Extensions of Empirical Characteristic Function (ECF) method for estimating parameters of affine jump-diffusions with unobserved stochastic volatility (SV) are considered. We develop a new approach based on a bias-corrected ECF for the Realized Variance (in the case of diffusions) and Bipower Variation or second generation jump-robust estimators of integrated stochastic variance (in the case of jumps in the underlying). Effective numerical implementation of Unconditional and Conditional ECF methods through a special configuration of grid points in the frequency domain is proposed. The method is illustrated based on a multifactor jump-diffusion SV model with exponential Poisson jumps in the volatility and underlying correlated by a new ''Gamma-factor copula'' that allows for analytically tractable joint characteristic function. A closed form Lauricella-Kummer-type density is derived for the stationary SV distribution. This distribution extends in a certain way a Generalized Gamma Convolution family of Thorin, and it is proven to be infinitely divisible, but not always self-decomposable. Numerical results for S&P 500 Index, VIX Index and rigorous Monte-Carlo study for a number of SV models are presented.

Finance, Econometrics and System Dynamics presents an overview of the concepts and tools for analyzing complex systems in a wide range of fields. The text integrates complexity with deterministic equations and concepts from real world examples, and appeals to a broad audience.

Thèse. HEC. 2006

This paper motivates and introduces a two-stage method for estimating diffusion processes based on discretely sampled observations. In the first stage we make use of the feasible central limit theory for realized volatility, as recently developed in Barndorff-Nielsen and Shephard (2002), to provide a regression model for estimating the parameters in the diffusion function. In the second stage the in-fill likelihood function is derived by means of the Girsanov theorem and then used to estimate the parameters in the drift function. Consistency and asymptotic distribution theory for these estimates are established in various contexts. The finite sample performance of the proposed method is compared with that of the approximate maximum likelihood method of Aiuml;t-Sahalia (2002).

In Volatility and Correlation 2nd edition: The Perfect Hedger and the Fox, Rebonato looks at derivatives pricing from the angle of volatility and correlation. With both practical and theoretical applications, this is a thorough update of the highly successful Volatility & Correlation – with over 80% new or fully reworked material and is a must have both for practitioners and for students. The new and updated material includes a critical examination of the ‘perfect-replication’ approach to derivatives pricing, with special attention given to exotic options; a thorough analysis of the role of quadratic variation in derivatives pricing and hedging; a discussion of the informational efficiency of markets in commonly-used calibration and hedging practices. Treatment of new models including Variance Gamma, displaced diffusion, stochastic volatility for interest-rate smiles and equity/FX options. The book is split into four parts. Part I deals with a Black world without smiles, sets out the author’s ‘philosophical’ approach and covers deterministic volatility. Part II looks at smiles in equity and FX worlds. It begins with a review of relevant empirical information about smiles, and provides coverage of local-stochastic-volatility, general-stochastic-volatility, jump-diffusion and Variance-Gamma processes. Part II concludes with an important chapter that discusses if and to what extent one can dispense with an explicit specification of a model, and can directly prescribe the dynamics of the smile surface. Part III focusses on interest rates when the volatility is deterministic. Part IV extends this setting in order to account for smiles in a financially motivated and computationally tractable manner. In this final part the author deals with CEV processes, with diffusive stochastic volatility and with Markov-chain processes. Praise for the First Edition: “In this book, Dr Rebonato brings his penetrating eye to bear on option pricing and hedging.... The book is a must-read for those who already know the basics of options and are looking for an edge in applying the more sophisticated approaches that have recently been developed.” —Professor Ian Cooper, London Business School “Volatility and correlation are at the very core of all option pricing and hedging. In this book, Riccardo Rebonato presents the subject in his characteristically elegant and simple fashion...A rare combination of intellectual insight and practical common sense.” —Anthony Neuberger, London Business School

This paper proposes a new approach to exploit the information in high frequency data for the statistical inference of continuous-time affine jump diffusion (AJD) models with latent variables. For this purpose, we construct unbiased estimators of the latent variables and their power functions based on the observed state variables over extended horizons. With the estimates of the latent variables, we propose a GMM procedure for the estimation of AJD models with the distinguishing feature that moments of both observed and latent state variables can be used without resorting to path simulation or discretization of the continuous-time process. Using high frequency return observations of the Samp;P 500 index, we implement our estimation approach to various continuous-time asset return models with stochastic volatility and random jumps.

In this paper, we study the problem of statistical inference of continuous-time diffusion processes and their higher-order analogues, and develop methods for modeling threshold diffusion processes in particular. The limiting properties of such estimators are also discussed. We also proposed the likelihood ratio test statistics for testing threshold diffusion process against its linear alternative. We begin in Chapter 1 with an introduction of continuous-time non-linear diffusion processes where I summarized the literature on model estimation. The most natural extension from affine to non-linear model would be piecewise linear diffusion process with piecewise constant variance functions. It can also be considered as a continuous-time threshold autoregressive model (CTAR), the continuous-time analogue of AR model for discrete-time time-series data. The order-one CTAR model is discussed in detail. The discussion is directed more toward the estimation techniques other than the mathematical details. Existing inferential methods (estimation and testing) generally assume known functional form of the (instantaneous) variance function. In practice, the functional form of the variance function is hardly known. So, it is important to develop new methods for estimating a diffusion model that does not rely on knowledge on the functional form of the variance function. In the second Chapter, we propose the quasi-likelihood method to estimate the parameters indexing the mean function of a threshold diffusion model without prior knowledge of its instantaneous variance structure. (and apply to other nonlinear diffusion models, which will be further investigated later.) We also explore the limiting properties of the quasi-likelihood estimators. We focus on estimating the mean function, after which the functional form of the instantaneous variance function can be explored and subsequently estimated from quadratic variation considerations.

Stochastic volatility is the main concept used in the fields of financial economics and mathematical finance to deal with time-varying volatility in financial markets. This work brings together some of the main papers that have influenced this field, andshows that the development of this subject has been highly multidisciplinary.

This thesis treats the problems of exact simulation and parameter inference for jump-diffusion processes. It has two parts. The first part develops a method for the exact simulation of a skeleton, a hitting time and other functionals of a one-dimensional jump-diffusion with state-dependent drift, volatility, jump intensity and jump size. The method requires the drift function to be C1, the volatility function to be C2, and the jump intensity function to be locally bounded. No further structure is imposed on these functions. The method leads to unbiased simulation estimators of security prices, transition densities, hitting probabilities, and other quantities. Numerical results illustrate its features. The second part develops and analyzes likelihood estimators for the parameters of a discretely-observed jump diffusion. We consider the case when the transition density of the process admits an expansion in terms of an infinite series. A randomization technique leads to an unbiased Monte Carlo estimator of the transition density and the likelihood function. We provide conditions under which resulting likelihood estimators are consistent and asymptotically normal. The method avoids the second-order bias of conventional discretization-based estimators. Unlike the estimators based directly on the density expansion, we do not require high-frequency observations. Numerical results confirm the method's properties.