Question

An investor has an option portfolio composed of 5,000 identical short call options. The call option’s delta is 0.4.

1. What does it mean intuitively that the delta is 0.4?
2. How can the portfolio be made delta-neutral?
3. How will the delta-neutral portfolio’s value change if there is a small (e.g. 0.1%) upward movement in the stock price?
4. Will the delta-neutral portfolio’s value change if there is a large (e.g. 10%) upward movement in the stock price? If no, why not. If yes, what hedge could protect against the change in the portfolio’s value?

Answer 1

a]

Intuitively, it means that for every 1% change in the price of the underlying stock, the price of the call option will change by 0.4%

b]

The portfolio can be made delta-neutral by buying a quantity of the underlying stock which would have a total delta equal to the total delta of the short call options.

Total delta of short call options = -5000 * 0.4 = -2000 (short call options have a negative delta)

The delta of the underlying stock is always equal to 1. A long position in the stock will have a positive delta.

Therefore, number of underlying stock to buy = 2000 / 1 = 2000.

Overall portfolio delta = (-5000 * 0.4) + (2000 * 1) = 0

If 2000 quantity of the underlying stock are bought, the portfolio will be delta-neutral

c]

If there is a small movement in the stock price, the overall portfolio value will not change because the change in value of call options will equal the change in value of stock

d]

If there is a large movement in the stock price, the overall portfolio value will change because the change in value of call options will not equal the change in value of stock. This is because a delta-neutral portfolio is delta-neutral for small changes in the stock price, and does not remain delta-neutral for large movements in the stock price.

To protect against changes in the portfolio value, a continuously rebalancing delta-netural hedge is required, which adjusts the number of shares / call options as the stock price changes, so that the overall portfolio remains delta-neutral.