Stochastic Covariance Research Articles

The times change: multivariate subordination. Empirical facts

The management of correlation risk is of the utmost importance in several areas of investment banking: multi-asset derivatives pricing and hedging, optimal asset allocation, risk management, statistical arbitrage and many others.§ However, the modeling of correlation as a time-dependent quantity—as opposed to that of the volatility, say—is still in its infancy. In this article, we present a mechanism for the presence of a stochastic covariance matrix in financial markets, and provide an explanation for the returns' heavy tailed joint distribution. This mechanism relies on sampling returns according to an appropriate event time. Using CAC 40 high-frequency data, we compare the predictions of our model with real-world statistics and find very good agreement. §Even single asset derivatives desks are sensitive to shifts in correlation if, for instance, they are focused on a specific sectorial or geographic exposure.

Multivariate asset price dynamics with stochastic covariation

Stochastic volatility models such as those of Heston [Rev. Financial Stud., 1993, 6(2), 327–343] and Hull and White [J. Finance, 1987, 42(2), 281–300] are often used to model volatility risk in the pricing and hedging of contingent claims on risky assets. Recent empirical evidence has shown that these models under general specifications often do not fully capture the volatility dynamics observed in situ. This paper provides an analytical demonstration of the consequences of multivariate stochastic covariation on the pricing of contingent claims and suggests a hedging strategy for full delta neutrality.

On Valuation with Stochastic Proportional Hazard Models in Finance

While the proportional hazard model is recognized to be statistically meaningful for analyzing and estimating financial event risks, the existing literature that analytically deals with the valuation problems is very limited. In this paper, adopting the proportional hazard model in continuous time setting, we provide an analytical treatment for the valuation problems. The derived formulas, which are based on the generalized Edgeworth expansion and give approximate solutions to the valuation problems, are widely useful for evaluating a variety of financial products such as corporate bonds, credit derivatives, mortgage-backed securities, saving accounts and time deposits. Furthermore, the formulas are applicable to the proportional hazard model having not only continuous processes (e.g., Gaussian, affine, and quadratic Gaussian processes) but also discontinuous processes (e.g., Levy and time-changed Levy processes) as stochastic covariates. Through numerical examples, it is demonstrated that very accurate values can be quickly obtained by the formulas such a closed-form formula.

Exponential Levy Models Extended by a Jump to Default

This paper proposes a new dynamically consistent framework for joint valuation of equity derivatives and credit products, in which uncertainness of the economy is represented by Levy processes. In the framework, the pre-default stock price of a given firm follows an extended exponential Levy model, while the default arrival rate is given by the Cox proportional hazard model with stochastic covariates driven by Levy processes. Under the models, we find the solution of the pricing generator for evaluating equity and credit derivatives, and then derive the pricing formulas of equity call options and credit default swaps by utilizing the pricing generator. In the numerical examples, setting the variance gamma process and the Brownian motion as driving factors of the models, we compute term structure of credit default swaps and equity implied volatility skews. We also examine the impact of the convexity adjustment on term structure of credit spreads both analytically and numerically.

Stochastic Covariance Models

A new class of stochastic covariance models based on Wishart distribution is proposed. Three categories of dynamic correlation models are introduced depending on how the time-varying covariance matrix is formulated and whether or not it is a latent variable. A stochastic covariance filter is also developed for filtering and predicting covariances. Extensions of the basic models enable the study of the long memory properties of dynamic correlations, threshold correlation effects and portfolio analysis. Suitable parameterization in the stochastic covariance models and the stochastic covariance filter facilitate efficient calculation of the likelihood function in high-dimensional problems, no matter whether the covariance matrix is observable or latent. Monte Carlo experiments investigating finite sample properties of the maximum likelihood estimator are conducted. Two empirical examples are presented. One deals with the realized covariance of high frequency exchange rate data, while the other examines daily stock returns.

Stochastic modelling of unresolved eddy fluxes

A subgrid-scale parameterization scheme motivated by statistical closure theory, but employing statistics obtained from high-resolution direct numerical simulations, is applied to large eddy simulations of two-level quasigeostrophic turbulence on the sphere. It is shown that these parameterizations are consistent with the phenomenology of quasigeostrophic turbulence. The parameterizations consist of 2 × 2 dissipation and stochastic forcing covariance matrices at each wavenumber, with the off-diagonal elements of the matrices representing vertical mixing. Two flow regimes, characterized by their deformation scales, are considered, namely atmospheric and oceanic. In the former, the deformation scale is fully resolved, and the truncation scale is within the enstrophy cascading interial range. In the latter, the deformation scale is not fully resolved, and the truncation scale is within the energy cascading inertial range. It is demonstrated through numerical experiments that both stochastic and deterministic variants of the scheme give comparable results for the energy spectra in the atmospheric regime. In the oceanic regime, the stochastic variant again gives excellent results, but the deterministic variant is found to be numerically unstable.

Multivariate COGARCH(1, 1) processes

Multivariate $\operatorname {COGARCH}(1,1)$ processes are introduced as a continuous-time models for multidimensional heteroskedastic observations. Our model is driven by a single multivariate L\'{e}vy process and the latent time-varying covariance matrix is directly specified as a stochastic process in the positive semidefinite matrices. After defining the $\operatorname {COGARCH}(1,1)$ process, we analyze its probabilistic properties. We show a sufficient condition for the existence of a stationary distribution for the stochastic covariance matrix process and present criteria ensuring the finiteness of moments. Under certain natural assumptions on the moments of the driving L\'{e}vy process, explicit expressions for the first and second-order moments and (asymptotic) second-order stationarity of the covariance matrix process are obtained. Furthermore, we study the stationarity and second-order structure of the increments of the multivariate $\operatorname {COGARCH}(1,1)$ process and their "squares".

Stochastic Covariance Models

A new class of stochastic covariance models based on Wishart distribution is proposed. Three categories of dynamic correlation models are introduced depending on how the time-varying covariance matrix is formulated and whether or not it is a latent variable. A stochastic covariance filter is also developed for filtering and predicting covariances. Extensions of the basic models enable the study of the long memory properties of dynamic correlations, threshold correlation effects and portfolio analysis. Suitable parameterization in the stochastic covariance models and the stochastic covariance filter facilitate efficient calculation of the likelihood function in high-dimensional problems, no matter whether the covariance matrix is observable or latent. Monte Carlo experiments investigating finite sample properties of the maximum likelihood estimator are conducted. Two empirical examples are presented. One deals with the realized covariance of high frequency exchange rate data, while the other examines daily stock returns.

Cholesky-Langevin Model for Biochemical Networks with Accurate Account of Stochastic Effects of Couplings between Molecular Species

The dynamics of stochastic biochemical networks are often modeled using Langevin equations as an approximation to the underlying Chemical Master Equation. However, the coupling effects between different molecular species are neglected in current Langevin models. These coupling effects are important when intrinsic stochasticity due to the small-copy number nature and the discrete nature of the reactions system is non-negligible. Here we develop a generalized Langevin approach, called the Cholesky-Langevin Model (CLM), to incorporate rigorously both the fluctuations of chemical species and the stochasticity of their pairwise-coupling effects. We exploit the property of positive semi-definiteness of the stochastic covariance structure of a biochemical network of arbitrary stoichiometry. By comparing the calculated probabilistic landscapes using CLM with that from the conventional Langevin Model and the exact landscape obtained by directly solving the Chemical Master Equation using a method recently developed in (Cao and Liang, 2008), we show through several examples that the CLM model significantly improves the accuracy in approximating the underlying Chemical Master Equation. This study shows the importance of the coupling effects between molecular species in biochemical networks, and points out an effective approach for their rigorous account. We also discuss the biological implications of our approach. (Cao and Liang, BMC Systems Biology 2008, 2:30)

Efficient Simulation of the Wishart Model

In financial mathematics, Wishart processes have emerged as an efficient tool to model stochastic covariance structures. Their numerical simulation may be quite challenging since they involve matrix processes. In this article, we propose an extensive study of financial applications of Wishart processes. First, we derive closed-form formulas for option prices in the single-asset case. Then, we show the relationship between Wishart processes and Wishart law. Finally, we review existing discretization schemes (Euler and Ornstein-Uhlenbeck) and propose a new scheme, adapted from Heston's QEM discretization scheme. Extensive numerical results support our comparison of these three schemes.

Pricing a CDO on stochastically correlated underlyings

In this paper, we propose a method to price collateralized debt obligations (CDO) within Merton's structural model on underlyings with a stochastic mean-reverting covariance dependence. There are two key elements in our development, first we reduce dimensionality and complexity using principal component analysis on the assets' covariance matrix. Second, we approximate this continuous multidimensional structure using a tree method. Trinomial-tree models can be developed for both the principal components and the eigenvalues assuming the eigenvectors are constant over time and the eigenvalues are stochastic. Our method allows us to compute the joint default probabilities for k defaults of stochastically correlated underlyings and the value of CDOs in a fast manner, without having lost much accuracy. Furthermore we provide a method based on moments to estimate the parameters of the model.

The Times Change: Multivariate Subordination, Empirical Facts

The management of correlation risk is of the utmost importance in several areas of investment banking: multi-asset derivatives pricing and hedging, optimal asset allocation, risk management, statistical arbitrage and many others. However, the modelling of correlation as a physical phenomenon - as opposed to that of the volatility, say - is still in its infancy. In this note, we present a mechanism for the arising of a stochastic covariance matrix in financial markets, and therefore, an explanation of the returns heavy tailed joint distribution. This mechanism relies on sampling returns according to an appropriate event time. Using CAC 40 high frequency data, we compare the predictions of our model to real world statistics and find a very good agreement between both.

The Term Structure of Inflation Expectations

We present estimates of the term structure of inflation expectations, derived from an affine model of real and nominal yield curves. The model features stochastic covariation of inflation with the real pricing kernel, enabling us to extract a time-varying inflation risk premium. We fit the model not only to yields, but also to the yields' variance-covariance matrix, thus increasing identification power. We find that model-implied inflation expectations can differ substantially from break-even inflation rates when market volatility is high. Our model's ability to be updated weekly makes it suitable for real-time monetary policy analysis.

The structure of dynamic correlations in multivariate stochastic volatility models

This paper proposes two types of stochastic correlation structures for Multivariate Stochastic Volatility (MSV) models, namely the constant correlation (CC) MSV and dynamic correlation (DC) MSV models, from which the stochastic covariance structures can easily be obtained. Both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices, and for calculating Value-at-Risk (VaR) forecasts and optimal capital charges under the Basel Accord through the use of covariance matrices. A technique is developed to estimate the DC MSV model using the Markov Chain Monte Carlo (MCMC) procedure, and simulated data show that the estimation method works well. Various multivariate conditional volatility and MSV models are compared via simulation, including an evaluation of alternative VaR estimators. The DC MSV model is also estimated using three sets of empirical data, namely Nikkei 225 Index, Hang Seng Index and Straits Times Index returns, and significant dynamic correlations are found. The Dynamic Conditional Correlation (DCC) model is also estimated, and is found to be far less sensitive to the covariation in the shocks to the indexes. The correlation process for the DCC model also appears to have a unit root, and hence constant conditional correlations in the long run. In contrast, the estimates arising from the DC MSV model indicate that the dynamic correlation process is stationary.

The specification of GARCH models with stochastic covariates

AbstractA number of studies investigate whether various stochastic variables explain changes in return volatility by specifying the variables as covariates in a GARCH(1, 1) or EGARCH(1, 1) model. The authors show that these models impose an implicit constraint that can obscure the true role of the covariates in the analysis. They illustrate the problem by reconsidering the role of contemporaneous trading volume in explaining ARCH effects in daily stock returns. Once the constraint imposed in earlier research is relaxed, it is found that specifying volume as a covariate does little to diminish the importance of lagged squared returns in capturing the dynamics of volatility. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:911–934, 2008

Portfolio single index (PSI) multivariate conditional and stochastic volatility models

The paper develops the structure of parsimonious portfolio single index (PSI) multivariate conditional and stochastic constant correlation volatility models, and methods for estimating the underlying parameters. These multivariate estimates of volatility can be used for more accurate portfolio risk management, to enable efficient forecasting of value-at-risk (VaR) thresholds, and to determine optimal Basel Accord capital charges. A parsimonious portfolio single index approach for modelling the conditional and stochastic covariance matrices of a portfolio of assets is developed, and estimation methods for the conditional and stochastic volatility models are discussed.

Robust Estimation and Hypothesis Testing of Linear Contrasts in Analysis of Covariance with Stochastic Covariates

Estimators of parameters are derived by using the method of modified maximum likelihood (MML) estimation when the distribution of covariate X and the error e are both non-normal in a simple analysis of covariance (ANCOVA) model. We show that our estimators are efficient. We also develop a test statistic for testing a linear contrast and show that it is robust. We give a real life example.

Multi-period corporate default prediction with stochastic covariates

We provide maximum likelihood estimators of term structures of conditional probabilities of corporate default, incorporating the dynamics of firm-specific and macroeconomic covariates. For US Industrial firms, based on over 390,000 firm-months of data spanning 1980 to 2004, the term structure of conditional future default probabilities depends on a firm's distance to default (a volatility-adjusted measure of leverage), on the firm's trailing stock return, on trailing S&P 500 returns, and on US interest rates. The out-of-sample predictive performance of the model is an improvement over that of other available models.

Binary Regression with Stochastic Covariates

In binary regression the risk factor X has been treated in the literature as a non-stochastic variable. In most situations, however, X is stochastic. We present solutions applicable to such situations. We show that our solutions are more precise than those obtained by treating X as non-stochastic when, in fact, it is stochastic.

The Specification of GARCH Models with Stochastic Covariates

A number of studies investigate whether various stochastic variables explain changes in return volatility by specifying the variables as covariates in a GARCH(1,1) or EGARCH(1,1) model. We show that these models impose an implicit constraint which can obscure the true role of the covariates in the analysis. To illustrate the problem, we reconsider the role of contemporaneous trading volume in explaining ARCH effects in daily stock returns. Once we relax the constraint imposed in prior research, we find that specifying volume as a covariate does little to diminish the importance of lagged squared returns in capturing the dynamics of volatility.