Unifying Black-Scholes type formulae which involve Brownian last passage times up to a finite horizon

The authors recently discovered some interesting relations between the Black-Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a finite horizon. This is achieved in the present paper, where Yuri's formula, as originally presented in Akahori, Imamura and yano (2008), is also derived