We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. In addition, we find that stocks with high idiosyncratic volatility relative to the Fama and French (1993) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility.

The Cross-Section of Volatility and Expected Returns

We examine the pricing of aggregate volatility risk in the cross-section of stock returns. Consistent with theory, we find that stocks with high sensitivities to innovations in aggregate volatility have low average returns. Stocks with high idiosyncratic volatility relative to the Fama and French (1993, Journal of Financial Economics 25, 2349) model have abysmally low average returns. This phenomenon cannot be explained by exposure to aggregate volatility risk. Size, book-to-market, momentum, and liquidity effects cannot account for either the low average returns earned by stocks with high exposure to systematic volatility risk or for the low average returns of stocks with high idiosyncratic volatility. IT IS WELL KNOWN THAT THE VOLATILITY OF STOCK RETURNS varies over time. While considerable research has examined the time-series relation between the volatility of the market and the expected return on the market (see, among others, Campbell and Hentschel (1992) and Glosten, Jagannathan, and Runkle (1993)), the question of how aggregate volatility affects the cross-section of expected stock returns has received less attention. Time-varying market volatility induces changes in the investment opportunity set by changing the expectation of future market returns, or by changing the risk-return trade-off. If the volatility of the market return is a systematic risk factor, the arbitrage pricing theory or a factor model predicts that aggregate volatility should also be priced in the cross-section of stocks. Hence, stocks with different sensitivities to innovations in aggregate volatility should have different expected returns. The first goal of this paper is to provide a systematic investigation of how the stochastic volatility of the market is priced in the cross-section of expected stock returns. We want to both determine whether the volatility of the market

https://www0.gsb.columbia.edu/faculty/rhodrick/crosssection.pdf

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen /Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatitilies perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quintile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

Modeling and Forecasting Realized Volatility

This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

/pdf/introduction-to-econophysics-correlations-and-complexity-in-4a1wviggsu.pdf

Introduction to Econophysics: Correlations and Complexity in Finance

This paper provides a general framework for integration of high-frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency volatility and return distributions. Most procedures for modeling and forecasting financial asset return volatilities, correlations, and distributions rely on restrictive and complicated parametric multivariate ARCH or stochastic volatility models, which often perform poorly at intraday frequencies. Use of realized volatility constructed from high-frequency intraday returns, in contrast, permits the use of traditional time series procedures for modeling and forecasting. Building on the theory of continuous-time arbitrage-free price processes and the theory of quadratic variation, we formally develop the links between the conditional covariance matrix and the concept of realized volatility. Next, using continuously recorded observations for the Deutschemark / Dollar and Yen / Dollar spot exchange rates covering more than a decade, we find that forecasts from a simple long-memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably compared to popular daily ARCH and related models. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal-normal mixture distribution implied by the theoretically and empirically grounded assumption of normally distributed standardized returns, gives rise to well-calibrated density forecasts of future returns, and correspondingly accurate quantile estimates. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation and financial risk management applications.

/pdf/modeling-and-forecasting-realized-volatility-2mqo5gygak.pdf

Modeling and forecasting realized volatility

Substantial progress has been made in developing more realistic option pricing models. Empirically, however, it is not known whether and by how much each generalization improves option pricing and hedging. We fill this gap by first deriving an option model that allows volatility, interest rates and jumps to be stochastic. Using S&P 500 options, we examine several alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant timeseries data, (2) out-of-sample pricing, and (3) hedging. Overall, incorporating stochastic volatility and jumps is important for pricing and internal consistency. But for hedging, modeling stochastic volatility alone yields the best performance. IN THE LAST TWO DECADES, option pricing has witnessed an explosion of new

/pdf/empirical-performance-of-alternative-option-pricing-models-18wa1txytu.pdf

Empirical Performance of Alternative Option Pricing Models

To keep the conditional variances generated by the GARCH (p, q) model nonnegative, Bollerslev imposed nonnegativity constraints on the parameters of the process. We show that these constraints can be substantially weakened and so should not be imposed in estimation. We also provide empirical examples illustrating the importance of relaxing these constraints.

/pdf/inequality-constraints-in-the-univariate-garch-model-4aa9akud9c.pdf

Inequality Constraints in the Univariate GARCH Model

Substantial progress has been made in extending the Black-Scholes model to incorporate such features as stochastic volatility, stochastic interest rates and jumps.On the empirical front, however, it is not yet known whether and by how much each generalized feature will improve option pricing and hedging performance. This paper fills this gap by first developing an implementable option model in closed form that allows volatility, interest rates and jumps to bestochastic and that is parsimonious in the number of parameters. The model includes many known ones as special cases. Delta-neutral and single-instrument minimum-variance hedging strategies are derived analytically. Using S&P 500 options, we examine a set of alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2)out-of-sample pricing and (3) hedging performance. The models of focus include the benchmark Black-Scholes formula and the ones that respectively allow for (i) stochastic volatility, (ii) both stochastic volatility and stochastic interest rates, and (iii) stochastic volatility and jumps.Overall, incorporating both stochastic volatility and random jumps produces the best pricing performance and the most internally-consistent implied-volatility process. Its implied volatility does not "smile" across moneyness. But, for hedging, adding either jumps or stochastic interest rates does not seem to improve performance any further once stochastic volatility is taken into account.

This paper examines the information embedded in both the stock and option markets prior to takeover announcements. During normal periods, buyer-seller initiated stock volume imbalances are significant predictors of next-day stock returns and option volume imbalances are uninformative. However, prior to takeover announcements, call volume imbalances are strongly positively related to next-day stock returns. Cross-sectional analysis shows that those takeover targets with the largest pre-announcement call-imbalance increases experience the highest announcement-day returns. The largest increase in buyer-initiated trading activity is in short-term out-of-the-money calls that subsequently experience the largest returns. Collectively, these findings are consistent with the hypothesis that, in the presence of pending extreme informational events, the options market plays an important role in price discovery.

Informational Content of Option Volume Prior to Takeovers

While recent studies document increasing idiosyncratic volatility over the past four decades, an explanation for this trend remains elusive. We establish a theoretical link between growth options available to managers and the idiosyncratic risk of equity. Empirically both the level and variance of corporate growth options are significantly related to idiosyncratic volatility. Accounting for growth options eliminates or reverses the trend in aggregate firm-specific risk. These results are robust for different measures of idiosyncratic volatility, different growth option proxies, across exchanges, and through time. Finally, our results suggest that growth options explain the trend in idiosyncratic volatility beyond alternative explanations.

/pdf/can-growth-options-explain-the-trend-in-idiosyncratic-risk-211wp97bi0.pdf

Charles Cao

Papers

Empirical Performance of Alternative Option Pricing Models

Inequality Constraints in the Univariate GARCH Model

Empirical Performance of Alternative Option Pricing Models

Informational Content of Option Volume Prior to Takeovers

Can Growth Options Explain the Trend in Idiosyncratic Risk