FN3142 Quantitative finance
Summer 2022
Question 1
Consider a process Xt that resembles an AR(1) process except for a small twist:
Xt = (−1)t δ0 + δ1Xt−1 + εt ,
where εt is a zero-mean white noise process with variance σ 2 and δ1 ∈ (−1, 1).
(a) Calculate the conditional and unconditional variances of Xt , that is, Vart−1 [Xt] and Var [Xt]. [20 marks]
(b) Derive the autocovariance and autocorrelation functions of this process for all lags as functions of the parameters δ0 and δ1 . [30 marks]
(c) Explain what covariance stationarity means. Does Xt satisfy the requirements? [15 marks]
(d) Calculate the 1- and 2-period ahead conditional and the unconditional expectations of Xt , i.e., Et−1 [Xt], Et−2 [Xt], and E[Xt]. Comment on your findings. [20 marks]
(e) Consider now another process given by
Wt = δ0 + δ1 (−1)t Wt−1 + εt .
Can this process be covariance stationary? Explain. [15 marks]
Question 2
(a) Define Value-at-Risk. What are its pros and cons relative to variance as a measure of risk? Explain in detail. [15 marks]
(b) Consider a portfolio consisting of a $50,000 position in asset K and a $150,000 position in asset L. Assume that returns on these two assets are i.i.d. Gaussian with mean zero, that the daily volatilities of these two assets are 1% for asset K and 2% for asset L, and that the coefficient of correlation between their returns is 0.2.
(i) What is the 10-day VaR at the 1% critical level for the portfolio?
(ii) Compare your answer above to the 1% critical level VaRs that we would have on investing in K and L assets separately. By how much does diversification reduce the VaR? [15 marks]
For parts (c) to (f), consider a dummy variable ut that takes value 1 when a daily loss is exceeding the VaR threshold on date t and 0 otherwise. Suppose that you build such a variable for three VaR models that are constructed using (i) a simple MA (moving average), (ii) an EWMA (a model called “exponentially weighted moving average”) and (iii) GARCH volatility estimates and use the critical value 1%. Suppose further that you use this dummy variable to run the following regressions:
ut = γ0 + εt
and obtain the (volatility-model dependent) estimates in the table below, with standard errors in parentheses:
(c) Explain how the above regression outputs can be used to test the accuracy of the VaR forecasts from these models (on their own). [20 marks]
(d) How do the empirical performances of the three methods for constructing the VaRs compare? Explain. [15 marks]
(e) Explain how you could compare the relative performances of the VaR forecasts. [20 marks]
Suppose now that you use the dummy variable to run the following regressions:
ut = γ1 + γ2ut−1 + εt
and obtain the (volatility-model dependent) estimates in the table below, with standard errors in parentheses:
(f) Briefly explain how the above regression outputs can be used to evaluate the accuracy of the VaR forecasts from these 3 models (on their own). [15 marks]
Question 3
(a) What is volatility clustering? Suggest tests for it. Explain your answers in detail. [20 marks]
(b) Explain Black’s observation about the “leverage efect,” i.e., the link between stock returns and changes in volatility, and provide an explanation for this efect. [20 marks]
(c) Does a simple GARCH(1,1) model capture the leverage efect? Explain. [20 marks]
(d) Describe two GARCH-type models that account for the leverage efect in your own words. Note: For full marks, write down the processes with equations and explain analytically how they work. [20 marks]
(e) Consider the volatility model called EWMA (“exponentially weighted moving average”) model of volatility:
σt(2) = λσt(2)−1 +(1 − λ)rt(2)−1 , (1)
which formally looks like a special case of the GARCH(1,1) with setting ω = 0, α = λ, and β = 1 − λ .
(i) Show why this formula corresponds to weights assigned to the rt(2) that decrease expo- nentially as we move back through time (rt is the percentage change in the market variable between day t − 1 and day t).
(ii) What undesirable property does this model have compared to GARCH(1,1) based on the parameter values? Explain. [20 marks]
Question 4
(a) What is the efficient market hypothesis statement according to Malkiel (1992)? Explain in your own words. [20 marks]
(b) Black (1986) gives an alternative definition of market efficiency. What is it and why is Black’s definition difficult to test? Explain in your own words. [20 marks]
(c) Suppose we are at time t, and we are interested in the efficiency of the market of a given stock. Let Ωt(w) denote the weak-form efficient markets information set at time t, Ωt(ss) denote the semi strong-form efficient markets information set at time t, and Ωt(s) denote the strong-form efficient markets information set at time t. To which information set, if any, do the following variables belong? Explain.
1. The nominal size of the short position Melvin Capital (a hedge fund) currently has in a given stock.
2. The size of the long position in Gamestop shares purchased today by DeepF, a user of the subreddit r/WallStreetBets
3. The current 3–month US Treasury bill rate.
4. The value of the stock at time t +2. [20 marks]
(d) Explain how “unit root” models and models with time trends look like. What are the similarities and differences between these models? [20 marks]
(e) Explain a way how one can test for unit root. [20 marks]