Vanna And Charm

Delta tells you how many shares to hold to hedge an option right now. It does not tell you that the same stock price, one day later, needs a different number of shares. If delta only changed because the stock moved, gamma alone would explain every rebalance. But delta also changes when nothing about the stock does — purely because a day passed, or implied volatility ticked up — and those two effects have their own names: charm for the drift from time passing, vanna for the shift from volatility moving.

A weather vane in changing wind

Think of delta as a weather vane, and the stock price as the wind direction it's pointing into. Gamma is how sharply the vane swings when the wind actually shifts direction. But a weather vane can also seize up or loosen with temperature, or its pivot can get slicker as a storm rolls in — its responsiveness itself changes before the wind moves, purely from conditions around it. Charm is the vane stiffening as the clock runs down toward a deadline; vanna is the vane's sensitivity changing as the "gustiness" (volatility) of the wind rises or falls.

The two formulas

Both are cross-partial derivatives, meaning they measure how one Greek responds to a second variable:

$$\text{vanna} = \frac{\partial \Delta}{\partial \sigma} = \frac{\partial \text{vega}}{\partial S}, \qquad \text{charm} = \frac{\partial \Delta}{\partial t}.$$

In plain English: vanna is how much your delta changes for a one-unit move in implied volatility (equivalently, how much vega changes for a one-unit stock move — mathematically the same quantity, read in different directions). Charm is how much delta drifts per day, holding stock and volatility fixed. For a European call, vanna and charm have closed forms:

$$\text{vanna} = -e^{-qT}\phi(d_1)\frac{d_2}{\sigma}, \qquad \text{charm} = -e^{-qT}\left[\phi(d_1)\frac{2(r-q)T - d_2\sigma\sqrt{T}}{2T\sigma\sqrt{T}} + q\,\text{sign}\right]$$

where $\phi$ is the standard normal density, $d_1, d_2$ are the usual Black-Scholes terms, $q$ is the dividend yield, and the sign term flips for calls versus puts. The formulas are dense; what matters is the shape: vanna is largest for options noticeably away from the strike with moderate time left, and charm accelerates sharply as expiry nears, because delta is then at its most decisive about snapping to 0 or 1.

Worked example 1: vanna on a 3-month 25-delta call

A stock at $100 with a call struck at $110, $\sigma = 25\%$, $T = 0.25$, $r=q=0$. Compute $d_1 = \frac{\ln(100/110) + 0.5(0.25)^2(0.25)}{0.25\sqrt{0.25}} = \frac{-0.0953 + 0.0078}{0.125} = -0.700$, and $d_2 = d_1 - \sigma\sqrt{T} = -0.700 - 0.125 = -0.825$. The normal density at $d_1$: $\phi(-0.700) = 0.3123$.

$$\text{vanna} = -\phi(d_1)\frac{d_2}{\sigma} = -0.3123 \times \frac{-0.825}{0.25} = 0.3123 \times 3.30 = 1.031.$$

This is vanna per 1.00 (100 percentage points) of volatility; scaled to "per 1 volatility point" it's $0.01031$. So if implied volatility jumps from 25% to 26%, delta rises by about $0.0103$ — on a position of 100 contracts (10,000 shares of exposure), that is roughly 103 shares worth of delta appearing with the stock not having moved at all.

Worked example 2: charm compounding into expiry week

Take a call struck $1 in the money, $\sigma = 20\%$, $r=q=0$, at two points: 30 days and 3 days to expiry. At 30 days out, delta might be $0.58$; at 3 days out, the same $1-in-the-money call's delta can be above $0.85$, because with so little time left, "$1 in the money" is close to a near-certain finish in the money. That swing — roughly $0.27$ of delta appearing over 27 days with the stock unchanged — is charm, and it accelerates: most of it happens in the final week, not spread evenly over the month.

Vanna is the reason moves in the surface below don't just change option prices — they change hedge ratios too. Explore how implied volatility varies by strike and tenor; any position with meaningful vanna has a delta that quietly rides up and down with this whole surface, even on days the stock itself barely moves.

What this means in practice

A market maker running a large book near expiry watches charm closely, because the hedge needed tomorrow, with the stock unchanged, differs from today's — ignoring it means starting every session already mis-hedged. Vanna matters most around known volatility events: a desk short vanna can find its hedge needs flipping violently the moment implied volatility jumps, even if the stock barely moves. Both show up prominently in dealer positioning analyses, since large vanna and charm exposures create predictable flows around expiries.