Implied Volatility Estimation via ℓ₁ Trend Filtering

Abstract

The Risk Neutral Density (RND) is a probability distribution over the stock price at option expiration, which can be extracted from a set of options prices. The VIX index extracted from SPX options with 30-day average maturity, for example, is an estimate of volatility under the RND for the S&P 500 Index. In recent years, there has been a sharp increase in research on RNDs to explore and exploit the risk preferences and returns expectations they contain. The standard extraction procedure involves fitting a (very) smooth function to the implied volatility (IV) smile using options that span the full range of strike prices at a given maturity, and then using interpolation to create a dense set of artificial IVs at prices between the strikes available in the market. There are different ways to do this, with no clear theoretical reason to pick one over another. In this article, Crespo and Huang introduce a new technique, ℓ₁ trend filtering, which they show to be easy to implement. In a Monte Carlo exercise and also using data on S&P index options and several individual stocks, ℓ₁ trend filtering demonstrates greater accuracy than more familiar methods.