  We derive sharp bounds for the prices of VIX futures using the full
information of S&P 500 smiles. To that end, we formulate the model-free
sub/superreplication of the VIX by trading in the S&P 500 and its vanilla
options as well as the forward-starting log-contracts. A dual problem of
minimizing/maximizing certain risk-neutral expectations is introduced and shown
to yield the same value.
  The classical bounds for VIX futures given the smiles only use a calendar
spread of log-contracts on the S&P 500. We analyze for which smiles the
classical bounds are sharp and how they can be improved when they are not. In
particular, we introduce a family of functionally generated portfolios which
often improves the classical bounds while still being tractable; more
precisely, determined by a single concave/convex function on the line.
Numerical experiments on market data and SABR smiles show that the classical
lower bound can be improved dramatically, whereas the upper bound is often
close to optimal.
