It is known that the stock prices or value of other financial instruments has its own tick size and so they are discrete variable in nature. But I read that the returns of stock market is a continuous variable and not sure how?
Because, the return is the difference of the same stock price. so, still it is the difference of the tick size which also remains to be a certain number. So how could a stock returns becomes a continuous variable.
source where i read stock returns are continuous: http://financial-dictionary.thefreedictionary.com/Discrete+random+variable
You should also read this page: https://en.wikipedia.org/wiki/Rate_of_return
Basically, you need to consider the rate of the prices (return) instead of the differences between two periods. Difference of 0.01 is equal to 1% change when the price changes from 1.00 to 1.01, but it is only 0.9901% when it changes from 1.01 to 1.02, etc... Furthermore, I propose to use logarithmic returns, because the returns can be summed over the time instead of multiplication.
As Dávid Zibriczky mentioned it is often useful to look at the logarithm of the price relative rather than the absolute difference. The reason for this is that the log price relative is proportional to the doubling time, i.e. the time that it takes before your welth is doubled. This also holds when one can invest in different assets if the moneyare invested according to a constant rebalancing portfolio. All this does not answer your question about discrete vs. continuous. The data will have certain specific values. Predictions about these values may be made according to a discrete model or a continuous model.
Strictly speaking, you are right. Since prices are discrete, stock returns are also discrete. However, this "discreteness" is not important.. If stock returns could have only 3 possible values, e.g. +1%, 0% or -1%, then assumption of continuous returns would be a huge mistake. However, stock returns can have so many possible values that simplifying assumption that they are continuous is very reasonable. Moreover, such an assumption makes a of models easier.
Dear all,
I agree with the explanations and i could just add tha you can think in terms of its instrinsic nature as being continous inspite of it is being measured in a dicrete way.
Have a nice time
Helena
PS: i do not undrstand why someone as down vote Ivica? He made his contribution in the best way he though...then he wanted to help...!
Dear all,
thank you soo much for insightful answers. really helped me understanding the nature of data.
I just need one more clarification. Can someone explain me in simple way. why using log values gives more advantage?
This definition of return Ln(Pt/Pt-1) is equivalent to the continuously compounded rate of return from theoretical finance. There it is often assumed that stock prices are lognormally distributed etc. Under this assumption, Ln(Pt/Pt-1) is normally distributed, what is sometimes useful. Additionally, log-returns have convenient property that return over n days is simple sum of the 1-day returns.
However, ordinary arithmetic average (Pt - Pt-1)/Pt-1 has some advantage too. Consider portfolio. If you know the weights of individual assets, then the return of the whole portfolio is s simply weighted average of the returns of the individual assets ...as long as you use arithmetic returns. This simple relationship does not hold if you use logarithmic returns.
Bottom line: In most of the practical applications, it makes no difference to use arithmetic or logarithmic returns. In the academic literature, it is more common to use logarithmic returns.
Dear Peter Molnar,
That is really very explanatory and thank you so much on helping me understanding this concept. Now I can understand the difference. thanks again
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