Why Implied-Volatility Convexity Does Not Ensure Butterfly Arbitrage Freedom
Summary
The document challenges the idea that fitting a convex curve in strike-implied-volatility space guarantees an arbitrage-free option surface. Implied volatility need not itself be convex for market prices to be free of arbitrage, and convexity of implied variance alone does not ensure the required no-butterfly condition. The relevant constraint in implied variance as a function of log-moneyness includes both first- and second-derivative terms, so it is not a simple convexity test.
For avoiding butterfly-spread arbitrage, convexity preservation in call price as a function of strike is the directly relevant condition. The note also cautions that constrained B-spline fitting is generally a least-squares fit rather than exact interpolation, and the resulting implied volatility can still look unusual. It supplies the form of the variance-based inequality but no worked numerical example; the discussion is limited to the arbitrage implications of interpolation constraints.
Key ideas
- Implied volatility need not be convex for option prices to be arbitrage-free.
- Convexity in implied variance does not by itself guarantee the no-butterfly condition.
- The variance-based arbitrage condition depends on both first and second derivatives.
- Convexity of call prices with respect to strike is the relevant shape condition for butterfly arbitrage.
- Constrained B-spline fitting may alter market quotes and can still produce unattractive implied-volatility shapes.
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# B-splines: convexity in IV/Price
# B-splines: convexity in IV/Price
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 ?
## Answer by jherek (score 6, accepted)
https://quant.stackexchange.com/a/48612
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.