# Delta-Gamma neutral vs Delta-Vega neutral

Imagine that the underlying stock price is 110. The call option has a strike price of 100. The annualized volatility is 25% and the interest rate is 10%. Finally, the time to maturity is 0.5 years. We sell this call option and we want to hedge it.

According to the Black-Scholes model, this option has a delta of 0.819, a gamma of 0.014 and a vega of 0.205.

For a delta-gamma neutral portfolio, we need another option and shares. Imagine we have the same call option but with a strike price of 110 (ATM), which has a delta of 0.644, a gamma of 0.019 and a vega of 0.290. Shares have a delta of 1 and gamma of 0. For the portfolio to be gamma-neutral: $$-1 * 0.014 + NrShares * 0 + NrATMoptions * 0.019 = 0$$

This results in 0.737 ATM call options. Then we make it delta-neutral: $$-1 * 0.819 + NrShares * 1 + 0.737 * 0.644 = 0$$

This gives 0.344 shares.

Now we want a delta-vega neutral portfolio. Therefore, we use the same ATM option.

$$-1 * 0.205 + NrShares * 0 + NrATMoptions * 0.290 = 0$$

This gives the same number (if we do not round the delta, gamma, vega) of ATM call options and shares as the delta-gamma neutral hedge portfolio. Does it mean that if we want a delta-gamma or a delta-vega neutral portfolio, we need exactly the same amount or am I doing something wrong?