Problem 1

A function is given by:

The inverse of the above function:

 a) is a continuous function;
 b) is a discrete function;
 c) is not a function;
 d) None of the above;

Problem 2

Here is a set of objects:

This set is connected by the mathematical concept of :

a)   singularity
b)   degeneracy
c)   infinity
d)   none of the above;

Explain your answer.

Problem 3

An operator "+" is used in the following manner in a sequence of code (computer programming language)

This is an example of:

a) overloading
b) isomorphism
c) inheritance
d) virtual function

Problem 4

A stock is currently trading at $50. A trader, who believes that the stock is more or less fairly priced, holds 5000 lots of the stock (say, 1000 shares a lot). Every time the stock hits $45 the trader buys the stock in small quantities, say 500 lots and every time the stock hits $30 the trader aggressively buys the stock, say 1000 lots. The trader identifies $20 as the support point (a turning point) and therefore between the price of $20 and $30 though the trader builds his position further by buying more aggressively, say 10,000 lots, but at the same time he starts to put limit orders with his broker (to sell his shares) at $19, $17 and $15, to sell almost 80% of his holding. Above $60 the trader starts to sell in small quantities and at $60 he sells a 1000 lots and at $65 he sells another 2000 lots. He puts a limit order at $80 to sell almost 90% of his holding.

However, there is one other factor that the trader considers very important, i.e. the volatility of the stock market index (where this particular stock trades). Currently the volatility level is at 19%. Regardless of the price of the stock he will do the following: if the volatility of the index (called vol) is at 10% or less, the trader will liquidate more than 80% of his portfolio and between 10% and 15% he will aggressively sell the stock (say, in multiples of 1000 lots); when the volatility is at or more that 35% he will aggressively buy the stock (in multiples of 1000 lots). Between 15% to 20% vol levels, he will sell selectively, no more than 1000 lots, and when the vol is between 20% to 30% he will sell between 2000 to 3000 lots.

Given the above scenario, build a simple rule based (no math) trading model for buying and selling the stock. You may make your assumptions about certain parameters and quantities.

Problem 5

The area of a triangle is given by its base times half of its height. If one of the points of the base is at the origin then create a generalized function for the area of this triangle such that the integral (area) of the function is one and the integral in the origin is equal to one.

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