Nonlinear Stochastic Volatility Research Articles

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A closed-form pricing formula for European options under a multi-factor nonlinear stochastic volatility model with regime-switching

A new nonlinear stochastic volatility model with regime switching stochastic mean reversion and its applications to option pricing

We establish a new nonlinear stochastic volatility model by modeling the volatility long-run mean with a particular stochastic process, whose parameters can jump between different regimes according to a Markov chain. Introducing the regime switching factor is consistent with the empirical evidence of the economic cycle and nonlinear volatility mean-reversion observed from market data. A closed-form solution that can be used to value European options has been successfully obtained based on the characteristic function approach, followed by some numerical examples comparing the models with and without regime switching.

Getting Started with Particle Metropolis-Hastings for Inference in Nonlinear Dynamical Models

This tutorial provides a gentle introduction to the particle Metropolis-Hastings (PMH) algorithm for parameter inference in nonlinear state-space models together with a software implementation in the statistical programming language R. We employ a step-by-step approach to develop an implementation of the PMH algorithm (and the particle filter within) together with the reader. This final implementation is also available as the package pmhtutorial from the Comprehensive R Archive Network (CRAN) repository. Throughout the tutorial, we provide some intuition as to how the algorithm operates and discuss some solutions to problems that might occur in practice. To illustrate the use of PMH, we consider parameter inference in a linear Gaussian state-space model with synthetic data and a nonlinear stochastic volatility model with real-world data.

Realized Non-linear Stochastic Volatility Models with Asymmetric Effects and Generalized Student's <i>t</i>-Distributions

This study proposes a class of realized non-linear stochastic volatility models with asymmetric effects and generalized Student's t-error distributions by applying three families of power transformation—exponential, modulus, and Yeo-Johnson—to lagged log volatility. The proposed class encompasses a raw version of the realized stochastic volatility model. In the Markov chain Monte Carlo algorithm, an efficient Hamiltonian Monte Carlo (HMC) method is developed to update the latent log volatility and transformation parameter, whereas the other parameters that could not be sampled directly are updated by an efficient Riemann manifold HMC method. Empirical studies on daily returns and four realized volatility estimators of the Tokyo Stock Price Index (TOPIX) over 4-year and 8-year periods demonstrate statistical evidence supporting the incorporation of skew distribution into the error density in the returns and the use of power transformations of lagged log volatility.

Bayesian Forecasting for Financial Risk Management, Pre and Post the Global Financial Crisis

ABSTRACTValue‐at‐risk (VaR) forecasting via a computational Bayesian framework is considered. A range of parametric models is compared, including standard, threshold nonlinear and Markov switching generalized autoregressive conditional heteroskedasticity (GARCH) specifications, plus standard and nonlinear stochastic volatility models, most considering four error probability distributions: Gaussian, Student‐t, skewed‐tand generalized error distribution. Adaptive Markov chain Monte Carlo methods are employed in estimation and forecasting. A portfolio of four Asia–Pacific stock markets is considered. Two forecasting periods are evaluated in light of the recent global financial crisis. Results reveal that: (i) GARCH models outperformed stochastic volatility models in almost all cases; (ii) asymmetric volatility models were clearly favoured pre crisis, while at the 1% level during and post crisis, for a 1‐day horizon, models with skewed‐terrors ranked best, while integrated GARCH models were favoured at the 5% level; (iii) all models forecast VaR less accurately and anti‐conservatively post crisis. Copyright © 2011 John Wiley & Sons, Ltd.

Bayesian Forecasting for Financial Risk Management, Pre and Post the Global Financial Crisis

Value-at-Risk (VaR) forecasting via a computational Bayesian framework is considered. A range of parametric models are compared, including standard, threshold nonlinear and Markov switching GARCH specifications, plus standard and nonlinear stochastic volatility models, most considering four error probability distributions: Gaussian, Student-t, skewed-t and generalized error distribution. Adaptive Markov chain Monte Carlo methods are employed in estimation and forecasting. A portfolio of four Asia-Pacific stock markets is considered. Two forecasting periods are evaluated in light of the recent global financial crisis. Results reveal that: (i) GARCH models out-performed stochastic volatility models in almost all cases; (ii) asymmetric volatility models were clearly favoured pre-crisis; while at the 1% level during and post-crisis, for a 1 day horizon, models with skewed-t errors ranked best, while IGARCH models were favoured at the 5% level; (iii) all models forecasted VaR less accurately and anti-conservatively post-crisis.

A Class of Nonlinear Stochastic Volatility Models

This paper proposes a class of nonlinear stochastic volatility models based on the Box-Cox transformation which offers an alternative to the one introduced in Andersen (1994). The proposed class encompasses many parametric stochastic volatility models that have appeared in the literature, including the well known lognormal stochastic volatility model, and has an advantage in the ease with which different specifications on stochastic volatility can be tested. In addition, the functional form of transformation which induces marginal normality of volatility is obtained as a byproduct of this general way of modeling stochastic volatility. The efficient method of moments approach is used to estimate model parameters. Empirical results reveal that the lognormal stochastic volatility model is rejected for daily index return data but not for daily individual stock return data. As a consequence, the stock volatility can be well described by the lognormal distribution as its marginal distribution, consistent with the result found in a recent literature (cf Andersen et al (2001a)). However, the index volatility does not follow the lognormal distribution as its marginal distribution.

Empirical Asset Pricing with Nonlinear Risk Premia

In this paper we introduce a simple continuous-time asset pricing framework, based on general multi-dimensional diffusion processes, that combines semi-analytic pricing with a nonlinear specification for the market price of risk. Our framework guarantees existence of weak solutions of the nonlinear SDEs under the physical measure, thus allowing to work with nonlinear models for the real world dynamics not considered in the literature so far. It emerges that the additional flexibility in the time series modelling is econometrically relevant: a nonlinear stochastic volatility diffusion model for the joint time series of the S&P 100 and the VXO implied volatility index data shows superior forecasting power over the standard specifications for implied and realized variance forecasting.

On the use of non-linear transformations in Stochastic Volatility models

Stochastic Volatility models have been considered as a real alternative to conditional variance models, assuming that volatility follows a process different from the observed one. However, issues like the unobservable nature of volatility and the creation of “rich” dynamics give rise to the use of non-linear transformations for the volatility process. The Box–Cox transformation and its Yeo–Johnson variation, by nesting both the linear and the non-linear case, can be considered as natural functions to specify non-linear Stochastic Volatility models. In this framework, a fully Bayesian approach is used for parametric and log–volatility estimation. The new models are then investigated for their within-sample and out-of-sample performance against alternative Stochastic Volatility models using real financial data series.

A class of nonlinear stochastic volatility models and its implications for pricing currency options

A class of stochastic volatility (SV) models is proposed by applying the Box–Cox transformation to the volatility equation. This class of nonlinear SV (N-SV) models encompasses all standard SV models, including the well-known lognormal (LN) SV model. It allows to empirically compare and test all standard specifications in a very convenient way and provides a measure of the degree of departure from the classical models. A likelihood-based technique is developed for analyzing the model. Daily dollar/pound exchange rate data provide some evidence against LN model and strong evidence against all the other classical specifications. An efficient algorithm is proposed to study the economic importance of the proposed model on pricing currency options.