Using BEKK Conditional Covariance for an Optimal Hedge Ratio
Summary
The document describes an application of a multivariate GARCH-BEKK model to spot and futures price returns. It presents estimated model parameters and asks how to derive the conditional variance and covariance matrix from them. The stated hedge ratio is the conditional covariance between spot and futures returns divided by the conditional variance of futures returns.
The displayed recursion illustrates how a constant term, lagged return shocks, and lagged conditional covariance contribute to the next covariance matrix. However, the parameter table includes off-diagonal terms, while the worked substitution uses simplified diagonal matrices, so the example does not fully show how all estimated parameters enter the BEKK calculation. No computed covariance matrix, hedge ratio, or hedge-efficiency result is provided; applying the method requires correctly specifying the fitted model and obtaining its conditional covariance forecasts.
Key ideas
- A BEKK model can estimate time-varying variances and covariance for spot and futures returns.
- The minimum-variance hedge ratio is conditional spot-futures covariance divided by conditional futures variance.
- The BEKK recursion uses a constant matrix, lagged shocks, and lagged conditional covariance.
- The displayed substitution omits off-diagonal estimated parameters and does not calculate a final hedge ratio.
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Full text
# To calculate the Hedge Efficiency and Optimal Hedge Ratio with BEKK in R
# To calculate the Hedge Efficiency and Optimal Hedge Ratio with BEKK in R
I estimated an MGARCH-BEKK model (using the R package `BEKK`, i.e. Baba, Engle, Kraft and Kroner; see Engle and Kroner (1995)) on time series of spot and futures prices. The estimated parameters are:
```
=====================================================
Estimate Std. Error t value Pr(> | t| )
-----------------------------------------------------
mu1.DLog_Base -0.002 0.001 -1.498 0.134
mu2.DLog_B3 0.0003 0.001 0.282 0.778
A011 0.004 0.003 1.047 0.295
A021 0.0004
A022 0.013 0.001 14.475 0
A11 0.008 0.027 0.314 0.754
A21 -0.096 0.089 -1.077 0.282
A12 -0.052 0.088 -0.588 0.557
A22 0.661 0.122 5.395 0.00000
B11 0.967 0.010 96.058 0
B21 0.124
B12 0.073 0.123 0.596 0.551
B22 0.011 0.185 0.058 0.953
-----------------------------------------------------
```
I don't now to calculate the conditional variance and covariance matrix.
$$ \left[ {\begin{array}{cc} \sigma_{ss} & \sigma_{sf} \\ \sigma_{fs} & \sigma_{ff} \\ \end{array} } \right] = \left[ {\begin{array}{cc} c_{11} & 0 \\ c_{21} & c_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} c_{11} & 0 \\ c_{21} & c_{22} \\ \end{array} } \right] + \left[ {\begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} \epsilon_{s,t-1}^2 & \epsilon_{s,t-1}\epsilon_{f,t-1} \\ \epsilon_{fs,t-1}\epsilon_{s,t-1} & \epsilon_{f,t-1}^2 \\ \end{array} } \right] \left[ {\begin{array}{cc} a_{11} & 0 \\ 0 & a_{22} \\ \end{array} } \right]$$
$$ + \left[ {\begin{array}{cc} b_{11} & 0 \\ 0 & b_{22} \\ \end{array} } \right] \left[ {\begin{array}{cc} \sigma_{ss,t-1} & \sigma_{sf,t-1} \\ \sigma_{fs,t-1} & \sigma_{ff,t-1} \\ \end{array} } \right] \left[ {\begin{array}{cc} b_{11} & 0 \\ 0 & b_{22} \\ \end{array} } \right] $$
My conditional variance and covariance matrix:
$$ \left[ {\begin{array}{cc} \sigma_{ss} & \sigma_{sf} \\ \sigma_{fs} & \sigma_{ff} \\ \end{array} } \right] = $$ $$ \left[ {\begin{array}{cc} 0.004 & 0 \\ 0.0004 & 0.013 \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.004 & 0 \\ 0.0004 & 0.013 \\ \end{array} } \right] + \left[ {\begin{array}{cc} 0.008 & 0 \\ 0 & 0.661 \\ \end{array} } \right] \left[ {\begin{array}{cc} \epsilon_{s,t-1}^2 & \epsilon_{s,t-1}\epsilon_{f,t-1} \\ \epsilon_{fs,t-1}\epsilon_{s,t-1} & \epsilon_{f,t-1}^2 \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.008 & 0 \\ 0 & 0.661 \\ \end{array} } \right]$$
$$ + \left[ {\begin{array}{cc} 0.967 & 0 \\ 0 & 0.011 \\ \end{array} } \right] \left[ {\begin{array}{cc} \sigma_{ss,t-1} & \sigma_{sf,t-1} \\ \sigma_{fs,t-1} & \sigma_{ff,t-1} \\ \end{array} } \right] \left[ {\begin{array}{cc} 0.967 & 0 \\ 0 & 0.011 \\ \end{array} } \right] $$
To calculate the optimal hedge ratio `BEKK`:
$$h_t = \frac{cov \left( \Delta S_t, \Delta f_t \mid \Omega_{t-1} \right) }{var \left( \Delta f_t \mid \Omega_{t-1} \right)}$$
$\Delta S_t$, $\Delta f_t$ is the return price spot and future, and $\Omega_{t-1}$ is conditional variance and covariance matrix.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.