Exercise 7.2 (Boundary conditions for the up-and-out call). In this exercise, it is verified that the up-and-out call price $v(t, x)$ given by $(7.3 .20)$ satisfies the boundary condition (7.3.6). Furthermore, the limit as $x \downarrow 0$ satisfies (7.3.5) and the limit as $t \uparrow T$ satisfies (7.3.7).
(i) Verify by direct substitution into (7.3.20) that (7.3.6) is satisfied.
(ii) Show that, for any positive constant $c$,
$$
\lim _{x \downarrow 0} \delta_{ \pm}\left(\tau, \frac{x}{c}\right)=-\infty, \lim _{x \downarrow 0} \delta_{ \pm}\left(\tau, \frac{c}{x}\right)=\infty .
$$
Use this to show that for any $p \in \mathbb{R}$ and positive constants $c_1$ and $c_2$, we have
$$
\begin{aligned}
\lim _{x \downarrow 0} x^p\left[N\left(\delta_{ \pm}\left(\tau, \frac{x}{c_1}\right)\right)-N\left(\delta_{ \pm}\left(\tau, \frac{x}{c_2}\right)\right)\right] & =0, \\
\lim _{x \downarrow 0} x^p\left[N\left(\delta_{ \pm}\left(\tau, \frac{c_1}{x}\right)\right)-N\left(\delta_{ \pm}\left(\tau, \frac{c_2}{x}\right)\right)\right] & =0 .
\end{aligned}
$$
If $p \geq 0,(7.8 .12)$ and (7.8.13) are immediate consequences of (7.8.11). However, if $p<0$, one should first use L'Hôpital's rule and then show that
$$
\lim _{x \downarrow 0} x^p \exp \left\{-\frac{1}{2} \delta_{ \pm}^2\left(\tau, \frac{x}{c_i}\right)\right\}=0, \lim _{x \downarrow 0} x^p \exp \left\{-\frac{1}{2} \delta_{ \pm}^2\left(\tau, \frac{c_i}{x}\right)\right\}=0 .
$$
To establish (7.8.14), you may wish to prove and use the inequality
$$
\frac{1}{2} a^2-b^2 \leq(a+b)^2 \text { for all } a, b \in \mathbb{R} .
$$
Conclude that $\lim _{x \downarrow 0} v(t, x)=0$ for $0 \leq t<T$.
(iii) Show that, for any positive $c$,
$$
\lim _{\tau \downarrow 0} \delta_{ \pm}(\tau, c)= \begin{cases}-\infty & \text { if } 0<c<1 \\ 0 & \text { if } c=1 \\ \infty & \text { if } c>1\end{cases}
$$
Use this to show that $\lim _{\tau \downarrow 0} v(t, x)=(x-K)^{+}$for $0<x<B$.