Fengler: non-parametric arb-free vol surface smoothing via Quadratic Programming (QP)
Given noisy quoted call prices, find the smoothest arb-free surface that's closest to them. A constrained Quadratic Program (QP) — minimise a quadratic objective subject to linear constraints — does the work; no parametric family is assumed.
An options market-maker at any serious flow shop marks hundreds of strike-maturity combinations every session. When raw market quotes contain butterfly or calendar violations, the desk needs a smoothing layer that enforces no-arbitrage before any model calibration can run. Parametric fits like SVI break when the real smile has a shape the family cannot reproduce. Fengler (2009) solves this without choosing a smile shape: represent the surface as a cubic spline, then run a Quadratic Program (QP) to find the spline coefficients that best fit the quotes while satisfying no-arb constraints. One QP per maturity slice, solvable in milliseconds — everyday infrastructure at any options desk. This tutorial works through the butterfly constraint as a linear inequality and sets up the QP.
✓ Intro · expand
Try first (productive failure)
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60s
✓ Try first · expand
Worked example
Write the butterfly arb-free constraint as a linear inequality on a vector of 5 adjacent call prices $C = (C_1, C_2, C_3, C_4, C_5)$ at strikes $K_1, K_2, K_3, K_4, K_5$ with uniform spacing $h$. (a) Write the second-difference matrix $D_2$ for an interior butterfly constraint at $K_3$. (b) Apply it to a test vector $C = (13, 10, 8, 6, 5)$ and check whether butterfly arb is satisfied at $K_3$.
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
Fengler's smoothing is non-parametric and arb-free by construction. So why does it not replace SVI/SABR in production? What does parametric fitting offer that QP smoothing doesn't?