Implied Volatility of Leveraged ETF Options

Abstract

This paper studies the problem of understanding implied volatilities from options written on leveraged exchanged-traded funds (LETFs), with an emphasis on the relations between LETF options with different leverage ratios. We first examine from empirical data the implied volatility skews for LETF options based on the S&P 500. In order to enhance their comparison with non-leveraged ETFs, we introduce the concept of moneyness scaling and provide a new formula that links option implied volatilities between leveraged and unleveraged ETFs. Under a multiscale stochastic volatility framework, we apply asymptotic techniques to derive an approximation for both the LETF option price and implied volatility. The approximation formula reflects the role of the leverage ratio, and thus allows us to link implied volatilities of options on an ETF and its leveraged counterparts. We apply our result to quantify matches and mismatches in the level and slope of the implied volatility skews for various LETF options using data from the underlying ETF option prices. This reveals some apparent biases in the leverage implied by the market prices of different products, long and short with leverage ratios two times and three times.

Key Words:

• Exchange-traded funds
• leverage
• implied volatility
• stochastic volatility
• moneyness scaling

Acknowledgements

We thank Andrew Ledvina for initial research assistance with the data, and Marin Nitzov for bringing the problem to our attention.

Notes

1 These are parameters that group together some of the stochastic volatility model parameters. We refer to Fouque et al. (2011).

2 See, for instance, ‘Volatility, Thy Name Is E.T.F.’ by Andrew Ross Sorkin, New York Times, 10 October 2011.