QUESTION $3(18$ marks$)$

$($a$)$ Assume that the stock price follows a geometric Brownian motion

$\frac{dS}{S}=(-)dt+dZ$

where $,$ and $$ are the expected return, volatility and continuous dividend yield on the stock.

Use delta$-$hedge and Ito's Lemma to derive the Black$-$Scholes partial differential equation.