A remark on Lin and Chang's paper 'Consistent modeling of S&P 500 and VIX derivatives'

Lin and Chang (2009, 2010) establish a VIX futures and option pricing theory when modeling S&P 500 index by using a stochastic volatility process with asset return and volatility jumps. In this note, we prove that Lin and Chang's formula is not an exact solution of their pricing equation. More generally, we show that the characteristic function of their pricing equation cannot be exponentially affine, as proposed by them. Furthermore, their formula cannot serve as a reasonable approximation. Using the (Heston, 1993) model as a special case, we demonstrate that Lin and Chang formula misprices VIX futures and options in general and the error can become substantially large.