We present empirical evidence showing that option-implied risk measures (OIRMs) are substantially impacted by bid–ask spreads in underlying options. Asking prices are more sensitive to shocks than bids, leading to highly skewed distributions of spreads. We derive and estimate a model of market making that empirically matches these asymmetric responses as well as the time-series properties of bid–ask spreads. Using these estimates to obtain bias-corrected option quotes, we compute several popular OIRMs. We find that fear and risk premia associated with market events that affect the center of the return distribution or unpredictable return jumps are on average overstated when relying on option mid-quotes, whereas risk associated with return-tail events is larger once the bias has been corrected.

Existing studies find conflicting estimates of the risk-return relation. We show that the trade-off parameter is inconsistently estimated when observed or estimated conditional variances measure risk. The inconsistency arises from misspecified, unbalanced, and endogenous return regressions. These problems are eliminated if risk is captured by the variance premium (VP) instead; it is unobservable, however. We propose a 2SLS estimator that produces consistent estimates without observing the VP. Using this method, we find a positive risk-return trade-off and long-run return predictability. Our approach outperforms commonly used risk-return estimation methods, and reveals a significant link between the VP and economic uncertainty.

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We develop an almost affine term-structure model with a closed-form solution for factor loadings in which the spot rate and the risk price are fractionally integrated processes with different integration orders. This model is used to explain two stylized facts. First, predictability of longterm excess bond returns requires sufficient volatility and persistence in the risk price. Second, the large volatility of long-term bond returns requires persistence in the spot rate. Decomposing long-term bond returns, we find that the expectations component from the level factor is more volatile than returns themselves and that the risk premium correlates negatively with level-factor innovations.

Predictive return regressions with persistent regressors are typically plagued by (asymptotically) biased/inconsistent estimates of the slope, non-standard or potentially even spurious statistical inference, and regression unbalancedness. We alleviate the problem of unbalancedness in the theoretical predictive equation by suggesting a data generating process, where returns are generated as linear functions of a lagged latent I(0) risk process. The observed predictor is a function of this latent I(0) process, but it is corrupted by a fractionally integrated noise. Such a process may arise due to aggregation or unexpected level shifts. In this setup, the practitioner estimates a misspecified, unbalanced, and endogenous predictive regression. We show that the OLS estimate of this regression is inconsistent, but standard inference is possible. To obtain a consistent slope estimate, we then suggest an instrumental variable approach and discuss issues of validity and relevance. Applying the procedure to the prediction of daily returns on the S&P 500, our empirical analysis confirms return predictability and a positive risk-return trade-off.

We propose a model for the term structure of interest rates that is a generalization of the discrete-time, Gaussian, affine yield-curve model. Compared to standard affine models, our model allows for general linear dynamics in the vector of state variables. In an application to real yields of U.S. government bonds, we model the time series of the state vector by means of a co-fractional vector autoregressive model. The implication is that yields of all maturities exhibit nonstationary, yet mean-reverting, long-memory behavior of the order d ˜ 0.87. The long-run dynamics of the state vector are driven by a level, a slope, and a curvature factor that arise naturally from the co-fractional modeling framework. We show that implied yields match the level and the variability of yields well over time. Studying the out-of-sample forecasting accuracy of our model, we find that our model results in good yield forecasts that outperform several benchmark models, especially at long forecasting horizons.

Univariate dependencies in market volatility, both objective and risk neutral, are best described by long-memory fractionally integrated processes. Meanwhile, the ex post difference, or the variance swap payoff reflecting the reward for bearing volatility risk, displays far less persistent dynamics. Using intraday data for the Standard & Poor's 500 and the volatility index (VIX), coupled with frequency domain methods, we separate the series into various components. We find that the coherence between volatility and the volatility-risk reward is the strongest at long-run frequencies. Our results are consistent with generalized long-run risk models and help explain why classical efforts of establishing a naïve return-volatility relation fail. We also estimate a fractionally cointegrated vector autoregression (CFVAR). The model-implied long-run equilibrium relation between the two variance variables results in nontrivial return predictability over interdaily and monthly horizons, supporting the idea that the cointegrating relation between the two variance measures proxies for the economic uncertainty rewarded by the market.

The dynamic dependencies in financial market volatility are generally well described by a long-memory fractionally integrated process. At the same time, the volatility risk premium, defined as the difference between the ex-post realized volatility and the market’s ex-ante expectation thereof, tends to be much less persistent and well described by a short-memory process. Using newly available intraday data for the S&P 500 and the VIX volatility index, coupled with frequency domain inference procedures that allow us to focus on specific parts of the spectra, we show that the existing empirical evidence based on daily and coarser sampled data carries over to the high-frequency setting. Guided by these empirical findings, we formulate and estimate a fractionally cointegrated VAR model for the two high-frequency volatility series and the corresponding high-frequency S&P 500 returns. Consistent with the implications from a stylized equilibrium model that directly links the realized and expected volatilities to returns, we show that the equilibrium variance risk premium estimated with the intraday data within the fractionally cointegrated system results in non-trivial return predictability over longer interdaily and monthly horizons. These results in turn suggest that much of the existing literature seeking to establish a risk-return tradeoff relationship between expected returns and expected volatilities may be misguided, and that the variance risk premium provides a much better proxy for the true economic uncertainty that is being rewarded by the market.

We use co-fractional models to evaluate the predictive relations between returns and a valuation ratio. The co-fractional model can handle situations where financial returns are predicted using persistent valuation ratios, like dividend to price. For our application we consider very long time series, covering 355 years of real-estate returns and rent to price ratios. We find robust evidence of a fractional root of d=0.75 in the rent to price ratio. The co-fractional model empirically outperforms the traditional triangular time-series model of return predictability. For annual data, the difference in predictive R-squared is about 8%. We conclude that the co-fractional VAR provides an alternative parsimonious model for the interaction between returns and valuation ratios. The long-memory properties of the fractional model have important implications for the fi t of present-value models and the term structure of risk.

We develop a term structure model that can match two stylized facts of excess returns on long-term bonds. The first stylized fact is the predictability of excess returns, which requires sufficient volatility in the price of risk. The second stylized fact is that yields are dominated by a level factor, which requires persistence in the spot interest rate or in the price of risk, or both. We calibrate a stochastic discount factor that is consistent with both stylized facts. Doing so we need time series processes that are outside the essentially affine class of term structure models. For parsimony we choose a fractionally integrated process, which leads to tractable analytical solutions. As an important implication we find that risk premiums of excess bond returns are very persistent.