代写FN3142 Quantitative Finance 2018调试SPSS

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]