Paper summary: arbitrage-free SVI volatility surfaces

Table of Contents

1. Summary

SVI is a five-parameter form for the implied variance smile of one expiry that fits market smiles well, but nothing about its parameters guarantees that a smile has a non-negative density or that two smiles do not cross. Gatheral and Jacquier show how to calibrate SVI so that static arbitrage is absent. Their central contribution is SSVI, a family that describes a whole surface by the ATM total variance of each expiry plus a skew parameter and a curvature function, and for which the conditions excluding calendar spread and butterfly arbitrage are explicit inequalities. They also give a method to remove calendar arbitrage from an SVI parameter set and an algorithm to remove butterfly arbitrage from a slice. For ORE Studio the result matters because a surface whose arbitrage conditions are inequalities on its parameters can be evolved through time without leaving the arbitrage-free region.

2. Source

Jim Gatheral and Antoine Jacquier, "Arbitrage-free SVI volatility surfaces", Quantitative Finance, 14(1), 59-71, 2014. Preprint: arXiv:1204.0646.

3. Problem being solved

A practitioner fitting smiles expiry by expiry wants three things at once: a good fit to each quoted smile, a density that is non-negative at every strike, and slices that do not cross across expiries. Raw SVI gives the first. The paper shows that it gives neither of the others reliably: Example 3.1 is a plausible parameter set with a visibly negative density, and the authors find general conditions excluding butterfly arbitrage in raw SVI "seemingly impossible" to state. Calendar arbitrage between two raw SVI slices can only be checked numerically, through the real roots of a quartic polynomial (Lemma 3.3).

4. The method

4.1. Three forms of one family

The paper first sets out three equivalent forms of SVI for one slice: the raw form \(\{a, b, \rho, m, \sigma\}\) (equation 3.1), the natural form \(\{\Delta, \mu, \rho, \omega, \zeta\}\) (equation 3.2, with the maps between them in Lemma 3.1), and the jump-wings form \(\{v_t, \psi_t, p_t, c_t, \tilde{v}_t\}\) (equation 3.5), whose parameters are the ATM variance, the ATM skew, the two wing slopes and the minimum variance. The jump-wings form is the one a trader can read.

4.2. SSVI

Definition 4.1 restricts the natural form to a surface:

\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}

where \(\theta_t\) is the ATM total variance of expiry \(t\) and \(\varphi\) a positive function. Theorem 4.1 gives necessary and sufficient conditions for the absence of calendar spread arbitrage, Theorem 4.2 sufficient conditions for the absence of butterfly arbitrage, and Corollary 4.1 combines them. Example 4.1 gives a Heston-like \(\varphi\) that is free of static arbitrage for all expiries when \(\lambda \ge (1 + |\rho|)/4\) (Remark 4.5); Example 4.2 gives the power law \(\varphi(\theta) = \eta\theta^{-\gamma}\), which is free of static arbitrage only up to a maximum expiry (Remark 4.4). The full statement of the conditions, and their simple form for the power law with \(\gamma = 1/2\), is in SVI Volatility Parameterisation.

4.3. Calibration without arbitrage

Section 5 of the paper turns the result into a calibration recipe. An SSVI fit to the market's slices gives an arbitrage-free first guess; each slice is then fitted with full SVI under a heavy penalty for crossing its neighbours, measured by the crossedness of Definition 5.1. Section 5.3 interpolates and extrapolates the calibrated slices so that the interpolated surface stays free of static arbitrage. Theorem 4.3 adds a related freedom: a non-negative function of time that never decreases may be added to an arbitrage-free SSVI surface without creating arbitrage.

5. Key assumptions and limitations

  1. Static arbitrage only. The conditions make one surface consistent at one instant. They say nothing about how the surface may move from one day to the next.
  2. One skew for all expiries. SSVI has one \(\rho\), so the term structure of the risk reversal is fixed by the shape of \(\varphi\). Hendriks and Martini (2019) relax this with an expiry-dependent \(\rho\).
  3. Sufficient, not necessary, for butterflies. Theorem 4.2 is sufficient; Lemma 4.2 shows it is close to necessary, but a surface outside the bound is not thereby shown to admit arbitrage.
  4. Equity framing. The examples are equity index smiles. Nothing in the conditions depends on the asset class, but FX quoting in delta needs a strike solve before a quote can be compared with the surface.

6. Applicability to synthetic market data generation

The paper's Remark 4.1 observes that the ATM variance curve \(\theta_t\) is directly observable and a natural place to impose dynamics on SSVI. That remark is the basis of the synthetic surface generator described in Synthetic market data generation: approach: the generator's state is the SSVI parameter set, each parameter follows a mean-reverting process bounded to the region of Corollary 4.1, and every surface the generator emits is therefore free of static arbitrage by construction. The paper does not consider such dynamics; the bound is what lets them be added without a check-and-repair step on each surface.

7. Open questions

  1. Expiry range. For the power-law curvature, the surface is arbitrage-free only below a total ATM variance of about two when \(\eta \le \sqrt{2}\). FX surfaces sit far below that bound; equity and commodity surfaces at long expiries and high volatility may not.
  2. Skew term structure. Whether one \(\rho\) reproduces the risk-reversal term structure of the major FX pairs, or whether eSSVI is needed, is untested on our data.

8. See also

Emacs 29.3 (Org mode 9.6.15)