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Vanna-Volga Method

Vanna-Volga builds a vol smile from three market quotes: ATM vol, risk reversal, and butterfly. It computes how much the Black-Scholes price needs adjusting to account for the smile. The adjustment equals the cost of hedging the option's skew and curvature exposure using three liquid benchmarks.

Built for FX options. This is the method behind most FX smile construction at banks. No optimization, no iteration -- closed-form.

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Hedge cost equals smile adjustment

Start with the Black-Scholes price. Measure the option's exposure to skew (vanna) and curvature (volga). Hedge that exposure using three liquid benchmarks whose market prices you know. The cost of the hedge is the smile adjustment. Invert to get implied volatility at any strike.

Try It: Build a Smile from Three Quotes​

Adjust the three market quotes below to see how they construct the full vol smile. Notice how ATM sets the level, the risk reversal tilts the smile (skew), and the butterfly lifts both wings (curvature).

Vanna-Volga Smile Builder

Typical crypto smile: moderate put skew, slight curvature. Reflects persistent demand for downside protection.
25Δ Put IV: 51.0%
ATM IV: 45.0%
25Δ Call IV: 45.0%
44%49%53%58%51.0%45.0%45.0%25ΔPATM25ΔCStrike (by delta)Implied Vol (%)
ATM vol+45.0%
ATM implied vol level
RR₂₅-6.0%
Negative = put skew (typical)
BF₂₅+3.0%
Higher = more curvature / fatter wings
σ(25ΔP) = σ_ATM + BF₂₅ - RR₂₅/2 = 45 + 3 - (-6)/2 = 51.0%
σ(25ΔC) = σ_ATM + BF₂₅ + RR₂₅/2 = 45 + 3 + (-6)/2 = 45.0%

The three sliders correspond to the three market quotes FX dealers publish. Together they fully determine the smile shape through the Vanna-Volga framework.

The Three Inputs​

Quote
What it measures
Smile feature it controls
ATM volatility
Implied vol at the at-the-money strike
Overall vol level (anchors the surface)
25-delta risk reversal
Call IV minus put IV at 25-delta
Skew (tilt of the smile)
25-delta butterfly
Average wing IV minus ATM IV
Curvature (wing elevation)
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One input, one smile dimension

ATM vol sets the level. Risk reversal sets the tilt. Butterfly sets the curvature. Change one input and you know exactly how the smile responds.

How the Method Works​

Step
What happens
Why
1. Price benchmarks
Compute market prices and flat-vol (Black-Scholes) prices for the three benchmark options
The difference is each benchmark's "smile cost"
2. Compute target Greeks
Calculate vanna and volga of the option you want to price
These measure how exposed it is to smile risk
3. Find hedge weights
Solve for weights so a portfolio of the three benchmarks matches the target's vanna and volga
Tells you how much of each benchmark's smile cost applies
4. Apply adjustment
Add the weighted smile costs to the Black-Scholes price
The adjusted price reflects the smile; invert to get IV
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Three quotes match three degrees of freedom

The vol surface smile has two second-order effects: vanna (spot-vol cross-sensitivity, controls skew) and volga (vol-of-vol sensitivity, controls curvature). Three quotes give exactly the degrees of freedom for level, skew, and curvature. FX dealers quote exactly these three quantities.

The Greeks Behind the Name​

Greek
What it measures
Smile dimension it controls
Hedged by
Vanna
How delta changes when vol changes
Skew
Risk reversal
Volga
How vega changes when vol changes
Curvature
Butterfly

Vanna maps to skew. Volga maps to curvature. Risk reversal hedges vanna risk. Butterfly hedges volga risk. ATM anchors the level. This decomposition carries over to any smile model. The delta of the target option determines skew exposure; vega determines overall vol sensitivity.

Strengths and Limitations​

Strength
What it means for you
Extremely fast -- closed-form, no optimization
Thousands of options per millisecond. No fitting step, no iteration.
Uses exactly what dealers quote
ATM, risk reversal, butterfly. No model risk from fitting.
Intuitive mapping
Each input maps to one smile feature. Easy to reason about.
Simple -- three inputs, no overfitting
No room for spurious wiggles.
Limitation
What it means for you
Poor wing behavior
Deep OTM wings (10-delta and beyond) are unconstrained. Can produce implausible values.
Only uses 3 quotes
Cannot incorporate additional strike data even when available.
Breaks down for severe smiles
Assumes Black-Scholes as the base. High RR or BF can cause issues.
FX conventions do not map cleanly to crypto
Premium-adjusted delta, forward delta -- different from crypto spot-delta.
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Fastest smile from three quotes, but limited

Vanna-Volga is the fastest way to build a smile from three quotes. With full strike grids (like on Deribit), SVI extracts more from the data and produces better wings. The method says nothing about term structure or calendar arbitrage -- each expiry is independent.

Relevance to Crypto​

Vanna-Volga is rarely used directly in crypto -- SVI is the standard because crypto exchanges provide full strike grids, not just three summary quotes. But the mental model is valuable:

Use case
Why it matters
Interpreting OTC quotes
When OTC desks quote ATM + RR + BF, Vanna-Volga tells you exactly what those numbers imply about the smile shape.
Quick sanity checks
Given ATM, RR, and BF, you can mentally approximate the smile without fitting a model.
Understanding skew decomposition
Skew comes from vanna, curvature from volga. This carries over to any smile model.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Test Your Understanding​

Test your understanding before moving on.

Q: If you increase the 25-delta butterfly while keeping ATM vol and risk reversal unchanged, what happens to the smile?
Q: Why can't Vanna-Volga use additional market quotes (e.g., 10-delta options) even when they're available?
Q: An ATM option has near-zero vanna and near-zero volga. What does Vanna-Volga predict for its smile adjustment?
Q: Why is Vanna-Volga rarely used in crypto options markets?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition​

Learn Vanna-Volga from scratchInteractive lesson · no prerequisites

This lesson starts from the three dealer quotes, then explains how ATM, risk reversal, and butterfly map to level, skew, and curvature through vanna and volga hedge costs.

See also:

  • SVI Parameterization -- The smile model Hypercall uses in production
  • SABR Model -- Stochastic vol model with dynamic interpretation
  • SSVI -- Surface-level SVI with calendar constraints
  • Vanna -- The cross-Greek that controls skew
  • Volga -- The vol convexity Greek that controls curvature
  • Skew -- How implied vol varies across strikes
  • Interpolation Methods -- All methods compared
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