A robust implementation of a Dupire type local volatility model is an
important issue for every option trading floor. Typically, this (inverse)
problem is solved in a two step procedure : (i) a smooth parametrization of the
implied volatility surface; (ii) computation of the local volatility based on
the resulting call price surface. Point (i), and in particular how to
extrapolate the implied volatility in extreme strike regimes not seen in the
market, has been the subject of numerous articles, starting with Lee (Math.
Finance, 2004).

We consider the maximization of the long-term growth rate in the
Black-Scholes model under proportional transaction costs as in Taksar, Klass
and Assaf [Math. Oper. Res. 13, 1988]. Similarly as in Kallsen and Muhle-Karbe
[Ann. Appl. Probab., 20, 2010] for optimal consumption over an infinite
horizon, we tackle this problem by determining a shadow price, which is the
solution of the dual problem. It can be calculated explicitly up to determining
the root of a deterministic function.

In the LIBOR market model, forward interest rates are log-normal under their
respective forward measures. This note shows that their distributions under the
other forward measures of the tenor structure have approximately log-normal
tails.