What Is SVI?

Last reviewed: May 17, 2026 by Options Analysis Suite Research.

What Is SVI?

For a single expiration, SVI represents the total implied variance as a function of log-moneyness k = log(K/F):

w(k) = a + b * (rho * (k - m) + sqrt((k - m)^2 + sigma^2))

Five parameters: a (vertical translation), b (smile slope), rho (smile asymmetry), m (smile center), sigma (smile width). The functional form is hyperbolic; it produces a smile shape that smoothly interpolates between linear-in-k far OTM and quadratic-in-k near ATM. The formula is closed-form and instant to evaluate.

SVI was introduced by Gatheral at the 2004 Global Derivatives conference (later refined in Gatheral 2006 textbook). Its design constraints were: (1) flexible enough to fit observed smiles, (2) closed-form so calibration is fast, (3) shaped so arbitrage-free conditions can be checked algebraically.

Why Does SVI Matter?

• Industry-standard per-expiration smile fit. SVI is the practitioner default for fitting a single-expiration smile. Even when downstream applications use eSSVI or stochastic-vol surfaces, the per-expiration SVI fit is often a sanity-check baseline.
• Foundation of arbitrage-free surface fitting. Gatheral and Jacquier (2014) extended SVI to SSVI (Surface SVI) and then eSSVI (extended SSVI) by enforcing the right inter-tenor parameter constraints. SVI is the conceptual root.
• Direct calibration from listed prices. Five parameters and closed-form pricing makes SVI calibration tractable on any option chain with at least 5-7 listed strikes.

How Do Raw SVI and Natural SVI Differ?

Two alternative parameterizations:

• Raw SVI (a, b, rho, m, sigma). The original five-parameter form. Simple to implement; some parameter combinations produce arbitrage violations.
• Natural SVI. A reparameterization in which the parameter constraints for arbitrage absence are easier to enforce algebraically. Uses different parameter symbols. Mathematically equivalent to raw SVI but more numerically stable.

Practitioners use raw SVI for daily fitting and natural SVI when arbitrage-checking is the priority.

Roper No-Arbitrage Conditions

An SVI smile must satisfy two conditions to be free of static arbitrage (Roper 2010):

• Butterfly arbitrage. The implied call-price function must be convex in strike, equivalently the second derivative w''(k) must satisfy a positivity condition. Algebraically: a complex inequality on (a, b, rho, m, sigma) that excludes parameter regions where the smile is too sharply curved.
• Calendar arbitrage. Across expirations, total variance must grow monotonically: w_T1(k) ≤ w_T2(k) for T1 ≤ T2 at every log-moneyness k. Per-expiration SVI does not enforce this; SSVI/eSSVI do.

SVI vs SSVI vs eSSVI

• SVI: per-expiration. Fits each expiration's smile independently. Calendar arbitrage between adjacent expirations is not enforced.
• SSVI (Surface SVI, Gatheral-Jacquier 2014): joint surface model with two parameters per smile (theta, phi(theta)) plus a shared rho. Calendar-arbitrage-free by construction. Stronger smile/term-structure consistency than independent per-expiration SVI fits.
• eSSVI (extended SSVI): SSVI with additional flexibility - the phi(theta) function is extended to a more general parametric form, allowing the smile width to vary across tenors more flexibly. The institutional standard for full-surface fitting on equity indices.

Calibration in Practice

1. Pre-clean. Filter illiquid contracts; compute mid-quote IVs; normalize to log-moneyness.
2. Fit per-expiration. Minimize sum-squared-IV-residual over the five SVI parameters via Levenberg-Marquardt or differential evolution. Closed-form pricing makes this tractable in milliseconds.
3. Validate arbitrage. Check the Roper butterfly inequality on the fitted parameters. Reject or re-fit if violated.
4. Smoothness check. Verify the second derivative is well-behaved across the strike grid.

Worked Example

Calibrated SVI for SPX 30-day expiration, log-moneyness range [-0.10, +0.05]:

• a = 0.024 (vertical offset)
• b = 0.18 (smile slope)
• rho = -0.62 (asymmetry; negative = put-side higher)
• m = 0.005 (smile center, slightly OTM call-side)
• sigma = 0.07 (smile width)

Fit residuals: average 25 basis points across the smile, worst at the deep-put wing (~50 bp). Total time-to-fit: roughly 50 ms on a single core. Roper inequality satisfied: butterfly arbitrage absent.

Limitations and Caveats

• Per-expiration only. Calendar arbitrage between independently fit SVI smiles is not enforced. Use SSVI or eSSVI for full-surface work.
• Parameter ambiguity at sparse data. With only 5-7 listed strikes, multiple parameter sets fit equally well. Bayesian priors or regularization help.
• No smile dynamics. SVI fits a static smile. If you need to know how the smile evolves with spot, you need a dynamic model (Heston, SABR with sticky-strike vs sticky-delta convention, or rough vol).

Related Concepts

eSSVI · Volatility Smile · Volatility Skew · Butterfly Arbitrage · SABR Model · Heston Model · Calibration · Pricing Model Landscape

References & Further Reading

• Gatheral, J. (2004). "A Parsimonious Arbitrage-Free Implied Volatility Parameterization with Application to the Valuation of Volatility Derivatives." Presentation at Global Derivatives & Risk Management. The original SVI parametrization.
• Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley. Extended treatment of SVI, parameter interpretation, and surface dynamics.
• Gatheral, J. and Jacquier, A. (2014). "Arbitrage-Free SVI Volatility Surfaces." Quantitative Finance, 14(1), 59-71. The SSVI and eSSVI extensions with explicit arbitrage constraints.
• Roper, M. (2010). "Arbitrage Free Implied Volatility Surfaces." Working paper, University of Sydney. Algebraic conditions for static-arbitrage-free implied volatility surfaces.

View live SVI fits across SPY expirations ->

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