Correlation update and volatility forcasting

[ziminli1228]

In the section of forecasting volatility, a few questions were asking updated correlations by first update volatility of A,B and cov of A,B, then get the updated and cov and correlation. Under GARCH(1,1), is there a reason why some questions are using different w for correlations and volatility while some are always using one ?

 

[David Harper CFA FRM]

Hi @ziminli1228 I almost didn't know to which problem etc you are referring; more specific references can be helpful. I figured it out really only because I've had to do these calcs several times.

In order to update the correlation, from yesterday ρ(n-1) to today ρ(n), needed are both (i) an updated covariance between X and Y and (ii) updated variances/StdDevs of each of X and Y so that we can get ρ(n) = cov(X_n, Y_n)/[σ(X_n)*σ(Y_n)].
So Hull is using GARCH to update both the numerator in the updated correlation--i.e., cov(X_n, Y_n)--and to update each of the multiplicands in the denominator; i.e., standard deviations σ(X_n) and σ(Y_n).

So it's really just that GARCH is being used three times (note that in theory he could use three omegas! or just one by assuming all three uses are the same!).

Here is Hull's EOC 11.6 for reference, with my XLS below (the omegas in extra yellow).

Hull EOC 11.6: Suppose that the current daily volatilities of asset X and asset Y are 1.0% and 1.2%, respectively. The prices of the assets at close of trading yesterday were $30 and $50 and the estimate of the coefficient of correlation between the returns on the two assets made at this time was 0.50. Correlations and volatilities are updated using a GARCH(1,1) model. The estimates of the model’s parameters are E = 0.04 and d = 0.94. For the correlation ^ = 0.000001 and for the volatilities ^ = 0.000003. If the prices of the two assets at close of trading today are $31 and $51, how is the correlation estimate updated?

Answer:
The most recent returns for X and Y are 1/30 = 0.03333 and 1/50 = 0.02, respectively. The previous covariance is 0.01×0.012×0.50 = 0.00006.
... The new estimate of the covariance is 0.000001 + 0.04×0.03333×0.02 + 0.94×0.00006 = 0.0000841

The new estimate of the variance of X is 0.000003 + 0.04 × 0.03333^2 + 0.94×0.01^2 = 0.0001414
... so that the new volatility of X is sqrt(0.0001414) = 0.01189 or 1.189%.

The new estimate of the variance of Y is 0.000003 + 0.04×0.02^2 + 0.94×0.012^2 = 0.0001544
... so that the new volatility of Y is sqrt(0.0001544) = 0.01242 or 1.242%.

The new estimate of the correlation between the assets is therefore 0.0000841 ∕ (0.01189×0.01242) = 0.569.

Solution in XLS https://www.dropbox.com/s/pglglb2j978pqy2/0310-hull-eoc-11-6.xlsx?dl=0