You are a market maker, and you just rebalance your inventory to make it delta and gamma neutral. Right after setting your inventory delta and gamma to zero, you are long 200 shares of the stock, short 400 calls (C1) on the stock with delta 0.40, and long 200 calls (C2) on the stock with delta 0.60. Which of the following statements is correct if your objective is to further make your inventory delta neutral?

Question

BlackTom AI · Accepted Answer

We start by restating the scenario and each answer option to ensure clarity.
Question setup: You are a market maker who has already rebalanced to delta and gamma neutral. Current positions: long 200 shares of the stock, short 400 calls (C1) with delta 0.40, and long 200 calls (C2) with delta 0.60.
Answer options:
- a. You will need to buy 160 shares of the stock.
- b. You will need to buy 266.67 calls C2.
- c. You will need to buy 400 calls C1.
- d. You will need to sell 160 shares of the stock.

Step-by-step analysis of each option:
Option a: Buy 160 shares. Since you already have a net delta exposure, adding +160 delta from 160 more shares would push your delta further positive, moving away from neutrality rather than toward it. This option would not help achieve delta neutrality; it would require an even larger opposite hedge later.

Option b: Buy 266.67 calls C2. Each C2 carries a delta of about +0.60. Purchasing 266.67 calls would add roughly 160 additional delta (266.67 × 0.60 ≈ 160). That would again increase your overall positive delta instead of reducing it to zero, making neutrality farther away. Moreover, the fractional number of options is practically awkward, signaling a mismatched hedge scale.

Option c: Buy 400 calls C1. Each C1 has a delta of about +0.40. Buying 400 calls would add about 160 delta (400 × 0.40 = 160). This would raise your net delta by 160, not reduce it, so you would not achieve neutrality and would still be left with a positive delta after this trade.

Option d: Sell 160 shares. Your current net delta is the sum of all positions’ deltas. With +200 from the stock, -160 from the 400 C1, and +120 from the 200 C2, the total delta equals +160. Offloading 160 shares reduces delta by 160, bringing the net delta to zero and achieving delta neutrality.

Putting it together, only the action in option d directly reduces the net delta by the exact amount needed to reach zero, whereas all other options either increase delta or mis-scale the hedge. The reasoning aligns with the goal of maintaining a delta-neutral portfolio by counterbalancing the existing positive delta with a corresponding negative delta hedge.

FINS3635-Options, Futures & Risk Mgmt - T3 2025

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