An Analytical Formula for Vix Futures and Its Applications: Error Corrections and Replication

Abstract

Zhu and Lian (Journal of Futures Markets, 2012) proposed the first closed-form formulas for VIX futures prices, whihch are conceptually appealing and easy to implement. Unfortunately, the paper is found to contain three kinds of errors. The main formula (9) for the price of futures misses the factor of 100 in two places. Consequently, the biggest reported RMSE of 3.230 by Zhu-Lian is significantly lower than the correct RMSE of 9.447; the biggest correct in-sample MPE of about 50% implies little practical value of Zhu-Lian. Further, the discretized equation set (14), which is used for estimating parameters, has variables with inconsistent units of time. As a result, the reported parameters could not be replicated. Lastly, the reported counts of VIX futures within three maturity-groups are bafflingly way off by between -26% and 48%. This short note corrects these mistakes, and tries to replicate the empirical work.