Conceptual explanation of the relationship between gamma and vega plotted against delta for a European call option

Question

I recently plotted Gamma and Vega against Delta for a European call option and found that the graphs look very similar. This makes sense to me mathematically since the two formulas are pretty much the same from Black Scholes, just with a few different constants

$$ \Gamma = Ke^{-rT}\phi(d_2)\frac{1}{S^2\sigma\sqrt{T}} $$

$$ \nu = Ke^{-rT}\phi(d_2)\sqrt{T} $$

However, I am uncertain how these apply specifically to delta and how it all conceptually comes together. Apologies if this is an obvious question and thank you for any assistance.

Answer 1

Gamma and vega have the same general shape , peaking at ATM and tapering to the tails. But gamma concentrate as the option gets closer to expiry (when vega is small). For options a long way from maturity, vega increases and gamma is small.

Consequently for short dated options,

• if the price is close to strike, the option will have to be rehedged often (peak gamma, ie the delta is very sensitive to price moves)
• if the option is far ITM or OTM the hedge does not have to be changed much.
• the option has little to no sensitivity to changes in implied vol.

For long-dated options

• once hedged, the hedges do not have to be readjusted often (gamma is low)
• the primary driver of PNL changes will be changes in implied vol (high vega).

In rates, the vol surface is usually divided into the gamma part (expiries less than say 1-2y) and the vega part (expiries longer than around 5y) although the terminology is a little loose.

Answer 2

I am uncertain how these apply specifically to delta and how it all conceptually comes together.

The point is that both greeks consist of a 2nd-order difference, with the subtlety that Vega is centered around the mean (by definition of volatility or variance) and Gamma is not.