Exercise 7.1 (Black-Scholes-Mertonequation for the up-and-out call). This exercise shows by direct calculation that the function $v(t, x)$ of $(7.3 .20)$ satisfies the Black-Scholes-Merton equation (7.3.4).
(i) Recall that $\tau=T-t$, so $\frac{d r}{d t}=-1$. Show that $\delta_{ \pm}(\tau, s)$ given by (7.3.18) satisfies
$$
\frac{\partial}{\partial t} \delta_{ \pm}(\tau, s)=-\frac{1}{2 \tau} \delta_{ \pm}\left(\tau, \frac{1}{s}\right) \text {. }
$$
(ii) Show that for any positive constant $c$,
$$
\frac{\partial}{\partial x} \delta_{ \pm}\left(\tau, \frac{x}{c}\right)=\frac{1}{x \sigma \sqrt{\tau}}, \quad \frac{\partial}{\partial x} \delta_{ \pm}\left(\tau, \frac{c}{x}\right)=-\frac{1}{x \sigma \sqrt{\tau}} .
$$
(iii) Show that
$$
\frac{N^{\prime}\left(\delta_{+}(\tau, s)\right)}{N^{\prime}\left(\delta_{-}(\tau, s)\right)}=\frac{e^{-r \tau}}{s}
$$
and hence
$$
e^{-r \tau} N^{\prime}\left(\delta_{-}(\tau, s)\right)=s N^{\prime}\left(\delta_{+}(\tau, s)\right)
$$
(iv) Show that
$$
\frac{N^{\prime}\left(\delta_{ \pm}(\tau, s)\right)}{N^{\prime}\left(\delta_{ \pm}\left(\tau, s^{-1}\right)\right)}=s^{-\left(\frac{2 r}{\sigma^2} \pm 1\right)}
$$
and hence
$$
N^{\prime}\left(\delta_{ \pm}\left(\tau, s^{-1}\right)\right)=s^{\frac{2 r}{\sigma^2} \pm 1} N^{\prime}\left(\delta_{ \pm}(\tau, s)\right) .
$$
(v) Show that
$$
\delta_{+}(\tau, s)-\delta_{-}(\tau, s)=\sigma \sqrt{\tau} .
$$
(vi) Show that
$$
\delta_{ \pm}(\tau, s)-\delta_{ \pm}\left(\tau, s^{-1}\right)=\frac{2}{\sigma \sqrt{\tau}} \log s
$$
(vii) Show that
$$
N^{\prime \prime}(y)=-y N^{\prime}(y) .
$$