Each of the following products pays a function of the spot price, S, of a non-dividend-paying stock one year from now. If there are no interest rates and spot is 100, give optimal upper and lower bounds on their prices today.
(i) The pay-off is l between 110 and 130 and zero otherwise;
(ii) The pay-off is S - 80;
(iii) The pay-off is zero below 80, increases linearly from zero at 80 to 20 at 120 and then is constant at 20 above 120;
(iv) The pay-off is (S - 100)2.