SVI (Stochastic Volatility Inspired): Gatheral's 5-parameter slice fit
Five parameters per maturity: ATM level, wing steepness, skew tilt, smile centre, smile curvature. Fits any reasonable single-slice equity smile in milliseconds.
An options market-maker marks hundreds of strike-maturity combinations every day and needs to refresh the whole surface with every tick. SVI (Gatheral 2004) solves the per-slice fitting problem: five numbers per maturity that fit any reasonable equity-index smile in milliseconds. The five parameters each have a direct meaning: $a$ pins the ATM vol level, $b$ sets how steeply the wings rise, $\rho$ tilts the skew left or right, $m$ shifts the smile centre, and $\sigma$ controls the curvature around ATM. The asymptotic wings are $b(1 + \rho)$ on the right and $b(1 - \rho)$ on the left — both must sit below Lee’s moment bound to avoid calendar arb. This tutorial calibrates those five parameters and checks the wing bounds.
β Intro Β· expand
Try first (productive failure)
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β Try first Β· expand
Worked example
An SPX-style SVI fit at maturity $T = 1$ year has parameters $a = 0.04$, $b = 0.10$, $\rho = -0.4$, $m = 0$, $\sigma = 0.10$. (a) Compute the total variance $w(0)$ at the at-the-money log-moneyness $k = 0$. (b) Convert to implied vol $\sigma_{BS}(0)$. (c) Compute the right-wing slope $b(1 + \rho)$ and check it sits well inside Lee's bound of $2$.
β Worked example Β· expand
Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
β Practice Β· expand
Reflection
SVI has five parameters; that's a lot of freedom for one smile slice. What's the trade-off between fitting accuracy and over-parameterisation, and how does the SSVI extension (next tutorial) reduce the parameter count?