ECMT6006 Applied Financial Econometrics 2020 S1 Final Exam
Note: This is an open-book online exam. There are in total four problems
and 25 questions. Please attempt all questions. The total mark is 50 and each
question is worth 2 marks. You will have 2 hours and 30 minutes to answer the
questions with handwritten solutions, and 30 minutes to scan and upload your
solutions into Canvas.
Problem 1. [14 marks] Consider a two-period model for returns Rt, t = 1, 2 of
an asset. Let ε0 = 1, and
Rt = µ+ εt,
εt = σtνt, σt = |εt−1|,
where µ = 1, and ν1, ν2 are independent and identically distributed as
νt =
{
1, with probability 2/3
−2, with probability 1/3
for t = 1, 2. Let Ft be the information set available at time t. Please answer
the following questions.
(i) What is the probability distribution of R1?
(ii) What is the probability distribution of R2?
(iii) Compute the conditional mean E1(R2) := E(R2|F1).
(iv) Compute the conditional variance Var1(R2) = Var(R2|F1).
(v) Verify the law of iterated expectation E(R2) = E [E1(R2)] using numbers
given in this problem.
(vi) What are the one-period ahead return point forecasts Rˆt for t = 1, 2?
(vii) What are the two standard deviation one-period ahead return interval
forecasts for t = 1, 2?
Problem 2. [10 marks] Consider the following decomposition of the return of
some financial asset:
Rt = µt + εt
where µt = Et−1(Rt) is the conditional mean of the return, and
εt = σtνt, νt|Ft−1 ∼ N(0, 1).
Answer the following questions.
(i) Show that εt is white noise process.
1
(ii) Show that ε2t is a conditionally unbiased proxy for σ
2
t .
(iii) Consider the following “asymmetric volatility” model for the conditional
variance:
σ2t = ω + βσ
2
t−1 + αε
2
t + δε
2
t−11{εt−1 ≤ 0}.
What empirical fact does this model try to capture? Explain why this
model can capture this empirical fact.
(iv) The model in (iii) is estimated on daily returns on some empirical financial
time series and the following results are obtained:
Estimate Std Error
ω 0.0519 0.0104
α 0.0001 0.0002
β 0.8427 0.0221
δ 0.2178 0.0387
Describe how you would test whether the “asymmetric” term in this volatil-
ity model is needed.
(v) How to construct the out-of-sample one-period ahead rolling window fore-
casts of the volatility using this model?
Problem 3. [14 marks] Answer the below questions on the Value-at-Risk (VaR)
and Expected Shortfall (ES) forecasting.
(i) Why VaR is often considered as a better risk measure than volatility?
(ii) Suppose a stock return follows a normal distribution with mean µ and
variance σ2, i.e., Rt ∼ N(µ, σ2), for all t. What is the relationship between
the 1% return VaR and the return volatility?
(iii) Explain the relationship and the major difference between VaR and ES.
Why is modelling ES useful?
(iv) How to use the “Historical Simulation” (HS) model and the “Filtered
Historical Simulation” (FHS) model to forecast 1% stock return VaR and
ES?
(v) Which method (between HS and FHS) would you prefer to use in practice?
Why?
(vi) How would you use the generalised Mincer-Zarnowitz regression to test
whether the 1% return VaR forecasts from a model is optimal? Be explicit
about the regression model, the null and alternative hypotheses, and how
to test the null hypothesis.
2
(vii) Suppose you have obtained two VaR forecasts, V aRHSt+1 and V aR
FHS
t+1 for
t = 1, . . . , T , from the Historical Simulation model and the Filtered Histor-
ical Simulation model. How would you compare these two forecasts using
the Diebold-Mariano test?
Problem 4. [12 marks] Let T be a set of a continuum of time. For any s ∈ T ,
consider the log asset price P (s) follow a continuous-time diffusion model
dP (s) = µ(s)ds+ σ(s)dW (s)
where µ(s) is the conditional mean (drift) process, σ(s) is the instantaneous/spot
volatility (diffusion) process, and W (s) is a standard Brownian motion. For any
s ∈ T , let Fs be the information set at time s.
(i) For t1, t2 ∈ T and t1 < t2, let R(t1, t2) be the log asset return from time
t1 to t2. What is the conditional distribution of R(t1, t2) given Ft1?
(ii) Let t = 1, 2, . . . denote the (discrete) number of days. Explain how to
define the integrated variance IVt of the asset return on day t, i.e., from
time t− 1 to t?
(iii) What is the difference between the integrated volatilty and GARCH-type
volatility in terms of the risk measurement?
(iv) Denote Rt := R(t − 1, t) as the log return on day t. Prove the below
relationship between the conditional mean of integrated variance IVt and
the conditional variance of the log return rt:
E(IVt|Ft−1) = Var(rt|Ft−1).
(v) Explain how to estimate the integrated variance using intra-day asset
prices.
(vi) Explain how the intra-day high frequency data can be used to build better
volatility forecasting models. Please give specific examples.