# Definition of defaults via unobserved assets

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 $$T$$: $$PD = P\left[A_T This has the advantage of being readily computable, but conceptually shouldn't one consider the counterparty defaulted if their assets dip below $$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.

• 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. Jul 14 '20 at 11:45
• @Magicisinthechain Thanks, I had suspected that as well. Would you have some good references where I could get started with exploring more advanced approaches? Jul 14 '20 at 11:47
• 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. Jul 15 '20 at 6:00
• 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). Jul 15 '20 at 11:31