On the Consistency of the Deterministic Local Volatility Function Model ('implied tree')

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.