Working Papers

Jumps, Realized Densities, and News Premia (pdf)

Announcements and other news continuously barrage financial markets causing asset prices to jump hundreds of times per day. Recursive utility implies that this jump-driven uncertainty will be priced differently than equivalent diffusive-driven uncertainty. I derive a tractable nonparametric continuous-time representation for the prices' jumps and derive the implied sufficient statistic for the jump dynamics. This statistic — the jump volatility — is the instantaneous variance of the jump part and measures news risk. I define the realized density as the daily return density conditional on its diffusion and jump volatilities. This solves the time-aggregation problem and reduces tracking the daily return density to forecasting its volatilities. I develop estimators for the volatilities and the realized density and estimate them using high-frequency data from SPY. This nonparametrically identifies the average curvature in investor's certainty equivalence functional. I then apply these methods to high-frequency data from the S&P 500 and show that total volatility commands a positive risk premium and the proportion of volatility driven by jumps commands a negative premium. This implies that investor's certainty equivalence function is quasiconvex. I further show that volatility premia are capable of explaining the large ex-post return on FOMC announcement days.

Smooth Priors and the Curse of Dimensionality:
Feasible Multivariate Density Estimation
with Minsu Chang

Since most economic data are multivariate, a classic problem in the literature is to estimate a multivariate density. When you have more than a couple of series, the curse-of-dimensionality makes nonparametric estimators imprecise. We provide a simple mixture representation for the conditional density of a multivariate Markov process. For any finite number of periods, the number of mixture components required to approximate the density well is a random variable. Consider an asymptotic experiment where the econometrician picks a small positive number \(\delta\), the number of series is fixed, and the number of periods \(T\) grows. We construct a bound on the number of mixture components as a function of \(T\) alone that holds with prior probability \(1-\delta\). Surprisingly, this estimator's convergence rate — \(\log(T) / \sqrt{T} \) — does not decline as the number of series. This bound exploits smoothness in the prior and does not require the likelihood to be smooth. We provide a computationally efficient Bayesian estimator using a Dirichlet process and analyze its performance in two empirical examples. The first is a monthly macroeconomic panel where our method shows consumption's conditional variance greatly increased during the Great Recession. The second is a daily financial panel where our method automatically detects the data's stylized features, including stochastic volatility and fat tails.

Work in Progress

Inference for Risk Prices using Equity Data, with Xu Cheng and Eric Renault

Jumps, Tail Risk, and the Distribution of Stock Returns