$($a$)$ The table below gives information about European options with a maturity date of

$6$ months.

$($i$)$ Devise the payoff profile of the hedging strategy from the above for an investor

betting on an increase in the stock price and calculate the payoff if the stock price

increases to $$66$ after $6$ months.

$(8$ marks$)$

$($ii$)$ Suppose that another investor expects a big stock price movement but is not sure

of the direction. She however bets that the downward movement is more likely.

Devise the corresponding trading strategy and calculate the payoff if the stock price

is $$55$ after $6$ months.

$(8$ marks$)$

$($b$)$ By analysing the pay off profiles of a protective put strategy and a straddle,

discuss in what ways these strategies shield the investor from potential losses.