(i) Suppose that $Z$ is a random variable having the normal distribution with $\mathbb{E} Z=\beta$ and $\operatorname{Var} Z=\tau^{2}$.

For positive constants $a, c$ show that

$$\mathbb{E}\left(a e^{Z}-c\right)_{+}=a e^{\left(\beta+\tau^{2} / 2\right)} \Phi\left(\frac{\log (a / c)+\beta}{\tau}+\tau\right)-c \Phi\left(\frac{\log (a / c)+\beta}{\tau}\right)$$

where $\Phi$ is the standard normal distribution function.

In the context of the Black-Scholes model, derive the formula for the price at time 0 of a European call option on the stock at strike price $c$ and maturity time $t_{0}$ when the interest rate is $\rho$ and the volatility of the stock is $\sigma$.

(ii) Let $p$ denote the price of the call option in the Black-Scholes model specified in (i). Show that $\frac{\partial p}{\partial \rho}>0$ and sketch carefully the dependence of $p$ on the volatility $\sigma$ (when the other parameters in the model are held fixed).

Now suppose that it is observed that the interest rate lies in the range $0<\rho<\rho_{0}$ and when it changes it is linked to the volatility by the formula $\sigma=\ln \left(\rho_{0} / \rho\right)$. Consider a call option at strike price $c=S_{0}$, where $S_{0}$ is the stock price at time 0 . There is a small increase $\Delta \rho$ in the interest rate; will the price of the option increase or decrease (assuming that the stock price is unaffected)? Justify your answer carefully.

[You may assume that the function $\phi(x) / \Phi(x)$ is decreasing in $x,-\infty<x<\infty$, and increases to $+\infty$ as $x \rightarrow-\infty$, where $\Phi$ is the standard-normal distribution function and $\phi=\Phi^{\prime}$.]