How Leverage Shifts and Scales a Volatility Skew: Asymptotics for Continuous and Jump Dynamics

Abstract

To model leveraged investments such as leveraged ETFs, define the beta-leveraged product on a positive semimartingale S to be the stochastic exponential of beta times the stochastic logarithm of S.

In various asymptotic regimes, we relate rigorously the implied volatility surfaces of the beta-leveraged product and the underlying S, via explicit shifting/scaling transformations. In particular, a family of regimes with jump risk admit a shift coefficient of -3/2, unlike the previously conjectured 1/2 shift. The 1/2, we prove, holds in a family of continuous (including fBm-driven) stochastic volatility regimes at short expiry and at small volatility-of-volatility. In another regime, which does not admit a simple spatial shifting/scaling rule, we find an expiry scaling together with a spatial transformation.