Free Mathematical Foundations of Risk Measurement – 2015 Edition Exam 8007 Exam Practice Test

Maximum likelihood estimation is a method for:

• Finding parameter estimates of a given density function
• Estimating the solution of a partial differential equation
• Solving a portfolio optimization problem
• Estimating the implied volatility of a simple European option

Which of the provided answers solves this system of equations?2y -- 3x = 3y +xy2 + x2 = 68

• x = 1; y = square root of 67
• x = 2; y = 8
• x = 2; y = -8
• x = -2; y = -8

In a quadratic Taylor approximation, a function is approximated by:

• a constant
• a straight line
• a parabola
• a cubic polynomial

Consider the linear regression model for the returns of stock A and the returns of stock B. Stock A is 50% more volatile than stock B. Which of the following statements is TRUE?

• The stocks must be positively correlated ( )
• Beta must be positive ( )
• Beta must be greater in absolute value than the correlation of the stocks ( )
• Alpha must be positive ( )

Suppose I trade an option and I wish to hedge that option for delta and vega. Another option is available to trade. To complete the hedge I would

• trade the underlying in such a way as to make the portfolio delta and vega neutral.
• trade the other option in such a way as to make the portfolio delta and vega neutral.
• trade the other option in such a way as to make the portfolio vega neutral, and then trade the underlying in such a way as to make the portfolio delta neutral.
• trade the underlying in such a way as to make the portfolio delta neutral, and then trade the other option in such a way as to make the portfolio vega neutral.

A linear regression gives the following output:Figures in square brackets are estimated standard errors of the coefficient estimates. What is the value of the test statistic for the hypothesis that the coefficient of is zero against the alternative that is less than zero?

• 0.125
• 2.5
• -1.25
• -2.5

The correlation between two asset returns is 1. What is the smallest eigenvalue of their correlation matrix?

• 1
• 0.5
• 0
• None of the above

Identify the type and common element (that is, common ratio or common difference) of the following sequence: 6, 12, 24

• arithmetic sequence, common difference 2
• arithmetic sequence, common ratio 2
• geometric sequence, common ratio 2
• geometric sequence, common ratio 3

A simple linear regression is based on 100 data points. The total sum of squares is 1.5 and the correlation between the dependent and explanatory variables is 0.5. What is the explained sum of squares?

• 0.75
• 1.125
• 0.3333
• 0.375

In a 2-step binomial tree, at each step the underlying price can move up by a factor of u = 1.1 or down by a factor of d = 1/u. The continuously compounded risk free interest rate over each time step is 1% and there are no dividends paid on the underlying. Use the Cox, Ross, Rubinstein parameterization to find the risk neutral probability and hence find the value of a European put option with strike 102, given that the underlying price is currently 100.

• 5.19
• 5.66
• 6.31
• 4.18