Exact Simulation of Wishart Multidimensional Stochastic Volatility Model

In this article, we propose an exact simulation method of the Wishart multidimensional stochastic volatility (WMSV) model, which was recently introduced by Da Fonseca et al. \cite{DGT08}. Our method is based onanalysis of the conditional characteristic function of the log-price given volatility level. In particular, we found an explicit expression for the conditional characteristic function for the Heston model. We perform numerical experiments to demonstrate the performance and accuracy of our method. As a result of numerical experiments, it is shown that our new method is much faster and reliable than Euler discretization method.Comment: 27 page

Wishart Stochastic Volatility: Asymptotic Smile and Numerical Framework

In this paper, a study of a stochastic volatility model for asset pricing is described. Originally presented by J. Da Fonseca, M. Grasselli and C. Tebaldi, the Wishart volatility model identifies the volatility of the asset as the trace of a Wishart process. Contrary to a classic multifactor Heston model, this model allows to add degrees of freedom with regard to the stochastic correlation. Thanks to its flexibility, this model enables a better fit of market data than the Heston model. Besides, the Wishart volatility model keeps a clear interpretation of its parameters and conserves an efficient tractability. Firstly, we recall the Wishart volatility model and we present a Monte Carlo simulation method in sight of the evaluation of complex options. Regarding stochastic volatility models, implied volatility surfaces of vanilla options have to be obtained for a short time. The aim of this article is to provide an accurate approximation method to deal with asymptotic smiles and to apply this procedure to the Wishart volatility model in order to well understand it and to make its calibration easier. Inspired by the singular perturbations method introduced by J.P Fouque, G. Papanicolaou, R. Sircar and K. Solna, we suggest an efficient procedure of perturbation for affine models that provides an approximation of the asymptotic smile (for short maturities and for a two-scale volatility). Thanks to the affine properties of the Wishart volatility model, the perturbation of the Riccati equations furnishes the expected approximations. The convergence and the robustness of the procedure are analyzed in practice but not in theory. The resulting approximations allow a study of the parameters influence and can also be used as a calibration tool for a range of parameters.Wishart processes; stochastic volatility models; stochastic; correlation; singular perturbation, asymptotic smile; Monte Carlo simulation

Nonparametric Bayes dynamic modeling of relational data

Symmetric binary matrices representing relations among entities are commonly collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being in inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lower-dimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the probability matrix space to the latent relational space, we obtain a flexible and computational tractable formulation. Employing P\`olya-Gamma data augmentation, an efficient Gibbs sampler is developed for posterior computation, with the dimension of the latent space automatically inferred. We provide some theoretical results on flexibility of the model, and illustrate performance via simulation experiments. We also consider an application to co-movements in world financial markets

On the valuation of fader and discrete barrier options in Heston's Stochastic Volatility Model

We focus on closed-form option pricing in Hestons stochastic volatility model, in which closed-form formulas exist only for few option types. Most of these closed-form solutions are constructed from characteristic functions. We follow this approach and derive multivariate characteristic functions depending on at least two spot values for different points in time. The derived characteristic functions are used as building blocks to set up (semi-) analytical pricing formulas for exotic options with payoffs depending on finitely many spot values such as fader options and discretely monitored barrier options. We compare our result with different numerical methods and examine accuracy and computational times. --exotic options,Heston Model,Characteristic Function,Multidimensional Fast Fourier Transforms

Stochastic Correlation and Risk Premia in Term Structure Models

This paper proposes and analyses a term structure model that allows for both stochastic correlation between underlying factors and an extended market price of risk specification. The issues of invariant transformation and different normalization are then considered so that a comparison between different restrictions can be made. We show that significant improvement in bond fitting is obtained by both allowing the market price of risk to have an extended affine form, and allowing the correlation between underlying factors to be stochastic as well as of variable sign. The overall model fit is more negatively impacted by the restriction on the market price of risk than the restriction of correlated factors. However, the stochastic correlation is priced significantly by market participants, though its impact on the risk premia reduces gradually as time to maturity increases. In addition, stochastic correlation is vital in obtaining good hedged portfolio positions. Certainly, the best hedged portfolio is the one that is built based on the model that takes into account both stochastic correlation and extended market price of risk.Term structure; Stochastic correlation, Risk premium; Wishart; Affine; Extended affine; Multidimensional CIR