A stock is trading at $42 and its volatility is

A stock is trading at $42 and its volatility is 35%. You own a put option with a strike price of $45 and one year to maturity. The risk-free rate is 2% per annum compounded continuously.

(a) How many stocks should you buy or short so that a small change in stock price has no effect on your portfolio (put and the position in stocks)?

(b) Given the portfolio you chose in (a), if stock price increases by a large amount (such as $42 to $50), would the value of the portfolio stay the same, increase or decrease?

(c) Given the portfolio you chose in (a), if stock price decreases by a large amount (such as $42 to $35), would the value of the portfolio stay the same, increase or decrease?