For each of the following positions IV-VI below, a) determine whether Bank A has vega 0, vega 0, vega = 0, or vega is indeterminate (?). b) Similarly, determine Bank As delta with respect to the stock price - is delta 0,0,=0, or ?. All options are European with maturity T on a stock that pays no dividends, and 0 < K₁ < K₂. Justify your answers.

IV. long a K₁, (K₁ + K₂)/2, K₂ call butterfly

V. long a K₁ put knocking out if S_T< K₂

VI. long a K₂ put knocking out if S_T > K₁