I see that the justification of the need to use cubic B-splines when interpolating in the strike-IV space is to impose a convexity constraint to get rid of potential arbitrage.

I could easily understand this argument if the fit was in the strike-price space. But is there any formal proof that a convexity constraint in the strike-IV space would necessarily mean convexity in the strike-price space ?


1 Answer 1


No, and this is wrong. The implied vols (from market prices) are actually not necessarily convex but yet may be still arbitrage-free, there are many examples of this for various equities. Furthermore, preserving convexity is not necessarily enough either. In terms of implied variance $w(y)=\sigma^2 T$ as a function of log-moneyness $y=\ln\frac{K}{F}$, the no butterfly arbitrage constraint becomes: $$1 - \frac{y}{w}\frac{\partial w}{\partial y} + \frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right)\left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^{2} w}{\partial y^2} \geq 0$$ and is not a nice linear constraint as in the case of call prices. In terms of implied vol, the expression is not all that different, and less readable. The above stems from Gatheral local vol derivation, and is also explained in my book.

Preserving convexity in the implied variance would mean that only the last term is positive, which does not guarantee that all is positive.

A convexity preserving interpolation on the call prices vs strike is what matters to avoid butterfly-spread arbitrage.

Furthermore, note that such a B-spline will not be an exact interpolation, but a least-square fit, precisely because of the convexity constraint. Furthermore, you it won't be perfect either, since the associated implied vol may look strange sometimes (see Peter Jackel paper "Clamping Down on Arbitrage", or Le Floc'h and Oosterlee Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II.


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