  The growth of the exhange-traded fund (ETF) industry has given rise to the
trading of options written on ETFs and their leveraged counterparts {(LETFs)}.
We study the relationship between the ETF and LETF implied volatility surfaces
when the underlying ETF is modeled by a general class of local-stochastic
volatility models. A closed-form approximation for prices is derived for
European-style options whose payoff depends on the terminal value of the ETF
and/or LETF. Rigorous error bounds for this pricing approximation are
established. A closed-form approximation for implied volatilities is also
derived. We also discuss a scaling procedure for comparing implied volatilities
across leverage ratios. The implied volatility expansions and scalings are
tested in three well-known settings: CEV, Heston and SABR.
