In essence, a position in the underlying stock has zero gamma and cannot be used to adjust the gamma of a portfolio due to linearity of returns. Also, a portfolio consisting of a stock and a single option can never be delta-neutral and gamma-neutral unless the option’s gamma happens to be zero. A trader can create a delta-neutral portfolio using a stock and a call or a stock and a put. It is also possible to create positions which are neutral to other types of greeks such as gamma, theta, or vega. A trader can control both the delta and the gamma of a portfolio by including a stock and two different options. This would provide a better hedge against larger moves in an future share price.

Consider a delta-neutral portfolio that has a gamma of Г_P, and a traded option that has a gamma equal to Г_T . If the number of options added to the portfolio is N_T, then the gamma of the portfolio is:

Portfolio gamma = N_T Г_T + Г_P

It follows that in order to make the portfolio gamma neutral, the trader should enter into a position in options to the tune of:

– Г_P / Г_T

However, the delta of the portfolio would change due to the addition of options, and the trader would need to readjust the position in the underlying asset so that delta neutrality is maintained. Gamma neutrality cannot be maintained for long. With the passage of time, it can be maintained by adjusting the position in the option so that it is always fixated at – Г_P / Г_T.

Delta neutrality provides protection against relatively small stock price moves between hedge rebalancing, while gamma neutrality protects a portfolio against larger movements in the price of component stock between hedge rebalancing.

A hypothetical example may help illustrate how these concepts fit into practice. Suppose that a portfolio is delta neutral and has a gamma of – 2,000. The delta and gamma of a given call option are 0.7 and 1.6, respectively. Gamma neutral can be attained by including in the portfolio a long position in the call option to the tune of:

portfolio gamma/ option gamma = 2,000 /1.6 = 1250

To maintain delta neutrality, the delta of the portfolio will have to be adjusted from zero to:

1250 x 0.7 = 875

Therefore, the trader has to sell 875 units of the underlying asset from the portfolio in order to maintain its delta neutrality.