FN3142 ZA
Quantitative Finance
Question 1
Consider the following MA(2) process:
zt = ut + Q1ut-1 + Q2ut-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 Var [zt]. [30 marks]
(c) Derive the autocorrelation function of this process for all lags as functions of the param- eters Q1 and Q2 . [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 Q = 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 tth 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]
FN3142 ZB
Quantitative Finance
Question 1
Consider a process Yt that resembles an MA(1) process except for a small change: Xt = ut + (-1)t δut-1 ,
where ut i~.i.d N(0, σu(2)) and 0 < δ < 1 constant.
Hint: (-1)t = 1 if t is an even number, and -1 if t is odd.
(a) Find Et [Xt+1], Et [Xt+2], and E[Xt]. Pay attention to t being odd or even. [30 marks]
(b) Find Var[Xt]. [20 marks]
(c) Derive the autocovariance for lags 1 and 2. [30 marks]
(d) Explain what covariance stationarity means, and relate it to your findings in parts (a), (b), and (c). [20 marks]
Question 2
There are three companies, called A, B, and C, and each has a 4% chance of going bankrupt. The event that one of the three companies will go bankrupt is independent of the event that any other company will go bankrupt.
Company A has outstanding bonds, and a bond will have a net return of r = 0% if the corporation does not go bankrupt, but it will have a net return of r = -100%, i.e., losing everything invested, if it goes bankrupt. Suppose an investor buys $1000 worth of bonds of company A, which we will refer to as portfolio P1 .
Suppose also that there exists a security whose payof depends on the bankruptcy of companies B and C in a joint fashion. In particular, if neither B nor C go bankrupt, this derivative will have a net return of r = 0%. If exactly one of B or C go bankrupt, it will have a net return of r = -50%, i.e., losing half of the investment. If both B and C go bankrupt, it will have a net return of r = -100%, i.e., losing the whole investment. Suppose an investor buys $1000 worth of this derivative, which is then called portfolio P2 .
(a) Calculate the VaR at the α = 10% critical level for portfolios P1 and P2 . [30 marks]
(b) Calculate the VaR at the α = 10% critical level for the joint portfolio P1 + P2 . [20 marks]
(c) Is VaR sub-additive in this example? Explain why the absence of sub-additivity may be a concern for risk managers. [20 marks]
(d) The expected shortfall ESα at the α critical level can be defined as
ESα = -Et [RjR < -VaRα +1] ,
where R is a return or dollar amount. Calculate the expected shortfall at the α = 10% critical level for portfolio P2 . Is this risk measure sub-additive? [30 marks]
Question 3
(a) Explain Black’s observation regarding the link between the stock returns and changes in volatility and provide a possible explanation for this efect. [25 marks]
(b) Does a simple GARCH(1,1) model capture the leverage efect? Explain. [25 marks]
(c) Describe two GARCH-type models that account for the leverage efect. Note: For full marks, write down the processes with equations and explain analytically how they work. [40 marks]
(d) For both GARCH-type models you mentioned in part (c), discuss whether they nest the GARCH(1,1) model. [10 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 tth 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]