How Vol Surfaces Are Built

Ever wondered how exchanges go from raw option prices to a smooth volatility surface? This page explains the process.

Key Insight

You don't actually see a continuous surface in the market. You see scattered quotes. The "surface" is constructed from those quotes using interpolation.

The Problem

Option markets don't quote every possible strike and expiry. You see something like:

Strike	7 DTE	14 DTE	30 DTE	60 DTE
$85k	$120	$340	--	--
$90k	$450	$820	$1,400	--
$95k	$1,200	--	$2,800	$4,200
$100k	$3,500	$4,800	$6,200	--
$105k	$1,800	$3,100	--	$5,800
$110k	$650	--	$2,100	--

Notice:

• Many cells are empty (--) - no trades at those strike/expiry combinations
• Even where prices exist, they're option prices in dollars, not IV

To get a vol surface, we need to:

1. Convert prices to IV
2. Fill in the gaps
3. Make sure it's internally consistent

Step 1: Price → IV

For each quoted option price, we solve for the volatility that makes Black-Scholes match:

Market Price = $3,500 (for ATM 7d call)

Black-Scholes(S=100k, K=100k, T=7/365, r=5%, σ=?) = $3,500

Solve for σ → σ = 52%

This is done numerically (there's no closed-form solution). The result is the implied volatility for that specific option.

After this step, we have scattered IV points:

Strike	7 DTE	14 DTE	30 DTE	60 DTE
$85k	68%	62%	--	--
$90k	58%	55%	52%	--
$95k	53%	--	50%	48%
$100k	52%	50%	49%	--
$105k	50%	48%	--	47%
$110k	51%	--	48%	--

Step 2: Interpolation

Now we need to fill the gaps. This is where it gets interesting.

Why Not Just Draw Lines?

Simple linear interpolation (drawing straight lines between points) can create problems:

Problem 1: Arbitrage opportunities

If the interpolated surface is inconsistent, traders could construct risk-free profits (violating no-arbitrage conditions). For example:

• If a $97k call is priced too cheap relative to$95k and $100k
• You could buy the $97k, sell a blend of$95k and $100k, and lock in profit

Problem 2: Nonsensical Greeks

Bad interpolation can produce:

• Negative gamma (should never happen for long options)
• Wildly oscillating delta
• Unrealistic skew

Professional Solutions

Most systems use parametric models that guarantee a well-behaved surface:

Method

How It Works

Pros

Cons

Linear

Straight lines between points

Simple, fast

Can create arbitrage, looks ugly

Cubic Spline

Smooth curves through points

Looks smooth

Can oscillate wildly, arbitrage issues

SVI

5-parameter model per expiry

Arbitrage-free (with constraints), flexible

Need to fit parameters

SABR

Stochastic alpha-beta-rho model

Captures dynamics, good for rates

Complex, 4 parameters

Local Vol

Derive local vol surface from options

Exact fit to market

Can be unstable, forward-looking issues

Hypercall uses SVI - it's the industry standard for crypto/equity options because it:

• Has a simple, interpretable parameterization
• Can be constrained to guarantee no arbitrage
• Extrapolates sensibly to wings

For each expiry, we fit 5 parameters $(a, b, \rho, m, \sigma)$ to the observed IV points:

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

Where $w$ is total variance and $k$ is log-moneyness.

The fitting process:

1. Collect all IV observations at this expiry
2. Convert to total variance: $w_i = \text{IV}_i^2 \times T$
3. Minimize weighted squared error:

$$\min_{a,b,\rho,m,\sigma} \sum_i \omega_i \left( w_i - w(k_i) \right)^2$$

4. Apply arbitrage constraints during optimization
5. Repeat for each expiry

Weights ($\omega_i$) are typically based on:

• Vega (more weight to ATM options)
• Bid-ask spread (more weight to tight spreads)
• Volume/open interest

Step 3: Extrapolation

What about deep OTM options where there's no market data?

The challenge: No one is trading the $50k put or$200k call. But we still need to show something.

Solutions:

• Use the fitted model parameters to extend smoothly
• Apply wing extrapolation rules (e.g., vol can't go below a floor)
• Show wider bid-ask spreads for extrapolated regions
• Cap how far extrapolation goes

What to watch for:

• Quotes in deep wings are uncertain
• Greeks in deep wings are approximations
• Don't over-trust pricing in illiquid strikes

Step 4: Calendar Consistency

The surface must be consistent across expiries. Specifically:

Total variance must increase with time:

$$\sigma(K, T_1)^2 \times T_1 \leq \sigma(K, T_2)^2 \times T_2 \quad \text{for } T_1 < T_2$$

If this is violated, you could trade calendar spreads for free money.

In practice:

• Fit each expiry's skew independently (SVI per slice)
• Then adjust to ensure calendar consistency
• Or use SSVI (surface SVI) which guarantees consistency by construction

What You Actually See

When you look at a vol surface on Hypercall or any exchange:

1. Quoted strikes/expiries: Direct market data converted to IV
2. Unquoted points: Interpolated using SVI or similar
3. Deep wings: Extrapolated with uncertainty
4. The surface: A smooth visualization of all of the above

The smoothness is constructed, not observed. Keep this in mind when:

• Trading illiquid strikes (the price you see may be modeled, not market)
• Analyzing wing behavior (less certainty in deep OTM)
• Comparing surfaces across platforms (different interpolation = different surfaces)

For Quants: Going Deeper

If you want to dive into the math of vol surface construction:

• SVI Parameterization: See SVI Reference for the full model
• Arbitrage Constraints: Butterfly, calendar, and call spread constraints
• SSVI: The surface extension that guarantees calendar-free by construction
• Local Volatility: The dual representation $\sigma_{\text{local}}(K,T)$

Most quant firms have proprietary surface construction methods that combine multiple models, add regularization, and incorporate real-time market microstructure.

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Related:

• Vol Surface - Reading and interpreting the surface
• SVI Parameterization - The mathematical model
• Implied Volatility - What IV actually is