$($i$)$ State the assumptions underlying the Black$-$Scholes option pricing formula

and discuss how realistic they are. $[6]$

An investment bank has written a number, $N,$ of European call options on a non$-$dividend

paying stock with strike price R$250,$ current stock price R$185,$ time to expiry

of $2$ years and an assumed continuously $-$ compounded interest rate of $6\%$ p$.$a$.$ The

bank is delta$-$hedging the option position assuming the Black$-$Scholes framework

holds and currently holds $125,000$ shares of the stock and is short R $3,800000$ in cash.

$($ii$)$ By using the hedging position and the Black$-$Scholes formula for the value of

the option, derive two equations satisfied by $N$ and $,$ the bank's assumed volatility. $[5]$

$($iii$)$ Estimate by interpolation. $[7]$

$($iv$)$ Deduce the value of $N.[2]$