Browsing by Author "Tauchen, George"

This dissertation develops new econometric techniques for use in estimating and conducting inference on parameters that can be identified from option prices. The techniques in question extend the existing literature in financial econometrics along several directions.

The first essay considers the problem of estimating and conducting inference on the term structures of a class of economically interesting option portfolios. The option portfolios of interest play the role of functionals on an infinite-dimensional parameter (the option surface indexed by the term structure of state-price densities) that is well-known to be identified from option prices. Admissible functionals in the essay are generalizations of the VIX volatility index, which represent weighted integrals of options prices at a fixed maturity. By forming portfolios for various maturities, one can study their term structure. However, an important econometric difficulty that must be addressed is the illiquidity of options at longer maturities, which the essay overcomes by proposing a new nonparametric framework that takes advantage of asset pricing restrictions to estimate a shape-conforming option surface. In a second stage, the option portfolios of interest are cast as functionals of the estimated option surface, which then gives rise to a new, asymptotic distribution theory for option portfolios. The distribution theory is used to quantify the estimation error induced by computing integrated option portfolios from a sample of noisy option data. Moreover, by relying on the method of sieves, the framework is nonparametric, adheres to economic shape restrictions for arbitrary maturities, yields closed-form option prices, and is easy to compute. The framework also permits the extraction of the entire term structure of risk-neutral distributions in closed-form. Monte Carlo simulations confirm the framework's performance in finite samples. An application to the term structure of the synthetic variance swap portfolio finds sizeable uncertainty around the swap's true fair value, particularly when the variance swap is synthesized from noisy long-maturity options. A nonparametric investigation into the term structure of the variance risk premium finds growing compensation for variance risk at long maturities.

The second essay, which represents joint work with Jia Li, proposes an econometric framework for inference on parametric option pricing models with two novel features. First, point identification is not assumed. The lack of identification arises naturally when a researcher only has interval observations on option quotes rather than on the efficient option price itself, which implies that the parameters of interest are only partially identified by observed option prices. This issue is solved by adopting a moment inequality approach. Second, the essay imposes no-arbitrage restrictions between the risk-neutral and the physical measures by nonparametrically estimating quantities that are invariant to changes of measures using high-frequency returns data. Theoretical justification for this framework is provided and is based on an asymptotic setting in which the sampling interval of high frequency returns goes to zero as the sampling span goes to infinity. Empirically, the essay shows that inference on risk-neutral parameters becomes much more conservative once the assumption of identification is relaxed. At the same time, however, the conservative inference approach yields new and interesting insights into how option model parameters are related. Finally, the essay shows how the informativeness of the inference can be restored with the use of high frequency observations on the underlying.

The third essay applies the sieve estimation framework developed in this dissertation to estimate a weekly time series of the risk-neutral return distribution's quantiles. Analogous quantiles for the objective-measure distribution are estimated using available methods in the literature for forecasting conditional quantiles from historical data. The essay documents the time-series properties for a range of return quantiles under each measure and further compares the difference between matching return quantiles. This difference is shown to correspond to a risk premium on binary options that pay off when the underlying asset moves below a given quantile. A brief empirical study shows asymmetric compensation for these return risk premia across different quantiles of the conditional return distribution.

This dissertations presents the estimation methods of financial models for which the density function is not known in closed form, but the characteristic function (or Laplace transform) is available in the analytical form. In this case the estimation is done via Generalized Method of Moments, where moment conditions are defined on frequency domain. The dissertation consists of three chapters. The first chapter develops a method of selection of finite set of moment conditions out of infinite number of possibilities using proposed optimality criterion. The second chapter describes the estimation of parametric asset price models using finite set of moment conditions based on characteristic function. The final chapter proposes a method for estimating parametric models for stochastic volatility using moment conditions based on the integrated Laplace transforms.

Chapter 2 develops a new estimator for the case where the moment function is the difference between model-implied and data-implied characteristic functions. There is an optimal GMM estimator that attains the Cramer-Rao lower bound and uses all continuum of moment conditions. However, the implementation of continuum moments GMM is not practical. I develop a practical and consistent procedure to select small finite subset of moment conditions that yields a nearly efficient estimator. The moment selection algorithm works by approximating the span of the continuum of moment conditions by an optimal finite subset. The method involves a metric to evaluate how close the asymptotic variance is to the maximum likelihood estimator asymptotic variance in relative terms. The Monte Carlo application for a jump-diffusion model indicates that it is enough to use 9 moment conditions to have almost efficient estimator.

Chapter 3 is co-authored with Professor George Tauchen. We present parametric estimation of models for stock returns by describing price dynamic as the sum of two independent L'{e}vy components. The increments (moves) are viewed as discrete-time log price changes that follow an infinitely divisible distribution, i.e. stationary and independent price changes (zero drift) that follow a L'evy-type distribution. We explore empirical plausibility of two parametric models: Jump-Diffusion (C-J) and pure jump model (TS-J). The first component of each model describes the dynamics of small frequent moves and is modeled by Brownian motion in C-J model and by tempered stable L'evy process in TS-J model. The second component is responsible for big rare moves in asset prices and is modeled by compound Poisson process in both models. Using high frequency data on 13 stocks of different market capitalization for 2006-2008 sample period we find that C-J model performs well only for large cap stocks, while medium cap stock dynamics are captured by TS-J model.

Chapter 4 is co-authored with Professor George Tauchen and Professor Viktor Todorov. We propose analytically tractable way to estimate parametric models for unobserved stochastic volatility. The estimation works my matching moments of the integrated joint Laplace Transform of volatility with those implied by parametric volatility model. We use so-called Realized Laplace Transform of volatility to receive a model-free and jump-robust estimate of integrated empirical Laplace transform of the unobservable volatility. Monte Carlo reveals the proposed estimation method is very efficient. Empirical application using the method reveals that market volatility has both persistent and transient volatility spikes and both of them are sufficiently volatile which necessitates rich jump specifications to capture the observed patterns in stochastic volatility.

This dissertation develops several techniques for conducting estimation and inference in non-standard econometric settings. It does so with a particular interest in applying these methods to applications in financial economics. By doing so this dissertation extends the literatures in financial economics and econometrics.

The first chapter develops econometric tools for studying the jump dependencies between the underlying or latent spot volatilities of two assets from high-frequency observations on a fixed time interval -- with a particular interest in the relationship between the individual volatilities of traded assets and the volatilities of aggregate risk factors such as the market volatility. The chapter derives an asymptotically valid test for the stability of a linear volatility jump relationship between these assets and proposes an asymptoticly valid and consistent likelihood based estimator for the beta in such relationships. The estimation context is made especially challenging because the error shrinks at a rate much slower than the standard root-n parametric rate. To conduct inference the chapter proposes a bootstrap procedure together with a justification of its asymptotic validity. Finally, the chapter considers three empirical applications of the methods and an extension on how to incorporate a Bayesian prior.

The second chapter, which represents joint work with George Tauchen, develops a method to select the threshold in threshold-based jump detection methods. The method is motivated by an analysis of threshold-based jump detection methods in the context of jump-diffusion models. The chapter shows that over the range of sampling frequencies a researcher is most likely to encounter that the usual in-fill asymptotics provide a poor guide for selecting the jump threshold. Because of this we develop a sample-based method. Our method estimates the number of jumps over a grid of thresholds and selects the optimal threshold at what we term the `take-off' point in the estimated number of jumps. The chapter show that this method consistently estimates the jumps and their indices as the sampling interval goes to zero. In several Monte Carlo studies we evaluate the performance of our method based on its ability to accurately locate jumps and its ability to distinguish between true jumps and large diffusive moves. In one of these Monte Carlo studies we evaluate the performance of our method in a jump regression context. Finally, we apply our method in two empirical studies. In one we estimate the number of jumps and report the jump threshold our method selects for three commonly used market indices. In the other empirical application we perform a series of jump regressions using our method to select the jump threshold.

The third chapter proposes a model and framework for forecasting the high-frequency analogue of two widely used tail risk measures: the value at risk and the expected shortfall. It does so by using the theory and literature on high-frequency asset price processes to motivate a particular pure-jump semimartingale forecasting model of high-frequency asset returns and thereby high-frequency value at risk and expected shortfall forecasts. Contrasted with methods developed to forecast daily values at risk and expected shortfalls this model provides a substantial improvement in the forecasting of high-frequency expected shortfalls and a modest, although still significant, improvement in the forecasting of high-frequency values at risk.

Finally, the fourth chapter develops a method for modeling high-frequency asset prices. It does so by showing how asset prices might be transformed into Levy processes. Once in the class of Levy processes this chapter develops a novel estimation procedure and a novel test of a model's specification by performing the estimation and testing over a suitably chosen family of weighting functions. An empirical study fits a selection of asset returns to two classes of Levy processes; and, finally, a detailed empirical exercise develops a flexible method to calculate intraday values at risk up to any within day horizon. A backtest of the intraday values at risk show their coverage to be right in line with the theoretical correct values.

This dissertation provides an empirical assesment of economic models based on high-frequency options data. Options data allows for the investigation of heterogeneous effects across moneyness and maturities, and the use of high-frequency data makes it possible to compute various estimates at higher frequencies and analyze the data behaviour around jumps.

The first chapter analyzes the implications of a disagreement model in which investors observe public information but agree to disagree about its interpretation. In this framework, the disagreement between investors can be recovered from the relationship between an asset’s trading volume and its volatility. Using high-frequency data for options on the S&P 500 index, I provide new empirical evidence of disagreement between investors in the market of S&P 500 index options. The options market provides two natural variables that are sharply related to disagreement: moneyness and tenor. I argue that these variables relate to disagreement about the distribution of the market index at different quantiles and at different times.

I extract the relationship between volume and volatility for options with different moneyness and expirations. I find evidence of little disagreement among investors, based on options that are at-the-money and near expiration. This evidence can be interpreted as investors’ having little disagreement about new information as it relates to the center of the distribution of returns over the short-run. From options that are far-from-the-money and with longer expirations, I find evidence of disagreement among investors. This disagreement increases as options get away from the money, indicating that investors have higher disagreement about rare events. There is also evidence that disagreement increases with the time to expiration of options, indicating that investors have higher disagreement about events farther into the future.

The second chapter examines a general 3-factor options pricing model. This model includes the usual factors that account for short-term and long-term volatility, but also a novel factor that drives the distribution of jump returns. The novel factor, known as the tail factor, impacts the probability of future jump returns, influencing not only the price of the underlying asset, but also the behaviour of its volatility. The second chapter investigates the plausibility of this tail factor.

To investigate the plausibility of the tail factor, I recover moments of the risk-neutral distribution of market returns. The risk-neutral moments are directly related to the implications of the 3-factor pricing model and the impact of the tail factor. The tail factor jumps whenever there are jumps in the underlying asset, but it can also have independent jumps. The model implies that these different types of jump will have different impacts on the risk-neutral moments. Indeed, the main implication is that price jumps lead to jumps in all risk-neutral moments of order two and higher. However, idiosyncratic jumps in the tail factor lead only to jumps in moments of order three and higher. I test these implications using high-frequency options data, which allow for the computation of the risk-neutral moments of the S&P 500 index in a model-free fashion and also at higher frequency. I find evidence of large and frequent moves in all of the computed risk-neutral moments, far in excess to the number of jumps in the market index. These large moves are in part due to jumps in the market index, that lead to co-jumps in all of the risk-neutral moments. However, the majority of large moves occur in the moments of order higher than three, consistent with the idea of jumps in the tail factor. The existence of the tail factor is further supported by co-jump tests, that reveal co-jumps between the higher order moments that are unrelated to jumps in the market index.

The last chapter examines a common assumption of many well known options pricing models: price jumps bear no impact on volatility. To examine the assumption, I analyze high-frequency moves in the market index and in the implied volatility from the options market. Specifically, I estimate jump regressions of changes in the implied volatility at times when the makret index jumps. To do so, I use high-frequency data on options and on the S&P 500 index, coupled with jump identification techniques and a valid inference for jump regressions.

The jump regressions indicate a negative correlation between jumps in the market index and changes in the implied volatility. The estimates indicate that a market crash of 100 basis points would lead to an increase of 450 basis points in the implied volatility of options. The negative correlation is stable for options with different moneyness and is stronger for out-of-the-money options. This indicates that the effect of price jumps is different for options that are close-to-the-money and for options that are out-of-the-money, inline with the idea of the volatility smirk. The empirical evidence from the jump regressions can be contrasted to the theoretical implications of options pricing models, such as Merton (1976) and, more recently, Andersen et al. (2015c). Merton’s model states that jumps in the underlying have no effect in options’ implied volatility, while the AFT model states that only positive price jumps do not have an impact. Both models are contradicted by the empirical evidence presented in this chapter: both positive and negative jumps in the market index have an impact on the implied volatility.