Implied risk neutral densities from option prices: hypergeometric, spline, lognormal and edgeworth functions

This work examines the stability and accuracy of four di¤erent methods to estimate Risk-Neutral Density functions (RNDs) using European options. These methods are the Double-Lognormal Function (DLN), the Smoothed Implied Volatility Smile (SML), the Density Functional Based on Con‡uent Hypergeometric function (DFCH) and the Edgeworth expansions (EE). These methodologies were used to obtain the RNDs from the option prices with the underlying USDBRL (price of US dollars in terms of Brazilian reals) for di¤erent maturities (1, 3 and 6 months), and then tested in order to analyze which method best …ts a simulated "true" world as estimated through the Heston model (accuracy measure) and which model has a better performance in terms of stability. We observed that in the majority of the cases the DFCH and DLN outperformed the SML and the EE methods in capturing the "true" implied skewness and kurtosis. However, due to the higher sensitivity of the skewness and kurtosis measures to the tails of the distribution (all the information outside the available strike prices is extrapolated and the probability masses outside this range can have in…nite forms) we also compared the tested models using the root mean integrated squared error (RMISE) which is less sensitive to the tails of the distribution. We observed that using the RMISE criteria, the DFCH outperformed the other methods as a better estimator of the "true" RND.