Colombia University

The growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). Motivated by a number of empirical market observations, we study the relationship between the ETF and LETF implied volatility surfaces under general stochastic volatility models. Analytic approximations for prices and implied volatilities are derived for LETF ​options, along with rigorous error bounds. In these price and IV expressions, we identify their non-trivial dependence on the leverage ratio. Moreover, we introduce a "moneyness scaling" procedure to enhance the comparison of implied volatilities across leverage ratios, and test it with empirical price data.

The Schramm-Loewner evolution (SLE(\kappa)) is a family of random fractal curves that arise in a natural way as scaling limits of interfaces in critical models in statistical physics. The SLE curves are constructed by solving the Loewner differential equation driven by a scaled Brownian motion. We will give an overview of some of the almost sure properties of SLE curves, concentrating in particular on properties that can be derived by studying the the geometry of growing curve locally at the tip. We will discuss a multifractual spectrum of harmonic measure at the tip, regularity in the capacity parameterization, and continuity of the curves as the \kappa-parameter is varied while the driving Brownian motion sample is kept fixed.

This is based on joint work with Greg Lawler, and with Steffen Rohde and Carto Wong.