On the one-sided maximum of Brownian and random walk fragments and its applications to new exotic options called “meander option”

Pacific Journal of Mathematics for Industry

Table 1 A list of interesting maxima

BM	$P (\cdot ≦ A)$	RW	P(·<A)
$sup_{u ≦ θ} B_{u} \sim \sqrt{θ} sup_{u ≦ 1} B_{u}$	1−e^−λA	$sup_{u ≦ θ} Z_{u}$	1−α^A
$sup_{u ≦ g_{θ}} B_{u} \sim \sqrt{g_{θ}} sup_{u ≦ 1} b_{u}$	1−e^−2λA	$sup_{u ≦ g_{θ}} Z_{u}$	1−α^2A
b:brownian bridge(b.b.)
$sup_{g_{θ} ≦ u ≦ θ} B_{u} \sim ε \sqrt{θ - g_{θ}} sup_{u ≦ 1} m_{u}$	$\frac{1}{1 + e^{- λA}}$	$sup_{g_{θ} ≦ u ≦ θ} Z_{u}$	$\frac{1}{1 + α^{A}}$
m:brownian meander
$sup_{u ≦ d_{θ}} B_{u}$	$1 - \frac{1 - e^{- 2 λA}}{2 λA}$	$sup_{u ≦ d_{θ}} Z_{u}$	$1 - \frac{1}{A} \frac{1}{α^{- 1} - α} (1 - α^{2 A})$
$sup_{θ ≦ u ≦ d_{θ}} B_{u}$	$1 - \frac{1 - e^{- λA}}{2 λA}$	$sup_{θ ≦ u ≦ d_{θ}} Z_{u}$	$1 - \frac{2}{α^{- 1} - α} \frac{1 - α^{A}}{A}$
$sup_{g_{θ} ≦ u ≦ d_{θ}} B_{u} \sim ε \sqrt{d_{θ} - g_{θ}} sup_{u ≦ 1} e_{u}$	$\frac{1}{1 - e^{- 2 λA}} - \frac{1}{2 λA}$	$sup_{g_{θ} ≦ u ≦ d_{θ}} Z_{u}$	$\frac{1}{1 - α^{2 A}} - \frac{1}{A} \frac{1}{α^{- 1} - α}$
e:normalized excursion

where
∙ for BM, θ∼Exp(λ²/2), i.e., its density is $f_{θ} (x) = 1_{(0, \infty)} (x) \frac{λ^{2}}{2} exp - \frac{λ^{2} x}{2}$ , and P(ε=1)=P(ε=0)=1/2.
∙ for RW, θ∼Geom(1−q), i.e., $P (θ = k) = (1 - q) q^{k}, (k = 0, 1, 2, \dots)$ , $α = \frac{1 - \sqrt{1 - q^{2}}}{q} .$
∙ for RW, $g_{t} = sup {u ≦ t : Z_{u} = 0}, d_{t} = inf {u > t : Z_{u} = 0} .$