Risk-Neutral Density Recovery via Spectral Analysis

In this paper, we propose a new method for estimating the conditional risk-neutral density (RND) directly from a cross-section of put option bid-ask quotes. More precisely, we propose viewing the RND recovery problem as an inverse problem. We first show that it is possible to define restricted put and call operators that admit a singular value decomposition (SVD), which we compute explicitly. We subsequently show that this new framework allows us to devise a simple and fast quadratic programming method to recover the smoothest RND whose corresponding put prices lie inside the bid-ask quotes. This method, termed the spectral recovery method (SRM), improves on other RND recovery methods in the sense that (1) it is fast and simple to implement since it requires solution of a single quadratic program, while being fully nonparametric; (2) it takes the bid-ask quotes as sole input and does not require any sort of calibration, smoothing, or preprocessing of the data; (3) it is robust to the paucity of price quotes; (4) it returns the smoothest density giving rise to prices that lie inside the bid-ask quotes (the estimated RND is therefore as well behaved as it can be); and (5) it returns a closed form estimate of the RND on the interval [0, B] of the positive real line, where B is a positive constant that can be chosen arbitrarily. We thus obtain the middle part of the RND together with its full left tail and part of its right tail. We apply this method to both real and simulated data and observe that it fares well in practice. The SRM is thus found to be a promising alternative to other RND recovery methods.