Problem 1
(20 points, each for 4 points) This question is about the larger VIX data set {vixlarge.mat} (this is a matlab format file, you can load it by installing package R.matlab and use commandrmat)that contains the VIX data and the associated dates. introduce and investigate in this proposal. The CBOE VIX is colloquially referred to as the fear index or the fear gauge. We choose to study the VIX not only on the widespread consensus that the VIX is a barometer of the overall market sentiment as to what concerns investors risk appetite, but also on the fact that there are many trading strategies that rely on the VIX index for hedging and speculative purposes.
Plot the VIX data against date. Clearly label the horizontal and vertical axises.
We know that volatility (ythearafter) exhibits a high degree of persistence and its likely thatytis better forecast by using more lags,yt1,yt3,.That makes us think of the model withJlags:
yt=0+1yt1+2yt2++JytJ+ut
whereutis the error term. But to capture long-range dependence might entailJ=10J=20,or higher. Let the dependent variableybe the VIX and the first and the 2nd to 23th columns of the independent variableXbe the intercept term and the 1-22 lag of VIX. Write your own code to imlpement AR(1) to AR(22) model, and pick out the best model by AIC and BIC, are the results same, genarate a table to illustrate your results? if not, why?
Set the window length at 3000 and make forecast on the next periodyt+1, start from the beginning and roll until the end. For each roll, we make forecast using AR(1) to AR(22). Compute the mean squared forecast errors and the mean absolute forecast errors for AR(1) to AR(22) and report them in a table.
In (b) and (c), estimating such a large number of coefficients could entail a lot of estimation error and lead to bad forecasting properties. Following Fernandes, Medeiros, and Scharth (2014), a very popular way to model the VIX index is the heterogeneous autoregressive (HAR) model by Corsi (2009) . The HAR model gains great popularity not only because the HAR model well approximates long memory and multiscaling properties of the VIX index, but it is also very easy to implement in practice. The standard HAR model in Corsi (2009) postulates thath-step-ahead daily volatilityyt+hcan be modeled by
yt+h=0+dy(1)t+wy(5)t+my(22)t+t+h
where we define
y(l)tl1s=1lyts
as the averages of the previouslperiods ofyfrom periodtand{t}is a zero mean innovation process. A typical choice in the literature for the lag index vectorlis[1,5,22]so as to mirror the daily, weekly, and monthly components of volatility process. Using HAR model to do forcasting exercise in (b) and (c) and comparing HAR model with the best AR model in (b) and (c) by AIC, BIC and rolling method, which one is the best?
Try to come up with an algorithm that can beat the best performing method stated in question (d). Clearly describe your motivation, the details of the algorithm, and the results.
Problem 2
(18 points, each for 3 points) (Commodity prices). Consider the daily gold price, London Bullion Market, price per Troy Ounce in U.S. Dollars at 10: 30 AM local time, from January2,1992to March 31,2015. See fileGoLDLBM1030. txt.
Obtain the time plot of the gold price.
Letrtbe the log return of the daily gold price. Obtain the time plot ofrt
Are there serial correlations in thertseries? You may useQ(10)to draw the conclusion.
Build an AR model forrt.Check the adequacy of the model.
Remove any parameter of the AR model witht-ratio less than 1.645 in absolute value. Write down the file model.
Use the final model to compute 1 -step to 3 -step ahead forecasts ofrtat the forecast origin March31,2015
Soution: First, we write a function to estimate.
Now,we can do simulation with the function
Problem 3
(32 points, each for 4) If two asset returnsR1,tandR2,thave correlationand time varying volatility1,tand2,tthen their covariance (GARCH covariance) is:
12,t=1,t2,t,
where we assume theis a constant. In the following, estimatebetween two series.
Get the close prices of MSFT and AMZN from 2017-01-04 to 2019-12-25, and compute the return as msftret and amznret.
## [1] MSFT
## [1] AMZN
Set GARCH specification as garchspec: mean function is a constant, variance function is standard garch, distribution form is specified as skew t distribution.
Estimate the GARCH model for each return series.
Compute the standardized returns for each return series andas the sample correlation of the standardized returns as msftamzncor, print the correlation
Compute the GARCH covariance and plot it.
An important application of GARCH cvariance is to optimize the variance of the portfolio which depends on the portfolio weights, the variance of all the assets and the covariance between the asset returns. The variance of portfolio of two assets (2p,t) with weightw1,tinvested in asset 1 and(1w1,t)in asset 2 is
2p,t=w21,t21,t+(1w1,t)222,t+2w1,t(1w1,t)12,t.
There are many ways to define optimal weightw1,t. One appraoch is to setw1,tsuch that the portfolio variance2p,tis minimized. Whats the first order condition to computew1,t? What is the solution ofw1,t?
If we use MSFT and AMZN to construct the portfolio, what is the optimal weight for MSFT? Plot your results.
Following CAPM model, a stocks beta is used to measured the systematic risk of a stock which is defined as the covariance of the stock return and the market return, divided by the variance of the market return. The higher it is, the more risky the stock and thus the higher the required rate of return. For US stocks, the market return is the return on the S&P 500. Compute the dynamic beta of MSFT and plot your results.
## [1] GSPC
Problem 4
(30 points, each for 5 points)
(MA)
(MA,Moving average
MA
MA = (C1+C2+C3+C4+C5+.+Cn)/n ,Cn
5103060100200n
A000009.XSHE(20191227252)
(1)51030,(3)
10301030 301030103010,303030
(3)
(3)
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