A large literature exists on techniques for extracting probability distributions for future asset prices from option prices. No definitive method has been developed however. The parametric ‘mixture of normals’, and nonparametric ‘smoothed implied volatility’ methods remain the most widespread approaches. These though are subject to estimation errors due to discretization, truncation, and noise. Recently, several authors have derived ‘model free’ formulae for computing the moments of the risk neutral density (RND) directly from option prices, without first estimating the full density. The accuracy of these formulae is studied here for the first time. The Black-Scholes formula is used to generate option prices, and error curves for the first 4 moments of the RND are computed using the ‘model-free’ formulae. It is found that, in practice, the formulae are prone to large and economically significant errors, because they contain definite integrals that can only be solved numerically. We show that without mathematically equivalent expressions with analytical solutions the formulae are difficult to deploy effectively in practice.
Bibliographical note© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
- Option pricing
- Risk neutral moments
- Risk neutral density
- Analytical solutions