$20\mathrm{.}10$ The formula for an infinitely lived call is given in equation $(12\mathrm{.}18).$ Suppose that $S$

follows equation $(20\mathrm{.}20),$ with $$ replaced by $r,$ and that $E^{*}(dV)=rVdt.$ Use It$\text{'}$s

Lemma to verify that the value of the call, $V(S),$ satisfies this equation:

$\frac{1}{2}{}^2{S}^2{V}_{SS}+(r-)S{V}_S-rV=0$ We will assume that the stock price, $S(t),$ follows the It$$ process given by

$dS(t)=\{hat()[S(t),t]-$hat$()[S(t),t]\}dt+$hat$()[S(t),t]dZ(t)$