  We show that the frequent claim that the implied tree prices exotic options
consistently with the market is untrue if the local volatilities are subject to
change and the market is arbitrage-free. In the process, we analyse -- in the
most general context -- the impact of stochastic variables on the P&L of a
hedged portfolio, and we conclude that no model can a priori be expected to
price all exotics in line with the vanilla options market. Calibration of an
assumed underlying process from vanilla options alone must not be overly
restrictive, yet still unique, and relevant to all exotic options of interest.
For the implied tree we show that the calibration to real-world prices allows
us to only price vanilla options themselves correctly. This is usually
attributed to the incompleteness of the market under traditional stochastic
(local) volatility models. We show that some `weakly' stochastic volatility
models without quadratic variation of the volatilities avoid the incompleteness
problems, but they introduce arbitrage. More generally, we find that any
stochastic tradable either has quadratic variation -- and therefore a
$\Ga$-like P&L on instruments with non-linear exposure to that asset -- or it
introduces arbitrage opportunities.
