  The Chicago Board Options Exchange Volatility Index (VIX) is calculated from
SPX options and derivatives of VIX are also traded in market, which leads to
the so-called "consistent modeling" problem. This paper proposes a time-changed
L\'evy model for log price with a composite change of time structure to capture
both features of the implied SPX volatility and the implied volatility of
volatility. Consistent modeling is achieved naturally via flexible choices of
jumps and leverage effects, as well as the composition of time changes. Many
celebrated models are covered as special cases. From this model, we derive an
explicit form of the characteristic function for the asset price (SPX) and the
pricing formula for European options as well as VIX options. The empirical
results indicate great competence of the proposed model in the problem of joint
calibration of the SPX/VIX Markets.
