FN3142 Quantitative Finance
Question 1
Consider the following MA(2) process:
zt = ut + α1ut−1 + α2ut−2,
where ut
is a zero-mean white noise process with variance σ
2
.
(a) Calculate the conditional and unconditional means of zt
, that is, Et
[zt+1] and E [zt
]. [20 marks]
(b) Calculate the conditional and unconditional variances of zt
, that is, V art
[zt+1] and V ar [zt
]. [30 marks]
(c) Derive the autocorrelation function of this process for all lags as functions of the param eters α1 and α2. [50 marks]
Question 2
(a) What are the two main problems in multivariate volatility modelling? Explain them briefly. [25 marks]
(b) Describe Bollerslev (1990)’s constant conditional correlation (CCC) model. [25 marks]
(c) Describe what tests you can use to test for volatility clustering. [25 marks]
(d) What information criteria can be used (as measures of performance) that penalise models for using a larger number of parameters? Describe their link with the log-likelihood function and the number of parameters. [25 marks]
Question 3
(a) Describe how one can determine Value-at-Risk (VaR) using models based on the normal distribution, and critically assess such procedure. [60 marks]
(b) Consider a position consisting of a $20,000 investment in asset X and a $20,000 in-vestment in asset Y. Assume that returns on these two assets are i.i.d. normal (Gaussian) with mean zero, that the daily volatilities of both assets are 3%, and that the correlation coefficient between their returns is 0.4. What is the 10-day VaR at the α = 1% critical level for the portfolio? [40 marks]
Question 4
(a) What does serial correlation mean? Explain. [10 marks]
(b) Suppose you have a fair coin, that is, the probability of seeing a ‘head’ is always equal to one-half, and coin tosses are independent of each other. Let xt take the value −1 or 1 depending on whether the t
th coin toss came up heads or tails.
Consider now a process yt that is given by
yt = xt + xt−1.
Calculate the autocorrelations of the process yt
. [30 marks]
(c) Describe the three types of market efficiency as defined by Roberts (1967). [30 marks]
(d) Does weak-form. market efficiency imply strong-form. market efficiency? What about the reverse? Explain. [10 marks]
(e) Under the Efficient Market Hypothesis (EMH), what should be the correlation coefficient between stock returns for two non-overlapping time periods? Can the process yt
from part
(b) describe a return process under EMH? Explain. [20 marks]