Use the Black$-$Scholes$-$Merton European put option pricing formula. Price the October $165$ put. Use the October $165$ call calculated in the previous question.

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

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?