Apologies to my readers for keeping them waiting like this. As of now, I am still out of station, with very limited connectivity and no books by my side to cross check whenever I am doubtful. But in that case, you and I will be betting, that my memory is better than what I think it is.

So moving forward, from the basic definition of a stochastic process, we try to garner some understanding of financial time series.[for those just landing- reading up on previous posts in the category: Quantitative Finance will prove helpful]

Now in its essence, if we go on the opposite side of the spectrum, and consider financial time series as a truly random signal, then in essence we have a stochastic process.

Now, let me consider $X$ as the daily return series of a financial price asset. Once you have that, consider a $dX$ change in the return as being governed by this equation

$dX = \mu dt + \sigma dW$

The rationale of this equation, I intend to clear up soon. But let me add a few words about it, $dt$ is the time factor into the equation. And here $dW$ is a standard stochastic process called Wiener Process.  This entire equation is called Brownian Motion(with a drift).

Now let me explain what is the rationale of each term. The time dependent term, is what incorporates a trend. A trend is the tendency of a stochastic process to deviate away from the mean. The $\mu$ factor is to account for the appreciation in value with time. The second fact $\sigma$ is a scaler. A Scaler of the Wiener Process, or more precisely, a factor which determines the level to which the entire process will be random. So if you make $\sigma =0$  you will get a purely deterministic process. You make $\mu =0$ you will get a purely random process.

Now, if we slightly modify the above equation, to reflect the following view:

$dX/X = \mu dt + \sigma dW$,

then what we have is a Geometrical Brownian Motion, i.e the next quantum of increment (or decrement) is going to depend on the current value of the random process.

This is an interesting thing to move forward with, and we will see quite a few developments from here extending to option theory, modelling etc.

Till then,

Soham

For more take a look at Category:Quantitative Finance

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