On Some Sample Path Properties of Intra-Day Futures Prices
利用国债期货的逐笔交易数据，构建连续时间资产价格模型，检验市场效率，发现日内价格不服从马尔可夫更新过程，但交易时间近似泊松过程。
This paper develops a time-series model for continuous time asset prices and then uses tick-by-tick data from Treasury bill futures to develop both a definition and test for efficiency in the continuous time case. The results suggest that intra-day data on futures prices do not behave like a Markov Renewal process; rather, lagged values of futures prices do have some predictive power. In addition, trading times are not useful in predicting futures prices. Finally, we estimate the bid-ask spread and show that even after adjusting for this spread, the serial dependence between current and lagged returns remains. The multitude of studies concerning efficiency in futures markets support the proposition that a Martingale approximation is reasonable for most commodity and capital asset markets, while the same data reject Gaussian processes as an appropriate model.1 All of these studies, however, use either close to close prices or open to open prices in the estimation process. The choice of daily data is arbitrary; a natural question concerns whether, based on continuous time data, futures prices can be shown to be realizations of continuous time Markov processes and whether they can be represented as stochastically linear processes. In this paper, we use intra-day, tick-by-tick, data on Treasury bills futures and develop both a definition and a test for efficiency in the continuous time case. Observations on continuous time prices yield two separate time series: the trading prices and times. As a result, we claim that the standard procedures for tests of market efficiency must be replaced by two separate necessary conditions; the first is the usual condition that successive price changes are independent; the second requires that the recurrence times between trades obey a Poisson process. We apply the above definitions to intra-day futures prices for Treasury bills for a 57 day period in 1983. The results indicate that the Markov model does not hold for intra-day futures prices but the trading times do seem to behave approximately as a Poisson process. Received for publication January 29, 1988. Revision accepted for publication July 19, 1989. * City University of New York and State University of New York at Stony Brook, respectively. We gratefully acknowledge financial support for this research from the Center for the Study of Futures Markets at Columbia University. Stephanie Dieringer provided invaluable help as a research assistant for this project. 1 Several studies have examined the martingale property. For a summary of these results see, for example, Kamara (1982). 2 See, for example, Fama (1965), Mandlebrot (1963), Stevenson and Bear (1970), and Neftci and Policano (1984). Some studies that do analyze intraday data include Feinstone (1985) and Hinich and Patterson (1985).