FE570 Homework Assignment #3 Due Date: In class on October 16 (Wednesday).
Problem 1. Realized volatility is normally calculated based on n data points using σt = 1 t (ri − r ̄)21/2. Consider the daily stock price
n i=t−n+1
for Netflix (NFLX) from October 22, 2013 to October 22, 2018 in the file
NFLX 2013−2018.csv.
Use the following volatility forecasting method to calculate “expected”
volatility using the given NFLX stock prices. Please use n = 22 for calculating σt and β = 2/(n + 1) for the following volatility forecasting methods. Note: when calculating return, please use ‘Close’ price from the given dataset, and also please use logarithmic return.
(a) The Random Walk forecast: σˆt = σt−1
(b) Exponential smoothing average (often called exponential moving average
or EMA):
σˆt = (1 − β)σt−1 + βσˆt−1, 0 < β < 1 (1)
(c) Exponentially weighted moving average (EWMA), which is a truncated
version of EMA:
nn
σˆt = βiσt−i/βi,0 < β < 1 (2)
i=1 i=1
(d) Compare and contrast the differences of these three estimation methods. Write your analysis and recommendation.
Problem 2. For volatility, the most frequently used model is the general- ized autoregressive conditional heteroskedasticity - GARCH (1,1) model. It
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can be expressed in the following form:
σ2 =ω+aε2 +bσ2 (3)
t t−1 t−1
The advantage of the stationary GARCH(1,1) model is that it can be easily used for forecasting. Namely, the conditional expectation of volatility at time (t + k) equals (please use logarithmic return):
E[σ2 ] = (a+b)k[σ2 −ω/(1−a−b)]+ω/(1−a−b) (4) t+k t
(a) Please use the same dataset (NFLX 2013−2018.csv) from the last prob- lem and estimate a GARCH(1,1) model, and then use Equation (4) to forecast the volatility for NFLX stock for the next 1, 2, ..., 5 days. (Note: You may use fGarch R package to build the model and perform forecast- ing.)
(b) Compare the EWMA realized volatility forecast and the 1 day ahead volatility forecast from the GARCH(1,1) model, and discuss your obser- vations.
Problem 3. In the Roll model, we define efficient price by mt, and we assume mt = mt−1 +ut, where ut are i.i.d. zero-mean random variables (ut ∼ N (0, σu2 )). The bid-ask spread is at − bt = 2c, a constant. At time t, there is a trade at transaction price pt, which may be expressed as: pt = mt + qtc where qt is a trade direction indicator set to +1 if the customer is buying and -1 if the customer is selling. We also assume that buys and sells are equally likely, serially independent (a buy this period does not change the probability of a buy next period), and that agents buy or sell independently of ut (a customer buy or sell is unrelated to the evolution of mt). By definition we have ∆pt = (qt−qt−1)c+ut, and we can write the variance and autocovariance
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as:
γ0 ≡ V ar(∆pt) = 2c2 + σu2. γ1 ≡ Cov(∆pt−1, ∆pt) = −c2.
Note: when calculating return, please use ‘Close’ price from the given dataset. For this assignment, you need to do the following:
(a) Please derive the value γl, l ≥ 2?
(b) Use the Roll model to estimate the bid-ask spread and fundamental volatility of NFLX stock during the period from October 22, 2013 to October 22, 2018. Please use the same dataset for the last two problems (NFLX 2013−2018.csv).
Homework Honor Policy: You are allowed to discuss the problems be- tween yourselves, but once you begin writing up your solution, you must do so independently, and cannot show one another any parts of your written solutions. The HW is to be pledged (that it adheres to this).
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