Sorry if my question is a bit basic. I am considering the default model as used eg in Vasicek (I think this goes back at least to Merton, though) that looks at an unobserved quantity modeling the assets of a counterparty through a logarithmic Wiener process $$dA_t = \mu A_tdt + \sigma A_tdB_t$$ with solution $$A_t = A_0e^{\left(\mu - \frac{1}{2}\sigma^2\right)t + \sigma\sqrt{t}B_t}.$$ Here, $B_t$ is a Brownian motion. I have see stated that the probability of default of the obligor over a certain time horizon $T$ is the probability that the assets of the obligor are below a certain threshold $D$ after time $T$: $$PD = P\left[A_T<D\right]\ .$$ This has the advantage of being readily computable, but conceptually shouldn't one consider the counterparty defaulted if their assets dip below $D$ at any time between $0$ and $T$?

Namely, I would define a stopping time by $$\tau:=\inf\left\{t\ge0:A_t\le D\right\}$$ and then define $$PD=P\left[\tau < T\right]\ .$$ Why is the first modeling of defaults used instead of the second one? What changes if we use this second definition? Do we still get a closed for solution for the PD (I haven't tried to work that out yet)?

NOTE: From a regulatory perspective for loans, even better would be to consider default if the counterparty's assets dip below a certain threshold and stay below it for at least a given period of consecutive time.

  • $\begingroup$ These simplifying assumptions are made to get analytical tractability. More advanced models do take the approaches suggested in your question, but then the analytics gets complicated. $\endgroup$ Jul 14 '20 at 11:45
  • $\begingroup$ @Magicisinthechain Thanks, I had suspected that as well. Would you have some good references where I could get started with exploring more advanced approaches? $\endgroup$ Jul 14 '20 at 11:47
  • $\begingroup$ My 50 cents: CR models are usually calibrated to PDs, rating transition data, and some dependence information (think: correlations). As all / most of the model parameters are calibrated, a Merton-style model should produce similar results as a knock-out-model, when calibrated to the same set of inputs. $\endgroup$ Jul 15 '20 at 6:00
  • $\begingroup$ A good starting point would be the intensity based default models (darrellduffie.com/uploads/MeasuringCorporateRiskDefault.pdf). And then continue on to doubly stochastic models ( capture heterogeneity over time), and the self exciting intensity based model (capture contagions kinda effects). $\endgroup$ Jul 15 '20 at 11:31

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