# Optimal Hedging Ratio using Copula Models

Let $$r_{s, t}$$ and $$r_{f, t}$$ be the return rates of the spot and futures of a commodity at time $$t$$. The hedging ratio based on variance minimization is calculated by finding the minimum of the variance of the combined returns:

$$\beta_t = \rho_{sf, t} \frac{\sigma_{s, t}}{\sigma_{f, t}},$$

where $$\rho_{sf, t}$$ is the time-varying correlation and $$\sigma_{s, t}$$ and $$\sigma_{f, t}$$ are the corresponding time-varying volatility of the spot and future returns respectively. From the several papers that I went through, the volatility measures are calculated using different GARCH-type models and the filtered standardized residuals are used to estimate various Copula models. My question is how the estimated static Copula models are then transformed to the time-varying correlation that is later used to calculate the hedge ratio? I would really be thankful for any kind of guidance.

Using a static copula model implies $$\rho_{s,f,t}\equiv\rho_{s,f}$$. In such case fitting a copula model to obtain $$\rho_{s,f}$$ is an overkill, since it can be estimated very simply by the empirical correlation of the two standardized residual series from the two GARCH models. Of course, a availability of the joint distribution via a copula-GARCH model facilitates all kinds of interesting calculations, so the model may well be worth fitting, just not for estimating $$\rho_{s,f}$$ alone.
If you want time-varying correlation, you may be tempted to consider using BEKK-GARCH or DCC-GARCH models; this is what many authors do. However, the models seem to be seriously flawed – except for the case of diagonal BEKK-GARCH; see Caporin & McAleer (2013), McAleer (2019a), McAleer (2019b), Allen & McAleer (2018). Other alternatives are time-varying copula GARCH and GO-GARCH, among other, though I am not sure how sound they are theoretically. In any case, the latter two as well as DCC-GARCH are available in the rmgarch package in R should you decide to try them out.