Leveraged {ETF} implied volatilities from {ETF} dynamics

TL;DR

This paper develops a theoretical framework for modeling and approximating implied volatilities of options on ETFs and LETFs using local-stochastic volatility models, providing practical scaling methods and testing in popular models.

Contribution

It introduces a closed-form approximation for option prices and implied volatilities on ETFs and LETFs within local-stochastic volatility models, including error bounds and scaling procedures.

Findings

The approximation performs well in CEV, Heston, and SABR models.

Scaling procedures effectively compare implied volatilities across leverage ratios.

Error bounds validate the approximation's accuracy.

Abstract

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.