Neutralizing a Portfolio’s Delta and Gamma with Options and Stock

Summary

The document explains how to adjust a portfolio with delta 450 and gamma 6,000 using an option with delta 0.6 and gamma 1.5, assuming stock is also available. To offset the portfolio’s gamma, take a short position of 4,000 options. That position contributes delta of -2,400, leaving the combined portfolio with delta of -1,950. Buying 1,950 shares then offsets that remaining delta, since stock has unit delta and zero gamma in the simplified setup.

The response gives a numerical hedge construction rather than a general treatment of Greek hedging. It assumes the stated Greeks apply to the relevant holdings and that the options and stock can be traded in the required quantities. It also assumes a stock-based underlying and does not address transaction costs, changing Greeks, discrete rebalancing, or constraints on short positions. A second answer prompts the reader to use the stock’s delta and gamma when setting up the two neutrality conditions.

Key ideas

• A short position of 4,000 options offsets gamma of 6,000 when each option has gamma 1.5.
• That option position contributes delta of -2,400, so the residual portfolio delta is -1,950.
• Buying 1,950 shares offsets the remaining delta under the unit-delta, zero-gamma stock assumption.
• The hedge relies on the assumed portfolio composition and available stock position.

Tags

# Delta and gamma neutral


# Delta and gamma neutral












A financial institution currently has a portfolio with delta of 450 and gamma of 6,000. A traded option is available with a delta of 0.6 and a gamma of 1.5. How could the portfolio be made both delta neutral and gamma neutral? Sorry for this easy question, however I want to know how to make the portfolio delta and gamma neutral. Thanks a lot.

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/18137

Your portfolio composition is not clear. To simplify, we assume that it consists of units of a stock and options on this stock. What you can do is to sell 4000 units of options that will bring it to gamma neutral, and then to balance the delta, you can buy 2,400-450=1,950 units of the stock.

## Answer by torbonde (score 0)

https://quant.stackexchange.com/a/17625

I assume that stock is also available. Consider then

- What is the delta of stock? What is the gamma of stock?

- What position should the financial institution take in the traded option and the stock so that the gamma and delta of the portfolio are both 0?

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