Short-Time Asymptotics for Options on Leveraged ETFs under Exponential Lévy Models with Local Volatility

In this talk, the speaker considers the small-time asymptotics of options on a Leveraged Exchange-Traded Fund (LETF) when the underlying Exchange-Traded Fund (ETF) exhibits both local volatility and Lévy jumps of either finite or infinite activity. We show that leverage modifies the drift, volatility, jump intensity, and jump distribution of an LETF in addition to inducing the possibility of default, even when the underlying ETF price remains strictly positive. The main results are closed-form expressions for the leading order terms of off-the-money European calls and put LETF option prices near expiration with explicit error bounds. These results show that the cost of an out-of-the-money European call on an LETF with positive (negative) leverage is asymptotically equivalent, in a 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. These observations, in turn, suggest a method to hedge off-the-money LETF options near expiration using options on the underlying ETF. Finally, the speaker derives a second-order expansion for the implied volatility of an off-the-money LETF option and shows both analytically and numerically how this is affected by leverage. This is the joint work with José E. Figueroa-López and Matthew Lorig.