Martingale

Written by Forex Automaton   
Wednesday, 16 April 2008 15:20

Fig.1:A martingale market synthesized from hourly returns in AUD/JPY, over time. Time axis is labeled in MM-YY format. By construction, there is no predictable trend other than the long-term trend created by the tiny positive deviation from zero of the average hourly return. Are you a chartist? Do you believe in moving averages or Elliott waves? Do you feel you could day-trade this market? This is a particular example of what we refer to as "fair game" and for all practical purposes take to represent the embodiment of the efficient market hypothesis.

To synthesize such a chart, you first obtain a distribution of returns from the real time series. Then you histogram the returns. Then you start with an arbitrary number (1 was used in this case) and generate a random number according to the distribution of returns you got (a software package such as ROOT lets you do that). Having a starting price and a return, you obtain the next price in the series. You can continue this random walk process as long as you want. It may be counterintuitive to some that a random walk looks like this (Fig.1). Indeed you see a chart where you might be tempted to identify trend lines, points where the trend changes, and possibly even lines of support and resistance. From this standpoint you can understand the source of our Olympian attitude towards all kinds of current news: the pseudo-random market in Fig.1 could generate a very rich stream of news reports and "current analysis" -- all totally content-free by construction. Chartists and reporters must admit: tools and concepts that let one distinguish between predictability and randomness are peripheral to their method of operation. However these tools and concepts are central to the Forex Automaton™ approach. Martingale is one of such concepts.

Martingale is a stochastic process (a time series like a forex exchange rate) where an expectation of any element does not depend on the prehistory (although properties of its distribution other than the expectation may depend on the prehistory).

By implication, the expectation for a change in an exchange rate to be recorded just now does not depend on a similar change   recorded time T before now. The same would apply to returns or logarithmic returns.

The efficient market hypothesis can be restated to read that time series elements x(t) and x(t+T) are statistically independent variables for any value of T (other than, of course, 0) and therefore the autocorrelation of the time series is zero (except for, of course, at the zero time lag). Therefore a martingale market is the "efficient market". Thus the efficient market hypothesis becomes falsifiable via observation of correlations.

In "Forecast of future prices, unbiased markets and martingale models" (Journal of Business, V.39, January 1966, 242-255) Benoit Mandelbrot defines martingale in a weaker and a stronger sense: "...to define a martingale, one may begin by postulating that it is possible to speak of a single value for E[Z(t+T)|Z(t)], without having to specify by which past values this expectation is conditioned. In a later stage, one will add the postulate that E[Z(t+T)|Z(t)]=Z(t)." I am bringing this up because the martingales constructed by random sampling from observed distributions of returns will never fall into the "later stage" category since the mean of the observed returns is never exactly zero. But they are certainly constructed to have Z(t+T) unconditioned by any specific past and therefore are martingales in the sense of the initial definition.

AUD/JPY autocorrelation compared with martingale autocorrelation

Fig.2:Autocorrelations of the martingale market from the Fig.1 (red) and of the actual AUD/JPY for the same time period (green).

You might be confused to believe that Fig.1 was a predictable market. But it is the Fig.2, autocorrelation, that tells the difference between actual (predictable to some extent) and pseudo-random behavior.

 

A spike in AUD/JPY hourly data of the kind that causes the autocorrelation feature specific to this market.

Fig.3:A spike in hourly AUD/JPY is the kind of pattern that causes the autocorrelation feature seen in Fig.2.

The remarkable feature of Fig.2 that distinguishes real data and is not present in the simulation is the downward spikes at one hour lag. Fig.3 illustrates what kind of real-market feature that corresponds to: a market jumping up and down (or down and up) on the hour-by-hour time scale. This happens when markets jump the gun or have a knee-jerk reaction to something, which they later come to regret, figurally speaking. The autocorrelation in Fig.2 tells you that this happens in real-life AUD/JPY a lot more often, or with a lot higher magnitude, than it does in the memory-free simulation (when you do not have memory, you do not regret!). Why does this make the market "predictable"? Because statistically, you can bet on market to regret the knee-jerk and be right more often than not. A forex trading system, such as the one under development here, can be built to continuously analyze the situation, learn the patterns like the one shown, and beat the market with confidence.

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Last Updated ( Tuesday, 05 October 2010 13:08 )