Moving Forward

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.

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

The rationale of this equation, I intend to clear up soon. But let me add a few words about it, is the time factor into the equation. And here 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 factor is to account for the appreciation in value with time. The second fact 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   you will get a purely deterministic process. You make you will get a purely random process.

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

,

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

Will be travelling

Expect the posts to get sparse and short, or possibly even non existent for the next 10-20 days, as I will be travelling.

Thanks

Covering Basics

This post, marks the official beginning of the series, where we will try to move forward first slowly, trying to build basics into simple probability theory, to more advanced concepts.

Now, lets get started like this. Consider the term, variable. What does it mean? Something which can vary. Lets try to finetune it a bit more, can it vary within bounds? We can’t necessarily say that.

What about, can we impose some limits on its behaviour? Depends on what kind of limit we are talking about.

Okay, so if we consider the equation , where are constants, then, will be a simple deterministic variable. Given we do know how will vary. Nice and good. We in our daily lives used to seeing and working with deterministic variables.
Now this set of variables have got more snobby cousins, called random variables. What are random variables?

Practically just think of an experiment, where there are n outcomes possible. And we assign a value to those outcomes, say 1,2,3….n. It doesnt mean a weight or something like,it just is a marker of the outcome which has come out. You can even call it, by the names of your kids, or your girlfriends. Now, if I ask, what is the outcome of the experiment, we dont know about it, accurately. Because, each time we repeat the experiment any one of them can pop up randomly. Hence we assign the result of the outcome as X and say, X is a random variable and can take values from 1,2,3…n

So here the values which it can take, is called sample space.So the outcome of the experiment of tossing a coin, or throwing a dice will have a sample space as {H,T} or {1,2,3…6}.

Plain and simple eh?

Now lets try to crank the quotient a bit.

Lets consider we have a signal source, an information source. Randomly it switches itself on, transmits something for some time period 0 to T, the information is not known before hand*, and switches itself off.

So because what is being transmitted is unknown,  and each time it switches itself on, is considered a process. Now the signal which is being transmitted, its contents (or information content we dont know and can’t predict).

In these cases, to model the situation  at hand, we deduce something, we assume a few things, and then try to forward our journey. As a sidenote, most of my education, I have tried, (not necessarily succeeded) to connect the dots which are coming towards me with the path I have traversed till now. So for example you know linear algebra, you will try to work with say equations, then slowly move to quadratic equations, all the while trying to form relationships between what you have studied, and trying to solve the challenge at hand by what you already know. Nothing hi-fi, simple plain things.

So now, we want to come back to our signal source, and want to study its characteristics. We notice a few things, in the signal source, the signal source, can generate anything, and can take any of the probable ways. Not all are equiprobable, not necessarily. So such kind of a setup, a system, a non deterministic situation we call as a stochastic process.

What is stochastic process?

Lets add some rigour to it, we dont know how it will evolve into future, it will have multiple paths of evolution, not all equiprobable, yet given the initial state, we still wont be able to deduce where it will go. (it reminds me as I write, of chaotic systems)

Now, consider Anna is talking with her best friend, Bethany. They are discussing some very juicy details of gossips, which the other counterparty doesnt know. And as happens, Anna doesnt know what Bethany is going to talk about, or how will the conversation evolve. So if you just consider that Anna doesnt talk at all (or rather, Anna’s talks doesnt disturb Bethany’s rantings and ravings) then Bethany will be transmitting an effectively stochastic signal. Now if I assume the same thing, being repeated multiple number of times, ad infinitum we have a stochastic process. Mighty boring for Anna, and Beth as well, but its interesting to note, that information transmission is considered a blue blood stochastic process. Had Anne and Beth both would have talked, and responded to each other they would have been a process with its information content dependent on each other.

So mathematically, given a probability space(“experiment”) , a stochastic process if sampled at any time period which lies between 0 and T,then we will have a randomvariable whose sample space is  each individual value in .

So that means,you sample a stochastic process over its numerous trials at time ‘t’, what we get is a random variable with its sample space as those value of sampled stochastic process.

Sampling a stochastic process yields a random variable

Now, I have in this figure, tried showing visually “driftless”(non trending) signals.Consider for example I have some finite drift involved, in the process. Just for kicks.

That just means, we have at some point of time, a series which keeps on growing large. Now, though we don’t have anything against trends, but consider if I then sample a time unit in neighbourhood of those instances where it goes arbitrarily large, what I will get is a random variable with some possible outcomes as very large. i.e I have some considerable chances of getting “fat tails” whhoops!!!

Take care,

Soham

* Information is defined as the randomness or entropy in a signal. The less we  know, about a particular signal, the more is the information content.

Detrending,Transforms and Time Series Analysis

I have much pleasure of being quoted by Prof Dhananjay, regarding the volatility.

Thy author in his limited wisdom, decided to enlighten his readers in his own small ways, the potential challenges and rewards one might get in this approach.

Considereth, O Thine Readership,thou hast a price series named and you want to analyse this. So the idea is, to first make it stationary.

A question might arise, why stationary and what is stationary?

Lets start our quest with the second question, what in the god’s name  is stationary? But before I broach that subject, I want, to hold your precious patience, and let me go about this in a holistic manner,

Do you remember, how easy is to solve an equation like this?

Its easy, simple and we can easily do it by transposing ; But what if you are asked to solve a system of equation like this:

So if we dont know how to solve a simultaneous equation, and what we know instead is how to solve a simple linear equation in one variable. Hence lets leverage that knowledge to solve our work. Lets decompose this equation into a simple linear equation and try to solve it like that.

Now, coming back to price series and return series etc, its very easy and we know how to deal with finite mean, finite standard deviations, nice looking bell curves, not so nice looking kurtic distributions etc. We know how to deal with it. But what we dont know, is how to deal with arbitrarily large variances etc,which is present highly in the day to day observation series we have.

Dont get me wrong, we do have finite variance, finite mean distributions as well. But when a series tends to trend(pun unintended) our work simply becomes unweildly. So we decided lets reduce it to something we know, we have worked with. Lets reduce it to those general characteristics? Why? Because its not stationary!

What? What is not stationary? The price series, if you try to investigate, it has time varying characteristics i.e the price can be thought as

Here the mean is not a constant but a time varying metric. Similarly And of ten in a trend the variance becomes arbitrarily large. That it means it will take an arbitrarily large observations for us to notice a reversion to the average.

So to cut these things off, lets detrend it. That is ,take away the time dependent components off and form a return series instead , where is the return of the price series from t-1st day to t th day.

It serves essentially two purpose: a. we essentially make the mean as finite from an unknown number(potentially large).  And secondly, if we assume that the variance is finite, then we have something we can work with.Of course how we treat the unknowns in our hand, underpins the challenge of the system.From here onwards our paths diverge into various forays, and I will try to bring forward some more development in the next few posts. I would be glad, to know if I have got your appetite whetted.

P.S: I will be trying to touch up more on these subjects, with some simulations and real life charts. At the very outset, it might not give that great a “feel” of things to come or how to use it, but trust me

Good one!

Although I dont agree everything written here, but nevertheless there are some good nuggets of information here.

“Five Ways to take smart risks in uncertain markets” is titled by Minyanville and is a synopsis of Dr. Doug Hirschhorn latest video blog on CNBC. He  is one of the premier trading coaches is the author of “8 Ways to Great: Peak Performance On The Job and In Your Life” and “The Trading Athlete”

If I could I would have titled it as, “Four Ways to train yourself before you make your first trade” [okay a popquiz: I said four ways, while there are five written, so which one I disagree with?]

The Link is here

Thanks goes to TheKirkReport