Definition
Implied Volatility (IV) is the volatility value that, when plugged into an option pricing model (like Black-Scholes), yields the market price of that option. The IV Surface is a 3D plot or function that shows the implied volatility for every available strike price (K) and expiry date (T).
Key Slices of the Surface
- Strike Dimension (Smile/Smirk): Shows how IV changes across different strike prices for a fixed expiry. Equity markets typically show a "smirk" or "skew," where IV is higher for lower strikes (out-of-the-money puts) due to higher demand for crash protection.
- Expiry Dimension (Term Structure): Shows how IV for a given moneyness (e.g., at-the-money) changes across different expiry dates.
Example (Equity Smirk)
For a 3-month option on a stock with a spot price of $100:
- Strike K=120 (OTM call) → IV = 18%
- Strike K=100 (ATM) → IV = 20%
- Strike K=80 (OTM put) → IV = 28% (This demonstrates the skew/smirk)
An IV surface cannot be arbitrary; it must satisfy certain conditions to prevent static arbitrage opportunities.
A. No Calendar Arbitrage (Across Maturities)
Total implied variance (σ²T) must be non-decreasing with time (T) for a fixed strike. If violated, one could execute a calendar spread for a risk-free profit.
Practical Test: For two maturities T1 < T2, you must ensure that (σ²(K, T1) * T1) ≤ (σ²(K, T2) * T2).
B. No Butterfly Arbitrage (Across Strikes)
The call price as a function of strike, C(K), must be convex. This means the price of a butterfly spread must be non-negative. Violations of this convexity in the implied volatility curve would allow for static butterfly arbitrage.
Since market quotes are only available for discrete strikes and expiries, a surface must be constructed by interpolating between points and extrapolating to the wings.
Popular Approaches
- Local Interpolation: Using methods like cubic splines on total variance versus log-moneyness. Naive splines can introduce arbitrage.
- Parametric Forms: Using arbitrage-free (or nearly so) models like SVI (Stochastic Volatility Inspired) or SABR to generate a smooth curve.
SVI (Stochastic Volatility Inspired)
SVI is a popular parametric model that expresses total variance as a function of log-moneyness. It is defined by 5 parameters that control the level, slope, skew, and curvature of the smile, and it tends to produce realistic wing behavior.
Goal
To choose model parameters (e.g., for Heston, SABR, or SVI) so that the model's output prices or implied volatilities match the market's prices or IVs as closely as possible.
Objective Function
The calibration process typically minimizes the sum of squared errors between model and market. Best practice is to use vega-weighted price errors, which gives more weight to options that are more sensitive to volatility (i.e., at-the-money options).
Optimization Methods
A hybrid approach is often used, starting with a global search algorithm (like differential evolution) to find a good starting point and then refining it with a local optimizer (like Levenberg-Marquardt).
SABR Calibration (Swaption Volatility)
The SABR model is widely used in interest rate markets to fit the smile of swaptions. It has parameters for volatility level (α), elasticity (β), correlation (ρ), and vol-of-vol (ν). Calibration involves fitting these parameters to the market-implied volatilities for a given expiry, often by fixing β and using the ATM vol to pin down α, then fitting ρ and ν to the skew and curvature.
Heston Calibration (Equity Volatility)
The Heston model is a stochastic volatility model used for equities. Calibration involves finding the set of model parameters (mean reversion speed κ, long-term variance θ, vol-of-vol ξ, correlation ρ, and initial variance v₀) that minimizes the pricing errors across all available strikes and expiries, often using fast pricing methods like FFT (Fast Fourier Transform).
These are two conventions describing how the volatility surface is assumed to move when the underlying spot price changes.
Sticky-Strike
The implied volatility for a fixed, absolute strike price (e.g., K=$100) remains constant as the spot price moves. The surface shifts horizontally with the spot, but the IV at a specific strike does not change.
Sticky-Delta
The implied volatility for a fixed moneyness level (often quoted by delta, e.g., a 25-delta put) remains constant as the spot price moves. As the spot moves, the strike corresponding to that delta also moves. This is common practice for quoting FX volatility.
Example
Spot is $100, and the 25-delta put is at strike K=$90 with IV=30%. If the spot moves to $105:
- Under sticky-strike: The IV at K=$90 remains 30%.
- Under sticky-delta: The IV for whatever the *new* 25-delta put strike is (e.g., K=$92) remains 30%. The IV at the fixed strike of K=$90 will have changed.
Practical Pitfalls
- Weighting: Always weight calibration errors by vega to avoid giving too much importance to deep out-of-the-money options.
- Local Minima: Use multi-start or global search methods before local refinement to avoid getting stuck in a suboptimal solution.
- Overfitting: Use regularization to penalize large day-to-day parameter changes and ensure stability.
- Illiquid Data: Down-weight or remove quotes for illiquid strikes that produce noisy IVs.
Final Interview Tips
Be able to explain what the vol surface is, state the key arbitrage constraints (calendar and butterfly), describe a parametric model like SVI, explain the calibration objective (vega-weighted errors), and distinguish between sticky-delta and sticky-strike conventions.