Volatility Interpolation and Extrapolation
Volatility interpolation fills gaps between quoted options to build a smooth, arbitrage free surface. Learn the main methods, SVI and the arbitrage checks.
Option markets quote implied volatilities only at listed strikes and expiries, and some of those quotes are stale or wide. But traders, risk systems and pricing models need a volatility for any strike and any date: to price an option at an unlisted strike, to mark a portfolio, or to feed a local volatility model. Volatility interpolation is the process of turning a scattered set of quotes into a smooth, consistent surface without creating arbitrage. Done badly, it produces prices that allow riskless profits, or Greeks that jump around for no reason.
The two directions#
A volatility surface has two dimensions:
- Strike (or moneyness): the smile at each expiry. See Volatility Smile and Skew.
- Time: the term structure at each strike. See Volatility Term Structure.
Interpolation is usually done in the strike direction at each quoted expiry first, then across expiries.
Choosing coordinates#
Interpolating directly in strike and volatility often works poorly. Common choices instead:
| Coordinate | Why |
|---|---|
| Log moneyness, ln(K / F) | Makes smiles at different forward levels comparable |
| Delta | Standard in FX markets, where quotes are by delta (e.g. 25 delta risk reversals) |
| Total implied variance, w = σ² × T | Interpolating in total variance across time helps avoid calendar arbitrage |
Methods across strikes#
- Linear interpolation: simple but creates kinks, which produce spikes in risk measures and in implied densities.
- Cubic splines: smooth, but can wiggle between points and create arbitrage if quotes are noisy.
- Parametric smile models: fit a formula with a few parameters to each expiry. Popular choices are SABR for rates and FX (see SABR Model) and SVI (stochastic volatility inspired), introduced by Jim Gatheral at Merrill Lynch, for equities.
- Model based fits: calibrate a full model such as Heston and read the surface from it.
The SVI parameterisation#
SVI writes total implied variance at log moneyness k as:
w(k) = a + b × [ρ × (k - m) + √((k - m)² + σ²)]Five parameters control level (a), angle of the wings (b), skew (ρ), horizontal shift (m) and the smoothness at the minimum (σ). SVI's wings grow linearly, matching a known theoretical result that total variance can grow at most linearly in log strike far from the money. Gatheral and Jacquier later published conditions under which SVI surfaces are free of arbitrage.
Arbitrage checks#
A valid surface must avoid two kinds of static arbitrage:
| Type | Condition | Meaning if violated |
|---|---|---|
| Butterfly (strike) arbitrage | Call prices are convex in strike; the implied density is non negative | A butterfly spread with negative cost |
| Calendar arbitrage | Total variance does not decrease with expiry at fixed moneyness | A calendar spread with negative cost |
Extrapolation#
Beyond the last quoted strikes, the surface must be extended. Extrapolating a spline can produce absurd values; parametric models like SVI and SABR extrapolate more sensibly, though wing behaviour still needs checks, because far wings drive prices of products like variance swaps. See Variance and Volatility Swaps.
Across time#
Between quoted expiries, traders typically interpolate total variance linearly in time at fixed moneyness, often adjusting for known events such as earnings or central bank meetings, which add extra variance on specific dates. See Earnings Trading.
Practical tips#
- Clean the data: remove stale quotes, wide spreads and options with almost no time value.
- Use mid prices carefully; weight fits by liquidity.
- Check the implied density by looking at the second derivative of call prices across strikes.
- Keep parameters stable day to day so risk numbers do not jump.
Frequently asked questions#
What is volatility interpolation?#
The process of estimating implied volatilities between and beyond quoted strikes and expiries to build a smooth volatility surface.
What is SVI?#
A five parameter formula for the implied volatility smile, introduced by Jim Gatheral, widely used for equity options because it fits well and can be made arbitrage free.
What is calendar arbitrage in a volatility surface?#
A situation where total implied variance falls with expiry at the same moneyness, implying that a calendar spread could be bought for a negative cost.
Next, move beyond standard options with Exotic Options Explained.
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