Bounds for VIX futures given S{\&}P 500 smiles
The authors study the problem of upper and lower bounds for future contracts having the popular VIX volatility index as underlying. Although the squared VIX may be replicated with calls and puts on S\&P500, VIX itself cannot, raising the need for sharp upper and lower bounds. This is typically done via affine payoff functions that produce a totally unsatisfactory zero lower bound. The authors exploit the classical duality approach to the problem which requires the proof of absence of a duality gap which is the main theoretical result of the paper. Since the replicating portfolio that arise from this approach are difficult to implement, the authors also develop what they call ``functionally generated portfolios'' which have an easier implementation and still produce a great improvement particularly for what concerns the lower bound on the VIX future.
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