Joint calibration to SPX and VIX Derivative Markets with Composite Change of Time Models

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.