At time $t$ a stock price is $S$. Suppose that the time interval between $t$ 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 10 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 $S$
(b) Suppose that $\bar{r}$ is the average interest rate between time $t$ and $T$, and that $\bar{V}$ is the average variance rate between times $t$ and $T .$ What is the stock price distribution at time $T$ in terms of $\bar{r}, \bar{V}, T-t,$ and $S ?$
(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 can be calculated using Equation (10.7) on the assumption that (1) the risk-free rate is constant and equal to the average value of $r ;$ and (2) the variance rate is constant and equal to the average value of $\sigma^{2}$
(e) Prove the result in Section 17.1