At time 0 the price of a non-dividend-paying stock is $S_0$. Suppose that the time interval between 0 and $T$ is divided into two subintervals of length $t_1$ and $t_2$. During the first subinterval, the risk-free interest rate and volatility are $r_1$ and $\sigma_1$, respectively. During the second subinterval, they are $r_2$ and $\sigma_2$, respectively. Assume that the world is risk neutral.
(a) Use the results in Chapter 15 to determine the stock price distribution at time $T$ in terms of $r_1, r_2, \sigma_1, \sigma_2, t_1, t_2$, and $S_0$.
(b) Suppose that $\bar{r}$ is the average interest rate between time zero and $T$ and that $\bar{V}$ is the average variance rate between times zero and $T$. What is the stock price distribution as a function of $T$ in terms of $\bar{r}, \bar{V}, T$, and $S_0$ ?
(c) What are the results corresponding to (a) and (b) when there are three subintervals with different interest rates and volatilities?
(d) Show that if the risk-free rate, $r$, and the volatility, $\sigma$, are known functions of time, the stock price distribution at time $T$ in a risk-neutral world is
$$
\ln S_T \sim \phi\left[\ln S_0+\left(\bar{r}-\frac{1}{2} \bar{V}\right) T, V T\right]
$$
where $\bar{r}$ is the average value of $r, \bar{V}$ is equal to the average value of $\sigma^2$, and $S_0$ is the stock price today and $\phi(m, v)$ is a normal distribution with mean $m$ and variance $v$.