Saturday, March 14, 2015

Flash Boy Velocity Problems.

Flash Boy Velocity Problems


The new book by Michael Lewis “Flash Boys” discusses how Wall Street is now using faster cables between Chicago and New Jersey to trade faster.     A millisecond or two does not seem like much time but Wall Street firms can make a lot of money buying and selling quickly.  

A firm that has knowledge of a large buy order at $20 per share may be able to snap up all available shares at $19.9 and screw their customer and that is what modern Wall Street is all about.

Here is the cite to Flash Boys on Amazon:


It is a highly entertaining and informative book.  

I have written a finance math problem that explains why Wall Street firms are willing to pay a lot of money to make quicker trades.

http://dailymathproblem.blogspot.com/2015/03/a-flash-boys-finance-problem.html


This post uses the issues raised by Flash Boys to motivate velocity and distance problems.   Velocity and distance problems are generally motivated by problems pertaining to falling balls or rockets so these problems are a bit different.

Question:   The new fiber line between Chicago and New Jersey was able to transmit trade information in 13 milliseconds.     The line was 827 miles long?   What is the speed of messages on the new line?   Express the answer in miles per second and kilometers per second.

Answer:    There are 1000 milliseconds in one second so it should take 13/10000 seconds to travel the 827 miles.    This means the trade is traveling at 827/0.013 miles per second.

827/0.013 is equal to 63,615 miles per second.

There are 1.60934 kilometers in a mile.   This means that the trades travel at 102,379 kilometers per second.


Question:  Prior to the construction of the new line trade information between Chicago and New Jersey took between 14.6 and 17.0 milliseconds to arrive depending upon what line was used?    Assume that all messages travel at the speed calculated in the previous question.  (I am not sure whether this assumption is correct but some lines are straighter and shorter than others so this problem is predicated on the assumption that differences in travel time are due to length of line.) 

How long is the line where it takes 14.6 milliseconds to make the trip?   How long is the line where it takes 17.0 milliseconds to make the trip?


Answer:   WE know that Distance is equal to speed x time.   Speed is 63,615 miles per second.  Time is either 0.146 seconds or 0.017 seconds.



Calculation of Distance Message Traveled In X Seconds
Seconds
0.0146
0.017
Speed Miles Per Second
63,615
63,615
Distarnce Miles
Seconds * Speed
Seconds * Speed
Distance Miles
929
1081

At 0.0146 seconds the cable is 929 miles.   At 0.017 seconds the cable is 1081 miles.

This could also be solved as a ratio problem.

D1/S = 827/0.013  =  D2/0.0146  =  D3/0.017


So

D2  =  827 x 0.146/0.013 = 929

D3  =827 x 0.017/0.013 = 1,081


Question:   The actual distance between New York and London is 5576 kilometers.   Assume there is a direct shortest distance cable between the two cities and that the message travels at the speed specified in the first question.  How long does it take for the message to travel from New York to London?   How long does it take for the message to arrive if the route is 10 percent longer than the direct route? 

Answer:  


Below is the calculation of the time it takes a message to travel from New York to London, assuming the same speed as in the Chicago to New Jersey problem.
Calculation of Time it Takes Message to Travel from New York to London
Direct Route
10% More than
 Direct Route
Distance In Kilometers
5576
6133.6
Kilometers Per Second
102,379
102,379
Time Traveled In Seconds
D/V
D/V
Time Traveled in Seconds
0.05446
0.05991


Note there are 1,331 kilometers in 827 miles.    The  ratio of kilometers for New York to London to Chicago to New Jersey should equal the ratio of the time traveled on the two message trips

(5576/1331)  =    (X/0.013)


So the direct New York to London trip should take



(5576/1331) x 0.013 = 0.0544 seconds.




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