SVI Volatility Model: Fitting Total Variance Without Static Arbitrage

For educational purposes only; not investment advice.

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.

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.

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.

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.

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.