SVI Parameterization

At one expiry a liquid index has maybe forty tradable strikes and forty implied volatilities. You need a curve through them, and not just any curve: you need one you can differentiate twice for Dupire, one that will not invent negative probabilities between quotes, and one that behaves itself far out in the wings where nothing trades but risk still lives. A cubic spline through the dots fails all three. It wiggles, its second derivative is noisy, and past the last quote it does whatever the polynomial feels like. What you want instead is a shape — a small family of curves that already look like smiles — and then you only have to pick which member of the family.

Two slopes and an elbow

Look at a smile plotted the right way and it is not really a curve at all. Far out to the left it is a straight line going up. Far out to the right it is a different straight line going up, usually less steeply. In the middle those two lines would meet at a sharp corner, except the market rounds the corner off into a smooth bowl. That is the entire picture: two rays and a rounded join.

There is a name for the smooth curve that hugs two straight asymptotes: a hyperbola. And there is a reason the wings really are straight rather than curving off. Roger Lee proved that total variance can grow at most linearly in log-moneyness at extreme strikes — grow faster and calls end up cheaper at a higher strike, which is free money. So straight wings are not a convenient approximation, they are the only thing allowed asymptotically.

SVI, "stochastic volatility inspired", is the hyperbola with those wings. Five parameters: how high the bowl sits, how wide the wings open, how much the whole thing tilts, where the bottom sits sideways, and how rounded the bottom is.

The formula

Two changes of variable first, both of which do real work.

Instead of the strike itself, use log-moneyness $k = \log(K / F)$, where $K$ is the strike and $F$ is the forward price. So $k = 0$ is at the money, $k = -0.1$ is a strike about 10 percent below the forward, and the scale is the same for a stock at 40 and an index at 4,000.

Instead of implied volatility, use total implied variance $w = \sigma_{\mathrm{imp}}^{2}\, T$, where $\sigma_{\mathrm{imp}}$ is the quoted Black-Scholes volatility and $T$ is the time to expiry in years. Total variance is the natural currency of options: it is what actually accumulates as time passes, and no-arbitrage conditions across expiries are simple statements about it. A 20 percent volatility for one year is $w = 0.04$.

The raw SVI slice is then

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

In plain English: take a V-shape centred at $m$, round off its point by an amount $\sigma$, tilt it by $\rho$, scale it by $b$, and lift the whole thing by $a$. Every parameter has a job you can see on the chart:

• $a$ — level. Shifts the whole slice up or down. Raise it and every option gets more expensive.
• $b$ — wing width. How fast total variance grows as you walk away from the middle. Bigger $b$ means fatter tails priced in.
• $\rho$ — tilt, between $-1$ and $+1$. Negative $\rho$ makes the left wing steeper than the right, which is the equity skew.
• $m$ — shift. Moves the bowl sideways in log-moneyness.
• $\sigma$ — roundedness. How blunt the elbow is. Small $\sigma$ gives an almost pointy V, large $\sigma$ a lazy bowl.

Two facts fall straight out of the algebra and are worth having. As $k$ runs off to the right the square root becomes $(k - m)$, so the right wing has slope $b(1 + \rho)$; running off to the left it becomes $-(k-m)$, so the left wing has slope $b(1 - \rho)$ in magnitude. And setting the derivative to zero puts the minimum at $k^{*} = m - \rho\sigma/\sqrt{1 - \rho^{2}}$, where the value is $a + b\sigma\sqrt{1 - \rho^{2}}$.

Worked example 1: reading a slice off the parameters

Take a one-year slice with $a = 0.02$, $b = 0.2$, $\rho = -0.5$, $m = 0$, $\sigma = 0.1$. Because $T = 1$, total variance and annual variance are the same number, so implied volatility is just $\sqrt{w}$.

At the money ($k = 0$): the bracket is $-0.5(0) + \sqrt{0 + 0.01} = 0.1$, so $w = 0.02 + 0.2(0.1) = 0.04$ and $\sigma_{\mathrm{imp}} = \sqrt{0.04} = 20.00$ percent.

Ten percent down ($k = -0.1$): the square root is $\sqrt{0.01 + 0.01} = \sqrt{0.02} = 0.14142$. The bracket is $(-0.5)(-0.1) + 0.14142 = 0.05 + 0.14142 = 0.19142$. So $w = 0.02 + 0.2(0.19142) = 0.05828$ and $\sigma_{\mathrm{imp}} = 24.14$ percent.

Ten percent up ($k = +0.1$): same square root, but the tilt now subtracts. The bracket is $-0.05 + 0.14142 = 0.09142$, so $w = 0.02 + 0.2(0.09142) = 0.03828$ and $\sigma_{\mathrm{imp}} = 19.57$ percent.

The slice runs 24.14 / 20.00 / 19.57 across those three strikes: a steep downside, a nearly flat upside. That is an equity smile, produced by five numbers.

Two cross-checks you can do in your head. The minimum should sit at $k^{*} = 0.5(0.1)/\sqrt{0.75} = 0.05/0.866 = 0.0577$, slightly above the money — and indeed the 105-ish strike came out lower than the 100 strike. And the wing slopes should be $b(1-\rho) = 0.2(1.5) = 0.3$ on the left and $b(1+\rho) = 0.2(0.5) = 0.1$ on the right: downside variance eventually grows three times as fast as upside variance.

Worked example 2: two slices that quietly arbitrage each other

Fit the one-year slice above, then fit next year's slice independently and get $a = 0.045$, $b = 0.13$, $\rho = -0.2$, $m = 0$, $\sigma = 0.12$ for $T = 2$. Each fit looks fine on its own chart. Now check the one rule that links expiries: total variance must never decrease as expiry increases, at any fixed log-moneyness. Otherwise you buy the two-year and sell the one-year at that strike and collect money for owning more time.

At the money, $k = 0$. One-year: $w_1 = 0.04$, computed above. Two-year: the square root is $\sqrt{0 + 0.0144} = 0.12$, so $w_2 = 0.045 + 0.13(0.12) = 0.045 + 0.0156 = 0.0606$. Fine, $0.0606 > 0.04$.

At $k = -0.2$, a strike about 18 percent below the forward. One-year: $\sqrt{0.04 + 0.01} = \sqrt{0.05} = 0.22361$, bracket $= 0.1 + 0.22361 = 0.32361$, so $w_1 = 0.02 + 0.2(0.32361) = 0.08472$. Two-year: $\sqrt{0.04 + 0.0144} = \sqrt{0.0544} = 0.23324$, bracket $= 0.04 + 0.23324 = 0.27324$, so $w_2 = 0.045 + 0.13(0.27324) = 0.08052$.

$w_2 = 0.0805$ is below $w_1 = 0.0847$. The slices have crossed, somewhere around $k = -0.16$, and past that point the surface pays you to own time.

Now the part that makes this dangerous. Convert back to volatilities: the one-year is $\sqrt{0.08472} = 29.11$ percent and the two-year is $\sqrt{0.08052 / 2} = \sqrt{0.04026} = 20.07$ percent. On a volatility chart that looks completely ordinary — a steep near-dated skew flattening out with maturity, which is what real surfaces do all day. The arbitrage is invisible in volatility units and obvious in total variance units. This is the whole reason practitioners work in $w$.

The fix is to fit the slices jointly, or to use a surface parameterisation that cannot cross by construction (SSVI Surface Parameterization).

The explorer below is the surface SVI is fitting — one row per expiry, one SVI slice each. Push skew up and watch the left column climb: that is $\rho$ going more negative and $b$ widening.

What this means in practice

SVI is the default smile interpolator on equity and index volatility desks. It is used to smooth noisy quotes before differentiating them, to fill in strikes nobody quotes, to extrapolate wings for a variance swap or a density extraction, and to compress a surface into a few dozen numbers for storage and risk reporting.

Fitting is a five-parameter least squares on each slice, and it is mildly ill-posed: $a$, $b$ and $\sigma$ trade off against each other, so different starting points land on visibly different parameter sets with nearly identical curves. The standard cure is Zeliade's quasi-explicit approach, which fixes $m$ and $\sigma$ on an outer grid and solves the remaining three linearly. Parameter values are therefore not stable enough to use as risk factors directly — track the fitted curve, not the fitted $b$.

Practice

1. With $a = 0.03$, $b = 0.25$, $\rho = -0.6$, $m = 0$, $\sigma = 0.2$ and $T = 1$, find the at-the-money implied volatility.
2. For the same parameters, what are the two wing slopes, and does either violate Lee's bound of 2?
3. A slice has one-year total variance 0.09 at $k = -0.3$. Next year's slice quotes 21 percent implied volatility at the same $k$. Is there a calendar arbitrage?

Answers. (1) Bracket $= 0 + \sqrt{0.04} = 0.2$, so $w = 0.03 + 0.25(0.2) = 0.08$ and $\sigma_{\mathrm{imp}} = \sqrt{0.08} = 28.3$ percent. (2) Left $0.25(1.6) = 0.40$, right $0.25(0.4) = 0.10$; both well inside the bound. (3) Two-year total variance is $0.21^{2} \times 2 = 0.0882$, which is below 0.09 — yes, a calendar arbitrage, despite the two-year volatility looking perfectly normal.

Key terms

• Log-moneyness $k = \log(K/F)$ — strike measured in percentage terms against the forward.
• Total implied variance $w = \sigma_{\mathrm{imp}}^{2} T$ — the accumulated variance an option prices in; the natural axis for arbitrage checks.
• Slice — one expiry's smile. SVI fits slices; SSVI fits the whole surface at once.
• Lee's moment formula — total variance grows at most linearly in $|k|$, so wing slopes cannot exceed 2.
• Calendar arbitrage — total variance decreasing with maturity at some strike; free money from owning time.