Mandelbrot Set

Mandelbrot set refers to the stock market theory explaining self-similarity within the assets traded in the financial market. It used fractals to apply this theory to stocks, currencies, and commodities. Its main purpose is to find correlations or similarities in the price movements of assets.

The entire Mandelbrot set function works on the fractal theory. It tries to find patterns in market and market segments. Also, it helps in identifying features like volatility and cylindrical movements of the financial markets. Moreover, fractal mathematics can determine the dependence of current prices on the past. However, this theory has some limitations in reality. 

Mandelbrot set is a proposed theory by Benoit Mandelbrot describing the presence of similarity in the markets using fractals. It explains how the price does not remain independent and is influenced by history. As a result, the entire financial market behavior depends on fractals (similar patterns). Thus, the assumption of a smooth bell curve in the market is also false. It falls in contrast to the existing or modern market theory.               

The origin of the fractals Mandelbrot set function was first given by mathematician Benoit Mandelbrot in 1961. He studied the cotton prices to find patterns or self-similarity in it. Soon, Benoit was successful and mentioned the same in the book "The Geometry of Nature" in 1963. As per Mandelbrot, the price movements do not follow a bell-curved graph. Instead, they fluctuate vividly and wildly, causing uncertain market volatility. Thus, Mandelbrot describes the market as risky, misleading, causing trouble strikes, and has its personality. Likewise, market timing is relative and can equally bring gains and losses in small packages. 

In addition, there are various other theories and observations listed by Mandelbrot in fractals Mandelbrot set function. It stated the markets in all geographic locations work alike and have akin similarity. They have a life of their own, meaning the patterns remain the same despite the scale of operation. Also, these markets may deceive at a point. At the same time, predicting volatility is much easier than prices since price movements are uncertain.

Another set of Mandelbrot theories defines financial theories that generally fail to capture market risk. Plus, the prices do not move steadily and slowly. They come with great risk, leading to a higher jump. Thus, bubbles tend to occur. Furthermore, the asset value usually perceived in markets is not absolute, just a difference between price movements.  

Mandelbrot has given an equation to generate a Mandelbrot set in nature. It is useful in understanding the correlation between fractals and chaos theory. Let us look at the formula:

 zn+1= zn2 + c 

Here, zn represents a complex number having a "n-th" iteration. However, the absolute value of zn does not exceed a certain number. Also, the "z" does not cross infinity; it stays within the set of c. Likewise, c is also a complex number, where its magnitude is less than 2. To find the magnitude, users can take the help of real and imaginary coefficients. Later, square them, subsequently add them, and finally take the square root of the answer obtained. 

Thus, if the c=0, the formation of Mandelbrot on the plane is a circle. But, when the values of c vary except zero, the resulting figure will be an alternative region, particularly not a circle. 

Let us look at some examples to understand the Mandelbrot set in nature of financial markets:

Example #1

Suppose Carol is a trader and analyst trying to understand the price movements in the past three months in the US market. She has been observing the trends and seems to find a price pattern. Therefore, Carol decides to generate a Mandelbrot set function to determine the price movements in Kuvin Ltd stock and gain more insights on the same. 

By applying the formula of a complex number (c), where c is the initial price of the stock ($100). Therefore, c = 50 + 0i. Here, the price is recorded at t=0. Later, when using the Mandelbrot equation (zn + 1= zn2 + c), the t=n. Now, Carol can apply the formula each time at "n," likewise updating the stock price. 

While plotting the prices, Carol may notice a divergence of a circle formation. If the price remains under a price range, there may not be many fluctuations. However, Carol noticed a divergence (spread) graph resulting in significant price movements.  

Example #2

Another example of the fractals Mandelbrot set includes a circled formation of the equation plotted. Samuel tried to perform three different iterations for the formula provided. In the first iteration (zn2+ c), the answer obtained is further inputs as zn for the next iteration. For instance, if zn=1, is 0.5 + 0.2i, then the upcoming iteration (zn=2) acts as zn2. 

Mandelbrot set has a wide prevalence among investors and traders. It provides multiple benefits at various points. Let us understand them in brief:

• This set helps traders make investment-related decisions by detecting price movements. 
• The fractals geometry serves as a tool for rebalancing the portfolio by analyzing the assets that exhibit the self-similarity in prices. 
• It also helps in identifying potential trade opportunities that serve as an aid to detect risky trends in advance. For instance, market movements at different time frames can potentially signal the risk associated with them. 
• Furthermore, it also models the probability of extreme events such as bubbles and financial crises. The patterns formed contribute to its modeling.