Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process
$dx=a(x_0-x)dt + sxdz$
where $a, x_0$ and $s$ are positive constants and $dz$ is the wiener process. What is the process followed by a bond price?
Solution: $dS=\mu Sdt+\sigma S dz$
where S is the bond price and $\mu$ and $\sigma$ are expected instantaneous return and instantaneous volatility respectively. Yield to maturity is the total return anticipated on a bond if the bond is held until the end of its lifetime.