FN3142 Quantitative Finance
Summer 2020
Question 1
Consider the following ARMA(1) process:
zt = γ + αzt−1 + εt + θεt−1, (1)
where εt
is a zero-mean white noise process with variance σ2, and assume |α|, |θ| < 1 and α + θ ≠ 0, which together make sure zt is covariance stationary.
(a) [20 marks] Calculate the conditional and unconditional means of zt
, that is, Et−1 [zt] and E [zt].
(b) [20 marks] Set α = 0. Derive the autocovariance and autocorrelation function of this process for all lags as functions of the parameters θ and σ.
(c) [30 marks] Assume now α ≠ 0. Calculate the conditional and unconditional variances of zt
, that is, V art−1 [zt
] and V ar [zt].
Hint: for the unconditional variance, you might want to start by deriving the uncondi-tional covariance between the variable and the innovation term, i.e., Cov [zt
, εt].
(d) [30 marks] Derive the autocovariance and autocorrelation for lags of 1 and 2 as functions of the parameters of the model.
Hint: use the hint of part (c).
Question 2
(a) [20 marks] Explain in your own words how one can conduct an unconditional coverage backtest for whether a Value-at-Risk measure is optimal, and relate this test to the so-called “violation ratio.”
(b) [20 marks] Suppose that after we have built the hit variable , i = 1, 2, for two particular Value-at-Risk measures and , the following simple regressions are run, with the standard errors in parentheses corresponding to the parameter estimates:
Describe how the above regression outputs can be used to test the accuracy of the VaR forecasts. Do these regression results help us decide which model is better? Explain.
(c) [20 marks] Using your own words, describe the conditional coverage backtest proposed by Christoffersen (1998) based on the fact that the hit variable is i.i.d. Bernoulli(α), where α is the critical level, under the null hypothesis that the forecast of the conditional Value-at-Risk measure is optimal.
(d) [20 marks] Give an example of a sequence of hits for a 5% VaR model, which has the correct unconditional coverage but incorrect conditional coverage.
(e) [20 marks] Discuss at least two approaches to VaR forecasting to deal with skewness and/or kurtosis of the conditional distribution of asset returns.
Question 3
Answer all five sub-questions.
(a) [20 marks] What is the definition of market efficiency for a fixed horizon? Is it possible to have deviations from efficiency in a market that is efficient? Explain.
(b) [20 marks] Describe collective data snooping and individual data snooping in your own words, and briefly discuss the differences between them.
(c) [20 marks] Forecast optimality is judged by comparing properties of a given forecast with those that we know are true. An optimal forecast generates forecast errors which, given a loss function, must obey some properties. Under a mean-square-error loss function, what three properties must the optimal forecast error for a horizon h possess?
For the remaining two sub-questions of the exercise, consider a forecast of a variable Yt+1. You have 100 observations of and Yt+1, and decide to run the following regression:
The results you obtain are given in Table I:
Table I. Regression results
(d) [20 marks] What null hypothesis should we set up in order to test for forecast optimality? Can this test be conducted with the information given?
(e) [20 marks] Explain what can be inferred from Table I.
Question 4
The probability density function of the normal distribution is given by
where µ is the mean and σ
2
is the variance of the distribution.
(a) [20 marks] Assuming that µ = 0, derive the maximum likelihood estimate of σ2 given the sample of i.i.d data (x1, x2, . . . , xT).
(b) [20 marks] Now assume that xt
is conditionally normally distributed as N(0, ), where
Write down the likelihood function for this model given a sample of data (x1, x2, . . . , xT).
(c) [15 marks] Describe how we can obtain estimates for {ω, α, β} for the GARCH(1,1) model and discuss estimation difficulties.
(d) [20 marks] Describe in your own words what graphical method and formal tests you can use to detect volatility clustering.
(e) [25 marks] Describe the RiskMetrics exponential smoother model for multivariate volatil-ity, and discuss the pros and cons of the constant conditional correlation model of Bollerslev (1990) versus the RiskMetrics approach.