Modelling High - frequency Economic Time Series

The minute-by-minute move of the Hang Seng Index (HSI) data over a four-year period is analysed and shown to possess similar statistical features as those of other markets. Based on a mathematical theorem [S. B. Pope and E. S. C. Ching, Phys. Fluids A 5, 1529 (1993)], we derive an analytic form for the probability distribution function (PDF) of index moves from fitted functional forms of certain conditional averages of the time series. Furthermore, following a recent work by Stolovitzky and Ching, we show that the observed PDF can be reproduced by a Langevin process with a move-dependent noise amplitude. The form of the Langevin equation can be determined directly from the market data. The availability of high-frequency economic time series, with a sampling rate of every few seconds, has generated a great deal of theoretical interest in the econometrics and the econophysics community[1–4]. Attempts have been made to devise models which produce time series with similar statistical characteristics as those of real markets. Many of these studies are based on variants of the Autogressive Conditional Heteroskedasticity (ARCH) process first introduced by Engle[5] to analyze the quarterly consumer price index in the UK over the period 1958 to 1977, and the generalised ARCH (GARCH) process which offers a more flexible description of the volatility memory effect (i.e., lag structure)[6]. The nonlinearity in the regression models makes it possible to generate probability distribution functions (PDF) with fat tails, a characteristic of financial data first noted by Mandelbrot[7]. However, since all these processes are discrete in time, an immediate question to ask is whether the quality of the modelling depends on the time unit chosen and if there is a time scale which is the most natural of all. Indeed, when the time step is not chosen properly, one has to either introduce many terms in the regression expression Preprint submitted to Elsevier Preprint 1 February 2008 0 10 20 30 τ (min) −0.2 0.0 0.2 0.4 0.6 0.8 1.0 C (τ )/ (0 ) HSI simulation Fig. 1. Normalised linear two-point correlation function of the minute-by-minute HSI moves over the four-year period 1994 to 1997. Note the weak oscillations of the correlation function, indicating a slightly under-damped behaviour. Also shown is the same correlation function calculated from the simulation using a Langevin equation. [the GARCH(p, q) model] to take into account memory effects, or to miss some of the important short-time statistics. An alternative approach, which partially circumvents the above difficulty, is to model the market price move as a continuous time process. Continuous time stochastic processes are quite familiar to physicists, ranging from simple Brownian motion to the fully-developed turbulence. In fact, the high-frequency market price movements have much in common with the velocity or temperature time series in turbulent flows[8–11], an analogy we exploit in this paper. To put this statement on more quantitative terms, let us first summarise two salient statistical features which seem to be universally true for all major stock indices[4]. (i) Short linear correlation of price moves — For a given stock index S(t), one may define the price move over a fixed time interval δ (say one minute), x(t) = S(t)− S(t− δ). (1) It has been shown that the “linear correlation” C(τ) = 〈x(t+ τ)x(t)〉 (2)