Multivariate Jump-diffusion Models Research Articles

Efficient estimation and filtering for multivariate jump–diffusions

This paper develops estimators of the transition density, filters, and parameters of multivariate jump–diffusion models. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. It is not necessary to diagonalize the volatility matrix. Our density and filter estimators converge at the canonical rate typically associated with exact Monte Carlo estimation. Our parameter estimators have the same asymptotic distribution as maximum likelihood estimators, which are often intractable for the class of models we consider. The results of this paper enable the empirical analysis of previously intractable models of asset prices and economic time series.

Goodness-of-Fit Test in Multivariate Jump Diffusion Models

In this article, we develop a new goodness-of-fit test for multivariate jump diffusion models. The test statistic is constructed by a contrast between an “in-sample” likelihood (or a likelihood of observed data) and an“out-of-sample” likelihood (or a likelihood of predicted data). We show that under the null hypothesis of a jump diffusion process being correctly specified, the proposed test statistic converges in probability to a constant that equals to the number of model parameters in the null model. We also establish the asymptotic normality for the proposed test statistic. To implement this method, we invoke a closed-form approximation to transition density functions, which results in a computationally efficient algorithm to evaluate the test. Using Monte Carlo simulation experiments, we illustrate that both exact and approximate versions of the proposed test perform satisfactorily. In addition, we demonstrate the proposed testing method in several popular stochastic volatility models for time series of weekly S&P 500 index during the period of January 1990 and December 2014, in which we invoke a linear affine relationship between latent stochastic volatility and the implied volatility index.

Multivariate Jump Diffusion Model with Markovian Contagion

Asset prices exhibit characteristics that significantly deviate from log-normality and display time-varying stochastics. There is ample evidence of jumps in one asset price or market leading to jumps in other assets' prices or markets. We propose a multivariate jump diffusion model with Markovian contagion to capture these asset price dynamics, where the channel of contagion is taken to periodically switch from an active to inactive state. We use a dynamic conditional correlation network approach to determine the contagion states and estimate the Markovian contagion model. Applying the model to an international equity and currency portfolio allocation shows the capabilities of the model in capturing fat tail characteristics of asset returns, as well as evaluate the extent of model risk, intra-asset class, inter-asset and inter-region contagion.

Efficient Parameter Estimation for Multivariate Jump-Diffusions

This paper develops unbiased Monte Carlo estimators of the transition and marginal densities of a multivariate jump-diffusion process. The drift, volatility, jump intensity, and jump magnitude are allowed to be state-dependent and non-affine. It is not necessary to diagonalize the volatility matrix. Our density estimators facilitate the parametric estimation of multivariate jump-diffusion models based on discretely observed data. The parameter estimators we propose have the same asymptotic behavior as maximum likelihood estimators under mild conditions. Our approach is found to be highly accurate and computationally efficient in a numerical case study of practical relevance.

RISK MANAGEMENT OF FINANCIAL CRISES: AN OPTIMAL INVESTMENT STRATEGY WITH MULTIVARIATE JUMP-DIFFUSION MODELS

AbstractThis paper provides an optimal asset allocation strategy to enhance risk management performance in the face of a financial crisis; this strategy entails constructing a good asset model – a multivariate jump-diffusion (MJD) model which includes idiosyncratic and systematic jumps simultaneously – and choosing suitable asset allocations and objective functions for fund management. This study also provides the dependence structure for the MJD model. The empirical implementation demonstrates that the proposed MJD model provides more detailed information about the financial crisis, allowing fund managers to determine an appropriate asset allocation strategy that enhances investment performance during the crisis.

Efficient Simulation of Value-at-Risk Under a Jump Diffusion Model: A New Method for Moderate Deviation Events

Importance sampling is a powerful variance reduction technique for rare event simulation, and can be applied to evaluate a portfolio’s Value-at-Risk (VaR). By adding a jump term in the geometric Brownian motion, the jump diffusion model can be used to describe abnormal changes in asset prices when there is a serious event in the market. In this paper, we propose an importance sampling algorithm to compute the portfolio’s VaR under a multi-variate jump diffusion model. To be more precise, an efficient computational procedure is developed for estimating the portfolio loss probability for those assets with jump risks. And the tilting measure can be separated for the diffusion and the jump part under the assumption of independence. The simulation results show that the efficiency of importance sampling improves over the naive Monte Carlo simulation from 9 to 277 times under various situations.

Pricing Mortality Securities with Correlated Mortality Indices

This paper proposes a stochastic model, which captures mortality correlations across countries and common mortality shocks, for analyzing catastrophe mortality contingent claims. To estimate our model, we apply particle filtering, a general technique that has wide applications in non-Gaussian and multivariate jump-diffusion models and models with nonanalytic observation equations. In addition, we illustrate how to price mortality securities with normalized multivariate exponential titling based on the estimated mortality correlations and jump parameters. Our results show the significance of modeling mortality correlations and transient jumps in mortality security pricing.

Optimal Filtering of Jump Diffusions: Extracting Latent States from Asset Prices

This paper provides an optimal filtering methodology in discretely observed continuoustime jump-diffusion models. Although the filtering problem has received little attention, it is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines time-discretization schemes with Monte Carlo methods. It is quite general, applying in nonlinear and multivariate jump-diffusion models and models with nonanalytic observation equations. We provide a detailed analysis of the filter’s performance, and analyze four applications: disentangling jumps from stochastic volatility, forecasting volatility, comparing models via likelihood ratios, and filtering using option prices and returns. (JEL C11, C13, C15, C51, C52, G11, G12, G17) This paper develops a filtering approach for learning about unobserved shocks and states from discretely observed prices generated by continuous-time jumpdiffusion models. The optimal filtering problem, which has received little attention, is the natural companion to the parameter estimation problem as filters allow researchers to analyze and use their models for practical applications. 1 Despite its usefulness, there is currently no general method for solving the filtering problem in continuous-time models.

MCMC maximum likelihood for latent state models

Abstract This paper develops a pure simulation-based approach for computing maximum likelihood estimates in latent state variable models using Markov Chain Monte Carlo methods (MCMC). Our MCMC algorithm simultaneously evaluates and optimizes the likelihood function without resorting to gradient methods. The approach relies on data augmentation, with insights similar to simulated annealing and evolutionary Monte Carlo algorithms. We prove a limit theorem in the degree of data augmentation and use this to provide standard errors and convergence diagnostics. The resulting estimator inherits the sampling asymptotic properties of maximum likelihood. We demonstrate the approach on two latent state models central to financial econometrics: a stochastic volatility and a multivariate jump-diffusion models. We find that convergence to the MLE is fast, requiring only a small degree of augmentation.