Arbitrage-Free Fits: SVI / Jump-Wing

In the last chapter you stopped treating the implied-volatility smile as a curve to admire and started treating it as something you have to fit. The trouble is that the market only hands you a fistful of points. A liquid name might give you a dozen strikes near the money and almost nothing in the deep wings, and those points are noisy, stale at the edges, and quoted on a grid that fights you. If you connect them with a naive spline, you will draw a curve that wiggles, that goes negative where it should not, and — worst of all — that implies a probability distribution which is not a probability distribution at all. You need a parameterization with the right shape baked in: few parameters, smooth, and honest about the tails. That is what SVI gives you, and it is what sits underneath the slice you see in SVQ.

SVI stands for Stochastic Volatility Inspired, and the name is a clue to why it works. Jim Gatheral noticed that under the long-maturity limit of the Heston stochastic-volatility model, the implied total variance — that is, implied variance multiplied by time to expiry — flattens into a particular hyperbola-shaped curve in log-moneyness. So rather than guess a functional form, SVI borrows the shape that a real stochastic-volatility model already produces, and fits that. The object we model is not the volatility itself but the total variance, written w of k, where k is the log-moneyness, the log of strike over forward.

Here is the raw SVI slice. The total variance is a plus b, times the quantity rho times the difference k minus m, plus the square root of that difference squared plus sigma squared. Five parameters, and each one earns its place. The parameter a raises or lowers the whole curve — it sets the overall variance level. The parameter b controls how steeply the two wings open up; it is the slope of the cone. The parameter rho, which lives between minus one and one, tilts the smile: a negative rho, which is what you almost always see in equities, lifts the left wing and presses down the right, encoding the fact that the market pays more for downside protection. The parameter m slides the whole smile left or right, locating the bottom. And sigma controls how rounded the bottom is — a large sigma gives you a gentle, smooth valley near the money; a small sigma gives a sharp kink. Five numbers, and you have a curve that can lean, shift, steepen and round itself to match almost any single-expiry smile you will ever meet.

Now, fitting the five numbers so the curve passes near your points is the easy half. The hard half — and the half that actually matters for whether a number is genuine evidence about how an asset can move — is making sure the curve is arbitrage-free. A volatility slice is not just a curve; it is a disguised probability density over where the asset can land at expiry. If your fit is too curved in the wrong place, that implied density goes negative, and a negative probability is a free lunch: a butterfly spread that pays you to hold it. This is butterfly arbitrage, and it is the failure mode that kills naive fits.

The condition that rules it out is precise. From a slice you can derive a function — call it g of k — built from w, its first derivative, and its second derivative. The implied risk-neutral density is proportional to g of k times a positive factor, so the slice is free of butterfly arbitrage when g of k stays non-negative for every log-moneyness. That is the real test, and nothing substitutes for it: you have to check g of k across the whole curve. There is a second, separate constraint that governs the far edges. Gatheral and Jacquier worked out, from Lee's moment formula, that no honest total-variance slice can let its wings grow faster than linearly in the absolute value of k. Concretely, the product b times the quantity one plus the absolute value of rho must not exceed four over the maturity — the famous wing bound. This is a necessary condition, not a sufficient one: it pins down the asymptotic slope so your far-strike extrapolation does not manufacture arbitrage out in the tails where you have no data to catch you. But it is no guarantee on its own — a slice can sit comfortably under the wing bound and still have g of k dip negative somewhere in the middle. A genuinely butterfly-free slice needs both: g of k non-negative everywhere and the wings held under the bound.

This is where the jump-wing parameterization earns its keep. The raw a, b, rho, m, sigma are mathematically clean but financially mute — nobody trades a "b of zero-point-one." Gatheral's jump-wing reparameterization rewrites the same curve in five quantities a trader can read: the at-the-money variance, the at-the-money skew, the slopes of the left and right wings, and the minimum variance. Now the no-arbitrage conditions translate into plain statements — the wing slopes must not be too steep, the skew must be consistent with them — and you can enforce them directly as you fit. Same curve, honest coordinates.