Journal of Risk Model Validation

Risk-neutral densities: advanced methods of estimating nonnormal options underlying asset prices and returns

André Santos and João Guerra

  • We tested the accuracy and stability of four non-structural models in estimating the "true" Risk-Neutral Density (RND) functions from option prices in demanding and realistic conditions.
  • The "true" RND was generated using the CGMY Gamma-OU model, a structural model able to generate realistic and flexible "true" RNDs.
  • We considered different scenarios that incorporate the characteristics of the USD/BRL currency options for each month between October 2006 and February 2010, a period characterized by important market events.
  • The Density Functional Based on Confluent Hypergeometric function (DFCH) was the most accurate "true" RND estimator during the stressed period (September 2008 to December 2008). However, the Mixture of Lognormal Distributions (MLN) was slightly better during the "normal period", before the subprime crisis became apparent in August 2007.
  • Additionally, we assessed the goodness of fit of the Subjective Probability Density Function (SPDF) to the observed returns of the USD/BRL and showed that the SPDF may be a useful risk management tool.

Option prices can be used to extract the implied risk-neutral density functions of the future underlying asset prices and returns. These market expectations provide valuable information that can be helpful to policy makers and investors. We tested the accuracy and stability of four nonstructural models in estimating the "true" risk-neutral density functions from option prices: the density functional based on the confluent hypergeometric function, the mixture of lognormal distributions, the smoothed implied volatility smile and Edgeworth expansions. The "true" risk-neutral density is unknown, so it was generated using the Carr–Geman–Madan–Yor (CGMY) Gamma-Ornstein–Uhlenbeck (Gamma-OU) model, a structural model able to generate flexible "true" risk-neutral densities. We observed that the density functional based on the confluent hypergeometric function and mixture of lognormal distributions outperformed the smoothed implied volatility smile and the Edgeworth expansion models in capturing the "true" risk-neutral density. The smoothed implied volatility smile had the best performance in terms of stability. This work aims to be more exhaustive than previous studies, by testing a wider variety of nonstructural models. In addition, we tested the effectiveness of these models in more demanding conditions that are closer to reality. In fact, the use of the CGMY Gamma-OU model to build the "true" risk-neutral density allows densities with a higher probability of extreme events and more leptokurtosis, and also generates sudden, discontinuous moves in prices, which has advantages over the diffusion models used in previous studies.

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