Suppose you hold LLL employee stock options representing options to buy $10,000$ shares of LLL stock. You wish to hedge your position by buying put options with three$-$month expirations and a $$22\mathrm{.}50$ strike price. Immediately after establishing your put options hedge, volatility for LLL stock suddenly jumps to $45$ percent. This changes the number of put options required to hedge your employee stock options. LLL accountants estimated the value of these options using the Black$-$Scholes$-$Merton formula and the following assumptions:

S $=$ current stock price $=$ $$20\mathrm{.}72$

K $=$ option strike price $=$ $$23\mathrm{.}15$

r $=$ risk$-$free interest rate $=\mathrm{.}043$

$\backslash$sigma $=$ stock volatility $=\mathrm{.}29$

T $=$ time to expiration $=3\mathrm{.}5$ years