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

For a valid (arbitrage-free) SVI surface, several constraints must hold. These extend the fundamental put-call parity relationship to the entire surface:

Butterfly constraint (no negative butterfly spreads):

$$b(1 + |\rho|) \leq \frac{4}{T}$$

Calendar constraint (variance must increase with time): For slices at $T_1 < T_2$:

$$w_1(k) \leq w_2(k) \quad \forall k$$

Roger Lee's moment formula (bounds on wings):

$$\limsup_{k \to \pm\infty} \frac{w(k)}{|k|} \leq 2$$

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

SSVI extends SVI to model the entire surface consistently:

$$w(k, \theta_t) = \frac{\theta_t}{2}\left( 1 + \rho\phi(\theta_t)k + \sqrt{(\phi(\theta_t)k + \rho)^2 + (1-\rho^2)} \right)$$

Where:

• $\theta_t$ = ATM total variance at time $t$
• $\phi(\theta)$ = function controlling term structure of skew

This guarantees calendar spread arbitrage-free by construction.

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