Keep vanna, vega and gamma in their own units
Write V for option value per share, S for spot and σ for annualized volatility in decimal form. With a smooth pricing function, vanna A = ∂Δ/∂σ = ∂²V/(∂S∂σ) = ∂vega/∂S. The last equality uses vega per decimal volatility. If a screen reports vega per IV point, its spot derivative is A/100.
- 30% to 31% IV is a change of 0.01 in σ, or one percentage point. A relative 1% rise from 30% reaches 30.3%, a change of 0.003, or 0.3 points. These scenarios produce different delta changes.
- Vanna is a local partial derivative. If spot and IV both move, delta also changes through gamma. A difference between two observed market deltas cannot isolate vanna when time, spot or other inputs changed too.
| Sensitivity | Derivative | What a small change estimates |
|---|
| Vega per decimal | ∂V/∂σ | Price change ≈ vega × change in σ |
| Vanna per decimal | ∂Δ/∂σ | Delta change ≈ A × change in σ |
| Vanna per IV point | A / 100 | Delta change ≈ (A/100) × IV-point change |
| Gamma | ∂Δ/∂S | Delta change ≈ gamma × spot change |
Specify the model before calculating the derivative
Use a European Black-Scholes-Merton option with continuous rate r and dividend yield q. Let T be remaining years, N the standard normal cumulative distribution, and φ(x) = exp(-x²/2) / √(2π). The example assumes S = $100, K = $105, T = 30/365, σ = 0.30 and r = q = 0. Prices are hypothetical dollars per share, not market quotes.
- d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T); d2 = d1 - σ√T. Call value C = S exp(-qT)N(d1) - K exp(-rT)N(d2). Put value P = C - S exp(-qT) + K exp(-rT).
- Call delta = exp(-qT)N(d1); put delta = exp(-qT)[N(d1) - 1]. Differentiating either with respect to σ gives A = -exp(-qT)φ(d1)d2/σ. Equivalent form: A = exp(-qT)φ(d1)[√T - d1/σ].
- Vega per decimal = S exp(-qT)φ(d1)√T. Differentiating it with respect to S produces the same A. Matching calls and puts have the same vanna under these inputs because their delta difference does not depend on σ.
- At the baseline, call price is $1.56643702, delta is 0.30004347 and vega is $9.96869087 per decimal volatility ($0.09968691 per IV point). Vanna is 0.70734913 per decimal, so a small +1-point IV move predicts a delta increase of 0.00707349.
Build a spot-by-IV grid for the same contract
Hold the $105 strike, 30 days, rates and dividend yield fixed. Change only spot across rows and IV across columns. Each cell shows call price / call delta. The center is the baseline; the four outer corners provide the price cross derivative. Calculate with full precision before rounding for display.
- At spot $100, central delta difference AΔ ≈ [Δ(0.31) - Δ(0.29)] / 0.02 = [0.3069400272 - 0.2927838129] / 0.02 = 0.70781071 per decimal. Divide by 100 for 0.00707811 per IV point.
- Use h = $0.10 and k = 0.01 decimal volatility. The price check is AC ≈ [C(S+h,σ+k) - C(S+h,σ-k) - C(S-h,σ+k) + C(S-h,σ-k)] / (4hk).
- The numerator is 1.6975319235 - 1.4967836779 - 1.6361433019 + 1.4382261491 ≈ 0.0028310928. Divide by 4 × 0.10 × 0.01 = 0.004 to obtain 0.70777320 per decimal, or 0.00707773 per IV point.
- Keep the strike fixed: adjacent strikes are different contracts. All nine cells use the same valuation instant and remaining time. Repeating a live quote request at different times is not this controlled experiment.
| Spot | IV 29% | IV 30% | IV 31% |
|---|
| $99.90 | $1.4382261491 / 0.28865982 | $1.5366344699 / 0.29601099 | $1.6361433019 / 0.30299698 |
| $100.00 | $1.4672981369 / 0.29278381 | $1.5664370191 / 0.30004347 | $1.6666399965 / 0.30694003 |
| $100.10 | $1.4967836779 / 0.29693080 | $1.5966438503 / 0.30409655 | $1.6975319235 / 0.31090156 |
Shrink both bumps and check convergence
The analytic answer is 0.70734913. Central differences approximate it over a finite rectangle, so they need not match exactly. Recalculate both checks with smaller bumps while retaining unrounded model prices and deltas.
- For a one-sided move from 30% to 31% at $100 spot, full repricing changes delta by 0.00689656. The baseline linear estimate is 0.00707349. The difference comes from changing vanna along the move; the symmetric 29%/31% slope answers a different question.
- Tiny bumps can amplify price rounding and floating-point cancellation. Four-decimal screen prices are insufficient to reproduce every digit of a small cross derivative. Compare several sensible bump sizes; do not assume the smallest one is best.
| Spot bump h | IV bump k | From delta / decimal | From price / decimal |
|---|
| $1.00 | 0.02 (2 points) | 0.70919718 | 0.70542452 |
| $0.10 | 0.01 (1 point) | 0.70781071 | 0.70777320 |
| $0.05 | 0.005 (0.5 point) | 0.70746450 | 0.70745514 |
| $0.01 | 0.001 (0.1 point) | 0.70735375 | 0.70735337 |
Scale delta exposure and price cross-effects separately
Assume one long contract has a 100-share deliverable and premium multiplier. Its baseline delta exposure is 30.004347 shares. Position vanna is 100 × 0.70734913 = 70.734913 shares of delta change per decimal volatility, or 0.707349 shares per IV point. A +1-point move therefore predicts about +0.707349 shares of delta exposure; full repricing gives +0.689656 shares.
- For n signed contracts and multiplier m, multiply A by n × m once. Three short contracts give about -2.122047 shares of delta change per IV point in this example. Verify the actual deliverable and quoting convention for adjusted contracts or non-equity options.
- The spot/volatility cross contribution to a local price approximation is A × change in S × change in σ. It has no extra factor of one-half: the two symmetric mixed terms in the second-order expansion combine.
- For +$1 spot and +1 IV point, that term is about $0.00707349 per share, or $0.707349 for the assumed contract. The exact rectangular interaction C(101,0.31) - C(101,0.30) - C(100,0.31) + C(100,0.30) is $0.00641013 per share. This interaction excludes the separate spot and IV price changes and is not the total P&L.
- A full price-change estimate also needs delta, vega, gamma, volatility curvature and any time or rate effects. Reprice the complete position for large shocks. Open interest alone does not reveal signed dealer vanna or predict hedging flows.
Check the sign and near-expiry behavior
The sign follows -d2 in this model, not whether the contract is a call or put. With all baseline inputs unchanged, vanna per decimal is +0.707349 at spot $100, +0.057134 at $105 and -0.558320 at $110. At d2 = 0 it is zero. A positive long-option vega therefore does not imply positive vanna.
- Near expiry, do not assume vanna always increases. At S = $100 and K = $105, shortening time from 30 days to 7 days raises vanna from 0.707349 to 0.817016; at 1 day it falls to 0.033998. At S = K = $105, the same sequence is 0.057134, 0.027618 and 0.010440 under r = q = 0.
- Use actual fractional time for an intraday expiry. The formula requires S > 0, K > 0, T > 0 and σ > 0. Require S - h > 0 and σ - k > 0 for the central differences. At expiry the payoff no longer depends on volatility, and delta is not a smooth function at the strike; do not evaluate the formula by dividing by zero.
- This is a fixed-input European model sensitivity. American exercise, discrete dividends, jumps, a different pricing model or a volatility surface that changes with spot can change the result. Agreement between derivatives checks the calculation, not the accuracy of a market-price forecast.
Use TradingView and Pineify to verify inputs
TradingView documents IV and standard Greeks in its options chain and time/IV scenarios in its Strategy Builder. Record one contract, quote timestamp, strike, expiry, exercise style and underlying price. For an IV-only comparison, hold time and spot fixed and record the resulting IV levels. A label such as IV +10% does not establish whether the shift is relative or in percentage points.
- In Pineify AI Stocks & Options Picker, select a ticker, open Options Chain, select the expiry and inspect the strike row. Available IV and delta appear in the table; the expanded row includes vega. The inspected implementation has no live Vanna field. Missing provider data must remain missing.
- For a manual check, open the Option Greeks Calculator and enter Call, spot 100, strike 105, 30 days, volatility 30, rate 0 and dividend yield 0. Repeat with volatility 29 and 31, then with spot 99.90 and 100.10. This calculator returns model price and delta; it does not return vanna directly.
- The calculator uses an approximate normal distribution and displays four decimal places, so it is useful for a coarse check. Use the formulas above with a higher-precision normal CDF to reproduce the detailed grid and convergence values. Never infer a missing Vanna value from a stock score or from rounded live deltas.
