The Relation between Implied and Realised Probability Density Functions
Professor Stewart Hodges, FORC, University of Warwick
(Joint work with Iliana Anagnou, Mascia Bedendo and Robert Tompkins)

Abstract:
A number of financial regulators [see Neuhaus (1995), Bahra (1996, 1997), McManus (1999) and Shiratsuka (2001)] have suggested that risk neutral densities (RND) associated with options markets could provide useful indicators of future market turbulence. Critical to this assumption is that such RNDs should provide an unbiased forecast of realised probability density functions. To date, this assumption has not been fully examined.

In this research, we test the ability of RNDs for options on the S&P 500 and the British Pound / US Dollar to predict future probability densities. We consider four approaches to estimate the RNDs, which are consistent with approaches proposed and used by financial regulators. We also provide a number of new testing procedures to assess the efficiency and unbiasness of the forecasts. These tests provide more power than the usual Komolgorov/Smirnov tests.

Using non-overlapping quarterly data from the mid 1980s to 2001, we find that we can reject the hypothesis that the RNDs for both the S&P 500 and British Pounds are unbiased forecasts. Even with a limited number of observations, the tests are powerful enough to allow rejection. However, when an adjustment for the risk premium is made, the results become more controversial. Depending on the nature of the adjustment, we can or cannot reject the accuracy of the adjusted implied densities. When a power utility adjustment is made, like Bliss and Panigirtzolou (2001), we are unable to reject the hypothesis that RNDs are unbiased forecasts of realised densities.

Overall, our results tend to support the conclusions of Shiratsuka (2001), that unadjusted RNDs should not be used by financial regulators as financial indicators, and that such use could prove counterproductive; actually increasing future market turbulence rather than alleviating it.

Moreover, we observe that return distributions simulated on the basis of historical volatility processes of some GARCH-type exhibit better forecasting performance than unadjusted implied RNDs. These findings seem to suggest that also in relative terms (unadjusted) implied densities do not constitute an efficient forecast of realised probability density.