Unfortunately, the standard second-order diffusion framework ( 3) is not tailored to capture these stylized facts. The empirical distribution of stock returns typically exhibits skewness and excess kurtosis, which could be induced by various macroeconomic shocks, such as the unemployment announcement, the Gulf war, and the oil crisis. The empirical analysis of real-world data sets demonstrates that it is in general reasonable to suppose that the volatility is a quadratic function in stock markets. Furthermore, Yan and Mei developed the generalized likelihood ratio test to check the empirical finding of Nicolau. In Nicolau’s empirical studies on the American stock markets, a regular pattern in all estimators of drift and diffusion can be observed: the drift is clearly linear, the volatility is a quadratic function with a minimum in the neighborhood of zero, and the specification fits the nonparametric estimators very well. In particular, Nicolau pointed out that is a promising model for stock returns and possesses some interesting properties. On the other hand, it has the advantage that the problem of its estimation can be reduced to the determination of some low-dimensional parameters by applying more efficient statistical methods (see for details). Therefore, much attention has been paid to the issue of specifying the functions forms for continuous-time diffusion models. Model misspecification may lead to misleading conclusions in estimation and hypothesis testing. Unfortunately, the existing economic theory generally provides little guidance about the precise specification of them. In financial markets, the correct specification of drift and volatility is essential and instructive among practitioners in obtaining valid conclusions. The model suggests directly modeling return rather than stock price and meets many general properties of stock returns such as stationarity in the mean, nonnormality of the distribution and weak autocorrelation. Second, in the context of stock prices, represents stock return and indicates the cumulation of. First, the model accommodates nonstationary integrated stochastic process that can be made stationary by differencing. Furthermore, Hanif studied the nonparametric estimation of the drift and diffusion functions using an asymmetric kernel and proved that the estimators are consistent and asymptotically normal.Īs pointed out by Nicolau, model ( 1) is especially useful in empirical finance. proposed the reweighted estimation of the diffusion function and investigated the consistency of the estimator. Thereafter, Wang and Lin presented the local linear estimation of the diffusion and drift functions and proved that the estimators are weakly consistent. In this model, can also be expressed as the integrated process For model ( 1), a nonparametric approach which is based on the infinitesimal generator and Taylor series expansion has been developed to estimate the drift and diffusion functions. is directly observable and differentiable. is a standard one-dimensional Brownian motion. Nicolau considered a second-order diffusion process which is defined by where and are the drift and diffusion functions, respectively. IntroductionĬontinuous-time stochastic processes have been widely used to model securities prices for option valuation. The empirical analysis of stock market data from North America, Asia, and Europe is provided for illustration. Furthermore, we consider a likelihood ratio test to identify the statistically significant presence of jump factor. A simulation study is conducted to evaluate the performance of the estimation method in finite samples. We develop an appropriate maximum likelihood approach to estimate model parameters. This paper proposes a second-order jump diffusion model to study the jump dynamics of stock market returns via adding a jump term to traditional diffusion model.
0 Comments
Leave a Reply. |