An examination of option-implied s&p 500 futures price distributions

Expected S&P 500 futures price distributions are derived using no-arbitrage option pricing models. These distributions are parameterized both as the lognormal and as a less restrictive three-parameter Burr-XII distribution. The resulting option-based probability assessments display some evidence of miscalibration very near to expiration and far from expiration, but are accurate over intermediate time ranges. The means of the implied price distributions correspond closely to the contemporaneous futures prices for both distributions, although marginally better with the Burr-XII. The Burr-XII distribution also performs better than the lognormal based on calibration statistics, and hence, is used to recalibrate estimated distributions.