At time 0 the price of a non-dividend-paying stock is S0. Suppose that the time interval between 0 and T is divided into two subintervals of length t1 and t2. During the first subinterval, the risk-free interest rate and volatility are r1 and σ1, respectively. During the second subinterval, they are r2 and σ2, respectively. Assume that the world is risk neutral.
(a) Use the results in Chapter 14 to determine the stock price distribution at time T in terms of 
(b) Suppose that 
is the average interest rate between time zero and T and that 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 
(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, α, are known functions of time, the stock price distribution at time T in a risk-neutral world is
where 
is the average value of r, 
is equal to the average value of σ2 , and S0 is the stock price today and ?(m, v) is a normal distribution with mean m and variance v.