Professor of Finance, London School of Economics

**New: ***The CME has introduced CVOL, a cross-asset class family of implied volatility indices based on the methodology of Simple Variance Swaps and What is the Expected Return on the Market?*

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We use option prices to derive bounds on the probability of a crash in an individual stock, and argue that the lower bound should be close to the truth. Empirically, the lower bound is highly statistically and economically significant; on its own, it outperforms 15 stock characteristics proposed by the prior literature combined. In a multivariate regression, a one standard deviation increase in the bound raises the predicted crash probability by 3 percentage points, whereas a one standard deviation increase in the next most important predictor (a measure of short interest) raises the predicted probability by only 0.3 percentage points. Long-Horizon Exchange Rate Expectations (with Lukas Kremens and Liliana Varela), June 2024

We study exchange rate expectations in surveys of financial professionals and find that they successfully forecast currency appreciation at the two-year horizon, both in and out of sample. Exchange rate expectations are also interpretable, in the sense that three macro-finance variables—the risk-neutral covariance between the exchange rate and equity market, the real exchange rate, and the current account relative to GDP—explain most of their variation. But there is no “secret sauce” in expectations: after controlling for the three macro-finance variables, the residual information in survey expectations does not forecast currency appreciation in our sample. Debt and Deficits: Fiscal Analysis with Stationary Ratios (with John Campbell and Can Gao), October 2023

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We introduce a new measure of a government’s fiscal position that exploits cointegrating relationships among fiscal variables and output. The measure is a loglinear combination of tax revenue, government spending and the market value of government debt that—unlike the debt-GDP ratio—is stationary in the US and the UK since World War II. Fiscal deterioration forecasts a long-run decline in spending rather than increased tax revenue or low returns for bondholders. Fiscal adjustment to tax and spending shocks occurs through mean-reversion in tax and spending growth, with a negligible contribution from debt returns. Sustainability in a Risky World (with John Campbell), May 2023

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How much consumption is “sustainable”, in the sense that welfare is not expected to decline over time? We impose a sustainability constraint on a standard consumption and portfolio choice problem. The constraint does not distort portfolio choice, but it imposes an upper bound on the sustainable consumption-wealth ratio, which must lie between the riskless interest rate and the expected return on optimally invested wealth (and if risky wealth evolves according to a geometric Brownian motion, it lies exactly halfway between the two). Sustainability requires an upward drift in wealth and consumption to compensate future generations for their risk exposure. The Forward Premium Puzzle in a Two-Country World, March 2013

NBER Working Paper 17564

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I explore the behavior of asset prices and the exchange rate in a two-country world. When the large country has bad news, the relative price of the small country’s output declines. As a result, the small country’s bonds are risky, and uncovered interest parity fails, with positive excess returns available to investors who borrow at the large country’s interest rate and lend at the small country’s interest rate. I use a diagrammatic approach to derive these and other results in a calibration-free way. How Much Do Financial Markets Matter? Cole–Obstfeld Revisited, November 2010

Cole and Obstfeld (1991) asked, “How much do financial markets matter?” Emphasizing the importance of intratemporal relative price adjustment as a risk-sharing mechanism that operates even in the absence of financial asset trade, their answer was: not much. I revisit their question and find that in calibrations featuring rare disasters that generate reasonable risk premia without implausibly high risk aversion, the cost of shutting down trade in financial assets is on the order of 3 to 20 per cent of wealth. Simple Variance Swaps, January 2013

NBER Working Paper 16884

The events of 2008‒9 disrupted volatility derivatives markets and caused the single-name variance swap market to dry up completely; it has never recovered. This paper introduces the

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We present a model featuring risk-averse investors with heterogeneous beliefs. Individuals who are correct in hindsight—whether through luck or judgment—get rich, so sentiment is bullish following good news and bearish following bad news. Sentiment makes extreme outcomes far more important for pricing and has asymmetric effects on left- and right-skewed assets. Investors take speculative positions that can conflict with their fundamental views. Moderate investors are contrarian: they trade against excess volatility created by extremists. All investors view speculation as socially costly; but they also think it is in their self-interest, and the market can collapse entirely if speculation is banned.

Market Efficiency in the Age of Big Data (with Stefan Nagel), *Journal of Financial Economics (2022), 145:1:154‒177*

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Modern investors face a high-dimensional prediction problem: thousands of observable variables are potentially relevant for forecasting. We reassess the conventional wisdom on market efficiency in light of this fact. In our equilibrium model, *N* assets have cash flows that are linear in *J* characteristics, with unknown coefficients. Risk-neutral Bayesian investors learn these coefficients and determine market prices. If *J* and *N* are comparable in size, returns are cross-sectionally predictable ex post. In-sample tests of market efficiency reject the no-predictability null with high probability, even though investors use information optimally in real time. In contrast, out-of-sample tests retain their economic meaning.

Volatility, Valuation Ratios, and Bubbles: An Empirical Measure of Market Sentiment (with Can Gao), *Journal of Finance (2021), 76:6:3211‒3254*

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We define a sentiment indicator based on option prices, valuation ratios and interest rates. The indicator can be interpreted as a lower bound on the expected growth in fundamentals that a rational investor would have to perceive in order to be happy to hold the market. The lower bound was unusually high in the late 1990s, reflecting dividend growth expectations that in our view were unreasonably optimistic. Our approach exploits two key ingredients. First, we derive a new valuation ratio decomposition that is related to the Campbell‒Shiller loglinearization but that resembles the Gordon growth model more closely and has certain other advantages. Second, we introduce a volatility index that provides a lower bound on the market’s expected log return.

Implied Dividend Volatility and Expected Growth (with Niels Gormsen and Ralph Koijen), *AEA Papers and Proceedings (2021), 111:361‒365*

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We study the behavior of implied dividend volatility, constructed from the prices of options on index-level dividends, during the Covid-19 pandemic. We use these data to construct a lower bound on expected excess returns on dividend claims and find that the bound moves significantly over time. However, most of the variation in dividend futures prices reflects changes in growth expectations rather than expected excess returns, making them valuable assets to uncover growth expectations. We conclude that the short-term economic outlook is uncertain and not expected to recover in the near term.

On the Autocorrelation of the Stock Market, *Journal of Financial Econometrics (2021), 19:1:39‒52*

I introduce an index of market return autocorrelation based on the prices of index options and of forward-start index options, and implement it empirically at a six-month horizon. The results suggest that the autocorrelation of the S&P 500 index was close to zero before the subprime crisis but was negative in its aftermath, attaining values around −20% to −30%. I speculate that this may reflect market perceptions about the likely reaction, via quantitative easing, of policymakers to future market moves.

Most of the literature on the economics of catastrophes assumes that such events cause a reduction in the stream of consumption, as opposed to widespread fatalities. Here we show how to incorporate death in a model of catastrophe avoidance, and how a catastrophic loss of life can be expressed as a welfare-equivalent drop in consumption. We examine how potential fatalities affect the policy interdependence of catastrophic events and “willingness to pay” (WTP) to avoid them. Using estimates of the “value of a statistical life” (VSL), we find the WTP to avoid major pandemics, and show it is large (10% or more of annual consumption) and partly driven by the risk of macroeconomic contractions. Likewise, the risk of pandemics significantly increases the WTP to reduce consumption risk. Our work links the VSL and consumption disaster literatures.

What is the Expected Return on a Stock? (with Christian Wagner), *Journal of Finance (2019), 74:4:1887‒1929*

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*Dimensional Fund Advisors Distinguished Paper Prize 2019*

*The Wharton School‒WRDS Best Paper Award in Empirical Finance, WFA 2017*

*Honorable Mention, AQR Insight Award 2017*

We derive a formula for the expected return on a stock in terms of the risk-neutral variance of the market and the stock’s excess risk-neutral variance relative to the average stock. These quantities can be computed from index and stock option prices; the formula has no free parameters. The theory performs well empirically both in and out of sample. Our results suggest that there is considerably more variation in expected returns, over time and across stocks, than has previously been acknowledged.

The Quanto Theory of Exchange Rates (with Lukas Kremens), *American Economic Review (2019), 109:3:810‒843*

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*Best Paper Award, IF2017 Annual Conference in International Finance*

*SIX Best Paper Award 2018*

We present a new identity that relates expected exchange rate appreciation to a risk-neutral covariance term, and use it to motivate a currency forecasting variable based on the prices of quanto index contracts. We show via panel regressions that the quanto forecast variable is an economically and statistically significant predictor of currency appreciation and of excess returns on currency trades. Out of sample, the quanto variable outperforms predictions based on uncovered interest parity, on purchasing power parity, and on a random walk as a forecaster of differential (dollar-neutral) currency appreciation.

Notes on the Yield Curve (with Steve Ross), *Journal of Financial Economics (2019), 134:689‒702*

We study the properties of the yield curve under the assumptions that (i) the fixed-income market is complete and (ii) the state vector that drives interest rates follows a finite discrete-time Markov chain. We focus in particular on the relationship between the behavior of the long end of the yield curve and the recovered time discount factor and marginal utilities of a pseudo-representative agent; and on the relationship between the “trappedness” of an economy and the convergence of yields at the long end.

Local version

This paper, a (very) slightly modified version of the one below, was solicited and republished by a sister journal of

Local version

I survey work of Steve Ross (1976) and of Douglas Breeden and Robert Litzenberger (1978) that first showed how to use options to synthesize more complex securities. Their results made it possible to infer the risk-neutral measure associated with a traded asset, and underpinned the development of the VIX index. The other main result of Ross (1976), which shows how to infer

Online Appendix

Data: Description of data SVIX2.xls epbound.xls crashprob.xls

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I derive a lower bound on the equity premium in terms of a volatility index, SVIX, that can be calculated from index option prices. The bound implies that the equity premium is extremely volatile and that it rose above 20% at the height of the crisis in 2008. The time-series average of the lower bound is about 5%, suggesting that the bound may be approximately tight. I run predictive regressions and find that this hypothesis is not rejected by the data, so I use the SVIX index as a proxy for the equity premium and argue that the high equity premia available at times of stress largely reflect high expected returns over the very short run. I also provide a measure of the probability of a market crash, and introduce simple variance swaps, tradable contracts based on SVIX that are robust alternatives to variance swaps. Averting Catastrophes: The Strange Economics of Scylla and Charybdis (with Robert Pindyck),

Faced with numerous potential catastrophes—nuclear and bioterrorism, mega-viruses, climate change, and others—which should society attempt to avert? A policy to avert one catastrophe considered in isolation might be evaluated in cost-benefit terms. But because society faces multiple catastrophes, simple cost-benefit analysis fails: even if the benefit of averting each one exceeds the cost, we should not necessarily avert them all. We explore the policy interdependence of catastrophic events, and develop a rule for determining which catastrophes should be averted and which should not. The Lucas Orchard,

Supplemental Material

This paper investigates the behavior of asset prices in an endowment economy in which a representative agent with power utility consumes the dividends of multiple assets. The assets are Lucas trees; a collection of Lucas trees is a Lucas orchard. The model generates return correlations that vary endogenously, spiking at times of disaster. Since disasters spread across assets, the model generates large risk premia even for assets with stable cashflows. Very small assets may comove endogenously and hence earn positive risk premia even if their cashflows are independent of the rest of the economy. I provide conditions under which the variation in a small asset’s price-dividend ratio can be attributed almost entirely to variation in its risk premium.

Consumption-Based Asset Pricing with Higher Cumulants, *Review of Economic Studies (2013), 80:2:745‒773*

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I extend the Epstein–Zin-lognormal consumption-based asset-pricing model to allow for general i.i.d. consumption growth. Information about the higher moments—equivalently, cumulants—of consumption growth is encoded in the *cumulant-generating function*. I use the framework to analyze economies with rare disasters, and argue that the importance of such disasters is a double-edged sword: parameters that govern the frequency and sizes of rare disasters are critically important for asset pricing, but extremely hard to calibrate. I show how to sidestep this issue by using observable asset prices to make inferences without having to estimate higher moments of the underlying consumption process. Extensions of the model allow consumption to diverge from dividends, and for non-i.i.d. consumption growth.

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I show that the pricing of a broad class of long-dated assets is driven by the possibility of extraordinarily bad news. This result does not depend on any assumptions about the existence of disasters, nor does it only apply to assets that hedge bad outcomes; indeed, it even applies to long-dated claims on the

We use equity index options to quantify the distribution of consumption growth disasters. The challenge lies in connecting the risk-neutral distribution of equity returns implied by options to the true distribution of consumption growth estimated from macroeconomic data. We attack the problem from three perspectives. First, we compare pricing kernels constructed from macro-finance and option-pricing models. Second, we compare option prices derived from a macro-finance model to those we observe. Third, we compare the distribution of consumption growth derived from option prices using a macro-finance model to estimates based on macroeconomic data. All three perspectives suggest that options imply smaller probabilities of extreme outcomes than have been estimated from international macroeconomic data. The third comparison yields a viable alternative calibration of the distribution of consumption growth that matches the equity premium, option prices, and the sample moments of US consumption growth. Disasters and the Welfare Cost of Uncertainty,