ⇐ xi^1_t} and tau_C = inf{t : X^{d+2}_t ⇐ xi^2_t}, where xi are deterministic barrier functions
- The first default time tau = tau_B ^ tau_C terminates the xVA BSDEs
- Asset processes are typically modelled as geometric Brownian motions in numerical experiments
- The approach can be extended to multiple counterparties via divide-and-conquer strategies
- A common criticism of structural models is that they underestimate short-term default risk because diffusion processes are predictable near the barrier; this can be addressed by introducing Levy jumps in the asset dynamics