## Mathematical finance

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In the earlier chapter we had this discussion about the range within which Nifty is likely to trade given that we know its annualized volatility. We arrived at an upper and lower end range for Nifty and even concluded that Nifty is likely to trade within the calculated range.

Fair enough, but how sure are we about this? Is there a possibility that Nifty would trade outside this range? If yes, what is the probability that it will trade outside the range and what is the probability that Nifty will trade within the range? If there is an outside range, then what are its values?

Finding answers to these questions are very important for several reasons. If not for anything it will lay down a very basic foundation to a quantitative approach to markets, which is very different from the regular fundamental and technical analysis thought process. The discussion we are about to have is extremely important and highly relevant to the topic at hand, and of course very interesting as well.

A Galton Board has pins stuck to a board. Technical analysis and investment ideas of binomial distribution bins are placed right below these pins.

The idea is to drop a small ball from above the pins. Moment you drop the ball, it encounters the first pin after which the ball can either turn left or turn right before it encounters another pin. The same procedure repeats until the ball trickles down and falls into one of the bins below.

Do note, once you drop the ball from top, you cannot do anything to artificially control the path that the ball takes before it finally rests in one of the bins. The path that the ball takes is completely natural and is not predefined or controlled. Now, can you imagine what would happen if you were to drop several such balls one after the other?

Obviously each ball will take a random walk before it falls into one of the bins. However what do you think about the distribution of these balls in the bins?. But this does not happen, there seems to be an order here. It appears that when you drop several balls on the Galton Board, with each ball taking technical analysis and investment ideas of binomial distribution random walk, they all get distributed in a particular way —.

You may have heard of the bell curve from your school days, technical analysis and investment ideas of binomial distribution curve is nothing but the normal distribution. Now here is the best part, irrespective of how many times you repeat this experiment, the balls always get distributed to form technical analysis and investment ideas of binomial distribution normal distribution.

This is a very popular experiment called the Galton Board experiment; I would strongly recommend you to watch this beautiful video to understand this discussion better —. This list can go on and on, however I would like to draw your attention to one more interesting variable that follows the normal distribution — the daily returns of a stock!

The daily returns of a stock or an index cannot be predicted — meaning if you were to ask me what will be return on TCS tomorrow I technical analysis and investment ideas of binomial distribution not be able to tell you, this is more technical analysis and investment ideas of binomial distribution the random walk that the ball takes. However if I collect the daily returns of the stock for a certain period and see the distribution of these returns — I get to see a normal distribution aka the bell curve!

Fair enough, but I guess by now you would be curious to know why is this important and how is it connected to Volatility? I think the following discussion could be a bit overwhelming for a person exploring the concept of normal distribution for the first time. So here is what I will do — I will explain the concept of normal distribution, relate this concept to the Galton board experiment, and then extrapolate it to the stock markets.

I hope this will help you grasp the gist better. So besides the Normal Distribution there are other distributions across which data can be distributed. Different data sets are distributed in different statistical ways. Some of the other data distribution patterns are — binomial distribution, uniform distribution, poisson distribution, chi square distribution etc.

However the normal distribution pattern is probably the most well understood and researched distribution amongst the other distributions. The normal distribution has a set of characteristics that helps us develop insights into the data set. The mean is the central value where maximum values are concentrated. This is the average value of the distribution. For instance, in the Galton board experiment the mean is that bin which has the maximum numbers of balls in it.

Keeping the average bin as a reference, the data is spread out on either sides of this average reference value. The way the data is spread out dispersion as it is called is quantified by the standard deviation recollect this also happens to be the volatility in the stock market context.

Likewise there is 2 nd standard deviation 2SD3 rd standard deviation SD etc. Now keeping the above in perspective, here is the general theory around the normal distribution which you should know —. Keeping the above in perspective, let us assume you are about to drop a ball on the Galton board and before doing so we both engage in a conversation —. Me — No, I cannot as each ball takes a random walk. However, I can predict the range of bins in which it may fall. Me — Most probably the ball will fall between the 4 th and the 6 th bin.

Me — Sure, I can. You — Nice, does that mean there is no chance for the ball to fall in either the 1 st or 10 th bin? Me — Well, there is certainly a chance for the ball to fall in one of the bins outside the 3 rd SD bins but the chance is very low. Probability wise, the chance is less than 0.

But one should be aware that black swan best day trading books to read have a non-zero probability and it can certainly occur — when and how is hard to predict.

In the picture below you can see the occurrence of a black swan event —. In the above picture there are so many balls that are dropped, but only a handful of them collect at the extreme ends.

Hopefully the above discussion should have given you a quick introduction to the normal distribution. For sake of this discussion, let us take up the case of Nifty and do some analysis. As we can see the daily returns are clearly distributed normally. Remember to calculate these values we need to calculate the log daily returns.

Do note, an average of 0. Now keeping this information in perspective let us calculate the following things —. The above calculation suggests that Nifty is likely to trade somewhere between and How confident I am about this? Also as you can notice when we want higher accuracy, the range becomes much larger. I would suggest you do the same exercise for Well it certainly can but the chance of going below is low, and if it really does go below then it can be termed as a black swan event.

You can extend the same argument to the upper end range as well. Since we are interested in calculating the range for next 30 days, we need to convert the same for the desired time period —. I hope the above calculations are clear to you. Of course you may have a very valid point at this stage — normal distribution is fine, but how do I get to use the information to trade? I guess as such this chapter is quite long enough to accommodate more concepts. Hence we will move the application part to the next chapter.

In the next chapter we will explore the applications of standard deviation volatility and its relevance to trading. Sir, What a surprise journey. Pleasant surprise because maths use to be my favourite subject and i did not expect that I will get chance to use my skill in share market also. Now after this I we are more curious for the next chapter.

I am eagerly waiting for it as you have said that this approach is different from the technical and fundamental approach. Congratulation for again simple explanation and Thanks for enlightening us.

Obviously using more data for calculation wil provide the best technical analysis and investment ideas of binomial distribution. But for precise caluclation how much data to be collected?

In fact you can try this for any time frame…1 year2 year etc…you will end up with a normal distribution! Sir, one more question: You have shown predicting the price movement for the year and month. But it will also be needed to calculate the range of price for a day.

Will you explain that also? The way you simplifying the things which are complex to most of us, is fabulous. Many such things i never paid any heed till now. Thank you very much and keep going. This is what you say at the end of the chapter. What about selecting strikes for an option buyer using normal distribution?

Sir, You r a doing a wonderful job teaching us the ABC of stock market trading. The brokerage rates of zerodha r also very low. It would make zerodha even more successful than it is now……. Pankit — **Technical analysis and investment ideas of binomial distribution** tips is like feeding a hungry man with 1 meal.

Hence we have all these educational initiatives like — http: Hello Mr Karthik, I have one query…in the calculation technical analysis and investment ideas of binomial distribution bin width, why did you divide difference of min and max by 50? When I divide it over 50, the intention is to create 50 bins, I can in fact divide it by or even No restriction on that.

Adding to the above query. Now my data range has changed in col E, so I need to change that.