On the Autocorrelation of the Stock Market, July, 2018
I introduce an index of market autocorrelation based on the prices of index options and of forward-start index options and implement it empirically at a six-month horizon. Forward-looking autocorrelation was close to zero before the subprime crisis but has been negative since late 2008, attaining a low of around -20% at the end of 2010 and remaining below -15% subsequently. I speculate that this may reflect market perceptions about the likely reaction, via quantitative easing, of policymakers to future market moves.
Notes on the Yield Curve (with Steve Ross), July, 2018
Note: a previous draft of this paper was circulated as The Long Bond
We study the properties of the yield curve under the assumptions that (i) the fixed-income market is complete, (ii) the state vector that drives interest rates follows a finite discrete-time Markov chain, and (iii) the pricing kernel is transition-independent.
The Forward Premium Puzzle in a Two-Country World, March, 2013 NBER Working Paper 17564 Slides
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
Note: this paper is largely subsumed by What is the Expected Return on the Market?
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 simple variance swap, a more robust relative of the variance swap that can be priced and hedged even if the underlying asset’s price can jump, and constructs SVIX, an index based on simple variance swaps that measures market volatility. SVIX is consistently lower than VIX in the time series, which rules out the possibility that the market return and stochastic discount factor are conditionally lognormal. The SVIX index points to an equity premium that—in contrast to the prevailing view in the literature—is extraordinarily volatile and that spiked dramatically at the height of the recent crisis.
The Quanto Theory of Exchange Rates (with Lukas Kremens), American Economic Review, forthcoming Slides 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.
What is the Expected Return on a Stock? (with Christian Wagner), Journal of Finance, forthcoming Slides The Wharton School‒WRDS Best Paper Award in Empirical Finance, WFA 2017 Honorable Mention, AQR Insight Award 2017
We derive a formula that expresses 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. We run panel regressions of realized stock returns onto risk-neutral variances, and find that the theory performs well at 6-month, 1-year, and 2-year forecasting horizons. The formula drives out beta, size, book-to-market and momentum, and outperforms a range of competitors in forecasting stock returns out of sample. Our results suggest that there is considerably more variation in expected returns, both over time and across stocks, than has previously been acknowledged.
Options and the Gamma Knife, Journal of Portfolio Management (2018), 44:6:47‒55 Local version
Note: this article was republished in slightly modified form by a sister journal, the Journal of Derivatives (2018), 25:4:71‒79
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 joint risk-neutral distributions from option prices, has been much less influential. I explain why, and propose an alternative approach to the problem. This paper is dedicated to Steve Ross, and was written for a special issue of the Journal of Portfolio Management in memory of him.
What is the Expected Return on the Market?, Quarterly Journal of Economics (2017), 132:1:367‒433 Online Appendix
Data: Description of dataSVIX2.xlsepbound.xlscrashprob.xls Slides
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), American Economic Review (2015), 105:10:2947‒2985
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, Econometrica (2013), 81:1:55‒111 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 Online Appendix
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.
On the Valuation of Long-Dated Assets, Journal of Political Economy (2012), 120:2:346‒358 Online Appendix
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 market in a lognormal world if the market’s Sharpe ratio is higher than its volatility, as appears to be the case in practice.
Disasters Implied by Equity Index Options (with David Backus and Mikhail Chernov), Journal of Finance (2011), 66:6:1969‒2012
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, American Economic Review (2008), Papers & Proceedings, 98:2:74‒78