8007 Exam II: Mathematical Foundations of Risk Measurement - 2015 Edition Questions and Answers

What is the indefinite integral of the function f(x) = ln(x), where ln(x) denotes the natural logarithmic function?

Let X be a random variable distributed normally with mean 0 and standard deviation 1. What is the expected value of exp(X)?

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?

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.

If the annual volatility of returns is 25% what is the variance of the quarterly returns?

What can be said about observations of random variables that are i.i.d. a normally distributed?

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

An underlying asset price is at 100, its annual volatility is 25% and the risk free interest rate is 5%. A European put option has a strike of 105 and a maturity of 90 days. Its Black-Scholes price is 7.11. The options sensitivities are: delta = -0.59; gamma = 0.03; vega = 19.29. Find the delta-gamma approximation to the new option price when the underlying asset price changes to 105

Let X be a random variable normally distributed with zero mean and let . Then the correlation between X and Y is:

A 2-step binomial tree is used to value an American put option with strike 105, given that the underlying price is currently 100. At each step the underlying price can move up by 10 or down by 10 and the risk-neutral probability of an up move is 0.6. There are no dividends paid on the underlying and the continuously compounded risk free interest rate over each time step is 1%. What is the value of the option in this model?

Which of the following is not a direct cause of autocorrelation or heteroskedasticity in the residuals of a regression model?

Evaluate the derivative of ln(1+ x2) at the point x = 1

Every covariance matrix must be positive semi-definite. If it were not then:

A biased coin has a probability of getting heads equal to 0.3. If the coin is tossed 4 times, what is the probability of getting heads at least two times?

A typical leptokurtotic distribution can be described as a distribution that is relative to a normal distribution

An operational risk analyst models the occurrence of computer failures as a Poisson process with an arrival rate of 2 events per year. According to this model, what is the probability of zero failures in one year?