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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?

          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?

Question 5 (Interest rate models)
(a) Suppose the instantaneous interest rate, rt, follows the Vasicek model
(i)
drt = α(μ - rt)dt + σ dWt,
(Equation 1)
where α > 0, μ and σ 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 rt for t given.
(iv) Derive the expected value and the variance of rt for t given.
(v) Describe the main drawback of the Vasicek model.
(b) Let the stochastic process Y follow the stochastic differential equation
dYt = (2α Yt + σ^2)dt + 2σ√(Yt)dWt,
where α and σ are constants and W is a standard Brownian motion. This is an example
of a Cox-Ingersoll-Ross process. Define a process Z by Zt = √(Yt). Derive the stochastic
differential equation that Zt will follow. What is the name of this process?

Added by Sandra B.

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