Sebastian Jaimungal (University of Toronto)

Hitting Time Problems with Applications to Finance and Insurance

The distribution of the first hitting time of a Brownian motion to a linear boundary is well known. However, if the boundary is no longer linear, this distribution is not in general identifiable. Nonetheless, the boundary and distribution satisfy a variety of beautiful integral equations due to Peskir. In this talk, I will discuss how to generalize those equations and lead to an interesting partial solution to the inverse problem: “Given a distribution of hitting times, what is the corresponding boundary?” By randomizing the starting point of the Brownian motion, I will show how a kernel estimator of the distribution with gamma kernels can be exactly replicated.
Armed with these tools, there are two natural applications: one to finance and one to insurance. In the financial context, the Brownian motion may drive the value of a firm and through a structural modeling approach I will show how CDS spread curves can be matched. In the insurance context, suppose an individual’s health reduces by one unit per annum with fluctuations induced by a Brownian motion and once their health hits zero the individual dies. I will show how life-table data can be nicely explained by this model and illustrate how to perturb the distribution for pricing purposes.
This is joint work with Alex Kreinin and Angelo Valov.