Second-Order Greeks: Gamma, Vanna, and Charm

Key Takeaways

• Second-order greeks are the rates of change of the first-order greeks; gamma, vanna, and charm all describe how an option's delta shifts.
• Gamma measures delta's response to the underlying price, vanna measures delta's response to implied volatility, and charm measures delta's response to the passage of time.
• Gamma creates convexity: a delta hedge is only accurate for small moves, and gamma tells you how fast that hedge goes stale.
• Ignoring vanna and charm leaves a nominally delta-neutral book quietly picking up directional risk as volatility or time changes.

What It Is

A first-order greek is a first derivative of the option price. Delta is the change in price per $1 move in the underlying; vega is the change per one-point move in implied volatility; theta is the change per day.

A second-order greek is a derivative of one of those. The three that matter most for a delta hedger all trace back to delta:

• Gamma is the change in delta per $1 move in the underlying (the second derivative of price with respect to spot).
• Vanna is the change in delta per one-point move in implied volatility (equivalently, the change in vega per $1 move in spot).
• Charm, also called delta decay, is the change in delta per day that passes.

Delta answers "how much do I make on the next dollar?" The second-order greeks answer "and how fast is that answer changing?"

The Intuition

Delta is a moving target. An at-the-money option starts near 0.50 delta, but if the stock rallies the option drifts toward 1.00, and if it falls the delta drifts toward 0. Gamma is the speed of that drift. High gamma means your hedge decays quickly and you must rebalance often.

Vanna and charm exist because delta also moves when the price does not. Raise implied volatility and an out-of-the-money option suddenly has a better chance of finishing in the money, so its delta rises; that sensitivity is vanna. Let a day pass with the stock unchanged and the same option's delta creeps toward 0 or 1 as uncertainty burns off; that drift is charm.

How It Works

Second-order greeks let you build a better local estimate of an option's behavior. A pure delta hedge assumes the price line is straight. In reality it curves, and the second-order terms capture that curvature.

For the profit and loss of a delta-hedged position, the standard approximation is:

• P&L per share ≈ 0.5 × gamma × (change in spot)²

Because the change in spot is squared, this term is always positive for long gamma and always negative for short gamma, regardless of direction. That is the convexity a long option holder is paying theta to own.

For the delta itself, the three second-order greeks combine to predict the new delta:

• New delta ≈ old delta + gamma × (change in spot) + vanna × (change in vol) + charm × (days elapsed)

Keeping a book neutral therefore means watching all three, not just re-buying stock when the price moves.

Worked Example

Start with one at-the-money call on a $100 stock. Its greeks are:

• delta = 0.50
• gamma = 0.05 per $1
• vanna = 0.02 per volatility point
• charm = -0.01 per day

Now three things happen at once. The stock rises $3, implied volatility rises 2 points, and one day passes.

• Gamma effect: 0.05 × 3 = +0.15
• Vanna effect: 0.02 × 2 = +0.04
• Charm effect: -0.01 × 1 = -0.01

New delta ≈ 0.50 + 0.15 + 0.04 - 0.01 = 0.68.

The delta hedger who sold 50 shares against this call is now short 50 but needs to be short 68, so the book is 18 shares net long and must be rebalanced. Meanwhile the convexity payoff from the $3 move is 0.5 × 0.05 × 3² = 0.225 per share, or about $22.50 on a 100-share contract. That gain is the reward for carrying positive gamma, funded by the theta paid each day.

Common Mistakes

1. Treating a delta hedge as static. Delta is accurate only for an instant. Gamma tells you how quickly it decays, and a large-gamma position left unhedged can swing from neutral to sharply directional on a modest move.
2. Forgetting vanna near an event. Around earnings or a data release, implied volatility can jump without the price moving. A book that looks neutral can pick up real delta purely from that vol shift.
3. Ignoring charm into expiration. Delta decay accelerates as expiration nears, especially for options near the strike, so an overnight gap in neutrality grows larger in the final week.
4. Confusing vanna's two definitions. Vanna is both the change in delta per unit of vol and the change in vega per unit of spot. They are the same number, but mixing up which input you are shocking leads to double counting.
5. Scaling per share versus per contract. Quoted greeks are usually per share; forgetting the 100 multiplier badly understates the true position risk.

Frequently Asked Questions

Q: What are second-order greeks in plain terms? They are the rates of change of the first-order greeks. Instead of telling you how an option's price reacts, second-order greeks such as gamma, vanna, and charm tell you how the sensitivities themselves move as price, volatility, and time change.

Q: Why do second-order greeks matter for delta hedging? A delta hedge is only exact for a single instant. Gamma, vanna, and charm each predict how the position's delta will drift as the market moves, so they tell a hedger how often and in which direction to rebalance.

Q: What is the difference between gamma, vanna, and charm? All three measure how delta changes, but with respect to different inputs. Gamma is delta's response to the underlying price, vanna is its response to implied volatility, and charm is its response to the passage of time.

Q: Is vega a second-order greek? No. Vega is a first-order greek because it is the first derivative of price with respect to volatility. Vanna, which is the derivative of vega with respect to spot, is the related second-order greek.

Q: How do professionals use second-order greeks day to day? Options market makers monitor gamma to schedule rebalancing, vanna to manage exposure around volatility shifts, and charm to correct for overnight delta drift.

Sources

1. Investopedia. "Gamma." https://www.investopedia.com/terms/g/gamma.asp
2. Investopedia. "Getting to Know the Greeks." https://www.investopedia.com/trading/getting-to-know-the-greeks/
3. Investopedia. "Delta." https://www.investopedia.com/terms/d/delta.asp
4. Corporate Finance Institute. "Option Greeks." https://corporatefinanceinstitute.com/resources/derivatives/option-greeks/

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.