$(4)$ Consider a call option with strike $K$ and maturity $T,$ on a lognormally distributed underlying

asset with spot price $S,$ volatility $,$ and paying dividends continuously at rate $q.$ Assume

that the risk$-$free interest rates are constant equal to $r.$

$($i$)$ Finding the value of the strike price such that the $$ of the call is $0\mathrm{.}5$ requires solving

$(C)=e^{-qT}N(d_1)=\frac{1}{2}$

for $K,$ where $N(x)$ be the cumulative density of the standard normal variable and

$d_1=\frac{ln(\frac{S}{K})+(r-q+\frac{{}^2}{2})T}{\sqrt[\phantom{2}]{T}}.$

Write down the Newton's method recursion for solving $(??)$ for $K.$

$($ii$)$ Consider a lognormally distributed asset with spot price $$30,$ volatility $30\%,$ and paying

$1\%$ dividends continuously. Assume that the risk$-$free interest rates are constant equal to

$2\mathrm{.}5\%.$ Find the strike at which the $$ of a three months call on this asset is $0\mathrm{.}5.$ Start with

an ATM strike and use Newton's method with tol $={10}^{-6}.$ Report all intermediate values

from Newton's method.