SVI and the No-Arbitrage Volatility Surface

An implied-volatility surface is a deceptively small object: a function from log-moneyness and maturity to a single number. The trouble is that not every such function corresponds to a market. A surface that prices a butterfly spread at a negative cost, or lets a longer-dated option be worth less than a shorter one, is not merely inaccurate — it is arbitrageable, and any model built on it will happily trade into the hole.

This post walks through how ferro-risk fits the SVI parametrization and, more importantly, how it keeps the fit honest.

The raw SVI slice

Work in total implied variance rather than volatility. For a single maturity slice, with $k = \log(K/F)$ the log-moneyness, Gatheral's raw SVI is

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

Five parameters carry a clear reading:

• $a$ shifts the overall variance level,
• $b \ge 0$ sets the angle between the two asymptotic wings,
• $\rho \in (-1, 1)$ is the skew — the counter-clockwise rotation,
• $m$ translates the smile along the moneyness axis,
• $\sigma > 0$ controls how rounded the vertex is.

Total variance scales with maturity, so working in $w = \sigma_{\text{BS}}^2\,t$ rather than $\sigma_{\text{BS}}$ keeps the slices on a comparable footing and makes the calendar condition below clean to state.

Butterfly arbitrage: the density must be a density

A slice is free of butterfly arbitrage exactly when the risk-neutral density it implies is non-negative everywhere. Following Gatheral and Jacquier, define the auxiliary function

$$g(k) \;=\; \left(1 - \frac{k\,w'(k)}{2\,w(k)}\right)^{2} \;-\; \frac{w'(k)^2}{4}\left(\frac{1}{w(k)} + \frac{1}{4}\right) \;+\; \frac{w''(k)}{2}$$

The implied density is then

$$p(k) \;=\; \frac{g(k)}{\sqrt{2\pi\,w(k)}}\; \exp\!\left(-\frac{d_-(k)^2}{2}\right), \qquad d_-(k) \;=\; -\frac{k}{\sqrt{w(k)}} \;-\; \frac{\sqrt{w(k)}}{2}$$

Every factor except $g(k)$ is manifestly positive, so the slice is butterfly-free iff $g(k) \ge 0$ for all $k$. That reduces a statement about option prices to a single scalar inequality on the fitted curve.

Calendar arbitrage: slices must not cross

The second condition is across maturities. Total variance must be non-decreasing in time at every fixed log-moneyness:

$$\frac{\partial w(k, t)}{\partial t} \;\ge\; 0 \qquad \text{for all } k$$

Geometrically: in $(k, w)$ space the slices must never cross. If a longer-dated slice dips below a shorter-dated one at some $k$, a calendar spread struck there has negative time value.

How ferro-risk fits a slice

Calibration minimizes a weighted squared error in total variance:

$$\min_{a,\,b,\,\rho,\,m,\,\sigma}\; \sum_{i} \omega_i \,\bigl(w(k_i) - w_i^{\text{mkt}}\bigr)^2$$

with weights $\omega_i$ taken from quoted size and bid-ask width — a tight, liquid quote pulls harder than a stale one. The five-parameter problem is too small to need anything exotic, but it is not convex, so the recipe matters:

1. Reduce the dimension. For fixed $(m, \sigma)$ the model is linear in $(a, b, b\rho)$. The inner problem is an ordinary constrained least squares; only the outer $(m, \sigma)$ search is genuinely nonlinear. This is the quasi-explicit calibration of Zeliade's note, and it makes the fit fast and repeatable.
2. Project onto the no-arbitrage set. After each candidate fit, evaluate $g(k)$ on a dense grid spanning the wings. If $\min_k g(k) < 0$, add a penalty proportional to the violation and re-solve. The accepted fit satisfies $g(k) \ge 0$ by construction.
3. Enforce calendars globally. Slices are not fitted in isolation. After all maturities are fit, a sweep checks the crossing condition and nudges offending slices until $\partial_t w \ge 0$ holds on the grid.

Diagnostics that ship with the fit

A number without an error bar is a liability, so every calibrated slice carries:

Diagnostic	Definition	Reading
RMSE (vol points)	$\sqrt{\frac1n \sum (\sigma_i - \sigma_i^{\text{mkt}})^2}$	Raw fit quality
$\min_k g(k)$	Butterfly margin on the grid	$\ge 0$ required; near 0 is fragile
Calendar gap	$\min_k \partial_t w$	$\ge 0$ required
Wing extrapolation	Slope of $w$ as $k \to \pm\infty$	Should match Lee's moment bounds

The last row is a reminder that SVI's wings are linear in $k$, which is exactly the Roger Lee moment-formula regime — a fitted slice that extrapolates sanely is not a coincidence, it is the parametrization doing its job.

Where this connects

A clean surface is an input, not an end. The local-volatility extraction (Dupire) that feeds the scenario engine differentiates $w$ twice — so any arbitrage left in the fit is amplified downstream into a negative local variance. Getting $g(k) \ge 0$ here is what makes the chain-risk and scenario-PnL work in the rest of the Ferro suite trustworthy. We will take up the Dupire step, and the SSVI surface-level parametrization that ties the slices together, in a follow-up.