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In this article, we present a new class of pricing models that extend the application of Wishart processes to the so-called stochastic local volatility (or hybrid) pricing paradigm. This approach combines the advantages of local and stochastic volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multidimensional specifications for the stochastic volatility component. Our work tries to fill the gap: we introduce two hybrid models in which the stochastic volatility dynamics is described by means of a Wishart process. The proposed parametrizations not only preserve the desirable features of existing Wishart-based models but significantly enhance the ability of reproducing market prices of vanilla options.

We extend the existing small-time asymptotics for implied volatilities under the Heston stochastic volatility model to the multifactor volatility Heston model, which is also known as the Wishart multidimensional stochastic volatility model (WMSV). More explicitly, we show that the approaches taken in Forde and Jacquier (2009) and Forde, Jacqiuer and Lee (2012) are applicable to the WMSV model under mild conditions, and obtain explicit small-time expansions of implied volatilities.

In this article, we focus upon a family of matrix valued stochastic processes and study the problem of determining the smallest time such that their Laplace transforms become infinite. In particular, we concentrate upon the class of Wishart processes, which have proved to be very useful in different applications by their ability in describing non-trivial dependence. Thanks to this remarkable property we are able to explain the behavior of the explosion times for the Laplace transforms of the Wishart process and its time integral in terms of the relative importance of the involved factors and their correlations., SCOPUS: ar.j, info:eu-repo/semantics/published

In this article, we obtain an equation for the high-dimensional limit measure of eigenvalues of generalized Wishart processes, and the results is extended to random particle systems that generalize SDEs of eigenvalues. We also introduce a new set of conditions on the coefficient matrices for the existence and uniqueness of a strong solution for the SDEs of eigenvalues. The equation of the limit measure is further discussed assuming self-similarity on the eigenvalues., 28 pages

International audience; We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

This paper proposes a multivariate stochastic volatility-in-vector autoregression model called the conditional autoregressive inverse Wishart-in-VAR (CAIW-in-VAR) model as a framework for studying the real effects of uncertainty shocks. We make three contributions to the literature. First, the uncertainty shocks we analyze are estimated directly from macroeconomic data so they are associated with changes in the volatility of the shocks hitting the macroeconomy. Second, we advance a new approach to identify uncertainty shocks by placing limited economic restrictions on the first and second moment responses to these shocks. Third, we consider an extension of the sign restrictions methodology of Uhlig (2005) to uncertainty shocks. To illustrate our methods, we ask what is the role of financial markets in transmitting uncertainty shocks to the real economy? We find evidence that an increase in uncertainty leads to a decline in industrial production only if associated with a deterioration in financial conditions.

In this paper, we introduce two new matrix stochastic processes: fractional Wishart processes and e -fractional Wishart processes with integer indices which are based on the fractional Brownian motions and then extend e -fractional Wishart processes to the case with non-integer indices. Both processes include classic Wishart processes (if the Hurst index H equals 1 2 ) and present serial correlation of stochastic processes. Applying e -fractional Wishart processes to financial volatility theory, the financial models account for the stochastic volatilities of the assets and for the stochastic correlations not only between the underlying assets’ returns but also between their volatilities and for stochastic serial correlation of the relevant assets.

In this article, we propose an exact simulation method of the Wishart multidimensional stochastic volatility (WMSV) model—a single asset model with a multidimensional Wishart variance process. Our method is based on analysis of the conditional characteristic function of the log-price given a terminal volatility level. In particular, we found an explicit expression for the conditional characteristic function for the Heston model. Numerical experiments demonstrate that our new method is much faster and reliable than the Euler discretization method.

Les processus de Dirichlet matriciels, en reference a leur mesure reversible, apparaissent de maniere naturelle dans de nombreux modeles differents en probabilite. En utilisant la langage des operateurs de diffusion et la theorie des equations de bord, nous decrivont les processus de Dirichlet sur le simplexe matriciel et proposont deux modeles pour les processus de Dirichlet matriciels, qui peuvent etre realises par les projections diverses, par le movement brownien sur le groupe unitaire special et par les processus de Wishart.

In this article, we deal with calibration and Monte Carlo simulation of the Wishart stochastic volatility model. Despite the analytical tractability of the considered model, being of affine type, t...

We consider stochastic partial differential equations appearing as Markovian lifts of matrix-valued (affine) Volterra-type processes from the point of view of the generalized Feller property (see, e.g., Dörsek and Teichmann in A semigroup point of view on splitting schemes for stochastic (partial) differential equations, 2010. arXiv:1011.2651). We introduce in particular Volterra Wishart processes with fractional kernels and values in the cone of positive semidefinite matrices. They are constructed from matrix products of infinite dimensional Ornstein–Uhlenbeck processes whose state space is the set of matrix-valued measures. Parallel to that we also consider positive definite Volterra pure jump processes, giving rise to multivariate Hawkes-type processes. We apply these affine covariance processes for multivariate (rough) volatility modeling and introduce a (rough) multivariate Volterra Heston-type model., Decisions in Economics and Finance, 42 (2)

Outperformance options allow investors to benefit from a view on the relative performance of two underlying assets without taking any directional exposure to the evolution of the market. These structures exhibit high sensitivity to the correlation between the underlying assets and are usually priced assuming constant instantaneous correlations. &nbsp This article considers a multi-asset model based on Wishart processes that accounts for stochastic volatility and for stochastic correlations between the assets returns, as well as between their volatilities. Under the assumptions of the model this article provides semi-closed form solutions for the price of outperformance options. The article shows that the price of these options depends crucially on the term structure of the correlation corresponding to the assets returns. Furthermore, the comparison of the prices obtained under this model and under other models with constant correlations commonly used by financial institutions reveals the existence of a stochastic correlation premium.

We develop a bivariate stochastic volatility model that allows for dynamic market exposure. The expected return on a stock depends on beta's co-movement with the stochastic discount factor and deviates from the standard security market line when beta risk is priced. When estimating the model on returns and options for a large number of firms we find that allowing for beta risk helps explain the expected returns on low and high beta stocks that are challenging for standard factor models. Overall, we find strong evidence that accounting for beta risk results in better model fit.

We consider affine Markov processes taking values in convex cones. In particular, we characterize all affine processes taking values in irreducible symmetric cones in terms of certain Levy–Khintchine triplets. This is the natural, coordinate-free formulation of the theory of Wishart processes on positive semidefinite matrices, as put forward by Bru (J Theor Probab 4(4):725–751, 1991) and Cuchiero et al. (Ann Appl Probab 21(2):397–463, 2011), in the more general context of symmetric cones, which also allows for simpler, alternative proofs.

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate., Comment: Accepted on: Journal of Applied Probability 51(3), 2014

Let Xe be a small perturbation Wishart process with values in the set of positive definite matrices of size m, i.e., the process Xe is the solution of stochastic differential equation with non-Lipschitz diffusion coefficient: \(dX_t^\varepsilon = \sqrt {\varepsilon X_t^\varepsilon } dB_t + dB'_t \sqrt {\varepsilon X_t^\varepsilon } + \rho I_m dt\), X0 = x, where B is an m×m matrix valued Brownian motion and B′ denotes the transpose of the matrix B. In this paper, we prove that \(\left\{ {{{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right. \kern-\nulldelimiterspace} {\sqrt {\varepsilon h^2 (\varepsilon )} ,\varepsilon > 0}}} \right\}\) satisfies a large deviation principle, and \({{\left( {X_t^\varepsilon - X_t^0 } \right)} \mathord{\left/ {\vphantom {{\left( {X_t^\varepsilon - X_t^0 } \right)} {\sqrt \varepsilon }}} \right. \kern-\nulldelimiterspace} {\sqrt \varepsilon }}\) converges to a Gaussian process, where h(ɛ) → +∞ and \(\sqrt \varepsilon h(\varepsilon ) \to 0\) as e → 0. A moderate deviation principle and a functional central limit theorem for the eigenvalue process of Xe are also obtained by the delta method.

International audience; In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.

In order to study beta risk, we develop a multivariate stochastic volatility model in which individual equity and market returns co-vary stochastically. In the model, the stochastic covariance matrix of an individual equity return with the market follows a bivariate Wishart process and the associated beta risk is priced. We estimate the model on returns and options jointly for a large cross-section of stocks. When analyzing the modelís empirical performance, we Önd that it outperforms the standard rolling regression approach that is prevalent in empirical asset pricing.

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or HJM modeling, can be consolidated. We model a num\'eraire process and multiplicative spreads between Libor rates and simply compounded OIS rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multi-curve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed-form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.

We study the optimal portfolio choice of international investors when variances and correlations are stochastic. We assume that the returns from the perspective of the domestic investor are driven by a Wishart Affine Stochastic Correlation (WASC) model. We show that this also holds from the perspective of the foreign investor and give the relations between the variance-covariance matrices and the parameters of its dynamics in both currencies. Stochastic second moments have an impact on risk and returns that characterize the domestic and the foreign investment opportunity sets. Optimal portfolios and hedging demands of international investors differ due to their dependence on exchange rate variances and correlations. The benefits from investing can be different for domestic and foreign investors, and can also react differently to changes in second moments. These findings hold both in complete and incomplete markets.

By exploiting the flexibility of the Wishart process, we propose an application of this framework to the pricing of Chicago Board Options Exchange (CBOE) volatility index (VIX) options. Our methodology is analytically tractable and yet flexible enough to efficiently price CBOE VIX options. In particular, the dynamics for the CBOE VIX is carried out in a linear affine way and the discounted Laplace transform exhibits an exponentially affine property. The tractable model structure lightens the computational burden and facilitates a fast identification of the parameter estimates. We empirically show that modeling the stochastic co-volatility factor can significantly improve the in-sample fitting results due to the improved modeling of higher conditional moments in the underlying transition probability density.

The paper develops two Dynamic Conditional Correlation (DCC) models, namely the Wishart DCC (WDCC) model and the Matrix-Exponential Conditional Correlation (MECC) model. The paper applies the WDCC approach to the exponential GARCH (EGARCH) and GJR models to propose asymmetric DCC models. We use the standardized multivariate t-distribution to accommodate heavy-tailed errors. The paper presents an empirical example using the trivariate data of the Nikkei 225, Hang Seng and Straits Times Indices for estimating and forecasting the WDCC-EGARCH and WDCC-GJR models, and compares the performance with the asymmetric BEKK model. The empirical results show that AIC and BIC favour the WDCC-EGARCH model to the WDCC-GJR and asymmetric BEKK models. Moreover, the empirical results indicate that the WDCC-EGARCH-t model produces reasonable VaR threshold forecasts, which are very close to the nominal 1% to 3% values.

We give formulas for the moments and the expectation of some functionals of Wishart process $X=(X_t)_{t\ge 0}$. These formulas are obtained via Ito calculus for matrix processes. Taking $t=1$ we can derive immediately the same results for centered and noncentered Wishart matrices.

The first part of this thesis is devoted to the study of an Affine Term Structure Model (ATSM) where we use Wishart-like processes to model the stochastic variance-covariance of interest rates. This work was initially motivated by some thoughts on calibration and model risk in hedging interest rates derivatives. The ambition of our work is to build a model which reduces as much as possible the noise coming from daily re-calibration of the model to the market. It is standard market practice to hedge interest rates derivatives using models with parameters that are calibrated on a daily basis to fit the market prices of a set of well chosen instruments (typically the instrument that will be used to hedge the derivative). The model assumes that the parameters are constant, and the model price is based on this assumption; however since these parameters are re-calibrated, they become in fact stochastic. Therefore, calibration introduces some additional terms in the price dynamics (precisely in the drift term of the dynamics) which can lead to poor P&L explain, and mishedging. The initial idea of our research work is to replace the parameters by factors, and assume a dynamics for these factors, and assume that all the parameters involved in the model are constant. Instead of calibrating the parameters to the market, we fit the value of the factors to the observed market prices.A large part of this work has been devoted to the development of an efficient numerical framework to implement the model. We study second order discretization schemes for Monte Carlo simulation of the model. We also study efficient methods for pricing vanilla instru- ments such as swaptions and caplets. In particular, we investigate expansion techniques for prices and volatility of caplets and swaptions. The arguments that we use to obtain the expansion rely on an expansion of the infinitesimal generator with respect to a perturbation factor. Finally we have studied the calibration problem. As mentioned before, the idea of the model we study in this thesis is to keep the parameters of the model constant, and calibrate the values of the factors to fit the market. In particular, we need to calibrate the initial values (or the variations) of the Wishart-like process to fit the market, which introduces a positive semidefinite constraint in the optimization problem. Semidefinite programming (SDP) gives a natural framework to handle this constraint.The second part of this thesis presents some of the work I have done on the hedging of interest rate risk in ALM. This work was motivated by the business at Cr ́edit Agricole S.A. and in particular by the Financial Division of the bank. The purpose of this part of the dis- sertation is twofold. First we want to communicate on a field of Finance which is less known by the mathematical finance community, and presents some interesting modeling challenges. Secondly we try to present an original approach to modeling and hedging interest rate risk.Chapter 6 is an attempt to formalize some of concepts that are used in practice in ALM. We recall some of the key concepts such as the schedule of an asset or a liability and the interest rate gap, and introduce a new concept : the notion of envelope. This concept will look familiar to people used to derivatives pricing, and the hedging of the interest rate riskof an asset or a liability (closing the gap using the language of ALM) is very similar to the hedging of an option. The remaining chapters present the results of the work we have done in three different projects.; Ce mémoire présente une partie du travail de recherche que j’ai effectué dans le cadre de ma thèse. Ce travail a été principalement effectué durant ma permanence au Groupe de Recherche Opérationnelle du Crédit Agricole. Etant donné le contexte dans lequel j’ai conduit mon travail de thèse, celui-ci a été principalement inspiré, et parfois meme directement motive par les besoins concrets des équipes opérationnelles. Le document contient deux par- ties qui sont indépendantes, et néanmoins représentent deux faces de la meme activité: la couverture du risque de taux d’intéret. Dans la première partie du document on étudie un modèle affine de la dynamique de la courbe des taux, ou un processus affine dans l’espace des matrices semidéfinies positives de type Wishart est utilis ́e pour modéliser la dynamique de la variance-covariance entre les taux d’intéret. L’ambition de notre travail est la construction d’un modèle qui fournisse une couverture globale et robuste des risques d’un book de produits exotiques de taux. Ce travail a conduit `a une pr ́e-publication [AAP14]. La deuxième partie du document est dédiée `a la couverture du risque de taux dans la gestion actif-passif du bilan d’une banque. L’objectif de cette seconde partie est de formaliser les principaux concepts qui sont utilisés en pratique dans ce domaine.

The first part of this thesis is devoted to the study of an Affine Term Structure Model (ATSM) where we use Wishart-like processes to model the stochastic variance-covariance of interest rates. This work was initially motivated by some thoughts on calibration and model risk in hedging interest rates derivatives. The ambition of our work is to build a model which reduces as much as possible the noise coming from daily re-calibration of the model to the market. It is standard market practice to hedge interest rates derivatives using models with parameters that are calibrated on a daily basis to fit the market prices of a set of well chosen instruments (typically the instrument that will be used to hedge the derivative). The model assumes that the parameters are constant, and the model price is based on this assumption; however since these parameters are re-calibrated, they become in fact stochastic. Therefore, calibration introduces some additional terms in the price dynamics (precisely in the drift term of the dynamics) which can lead to poor P&L explain, and mishedging. The initial idea of our research work is to replace the parameters by factors, and assume a dynamics for these factors, and assume that all the parameters involved in the model are constant. Instead of calibrating the parameters to the market, we fit the value of the factors to the observed market prices.A large part of this work has been devoted to the development of an efficient numerical framework to implement the model. We study second order discretization schemes for Monte Carlo simulation of the model. We also study efficient methods for pricing vanilla instru- ments such as swaptions and caplets. In particular, we investigate expansion techniques for prices and volatility of caplets and swaptions. The arguments that we use to obtain the expansion rely on an expansion of the infinitesimal generator with respect to a perturbation factor. Finally we have studied the calibration problem. As mentioned before, the idea of the model we study in this thesis is to keep the parameters of the model constant, and calibrate the values of the factors to fit the market. In particular, we need to calibrate the initial values (or the variations) of the Wishart-like process to fit the market, which introduces a positive semidefinite constraint in the optimization problem. Semidefinite programming (SDP) gives a natural framework to handle this constraint.The second part of this thesis presents some of the work I have done on the hedging of interest rate risk in ALM. This work was motivated by the business at Cr ́edit Agricole S.A. and in particular by the Financial Division of the bank. The purpose of this part of the dis- sertation is twofold. First we want to communicate on a field of Finance which is less known by the mathematical finance community, and presents some interesting modeling challenges. Secondly we try to present an original approach to modeling and hedging interest rate risk.Chapter 6 is an attempt to formalize some of concepts that are used in practice in ALM. We recall some of the key concepts such as the schedule of an asset or a liability and the interest rate gap, and introduce a new concept : the notion of envelope. This concept will look familiar to people used to derivatives pricing, and the hedging of the interest rate riskof an asset or a liability (closing the gap using the language of ALM) is very similar to the hedging of an option. The remaining chapters present the results of the work we have done in three different projects.; Ce mémoire présente une partie du travail de recherche que j’ai effectué dans le cadre de ma thèse. Ce travail a été principalement effectué durant ma permanence au Groupe de Recherche Opérationnelle du Crédit Agricole. Etant donné le contexte dans lequel j’ai conduit mon travail de thèse, celui-ci a été principalement inspiré, et parfois meme directement motive par les besoins concrets des équipes opérationnelles. Le document contient deux par- ties qui sont indépendantes, et néanmoins représentent deux faces de la meme activité: la couverture du risque de taux d’intéret. Dans la première partie du document on étudie un modèle affine de la dynamique de la courbe des taux, ou un processus affine dans l’espace des matrices semidéfinies positives de type Wishart est utilis ́e pour modéliser la dynamique de la variance-covariance entre les taux d’intéret. L’ambition de notre travail est la construction d’un modèle qui fournisse une couverture globale et robuste des risques d’un book de produits exotiques de taux. Ce travail a conduit `a une pr ́e-publication [AAP14]. La deuxième partie du document est dédiée `a la couverture du risque de taux dans la gestion actif-passif du bilan d’une banque. L’objectif de cette seconde partie est de formaliser les principaux concepts qui sont utilisés en pratique dans ce domaine.

Wishart processes have been first introduced by Bru [24] for some applications in biology on the perturbation of experimental data. Their definition and main mathematical properties are described in her paper [25]. They are also named because, as we will see, their marginal laws follow Wishart distributions. These distributions have been introduced by Wishart [124] in 1928. They arise naturally in statistics when estimating the covariance matrix of a Gaussian vector.

We analyze the optimal portfolio choice in a multi-asset Wishart-model in which return variances and correlations are stochastic and subject to jump risk. The optimal portfolio is characterized by the positions in stock diffusion risk, variance-covariance diffusion risk, and jump risk. We find that including jumps in the second moments changes the optimal positions and particularly variance-covariance hedging demands significantly. Erroneously omitting these jumps gives rise to substantial model risk. Furthermore, variance-covariance jump risk can have a significant impact on potential utility gains when the market is completed by adding derivatives. As a robustness check, we compare our results to those obtained for other parametrizations of Wishart-models from the literature as well as to various single-asset models.