The course Financial Math 2 includes 70 exercises that are performed individually or in small groups. Sample exercises are indicated here. In the course, you will learn how to solve problems like these—and many more! Note that some of the more challenging exercises address non-financial applications. These anticipate financial applications covered in Math 3. If these sample exercises are too easy, you may want to skip Math 2 and go straight to Math 3.

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Sample 1. The probability that a flight will have a delayed departure and/or a delayed arrival is .30. The probability that a flight will have a delayed departure is .20. The probability that it will have a delayed arrival is .15. If a flight has just had a delayed departure, what is the probability that it is going to arrive late?

Sample 2. Suppose you roll two dice. Consider the two events:

• X: the sum is even.
• Y: the white die exceeds the black die (W > B).

Are these two events independent?

Sample 3.
a. Suppose you roll a fair 6-sided die. Let X be a random variable for the result squared. Calculate E(X).

b. A random variable Z has probability density function:

• Graph the PDF.
• Calculate E(Z).

Sample 4. Consider the linear function q = Mp defined by:

What is ?

Sample 5. Consider the function: h(x,y) = xy². Calculate its gradient at the point (2,5).

Sample 6. Fit a tangent hyper-plane approximation to the function f(x,y,z) = x + y² + z³ at the point (1,2,3). (Hint: You will be using either a derivative, gradient or Jacobian based upon the dimensions of the domain and the dimensions of the range.)

Sample 7. Suppose random variable X is lognormal with mean 1.1 and standard deviation 0.25. Use a standard normal table to calculate Pr(X < .9).

Sample 8. If an asset’s log return Q is normally distributed, what distribution must its future price C have? Explain your answer.

Sample 9. A portfolio’s market value depends upon an underlier U. Suppose the current value of the underlier is $12.50, and at that value, the portfolio’s market value is $56MM. Assume the portfolio’s delta and gamma are, respectively, 4MM and 1MM.

a. Construct a delta-gamma approximation for the portfolio’s value P(U).

b. Use your delta-gamma approximation to approximate what the portfolio’s value would be if the underlier’s value immediately rose by $1.5.