Risk Neutral Density Estimation: Looking at the Tails

Abstract

Previous estimation results of risk neutral densities explain in rather general terms that the tails of the resulting distribution “look fat,” and a way has to be found to model the tails of the estimated distribution. We use deep out of the money S&P 500 index options to examine model mispricing of the tails of daily estimated risk neutral densities. Out-of-sample tests show that model mispricing increases as one moves farther into the tails of the distribution. Across most moneyness groups, model mispricing increases as the option reaches maturity. We compare two curve fitting methods that have been proposed in the literature to estimate risk neutral densities. The first method interpolates with a fourth order spline and attaches tails from the General Extreme Value distribution (Figlewski 2010). The second method extends the available implied volatility space by balancing smoothness and fit of the estimated risk neutral density (Jackwerth 2004). Fitting a fourth order spline produces a closer fit to the observed implied volatilities. Examining the ability to replicate the implied volatility with the complete estimated option-implied risk neutral density by looking at mean root-mean-squared error, the method by Jackwerth (2004) resulted in lower in- and out-of-sample model mispricing, except for the deepest out of the money put options.

TOPICS: Tail risks, options

Key Findings

• • This paper compares two methods from the curve fitting literature to estimate option-implied risk neutral densities and looks at the accuracy to recover implied volatilities.

• • Model mispricing, measured by the root-mean-squared error, increases for deeper out of the money options.

• • Model mispricing increases as the option reaches its maturity across most out-of-sample moneyness groups.