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We propose a novel no-arbitrage framework, which exploits convex asset pricing constraints to study investors’ marginal utility of wealth or, more generally, Stochastic Discount Factors (SDFs). We establish a duality between minimum dispersion SDFs and penalized portfolio selection problems, building the foundation for characterizing the feasible tradeoffs between a SDF’s pricing accuracy and its comovement with systematic risks. Empirically, a minimum variance CAPM–SDF produces a Pareto optimal tradeoff. This SDF only depends on two distinct risk factors: A traded market factor and a minimum variance excess return that bounds the mispricing of risks unspanned by market shocks.

We propose a model-free method for measuring the jump skewness risk premium via a trading strategy. We find that in the S&P 500 option market, the premium is positive and greater in absolute terms than the variance premium. The trading strategy allows for examining the premium in different holding periods: daytime, when markets are open, and overnight, outside of trading hours. We demonstrate that both premia vary considerably between trading and non-trading periods, and also increase in periods of market distress. The daytime return on jump skewness is not spanned by other systematic risk factors, suggesting it is a systematic risk factor itself. Outside of trading hours, skewness risk does not seem to be distinguishable from variance risk. We also decompose total return skewness into a jump component, and a signed variance component, and demonstrate that only the jump component is priced. Our work also sketches a new set of stylized facts about variance and jump skewness premia that option pricing models should match.

We address potential strengths and weaknesses of alternative protection schemes, which can be adopted as a ‘default option’ in a private, third pillar, pension product. In light of the observed behavior of savers adopting the ‘default option’ at international level, we perform a comparative analysis aimed at quantifying the costs and the benefits of two different risk mitigation techniques and market-standard investment products available to European consumers. We make the case for eligibility of life-cycle target-date funds as default option for the pan-European pension products.

We provide a theoretical characterization of international stochastic discount factors (SDFs) in incomplete markets under different degrees of market segmentation. Using 40 years of data on a cross-section of countries, we estimate model-free SDFs and factorize them into permanent and transitory components. We find that large permanent SDF components help to reconcile the low exchange rate volatility, the exchange rate cyclicality, and the forward premium anomaly. However, integrated markets entail highly volatile and almost perfectly comoving international SDFs. In contrast, segmented markets can generate less volatile and more dissimilar SDFs. In quest of relating the SDFs to economic fundamentals, we document strong links between proxies of financial intermediaries' risk-bearing capacity and model-free international SDFs. We interpret this evidence through the lens of an economy with two building blocks: limited participation by households and financiers who face an intermediation friction.

This paper contains comments on Nonparametric Tail Risk, Stock Returns and the Macroeconomy.

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.

Under mild assumptions, we recover the conditional moments of market returns from a model-free pricing kernel projection on tradeable realized moments that minimizes dispersion. Recovered moments predict realized moments and reveal the time-varying properties of horizon-dependent equity premia, variance risk premia and market Sharpe ratios. Projected kernels tend to be monotonic at short horizons and U-shaped at long horizons. They induce optimal trading strategies with counter-cyclical Sharpe ratios and plausible nonlinear market timing properties.

Testing procedures for predictive regressions with lagged autoregressive variables imply a suboptimal inference in presence of small violations of ideal assumptions. We propose a novel testing framework resistant to such violations, which is consistent with nearly integrated regressors and applicable to multi-predictor settings, when the data may only approximately follow a predictive regression model. The Monte Carlo evidence demonstrates large improvements of our approach, while the empirical analysis produces a strong robust evidence of market return predictability hidden by anomalous observations, both in- and out-of-sample, using predictive variables such as the dividend yield or the volatility risk premium.

This internet appendix provides additional derivations, results, and robustness checks to support the findings of the main paper.The paper "Ambiguity and Reality" to which this appendix applies is available at the following URL: http://ssrn.com/abstract=1668569

Testing procedures for predictive regressions with lagged autoregressive variables imply a suboptimal inference in presence of small violations of ideal assumptions. We propose a novel testing framework resistant to such violations, which is consistent with nearly integrated regressors and applicable to multi-predictor settings, when the data may only approximately follow a predictive regression model. The Monte Carlo evidence demonstrates large improvements of our approach, while the empirical analysis produces a strong robust evidence of market return predictability, using predictive variables such as the dividend yield, the volatility risk premium or labor income.

Conventional tests of present-value models over-reject the null of no predictability. In order to better account for the intrinsic probability of detecting predictive relations by chance alone, we develop a new nonparametric Monte Carlo testing method, which does not rely on distributional assumptions to aggregate the information from the time series of price-dividend ratios and dividend growth. We find evidence of return predictability, but no apparent evidence of dividend growth predictability in postwar US data, thus reconciling the diverging conclusions in the literature. Our findings are robust to the specification of the predictive information set, the choice of the sample period and the use of different cash-flow proxies.

Model builders face ambiguity about the true data generating process. Consequently, they need to deal with ambiguity attitudes (inside uncertainty) and ambiguous financial reality (outside uncertainty) when developing and estimating financial models. We introduce a novel approach for systematically dealing with outside uncertainty in addition to inside uncertainty in a tractable way. By bounding the effects of ambiguous data features, we avoid the adverse consequences of outside uncertainty, such as strongly biased equity premiums and investment policies. In a real data application, we show that asset managers can be more reliably evaluated using our bounded-influence approach. The internet appendix for this paper is available at the following URL: http://ssrn.com/abstract=2218212

Using a novel data set on correlation swaps, we study the relation between correlation risk, hedge fund characteristics, and their risk-return profile. We find that the ability of hedge funds to create market-neutral returns is often associated with a significant exposure to correlation risk, which helps to explain the large abnormal returns found in previous models. We also estimate a significant negative market price of correlation risk, which accounts for the cross-section of hedge fund excess returns. Finally, we detect a pronounced nonlinear relation between correlation risk exposure and the tail risk of hedge fund returns.

We study how the equilibrium risk-sharing of agents with heterogenous perceptions of aggregate consumption growth affects bond and stock returns. While credit spreads and their volatilities increase with the degree of heterogeneity, the decreasing risk premium on moderately levered equity can produce a violation of basic capital structure no-arbitrage relations. Using bottom-up proxies of aggregate belief dispersion, we give empirical support to the model predictions and show that risk premia on corporate bond and stock returns are systematically explained by their exposures to aggregate disagreement shocks.

We produce novel evidence for an equilibrium link between investors' disagreement, the market price of volatility and correlation, and the differential pricing of index and individual equity options. We show that belief disagreement is positively related to (i) the wedge between index and individual volatility risk premia, (ii) the different slope of the smile of index and individual options and (iii) the correlation risk premium. Priced disagreement risk also explains returns of option volatility and correlation trading strategies in a way that is robust to the inclusion of other risk factors and different market conditions.

We study a new class of three-factor affine option pricing models with interdependent volatility dynamics and a stochastic skewness component unrelated to volatility shocks. These properties are useful in order (i) to model a term structure of implied volatility skews more consistent with the data and (ii) to capture comovements of short and long term skews largely unrelated to the volatility dynamics. We estimate our models using about fourteen years of S&P 500 index option data and find that on average they improve the out-of-sample pricing accuracy of benchmark two- and three-factor affine models by 20%. Using an appropriate decomposition of volatility and skewness, highlighting the main directions of improvements produced by our setting, we show that the enhanced fit results from an improved modeling of the term structure of implied-volatility skews. The largest pricing improvements tend to concentrate during periods of financial crises or market distress, suggesting volatility- unrelated skewness as a potentially useful reduced-form risk factor for reproducing some of the crisis-related dynamics of index option smiles.

This appendix extends simulation and empirical results reported in Mancini and Trojani (2010). It discusses the choice of the robustness tuning constants; describes the unconditional, independence and conditional coverage tests for VaR forecast evaluation; provides additional Monte Carlo simulation results on GARCH model estimation and VaR prediction; extends the empirical analysis on backtesting. Notation is the same as in Mancini and Trojani (2010).

We study an equilibrium asset pricing model with several Lucas (1978) trees subject to persistent distress events, where the agent has incomplete information about the state of an underlying common factor and learns from the events occurring to each tree. Contrary to similar asset pricing models with learning in one-tree economies, we find that cross-sectional learning and distress events can reverse several implications and help to explain empirical equity premia and risk-free rate dynamics. We also find that learning helps to generate more realistic dispersion of cross-sectional expected returns, relative to pure aggregate consumption risk models with complete information and disaster risk. The model provides a simple setting to study the asset pricing implications of orchards in which the cash flow links among different trees are asymmetric and some trees are more exogeneous than others. This allows, among other things, to link reduced-form assumptions of cash-flow risk heterogeneity to the structural properties of the orchard. Finally, we show that the cash-flow connectivity of a firm in the orchard is linked to the slope of the dividend strip curve. Sectors whose dividend process is exogenous in the orchard have negatively sloped term structures of dividend swaps. The opposite holds for endogenous sectors.

We study the robustness of block resampling procedures for time series. We first derive a set of formulas to characterize their quantile breakdown point. For the moving block bootstrap and the subsampling, we find a very low quantile breakdown point. A similar robustness problem arises in relation to data-driven methods for selecting the block size in applications. This renders inference based on standard resampling methods useless already in simple estimation and testing settings. To solve this problem, we introduce a robust fast resampling scheme that is applicable to a wide class of time series settings. Monte Carlo simulations and sensitivity analysis for the simple AR(1) model confirm the dramatic fragility of classical resampling procedures in presence of contaminations by outliers. They also show the better accuracy and efficiency of the robust resampling approach under di®erent types of data constellations. A real data application to testing for stock return predictability shows that our robust approach can detect predictability structures more consistently than classical methods.

We study an equilibrium asset pricing model with several Lucas (1978) trees subject to event risk, that is, the possibility that trees experience unexpected disasters. We exploit the market clearing mechanism, in the presence of multiple positive net supply assets, to show that the implications of disasters for some cash-flows extend to the valuation of seemingly unrelated ones. Price-dividend ratios, risk premia, credit-spreads depend on the share of aggregate supply of each tree, but the endogeneity of risk neutral probabilities of disaster implies that the asset pricing implications of event risk go beyond the effects analyzed by the ‘multiple tree’ literature so far.

Writers of index options earn high returns due to a significant and high volatility risk premium, but writers of options in single-stock markets earn lower returns. Using an incomplete information economy, we develop a structural model with multiple assets and explain endogenously the different volatility risk premia of index and single-stock options. We show that higher investors' disagreement increases the volatility of stock returns and the volatility premia of individual options. At the same time, it generates a higher endogenous correlation of stock returns that further increases the volatility premium of index options relative to single-stock options due to an additional correlation risk premium. In equilibrium, this different exposure to disagreement risk is compensated in the cross-section of options and model-implied trading strategies exploiting differences in disagreement earn substantial excess returns. We test the model predictions in a set of panel regressions, by merging three datasets of firm-specific information on analysts' earning forecasts, options data on S&P 100 index options, options on all constituents, and stock returns. We find that belief disagreement is the most powerful determinant of volatility risk premia in individual and index options. Sorting stocks based on differences in beliefs, we find that volatility trading strategies exploiting different exposures to disagreement risk in the cross-section of options earn high Sharpe ratios. The results are robust to different standard control variables and transaction costs and are not subsumed by other theories explaining the volatility risk premia.

A random variable dominates another random variable with respect to the covariance order if the covariance of any two monotone increasing functions of this variable is smaller. We characterize completely the covariance order, give strong sufficient conditions for it, present a number of examples in concrete economic applications, and provide natural extensions for the multivariate context. In analogy to mean preserving spreads in standard stochastic dominance, we show that the covariance order is intimately linked to a comparison of median preserving spreads of random variables. Moreover, it arises naturally in a variety of important economic questions like, e.g., Hansen-Jagannathan stochastic discount factor bounds, the efficient portfolios implied by semi-variance optimization problems, or the measurement of macroeconomic inequality and dispersion in beliefs.

We develop infinitesimally robust statistical procedures for general diffusion processes. We first prove existence and uniqueness of the times series influence function of conditionally unbiased M-estimators for ergodic and stationary diffusions, under weak conditions on the (martingale) estimating function used. We then characterize the robustness of M-estimators for diffusions and derive a class of conditionally unbiased optimal robust estimators. To compute these estimators, we propose a general algorithm, which exploits approximation methods for diffusions in the computation of the robust estimating function. Monte Carlo simulation shows a good performance of our robust estimators and an application to the robust estimation of the exchange rate dynamics within a target zone illustrates the methodology in a real-data application.

We study how investors' disagreement on firm's future cash flows affect the joint behavior of credit spreads, option implied volatilities, and stock returns. We find several important features that can help to explain the dynamics of credit spreads. First, beliefs heterogeneity impacts on the pricing kernel in a way that supports more realistic corporate credit spreads. Second, it induces a positive co--movement between credit spreads, stock returns volatility and the option implied volatility. Third, we obtain an endogenous interaction between option implied volatility skew and leverage, which offers a structural explanation for the inverted volatility skew of some single-stock options. Fourth, the model helps to rationalize both the negative (positive) delta of some call (put) options and the positive link between credit spreads and some individual stock prices. We test the main model predictions in a set of panel regressions, by matching four data sets of firm--specific information on professional earning forecasts, balance-sheet data, corporate bond spreads, stock returns and individual option prices. We find that disagreement among investors impacts on credit spreads, the volatility smile of individual options and stock returns in a way that is consistent with our model, even after controlling for other control variables. Moreover, it has explanatory power for the violations of arbitrage--free restrictions of single--factor models in the data, which are found to have a similar leverage-dependent structure as in our model.

We study the relation between beliefs disagreement among investors and the cross-sectional differences in option returns. Writers of index options earn high returns due to a significant and high volatility risk premium but writers of options in single-stock markets earn lower returns. We develop a structural model using an incomplete information economy with multiple assets and explain endogenously the different volatility risk premia of index and single-stock options. We show that higher disagreement increases the volatility of stock returns and the volatility premia of individual options. At the same time, it generates a higher endogenous correlation of stock returns that further increases the volatility premium of index options relative to single-stock options. In equilibrium, this different exposure to disagreement risk is compensated for in the cross-section of options and model-implied trading strategies exploiting differences in disagreement earn substantial excess returns. We test the model predictions in a set of panel regressions, by merging three datasets of firm-specific information on analysts' earning forecasts, options data on S&P 100 index options, options on all constituents, and stock returns. We find that beliefs disagreement is the most powerful determinant of volatility risk premia in individual and index options. Sorting stocks based on differences in beliefs, we find that volatility trading strategies exploiting different exposures to disagreement risk in the cross-section of options earn high Sharpe ratios. The results are robust to stock risk-characteristics and are not subsumed by other theories explaining the volatility risk premia.

We propose a multivariate nonparametric technique for generating reliable shortterm historical yield curve scenarios and confidence intervals. The approach is based on a Functional Gradient Descent (FGD) estimation of the conditional mean vector and covariance matrix of a multivariate interest rate series. It is computationally feasible in large dimensions and it can account for non-linearities in the dependence of interest rates at all available maturities. Based on FGD we apply filtered historical simulation to compute reliable out-of-sample yield curve scenarios and confidence intervals. We back-test our methodology on daily USD bond data for forecasting horizons from 1 to 10 days. Based on several statistical performance measures we find significant evidence of a higher predictive power of our method when compared to scenarios generating techniques based on (i) factor analysis, (ii) a multivariate CCC-GARCH model, or (iii) an exponential smoothing covariances estimator as in the RiskMetricsTM approach.

This paper studies the term structure implications of a simple structural economy in which the representative agent displays ambiguity aversion, modeled by Multiple Priors Recursive Utility. Bond excess returns reflect a premium for ambiguity, which is observationally distinct from the risk premium of affine yield curve models. The ambiguity premium can be large even in the simplest logutility model and is non zero also for stochastic factors that have a zero risk premium. A calibrated low-dimensional two-factor economy with ambiguity is able to reproduce the deviations from the expectations hypothesis documented in the literature, without modifying in a substantial way the nonlinear mean reversion dynamics of the short interest rate. In this economy, we do not find any apparent tradeoffs between fitting the first and second moments of the yield curve and the large equity premium.

Control problems with Recursive Multiple-Priors Utility (RMPU) are higly non-linear so that RMPU asset prices have been studied in very simple exchange economies only. We identify a continuous-time exchange equilibrium with Locally-Constrained-Entropy RMPU (LCE-RMPU) that is tractable even in the presence of a stochastic opportunity set and incomplete markets. We find that time variation in the LCE-based ambiguity set is able to capture important features of consumption and asset markets data.

Revised version of paper no. 2005-04. We propose a new multivariate GARCH model with Dynamic Conditional Correlations that extends previous models by admitting multivariate thresholds in conditional volatilities and correlations. The model estimation is feasible in large dimensions and the positive deniteness of the conditional covariance matrix is easily ensured by the structure of the model. Thresholds in conditional volatilities and correlations are estimated from the data, together with all other model parameters. We study the performance of our model in three distinct applications to US stock and bond market data. Even if the conditional volatility functions of stock returns exhibit pronounced GARCH and threshold features, their conditional correlation dynamics depends on a very simple threshold structure with no local GARCH features. We obtain a similar result for the conditional correlations between government and corporate bond returns. On the contrary, we ¯nd both threshold and GARCH structures in the conditional correlations between stock and government bond returns. In all applications, our model improves signi¯cantly the in-sample and out-of-sample forecasting power for future conditional correlations with respect to other relevant multivariate GARCH models.

We develop a new framework for intertemporal portfolio choice when the covariance matrix of returns is stochastic. An important contribution of this framework is that it allows to derive optimal portfolio implications for economies in which the degree of correlation across different industries, countries, and asset classes is time-varying and stochastic. In this setting, markets are incomplete and optimal portfolios include distinct hedging components against both stochastic volatility and correlation risk. The model gives rise to simple optimal portfolio solutions that are available in closed-form. We use these solutions to investigate, in several concrete applications, the properties of the optimal portfolios. We find that the hedging demand is typically four to five times larger than in univariate models and it includes an economically significant correlation hedging component, which tends to increase with the persistence of variance covariance shocks, the strength of leverage effects and the dimension of the investment opportunity set. These findings persist also in the discrete-time portfolio problem with short-selling or VaR constraints.

We solve analytically the Merton's problem of an investor with timeadditive power utility. For general state dynamics, we prove existence of two power series representations of the relevant optimal policies and value functions, which hold for all admissible risk aversion parameters. We characterize all terms in the power series by a recursive formula, allowing analytical computations to arbitrary order. Some applications to explicit model settings highlight a very satisfactory accuracy of nite order approximations provided by our power series solution approach.

We develop a continuous time general equilibrium yield curve model under ambiguity aversion. Even a moderate level of 'aggregate ambiguity' affects significantly the term structure and can drive the prices of common interest rate derivatives toward the patterns observed in fixed income markets. Equilibrium term premia and interest rates have rich functional forms, with random factors unpriced under the 'standard' paradigm that pay a premium for ambiguity. Explicit descriptions of the impact of ambiguity aversion on popular term structure factor models are provided both for cases where ambiguity is time varying and for cases where it is not.