SVI Volatility Parameterisation

Table of Contents

1. Summary

SVI, the stochastic volatility inspired parameterisation, writes the total implied variance of one expiry as a function of log-moneyness with five parameters. Gatheral introduced it in 2004, and it is widely used because it fits listed and OTC smiles well and has the right asymptotic shape in both wings. Its weakness is that no simple condition on the five parameters rules out arbitrage: a raw SVI smile can imply a negative density, and two SVI slices can cross. SSVI, the surface SVI of Gatheral and Jacquier (2014), restricts the family so that a whole surface is described by the ATM total variance of each expiry and two or three further parameters, and for that restriction the absence of static arbitrage reduces to inequalities on the parameters. The extended SSVI of Hendriks and Martini (2019) lets the skew parameter vary by expiry and keeps those inequalities. This note defines each form and states the conditions; the surfaces built on it are described in FX Volatility Surface.

2. Layperson's mental model

Picture the smile of one expiry as a length of bendable wire shaped like a shallow bowl: its height is how expensive options are in general, and its two rising arms are how much dearer far-from-the-money options are. Raw SVI gives you five knobs to bend that wire: raise or lower the whole bowl (\(a\)), steepen both arms (\(b\)), tilt it so one arm rises faster than the other (\(\rho\)), slide it sideways (\(m\)), and round off its bottom (\(\sigma\)). Five knobs fit almost any smile. But nothing stops you bending the wire into a shape no real market could produce, with a kink that implies a negative probability, and when you bend one wire per expiry, nothing stops two of them from crossing.

SSVI swaps the separate wires for one frame that holds them all. Each expiry still has its own height, its ATM total variance \(\theta_t\), but every wire shares one tilt (\(\rho\)) and one stiffness (\(\eta\)), and the frame bends each wire from those few numbers. The frame comes with marks on its knobs: keep the heights rising with expiry and the tilt and stiffness inside their marked ranges, and no wire can kink or cross another, whatever values you pick inside the marks. That is what lets a generator turn the knobs continuously without ever producing an impossible surface. 

// total implied variance at log-moneyness k, for an expiry whose ATM total variance is theta
double ssvi_total_variance(double k, double theta, double rho, double eta) {
    const double phi = eta / std::sqrt(theta);
    const double x = phi * k + rho;
    return 0.5 * theta * (1.0 + rho * phi * k + std::sqrt(x * x + 1.0 - rho * rho));
}

3. Detail

3.1. Coordinates

Every form below is written in two quantities. The log-moneyness against the forward to expiry \(t\) is

\begin{equation} k = \ln\left(\frac{K}{F_t}\right) \end{equation}

and the total implied variance is

\begin{equation} w(k, t) = \sigma_{BS}^2(k, t)\, t \end{equation}

where \(\sigma_{BS}\) is the Black-Scholes implied volatility. Total variance is the natural coordinate because it adds across time: a surface with no calendar spread arbitrage is one whose total variance does not fall as \(t\) rises at any fixed \(k\). The no-arbitrage conditions themselves are in Vol Surface No-Arbitrage Conditions.

3.2. Raw SVI

For one expiry and a parameter set \(\chi_R = \{a, b, \rho, m, \sigma\}\), raw SVI is (Gatheral and Jacquier, 2014, equation 3.1):

\begin{equation} w(k; \chi_R) = a + b\left(\rho\,(k - m) + \sqrt{(k - m)^2 + \sigma^2}\right) \end{equation}

with \(b \ge 0\), \(|\rho| < 1\), \(\sigma > 0\) and \(a + b\,\sigma\sqrt{1 - \rho^2} \ge 0\), the last ensuring the variance is never negative. The smile is a hyperbola in total variance, so each wing is asymptotically linear in \(k\), as Lee's moment formula requires of any arbitrage-free smile. Each parameter has one visible effect:

Parameter Effect on the smile
\(a\) Raises or lowers the general level of variance
\(b\) Steepens both wings together
\(\rho\) Rotates the smile: a lower \(\rho\) steepens the put wing against the call wing
\(m\) Translates the smile along \(k\)
\(\sigma\) Widens the rounded minimum, lowering the ATM curvature

The natural SVI of the same paper (equation 3.2) is the same family reparameterised, with an explicit map between the two (Lemma 3.1). The jump-wings SVI-JW (equation 3.5) rewrites the parameters as quantities a trader reads off a smile: the ATM variance \(v_t\), the ATM skew \(\psi_t\), the slopes \(p_t\) and \(c_t\) of the put and call wings, and the minimum variance \(\tilde{v}_t\). If smiles scaled exactly as \(1/\sqrt{w_t}\), the SVI-JW parameters would not change with the expiry.

3.3. Why raw SVI admits arbitrage

Raw SVI parameterises one slice, and a surface built from it is a separate set of five parameters per expiry. Two failures follow. A single slice can carry butterfly arbitrage, a negative risk-neutral density at some strike, and Gatheral and Jacquier give an example with plausible parameters (Example 3.1). They state that general conditions on the five parameters that exclude it appear impossible to find, because the density is a highly non-linear function of them. And two slices can cross, which is calendar spread arbitrage; the paper reduces the check to whether a quartic polynomial has real roots (Lemma 3.3), which is a numerical test rather than a constraint a parameter can be held to. Martini and Mingone (2020) later characterise the absence of butterfly arbitrage in raw SVI completely, as conditions that are exact but not simple.

3.4. SSVI: one surface, few parameters

SSVI describes the whole surface at once (Gatheral and Jacquier, 2014, Definition 4.1):

\begin{equation} w(k, \theta_t) = \frac{\theta_t}{2}\left(1 + \rho\,\varphi(\theta_t)\,k + \sqrt{\left(\varphi(\theta_t)\,k + \rho\right)^2 + 1 - \rho^2}\right) \end{equation}
  • \(\theta_t = \sigma_{BS}^2(0, t)\,t\) is the ATM total variance of expiry \(t\). It is the only quantity that varies by expiry, and it is directly observable from the market; the authors note it as the natural place to impose dynamics on the surface (Remark 4.1).
  • \(\rho \in (-1, 1)\) sets the skew. The smile is symmetric if and only if \(\rho = 0\), as in a stochastic volatility model whose spot and volatility are uncorrelated.
  • \(\varphi\) is a positive function of \(\theta\) that sets the curvature.

The ATM volatility skew is (equation 4.2)

\begin{equation} \left.\partial_k \sigma_{BS}(k, t)\right|_{k=0} = \frac{\rho\sqrt{\theta_t}}{2\sqrt{t}}\,\varphi(\theta_t) \end{equation}

The paper gives two forms of \(\varphi\):

  • The Heston-like form (Example 4.1), \(\varphi(\theta) = \frac{1}{\lambda\theta}\left(1 - \frac{1 - e^{-\lambda\theta}}{\lambda\theta}\right)\), consistent with the implied variance skew of the Heston model. Its short-expiry smile flattens, as that of a pure diffusion does.
  • The power-law form (Example 4.2), \(\varphi(\theta) = \eta\,\theta^{-\gamma}\) with \(\eta > 0\) and \(0 < \gamma < 1\). Its short-expiry smile is V-shaped, which is what observed surfaces and models with jumps show. With \(\gamma = 1/2\) the SVI-JW parameters of every slice are constant (Lemma 4.1) and the ATM skew is \(\rho\eta/(2\sqrt{t})\), falling as \(1/\sqrt{t}\).

The Heston-like form has no amplitude. Because \(1 - (1 - e^{-x})/x < x/2\) for \(x > 0\), it is always below \(1/2\), and that cap can stop it reaching the skew of a surface; an experiment fitting a simulated Heston surface missed by 4 volatility points RMSE until a scale factor was added (riskhub.org, below).

3.5. The absence of static arbitrage in SSVI

Corollary 4.1 of the paper: an SSVI surface is free of static arbitrage if, for all \(\theta > 0\),

  1. \(\partial_t \theta_t \ge 0\);
  2. \(0 \le \partial_\theta\left(\theta\varphi(\theta)\right) \le \frac{1}{\rho^2}\left(1 + \sqrt{1 - \rho^2}\right)\varphi(\theta)\);
  3. \(\theta\varphi(\theta)\,(1 + |\rho|) < 4\);
  4. \(\theta\varphi(\theta)^2\,(1 + |\rho|) \le 4\).

The first two exclude calendar spread arbitrage (Theorem 4.1) and are necessary as well as sufficient. The last two exclude butterfly arbitrage (Theorem 4.2); Lemma 4.2 shows the third is necessary and the fourth tight. Unlike raw SVI, every condition is an inequality on the parameters, so a surface can be held inside the region by bounding them.

For the power law with \(\gamma = 1/2\) the conditions take a particularly simple form. Condition 2 holds for every \(\rho\), because \(\partial_\theta(\theta\varphi)/\varphi = 1/2\) never exceeds \((1 + \sqrt{1 - \rho^2})/\rho^2\). Condition 4 becomes \(\eta^2(1 + |\rho|) \le 4\), which holds for any \(\rho\) when \(\eta \le \sqrt{2}\). Condition 3 becomes \(\eta\sqrt{\theta}\,(1 + |\rho|) < 4\), which, with \(\eta \le \sqrt{2}\), holds whenever \(\theta < 2\), that is whenever \(\sigma_{ATM}\sqrt{t} < \sqrt{2}\). So the region is: \(\theta_t\) rising in \(t\), \(|\rho| < 1\), \(0 < \eta \le \sqrt{2}\), and total ATM variance below two. The bound \(\theta < 2\) is sufficient, not exact: it is condition 3 at the largest \(\eta\) and \(|\rho|\) allowed, and for a given pair the exact limit is \(\theta < \left(4 / (\eta(1 + |\rho|))\right)^2\), which is looser whenever \(\eta < \sqrt{2}\) or \(|\rho| < 1\). Remark 4.4 of the paper notes the general point behind the last bound: a power-law SSVI is free of static arbitrage only up to some maximum expiry.

3.6. Extended SSVI

A single \(\rho\) ties the skew of every expiry to one number, which limits the term structures of risk reversal an SSVI surface can represent. Hendriks and Martini (2019) make \(\rho\) a function of the expiry, the extended SSVI (eSSVI), and give necessary and sufficient conditions for that surface to be free of calendar spread arbitrage. Its authors report that the extension can improve the accuracy of a calibration at short maturities. A later paper on a global parametrisation of eSSVI that keeps it free of arbitrage is listed below and has not been read in full.

3.7. In ORE Studio

SSVI with the power-law curvature and \(\gamma = 1/2\) is the surface the synthetic volatility generator evolves, as Synthetic market data generation: approach describes; its parameters are the generator's state, held inside the region above. A tenant's surface blueprint may fit SSVI, SABR, a spline or another smile model to the quotes it reads, so the generator's choice does not bind the tenant's.

The calibration engine, ssvi_calibration_engine in ores.analytics.quant, fits the power-law form with \(\gamma\) free in \((0, 1)\), as ORE's QuantExt does, so it does not assume the generator's \(\gamma = 1/2\). With \(\gamma\) free, condition 4 depends on \(\theta\), so it is checked expiry by expiry. A fitted surface can be tested against the closed form of the ATM skew, which for any \(\varphi\) is \(\rho\sqrt{\theta_t}\,\varphi(\theta_t)/(2\sqrt{t})\).

4. See also

4.1. Further reading

  • Gatheral, J. (2004). "A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives." Presentation, Global Derivatives & Risk Management, Madrid. Slides. The origin of SVI.
  • Gatheral, J. and Jacquier, A. (2014). "Arbitrage-free SVI volatility surfaces." Quantitative Finance, 14(1), 59-71. arXiv:1204.0646.
  • Martini, C. and Mingone, A. (2020). "No arbitrage SVI." arXiv:2005.03340. The complete characterisation of butterfly arbitrage in raw SVI.
  • Hendriks, S. and Martini, C. (2019). "The extended SSVI volatility surface." Journal of Computational Finance, 22(5), 25-39. risk.net; working paper (2017), doi:10.2139/ssrn.2971502.
  • SSVI and Surface-Wide Parameterization (riskhub.org). A worked fit to a simulated Heston surface, comparing raw SVI, per-slice SSVI and global SSVI, and testing out of sample in maturity.
  • ORE's QuantExt, qle/termstructures/svimodeltraits.cpp and test/sviparametricvolatility.cpp. An open-source implementation of these forms, and the source of the calibration engine's test data.
  • "No arbitrage global parametrization for the eSSVI volatility surface." arXiv:2204.00312.

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