MSc Thesis submitted!

September 30, 2010 in Research, Research Updates

The Cut-Off Phenomenon in Random Walks on Finite Groups

2 comments

February 12, 2015 at 12:29 pm

Hi J.P.,

I was a student in your Maths Studies class last year. I am currently doing a MSc in Financial Economics and my thesis is aimed at examining the reliability of mathematical models as stock market predictors. One of the methods I am applying is Markov Chains.

I remembered you mentioned that your area of study involves this method. I was wondering would you be able to clear up a question i have regarding the Markov property.

I fully understand that it states that the future state of a subject is determined by the current state and so the past has no effect on the future state. However, seeing as yesterdays state determined the current state, does this not suggest that the past does indeed have an indirect effect on the future?

i.e. If the stock price of a company today affects the price tomorrow. Do past prices in turn affect future prices?

I would really appreciate your help,

Thanks.

February 12, 2015 at 1:03 pm

J.P. McCarthy

I think you are getting caught up in semantics, and I think I can help you using Dynamical Systems (who would have thunk it?)

Consider,

Past $\longrightarrow$ Current $\longrightarrow$ Future

Yesterday $\longrightarrow$ Today $\longrightarrow$ Tomorrow

You state that

“tomorrow’s state is determined by today’s state.”

I think you are thinking of the state as the price of the stock. For ease of explanation, we will suppose the stock price is a whole number so now the price of a stock can be modelled by a natural number so, in this framework, the set of states is $S:=\mathbb{N}$ . If tomorrow’s state (price) is always determined by today’s state (price) then there is a rule that tells you how to go from one state (price) to another:

Yesterday $\overset{f}{\longrightarrow}$ Today $\overset{f}{\longrightarrow}$ Tomorrow

That is the price of the stocks is a dynamical system $(f,S)$ as to get from one state to the next you apply a rule (function).

Now, denote yesterday’s price by $x_0$ , today’s by $x_1$ and tomorrow’s by $x_2$ . What you are inferring is that

$f(x_0)=x_1$ and $f(x_1)=x_1$ so that

$x_2=f(x_1)=f(f(x_0)),$

so that tomorrow’s price depends on yesterday’s and so how is this Markov… clearly this is a terrible model of financial stocks. For the remainder of this we will fix an initial date and define

$x_0=$ stock price on morning of initial date
$x_i=$ stock price at close of trading after $i$ days of trading

The behaviour of the stocks is given by an orbit:

$\{x_0,x_1,x_2,x_3,x_4,\dots\}$

Consider stocks that go as

$\{10,9,7,6,5,8,12,9,11,14,\}$

so a fall followed by a bounce-back, a mini-set back and then growth. This simple behaviour can’t be captured by a dynamical system because we have

$x_2=f(x_1)=f(9)=7$ AND $x_8=f(x_7)=f(9)=11$ ,

and this is a contradiction because we can’t have the rule, $f$ , sending nine to 7 AND 11: a function will always send an input to the same output. So this model is not only unrealistic but empirically impossible.

The problem you are having is understanding “determined by today’s state”… or rather the set of states. Let us model a stock again. Define

$\xi_0=$ stock price on morning of initial date
$\xi_k=$ stock price at close of trading after $i$ days of trading.

Now there are a whole range of things that determines the price of the stock, $\xi_k$ . Let us call this whole range of things the probability space or the universe $\Omega$ . Now the stock price depends on the circumstances of the universe at that time so we have

$\xi_k:\Omega\rightarrow \mathbb{N}$ .

Now the thing is that we do not and can not know the entire state of the ‘universe’. So $\xi_k$ is a random variable. We can only talk about the probability of it being equal to something. So for example, for a stock that has an orbit:

$\{\xi_0,\xi_1,\xi_2\,\dots\}=\{10,9,7,6,5,8,12,9,11,14,\dots\}$

we can’t say what $\xi_{10}$ is going to be equal to instead we can talk about the probability of $\xi_{10}$ being, e.g. more than 14, less than 14, equal to 14, between 12 and 16, even, odd, prime, etc.

All of these outcomes are events and we can talk about the probability of an event. Probability then is a function:

$\mathbb{P}:\mathcal{F}\rightarrow [0,1]$ ,

from the set of events, $\mathcal{F}$ , to a number between 0 and 1 (i.e. the event’s probability).

Now, in particular, you are interested in events such as $\xi_{10}=k$ , a specific number, e.g. $\mathbb{P}[\xi_{10}=13]$ — the probability that $\xi_{10}$ is 13. You can make a(n infinite) row vector of these probabilities:

$(\mathbb{P}[\xi_{10}=0],\mathbb{P}[\xi_{10}=1],\mathbb{P}[\xi_{10}=2],\dots,\mathbb{P}[\xi_{10}=14],\dots)$ ,

and this is called the distribution of the random variable $\xi_{10}$ .

Now, enter conditional probability. When you are talking about $\xi_{10}$ , you already know the stock price on the first nine closes… now the probability distribution of $\xi_{10}$ is conditioned by the fact that you know the history up to that point… so rather than looking at the distribution $(\mathbb{P}[\xi_{10}=k])_{k\geq 0}$ , you look at the conditional distribution (this $\mid$ means “given that I know”)

$\displaystyle (\mathbb{P}[\xi_{10}=k\,\mid\,\{\xi_0,\xi_1,\dots,\xi_9\}=\{10,9,\dots,14\}])_{k\geq 10}$

Now… the Markov property doesn’t say that the past can’t affect the future exactly — it just says that the distribution of $\xi_{10}$ depends on $\xi_9$ only… and knowing about $\xi_0,\xi_1,\xi_8$ doesn’t tell you any extra. We write this as, for a given $k$ ,

$\displaystyle \mathbb{P}[\xi_{10}=k\,\mid\,\{\xi_0,\xi_1,\dots,\xi_9\}=\{10,9,\dots,14\}]=\mathbb{P}[\xi_{10}=k\,\mid\,\xi_9=14]$

So the proper statement you should have for the Markov Property is

“The distribution of the next state is determined by the present state.”

This is more correct and the states are the random variables $\xi_k$ . This is the answer to the question you asked.

There is actually a way though to make a dynamical system out of this where the set of states isn’t the stock prices but their distributions. E.g. the states are sequences, i.e. the sequence $\nu_k$ is the distribution of $\xi_k$ . Then you can define a function $P:\text{ distributions }\rightarrow \text{ distributions}$ called a stochastic operator that takes one distribution to the next:

$\nu_{k+1}=P(\nu_k).$

Then the orbit of the distributions of the stock prices
$\{\nu_0,\nu_1,\nu_2,\dots\}$

is a dynamical system. The set of states $S$ is the set of distributions and the iterator function is this $P$ . For more on this you can look in my MSc thesis or indeed at Q.2 of the homework I gave ye (https://jpmccarthymaths.files.wordpress.com/2012/02/homework1.pdf).

Now what this means for you is your project is that you need to swallow the following. Suppose you have a (time-homogenous (Google this)) Markov stock that has the following orbit:

$\{20,19,18,17,\xi_{4},\dots,14,15,16,17,\xi_{104}\}$ .

Now, because $\xi_{3}=\xi_{103}=17$ , because the stock is Markov, you have to accept that
$\mathbb{P}[\xi_4=18]=\mathbb{P}[\xi_{104}=18]$

because the distribution of the next state only depends on the present state: it doesn’t matter that $\xi_4$ is after a slump and $\xi_{104}$ after a rise. Is this a good model for stock prices? This is your project.

Regards,
J.P.

	J.P. McCarthy on MATH6040: Spring 2022, Week…
	J.P. McCarthy on MATH6040: Spring 2022, Week…
	J.P. McCarthy on MATH7019: Winter 2020, Week…
	J.P. McCarthy on MATH7019: Winter 2020, Week…
	A Sufficient Conditi… on Almost All Trees have Quantum…

MSc Thesis submitted!

Recent Comments

Categories

J.P. McCarthy Maths

2 comments

Leave a Reply Cancel reply

MSc Thesis submitted!

Share this:

Like this:

Related

Recent Comments

Categories

J.P. McCarthy Maths

2 comments

Leave a Reply Cancel reply