  Given a finite honest time, we first show that the associated Az\'ema
optional supermartingale can be expressed as the drawdown and the relative
drawdown of some local optional supermartingales with continuous running
supremum. The relative drawdown representation then allows us to provide a
characterisation of finite honest times using a family of non-negative local
optional supermartingales with continuous running supremum which converges to
zero at infinity. Then we extend the notion of semimartingales of
class-$(\Sigma)$ by allowing for jumps in its finite variation part of the
semimartingale decomposition. This enables one to establish the
Madan-Roynette-Yor option pricing formula for a larger class of processes, and
finally, we apply the extended formula to the construction of finite honest
times.
