Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / B / 4 / a / Solution

Apply Itô's formula to $\log X$ for positive initial value $x_0$. The drift correction is $-{\sigma }^2/2$, soThis is geometric Brownian motion. Under the physical probability measure, $W_T/\sqrt{T}$ has the standard normal distribution. For positive volatility,The drift here is $\mu$, because this is a physical bankruptcy probability. The risk-neutral measure will instead give drift $r$ for pricing. If $\sigma =0$, the value is deterministic and the default probability is the indicator of $x_0{e}^{\mu T}<m$; the usual positive-volatility convention is used in the remaining formulas.