(Boundary conditions for lookback option). The lookback option price $v(t, x, y)$ of $(7.4 .35)$ must satisfy the boundary conditions
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7 Exotic Options
(7.4.7)-(7.4.9). As we saw in Subsection 7.4.3, this is equivalent to the function $u(t, z)$ of $(7.4 .16)$ given by $(7.4 .36)$,
$$
\begin{aligned}
u(t, z)=\left(1+\frac{\sigma^2}{2 r}\right) z N\left(\delta_{+}(\tau, z)\right)+e^{-r \tau} N\left(-\delta_{-}(\tau, z)\right) & \\
& -\frac{\sigma^2}{2 r} e^{-r \tau} z^{1-\frac{2 r}{\sigma^2}} N\left(-\delta_{-}\left(\tau, z^{-1}\right)\right)-z, 0 \leq t<T, 0<z \leq 1,
\end{aligned}
$$
satisfying the boundary conditions (7.4.19)-(7.4.21). This function was shown to satisfy boundary condition (7.4.20) in Exercise 7.5(v). Here we verify by direct computation that the limit of $u(t, z)$ as $z \downarrow 0$ satisfies (7.4.19) and the limit of $u(t, z)$ as $t \uparrow T(\tau \downarrow 0)$ satisfies (7.4.21).