The Default Density Process Under a Lognormal Intensity Model

Summary

The document studies the stochastic behavior of marginal default density when default intensity follows a lognormal diffusion. It writes default density as intensity multiplied by survival probability, then applies Itô’s product rule to derive a stochastic differential equation for that quantity. The drift includes the intensity itself, while the diffusion term inherits the intensity process’s volatility.

The author reports that simple numerical examples appear roughly lognormal after shifts in mean and volatility, and asks how generally that approximation holds. No numerical results, proofs, or literature references are supplied, so the extent of deviation from lognormality remains unresolved. The equation provides a starting point for analysis, but the survival factor depends on the accumulated path of intensity, which limits conclusions based only on the marginal intensity distribution. Further study or simulation would be needed to characterize the resulting density distribution.

Key ideas

• Marginal default density is intensity multiplied by the probability of survival to that time.
• Applying Itô’s product rule yields a drift term adjusted by the current intensity.
• The resulting process also retains a diffusion term driven by the intensity’s Brownian motion.
• Numerical examples suggest approximate lognormality, but the document provides no general result or evidence on its accuracy.

Tags

# Marginal default distribution when default intensity is log-normal


# Marginal default distribution when default intensity is log-normal












I'm interested in the stochastic process followed by the marginal default density $e^{-\int_0^th(s)ds}h(t)$ in the case where the default intensity $h(t)$ follows a log-normal process.

Assuming

$$\frac{dh(t)}{h(t)} = \mu(t)dt+\sigma(t)dB(t)$$

an application of Ito's product rule shows that

$$ d\left(e^{-\int_0^th(s)ds}h(t)\right)=e^{-\int_0^th(s)ds}h(t)\left((\mu(t)-h(t))dt+\sigma(t)dB(t)\right) $$

I have generated some simple numerical examples which give the impression that the resulting process is not too far from log-normal (with shifted mean and volatility). But I'm interested in what might be known about the process more generally, and how far it diverges from log-normality. I imagine this must be a well-studied problem, and would be grateful if someone could direct me to where it has been studied in the literature.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.