From: On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called “meander option”
BM
P ( · ≦ A )
RW
P(·<A)
sup u ≦ θ B u ∼ θ sup u ≦ 1 B u
1−e−λA
sup u ≦ θ Z u
1−αA
sup u ≦ g θ B u ∼ g θ sup u ≦ 1 b u
1−e−2λA
sup u ≦ g θ Z u
1−α2A
b:brownian bridge(b.b.)
sup g θ ≦ u ≦ θ B u ∼ε θ − g θ sup u ≦ 1 m u
1 1 + e − λA
sup g θ ≦ u ≦ θ Z u
1 1 + α A
m:brownian meander
sup u ≦ d θ B u
1− 1 − e − 2 λA 2 λA
sup u ≦ d θ Z u
1− 1 A 1 α − 1 − α 1 − α 2 A
sup θ ≦ u ≦ d θ B u
1− 1 − e − λA 2 λA
sup θ ≦ u ≦ d θ Z u
1− 2 α − 1 − α 1 − α A A
sup g θ ≦ u ≦ d θ B u ∼ε d θ − g θ sup u ≦ 1 e u
1 1 − e − 2 λA − 1 2 λA
sup g θ ≦ u ≦ d θ Z u
1 1 − α 2 A − 1 A 1 α − 1 − α
e:normalized excursion