Dynamics of state price densities

State price densities (SPDs) are an important element in applied quantitative finance. In a Black-Scholes world they are lognormal distributions but in practice volatility changes and the distribution deviates from log-normality. In order to study the degree of this deviation, we estimate SPDs using EUREX option data on the DAX index via a nonparametric estimator of the second derivative of the (European) call pricing function. The estimator is constrained so as to satisfy no-arbitrage constraints and corrects for the intraday covariance structure in option prices. In contrast to existing methods, we do not use any parametric or smoothness assumptions

Aplikace Kalmanova filtru

The aim of this work is to discuss the use of the Kalman lter in some economical problems. Generally taken, the Kalman lter is a mathematical method (an algorithm) used for estimation of the non-observable component of a state. Especially, this approach will be applied to estimate the risk-neutral state price density of CALL options. In such case a non-linear relation between state and observed variables may be assumed, and the problem has to be linearized by Taylor expansion. In detail, the main Kalman ltering in the simple linear case will be presented in the rst chapter. In the second chapter, you can nd some application of that Kalman ltering in case of CALL options. The study of the extended Kalman lter and its application in case of a nonlinear state model and the use of the Taylor expansion can be found in Chapter 3. In the fourth chapter, we will be talking about estimating the risk-neutral price density of a CALL option. The corresponding outputs from the program R and the most important results of this work are summarized in the last Chapter 5.Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review

In this selective literature review, we start by observing that in efficient markets, there is information incorporated in option prices that might help us to design option pricing models. To this end, we review the numerous methods of recovering risk-neutral probability distributions from option prices at one particular time to expiration and their applications. Next, we move beyond one time to expiration to the construction of implied binomial trees, which model the stochastic process of the underlying asset. Finally, we describe extensions of implied binomial trees, and other non-parametric methods.Binomial Trees; Risk-Neutral

Three Essays on Estimation and Testing of Nonparametric Models

In this dissertation, I focus on the development and application of nonparametric methods in econometrics. First, a constrained nonparametric regression method is developed to estimate a function and its derivatives subject to shape restrictions implied by economic theory. The constrained estimators can be viewed as a set of empirical likelihood-based reweighted local polynomial estimators. They are shown to be weakly consistent and have the same first order asymptotic distribution as the unconstrained estimators. When the shape restrictions are correctly specified, the constrained estimators can achieve a large degree of finite sample bias reduction and thus outperform the unconstrained estimators. The constrained nonparametric regression method is applied on the estimation of daily option pricing function and state-price density function. Second, a modified Cumulative Sum of Squares (CUSQ) test is proposed to test structural changes in the unconditional volatility in a time-varying coefficient model. The proposed test is based on nonparametric residuals from local linear estimation of the time-varying coefficients. Asymptotic theory is provided to show that the new CUSQ test has standard null distribution and diverges at standard rate under the alternatives. Compared with a test based on least squares residuals, the new test enjoys correct size and good power properties. This is because, by estimating the model nonparametrically, one can circumvent the size distortion from potential structural changes in the mean. Empirical results from both simulation experiments and real data applications are presented to demonstrate the test's size and power properties. Third, an empirical study of testing the Purchasing Power Parity (PPP) hypothesis is conducted in a functional-coefficient cointegration model, which is consistent with equilibrium models of exchange rate determination with the presence of trans- actions costs in international trade. Supporting evidence of PPP is found in the recent float exchange rate era. The cointegration relation of nominal exchange rate and price levels varies conditioning on the real exchange rate volatility. The cointegration coefficients are more stable and numerically near the value implied by PPP theory when the real exchange rate volatility is relatively lower

A Contribution to Functional Data Analysis

Functional Principal Component Analysis (FPCA) approximates a sample curve as a linear combination of orthogonal basis functions. It is often possible to describe the essential parts of the variations of functional data by looking only at a usually very small set of principal components and the corresponding principal scores. Two approaches based on FPCA to estimate smooth derivatives of noisy and discretely observed high-dimensional spatial curves are presented. To handle observed data, both approaches rely on local polynomial regressions. The requirements under which the methods are asymptotically equivalent are evaluated. If the curves are contained in a finite-dimensional function space, it is shown that both methods providing better rates of convergence than estimating the curves individually. The methodology is illustrated in a simulation and empirical study, in which state price density (SPD) surfaces from call option prices are estimated. Serious issues deriving to an FPCA decomposition arise in presence of a Registration problem. Registration aims to decompose amplitude and phase variation of samples of curves. Phase variation is captured by warping functions which monotonically transform the domains. Resulting registered curves should then only exhibit amplitude variation. Most existing registration method rely on aligning typical shape features like peaks or valleys to be found in each sample function. It is shown that this is not necessarily an optimal strategy for subsequent statistical data exploration and inference. In this context a major goal is to identify low dimensional linear subspaces of functions that are able to provide accurate approximations of the observed functional data. Problems of identifiability are discussed in detail, and connections to established registration procedures are analyzed. The methodology is applied to simulated and real data for example an analysis of the juggling dataset. Here an elementary landmark registration is used to extract the juggling cycles from the data. The resulting cycles are then registered to functional principal components. After the registration step the focus is then at the functional principal component analysis to explain the amplitude variation of the cycles. More results about the behavior of the juggler's movements of the hand during the juggling trials are obtained by a further investigation of the principal scores

Essays in applied financial economics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Economics, 2007.Includes bibliographical references.This dissertation is composed of three chapters. The first demonstrates that natural gas violates many of the simplifying assumptions frequently used in modeling its behavior. Careful analysis of futures contracts written on gas suggests that gas prices are seasonal while returns are non-Gaussian and evidence stochastic volatility. In addition, examination of options prices indicates the intermittent presence of jumps. We find that models which disregard these properties struggle to recover options prices with any precision. Thus, we propose an alternative nonparametric approach to gas options pricing that captures these salient features while also shedding light on the nature of risk aversion embedded in gas markets. The second chapter offers a parametric approach to pricing derivatives written on natural gas futures designed to overcome the shortcomings of existing parametric schemes. First, it proposes a model of the underlying futures prices that admits stochastic volatility. Second, it makes use of a state-of-the-art Bayesian particle filtering technique to estimate the underlying process parameters along with a simulation-based technique for option pricing. While it trades off some performance relative to nonparametric approaches, such as the kernel scheme employed in the first chapter, the strategy employed is very general and allows for the pricing of more complex derivatives. The final chapter presents new estimates and approaches to estimating the home bias puzzle. It uses micro-level data to calculate households' foreign equity exposure as a function of wealth. We find simple estimates have significant errors-in-variables problems and we construct an estimator using grouping to account for this issue. Our estimates still imply low aggregate investment in foreign equity.(cont) Finally, we disaggregate the investment decision by incorporating two step decisions that allow households to forgo participating in the market. As a result of the decoupling, we find foreign equity levels closer to that of standard portfolio theories.by Erik Charles Ruben.Ph.D

Essays on Applied Machine Learning for Implied Volatility Interpolation and Artificial Counterfactuals

This dissertation consists of two chapters. Chapter 1: Volatility estimates under the risk neutral density have become a much revisited topic of interest in recent years. The density proves itself a powerful tool for sentiment analysis, since its moments provide insights about expectations in price trends. A standard procedure for its extraction utilizes artificial volatility predictions to form a dense enough grid for approximating a complete probability distribution. This paper proposes two common machine learning technique variations to produce implied volatility predictions when data is very scarce. First, a model using regularization through a variation of a generalized LASSO path combined with signal processing called ‘1 trend filtering. Second, a model averaging strategy by creating an ensemble model from weak predictors from past literature via random forests. These models suggest good interpolating capabilites under stringent conditions, hence serving as a good complement to other methodologies preferred for more abundant data sets. Chapter 2: Synthetic control is an important tool in the set of methodologies for estimating treatment effects. It is, however, dependent on the assumption of trend stationarity. In order to relax the assumption, this paper proposes an alternative approach based on modern v techniques for automatic forecasting which learn the trend and seasonality components from the treated unit and correct them using candidate controls in the same spirit as synthetic control when candidate controls are cointegrated. Monte Carlo simulations show that this method is more robust than synthetic control in the presence of non-stationary cointegrated series, and able to identify treatment effects in a variety of forms. An empirical application reexamines the work of Abadie and Gardeazabal (2003) demonstrating the method’s ability to replicate their results

Essays on the Econometrics of Option Prices

<p>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.</p><p>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.</p><p>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.</p><p>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.</p>Dissertatio