Variance Risk Premium Research Articles

Economic Uncertainty and Bank Lending

AbstractThis paper shows that economic uncertainty proxied by the variance risk premium (VRP) has an important explanatory power for bank credit supply and demand. Higher VRP exerts a negative influence on both loan demand and supply, increases loan spreads, reduces loan size and maturity, and increases the proportion of collateralized loans. These effects are substantive in the syndicated loan market and are worse off during economic recessions. Upon isolating the effect of uncertainty on bank credit supply from loan demand, we find that VRP dominates credit supply, which suggests that bank credit shortages can potentially aggravate economic downturn during recessions.

Financial Uncertainty with Ambiguity and Learning

We examine a production-based asset pricing model with regime-switching productivity growth, learning, and ambiguity. Both the mean and volatility of the growth rate of productivity are assumed to follow a Markov chain with an unobservable state. The agent’s preferences are characterized by the generalized recursive smooth ambiguity utility function. Our calibrated benchmark model with modest risk aversion can match moments of the variance risk premium in the data and reconcile empirical relations between the risk-neutral variance and macroeconomic quantities and their respective volatilities. We show that the interplay between productivity volatility risk and ambiguity aversion is important for pricing variance risk in returns. This paper was accepted by Tomasz Piskorski, finance.

Does liquidity drive stock market returns? The role of investor risk aversion

In this paper, we explore the relations between liquidity, stock returns, and investor risk aversion as captured by the variance risk premium (VRP). This is motivated by theoretical and empirical evidence in the literature which suggests that investor risk aversion negatively correlates with asset liquidity, and ample empirical evidence documenting liquidity risk premium. We use monthly US data from January 1999 to December 2018 and show that innovations in the VRP Granger-cause stock returns, which in turn drive liquidity. Our findings are consistent with predictions of prior theories and highlight the predictability of the VRP. They also contribute to the on-going debate on the causal relation between stock returns and liquidity. Finally, we explore the channels through which the VRP impacts liquidity and find that the VRP influences market and momentum factors, and that movements in these factors lead to changes in liquidity.

Same firm, two volatilities: How variance risk is priced in credit and equity markets

Variance risk premia (VRP) based on equity and credit market information for the same firm differ substantially in magnitude. VRP is strongly dependent on firm characteristics. Higher-leveraged and larger firms have lower VRP. The smirk in the plot of VRP vs. leverage is higher for low-levered firms than for high-levered firms. This smirk is more pronounced in the credit market than in the equity market. VRP, and especially credit VRP, correlates with higher future returns and is a priced source of risk in both markets.

Predictability and the Cross-Section of Expected Returns: A Challenge for Asset Pricing Models

Many modern macro finance models imply that excess returns on arbitrary assets are predictable via the price-dividend ratio and the variance risk premium of the aggregate stock market. We propose a simple empirical test for the ability of such a model to explain the cross-section of expected returns by sorting stocks based on the sensitivity of expected returns to these quantities. Models with only one uncertainty-related state variable, like the habit model or the long-run risks model, cannot pass this test. However, even extensions with more state variables mostly fail. We derive conditions under which models would be able to produce expected return patterns in line with the data and discuss various examples. This paper was accepted by David Simchi-Levi, finance.

Option Returns, Risk Premiums, and Demand Pressure in Energy Markets

We study energy futures option returns for crude oil, natural gas, heating oil, and gasoline. Average call and put returns are negative at short maturities, more so for OTM options, and increase with maturity. Put returns are less negative than call returns, but this is not the case for delta-hedged returns, indicating that the aggregate risk of the underlying energy futures is priced in the raw option returns. Moneyness patterns in raw and delta-hedged returns are similar to patterns for index option returns. Significant differences between the four commodities remain after removing the effect of the underlying futures returns, with natural gas as the main outlier. Variance risk premiums are negative and explain some maturity patterns in returns, but they cannot account for return differences across markets. Energy producers are net short the underlying through their option positions, and speculators net long. The larger the net long position of the speculators, the lower the returns on call options, which suggests that demand from speculators may affect option returns in energy markets.

Recovering the Variance Premium

This paper generalizes Ross (2015) recovery theory to accommodate growth, including the Black-Scholes and stochastic volatility option models. The new theory recovers information about equity risk premia and variance risk premia from options prices. In the Heston (1993) stochastic volatility model, the theory predicts an exact (negative) value for the variance risk premium as a function of the equity premium. Recovery theory also predicts that the stochastic discount factor is the reciprocal return on a model-free portfolio of index options. This paper tests the theory on returns from one-month VIX and three-month VIX3M option portfolios from 2007-2018. Recovery theory links the equity pre- mium to the values of both conditional and unconditional variance premia. It also predicts how VIX3M option variance is a biased predictor of future one-month VIX variance. Empirically, recovery theory simultaneously matches the average S&P 500 equity premium, the average variance premium, and observed biases in the variance expectations hypothesis. Autocorrelation properties of VIX indices imply a 12% annual equity premium.

Does Time Varying Risk Premia Exist in the International Bond Market? An Empirical Evidence from Australian and French Bond Market

The presence of risk premium is an issue that weakens the rational expectation hypothesis. This paper investigates changing behavior of time varying risk premium for holding 10 year maturity bond using a bivariate VARMA-DBEKK-AGARCH-M model. The model allows for asymmetric risk premia, causality and co-volatility spillovers jointly in the global bond markets. Empirical results show significant asymmetric partial co-volatility spillovers and risk premium exist in the bond markets. The estimates of the bivariate risk premia show bi-directional causality exist between the Australia and France Bond markets. Overall results suggest nonexistence of pure rational expectation theory in the risk premium model. This information is useful for the agents’ strategic policy decision making in global bond markets.

Variance risk premium and expected currency return: the story is different at the tails of the distribution

ABSTRACT This paper investigates the prediction power of variance risk premium of domestic currency over the currency return for five major emerging market currencies controlling after the impact of past exchange rate developments and asymmetries in the variance risk premium, and the dependence between variance risk premium and currency return at different quantiles. The OLS regression results show that higher (lower) variance risk premium leads appreciation (depreciation) of domestic currency against US Dollar and this relationship holds in after controlling asymmetries in the variance risk premium and past currency returns. Besides, the results don’t confirm an asymmetric effect of the variance risk premium and a linearly increasing effect of variance risk premium conditional on different quantiles of past currency return. Finally, quantile-on-quantile regression (QQR) results show that the negative association of variance risk premium and currency return reverse to be positive at the tails.

GARCH option pricing and implied FX volatility indices

The focus of this paper is the foreign exchange (FX) variance risk premium and the modelling of FX volatility indices of both a developed (United States Dollar) and an emerging market (South African Rand). Regarding the methodology, the variance risk premium is estimated as the difference between realised variance and the variance obtained from the volatility index, and different univariate Generalised Autoregressive Conditional Heteroskedasticity (GARCH) models are considered for the modelling of the volatility index. Empirical results indicate that the variance risk premium is negative on average, this is consistent with previous findings in the literature. Furthermore, asymmetric GARCH models outperform the symmetric GARCH model when modelling FX implied volatility indices.

Uncertainty, sentiments and time-varying risk premia

Why are stock prices much more volatile than the underlying dividends? The excess volatility of prices can in principle be attributed to two different causes: time-varying discount rates for expected future dividends, arising from variation in risk premia; or the irrational exuberance of investors, bidding prices up and down even in the absence of changes in the underlying value of the asset. No consensus has so far emerged among economists as to the prevalence of one or the other source of price variation. I propose in this paper a novel way to approach this problem, by identifying changes in the uncertainty faced by investors regarding the fundamental value of an asset and exploiting the different response in prices that such changes in uncertainty would generate through sentiments or risk premia. I then apply this framework to the S&P 500 index from 1872 till 2019: the positive correlation found between uncertainty and prices (or, equivalently, the negative correlation between uncertainty and implied risk premia) is not compatible with rational investors' behavior and suggests instead the presence of a significant sentiments component in stock prices.

Uncertainty Network Risk and Currency Returns

We examine the pricing of a horizon specific uncertainty network risk, extracted from option implied variances on exchange rates, in the cross-section of currency returns. Buying currencies that are receivers and selling currencies that are transmitters of short-term shocks exhibits a high Sharpe ratio and yields a significant alpha when controlling for standard dollar, carry trade, volatility, variance risk premium and momentum strategies. This profitability stems primarily from the causal nature of shock propagation and not from contemporaneous dynamics. Shock propagation at longer horizons is priced less, indicating a downward-sloping term structure of uncertainty network risk in currency markets.

Information-Driven Volatility

Modern asset pricing theory predicts an unambiguously positive relationship between volatility and expected returns. Empirically, however, realized volatility in the past often predicts expected returns in the future with a negative sign, as exemplified by the volatility-managed portfolios of Moreira and Muir (2017). Theoretically, we show that information-driven volatility induces a negative correlation between past realized volatility and expected volatility and expected returns in the future. We develop a simple asset pricing model based on this intuition and demonstrate that our model can account for several volatility-related asset pricing puzzles such as the return on volatility managed portfolios, the “variance risk premium” return predictability (Bollerslev, Tauchen, and Zhou, 2009), and the predictability of returns by implied volatility reduction on macroeconomic announcement days.

Time is Money: The Equilibrium Trading Horizon and Optimal Arrival Price

Executing even moderately large derivatives orders can be expensive and risky; it’s hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypothetical at-the-money option within Arithmetic Brownian Motion and invert the Bachelier model to compute an inflection point between implied variance and liquidity cost as governed by a central limit order book, each in real time as they evolve. As a result, we demonstrate a novel, continuous-time Arrival Price framework. Further, we argue that traders should be indifferent to choosing between variance risk and liquidity cost, unless they have a predetermined bias or an exogenous position with a convex payoff. We, therefore, introduce half-life factor asymptotics to the model based on a convexity factor and compare results to existing models. We also describe a specialization of the model for trading a basket of correlated instruments, as exemplified by a futures calendar spread. Finally, we establish groundwork for microstructure optimizations as well as explore short term drift and conditional expected slippage within the Equilibrium Horizon framework.

Modeling Conditional Factor Risk Premia Implied by Index Option Returns

We propose a novel factor model for option returns. Option exposures are estimated nonparametrically and factor risk premia can vary nonlinearly with states. The model is estimated using regressions, with minimal assumptions on factor and option return dynamics. Using index options, we characterize the conditional risk premia for the market return, market variance, and tail and intermediary risk factors. All average risk premia have the expected sign and meaningful magnitudes. Market and variance risk premia display pronounced time-variation, spike during crises, and always have the expected sign. Combined, market return and variance explain more than 90% of option return variation.

The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications

In this paper, we propose two new representation formulas for the conditional marginal probability density of the multi-factor Heston model. The two formulas express the marginal density as a convolution with suitable Gaussian kernels whose variances are related to the conditional moments of price returns. Via asymptotic expansion of the non-Gaussian function in the convolutions, we derive explicit formulas for European-style option prices and implied volatility. The European option prices can be expressed as Black–Scholes style terms plus corrections at higher orders in the volatilities of volatilities, given by the Black–Scholes Greeks. The explicit formula for the implied volatility clearly identifies the effect of the higher moments of the price on the implied volatility surface. Further, we derive the relationship between the VIX index and the variances of the two Gaussian kernels. As a byproduct, we provide an explanation of the bias between the VIX and the volatility of total returns, which offers support to recently proposed methods to compute the variance risk premium. Via a series of numerical exercises, we analyse the accuracy of our pricing formula under different parameter settings for the one- and two-factor models applied to index options on the S&P500 and show that our approximation substantially reduces the computational time compared to that of alternative closed-form solution methods. In addition, we propose a simple approach to calibrate the parameters of the multi-factor Heston model based on the VIX index, and we apply the approach to the double and triple Heston models.

Volatility Dependent Structured Products

We construct a derivative that depends on the SPY and VIX and, in this way, incorporates both the market risk premium and the variance risk premium. We show that the product’s Sharpe ratio is higher than the SPY Sharpe ratio. If we had invested $10,000 into the product, the product’s payoff would have been about $60,000 at the end of 2018. In comparison, if we invested $10,000 into the SPY, the SPY payoff would be only about $30,000. TOPICS:Asset-backed securities, real assets/alternative investments/private equity, CLOs, CDOs, and other structured credit, analysis of individual factors/risk premia, derivatives Key Findings ▪ We construct a volatility dependent derivative (product) that includes the SPY and the in-the-money VIX put option. ▪ The product incorporates both the market risk premium and the variance risk premium. ▪ The Sharpe ratio of the product is higher than the Sharpe ratio of the SPY.

Risk aversion connectedness in developed and emerging equity markets before and after the COVID-19 pandemic

This study investigates the dynamic connectedness across the variance risk premium in international developed and emerging equity markets based on a Bayesian time-varying parameter vector autoregressive methodology. The empirical results indicate that the total spillover index is on average 65.6%, indicating a high, albeit declining, level of interconnectedness across the investor sentiment in the three markets under review until early 2020. Following the COVID-19 outbreak though, the total investors' risk aversion connectedness – as expected – strengthens, but more importantly, its dynamics alter, indicating that the risk aversion of emerging markets is an important contributor to the connectedness of international markets.

Moment Risk Premia and Stock Return Predictability

AbstractWe study the predictive power of option-implied moment risk premia embedded in the conventional variance risk premium. We find that although the second-moment risk premium predicts market returns in short horizons with positive coefficients, the third-moment (fourth-moment) risk premium predicts market returns in medium horizons with negative (positive) coefficients. Combining the higher-moment risk premia with the second-moment risk premium improves the stock return predictability over multiple horizons, both in sample and out of sample. The finding is economically significant in an asset-allocation exercise and survives a series of robustness checks.

Equilibrium Asset Pricing with Leverage and Default

ABSTRACTWe develop a general equilibrium model linking the pricing of stocks and corporate bonds to endogenous movements in corporate leverage and aggregate volatility. The model features heterogeneous firms making optimal investment and financing decisions and connects fluctuations in macroeconomic quantities and asset prices to movements in the cross section of firms. Empirically plausible movements in leverage produce realistic asset return dynamics. Countercyclical leverage drives predictable variation in risk premia, and debt‐financed growth generates a high value premium. Endogenous default produces countercyclical aggregate volatility and credit spread movements that are propagated to the real economy through their effects on investment and output.