## 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 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. (slides)

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

#### with Minsu Chang, PhD Candidate (University of Pennsylvania)

Most economic data are multivariate and so estimating multivariate densities is a classic problem in the literature. However, given vector-valued data — $\lbrace x_t \rbrace_{t=1}^T$ — the curse of dimensionality makes nonparametrically estimating the data’s density infeasible if the number of series, $D$, is large. Hence, we do not seek to provide estimators that perform well all of the time (it is impossible), but rather seek to provide estimators that perform well most of the time. We adapt the ideas in the Bayesian compression literature to density estimation by randomly binning the data. The binning randomly determines both the number of bins and which observation is placed in which bin. This novel procedure induces a simple mixture representation for the data’s density. For any finite number of periods, $T$, the number of mixture components used is random. We construct a bound for this variable as a function of $T$ that holds with high probability. We adopt the nonparametric Bayesian framework and construct a computationally efficient density estimator using Dirichlet processes. Since the number of mixture components is the key determinant of our model’s complexity, our estimator’s convergence rates — $\sqrt{\log(T)} / \sqrt{T}$ in the unconditional case and $\log(T) / \sqrt{T}$ in the conditional case — depend on $D$ only through the constant term. We then analyze our estimators' performance in a monthly macroeconomic panel and a daily financial panel. Our procedure performs well in capturing the data’s stylized features such as time-varying volatility and fat-tails.

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

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

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.

## Work in Progress

### Jumps, Tail Risk, and the Distribution of Stock Returns

Time-varying tail risk is a fundamental underlying determinant of investors' risk and the key object of interest in financial risk management. Measuring tail risk is difficult because it requires tracking the probability of rare, extreme events. One key driver of tail risk is time-varying jump risk. I build upon my work in Jumps, Realized Densities, and News Premia by jointly modeling diffusion volatility, jump volatility, and returns in the S&P 500. I allow for skewness arising from both instantaneous and lagged correlations between volatility and returns. I show that the data's diffusion and jump volatilities share a common long-memory component. I then develop daily return density forecasts that perform remarkably well in practice. That is, the ensuing quantile-quantile plot is visually indistinguishable from the 45° line. The tail risk measures used by practitioners, such as Value-at-Risk and Expected Shortfall, are statistics of this density and can be easily computed.