Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black–Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Inference for a Class of Stochastic Volatility Models Using Option and Spot Prices: Application of a Bivariate Kalman Filter

In this paper Bayesian methods are applied to a stochastic volatility model using both the prices of the asset and the prices of options written on the asset. Posterior densities for all model parameters, latent volatilities and the market price of volatility risk are produced via a Markov Chain Monte Carlo (MCMC) sampling algorithm. Candidate draws for the unobserved volatilities are obtained in blocks by applying the Kalman filter and simulation smoother to a linearization of a nonlinear state space representation of the model. Crucially, information from both the spot and option prices affects the draws via the specification of a bivariate measurement equation, with implied Black-Scholes volatilities used to proxy observed option prices in the candidate model. Alternative models nested within the Heston (1993) framework are ranked via posterior odds ratios, as well as via fit, predictive and hedging performance. The method is illustrated using Australian News Corporation spot and option price data.     

Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

This article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.     

An FFT approach for option pricing under a regime-switching stochastic interest rate model

In this article, we investigate the pricing of European-style options under a Markovian regime-switching Hull–White interest rate model. The parameters of this model, including the mean-reversion level, the volatility of the stochastic interest rate, and the volatility of an asset’s value, are modulated by an observable, continuous-time, finite-state Markov chain. A closed-form expression for the characteristic function of the logarithmic terminal asset price is derived. Then, using the fast Fourier transform, a price of a European-style option is computed. In a two-state Markov chain case, numerical examples and empirical studies are presented to illustrate the practical implementation of the model.     

Option pricing under the Merton model of the short rate in subdiffusive Brownian motion regime

In this paper, we propose an extension of the Merton short rate model, which reflects the subdiffusive nature of the short rate dynamics. The subdiffusive property is manifested by the random periods of time, during which the asset price does not change. We derive explicit formulas for European call and put options and present some simulation results for the case of α stable. Moreover, we discuss the implied volatility of this model.     

Empirical Performance and Asset Pricing in Hidden Markov Models

Abstract

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address leptokurtic feature, volatility smile, and volatility clustering effects of the asset return distributions. However, analytical tractability remains a problem for most alternative models. In this article, we study a class of hidden Markov models including Markov switching models and stochastic volatility models, that can incorporate leptokurtic feature, volatility clustering effects, as well as provide analytical solutions to option pricing. We show that these models can generate long memory phenomena when the transition probabilities depend on the time scale. We also provide an explicit analytic formula for the arbitrage-free price of the European options under these models. The issues of statistical estimation and errors in option pricing are also discussed in the Markov switching models.     

Comparison of asymmetric stochastic volatility models under different correlation structures

This paper conducts simulation-based comparison of several stochastic volatility models with leverage effects. Two new variants of asymmetric stochastic volatility models, which are subject to a logarithmic transformation on the squared asset returns, are proposed. The leverage effect is introduced into the model through correlation either between the innovations of the observation equation and the latent process, or between the logarithm of squared asset returns and the latent process. Suitable Markov Chain Monte Carlo algorithms are developed for parameter estimation and model comparison. Simulation results show that our proposed formulation of the leverage effect and the accompanying inference methods give rise to reasonable parameter estimates. Applications to two data sets uncover a negative correlation (which can be interpreted as a leverage effect) between the observed returns and volatilities, and a negative correlation between the logarithm of squared returns and volatilities.     

Conditional correlation in asset return and GARCH intensity model

In an asset return series, there is a conditional asymmetric dependence between current return and past volatility depending on the current return’s sign. To take into account the conditional asymmetry, we introduce new models for asset return dynamics in which frequencies of the up and down movements of asset price have conditionally independent Poisson distributions with stochastic intensities. The intensities are assumed to be stochastic recurrence equations of the GARCH type to capture the volatility clustering and the leverage effect. We provide an important linkage between our model and existing GARCH, explain how to apply maximum likelihood estimation to determine the parameters in the intensity model and show empirical results with the S&P 500 index return series.     

Derivative Pricing With Wishart Multivariate Stochastic Volatility

This paper deals with the pricing of derivatives written on several underlying assets or factors satisfying a multivariate model with Wishart stochastic volatility matrix. This multivariate stochastic volatility model leads to a closed-form solution for the conditional Laplace transform, and quasi-explicit solutions for derivative prices written on more than one asset or underlying factor. Two examples are presented: (i) a multiasset extension of the stochastic volatility model introduced by Heston (1993), and (ii) a model for credit risk analysis that extends the model of Merton (1974) to a framework with stochastic firm liability, stochastic volatility, and several firms. A bivariate version of the stochastic volatility model is estimated using stock prices and moment conditions derived from the joint unconditional Laplace transform of the stock returns.     

Bounds for the expected value of the stochastic Divisia's price index

In this article, we estimate bounds for the expected value of the stochastic Divisia's price index, that is, we assume that prices and quantities of the given commodities are stochastic processes with continuous time. We consider some special case of the stochastic model in which prices and quantities are described by the geometric Brownian motion. It is shown that the precision of this estimation depends rather on the volatility of prices than quantities volatilities.     

On the arbitrage price of European call options

We show that in a discrete price and discrete time model for option pricing, specifically that given by the Cox–Ross–Rubinstein model, the arbitrage price of a European call option can depend on parameters other than volatility (the standard deviation of the log asset price). We provide two theorems to illustrate this phenomenon. Our first theorem considers two securities with the same volatility so that at a specified time n₀, with probability near 1, the two securities are equal. If their call options differ, both the discounted securities will be martingales. Our second theorem considers two securities with the same volatility so that at times n = 0, ..., N ? 1 the securities are equal with probability near 1. If their call options differ, one of the discounted securities will be a martingale and the other discounted security will be a supermartingale.     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama–French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

A Standardized Normal-Laplace Mixture Distribution Fitted to Symmetric Implied Volatility Smiles

This article proposes to use a standardized version of the normal-Laplace mixture distribution for the modeling of tail-fatness in an asset return distribution and for the fitting of volatility smiles implied by option prices. Despite the fact that only two free parameters are used, the proposed distribution allows arbitrarily high kurtosis and uses one shape parameter to adjust the density function within three standard deviations for any specified kurtosis. For an asset price model based on this distribution, the closed-form formulas for European option prices are derived, and subsequently the volatility smiles can be easily obtained. A regression analysis is conducted to show that the kurtosis, which is commonly used as an index of tail-fatness, is unable to explain the smiles satisfactorily under the proposed model, because the additional shape parameter also significantly accounts for the deviations revealed in smiles. The effectiveness of the proposed parsimonious model is demonstrated in the practical examples where the model is fitted to the volatility smiles implied by the NASDAQ market traded foreign exchange options.     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama-French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

A Comparison of the Runs Test for Volatility Forecastability and the LM Test for GARCH Using Aggregated Returns

Christoffersen and Diebold (2000) have introduced a runs test for forecastable volatility in aggregated returns. In this note, we compare the size and power of their runs test and the more conventional LM test for GARCH by Monte Carlo simulation. When the true daily process is GARCH, EGARCH, or stochastic volatility, the LM test has better power than the runs test for the moderate-horizon returns considered by Christoffersen and Diebold. For long-horizon returns, however, the tests have very similar power. We also consider a qualitative threshold GARCH model. For this process, we find that the runs test has greater power than the LM test. Theresults support the use of the runs test with aggregated returns.     

Common Drifting Volatility in Large Bayesian VARs

The general pattern of estimated volatilities of macroeconomic and financial variables is often broadly similar. We propose two models in which conditional volatilities feature comovement and study them using U.S. macroeconomic data. The first model specifies the conditional volatilities as driven by a single common unobserved factor, plus an idiosyncratic component. We label this model BVAR with general factor stochastic volatility (BVAR-GFSV) and we show that the loss in terms of marginal likelihood from assuming a common factor for volatility is moderate. The second model, which we label BVAR with common stochastic volatility (BVAR-CSV), is a special case of the BVAR-GFSV in which the idiosyncratic component is eliminated and the loadings to the factor are set to 1 for all the conditional volatilities. Such restrictions permit a convenient Kronecker structure for the posterior variance of the VAR coefficients, which in turn permits estimating the model even with large datasets. While perhaps misspecified, the BVAR-CSV model is strongly supported by the data when compared against standard homoscedastic BVARs, and it can produce relatively good point and density forecasts by taking advantage of the information contained in large datasets.     

Option pricing for a stochastic volatility Lévy model with stochastic interest rates

An alternative option pricing model under a forward measure is proposed, in which asset prices follow a stochastic volatility Lévy model with stochastic interest rate. The stochastic interest rate is driven by the Hull–White process. By using an approximate method, we find a formulation for the European option in term of the characteristic function of the tail probabilities.     

Common-factor stochastic volatility modelling with observable proxy

Multi-asset modelling is of fundamental importance to financial applications such as risk management and portfolio selection. In this article, we propose a multivariate stochastic volatility modelling framework with a parsimonious and interpretable correlation structure. Building on well-established evidence of common volatility factors among individual assets, we consider a multivariate diffusion process with a common-factor structure in the volatility innovations. Upon substituting an observable market proxy for the common volatility factor, we markedly improve the estimation of several model parameters and latent volatilities. The model is applied to a portfolio of several important constituents of the S&P500 in the financial sector, with the VIX index as the common-factor proxy. We find that the prediction intervals for asset forecasts are comparable to those of more complex dependence models, but that option-pricing uncertainty can be greatly reduced by adopting a common-volatility structure. The Canadian Journal of Statistics 48: 36–61; 2020 © 2020 Statistical Society of Canada     

Splines for financial volatility

Summary.  We propose a flexible generalized auto-regressive conditional heteroscedasticity type of model for the prediction of volatility in financial time series. The approach relies on the idea of using multivariate B -splines of lagged observations and volatilities. Estimation of such a B -spline basis expansion is constructed within the likelihood framework for non-Gaussian observations. As the dimension of the B -spline basis is large, i.e. many parameters, we use regularized and sparse model fitting with a boosting algorithm. Our method is computationally attractive and feasible for large dimensions. We demonstrate its strong predictive potential for financial volatility on simulated and real data, and also in comparison with other approaches, and we present some supporting asymptotic arguments.     

DYNAMIC FACTOR MODELS

This paper introduces nonlinear dynamic factor models for various applications related to risk analysis. Traditional factor models represent the dynamics of processes driven by movements of latent variables, called the factors. Our approach extends this setup by introducing factors defined as random dynamic parameters and stochastic autocorrelated simulators. This class of factor models can represent processes with time varying conditional mean, variance, skewness and excess kurtosis. Applications discussed in the paper include dynamic risk analysis, such as risk in price variations (models with stochastic mean and volatility), extreme risks (models with stochastic tails), risk on asset liquidity (stochastic volatility duration models), and moral hazard in insurance analysis.

We propose estimation procedures for models with the marginal density of the series and factor dynamics parameterized by distinct subsets of parameters. Such a partitioning of the parameter vector found in many applications allows to simplify considerably statistical inference. We develop a two- stage Maximum Likelihood method, called the Finite Memory Maximum Likelihood, which is easy to implement in the presence of multiple factors. We also discuss simulation based estimation, testing, prediction and filtering.