Jump-Diffusion Long-Run Risks Models, Variance Risk Premium, and Volatility Dynamics

The variance risk premium defined as the difference between risk neutral variance and physical variance is one of the most crucial information recovered from option prices. It does not, however, reflect the asymmetry in upside and downside movements of underlying asset returns, and also has limitation in reflecting asymmetric preference of investors over gains and losses. In this sense, this paper decomposes variance risk premium into downside - and upside-variance risk premium, and then derives the skewness risk premium and examines its effectiveness in predicting future underlying asset returns. Using KOSPI200 option prices, we obtained the following results. First, we found out that the estimated skewness risk premium has meaningful forecasting power for future stock returns, while the estimated variance risk premium has little forecasting power. Second, by utilizing our results of skewness risk premium, we developed a profitable investment strategy, which verifies the effectiveness of skewness risk premium in predicting future stock returns. In conclusion, the empirical results of this paper can contribute to the literature in that it helps us understand why variance risk premium, in most global markets except the US market, has not been successful in forecasting future stock returns. In addition, our results showing the profitability of investment strategies based on skewness risk premium can also give important implications to practitioners.

PurposeThe purpose of this paper is to study the variance risk premium in corn and soybean markets, where the variance risk premium is defined as the difference between the historical realized variance and the corresponding risk-neutral expected variance.Design/methodology/approachThe authors compute variance risk premiums using historical derivatives data. The authors use regression analysis and time series econometrics methods, including EGARCH and the Kalman filter, to analyze variance risk premiums.FindingsThere are moderate commonalities in variance within the agricultural sector, but fairly weak commonalities between the agricultural and the equity sectors. Corn and soybean variance risk premia in dollar terms are time-varying and correlated with the risk-neutral expected variance. In contrast, agricultural commodity variance risk premia in log return terms are more likely to be constant and less correlated with the log risk-neutral expected variance. Variance and price (return) risk premia in agricultural markets are weakly correlated, and the correlation depends on the sign of the returns in the underlying commodity.Practical implicationsCommodity variance (i.e. volatility) risk cannot be hedged using futures markets. The results have practical implications for US crop insurance programs because the implied volatilities from the relevant options markets are used to estimate the price volatility factors used to generate premia for revenue insurance products such as “Revenue Protection” and “Revenue Protection with Harvest Price Exclusion.” The variance risk premia found implies that revenue insurance premia are overpriced.Originality/valueThe empirical results suggest that the implied volatilities in corn and soybean futures market overestimate true expected volatility by approximately 15 percent. This has implications for derivative products, such as revenue insurance, that use these implied volatilities to calculate fair premia.

The intertemporal CAPM model of Merton (Econometrica, 41:867–887, 1973) demonstrates that the aggregate market risk is determined by the uncertainty of the underlying return, quantified by the return variance. When holding the market portfolio, however, an investor is also bearing the uncertainty of the variance itself. Just like the equity premium demanded by investors is the result of fear to the uncertainty of future returns, a variance risk premium, defined as the difference between risk-neutral and physical expected variances is also required to compensate for the randomness of the unknown variance. Many recent studies document the link between risk preference and variance risk premium. In particular, Bakshi and Madan (Manag Sci, 52:1945–1956, 2006) argue that the variance risk premium is approximately determined by the risk aversion parameter and the higher order moments of underlying returns. In a similar vein, Bollerslev (J Financ Quant Anal, 2013) assume an affine version of the stochastic volatility model as in Heston (Rev Financ Stud, 6:327–343, 1993) and show that the variance risk premium is linearly related to the risk aversion parameter. Other investigations in this area includes Bekaert and Engstrom (Asset return dynamics under bad environment-good environment fundamentals, Working Paper, NBER, 2010), Todorov (Rev Financ Stud, 23:345–383, 2010), and Gabaix (Q J Econ, 127:645–700, 2012).

Variance risk premium and equity returns

We illustrate the role of left tail dependence—left tail mean (LTM)—in equity risk premium (ERP) predictability. LTM measures the average of pairwise left tail dependency among major equity sectors incorporating shocks imperceptible at the aggregate level. LTM, as well as the variance risk premium, significantly predicts the ERP in and out of sample, which is not the case with commonly used predictors. We find this predictability is the result of procyclical shocks’ reversals in a stable business cycle. This paper contributes to the ongoing debate on ERP predictability. (JEL G10, G12, G14)

Moment spreads in the energy market

Variance risk in aggregate stock returns and time-varying return predictability

Empirical evidence suggests that excess bond returns are forecastable by financial indicators such as forward spreads and yield spreads, a violation of the expectations hypothesis based on constant risk premia. But existing evidence does not tie the forecastable variation in excess bond returns to underlying macroeconomic fundamentals, as would be expected if the forecastability were attributable to time variation in risk premia. We use the methodology of dynamic factor analysis for large datasets to investigate possible empirical linkages between forecastable variation in excess bond returns and macroeconomic fundamentals. We find that several common factors estimated from a large dataset on U.S. economic activity have important forecasting power for future excess returns on U.S. government bonds. Following Cochrane and Piazzesi (2005), we also construct single predictor state variables by forming linear combinations of either five or six estimated common factors. The single state variables forecast excess bond returns at maturities from two to five years, and do so virtually as well as an unrestricted regression model that includes each common factor as a separate predictor variable. The linear combinations we form are driven by both "real" and "inflation" macro factors, in addition to financial factors, and contain important information about one year ahead excess bond returns that is not captured by forward spreads, yield spreads, or the principal components of the yield covariance matrix.

Characterizing the Variance Risk Premium in Consumption-Based Models

The conditional covariance between the market return and its variance, which we refer to as the leverage effect, is positively related to the variance risk premium. It contains incremental information about the variance risk premium after controlling for other return moments and additional variables suggested by the literature as determinants of the variance risk premium. This empirical finding is supported by theory: the pricing of volatility risk is the economic channel behind the strong positive relation between the two variables. We use this relation to construct a time series of the variance risk premium dating back to 1926. (JEL G12, G13) Received February 7, 2020; editorial decision September 01, 2021.

Variance Risk Premia in Energy Commodities

Variance trading and market price of variance risk

VIX Decomposed Tail Risk Premia and the Tail Risk Factor

This study explores the relationship between realized variance jump risk and conditional equity risk premium. Using high frequency records of the Standard & Poor’s 500 index, we construct a realized variance measure and estimate its jump component using a Heterogeneous Autoregressive model augmented by a jump component. We find strong evidence that the realized variance jump risk measure significantly relates to excess stock market returns in-sample and out-of-sample from 1998 to 2010. Further, the predictive power of the variance jump remains both statistically and economically significant after controlling for commonly-used return predictors, and is also independent from variance risk premium and price jump risk. Calibration-based evidence is also consistent with our empirical findings.

Variance risk: A bird’s eye view