Assume that a non$-$dividend$-$paying stock has an expected return of $$ and a volatility of $.$ An innovative financial institution has just announced that it will trade a derivative that pays off a dollar amount equal to $ln{S}_T$ at time $T$ where $S_T$ denotes the values of the stock price at time $T.$

a$)$ Use risk$-$neutral valuation to calculate the price of the derivative at time $t$ in term of the stock price, $S,$ at time $t$

b$)$ Confirm that your price satisfies the differential equation $(15\mathrm{.}16)$