Moving averages are probably the most popular and elemental analysis tool in finance, widely known and applied both by professional and amateur traders. Such a large favour is due to their simplicity and intuitive meaning, which can help to understand the more or less hidden trend of an evolving dataset, filtering some of the noise. Moving averages are often studied and taken into account in modeling price dynamics both in financial literature and in textbooks for market traders (technical analysis) [1–6].
In this paper we introduce a model for price dynamics where a moving average plays a central role (in the following with ’price’ we mean a more general class of financial objects, including share quotes, indices, exchange rates, and so on). We focus our attention on the logarithm of price, not the price itself, since the di erence of two consecutive logarithmic prices corresponds to price return, being a universal measure of a price change, not a ected by size factors.
The moving average is therefore computed with logarithmic prices, but we do not simply look a time window in the past, giving the same weight to each past quote. On the contrary, we take into account every quote in the past, but its weight decays exponentially with time [7]. This choice is suggested by several evidences that correlations between price changes rapidly go to zero [8,9], at variance with absolute price changes (see for example [10,11]).
In our price dynamics the di erence between the logarithmic price and the moving average at a given time linearly influences the future price, together with noise. Let us stress that the moving average can either attract or repulse the future price, being this a peculiar feature of the market considered. In turns out that the logarithmic price plus its moving average evolves following a one-step Markov process. We test our model analyzing three datasets:
a) Nasdaq index: 4032 daily closes from 1st Nov 1984 to 25th Sep 2000.
b) Italian Mibtel index: 1700 daily closes from 4th Gen 1994 to 2nd Oct 2000.
c) 1998 US Dollar - Deutsche Mark (USD-DM) bid exchange rate: 1620861 high frequency data, that represent all worldwide 1998 bid quotes, made available by Olsen & Associated.
The average time di erence between two consecutive data is about 20 seconds. The results give a clear evidence that our model for price dynamics is fully consistent with all the above datasets. We also study the problem of finding an optimal strategy for repeated trading operations in the context of our model, giving an analytic solution for a realistic situation (small di erences between logarithmic price and moving average). In particular, we show that a simple suboptimal strategy implies a capital growth rate larger than the growth rate of the underlying asset.
This simple result highlights a certain degree of market inefficiency. Let us briefly sum up the contents of the paper. In sect. II the model of price dynamics is introduced, and the exact solution is derived. In sect. III the model is tested against the three datasets, fixing the free parameters of the theory for each case. In sect. IV strategies for repeated trades are investigated, in order to find the optimal one. In sect. V a suboptimal trading strategy is applied to the three datasets, showing that markets hide some inefficiency. In sect. VI some conclusions and some final remarks are presented.
By Prof. R. Baviera, Prof. M. Pasquini, Prof. J. Raboanary and Prof. M. Serva
Next: The Model
Summary: Index