Let's say I have two equity indices X and Y. Assume they are negatively correlated with some leverage. I want to hedge X with Y.
I have seen many ways of computing a beta to describes the relationship
$(X_t)$ and $(Y_t)$ are strictly positive. $t\in\{0,1,..,n\}$ daily,
$D^X_t = X_t - X_{t-1}$ (difference series)
$R^X_t = \frac{X_t-X_{t-1}}{X_{t-1}}$ (return series)
Let's say today is t=n, I observe all the past values up to t=0. The goal of my daily hedging is to find at time n the beta, the quantity of Y to buy for each dollar of X, such that $E[D^X_{n+1}-\beta_{n+1} D^Y_{n+1} ] =0 $
The CAPM Beta could be defined as: $\beta_t=\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^Y]}$ (which I could estimate in many ways let's ignore that here unless someone has a better method than a Kalman filter).
My questions are:
could I define $\beta_t=\bigg (\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^X]} \bigg )^{-1} $
$\frac{{cov}(R_t^X,R_t^Y)}{ V[R_t^X]}$ makes more sense to me because I make the market of X, I think in terms of move of X.
- could I define $\beta_t = \frac{{cov}(D_t^X,D_t^Y)}{ V[D_t^Y]}$? If I estimate it dynamically with a kalman filter for example I understand that the series $D$ and therefore this beta strongly depends on the level of X and Y whereas $R$ is rescaled.
I could further expand on my studies but I do not want to loose 90% of the potential audience nor skew the answer.