FN3142 Quantitative Finance

Summer 2020

Question 1

Consider the following ARMA(1) process:

zt  = √ + Qzt-1 + εt + θεt-1 ,                                            (1)

where εt  is a zero-mean white noise process with variance σ 2 , and assume  j Qj , jθj  <  1  and Q + 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 Q = 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 Q ≠ 0.  Calculate the conditional and unconditional variances of zt , that is, V art-1  [zt] and Var [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 Hitt(i)  =  1{rt   ≤  Var t(-i)},i =  1, 2, for two particular Value-at-Risk measures Var t(-1)  and Var t(-2), the following simple regressions are run, with the standard errors in parentheses corresponding to the parameter estimates:

Hitt(1) = 0.06151 + ut        (0.00432)

Hitt(2) = 0.04372 + ut        (0.00589)

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 Christofersen (1998) based on the fact that the hit variable is i.i.d.  Bernoulli(Q), where Q is the critical level, under the null hypothesis that the forecast of the conditional Value-at-Risk measure VaRt  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 diferences 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 et+hjt  = Yt+h - Y(ˆ)thjt for a horizon h possess?

For the remaining two sub-questions of the exercise, consider a forecast Y(ˆ)t+1jt   of a variable Yt+1 . You have 100 observations of Y(ˆ)t+1jt and Yt+1, and decide to run the following regression:

Yt+1  = Q + βY(ˆ)t+1jt + ετ

The results you obtain are given in Table I:

           Table I. Regression results

(d) [20 marks] What null hypothesis should we setup 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 σ 2   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, σt(2)), where

σt(2) = ω + βσt(2)-1 + αxt(2)-1

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 di伍culties.

(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.