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PRELIMINARY EXAM 2023
MODULE CODE : FN3142
MODULE TITLE : Quantitative Finance
Question 1
Recall that the probability density function for a normally distributed random variable, with mean μ and variance σ2 is:
(a) Show that a stationary GARCH(1,1) model can be re-written as a function of the unconditional variance and the deviations of the lagged conditional variance and lagged squared residual from the unconditional variance. 40 marks
(b) Now assume that xt is conditionally normally distributed N(0, o2) where 
. Write down the log-likelihood function for this model given a sample of data (x1; x2; …; xT ). 40 marks
(c) Describe and explain how we can obtain estimates of (o,a, p) for the GARCH(1,1) model and discuss any issues that arise. 20 marks
Total = 100 marks
Question 2
Suppose that for a given set of data VaR forecasts are calculated with historical simulation and GARCH methods.
(a) Show how to construct a sequence of ‘hit’ variables and for testing the accuracy of the VaR forecasts. 40 marks
(b) The following regression was run (standard errors are in parentheses below the parameter estimates):
Hits=0.095+ ur
(0.025)
Hit fdR ci=-0.2825+ ur
(0.35)
Explain how the above information can be used to test the accuracy of the VaR forecasts from these two models. 40 marks
(c) Describe a method based on the chi-squared statistic that can be used to test for the serial correlation in hits. 20 marks
Total = 100 marks
Question 3
(a) Describe how one can test forecast optimality with a Mincer-Zarnowitz regression? 40 marks
(b) Consider a forecast of a variable, Yt. You have 100 observations of and Yt and you run the following regression:
The following results are obtained:
|
|
Estimate |
std error |
t-statistic |
|
β0 |
-0.008 |
0.0052 |
-2.3329 |
|
β1 |
1.6135 |
1.0399 |
0.1468 |
(i) What can be inferred from this output? 20 marks
(ii) What hypothesis do you need to test in relation to a Mincer-Zarnowitz regression and what is your test and conclusion? 40 marks Total = 100 marks
Question 4
(a) What is the “efficient market hypothesis” ? 30 marks
(b) Discuss two of the modifications/extensions/refinements of the original definition of the efficient market hypothesis. 40 marks
(c) How does “collective data snooping” relate to the efficient market hypothesis? 30 marks
Total = 100 marks