We give a general formula for the dynamics of beliefs -- the moments of agents' conditional distributions -- for very general dynamic processes for fundamentals. The implied belief dynamics fit the behavior of US aggregate stock returns surprisingly well, including the joint behavior of prices, volatility, and skewness. The model additionally shows that long memory in volatility should be pervasive, and explains why.

We establish four empirical facts or regularities:
1. Economic activity is skewed left in both levels and growth rates
2. Skewness is greater at the aggregate than the sector level
3. The cross-sectional volatility of sector growth rates is countercyclical
4. Sectors become more central following negative shocks
Those facts are naturally generated in a production network that features complementarity across inputs. Three leading alternative models -- skewed aggregate shocks, concave decision rules, and fixed adjustment costs with uncertainty shocks -- fail to match all four facts.

We show how state prices and risk preferences can be inferred from returns on synthetic options, which are just dynamic portfolios on the market and risk-free asset. While exchange-traded options earn significantly negative alphas, implying risk aversion rises as wealth falls, synthetic options have zero or positive alphas, implying roughly constant risk aversion. Over the last two decades, returns on exchange-traded options have converged (up) to those on synthetic options, consistent with a model in which prices are driven by intermediary frictions that have shrunk over time.

This paper provides the first time-series of option-implied conditional skewness at the firm and market levels. Firm skewness is significantly procyclical and leads the cycle. Market skewness is acyclical. Firm skew and aggregate volatility are closely linked, implying they are driven by a common factor.

This data is from the current draft of the paper and is subject to revision until final publication. Version date: 5/31/2022.

• Monthly firm-level, market-level, and idiosyncratic skewness, 1980–2021: Data

This paper describes the response of the economy to large shocks in a nonlinear production network. A sector's tail centrality measures how a large negative shock transmits to GDP. In a benchmark case, it is measured as a sector's average downstream closeness to final production. Increases in interconnectedness can simultaneously reduce the sensitivity of the economy to small shocks while increasing the sensitivity to large shocks. Tail risk is related to conditional granularity, where some sectors become highly influential following negative shocks.

This is a review article. It discusses recent research on uncertainty in financial markets and the real economy. The three major areas of focus are the term structure of uncertainty, time-variation in the variance risk premium -- including evidence that it has been declining since 2010 -- and the behavior of conditional skewness. We also present novel data on commodity options in the 1920's and crash cliquets in recent times.

We use options on individual stocks to construct a measure of cross-sectional (as opposed to aggregate) uncertainty back to 1980. The index is much less variable than aggregate uncertainty, it is acyclical and it does not forecast aggregate output, all of which are inconsistent with recent models emphasizing the importance of uncertainty shocks. The results imply firm-level uncertainty is not a major "headwind" policymakers should worry about.

• Published replication files
• Updated US firm uncertainty:
  • Daily available on request
  • Monthly (1/1980–12/2021)
• Updated European firm uncertainty:
  • Daily (1/1/2002–12/31/2018)
  • Monthly (1/2002–3/2020)

We construct option portfolios that directly hedge uncertainty about the macroeconomy, as opposed to the financial sector (e.g. the VIX). Those portfolios historically earn positive average returns, which is inconsistent with the view that uncertainty shocks are "bad" in the sense of being associated with high marginal utility.

• Archive with returns on straddles and strangles for each market plus at-the-money IVs

Uncertainty shocks can be identified as news about second moments. Our identified uncertainty shocks are not associated with recessions. The realization of volatility, however, as opposed to its expectation, is.

• Replication files
• Data for replicating just the main VAR
• Data on monthly S&P 500 implied volatility (VIX) extended to 1983 (daily is available on request)

The only significantly priced risk in the variance market over the period 1996–2014 was transitory realized variance. News about future variance was unpriced, suggesting it is not an important driver of the real economy. The results allow us to distinguish among consumption-based models.

• Variance swap price data (reported as variance forwards)

We analyze restrictions on short-term investment in a noisy rational expectations model of a futures market. In our simple frictionless setting, the restriction has zero effect on the information content of prices at lower frequencies. While the entrance of high-frequency investors hurts those who invest at lower frequencies, restricting high-frequency investment does not restore the old equilibrium – in fact it makes the situation even worse.