Question at position 6 You buy one European call option with strike K1, one European call option with strike K3, and short two European call options with strike K2=K1+K32 on the same underlying asset with the same expiration at T . Note that K1

Question

Question at position 6 You buy one European call option with strike K1, one European call option with strike K3, and short two European call options with strike K2=K1+K32 on the same underlying asset with the same expiration at T . Note that K1<K2<K3. When you set up the position at t, do you have a cash inflow or a cash outflow? Hint: Think about no arbitrage arguments if a strategy gives you a positive or negative cashflow in the future.You have a cash inflowYou have a cashflow of exactly zeroYou have a cash outflowThere is not enough information to answer this question单项选择题

A

You have a cash inflow

B

You have a cashflow of exactly zero

C

You have a cash outflow

D

There is not enough information to answer this question

BlackTom AI · Accepted Answer

For K2 = (K1 + K3)/2 the portfolio C(K1) + C(K3) − 2C(K2) is the standard (long) butterfly. Compute its payoff at expiration ST piecewise.

1) ST ≤ K1: all calls expire worthless, payoff = 0.

2) K1 < ST ≤ K2: only the K1 call is in the money:
   payoff = (ST − K1).

3) K2 < ST ≤ K3: K1 and K2 calls are in the money (K3 still out):
   payoff = (ST − K1) − 2(ST − K2) = −ST + (2K2 − K1).
   Using K2 = (K1 + K3)/2 gives 2K2 − K1 = K3, so payoff = K3 − ST.

4) ST > K3: all three calls in the money:
   payoff = (ST − K1) + (ST − K3) − 2(ST − K2)
          = −K1 − K3 + 2K2 = 0  (again because 2K2 = K1 + K3).

Thus the payoff is:
- 0 for ST ≤ K1,
- rises linearly to a maximum at ST = K2 of value K2 − K1 (which equals K3 − K2),
- then falls linearly to 0 at ST = K3,
- and is 0 for ST ≥ K3.

This payoff is nonnegative for all ST and strictly positive for ST in (K1, K3) (except endpoints). By no-arbitrage, a claim that yields a nonnegative payoff in every state and a strictly positive payoff in some states cannot have zero or negative initial price. If it had a negative initial price (i.e., you receive money now), you could take it and guarantee a nonnegative future payoff with some positive probability of gain — an arbitrage. If it had zero initial price, you would have a free positive-probability payoff, also an arbitrage. Therefore the initial cost must be positive: you must pay to establish the position (a cash outflow).

Why other choices are wrong:
- "You have a cash inflow": would imply receiving money now for a portfolio that never loses and sometimes pays positive later, which would create arbitrage — impossible.
- "Cashflow of exactly zero": only possible if the payoff were identically zero for all ST, which it is not.
- "Not enough information": we have enough — the payoff structure and no-arbitrage determine the sign of the initial cashflow.