Research Article: Modeling financial interval time series

Date Published: February 14, 2019

Publisher: Public Library of Science

Author(s): Liang-Ching Lin, Li-Hsien Sun, Cathy W.S. Chen.


In financial economics, a large number of models are developed based on the daily closing price. When using only the daily closing price to model the time series, we may discard valuable intra-daily information, such as maximum and minimum prices. In this study, we propose an interval time series model, including the daily maximum, minimum, and closing prices, and then apply the proposed model to forecast the entire interval. The likelihood function and the corresponding maximum likelihood estimates (MLEs) are obtained by stochastic differential equation and the Girsanov theorem. To capture the heteroscedasticity of volatility, we consider a stochastic volatility model. The efficiency of the proposed estimators is illustrated by a simulation study. Finally, based on real data for S&P 500 index, the proposed method outperforms several alternatives in terms of the accurate forecast.

Partial Text

There are a large number of models to develop in order to analyze financial data. Conventionally, most of well-proposed models are constructed by daily closing price. By doing so, some important valuable intra-daily information may be discarded such as maximum and minimum prices. According to the recent literature, we can treat the maximum and minimum prices as an interval valued observations. Symbolic data methodologies are applied to deal with this approach. For instance, Billard and Diday [1, 2] propose the evaluation of mean, variance, and covariance along with regression analysis based on interval valued observations. By integrating the time dependency factor, their method evolves into the analysis of interval time series. The recent research pays more attention to model and forecast the interval time series process. In this study, we propose an interval time series model, and apply the proposed model to forecast the consecutive interval.

Referring to Andersen et al. [13] and Aït-Sahalia et al. [14], the intra-daily log price, a.k.a. the high frequency data, on the i-th day follows the stochastic differential equation
dYt=μdt+σdWt,i−10 is a two-dimensional standard Brownian motion, and O is the initial log price and ζ is a random variable from the stationary distribution of σt2 and independent of (Bt, Wt). Referred to Bibby et al. [20], we assume the drift function b(⋅) to satisfy the mean reverting function, that is, b(σt2)=ρ(θ−σt2). Then, the non-negative diffusion function v(⋅) is uniquely specified by the invariant density of σt2. For example, if v(x) is proportion to a constant, x, or x2, the invariant density of σt2 is respectively normal, gamma, or inverse gamma distributions. However, if the intra-daily volatility is a stochastic processes, the Girsanov theorem can not be applied straightforwardly. In this section, we consider that σt2 is stochastic on the discrete time i = 1, 2, …, n, but has a stationary distribution during a fixed time interval t ∈ [i − 1, i].

We construct the observations as follows. Set the i-th intra-daily log price to satisfy
where WΔ is normally distributed with mean 0 and variance Δ, and the sampling frequency is Δ = 1/5000. The log opening, maximum, minimum, and closing prices are denoted by Yi = (Oi, Ui, Li, Ci), where 1 ≤ i ≤ n with n = 250, say. Set Ci = Oi+1, and repeat the above procedure for i = 1, 2, …, n − 1. We consider three practically oriented experiments based on the real observable data. According to the empirical evidence, the higher annualized market volatility is around 0.24, in contrast, the lower one is around 0.04. We also consider one particular case of the moderate volatility with the annualized market volatility being 0.12, and two cases of more violent volatilities with the annualized market volatilities being 0.36 and 0.48. So the daily volatilities are given by 0.04/250, 0.12/250, 0.24/250, 0.36/250, and 0.48/250. In addition, for the setting of drift term, we study two cases for the coefficient of variation: σ/μ = 1 (unit dispersion) and σ/μ = 2 (over dispersion).

We present the one-step predictions of an interval valued time series for the S&P 500 index. According to Arroyo et al. [4], the daily high/low prices of the S&P 500 index are utilized to compare the prediction performances of various methods. We make an one-step prediction by applying the rolling window where the historical data of previous year is used to estimate the parameters. The most challenging period is the financial crisis occurred on year 2008. Therefore, we first study the performances of various methods in one-step prediction on year 2008. Besides, we want to investigate the effect on the historical data. We select the periods of 2006 and 2017. For the former, the historical data of previous year (2005) has the similar pattern as the current year (2006). For the latter, the volatility in the historical data (2016) is violent compared to the predicted period (2017). Therefore, the prediction and estimation time periods are set to be
similar volatility period: prediction from January 2006 to December 2006; estimation from January 2005 to December 2005.High volatility period: prediction from January 2008 to December 2008; estimation from January 2007 to December 2007.dissimilar volatility period: prediction from January 2017 to December 2017; estimation from January 2016 to December 2016.

We propose the joint densities of daily log opening, maximum and closing prices and daily log opening, minimum and closing prices based on stochastic differential equations. Simulation studies show that the proposed estimators have higher efficiency than the conventional one using RE. In the real data analysis for S&P 500 index, the one-step forecasts of proposed method outperforms than several alternatives in terms of MDE, RE, and RN.




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