a)Suppose there are 256 business days in a year. You buy today an ATMF put written on the CAC40 price index, of maturity 1 year. You and the market anticipate that the average annualized volatility of the CAC40 price index will be 16% next year. You "manage your gamma" so that your portfolio is always delta-neutral. What is the average realized daily volatility of the CAC40 price index such that your P&L will be (approximately) zero at the end of the year? Explain precisely why. Under what circumstances will your final P&L be positive? Negative?

b)You buy a bull spread involving 2 calls with strikes K1 and K2 (> K1), such that (K1+K2)/2 = S(0), where S(0) is the underlying asset price today. What should you do to be delta-neutral? If you do that, what will be the approximate values of your gamma, theta, and vega?