Problem 7.5 This problem is based on the data contained in the text file monthly.asc. The first column represents the values of the short interest rate as given by the one year T-bill between May 1986 and November 1999, the second column gives the long interest rate as given by the 30 years US Government Bonds, and the third column gives the values of the S&P 500 index during the same period.

1. Use the equation:

St+Δt − St = St[μΔt + σ

√

Δtt]

used in the text as equation (7.21) to model the monthly S&P 500 values, to estimate the parameters μ and σ from the data. Remember, Δt = 1/12 stands for one month. Explain your work.

2. Similarly, use the data and the equation:

rt+Δt = rt − λr(rt − r)Δt + σr

√

Δt (r)

t used in the text as equation (7.19) to model the monthly values of the short interest rate, to estimate the parameters λr, r, and σr. Be imaginative and explain your work.

3. Finally, use the data and the equation:

st+Δt = st − λs(st − s)Δt + σs

√

Δt (s)

t used in the text as equation (7.21) to model the monthly values of the interest rate spread, to estimate the parameters λs, s, and σs. Again, explain your work.

4. Assuming that the three white noise sequences {t}t, {(r)
t }t, and {(s)
t }t are independent, use the above equations and parameter estimates to generate a 100×120×3 array containing 100 scenarios of 120 possible monthly values of the short interest rate, the long interest rate and the S&P 500 index, starting from the last entries of the data in the file monthly.asc.
Save these three sets of 100 scenarios in text files named IndepShort, IndepLong and IndepSP500 respectively.
5. From the data, compute the variance/covariance matrix of the log-return of the S&P 500, the first difference of the short interest rate, and the first difference of the spread. Assuming now that the variance covariance matrix of the three white noise sequences {t}t, {(r)
t }t, and {(s)
t }t is the same as the empirical variance/cavariance matrix you just computed, generate a new set of 100 × 120 × 3 array containing 100 scenarios of 120 possible monthly values of the short interest rate, the long interest rate and the S&P 500 index. Save the 3 sets of scenarios in text files named CorrelShort, CorrelLong and CorrelSP500 this time.