For each of the pairs of a and b in table choose the most

Question

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, K_i+1 = K_i + 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 S_T ≥ 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 S_T ≤ K₁

IX A portfolio of +1 K₁ call and -2 K₂ calls, all of which knock out if S_t 3 for any 0 ≤ t ≤ T

X A portfolio of +1 K₁ call and -2 K₂ calls, all of which knock out if S_T > 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.