This thesis studies ex ante risk-neutral volatility and skewness indices in the S&P 500 (SPX) market. In the first essay, motivated by the Arrow-Debreu state pricing model, we develop a forward-looking state-price volatility index, SVX, as a forecast for future market realized volatility. We argue that the current Volatility Index (VIX) published by Chicago Board Options Exchange (CBOE) is not a model-free measure. Since SVX and VIX are constructed under the same diffusion setting, we hypothesise that these two measures should be highly correlated. Moreover, VIX may incorporate too much noise from deep out-of-the-money (OTM) options and in such cases it overestimates the future realized volatility. Both hypotheses are supported by the results. Using SPX option prices from 1996 to 2010, while we find that SVX is 99% correlated with VIX, it is a more efficient predictor for the short-term future realized volatility of SPX returns than VIX. This result is robust to different measures of market realized volatilities. We also show that SVX provides a better volatility forecast than alternative measures, such as the square root of GARCH (1, 1) variance and model-free implied volatilities obtained from Investment Banks.
In the second essay, we extend the analysis to asymmetric considerations, with a particular focus on partial moment volatilities. We develop a forward-looking lower partial moment volatility index as a forecast for market downturns, which we denote the bear index (BEX), and an upper partial moment counterpart, which we denote the bull index (BUX). We first find that BEX and BUX are superior estimators for lower and upper partial moments of future market realized volatility, respectively, than VIX. We then examine the contemporaneous relation between SPX returns and changes in BEX, BUX and VIX. By dividing volatility into upside and downside components, our results provide further insights into understanding the `volatility feedback effect'. We show that when the increase in downside volatility is greater than that of the upside counterpart, this excess from the downside contributes to an associated decline in returns. Lastly, we show that BEX might be more suitable as an `investor fear gauge' than VIX.
The third essay studies the dynamics of risk-neutral skewness in the SPX market. We use the same state-preference pricing approach applied in the previous two essays to develop a forward-looking market skewness index SIX, which estimates how negatively skewed the SPX market return distribution is in the next 30 days. We conduct a horse race between SIX and the recently published CBOE SKEW index. We confirm that the risk-neutral market return distribution exhibits negative skewness. Moreover, we document a negative relationship between risk-neutral volatility and skewness, where the former is modelled by VIX and the latter modelled by SIX. We also find an opposing relationship between VIX and SKEW. We then study the contemporaneous relationship between daily SPX returns and changes in the skewness measures. We find that the rate of change in daily market returns is negatively related to changes in SIX. The opposite result is found for SKEW. Specifically, only SIX has a statistically significant asymmetric response to a market drop compared to a market advance. Lastly, we investigate the predictability of these ex ante skewness measures on future SPX market returns and document a negative (positive) relationship between the daily changes of SIX (SKEW) with those of expected SPX returns in the daily, weekly and monthly horizons. More specifically, we find that SIX provides a biased but efficient forecast of future physical skewness, while there is no statistically significant relationship for SKEW. Our results suggest that SIX is an important indicator of institutional anxiety regarding stock market uncertainty, and is a useful complement to the existing VIX index.