# Empirical Returns of Leveraged ETF Options

The popularity of ETFs has led to increased trading of options written on ETFs and leveraged ETFs (LETFs).

Traded options have different strikes and expiration dates. For each expiration date, we can compare returns of LETF calls and puts with different strikes and *leverage ratios*.

Option prices can be considered as a function of *moneyness*, which is defined by the ratio of the strike over the spot LETF price. As such, at-the-money options have moneyness of value 1.

The prices of options on LETFs with different leverage ratios are generally not comparable at the same strike or moneyness. The reason is that LETFs can differ greatly in their dynamics and dependence on the reference index. For example,

- an out-of-the-money (OTM) put on an LETF with leverage ratio β ≥ 1 is a bearish position (on the reference index) with low moneyness
- an OTM put on an LETF with β ≤ −1 also has low moneyness but is a bullish position on the reference index.

The availability of multiple leverage ratios for the LETF options leads us to investigate the link between these markets. Which pair of options represents the same bet on the reference index, measured in terms of expected returns?