Variance Risk Premium Research Articles

Optimal Volatility Dependent Derivatives in the Stochastic Volatility Model

We consider derivatives that maximize an investor’s expected utility in the stochastic volatility model. We show that the optimal derivative that depends on the stock and its variance significantly outperforms the optimal derivative that depends on the stock only. Such derivatives yield a much higher certainty equivalent return. This result implies that investors could benefit from structured financial products constructed along these ideas. TOPICS:Derivatives, fixed income and structured finance Key Findings ▪ A derivative is optimal if it maximizes an investor’s expected utility. In the stochastic volatility model, the optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility incorporates both the market risk premium and the variance risk premium. ▪ The optimal buy-and-hold derivative with the payoff that depends on the stock price and its volatility usually outperforms significantly both the optimal buy-and-hold derivative with the payoff that depends on the stock price only and the optimal buy-and-hold portfolio made up of the stock and the risk-free bond. ▪ Investors could benefit from derivatives with payoffs that depend on the stock price and its volatility.

Mutual Fund Hedging Demands and the Cross-Section of Variance Risk Premiums

This paper empirically studies whether institutional stock holdings predict equity option returns. I find that institutional concentration in the underlying stock negatively predicts the cross-section of corresponding option returns. Evidence is consistent with a hedging and demand pressure channel: For stocks with more concentrated ownership, some institutional holders are more likely to overweight them and demand more of their options to hedge. To absorb the order imbalances, dealers sell options and charge higher prices, leading to lower option returns. Using option holdings of U.S. equity mutual funds, I document a positive correlation between funds' stock concentration and their option share in the same firms.

Option-Based Estimation of the Price of Coskewness and Cokurtosis Risk

AbstractWe show that the prices of risk for factors that are nonlinear in the market return can be obtained using index option prices. The price of coskewness risk corresponds to the market variance risk premium, and the price of cokurtosis risk corresponds to the market skewness risk premium. Option-based estimates of the prices of risk lead to reasonable values of the associated risk premia. An analysis of factor models with coskewness risk indicates that the new estimates of the price of risk improve the models’ performance compared with regression-based estimates.

The Bitcoin VIX and Its Variance Risk Premium

The authors acquire a unique dataset of high-frequency traded prices for bitcoin call and put options from the Deribit cryptocurrency derivatives exchange, by 15-minute sampling via the application programming interface They use these prices to construct a term structure of bitcoin implied volatility indices using a variance swap fair-value formula that is employed by the CBOE for the VIX, an index commonly referred to as the “investor fear gauge” for the US stock market Employing over seven million option prices, they construct the bitcoin implied volatility indices with maturities from one week to three months, sampled every 15 minutes from March 2019 to March 2020 They discuss the features of the index and the associated bitcoin variance risk premia, with three different regular time partitions for realized variance, viz 15-minutes, hourly, and daily They also examine the relationship between bitcoin’s 30-day realized variance, volatility index, and variance risk premium, with their equivalent for US equities, oil, gold, the USD/EUR exchange rate, and the 10-year US Treasury note TOPICS: Options, currency, statistical methods Key Findings • The authors novel dataset of bitcoin option prices from Deribit is applied to quote bitcoin VIX indices with maturities from one-week to three months The short-maturity bitcoin VIX exceeded 200% after the global outbreak of Covid-19 • The indices provide indicative fair values for bitcoin variance swaps, which are traded on-chain Using realized volatility monitored at 15-minute, hourly, and daily frequencies, they examine the bitcoin variance risk premium at different maturities • Bitcoin’s 30-day realized volatility, volatility index, and variance risk premium are correlated with their equivalent for US equities, oil, gold, the USD/EUR exchange rate, and the 10-year US Treasury note Diversification potential decreased dramatically after the outbreak of Covid-19

SPAC IPO waves

We examine the wave pattern of U.S. SPAC IPOs using a hand-collected data set of the entire SPAC population since their emergence in 2003. We find that both the SPAC volume and SPAC share of total IPOs are negatively related to market-wide uncertainty (VIX) and time-varying risk aversion (variance risk premium). We attribute our findings to risk-averse investors’ reluctancy to invest in opaque securities. In response, the SPAC sponsor can credibly signal the issue’s quality by increasing their “skin in the game” through the purchase of additional warrants.

Extracting expected stock risk premia from option prices and the information contained in non-parametric-out-of-sample stochastic discount factors

This paper analyzes the factor structure and cross-sectional variability of a set of expected excess returns extracted from option prices and a non-parametric and out-of-sample stochastic discount factor. We argue that the existing potential segmentation between the equity and option markets makes it advisable to avoid using only option prices to extract expected equity risk premia. This set of expected risk premia significantly forecasts future realized returns, and the first two principal components explain 94.1% of the variability of expected returns. A multi-factor model with the market, quality, funding illiquidity, the default premium and the market-wide variance risk premium as factors significantly explains the cross-sectional variability of expected excess returns. The (asymptotically) different from zero adjusted cross-sectional R-squared statistic is 83.6%.

The Price of the Smile and Variance Risk Premia

Using a new specification of multifactor volatility, we estimate the hidden risk factors spanning S&P 500 index (SPX) implied volatility surfaces and the risk premia of volatility-sensitive payoffs. SPX implied volatility surfaces are well-explained by three dependent state variables reflecting (i) short- and long-term implied volatility risks and (ii) short-term implied skewness risk. The more persistent volatility factor and the skewness factor support a downward sloping term structure of variance risk premia in normal times, whereas the most transient volatility factor accounts for an upward sloping term structure in periods of distress. Our volatility specification based on a matrix state process is instrumental to obtaining a tractable and flexible model for the joint dynamics of returns and volatilities, which improves pricing performance and risk premium modeling with respect to recent three-factor specifications based on standard state spaces. This paper was accepted by Gustavo Manso, finance.

The Variation in Variance Risk Premium and its Predictive Power: Evidence from Option Market Sentiments

We show that the highly volatile variance risk premium (VRP) can be theoretically and empirically reconciled with investor sentiment captured by temporary variation in risk aversion. In an effort to understand the poor predictive power of the VRP in non-U.S. markets, we propose a new investor sentiment index, the Variance Sentiment Index(VSI), obtained from the trading behavior of individual investors. We show that the VSI predicts local return dynamics, in a similar way to what the VRP does in the U.S. market. Moreover, the VSI does not lose its predictive power even in the presence of the global VRP.

SPAC IPO Waves

We examine the wave pattern of U.S. SPAC IPOs using a hand-collected data set of the entire SPAC population since their emergence in 2003. We find that both the SPAC volume and SPAC share of total IPOs are negatively related to market-wide uncertainty (VIX) and time-varying risk aversion (variance risk premium). We attribute our findings to risk-averse investors' reluctancy to invest in opaque securities. In response, the SPAC sponsor can credibly signal the issue’s quality by increasing their “skin in the game” through the purchase of additional warrants.

Decomposing the VIX: Implications for the predictability of stock returns

AbstractThe VIX index is not only a volatility index but also a polynomial combination of all possible higher moments in market return distribution under the risk‐neutral measure. This paper formulates the VIX as a linear decomposition of four fundamentally different elements: the realized variance (RV), the variance risk premium (VRP), the realized tail (RT), and the tail risk premium (TRP), respectively. Using an innovative and nonparametric tail risk measure, we find that approximately one‐third of the VIX's formation is attributed to the TRP. In addition to VRP, RT and TRP are crucial components for predicting future returns on equity portfolios.

Inference for variance risk premium

PurposeWith the presence of pricing errors, the authors consider statistical inference on the variance risk premium (VRP) and the associated implied variance, constructed from the option prices and the historic returns.Design/methodology/approachThe authors propose a nonparametric kernel smoothing approach that removes the adverse effects of pricing errors and leads to consistent estimation for both the implied variance and the VRP. The asymptotic distributions of the proposed VRP estimator are developed under three asymptotic regimes regarding the relative sample sizes between the option data and historic return data.FindingsThis study reveals that existing methods for estimating the implied variance are adversely affected by pricing errors in the option prices, which causes the estimators for VRP statistically inconsistent. By analyzing the S&P 500 option and return data, it demonstrates that, compared with other implied variance and VRP estimators, the proposed implied variance and VRP estimators are more significant variables in explaining variations in the excess S&P 500 returns, and the proposed VRP estimates have the smallest out-of-sample forecasting root mean squared error.Research limitations/implicationsThis study contributes to the estimation of the implied variance and the VRP and helps in the predictions of future realized variance and equity premium.Originality/valueThis study is the first to propose consistent estimations for the implied variance and the VRP with the presence of option pricing errors.

Dynamics of variance risk premium: Evidence from India

Extant empirical evidence suggests that variance risk might not be priced in markets that witness high retail participation. We examine and reject this hypothesis using data from Indian markets. We employ both model-dependent and model-free approaches in our analysis. Our result is robust to alternate specifications of volatility, sampling frequencies and sample periods. We then examine the dynamics of variance risk premium. Specifically, we split realized variance into its two components: jumps and continuous variance. We find that only past continuous variance is significant in forecasting short-term synthetic variance swap returns; realized jumps do not have any predictive power. These results survive inclusion of classical risk factors. These findings suggest that only the continuous component of realized variance has significant impact on variance risk premium.

Volatility Estimated Based on the Holding-Period Return versus the Logarithmic Return: Their Difference Can Make a Difference

Estimation of true volatility is a crucial part of equity investment, and there are many strategies for realizing this objective (e.g., the Parkinson realized volatility, the GARCH projected volatility, and stochastic implied volatility). Instead of treating volatility estimates as alternatives to one another, a recent series of studies has emphasized the return prediction performances of their paired differences, such as the variance risk premium, which subtracts the implied variance from the realized variance under a nonparametric framework. This article proposes a novel method of differencing variances that are computed on the basis of the holding-period return versus the logarithmic return. Via this approach, the authors provide a satisfactory substitute for the variance risk premium in cases in which its requirements for intraday and options data are not satisfied. They argue that the volatility differences can capture the information contained in logarithmic returns that involve continuous transactions. This argument is supported by follow-up empirical results—the proxy is demonstrated to be able to foreshadow the next-month market return and to perform well in adjusting holdings in long–short portfolio construction strategies. TOPICS:Volatility measures, exchanges/markets/clearinghouses, risk management Key Findings • The authors build a variance difference measure that is based on the holding-period versus logarithmic return for capturing information obtained from continuous trading activities. • This measure retains the return predictability of differencing volatility estimates, thereby providing a satisfactory alternative to the variance risk premium when its stringent data requirements cannot be satisfied. • The long–short portfolio strategies that are developed according to their variance difference are shown to outperform the market index, especially during market downtrend periods.

Asset Variance Risk Premium and Capital Structure

AbstractThis article investigates how the asset-return variance risk premium changes leverage. I find that the premium reduces leverage by increasing risk-neutral bankruptcy probability and costs in a model where asset returns have stochastic variance with the risk premium. Empirically, the model calibrations verify a significant reduction in optimal leverage, closer to observed leverage than the model without the premium. In model-free regressions, I document that leverage correlates negatively with the variance premium. The highest negative correlation is among investment-grade firms with low asset beta and historical variance but high variance premiums because their assets have high exposure to the market’s variance premium.

Nonlinear Dynamics in Conditional Volatility

Investors pay a substantial premium to hedge against fluctuations in volatility—the variance risk premium (VRP). The asset-pricing literature has presented numerous models with jumps in economic fundamentals to reproduce the properties and the time variation of the VRP. This paper shows that these quantitative results are almost exclusively driven by an inaccurate measure of conditional volatility. Solved accurately, conditional volatility exhibits—counterfactually—a strong procyclical pattern and the models do not deliver a sizeable VRP in response to jumps in state variables. The notion that the VRP is purely a compensation for fluctuations in macroeconomic uncertainty does not hold.

Option-implied information: What\u2019s the vol surface got to do with it?

We find that option-implied information such as forward-looking variance, skewness and the variance risk premium are sensitive to the way the volatility surface is constructed. For some state-of-the-art volatility surfaces, the differences are economically surprisingly large and lead to systematic biases, especially for out-of-the-money put options. Estimates for risk-neutral variance differ across volatility surfaces by more than 10% on average, leading to variance risk premium estimates that differ by 60% on average. The variations are even larger for risk-neutral skewness. To overcome this problem, we propose a volatility surface that is built with a one-dimensional kernel regression. We assess its statistical accuracy relative to existing state-of-the-art parametric, semi- and non-parametric volatility surfaces by means of leave-one-out cross-validation, including the volatility surface of OptionMetrics. Based on 14 years of end-of-day and intraday S&P 500 and Euro Stoxx 50 option data we conclude that the proposed one-dimensional kernel regression represents option market information more accurately than existing approaches of the literature.

Predicting the VIX and the volatility risk premium: The role of short-run funding spreads Volatility Factors

This paper presents an innovative approach to extract Volatility Factors which predict the VIX, the S&P500 Realized Volatility (RV) and the Variance Risk Premium (VRP). The approach is innovative along two different dimensions, namely: (1) we extract Volatility Factors from panels of filtered volatilities — in particular large panels of univariate ARCH-type models and propose methods to estimate common Volatility Factors in the presence of estimation error and (2) we price equity volatility risk using factors which go beyond the equity class namely Volatility Factors extracted from panels of volatilities of short-run funding spreads. The role of these Volatility Factors is compared with the corresponding factors extracted from the panels of the above spreads as well as related factors proposed in the literature. Our monthly short-run funding spreads Volatility Factors provide both in- and out-of-sample predictive gains for forecasting the monthly VIX, RV as well as the equity premium, while the corresponding daily volatility factors via Mixed Data Sampling (MIDAS) models provide further improvements.

Bid and ask prices of index put options: Which predicts the underlying stock returns?

AbstractIn this study, we separately estimate the implied volatility from the bid and ask prices of deep out‐of‐the‐money put options on the S&P500 index. We find that the implied volatility of ask prices has stronger predictive power for stock returns than does the implied volatility of bid prices. We identify two sources of the better performance of the ask price implied volatility: one is its stronger predictive power during economic recessions and in the presence of increasing intermediary capital risk, and the other is its richer information about the future market variance risk premium.

Measuring Granger Causality in Quantiles

ABSTRACT We consider measures of Granger causality in quantiles, which detect and quantify both linear and nonlinear causal effects between random variables. The measures are based on nonparametric quantile regressions and defined as logarithmic functions of restricted and unrestricted expectations of quantile check loss functions. They can consistently be estimated by replacing the unknown expectations of check loss functions by their nonparametric kernel estimates. We derive a Bahadur-type representation for the nonparametric estimator of the measures. We establish the asymptotic distribution of this estimator, which can be used to build tests for the statistical significance of the measures. Thereafter, we show the validity of a smoothed local bootstrap that can be used in finite-sample settings to perform statistical tests. A Monte Carlo simulation study reveals that the bootstrap-based test has a good finite-sample size and power properties for a variety of data-generating processes and different sample sizes. Finally, we provide an empirical application to illustrate the usefulness of measuring Granger causality in quantiles. We quantify the degree of predictability of the quantiles of equity risk premium using the variance risk premium, unemployment rate, inflation, and the effective federal funds rate. The empirical results show that the variance risk premium and effective federal funds rate have a strong predictive power for predicting the risk premium when compared to that of the predictive power of the other two macro variables. In particular, the variance risk premium is able to predict the center, lower, and upper quantiles of the distribution of the risk premium; however, the effective federal funds rate predicts only the lower and upper quantiles. Nevertheless, unemployment and inflation rates have no effect on the risk premium.

Characteristics of Model-Free Pricing Kernels

This study combines model-free conditional estimators for the risk-neutral and the physical distribution of equity returns to obtain daily measures for the pricing kernel at the monthly time horizon. Despite their time-varying nature, our pricing kernels are non-parametric, forward-looking, agnostic about preferences, economic state variables or their dynamics and rely only on minimal technical constraints. Still, our realized pricing kernel estimates are clearly linked to economic state variables like the term spread, the credit spread or liquidity. We decompose the expected variance of the log pricing kernel and find that jumps contribute a considerable portion to overall pricing kernel risk. Building on statistical tests, we confirm an U-shape in the pricing kernel at all times and find a strong link between variations in its magnitude and the variance risk premium. A central hump in the pricing kernel can be confirmed unconditionally, but appears to fade during crisis times.