Question 5 (Interest rate models)
(a) Suppose the instantaneous interest rate, $r_t$, follows the Vasicek model
(i)
$dr_t = \alpha(\mu - r_t)dt + \sigma dW_t$,
(Equation 1)
where $\alpha > 0$, $\mu$ and $\sigma$ are constants and $W$ is a standard Brownian motion.
State the type of process of which the Vasicek model is an example.
(ii) Solve the stochastic differential equation (1).
(iii) State the distribution of $r_t$ for $t$ given.
(iv) Derive the expected value and the variance of $r_t$ for $t$ given.
(v) Describe the main drawback of the Vasicek model.
(b) Let the stochastic process $Y$ follow the stochastic differential equation
$dY_t = (2\alpha Y_t + \sigma^2)dt + 2\sigma \sqrt{Y_t}dW_t$,
where $\alpha$ and $\sigma$ are constants and $W$ is a standard Brownian motion. This is an example
of a Cox-Ingersoll-Ross process. Define a process $Z$ by $Z_t = \sqrt{Y_t}$. Derive the stochastic
differential equation that $Z_t$ will follow. What is the name of this process?
