# SVI Volatility Model: Fitting Total Variance Without Static Arbitrage

Understand Raw SVI's five parameters, calculate a smile in log-forward moneyness, calibrate executable quotes, and test positivity, butterfly, calendar, and wing constraints.

Canonical: https://wiki.fcontext.com/options/svi-volatility-model/
Fact checked: 2026-07-22

> For educational purposes only; not investment advice.

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## Direct answer

**SVI**, or Stochastic Volatility Inspired, is a parametric representation of implied **total variance** across strikes for an expiration. In its Raw SVI form:

`w(k) = a + b{ρ(k−m) + √[(k−m)² + σ²]}`

where `w(k) = σ_imp²(k,T)T` and `k = ln(K/F_T)` is log-forward moneyness. Raw SVI uses five parameters to turn discrete option quotes into a smooth smile for interpolation, marking, Greeks, and relative-value diagnostics.

Despite its name, a fitted SVI slice is not by itself a stochastic process or a forecast of future volatility. It describes a static risk-neutral price surface. Separate choices determine how slices connect across maturities and how the surface moves through time.

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## What the five parameters control

- `a`: vertical total-variance level, but not necessarily ATM variance.
- `b ≥ 0`: overall scale of the left and right wing slopes.
- `−1 < ρ < 1`: asymmetry; negative `ρ` makes the left asymptotic slope larger than the right.
- `m`: horizontal location of the smile's center in log-moneyness.
- `σ > 0`: smoothness and width around the center; this parameter is not implied volatility.

The asymptotic total-variance slopes are `b(1−ρ)` on the left and `b(1+ρ)` on the right. The minimum total variance is:

`w_min = a + bσ√(1−ρ²)`

so nonnegative total variance requires `a + bσ√(1−ρ²) ≥ 0`. These basic restrictions do **not** guarantee absence of butterfly arbitrage.

For a twice-differentiable total-variance smile, a widely used density condition is:

`g(k) = [1 − kw'(k)/(2w(k))]² − [w'(k)²/4][1/w(k)+1/4] + w''(k)/2 ≥ 0`

for all relevant `k`, together with appropriate wing behavior. A negative `g(k)` corresponds to a negative risk-neutral density in the continuous-strike construction. Robust implementations enforce analytic no-arbitrage regions or test a sufficiently wide, dense grid and the wings; visual smoothness and low RMSE are insufficient.

### One slice is not a surface

Calibrating every expiration independently can create crossings through time. Under consistent forward and discount conventions, total variance should not decrease with maturity at fixed log-forward moneyness in a standard no-calendar-arbitrage construction. SSVI and related parameterizations link slices with explicit restrictions. Event expirations can create genuine steps in total variance; no-arbitrage control should preserve valid event information rather than cosmetically smoothing it away.

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## Example: reading one Raw SVI slice

Let `T = 0.25` year and:

`a = 0.01, b = 0.10, ρ = −0.5, m = 0, σ = 0.20`

The positivity minimum is:

`w_min = 0.01 + 0.10×0.20×√(1−0.5²) = 0.02732 > 0`

Evaluate three log-moneyness points:

| `k` | Total variance `w(k)` | `IV = √[w(k)/T]` |
| ---: | ---: | ---: |
| `−0.20` | `0.04828` | `43.95%` |
| `0` | `0.03000` | `34.64%` |
| `+0.20` | `0.02828` | `33.64%` |

At `k = 0`:

`w(0) = 0.01 + 0.10×0.20 = 0.03000`

At `k = −0.20`:

`w(−0.20) = 0.01 + 0.10[0.10 + √(0.20²+0.20²)] = 0.04828`

The negative `ρ` produces a steeper left wing. Its asymptotic slopes are `0.10(1−(−0.5)) = 0.15` on the left and `0.10(1+(−0.5)) = 0.05` on the right.

This calculation proves neither a good market fit nor absence of static arbitrage. The full slice still needs the `g(k)`/density and wing checks, and neighboring expirations need calendar checks. Also, the values are annualized IV derived from total variance; comparing `w` rather than raw IV is essential across maturities.

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## Calibration and production checklist

- Fix one expiration and derive a synchronized forward and discount factor from reliable inputs or put-call parity.
- Convert every strike to `k = ln(K/F_T)` and every valid IV to `w = IV²T` using one clock.
- Prefer executable OTM quotes; document treatment of zero Bids, crossed markets, stale quotes, corporate actions, and deep wings.
- Fit price, IV, or total-variance errors with weights tied to Vega, Bid-Ask width, and liquidity; report the chosen objective.
- Use parameter bounds and multiple starting points; Raw SVI parameters can compensate for one another.
- Check fitted option prices against Bid/Ask, not only a statistical average error.
- Enforce nonnegative total variance, butterfly/density constraints, and valid asymptotic wings.
- Test calendar monotonicity across all expirations in consistent coordinates and preserve explainable event steps.
- Inspect interpolation between expirations and extrapolation beyond quoted wings; most risk can sit outside the fit region.
- Recompute prices and Greeks on dense strike/maturity grids and compare numerical derivatives across grid sizes.
- Mark missing-node estimates as model values, not market observations.
- Monitor parameter jumps and residuals through time; a stable price surface can have unstable Raw parameters.
- Do not treat a quote's distance from SVI as arbitrage until executable package prices, fees, dividends, and constraints are checked.
- Version the data, forward curve, solver, constraints, weights, and fallback rules so a mark can be reproduced.

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## Common misconceptions

- **“SVI is a stochastic-volatility forecasting model.”** A calibrated slice is a static implied-total-variance parameterization.
- **“`σ` is the option's implied volatility.”** It is Raw SVI's center-width parameter; IV is `√(w/T)`.
- **“`a` is ATM variance.”** Other parameters and `m` also determine the value and minimum near ATM.
- **“Five valid parameter bounds make the smile arbitrage-free.”** Butterfly, wings, and cross-maturity calendar conditions remain.
- **“A lower RMSE is always a better surface.”** It can fit noise or violate economically critical constraints.
- **“Each expiry can be fit independently.”** Good slices can cross and imply calendar arbitrage when combined.
- **“An SVI residual is a trade signal.”** Bad data, illiquidity, special dividends, and different execution costs can explain it.

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## Related topics

- [Implied volatility](/options/implied-volatility/)
- [Local volatility model](/options/local-volatility-model/)
- [Skew dynamics](/options/skew-dynamics/)
- [Option model risk](/options/option-model-risk/)

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## Academic sources

- [Gatheral and Jacquier: Arbitrage-Free SVI Volatility Surfaces](https://doi.org/10.1080/14697688.2013.819986)
- [Martini and Mingone: No Arbitrage SVI](https://arxiv.org/abs/2005.03340)
- [Guo et al.: Generalized Arbitrage-Free SVI Volatility Surfaces](https://doi.org/10.1137/120900320)
- [Gatheral and Jacquier: Convergence of Heston to SVI](https://arxiv.org/abs/1002.3633)