For Questions assume all options are European style with maturity T.A'K call' is call option with strike price K. A 'knockout' option has payout zero if the defined event occurs.
Ordering option prices by monotonicity theorem II
Consider the following ten options I-X. Here, Ki+1 = Ki + β for i = 1, 2, for a constant β > 0.
I K1 call
II K1 put
III K3, K2 put spread, that is, +1 K3 put and -1 K2 put
IV K1, K2, K3 call butterfly, that is, +1 K1 call, -2 K2 calls and +1 K3 call
V K1 call that knocks out if ST ≥ K2
VI Digital call with strike K1 and payout β
VII K1, K3 digital call spread, that is, a portfolio of +1 digital call with strike K1 and payout β, and -1 digital call with strike K3 and payout β
VIII Digital put with strike K3 and payout β that knocks out if ST ≤ K1
IX A portfolio of +1 K1 call and -2 K2 calls, all of which knock out if St 3 for any 0 ≤ t ≤ T
X A portfolio of +1 K1 call and -2 K2 calls, all of which knock out if ST > K3
For each of the pairs of A and B in Table, choose the most appropriate relationship between prices at time t ≤ T out of =, ≥, ≤ and ?, where ? means the relationship is indeterminate.