Problem #1

The quantitative analysis group (in the Capital Markets Division) of a Bank is analyzing the price data of a particular stock over the past thirty years. It is looking at the daily price on all trading days over the last thirty years and recording it as a zero (0) if the price went down from the previous day and one (1) if the price went up from the previous day. The following is the sequence generated:

1 0 1 0 0 0 0 1 1 1 0 1 0 11 0 0 1 0 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 0 0 0 0 1 0 0 1 1...

The sequence goes on for 7,560 bits (each 0 or 1 is a bit representing one trading day). The sequence seems to completely random and there seems to be no discernible pattern in the above sequence. A computer programmer is given the task to write a source code in say, C++ language, to exactly mimic the above sequence and generate an output. The programmer realizes that there is a major problem writing the code. What could be a major problem?

1. the computer program to generate the above sequence will be as long as the sequence itself (simply too long to be manageable);
2. C++ may not be an adequate language to write the program;
3. the computer program may not be able to capture the randomness in the series;
4. the sequence may exhibit minor patterns for some periods of time while at others it may become totally random creating problem in writing the algorithm;
5. None of the above;

Problem #2

The programmer may tell the head of the quantitative analysis team that:

1. the problem is beyond his "complexity horizon";
2. the problem cannot be coded in C++ or any other high level languages;
3. the above sequence has a faulty data which needs to be cleaned or removed if a program has to be written to capture the pattern;
4. None of the above;

Explain your answers for questions 1 and 2.

Problem #3

A stock is trading at $100 and you buy a "one year option " to buy this stock from your broker for $100. That is regardless of the price of the stock after one year, you shall have the right to buy this stock for $100. You may choose to buy it, if it is profitable to you (i.e. if the price of the stock is more than $100) or forget about it and throw the option in the trash bin if it is not profitable to you (i.e. if the price of the stock is less than $100). Your broker is your best friend and he lets you on to a secret: that the stock can either be $120 or $80 at the end of the year but he does not know what the probability of each happening is. He lets you on to another secret: that he can borrow any amount of money from a bank for zero interest rate.

Your broker asks a price of $15 from you. Do you think this is the fair price? Explain your answer (correct answer is not important but the method of reasoning is).

Problem #4

A study done by a couple of physicists in 1999 found that the average diameter of the world wide web (i.e. the shortest path between two randomly chosen web pages on the internet or the smallest number of URL links that must be followed to navigate from one web page to the other on the internet) is given by a power law . If N is the total number of web pages on the internet then this average diameter, d, was most likely explained by the following power law:

Problem #5

In the above problem, if the above law holds today and if the total number of web pages on the internet is say, 2 billion (i.e. N = 2 billion) then the average diameter of the world wide web would be:

1. 4.12 billion
2. 3.28 billion
3. 64,187
4. 19.51

Any comments and queries can be sent through our web-based form.

Sources & Inspiration: Question No. 1 and 2 are inspired from the writings of John Allen Paulos (A Mathematician Plays the Market), Question No. 3 is inspired by writings of John Norstad and can be found on his site http://homepage.mac.com/j.norstad/index.html and also in the writings of Jeremy Bernstein (The Einsteins of Wall Street). Question No. 4 and 5 are based on actual research done by Reka Albert, Hawoong Jeong and Albert-Laszlo Barabasi of the Department of Physics, University of Notre Dame, Indiana, USA

More FEAT>>

back to top