Right here goes...
Assuming that you correctly demonstrated the pricing for the State-Price derivatives in the first part.
The new fee can be represented as:
0.1% x E(s1)
+0.4%x(C1+SPD1)
+05%x(C2+SPD2)
where C1 is a call-option with strike price S0, and SPD1 is a State-Price Derivative paying S0, if S1 is between S0 and infinity.
C2 is a call-option with Strike-Price U, and SPD2 is a State-Price Derivative paying S0 if S1 is between U and infinity.
SPD1 and C1 can be calculated directly (C1 is about 12p I believe).
Equate:
0.5%S0 = 0.1% S0 + 0.4%(C1+SPD1) + 0.5%(C2+SPD2)
=> C2+SPD2=(0.4%/0.5%)*(S0-(C1+SPD1)
Modify the Garman-Kohlhagen formula for C2 to include the value for SPD2 (from part iii). This gives us a formula relating U to C2+SPD2. Solve this formula for the given value of C2+SPD2, using interpolation.
There....
Wish I'd thought of it in the exam, I ignored the SPD's completely, and so got ridiculous answers.
Still annoyed about the lagrangian part of q6 though. Completely didn't see that.