A Note on "A Family of Maximum Entropy Densities Matching Call Option Prices"

TL;DR

This paper revisits the problem of recovering maximum entropy densities from option prices, introducing a more stable numerical approach using the Langevin function and providing convergence estimates.

Contribution

It offers a new, more stable numerical method for density recovery from option prices, improving upon previous approaches with detailed convergence analysis.

Findings

The inversion problem can be reformulated using the Langevin function.

The new method demonstrates enhanced numerical stability.

Convergence estimates depend on a parameter m, with sharper bounds provided.

Abstract

In Neri and Schneider (2012) we presented a method to recover the Maximum Entropy Density (MED) inferred from prices of call and digital options on a set of n strikes. To find the MED we need to numerically invert a one-dimensional function for n values and a Newton-Raphson method is suggested. In this note we revisit this inversion problem and show that it can be rewritten in terms of the Langevin function for which numerical approximations of its inverse are known. The approach is very similar to that of Buchen and Kelly (BK) with the difference that BK only requires call option prices. Then, in continuation of our first paper, we presented another approach which uses call prices only and recovers the same density as BK with a few advantages, notably, numerical stability. This second paper provides a detailed analysis of convergence and, in particular, gives various estimates of how far (in different senses) the iterative algorithm is from the solution. These estimates rely on a constant m > 0. The larger m is the better the estimates will be. A concrete value of m is suggested in the second paper, and this note provides a sharper value.