How does delta change as time passes or volatility changes? Are there separate Greeks for those effects?

Yes — they have names, just less famous ones. The full taxonomy of "Greek-of-a-Greek" terms includes charm and vanna alongside gamma.

The three primary effects on delta:

Gamma ($\partial \Delta / \partial S$):

The rate of change of delta with respect to the underlying. For long options, gamma is positive — delta increases as the stock rises. This is the most-watched second-order Greek.

Formula: $\Gamma = N'(d_1) e^{-qT} / (S \sigma \sqrt{T})$

Gamma is largest for at-the-money options near expiry. For deep ITM or OTM options, $\Gamma \to 0$.

Charm ($\partial \Delta / \partial T$):

The rate of change of delta with respect to time to expiry. As $T \to 0$, delta of an ITM call snaps to 1 and OTM call snaps to 0 — that snapping behaviour is charm.

Charm matters most for at-the-money options near expiry. Market-makers pay close attention to "weekend charm" (delta drift over the closed-market period) when managing positions through Friday close.

Vanna ($\partial \Delta / \partial \sigma$):

The rate of change of delta with respect to volatility. Equivalently, $\partial^2 V / \partial S \partial \sigma$. Counter-intuitive direction: when implied vol rises, the delta of out-of-the-money calls INCREASES (the call has more chance of finishing ITM, so its sensitivity rises).

Vanna matters most for skew-trading desks. If your book is short OTM puts (negative vanna), a vol spike will increase your effective short delta beyond what your hedges anticipated.

The three put-call relationships:

For European options on the same underlying with same strike and expiry:

• $\Gamma_{\text{put}} = \Gamma_{\text{call}}$ (gamma is symmetric)
• $\text{Charm}_{\text{put}} \neq \text{Charm}_{\text{call}}$ (signs differ)
• $\text{Vanna}_{\text{put}} = \text{Vanna}_{\text{call}}$ (vanna is symmetric for vanilla options)

Practical example combining the three:

You sell 100 ATM call contracts on a $50 stock with $\Delta=0.55$, $\Gamma=0.045$, $\text{vanna}=0.12$, $\text{charm}=-0.04$. You initially hedge with 5,500 shares.

Over a weekend:

• Stock unchanged at $50.
• Implied vol jumps from $30\%$ to $33\%$.
• Two days pass.

Your delta drift is:

• Vanna effect: $+0.12 \times 0.03 = +0.0036$ per option
• Charm effect: $-0.04 \times 0.00548$ ($2/365$years)$= -0.000219$ per option
• Net: $+0.00338$per option$\times 100$contracts$\times 100$shares$= +33.8$ share equivalents

You are now slightly short-delta. Need to buy $\approx 34$ more shares to re-neutralise. Most of the drift came from vanna (the vol jump), not charm (the time passage).

Exam relevance:

CFA Level II tests gamma directly. Charm and vanna appear only in conceptual questions about "what other factors change delta" rather than numerical calculations. Knowing the names and signs will earn you the conceptual mark; knowing the numerical formulas is not required.

Trader takeaway:

For risk reports, the holy trinity is delta, gamma, vega. For market-making in options, you also watch vanna and charm. For exotic-options trading, you watch everything — including volga ($\partial^2 V / \partial \sigma^2$), color ($\partial^2 \Delta / \partial S \partial T$), and speed ($\partial^3 V / \partial S^3$).