Harry on Economy

(Read from the beginning) OR
(If you have not read article "6 of n" already then please consider reading it before continuing here)

Markov processes

Let us say we are playing head or tails game. If the toss is head then we double the amount. If the toss is tail then our amount is halved. Let us see what happens with 3 tosses assuming we start with $ 200.

Say we are trying to solve a problem that states that at “Time 2” we have $ 200 and the next toss (“Time 3”) is head then how much money will we have after the toss. The answer is simple, we will have $ 400 (either THH or HTH).

Note that we could have had $ 200 either by path HT or by path TH. However, in order to know the value of our funds at Time 3 we do not need to know / remember how we came into the possession of $ 200. All we need to know is that we have $ 200 and whether next toss is a head or tail. This is called a Markov process. A process is a Markov process if all we need to know is current state of the game and value of the random variable.

Consider this against another version of the same game where we are told that for the first toss we double our money if we get a head and money is halved if we get a tail (same as previous version) but then on we double our money only if the next toss is same as the first toss. Now let us see what happens with 3 tosses.

Say we are trying to solve the same problem that states that at “Time 2” we have $ 200 and the next toss (“Time 3”) is head then how much money will we have after the toss. Now the answer is not so simple!!!! We will have to know whether we came into possession of $ 200 as path HT or path TT. Hence this is not a Markov process.

Computational requirements of a process have a major impact on performance depending on the fact whether a process is a Markov process or not. If we are tracking price of option over 360 days and each day being a single period then the computation will be massive if we need to keep track of each and every state on the previous dates.

So you might start wondering what has this “Markov” discussion got to do with stock market. This is something that ties up with Efficient Market Hypothesis. Efficient market hypothesis claims that stock price reflect all information available in public domain about a particular stock. Hence the future stock price is essentially the function of future news about the stock. Hence stock price is a Markov process. Note that this is in direct conflict with most of the technicals. For e.g. Future price of a stock as per head and shoulders pattern will depend on whether stock price had a left shoulder and head before the formation of right shoulder. Hence, stock price is not a Markov process as per head and shoulders pattern.

Finally, a formal definition to bore you to death

Consider the binominal asset-pricing model. Let X0, X1,…., XN be an adapted process. If, for every n between 0 and N-1 and for every function f(x), there is another function g(x) (depending on n and f) such that

En[f(Xn+1)] = g(Xn),

We say that X0, X1,….,XN is a Markov process.

(Read from the beginning) OR
(If you have not read article “5 of n” already then please consider reading it before continuing here)

Martingales

This can easily be explained with the help of an example:

Let us say bank interest rate on deposits is 10%. If you start with $10,000 of deposits then your deposits are going to grow as follows

Year 0 : 10,000

Year 1:  11,000

Year 2:  12,100

Year 3:  13,310

Year 4:  14,641

Notice that

(a)  The deposit balance for year 3 can be calculated from deposit balance of year 4.

(b)  Deposit balance of year 2 can be calculated from deposit balance of year 3.

From (a) and (b) it should be easy to deduce that

(c)  Deposit balance for year 2 can be calculated from deposit balance of year 4.

To generalize:

The deposit balance at year n can be calculated from deposit balance at year n+1 or n+2 or n + 3. This is called Martingale.

To bore you to death:

Mn =  En[Mn+1], n = 0,1,…….,N-1 is a martingale process. Remember that ‘E’ in this equation represents expected value.

Super and sub-martingale

Let us say prevailing interest rate on deposits is 10%. Due to this reason stock market must pay returns higher than interest rate and hence stock market is a super-martingale.

 Of course these days stock market has become a sub-martingale since stock market returns are way less than bank returns.

Next: Markov process