With regards to my previous post on a volatility estimator(Rolling variance estimator), based on the risk metrics estimator by JP morgan

I have identified the problem to be that the variables should be raised to the power E in order to work. The variables are used during the stochastic simulation.

But what would be the proper way of writing this, which combination of variables should be written like exp(…)?

The goal is to rewrite:

mean_y_1(t) =alpha_1*mean_y(t-1) + (1-alpha_1)*y(t-1)
mean_y_2(t) =alpha_2*mean_y(t-1) + (1-alpha_2)*y(t-1)

Var_y_1(t) = alpha_1*Var_y_1(t-1) + (1-alpha_1)*(y-mean_y_1(t))*(y-mean_y_1(t-1))
Var_y_2(t) = alpha_2*Var_y_2(t-1) + (1-alpha_2)

*(y-mean_y_2(t))*(y-mean_y_2(t-1))

Cov_y_1_2(t) = (1-alpha_3)*(Cov_y_1_2(t-1) +alpha_3*(y-mean_y_1(t))*(y-mean_y_2(t)))

Cor_y(t) =Cov_y_1_2(t)/(sqrt(var_y_1)*sqrt(var_y_2))

to estimators where the variables are expressed in terms of exponentials, but how should this be done?

So, should

mean_y(t) =alpha*mean_y(t-1) + (1-alpha)*y(t-1)
Be expressed as:
exp(mean_y(t)) =alpha*exp(mean_y(t-1)) + (1-alpha)*exp(y(t-1))

?

And

Var_y_1(t) = alpha_1*Var_y_1(t-1) + (1-alpha_1)*(y-mean_y_1(t))*(y-mean_y_1(t-1))
Be expressed as:
exp(Var_y_1(t)) = alpha_1*exp(Var_y_1(t-1)) + (1-alpha_1)

*(exp(y)-exp(mean_y_1(t)))*(exp(y)-exp(mean_y_1(t-1)))

or

exp(Var_y_1(t)) = alpha_1

*exp(Var_y_1(t-1)) + (1-alpha_1)*(exp(y-mean_y_1(t))*(y-mean_y_1(t-1)))

or something else?

And should the correlation coefficient be written like:

exp(Cov_y_1_2(t)) = (1-alpha_3)*(exp(Cov_y_1_2(t-1)) +alpha_3*(exp(y)-exp(mean_y_1(t)))*(exp(y)-exp(mean_y_2(t))))
or:
exp(Cov_y_1_2(t)) = (1-alpha_3)*(y-mean_y_2(t)))

*exp(Cov_y_1_2(t-1) +alpha_3*exp(y-mean_y_1(t))

or maybe even(although this is probably incorrect)

exp(Cov_y_1_2(t)) = (1-alpha_3)*(exp(Cov_y_1_2(t-1)) +alpha_3*(exp(y-mean_y_1(t)))*(exp(y-mean_y_2(t))))

and should the correlation coefficient be written like:

Cor_y(t) =Cov_y_1_2(t)/(sqrt(var_y_1)*sqrt(var_y_2))

or

exp(Cor_y(t) ) =exp(Cov_y_1_2(t))/(sqrt(exp(var_y_1))*sqrt(exp(var_y_2)))

and how would this impact the choice of alpha? since now the alpha’s originally were around 1/10?

Any help would be greatly appreciated!