Crooked Volatility Smiles: Evidence from Leveraged and Inverse ETF Options

We find that leverage in exchange traded funds (ETFs) can affect the “crookedness” of volatility smiles. This observation is consistent with the intuition that return shocks are inversely correlated with volatility shocks – resulting in more expensive out-of-the-money put options and less expensive out-of-the-money call options. We show that the prices of options on leveraged and inverse ETFs can be used to better calibrate models of stochastic volatility. In particular, we study a sextet of leveraged and inverse ETFs based on the S&P 500 index. We show that the Heston model (Heston , 1993) can reproduce the crooked smiles observed in the market price of options on leveraged and inverse leveraged ETFs. We show further that the model predicts a leverage dependent moneyness, consistent with empirical data, at which options on positively and negatively leveraged ETFs have the same price. Finally, by analyzing the asymptotic behavior for the implied variances at extreme strikes, we observe an approximate symmetry between pairs of LETF smiles empirically consistent with the predictions of the Heston model.