Abstract
This thesis is devoted to the study of the higher-moment risk, in particular, the skewness risk.
In Chapter 2, we provide an exact formula for the skewness of stock returns implied in the Heston (1993) model by using a moment-computing approach. We compute the moments of Ito integrals by using Ito’s Lemma skillfully. The model’s affine property allows us to obtain analytical formulas for cumulants. The formulas for the variance and the third cumulant are written as time-weighted sums of expected instantaneous variance, which are neater and more intuitive than those obtained with the characteristic function approach. Our skewness formula is then applied in calibrating Heston’s model by using the market data of the CBOE VIX and SKEW.
The CBOE SKEW is an index launched by the Chicago Board Options Exchange (CBOE) in February 2011. Its term structure tracks the risk-neutral skewness of the S&P 500 index (SPX) for different maturities. In Chapter 3, we develop a theory for the CBOE SKEW by modelling SPX using a jump-diffusion process with stochastic volatility and stochastic jump intensity. With the term structure data of VIX and SKEW, we estimate model parameters and obtain the four processes of variance, jump intensity and their long-term mean levels. Our results can be used to describe the risk-neutral distribution of the SPX returns and to price SPX options.
In Chapter 4, We measure the jump magnitude of the SPX index by using sum of cubed returns (i.e., realized cubic variation). We further detect the existence of jumps if the jump magnitude is higher than a given threshold. Both option-implied and time-series information are used to forecast future one-month jump magnitude and jump existence likelihood. Our results show that option-implied information, coupled with past diffusive variance, is more efficient in forecasting jump magnitude than is time-series information, whereas past realized variance outperforms option-implied information in forecasting jump existence likelihood. We also find that the realized cubic variation and its risk-neutral expectation are significantly negatively correlated.