A Cross-Sectional Characterization of Risk-Neutral Option-Implied Probability Distributions

Abstract

A fundamental problem in financial engineering is modelling the distribution of future stock returns. The widely accepted Black-Scholes-Merton model for option pricing assumes underlying stock prices follow a Geometric Brownian Motion (GBM) in con- tinuous time. This price model itself assumes the price of underlying assets move with constant drift and volatility. This implies that future stock returns follow a nor- mal distribution while future stock prices follow a lognormal distribution. However, this is not necessarily what we observe in the market. First, because empirical stock return distributions deviate from normal distributions. And second, because options and derivatives markets imply separate probability distributions for the underlying stocks that frequently contrast with distributions under GBM. These are known as option-implied probability distributions. In this paper, we do a comparative and cross-sector analysis of option-implied distributions for US equities with high option liquidity over a two-year period from 2014-2016. We characterize the distributions using summary statistics, higher-order moments, and parameters based on distribution fitting. Additionally, we use machine learning techniques to cluster and classify the stocks into major sectors and super sectors of the economy. Our main contributions are a) the robust modelling of the Burr distribution family as a close fit for option-implied distributions, and b) an identification of sectors that can be well classified using these distributions.