SVI Parameterization

Advanced Topic

This page covers the mathematical details of SVI - the industry-standard model for vol surface interpolation. If you're looking for intuition on how vol surfaces work, start with Vol Surface or How Surfaces Are Built.

SVI (Stochastic Volatility Inspired) is a parametric model for the volatility smile, widely used for interpolation and extrapolation.

The SVI Formula

The raw SVI parameterization expresses total implied variance as:

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

Where:

• $w(k) = \sigma^2(k) \cdot T$ is total implied variance
• $k = \ln(K/F)$ is log-moneyness
• $a, b, \rho, m, \sigma$ are the five parameters

Parameter Interpretation

Parameter	Range	Controls
$a$	$a > 0$	Variance level (vertical shift)
$b$	$b \geq 0$	Slope magnitude
$\rho$	$-1 < \rho < 1$	Skew direction and magnitude
$m$	Any real	Horizontal shift of minimum
$\sigma$	$\sigma > 0$	Curvature (ATM smile convexity)

Skew ($\rho$)

• $\rho < 0$: Put skew (left wing higher)
• $\rho > 0$: Call skew (right wing higher)
• $\rho = 0$: Symmetric smile

Curvature ($\sigma$)

• Small $\sigma$: Sharp V-shape
• Large $\sigma$: Smooth U-shape

Converting to IV

From total variance $w(k)$ to implied volatility:

$$\sigma(k, T) = \sqrt{\frac{w(k)}{T}}$$

Jump-Wing (JW) Parameterization

An alternative parameterization more intuitive for traders:

$$v_t, \psi_t, p_t, c_t, \tilde{v}_t$$

Where:

• $v_t$ = ATM variance
• $\psi_t$ = ATM skew
• $p_t$ = Put wing slope
• $c_t$ = Call wing slope
• $\tilde{v}_t$ = Minimum variance

Fitting SVI

To Market Data

1. Collect IV observations at multiple strikes
2. Convert to total variance: $w_i = \sigma_i^2 \cdot T$
3. Minimize weighted least squares:

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

Weights $\omega_i$ often based on vega or bid-ask spread.

Practical Tips

• Initialize with SABR or simple estimates
• Enforce constraints during optimization
• Check for calendar arbitrage across expiries

Why SVI?

Advantage	Explanation
Parsimonious	Only 5 parameters per slice
Flexible	Can fit most observed smile shapes
Extrapolates well	Behaves sensibly in wings
Arbitrage control	Constraints are well-understood

See Also

• Vol Surface
• Skew
• Reading Volatility: Smile & Smirk