Volatility Surface Construction Techniques
Volatility Surface Construction Techniques
The implied volatility surface maps option prices across strikes and maturities into a coherent framework for pricing and risk management. Constructing an arbitrage-free surface requires careful interpolation between observed prices, extrapolation to illiquid regions, and calibration to ensure consistency with market dynamics.
Definition and Key Concepts
Volatility Surface Dimensions
| Dimension | Description | Common Parameterization |
|---|---|---|
| Strike | Option exercise price | Moneyness, delta, log-strike |
| Maturity | Time to expiration | Days, months, years |
| Spot | Current underlying price | Reference for moneyness |
Surface Characteristics
| Feature | Description |
|---|---|
| Smile | Volatility higher for OTM puts and calls |
| Skew | Asymmetric smile (usually negative for equities) |
| Term structure | How ATM vol changes with maturity |
| Wings | Volatility behavior for deep OTM options |
Moneyness Measures
| Measure | Formula | Use Case |
|---|---|---|
| Simple moneyness | K/S | Basic comparison |
| Log moneyness | ln(K/F) | More symmetric |
| Standardized | ln(K/F) / (σ√T) | Normalized across tenors |
| Delta | Option delta | Market convention |
How It Works in Practice
Raw Data Collection
Market inputs:
- Listed option prices (bid/ask)
- Underlying spot price
- Interest rates
- Dividend estimates
Data cleaning:
- Filter illiquid options (wide bid-ask, low volume)
- Remove arbitrage violations
- Check put-call parity
- Flag stale prices
Implied Volatility Extraction
For each option price: Invert Black-Scholes to find σ where: Market Price = BS(S, K, T, r, σ)
Numerical methods:
- Newton-Raphson iteration
- Bisection method
- Rational approximation (faster)
Interpolation Methods
| Method | Description | Pros | Cons |
|---|---|---|---|
| Linear | Straight line between points | Simple | Not smooth |
| Cubic spline | Piecewise cubic polynomials | Smooth | Can oscillate |
| SABR | Stochastic vol model | Market standard | Calibration needed |
| SVI | Parametric smile formula | Efficient, stable | May need adjustment |
SABR Model
Parameters:
- α (alpha): ATM volatility level
- β (beta): CEV parameter (often fixed at 0.5 or 1)
- ρ (rho): Correlation between spot and vol
- ν (nu): Volatility of volatility
SABR formula (simplified): σ(K) ≈ α × [1 + (ρν/α × ln(F/K) + ...)]
Calibration: Fit α, ρ, ν to match market smile at each tenor.
Worked Example
Building an Equity Vol Surface
Market data (S&P 500 at 5,000):
1-month options:
| Strike | Market IV |
|---|---|
| 4,500 (90%) | 22% |
| 4,750 (95%) | 18% |
| 5,000 (100%) | 15% |
| 5,250 (105%) | 14% |
| 5,500 (110%) | 16% |
3-month options:
| Strike | Market IV |
|---|---|
| 4,500 (90%) | 24% |
| 4,750 (95%) | 20% |
| 5,000 (100%) | 17% |
| 5,250 (105%) | 16% |
| 5,500 (110%) | 18% |
Step 1: Fit smile at each tenor
1-month SABR calibration:
- α = 0.15
- ρ = -0.40
- ν = 0.80
3-month SABR calibration:
- α = 0.17
- ρ = -0.35
- ν = 0.60
Step 2: Interpolate between tenors
For 2-month, 95% strike: 1-month IV at 95%: 18% 3-month IV at 95%: 20%
Linear time interpolation: 2-month IV ≈ 18% + (1/2) × (20% - 18%) = 19%
Variance interpolation (preferred): σ²(2m) × 2m = σ²(1m) × 1m + (σ²(3m) × 3m - σ²(1m) × 1m) × (2-1)/(3-1) σ²(2m) = [0.18² × 1 + 0.5 × (0.20² × 3 - 0.18² × 1)] / 2 = 0.0361 σ(2m) = 19.0%
Surface Arbitrage Checks
Calendar spread arbitrage: Total variance must increase with time. σ₁²T₁ < σ₂²T₂ for T₁ < T₂
Butterfly arbitrage: Second derivative of price with respect to strike must be positive. ∂²C/∂K² > 0 (no negative butterflies)
Vertical spread arbitrage: Call prices decrease with strike. C(K₁) > C(K₂) for K₁ < K₂
Full Surface Output
Interpolated surface (selected points):
| Strike \ Tenor | 1M | 2M | 3M | 6M | 1Y |
|---|---|---|---|---|---|
| 85% | 26% | 25% | 26% | 26% | 27% |
| 90% | 22% | 22% | 24% | 24% | 25% |
| 95% | 18% | 19% | 20% | 21% | 22% |
| 100% | 15% | 16% | 17% | 18% | 19% |
| 105% | 14% | 15% | 16% | 17% | 18% |
| 110% | 16% | 17% | 18% | 19% | 20% |
Risks, Limitations, and Tradeoffs
Model Risk
| Issue | Impact |
|---|---|
| Smile dynamics | Surface may not move as predicted |
| Extrapolation | Deep OTM vols are uncertain |
| Jump risk | Continuous models miss gaps |
| Calibration instability | Parameters can be noisy |
Data Quality Issues
| Issue | Description | Solution |
|---|---|---|
| Stale quotes | Old prices mislead calibration | Filter by timestamp |
| Wide spreads | Mid price may not be tradable | Weight by liquidity |
| Low volume | No price discovery | Use nearby strikes |
| Put-call violations | Bid-ask overlap | Arbitrage filter |
Interpolation Tradeoffs
| Method | Smoothness | Stability | Accuracy |
|---|---|---|---|
| Linear | Poor | Excellent | Moderate |
| Cubic spline | Good | Moderate | Good |
| SABR | Good | Good | Excellent (near ATM) |
| SVI | Good | Excellent | Good |
Common Pitfalls
| Pitfall | Description | Prevention |
|---|---|---|
| Over-fitting | Too many parameters | Use parsimonious models |
| Negative density | Arbitrage in surface | Check butterfly constraint |
| Wing explosion | Unrealistic deep OTM vols | Apply wing dampening |
| Time decay | Using old calibration | Recalibrate daily |
| Wrong forward | Dividend error affects moneyness | Verify forward prices |
Advanced Techniques
Local Volatility
Dupire formula: Extract local vol from surface: σ_local²(K,T) = (∂C/∂T + rK∂C/∂K) / (½K²∂²C/∂K²)
Use: Exotic option pricing requires local vol.
Stochastic Volatility
Beyond SABR:
- Heston model (mean-reverting vol)
- SABR-LMM (interest rate markets)
- Rough volatility (fractional processes)
Machine Learning Approaches
Modern techniques:
- Neural network interpolation
- Gaussian process regression
- Arbitrage-free neural networks
Checklist and Next Steps
Data preparation checklist:
- Collect bid/ask prices
- Filter by liquidity metrics
- Check put-call parity
- Verify dividend assumptions
- Calculate forwards for each tenor
Calibration checklist:
- Select model (SABR, SVI, etc.)
- Fit parameters at each tenor
- Check arbitrage constraints
- Verify smile shape
- Document calibration results
Validation checklist:
- Backtest pricing accuracy
- Compare to market prices
- Check extrapolation reasonableness
- Monitor stability over time
- Review with trading desk
Related articles:
- For dispersion trades, see Dispersion Trades Using Options
- For vol selling, see Managing Volatility Premium Selling Strategies