SVI and SSVI Volatility Surfaces: Static Arbitrage and Dupire Local Volatility

1. Start with clean quotes and the right coordinates

A surface calibration is only as coherent as its inputs. Reconcile spot, discount factors, forwards, expiries, option conventions, bid-ask spreads, and delta-to-strike conversions before fitting. For equity-style smiles, a common coordinate is log-forward moneyness k = log(K/F), with total implied variance w(k,T) = sigma_imp(k,T)^2 T.

Total variance is useful because no-calendar-arbitrage conditions and maturity behavior are easier to express in w than in raw implied volatility. Weight observations by liquidity or spread rather than treating a stale wing quote as equal to an at-the-money quote.

• Use one consistent forward and discounting convention per expiry.
• Keep bid, ask, mid, liquidity, and source timestamps available for diagnostics.
• Convert delta quotes only with the market convention used by the instrument.
• Do not extrapolate illiquid wings silently; state the rule and test its risk impact.

2. Fit SVI slices without losing economic shape

Raw SVI represents total variance as a five-parameter function of log-moneyness. The parameters control overall level, slope, skew orientation, horizontal shift, and curvature. Its flexibility makes it practical, but unconstrained least squares can produce visually good fits with poor arbitrage properties.

Calibrate with sensible parameter domains and multiple starts, inspect residuals in price or volatility space, and compare fitted wings with observed spreads. A low objective value is not sufficient if parameters jump between nearby expiries or the resulting density becomes negative.

3. Use SSVI and explicit static-arbitrage checks

SSVI parameterizes the whole surface through maturity-dependent at-the-money total variance and a controlled smile shape. Its attraction is not merely fewer parameters: suitable restrictions can make static-arbitrage control more transparent than calibrating unrelated slices.

Check calendar arbitrage by verifying that total variance behaves consistently across maturity for fixed log-moneyness. Check butterfly arbitrage through convexity of option prices or equivalent density conditions. Run these diagnostics on a dense grid, not only at quoted strikes.

4. Differentiate carefully before using Dupire local volatility

Dupire local volatility extracts a state- and time-dependent diffusion coefficient from the option-price or implied-variance surface. The formula depends on derivatives across strike and maturity, so noise that looks harmless in implied volatility can become severe after differentiation.

Validate local volatility by checking positivity, smoothness, boundary behavior, and repricing. A local-vol Monte Carlo or PDE engine should reproduce the vanilla surface within numerical tolerance when fed the extracted local volatility. Large repricing errors usually reveal derivative, interpolation, extrapolation, or convention problems.

• Differentiate the fitted representation, not raw noisy quotes.
• Stress maturity interpolation near short expiries where derivatives are unstable.
• Inspect local variance for negative or explosive regions before simulation.
• Reprice the calibration vanillas as an end-to-end consistency test.

5. Treat the surface as a production model, not a chart

A desk-quality surface needs fallback logic, stale-data controls, calibration monitoring, parameter-jump alerts, arbitrage diagnostics, and reproducible snapshots. Changes in quote coverage or convention should be visible to users rather than absorbed silently by the optimizer.

For model validation, challenge the full chain: market-data selection, forward construction, parameterization, objective weights, constraints, interpolation, extrapolation, numerical derivatives, local-vol extraction, and downstream pricing impact.

• Track fit error alongside arbitrage violations and parameter stability.
• Compare against a simpler interpolation or previous-day surface as a challenger.
• Test sparse markets, crossed quotes, missing wings, and stressed skew regimes.
• Version inputs and calibration settings so historical prices can be reproduced.

Practise aloud

Interview drills with answer direction

Turn reading into practice

A focused study plan

1. Step 1

   Normalize

   Build forwards, convert quotes consistently, and move into log-moneyness and total variance.

2. Step 2

   Calibrate

   Fit SVI slices with bounds, multiple starts, spread-aware weights, and residual diagnostics.

3. Step 3

   Constrain

   Introduce SSVI or explicit cross-maturity constraints and run dense-grid arbitrage tests.

4. Step 4

   Differentiate

   Extract Dupire local volatility and test positivity, smoothness, and sensitivity.

5. Step 5

   Reprice

   Use PDE or Monte Carlo repricing to close the validation loop and document fallbacks.

Self-review

Frequent mistakes to catch early

• Fitting every expiry independently and ignoring cross-maturity arbitrage.
• Optimizing only a least-squares error while ignoring bid-ask spreads and density checks.
• Differentiating raw implied-volatility quotes inside Dupire.
• Treating successful optimizer termination as proof that the surface is production-ready.

Keep building

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