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Volatility Interpolation: Connecting Sparse Option Quotes

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

Volatility interpolation estimates option values or implied volatility between quoted strikes and expiries. Markets provide discrete, noisy nodes, while valuation and risk systems need a continuous surface. A defensible method does not merely average two IVs: it chooses economically comparable coordinates, uses tradable inputs, and verifies that interpolated option prices preserve no-arbitrage shape.

Across expiries, interpolate cumulative total variance w = σ²T on a consistent moneyness point rather than annualized IV by habit. Across strikes, use log-forward moneyness k = ln(K/F_T), Delta, or an arbitrage-constrained price representation. Interpolation is between observations; fitting summarizes observations; extrapolation extends beyond them and carries materially greater model risk.

Coordinates, data, and constraints

Section titled “Coordinates, data, and constraints”

Absolute strike is not comparable across expiries because rates, dividends, borrow, and time move the forward F_T. A strike of 100 can be ATM for one expiry and off-center for another. k = ln(K/F_T) expresses strike relative to that expiry’s forward.

Term uncertainty accumulates more naturally as total variance:

w(k,T) = σ_imp(k,T)^2 × T

At a fixed, consistent moneyness, a negative interval forward variance is a warning of inconsistent quotes, forwards, event treatment, or interpolation. Along strikes, undiscounted Call prices should be non-increasing and convex in strike. Violating convexity can imply a negative-cost butterfly with nonnegative payoff; violating valid term relationships can create calendar inconsistency.

Clean inputs before fitting. Use synchronized two-sided quotes, reject crossed markets and basic price-bound violations, treat zero bids and wide wings cautiously, estimate forwards and discount factors consistently, and retain Bid/Ask intervals. A smooth algorithm cannot repair a bad node; it spreads the error.

Linear methods are transparent but produce slope kinks. Unconstrained cubic splines can overshoot. Delta coordinates match common market quoting but Delta itself depends on IV. Parametric surfaces such as SVI can be compact and smooth but still need admissible parameters and cross-expiry checks. Price-space interpolation can enforce monotonicity and convexity more directly, then invert back to IV.

Total-variance example

Section titled “Total-variance example”

Suppose comparable ATM IV is 20% at 30 days and 24% at 60 days, using a 365-day year:

w30 = 0.20^2 × 30/365 ≈ 0.00329

w60 = 0.24^2 × 60/365 ≈ 0.00947

Linear interpolation of total variance to 45 days gives:

w45 = w30 + (45−30)/(60−30) × (w60−w30) ≈ 0.00638

σ45 = sqrt[w45/(45/365)] ≈ 22.75%

Directly averaging IV gives 22%, a different answer because annualized volatility is not cumulative variance. Neither calculation is automatically valid if an earnings event lies between the maturities; event variance should be assigned to horizons that actually contain it, not smoothed across the wrong dates.

For a strike illustration, let the forward be $100, with IV 28% at strike 90, 22% at 100, and 24% at 110. A preliminary straight-line estimate at 95 is 25%. It is only a local estimate. Reprice the resulting smile and confirm Calls fall and remain convex across strike; additional neighboring nodes may reveal curvature that the two-point line misses.

Validation and model controls

Section titled “Validation and model controls”
  • Version raw quotes, timestamps, forward inputs, discount curves, dividends, filters, coordinates, knots, parameters, and solver settings.
  • Label every risk number as observed-node, interpolated, or extrapolated; reserve more model risk for wings and distant maturities.
  • Reprice the full grid and test intrinsic/upper bounds, strike monotonicity, butterfly convexity, and cross-expiry consistency.
  • Compare fitted values with executable Bid/Ask, not only midpoint residuals.
  • Use leave-one-out checks: remove a liquid node, refit, and test whether the estimate falls within its market interval.
  • Report errors separately for ATM, downside wing, upside wing, and expiry buckets; a whole-surface average hides tail failure.
  • Perturb one quote, forward, knot, or method and inspect changes in price, Delta, Vega, and higher-order Greeks.
  • Model event variance explicitly where possible; a genuine term-structure step should not be erased merely to look smooth.
  • Compare several extrapolation rules and stress tail positions. Data on both sides constrain interpolation, but not extrapolation.
  • Stop publication when small input changes cause large surface jumps or negative forward variance without an explained market structure.

Common misconceptions

Section titled “Common misconceptions”
  • “The middle IV is the average of two IVs.” Tenor interpolation should respect total variance; strike interpolation must respect shape.
  • “The smoothest curve is the most accurate.” Smoothness can erase events or hide arbitrage.
  • “Every midpoint is a valid node.” Wide, stale, or one-sided markets may have no executable midpoint.
  • “Perfect node fit proves no arbitrage.” Violations can occur between nodes or across expiries.
  • “Interpolation and extrapolation have equal confidence.” Extrapolation lacks observations on one side or both.
  • “Absolute strike is stable across maturities.” Forward carry changes economic moneyness.
  • “One good price fit guarantees stable Greeks.” Derivatives amplify quote noise, knots, and smoothing choices.
  • “A negative forward variance is a trading signal.” First investigate data, forward, units, events, and curve construction.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”