Vanna-Volga from zero
1/5Three liquid options price everything
Vanna-Volga uses exactly three market quotes to construct an entire smile. The 25Δ put, the ATM straddle, and the 25Δ call. That is the entire input. Everything else is derived.
In FX markets, dealers do not quote option prices by strike. They quote three numbers:
ATM vol (σATM). The at-the-money straddle volatility. This sets the overall level of the smile.
25Δ risk reversal (RR). The difference between the 25-delta call vol and the 25-delta put vol. This captures the skew -- how much the smile tilts.
25Δ butterfly (BF). The average of the 25-delta put and call vols minus ATM vol. This captures the curvature -- how much both wings lift above ATM.
From these three numbers, you can recover the individual vols:
The Vanna-Volga method takes these three liquid reference points and builds a full smile by asking: what is the cheapest way to hedge the smile risk of any target option using these three instruments?
Think of ATM, RR, and BF as three knobs on a mixing board. ATM is the master volume. RR is the balance control (left vs right). BF is the loudness boost (both sides). Three knobs, one smile.
What are vanna and volga?
Vanna and volga are the two second-order Greeks that Black-Scholes pretends do not exist. They measure sensitivity to smile risk -- the cross-effect between spot and vol (vanna) and the convexity in vol (volga).
Vanna = ∂²V / ∂S∂σ. This is the sensitivity of delta to changes in vol, or equivalently, the sensitivity of vega to changes in spot. When vol moves, delta shifts. When spot moves, vega shifts. Both effects are vanna.
Vanna is largest near ATM and is anti-symmetric around the forward. For puts (left wing), vanna is positive: when vol rises, the put's delta becomes more negative (further in the money in probability space). For calls (right wing), vanna is negative.
Volga = ∂²V / ∂σ². This is the vol gamma -- the convexity of the option price with respect to vol. An option with positive volga benefits from vol moving in either direction.
Volga is largest in the wings and is symmetric around the forward. Deep OTM puts and deep OTM calls both have large positive volga. ATM options have near-zero volga.
The chart above shows the two profiles across strikes. Black-Scholes assumes a flat smile, so it prices vanna and volga exposure at zero cost. But in a real market with a smile, holding vanna and volga exposure is not free -- it has a cost, and that cost is exactly the smile adjustment that Vanna-Volga computes.
Delta and gamma are first-order effects that Black-Scholes handles. Vega is a first-order vol sensitivity that BS also handles (even though it assumes vol is constant, the model still has a vega). The second-order vol effects -- how delta changes with vol (vanna) and how vega changes with vol (volga) -- are precisely what the smile encodes. A smile is nothing more than the market's pricing of vanna and volga risk.
The replication argument
Here is the key idea: build a portfolio of the three liquid benchmark options that matches the target option's vanna and volga. The cost of this hedging portfolio -- over and above its Black-Scholes value -- is the smile correction.
Start with a target option at some arbitrary strike K. Compute its vanna and volga under Black-Scholes (using ATM vol). Now find weights (x₁, x₂, x₃) on the three benchmark options such that:
Once you have the weights, the VV price is:
Drag the target strike in the widget above. Watch how the replication weights shift:
Target near 25Δ put: Almost all weight goes to the put benchmark. The ATM and call benchmarks contribute little.
Target near ATM: The ATM benchmark dominates. The correction is small because BS is nearly correct at ATM.
Target between benchmarks: Weights interpolate smoothly. The smile at any intermediate strike is a weighted combination of the three reference points.
The formula
When you work out the replication weights explicitly, the VV correction splits cleanly into two terms: a vanna correction that creates skew and a volga correction that creates curvature.
Volga term: proportional to the butterfly. Symmetric -- adds to both wings equally.
This decomposition is the reason the method is called Vanna-Volga. The entire smile correction is explained by two effects:
The vanna correction is anti-symmetric around ATM. It is driven by the risk reversal quote. When the market quotes a large negative RR (puts richer than calls), the vanna correction tilts the smile leftward. For deep OTM puts, the correction is largest and positive (adds premium). For deep OTM calls, it is negative (removes premium).
The volga correction is symmetric around ATM. It is driven by the butterfly quote. When the market quotes a large BF, the volga correction lifts both wings. ATM is unaffected (volga is near zero there). The further you go into the wings, the larger the correction.
The stacked bar chart above shows both corrections across strikes. Notice:
The blue bars (vanna) are negative on the left and positive on the right -- this is the skew component.
The orange bars (volga) are positive everywhere in the wings -- this is the curvature component.
The green line is the total correction. On the put side, vanna and volga reinforce each other (both add premium). On the call side, they partially cancel (vanna subtracts, volga adds). This is why put wings are typically steeper than call wings.
FX desks love it, equity desks don't
Vanna-Volga is the dominant smile model in foreign exchange because the FX market literally quotes ATM, RR, and BF. The model's inputs are the market's native language. In equity and crypto, the market quotes strikes directly, and VV's three-point assumption is too rigid.
Why FX loves it: The interbank FX options market has standardized on quoting conventions that map directly to Vanna-Volga inputs. A dealer sees ATM = 8.2, RR = -1.3, BF = 0.4 and immediately has the three vols needed for VV. No calibration step. No optimizer. Just algebra.
For FX vanillas at standard deltas, VV is fast, accurate, and arbitrage-free. For first-generation exotics (one-touch, double no-touch), VV gives quick and dirty prices that are remarkably close to full-model answers.
Why equity/crypto does not: Listed equity and crypto options provide a full grid of prices across many strikes and expiries. You have far more than three data points. Fitting a three-parameter model to thirty strikes throws away information.
Worse, the VV smile is not flexible enough to match the actual shapes seen in equity and crypto markets. Steep short-dated skews, wing convexity that varies with expiry, term structure of the butterfly -- none of these are captured by three numbers.
In those markets, SVI, SSVI, or SLV are better choices because they can absorb the full richness of the observed surface.
Even on desks that do not use VV for production pricing, it is valuable as a mental model. "This option costs more than BS because of vanna and volga" is a complete explanation of why smiles exist. The decomposition into skew (vanna) and curvature (volga) helps traders reason about what drives the price of any option -- even when the actual pricing uses a more sophisticated model.
Extensions: The basic VV method uses 25-delta benchmarks. Some desks extend it to five points (adding 10-delta put and call) to capture wing behavior better. Others use a "second-order VV" that includes higher-order Greeks. But at that point, you are building a more complicated model and might as well use SVI.
In crypto: The VV framework is occasionally used for quick mental math -- "how much should this OTM put cost given the market RR and BF?" -- but it is not a production model. Crypto vol surfaces are too noisy and too steep for three-point interpolation. The value is conceptual, not operational.
Where to go next:
Black-Scholes -- the baseline model that VV adjusts
Greeks Reference -- full treatment of vanna, volga, and other second-order sensitivities
SVI Parameterization -- the strike-based alternative for equity/crypto smiles
Stochastic Local Vol -- the production exotic pricing model