We propose a non-parametric method based on a model-free formula to evaluate the tails of a risk-neutral distribution using the full cross-section of option prices at a fixed horizon. The method leads to the joint estimation of risk-neutral tail probabilities and tail expectations beyond the minimum and maximum strike prices. We confirm the accuracy of the risk-neutral tail measures using simulated data. We extract time series of left and right option implied tail risk measures from S&P 500 index options. We find the ratio of risk-neutral left tail conditional expectation to a physical measure of tail risk significantly predicts the equity risk premium at longer return horizons of six months to twelve months with a significant improvement in ex- planatory power when compared to using the physical tail risk measure alone. We also find that both the risk-neutral left and right tail conditional expectations significantly predict the one-month ahead variance risk premium.

We compile option-implied tail loss and gain measures based on a deep out-of-the- money option pricing formula derived by applying 'extreme value theory', and then use these measures to investigate the information content of option-implied tail risk on the future returns of the underlying assets. Our empirical analysis shows that both tail measures implied by S&P 500 and VIX options can predict future changes in the corresponding underlying assets and are informative on the future returns of the S&P 500 index. The relationships are particularly strong during periods of economic recession and driven by the tail-risk premium.

We study short-term market risks implied by weekly S&P 500 index options. The introduction of weekly options has dramatically shifted the maturity profile of traded options over the last five years, with a substantial proportion now having expiry within one week. Economically, this reflects a desire among investors for actively managing their exposure to very short-term risks. Such short-dated options provide an easy and direct way to study market volatility and jump risks. Unlike longer-dated options, they are largely insensitive to the risk of intertemporal shifts in the economic environment, i.e., changes in the investment opportunity set. Adopting a novel general semi-nonparametric approach, we uncover variation in the shape of the negative market jump tail risk which is not spanned by market volatility. Incidents of such tail shape shifts coincide with serious mispricing of standard parametric models for longer-dated options. As such, our approach allows for easy identification of periods of heightened concerns about negative tail events on the market that are not always "signaled" by the level of market volatility and elude standard asset pricing models.

We study whether prices of traded options contain information about future extreme market events. Our option-implied conditional expectation of market loss due to tail events, or tail loss measure, predicts future market returns, magnitude, and probability of the market crashes, beyond and above other option-implied variables. Stock-specific tail loss measure predicts individual expected returns and magnitude of realized stock-specific crashes in the cross-section of stocks. An investor that cares about the left tail of her wealth distribution benefits from using the tail loss measure as an information variable to construct managed portfolios of a risk-free asset and market index.

The study of heavy-tailed distributions allows researchers to represent phenomena that occasionally exhibit very large deviations from the mean. The dynamics underlying these phenomena is an interesting theoretical subject, but the study of their statistical properties is in itself a very useful endeavor from the point of view of managing assets and controlling risk. In this book, the authors are primarily concerned with the statistical properties of heavy-tailed distributions and with the processes that exhibit jumps. A detailed overview with a Matlab implementation of heavy-tailed models applied in asset management and risk managements is presented. The book is not intended as a theoretical treatise on probability or statistics, but as a tool to understand the main concepts regarding heavy-tailed random variables and processes as applied to real-world applications in finance. Accordingly, the authors review approaches and methodologies whose realization will be useful for developing new methods for forecasting of financial variables where extreme events are not treated as anomalies, but as intrinsic parts of the economic process.

Implied correlation and variance risk premium stand out in predicting market returns. However, while the predictive ability of implied correlation lasts for up to a year, the variance risk premium predicts market returns only for one quarter ahead. Contrary to the accepted view, implied correlation predicts the market return not through a diversification risk (average correlation) channel, but by predicting a concentration of market exposure, which defines the level of non-diversifiable market risk, or systematic diversification. Economy-wide implied correlation built exclusively from option prices of nine sector ETFs and the S&P500 efficiently predicts future market returns and systematic diversification risk in the form of market betas dispersion. Newly developed implied correlations for economic sectors provide industry-related information and are used to extract option-implied risk factors from sector-based covariances.