Hedging option delta with futures instead of shares

Question

You run a book of index options that is net long $2{,}500$ index-units of delta. Instead of trading a basket of stocks, you hedge with the index future, which has a multiplier of $\$ 50 $per index point (one future$ =50$ units of index exposure).

How many futures do you trade, and what subtlety does using futures introduce versus using the cash index?

Accepted Answer

The hedge count. Each future carries 50 index-units of exposure. To offset +2{,}500 units of long delta, sell \frac{2{,}500}{50} = 50 	ext{ futures.} Selling 50 futures neutralizes the directional risk of the option book, and it is far cheaper and more liquid than trading hundreds of underlying stocks. The subtlety, basis and carry. A future is not the spot index; it trades at the forward price, F = S\,e^{(r-q)(Tf - t)}, where q is the dividend yield and Tf the future's expiry. Two consequences: 1. Delta of the future. A one-point move in spot moves the future by \approx e^{(r-q)	au} points…

Hedging option delta with futures instead of shares

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