The hazard rate tangent approximation for boundary hitting times (Q1901084)

The approximation of the distribution of the hitting time \(\tau_f = \inf_{t > 0} \{t : X_t \geq f(t)\}\) of a diffusion process \(X_t\) is considered. When \(X_t\) is the standard Brownian motion \(B_t\), the tangent approximation provides an inaccurate density approximation of \(\tau_f\) by using the Bachelier-Levy formula: \(p^{a,b} (t) = {a \over t^{3/2}} \varphi ({a + bt \over t^{1/2}})\), where \(\varphi\) is the standard Brownian density, \(a = f(t) - tf'(t)\), \(b = f'(t)\). Authors introduce the hazard rate tangent approximation, which considerably improves on the tangent approximation when \(f\) is convex or concave. Numerical examples are also given.