The Impact of the Prior Density on a Minimum Relative Entropy Density: A Case Study with SPX Option Data

TL;DR

This paper investigates how the choice of prior density affects the maximum relative entropy density estimation for option prices, demonstrating that in liquid markets the prior's influence diminishes with more data.

Contribution

It generalizes existing algorithms to the relative entropy case and analyzes the impact of prior densities in different market scenarios, including liquid markets with many strikes.

Findings

Prior choice significantly affects densities with limited data.

In liquid markets, the prior's influence on digital prices vanishes.

The fair variance swap rate becomes independent of the prior as data increases.

Abstract

We study the problem of finding probability densities that match given European call option prices. To allow prior information about such a density to be taken into account, we generalise the algorithm presented in Neri and Schneider (2011) to find the maximum entropy density of an asset price to the relative entropy case. This is applied to study the impact the choice of prior density has in two market scenarios. In the first scenario, call option prices are prescribed at only a small number of strikes, and we see that the choice of prior, or indeed its omission, yields notably different densities. The second scenario is given by CBOE option price data for S&P500 index options at a large number of strikes. Prior information is now considered to be given by calibrated Heston, Schoebel-Zhu or Variance Gamma models. We find that the resulting digital option prices are essentially the same as those given by the (non-relative) Buchen-Kelly density itself. In other words, in a sufficiently liquid market the influence of the prior density seems to vanish almost completely. Finally, we study variance swaps and derive a simple formula relating the fair variance swap rate to entropy. Then we show, again, that the prior loses its influence on the fair variance swap rate as the number of strikes increases.