Short-Time Expansions for Call Options on Leveraged ETFs Under Exponential L\'evy models With Local Volatility

TL;DR

This paper derives short-time asymptotic formulas for pricing out-of-the-money European options on leveraged ETFs under models with local volatility and jumps, providing explicit error bounds and implications for hedging strategies.

Contribution

It provides closed-form leading order expressions for LETF option prices near expiration under complex jump and volatility models, including implied volatility expansions.

Findings

Asymptotic equivalence of LETF and ETF option prices with modified parameters

Explicit error bounds for the approximations

A second order implied volatility expansion

Abstract

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.