a) Assuming European options on non-dividend paying stocks, state the exact formulas for the delta, gamma and theta for a written (short) position in: 

i) a strangle spread (X for put less than for X call, same T); 

ii) a butterfly using puts (3 different X and same T); 

iii) a vertical spread using calls (different X and same T); iv) a straddle spread (same X for put and calls); and, a strap spread (2 calls, 1 put, same).

b) Using the parameters: S = 53; r = 0.025; σ = 0.38; T = 0.5 if each spread is constructed to be delta neutral and have the same initial value (the value of the positions, V*, is the same), then calculate the relative gamma and theta of the spread positions and explain the reasons for the differences in the Greeks?