A stochastic correlation extension of the Vasicek credit risk model

Abstract

In the Vasicek credit portfolio model, tail risk is driven primarily by the asset-correlation parameter, yet empirically is subject to correlation risk. We propose a stochastic correlation extension of the Vasicek framework in which the correlation state evolves as a diffusion on the circle. This representation accommodates both non-mean-reverting and mean-reverting dependence regimes via circular Brownian motion and von Mises process, while retaining tractable transition densities. Conditionally on a fixed correlation state, we derive closed or semi-closed form expressions for the joint distribution of two assets, the joint first-passage (default) time distribution, and the joint survival probability. A simulation study quantifies how correlation volatility and persistence reshape joint default-at-horizon, survival, and joint barrier-crossing probabilities beyond marginal volatility effects. An empirical illustration using U.S. bank charge-off rates demonstrates economically interpretable time-variation in a dependence index and shows how inferred stochastic dependence translates into materially different joint tail-event probabilities. Overall, circular diffusion models provide a parsimonious and operationally tractable route to incorporating correlation risk into Vasicek structural credit calculations.

Figures (10)

• Figure 1: Joint default probability $p_{\mathrm{JD}}(t)$ as a function of maturity $t$ for varying GBM volatility $\sigma$. Higher marginal volatility increases joint default risk and leads to earlier saturation in $t$.
• Figure 2: Joint survival probability $p_{\mathrm{Surv}}(t)=\mathbb{P}(\tau_1>t,\tau_2>t)$ for varying GBM volatility $\sigma$. Survival decays faster and to lower levels as marginal volatility increases.
• Figure 3: Joint default probability $p_{\mathrm{JD}}(t)$ for varying correlation-volatility parameter $\sigma_{\mathrm{von}}$ in the von Mises correlation model. Higher $\sigma_{\mathrm{von}}$ reduces the level of joint default probabilities.
• Figure 4: Joint default probability $p_{\mathrm{JD}}(t)$ for varying mean-reversion speed $\lambda$ of the correlation dynamics. Higher $\lambda$ increases the level of joint default probabilities, consistent with more persistent dependence.
• Figure 5: Joint first-passage probability $p_{\mathrm{FPT}}(t)=\mathbb{P}(\tau_1\le t,\tau_2\le t)$ for varying GBM volatility $\sigma$. Higher volatility substantially increases short-horizon joint barrier-crossing likelihood.

Theorems & Definitions (12)

• Proposition 2.1
• proof
• Theorem 2.1: Joint distribution of asset values
• proof
• Corollary 2.1.1
• Theorem 2.2
• proof
• Corollary 2.2.1
• Lemma 2.3
• proof