Using Markov Chain Monte Carlo Simulation to Overcome the Lag Time Associated With Triple Exponential Smoothing of Financial Time Series

Authors: I. Ilkay Boduroglu, Mustafa Karakas (Istanbul Technical University, Informatics Institute, Computational Science & Engineering Graduate Program)

Abstract

Clarence Darrow, the famed lawyer of the Monkey Trial, makes an amazing gesture at the final scene of the play "Inherit the Wind". He picks up the Bible and Darwin's 1859 book Origin of the Species from the table where they have been waiting for the verdict, shows them to the audience, puts them together under his arm and walks out of the courtroom.

This is, in a way, a gesture of combining two opposing schools of thought, a task still difficult for most of us, in other contexts as well. Take the "secular" efficient market believers and their insecular counterparts, the technical analysis people, as an example. If the efficient market hypothesis holds, then technical analy- sis of financial time series would be meaning- less. This is because efficient market people believe that historical data is useless since all the information about a security is reflected at its current daily log return, D(n) where

D(n) = log(S(n)) - log(S(n - 1))

Note that S(n) is the closing price of the security at time n. Furthermore, the efficient market believers assume that the random variable D(n) is normally distributed. "Well, not so!" says the technical analysis people. They have proof that the normality of log returns assumption fails miserably under the Kolmogorov-Smirnov test for normality. Plus, they say, "historical data do matter!" That is, there is no such thing as "random walk" when price of securities is the issue at hand. There also, this school of thought produces hard evidence. The test for independence in a contingency table clearly shows that the Markov Transition Matrix, which is created using a number of discretized states of log returns, does have rows that are not identically distributed. There goes the random walk assumption. This is because if the random walk assumption held, then the previous log return would not be able to determine the next log return in any way.

Having listed the shortcomings of the efficient markets school, let us now discuss the shortcomings of technical analysis:

• Technical indicators suffer from the well-known time-lag problem: That is, buy (or sell) signals given by a technical indicator significantly lag the bottoms (or tops) of price time series.
• Technical indicators use raw price data which is a badly auto-correlated time series: Thus, technical indicators fool the people who use them. That is, if you are trying to predict a time series that has high auto-correlation using whatever means, even if you are successful during the training period, this success could be due to auto-correlation and not due to the technical indicator itself. There is no rea- son to expect the auto-correlation to continue to help the technical indicator dur- ing trading. Hence, the well-known disclaimer with which all technical analysis software tools come: "Past performance of technical analysis tools do not guarantee future performance."

We shall try to solve in this paper the two problems of technical analysis listed above in the special case of the well-known technical analysis tool, TRIX, or triple exponential smoothing.

How do we overcome problem (a)? Let us say the lag time is L units of time (or L bars as in technical indicator jargon.) Since this parameter may easily be approximated, we use the following simple solution to overcome problem (a): We create aMarkov Chain model, which captures the behavior of the technical indicator and price movement. We simulate this Markov Chain into the future for, say, 3L bars. We find the epoch, that is the point in time, where the indicator gives a buy signal. We then go back L bars and set the modified buy signal at that point.

How do we overcome problem (b)? We use log returns instead of raw price time series. The auto-correlation coefficient with lag=1 of the series S is in the neighborhood of 0.95, where it is about 0.05 in the series D.