Tempered stable and tempered infinitely divisible GARCH models

In this paper, we introduce a new GARCH model with an infinitely divisible distributed innovation, referred to as the rapidly decreasing tempered stable (RDTS) GARCH model. This model allows the description of some stylized empirical facts observed for stock and index returns, such as volatility clustering, the non-zero skewness and excess kurtosis for the residual distribution. Furthermore, we review the classical tempered stable (CTS) GARCH model, which has similar statistical properties. By considering a proper density transformation between infinitely divisible random variables, these GARCH models allow to find the risk-neutral price process, and hence they can be applied to option pricing. We propose algorithms to generate scenario based on GARCH models with CTS and RDTS innovation. To investigate the performance of these GARCH models, we report a parameters estimation for Dow Jones Industrial Average (DJIA) index and stocks included in this index, and furthermore to demonstrate their advantages, we calculate option prices based on these models. It should be noted that only historical data on the underlying asset and on the riskfree rate are taken into account to evaluate option prices. --tempered infinitely divisible distribution,tempered stable distribution,rapidly decreasing tempered stable distribution,GARCH model option pricing

Time series analysis for financial market meltdowns

There appears to be a consensus that the recent instability in global financial markets may be attributable in part to the failure of financial modeling. More specifically, current risk models have failed to properly assess the risks associated with large adverse stock price behavior. In this paper, we first discuss the limitations of classical time series models for forecasting financial market meltdowns. Then we set forth a framework capable of forecasting both extreme events and highly volatile markets. Based on the empirical evidence presented in this paper, our framework offers an improvement over prevailing models for evaluating stock market risk exposure during distressed market periods. --ARMA-GARCH model,»-stable distribution,tempered stable distribution,value-at-risk (VaR),average value-at-risk (AVaR)

Auxiliary Likelihood-Based Approximate Bayesian Computation in State Space Models

A computationally simple approach to inference in state space models is proposed, using approximate Bayesian computation (ABC). ABC avoids evaluation of an intractable likelihood by matching summary statistics for the observed data with statistics computed from data simulated from the true process, based on parameter draws from the prior. Draws that produce a 'match' between observed and simulated summaries are retained, and used to estimate the inaccessible posterior. With no reduction to a low-dimensional set of sufficient statistics being possible in the state space setting, we define the summaries as the maximum of an auxiliary likelihood function, and thereby exploit the asymptotic sufficiency of this estimator for the auxiliary parameter vector. We derive conditions under which this approach - including a computationally efficient version based on the auxiliary score - achieves Bayesian consistency. To reduce the well-documented inaccuracy of ABC in multi-parameter settings, we propose the separate treatment of each parameter dimension using an integrated likelihood technique. Three stochastic volatility models for which exact Bayesian inference is either computationally challenging, or infeasible, are used for illustration. We demonstrate that our approach compares favorably against an extensive set of approximate and exact comparators. An empirical illustration completes the paper.Comment: This paper is forthcoming at the Journal of Computational and Graphical Statistics. It also supersedes the earlier arXiv paper "Approximate Bayesian Computation in State Space Models" (arXiv:1409.8363

An alternative model for multivariate stable distributions

Includes bibliographical references (leaves 52-55).As the title, "An Alternative Model for Multivariate Stable Distributions", depicts, this thesis draws from the methodology of [J36] and derives an alternative to the sub-Gaussian alpha-stable distribution as another model for multivariate stable data without using the spectral measure as a dependence structure. From our investigation, firstly, we echo that the assumption of "Gaussianity" must be rejected, as a model for, particularly, high frequency financial data based on evidence from the Johannesburg Stock Exchange (JSE). Secondly, the introduced technique adequately models bivariate return data far better than the Gaussian model. We argue that unlike the sub-Gaussian stable and the model involving a spectral measure this technique is not subject to estimation of a joint index of stability, as such it may remain a superior alternative in empirical stable distribution theory. Thirdly, we confirm that the Gaussian Value-at-Risk and Conditional Value-at-Risk measures are more optimistic and misleading while their stable counterparts are more informative and reasonable. Fourthly, our results confirm that stable distributions are more appropriate for portfolio optimization than the Gaussian framework

Three Essays in Theoretical and Empirical Derivative Pricing

The first essay investigates the option-implied investor preferences by comparing equilibrium option pricing models under jump-diffusion to option bounds extracted from discrete-time stochastic dominance (SD). We show that the bounds converge to two prices that define an interval comparable to the observed option bid-ask spreads for S&P 500 index options. Further, the bounds' implied distributions exhibit tail risk comparable to that of the return data and thus shed light on the dark matter of the divergence between option-implied and underlying tail risks. Moreover, the bounds can better accommodate reasonable values of the ex-dividend expected excess return than the equilibrium models' prices. We examine the relative risk aversion coefficients compatible with the boundary distributions extracted from index return data. We find that the SD-restricted range of admissible RRA values is consistent with the macro-finance studies of the equity premium puzzle and with several anomalous results that have appeared in earlier option market studies. The second essay examines theoretically and empirically a two-factor stochastic volatility model. We adopt an affine two-factor stochastic volatility model, where aggregate market volatility is decomposed into two independent factors; a persistent factor and a transient factor. We introduce a pricing kernel that links the physical and risk neutral distributions, where investor's equity risk preference is distinguished from her variance risk preference. Using simultaneous data from the S&P 500 index and options markets, we find a consistent set of parameters that characterizes the index dynamics under physical and risk-neutral distributions. We show that the proposed decomposition of variance factors can be characterized by a different persistence and different sensitivity of the variance factors to the volatility shocks. We obtain negative prices for both variance factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if those increases have little persistence. We also obtain negative correlations between shocks to the market returns and each volatility factor, where correlation is less significant in transient factor and therefore has a less significant effect on the index skewness. Our empirical results indicate that unlike stochastic volatility model, join restrictions do not lead to the poor performance of two-factor SV model, measured by Vega-weighted root mean squared errors. In the third essay, we develop a closed-form equity option valuation model where equity returns are related to market returns with two distinct systematic components; one of which captures transient variations in returns and the other one captures persistent variations in returns. Our proposed factor structure and closed-form option pricing equations yield separate expressions for the exposure of equity options to both volatility components and overall market returns. These expressions allow a portfolio manager to hedge her portfolio's exposure to the underlying risk factors. In cross-sectional analysis our model predicts that firms with higher transient beta have a steeper term structure of implied volatility and a steeper implied volatility moneyness slope. Our model also predicts that variances risk premiums have more significant effect on the equity option skew when the transient beta is higher. On the empirical front, for the firms listed on the Dow Jones index, our model provides a good fit to the observed equity option prices

Zobecněná stabilní rozdělení a jejich aplikace

Title: Generalized stable distributions and their applications Author: Mgr. Lenka Slámová, MSc. Department: Department of probability and mathematical statistics Supervisor: Prof. Lev Klebanov, DrSc. Abstract: This thesis deals with different generalizations of the strict stability property with a particular focus on discrete distributions possessing some form of stability property. Three possible definitions of discrete stability are introduced, followed by a study of some particular cases of discrete stable distributions and their properties. The random normalization used in the definition of discrete stability is applicable for continuous random variables as well. A new concept of casual stability is introduced by replacing classical normalization in the definition of stability by random normalization. Examples of casual stable distributions, both discrete and continuous, are given. Discrete stable distributions can be applied in discrete models that exhibit heavy tails. Applications of discrete stable distributions on rating of scientific work and financial time series modelling are presented. A method of parameter estimation for discrete stable family is also introduced. Keywords: discrete stable distribution, casual stability, discrete approximation of stable distributionNázev práce: Zobecněná stabilní rozdělení a jejich aplikace Autor: Mgr. Lenka Slámová, MSc. Katedra: Katedra pravděpodobnosti a matematické statistiky Vedoucí doktorské práce: Prof. Lev Klebanov, DrSc. Abstrakt: Tato práce se zabývá různými zobecněními vlastnosti striktní stability s důrazem na diskrétní rozdělení s určitou formou stability. Zavedeny jsou tři možné definice diskrétní stability, následovány studií vlastností několika speciálních případů diskrétních stabilních rozdělení. Náhodná normalizace, která je použita v definici diskrétní stability, funguje i v případě spojitých náhodných veličin. Záměnou klasické normalizace v definici stability za náhodnou normalizaci je zaveden nový koncept ležérní stability. Jsou prezentovány příklady jak spojitých, tak diskrétních ležérně stabilních rozdělení. Disrétní stabilní rozdělení mají využití v diskrétních modelech, které vykazují těžké chvosty. Využití těchto rozdělení je ukázáno na modelu hodnocení vědecké práce a na modelu pro finanční časové řady. Metoda odhadu parametrů diskrétních stabilních rozdělení je v práci rovněž prezentována. Klíčová slova: diskrétní stabilní rozdělení, ležérní stabilita, diskrétní aproximace stabilních rozděleníKatedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Parametric Estimation of Tempered Stable Laws

Tempered stable distributions are frequently used in financial applications (e.g., for option pricing) in which the tails of stable distributions would be too heavy. Given the non-explicit form of the probability density function, estimation relies on numerical algorithms which typically are time-consuming. We compare several parametric estimation methods such as the maximum likelihood method and different generalized method of moment approaches. We study large sample properties and derive consistency, asymptotic normality, and asymptotic efficiency results for our estimators. Additionally, we conduct simulation studies to analyze finite sample properties measured by the empirical bias, precision, and asymptotic confidence interval coverage rates and compare computational costs. We cover relevant subclasses of tempered stable distributions such as the classical tempered stable distribution and the tempered stable subordinator. Moreover, we discuss the normal tempered stable distribution which arises by subordinating a Brownian motion with a tempered stable subordinator. Our financial applications to log returns of asset indices and to energy spot prices illustrate the benefits of tempered stable models