free rates are $0\mathrm{.}0503($July $17$th$),0\mathrm{.}0535($August ${21}^{st}),0\mathrm{.}0571($October ${16}^{th}).$ The stock price is $$165\mathrm{.}13$

$\backslash$table$[[,$Calls,Puts$],[$Strike$,$Jul,Aug,Oct,Jul,Aug,Oct$],[155,10\mathrm{.}5,11\mathrm{.}75,14\mathrm{.}00,0\mathrm{.}19,1\mathrm{.}25,2\mathrm{.}75],[160,6\mathrm{.}00,8\mathrm{.}13,11\mathrm{.}13,0\mathrm{.}75,2\mathrm{.}75,4\mathrm{.}50],[165,2\mathrm{.}69,5\mathrm{.}25,8\mathrm{.}13,2\mathrm{.}38,4\mathrm{.}75,6\mathrm{.}75],[170,0\mathrm{.}81,3\mathrm{.}25,6\mathrm{.}00,5\mathrm{.}75,7\mathrm{.}5,9\mathrm{.}00]]$

Let the standard deviation of the continuously compounded return on the stock be $21$ percent. Ignore dividends. Respond to the following

a$.$ What is the theoretically fair value of the October $165$ call?

b$.$ Based on your answer, recommend a risk$-$less strategy using the delta$-$hedge.

c$.$ If the stock price decreases by $$1,$ how will the option position offset the loss on the stock?