(i) Suppose that there are three European Call options with same maturity on the same

underlying asset. The underlying asset will have a spot price of sT at maturity. The call

option prices are c1, c2 & c3, and strike prices are x1, x2 & x3. Also, suppose that the strike

prices have the following relationship: x1 < x2 < x3 and x3 x2 = x2 x1. Construct a

portfolio with the three call options, and show that

c2 0.5(c1 + c3)

Hint: the key to solve this problem is that you must construct a portfolio with the above

three call options first, and do some scenario analysis (i.e. compare strike prices with the

spot price and work out the portfolio value at maturity). The no-arbitrage pricing condition

also applies.

(ii) Using the result in part (i) and put-call parity, show that

p2 0.5(p1 + p3)

In this case, p1, p2 & p3 are option prices for three European Put options with strike prices

x1, x2 & x3. Note that x1, x2 & x3 follow the same relationship as given in the part (i)