  In this article, we consider the small-time asymptotics of options on a
\emph{Leveraged Exchange-Traded Fund} (LETF) when the underlying Exchange
Traded Fund (ETF) exhibits both local volatility and jumps of either finite or
infinite activity. Our main results are closed-form expressions for the leading
order terms of off-the-money European call and put LETF option prices, near
expiration, with explicit error bounds. We show that the price of an
out-of-the-money European call on a LETF with positive (negative) leverage is
asymptotically equivalent, in short-time, to the price of an out-of-the-money
European call (put) on the underlying ETF, but with modified spot and strike
prices. Similar relationships hold for other off-the-money European options. In
particular, our results suggest a method to hedge off-the-money LETF options
near expiration using options on the underlying ETF. Finally, a second order
expansion for the corresponding implied volatility is also derived and
illustrated numerically.
