A spread is a combination of option positions that involves four strike prices (see figure below, which shows the gross payoffs). Assume options are European with the same maturity and a > 0.
(a) Find the option positions and strike prices necessary to obtain the the spread. Show that your portfolio replicates the spread payoff and assume that the underlying options are all call options.
(b) Same as (a) but assuming underlying options are all put options
(c) Consider: (i) buy a call with a strike price of X, and (ii) buy a put with a strike price of X+3a. Assuming no arbitrage, show mathematically that initial investment (i.e. the amount of money you need to pay at t = 0) to create the strategy is higher than initial investment to create the spread.
ST on horizontal axis and payoff (a) on vertical axis with (X, X+a, X+2a, X+3a) on horizontal axis at each point of 3-sided trapezoid (X has 0 payoff, X+a has 'a' payoff, X+2a has 'a' payoff, and X+3a has 0 payoff)