How Frequently Does the Stock Price Jump? – An Analysis of High-Frequency Data with Microstructure Noises

The stock price is assumed to follow a jump-diffusion process which may exhibit time-varying volatilities. An econometric technique is then developed for this model and applied to high-frequency time series of stock prices that are subject to microstructure noises. Our method is based on first devising a localized particle filter and then employing fixed-lag smoothing in the Monte Carlo EM algorithm to perform the maximum likelihood estimation and inference. Using the intra-day IBM stock prices, we find that high-frequency data are crucial to disentangling frequent small jumps from infrequent large jumps. During the trading sessions, jumps are found to be frequent but small in magnitude, which is in sharp contrast to infrequent but large jumps when the market is closed. We also find that at the 5- or 10-minute sampling frequency, the conclusion will critically depend on whether heavy-tailed microstructure noises have been accounted for. Ignoring microstructure noises can, for example, lead to an overestimation of the jump intensity of 50% or more.Particle filtering, jump-diffusion, maximum likelihood, EM-algorithm.

Estimating the Structural Credit Risk Model When Equity Prices Are Contaminated by Trading Noises

The transformed-data maximum likelihood estimation (MLE) method for struc- tural credit risk models developed by Duan (1994) is extended to account for the fact that observed equity prices may have been contaminated by trading noises. With the presence of trading noises, the likelihood function based on the observed equity prices can only be evaluated via some nonlinear filtering scheme. We devise a particle filtering algorithm that is practical for conducting the MLE estimation of the structural credit risk model of Merton (1974). We implement the method on the Dow Jones 30 firms and on 100 randomly selected firms, and find that ignoring trading noises can lead to significantly over-estimating the firm's asset volatility. A simulation study is then conducted to ascertain the performance of the estimation method.Particle filtering, maximum likelihood, option pricing, credit risk, simulation

Estimating the Structural Credit Risk Model When Equity Prices Are Contaminated by Trading Noises

The transformed-data maximum likelihood estimation (MLE) method for structural credit risk models developed by Duan (1994) is extended to account for the fact that observed equity prices may have been contaminated by trading noises. With the presence of trading noises, the likelihood function based on the observed equity prices can only be evaluated via some nonlinear filtering scheme. We devise a particle filtering algorithm that is practical for conducting the MLE estimation of the structural credit risk model of Merton (1974). We implement the method on the Dow Jones 30 firms and on 100 randomly selected firms, and find that ignoring trading noises can lead to significantly over-estimating the firm’s asset volatility. The estimated magnitude of trading noise is in line with the direction that a firm’s liquidity will predict based on three common liquidity proxies. A simulation study is then conducted to ascertain the performance of the estimation method.Credit Risk; Maximum Likelihood; Microstructure; Option Pricing; Particle Filtering

Estimating and Testing Exponential Affine Term Structure Models by Kalman Filter

This paper proposes a unified state-space formulation for parameter estimation of exponential-affine term structure models. This class of models, charaterized by Duffie and Kan (1993), contains models such as Vasicek (1977), Cox, Ingersoll and Ross (1985) and Chen and Scott (1992), among others. The proposed method uses an approximate linear Kalman filter which only requires specifying the conditional mean and variance of the system in an approximate sense. The method allows for measurement errors in the observed yields to maturitiy, and can simultaneously deal with many yields on bonds with different maturities. A Monte Carlo study indicates thet the proposed method is a reliable procedure for moderate sample sizes. An empirical analysis of three existing exponential-affine term structure models is carried out using monthly U.S. Treasury yield data with four different maturities. Our test results indicate a strong rejection of all three models. Cette recherche propose une approche unificatrice pour l'estimation des paramètres de modèles de structure de taux d'intérêt de la classe exponentielle-affine. Cette famille de modèles, caractérisée par Duffie et Kan (1993), contient entre autres les modèles de Vasicek (1977), Cox, Ingersoll et Ross (1985) et Chen et Scott (1992). La méthode proposée utilise un filtre de Kalman approximatif qui requiert la spécification de l'espérance et de la variance conditionnelle du système. La méthode utilise simultanément plusieurs séries de rendements et permet l'ajout d'erreurs de mesure pour chaque serie. Une étude de simulation indique que la méthode proposée est fiable pour des échantillons de taille modérée. Une étude empirique utilisant trois modèles différents de la classe exponentielle-affine est présentée.Term Structure, Kalman Filter, Exponential Affine, State Space Model, Quasi Maximum Likelihood, Lagrange Multiplier Test, Structure à Terme, Filtre de Kalman, Exponentielle-affine, Modèle State-Space, Quasi-maximum de vraisemblance, Test du Multiplicateur de Lagrange

Empirical Martingale Simulation for Asset Prices

This paper proposes a simple modification to the standard Monte Carlo simulation procedure for computing the prices of derivative securities. The modification imposes the martingale property on the simulated sample paths of the underlying asset price. This procedure is referred to as the empirical martingale simulation (EMS). The EMS ensures that the price estimated by simulation satisfies rational option pricing bounds. The EMS also yields a substantial error reduction for the price estimate. The EMS can be easily coupled with the standard variance reduction methods to obtain greater computational efficiency. Simulation studies are conducted for European and Asian call options using both the Black and Scholes and GARCH option pricing frameworks. The results indicate that the EMS yields substantial variance reduction particularly for in- and at-the-money options. Cette étude propose une modification simple aux procédures traditionnelles de calcul de prix des produits dérivés par simulation de Monte Carlo. La modification impose la propriété de martingale aux trajectoires simulées de la variable d'état sous-jacente. L'utilisation de cette procédure assure que l'estimé de prix respecte les bornes rationnelles d'option tout en diminuant de façon substantielle la variance des estimés de prix. La procédure peut aisément être jumelée aux méthodes traditionnelles de réduction de variance afin d'obtenir une plus grande efficacité. Une étude de simulation est présentée pour des options d'achat Européennes et Asiatiques. Les résultats indiquent que la méthode obtient des réductions substantielle de la variance des estimés de prix et ce, particulièrement pour les options in et at the money .Martingale, Option Pricing, Monte Carlo Simulation, GARCH, Asian Options, Martingale, Evaluation des Options, Simulation de Monte Carlo, GARCH, Options Asiatiques

Jump starting GARCH: pricing and hedging options with jumps in returns and volatilities

This paper considers the pricing of options when there are jumps in the pricing kernel and correlated jumps in asset returns and volatilities. Our model nests Duan’s GARCH option models, where conditional returns are constrained to being normal, as well as mixed jump processes as used in Merton. The diffusion limits of our model have been shown to include jump diffusion models, stochastic volatility models and models with both jumps and diffusive elements in both returns and volatilities. Empirical analysis on the S&P 500 index reveals that the incorporation of jumps in returns and volatilities adds significantly to the description of the time series process and improves option pricing performance. In addition, we provide the first-ever hedging effectiveness tests of GARCH option models.Options (Finance) ; Hedging (Finance)