What is the Fisher Transform?

The Fisher transform can be used to identify price reversals on its own or trends when two Fisher transform lines cross each other. Ehler wrote about the Fisher transform indicator in Stocks and Commodities Magazine article, "Using the Fisher Transform" published on November 2002.

This article explains the Fisher transform. If you are looking for VBA code to implement the Fisher transform, you can view it here

1) Normalizing your prices

You start by normalizing your price series with the following formula

Today’s Median = (Today’s High + Today’s Low) / 2

Today’s Normalized Price

= [ {(Today’s Median – Minimum Median over last n days)

/ (Maximum Median over last n days – Minimum Median over last n days)} - 0.5 ] x 2

, where n is specified by the user and includes today.

Your normalized price will now range from -1 to 1. (See Chart 1 below) This is necessary as the Fisher Transform formula requires its inputs to be between -1 to 1 to work.

2) Smooth the normalized price

Smoothed Normalized Price, SNP = 0.5 x Normalized Price + 0.5 x Yesterdays’ Smoothed Normalized Price.

Make sure that your SNP capped between -0.9999 to 0.9999 to prevent calculation errors.

3) Fisher trsanform

Fisher transform = 0.5 ln[ (1 + SNP )/(1 – SNP)]

4) Smooth your Fisher transform

Smoothed Fisher transform = 0.5 x Fisher transform + 0.5 x Yesterday’s Smoothed Fisher transform

The unsmoothed Fisher transform converts your Smoothed Normalized Price to something like in Chart 2 below.

Notice that Chart 2 above is very similar to Chart 3 below. Chart 3 is actually the cumulative probability distribution function of a standard normal random variable. Hence, the Fisher transform converts your prices around the High and Low to values that represent the extreme ends of an approximately normal distribution. In Chart 2 and 3, points in the extreme two ends have very small gradients or slope of close to 0, implying that these values happen rarely. The larger the Fisher transform value, the rarer it is supposed be and may signal an impending turnaround. On the other hand, you can plot 2 lines of the Fisher transform for different periods to detect trends.

Ehler also proposed a trading rule to profit from market trends. He suggested to buy on the next period open if both the last Fisher transform indicator value is greater than 3 and the current value is less than 0.9 of the last value and vice versa.

What is a "cumulative probability distribution function"?

Suppose the return of a stock is random and follows a normal distribution. You record the stock's return for 1000 days. You notice that the return ranges from -4 to 4 in Chart 3 above. Next you count the number of days return equals -4 or -3 or -2, .. or 4 respectively. You plot them out with the horizontal axis representing stock return and the vertical axis representing the number of days.

For example, you find out that the stock returned -4 for 3 days, -3 for 5 days and you give the value of -4 on the horizontal axis a height of 3 and the value of -3 a height of 5 respectively. You will realize that you will obtain something like Chart 4. If you sum the number of days and give the return -4 a height of 3 and the return -3 a height of 8, you will obtain Chart 3. This is why the slope or gradient of points in Chart 3 form Chart 4.

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