## Teaching Experience

### Jumps, Realized Densities, and News Premia (pdf)

Announcements and other news continuously barrage financial markets, causing asset prices to jump hundreds of times each day. If price paths are continuous, the diffusion volatility nonparametrically summarizes the return distributions' dynamics, and risk premia are instantaneous covariances. However, this is not true in the empirically-relevant case involving price jumps. To address this impasse, I derive both a tractable nonparametric continuous-time representation for the price jumps and an implied sufficient statistic for their dynamics. This statistic — jump volatility — is the instantaneous variance of the jump part and measures news risk. The realized density then depends, exclusively, on the diffusion volatility and the jump volatility. I develop estimators for both and show how to use them to nonparametrically identify continuous-time jump dynamics and associated risk premia. I provide a detailed empirical application to the S&P 500 and show that the jump volatility premium is less than the diffusion volatility premium.

### Bypassing the Curse of Dimensionality: Feasible Multivariate Density Estimation (pdf)

#### with Minsu Chang, (Georgetown University)

Given vector-valued data — {x_{t}} — the *curse of dimensionality* makes nonparametrically estimating the data’s density infeasible when the number of series, D, is large. Because estimators that always converge rapidly do not exist, we construct estimators that converge rapidly most of the time. We adapt ideas from the Bayesian compression literature to represent the density as a parsimonious mixture. For a finite number of periods, T, the number of mixture components is random. We bound this variable as a function of T with high probability. We adopt the nonparametric Bayesian framework, constructing a computationally efficient estimator using Dirichlet processes. The number of mixture components governs our model’s complexity, and our estimator’s convergence rates — log(T) / √T in the unconditional case and log(T) / √T in the conditional case — depend on D only through the constant term. We show our procedure produces a well-calibrated joint predictive density for a monthly macroeconomic panel.

### Identification Robust Inference for Risk Prices in Structural Stochastic Volatility Models (pdf)

#### with Xu Cheng (University of Pennsylvania) and Eric Renault (University of Warwick)

In structural stochastic volatility asset pricing models, changes in volatility affect risk premia through two channels: (1) the investor's willingness to bear high volatility in order to get high expected returns as measured by the market return risk price, and (2) the investor's direct aversion to changes in future volatility as measured by the volatility risk price. Disentangling these channels is difficult and poses a subtle identification problem that invalidates standard inference. We adopt the discrete-time exponentially affine model of Han, Khrapov and Renault (2018), which links the identification of volatility risk price to the leverage effect. In particular, we develop a minimum distance criterion that links the market return risk price, the volatility risk price, and the leverage effect to well-behaved reduced-form parameters that govern the return and volatility's joint distribution. The link functions are almost flat if the leverage effect is close to zero, making estimating the volatility risk price difficult. We adapt the conditional quasi-likelihood ratio test Andrews and Mickusheva (2016) develop in a nonlinear GMM framework to a minimum distance framework. The resulting conditional quasi-likelihood ratio test is uniformly valid. We invert this test to derive robust confidence sets that provide correct coverage for the prices regardless of the leverage effect's magnitude.