Instructions: You can submit one solution per group of five. Please clearly indicate the names
of the group members. Answers should be clear. Fuzzy logic will be not accepted. Any additional
assumption in the analysis should be clearly stated. Data should be downloaded from Bloomberg
or your favorite data source (WRDS or Yahoo Finance). There is one bonus question to help to
hedge your midterm grade.
Question 1 (25 Points)
Suppose that the monthly log returns, in percentages, of a stock follow the following Markov
switching model:
rt = 1.25 + at
; at = stt
s
2
t =
0.10a
2
t-1 + 0.93s
2
t-1
if st = 1
4.24 + 0.10a
2
t-1 + 0.78s
2
t-1
if st = 2
where the transition probabilities are
P (st = 2|st-1 = 1) = 0.15; P (st = 1|st-1 = 2) = 0.05
Suppose that a100 = 6.0, s
2
100 = 50.0, and s100 = 2 with probability 1. What is the 1-step ahead
volatility forecast at the forecast origin t = 100? Also, if the probability of s100 = 2 is reduced to
0.8., what is the 1-step ahead volatility forecast at the forecast origin t = 100?
1
Question 2 (25 Points)
Consider the monthly simple returns of GE from January 1926 to December 2008. Use the last
three years of data for forecasting evaluation.
(a) Using lagged returns rt-1, rt-2, rt-3 as input, build a 3-2-1 feed-forward network to forecast
1-step-ahead returns. Calculate the mean squared error of forecasts.
(b) Again using lagged returns rt-1, rt-2, rt-3 and their sign (directions) to build a 6-5-1 feed forward network to forecast 1-step ahead direction of GE stock price movement with 1 denoting
upward movement. Calculate the mean squared error of forecasts.
Note: Let rtn denote a time series in R. To create a direction variable for rtn, use the
command
drtn = ifelse(rtn > 0, 1, 0)
Question 3 (25 Points)
Download the CBOE S&P 500 BuyWrite Index (BXM), the S&P 500 Total return (SPTR) the
CBOE Volatility Index (VIX) and the CBOE Crude Oil Volatility index from http://www.cboe.
com/micro/buywrite/dailypricehistory.xls Suppose each of these time series follows a diffusion model
dXt = µ(Xt)dt + s(Xt)dWt
(a) Estimate the drift and volatility nonparametrically using all available except the last 30 days.
Comment on the shape of the drift and the volatility. Is their evidence for mean-reversion?
Is the volatility function convex or concave?
(b) Use the estimated model in (b) to predict the volatility and the drift for the last 30 days.
Calculate the mean-squared error of your forecasts. Comment on the accuracy of the forecasts
Question 4 (25 Points)
The file ibm-d2-dur.txt contains the adjusted duration between trades of IBM stock on November
2, 1990. The file has three columns consisting of day, time and trade measured in seconds from
midnight, and adjusted durations.
(a) Build an EACD model for the adjusted duration and check the fitted model. Comment on
your findings.
(b) Build a WACD model for the adjusted duration and check the fitted model. Comment on
your findings.
(c) Build a GACD model for the adjusted duration and check the fitted model. Comment on
your findings.
(d) Compare the prior three duration models.
2
Bonus Question 5 (25 Points)
Because of the existence of inverted yield curves in the the term structure of interest rates, the
spread of interest rates should be nonlinear. To verify this, consider the weekly U.S. interest rates of
(i) Treasury 1-year constant maturity rate and (ii) Treasury 3-year constant maturity rate. Denote
the the two interest rates by r1t and r3t
, respectively, and the data span is from January 5, 1962,
to April 10, 2009. The data can be obtained from the Federal Reserve Bank of St. Louis at
http://research.stlouisfed.org/fred2/categories/115.
(a) Let st = r3t - r1t the term spread in log-interest rates. Is {st} linear? Perform some nonlinearity tests and draw the conclusion using the 5% significance level.
(b) Let s
*
t = (1 - B)st the change in the term spread. Is {s
*
t } linear? Perform some nonlinearity
tests and draw the conclusion using the 5% significance level.
(c) Build a threshold model for the st series and check the fitted model
(d) build a threshold model for the s
*
t
series and check the fitted model.