Delta-Gamma neutral vs Delta-Vega neutral

Question

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?

Answer 1

For a flat IV skew (not surface), the gamma neutral portfolio == the vega neutral portfolio. This is because Vega = s^2 * σ * T * gamma, so when the net gamma exposure is 0, the net vega exposure is 0, and so is the net theta exposure.

When there is a skew present, portfolio's can be constructed that are locally gamma, vega, or theta neutral. In the gamma/theta cases, these represent local arbitrages of long gamma or theta (long gamma without theta risk, and vice versa). These quickly change as the higher order greeks take effect through time/spot/vol.

In your case, yes. You need the same ratio of options to achieve gamma, vega, or theta neutrality. But in the real world this rarely holds because of skew.