Page **1** of **1**

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 21st, 2015, 4:58 am**

by **EdisonCruise**

Suppose the spot price is S, forward price is F. S and F are the prices of two correlated asset. John Hull?s book directly gives a method to calculate Minimum Variance Hedge Ratio

http://financetrain.com/minimum-variance-hedge-ratio/as h=rho*sigma_S/sigma_Fwhere rho is correlation between dS and dF during the hedging period.The number of optimal Contract N is N= h*QA/QFWhere QA is Size of position being hedged (units) and QF is Size of one futures contract (units).John Hull?s book doesn?t give details on the derivation process of both equations. And I find some problem on them.Suppose one long the spot S, and short forward F for hedging by holding N contracts, then the portfolio return Rh should be:Rh=(QA*dS-QF*dF*N)/(QA*S)=dS/S - (QF*N*F)/(QA*S)*(dF/F)dS/S and dF/F are the spot return Rs and forward return RF respectively.And leth=(QF*N*F)/(QA*S) then Rh= Rs - h* RFSo VAR(Rh)= VAR(Rs) + h^2*VAR(RF)-2h*COV(Rs,RF)Let dVAR(Rh)/dh=0, so that 2h*VAR(RF)-2COV(Rs,RF)=0h=COV(Rs,RF)/VAR(RF)= rho*sqrt(VAR(RF)* VAR(Rs))/ VAR(RF)=rho*sigma_S/sigma_FThe hedge ratio here is the same as John Hull, but optimal contracts N becomesN = h*QA*S/(QF*F)Which is different. The calculation of optimal contracts here requires S and F.So what?s wrong here?

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 21st, 2015, 5:14 am**

by **EdisonCruise**

I think there is some doubt here. Does hedge ratio here means money amount ratio or unit amount ratio?

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 21st, 2015, 8:39 am**

by **daveangel**

h is a weighting and n is number of contracts.

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 12:13 am**

by **EdisonCruise**

Thank you. But h is a weighting of what? money amount or asset unit?

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 7:30 am**

by **daveangel**

money amountlet's say S and F were perfectly correlated with the same volatility. but for some reason F = 2*S then h = 1 but n = 0.5

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 7:42 am**

by **EdisonCruise**

Thank you daveangel. I agree with you. But the John Hull's book uses asset size amount.

http://financetrain.com/minimum-variance-hedge-ratio/In your example, I think F and S should be future and spot price. But these two prices are not needed in John Hull's book when calculating optimal contracts. It only require sizes. So is there anything wrong in that book?

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 7:58 am**

by **daveangel**

no - I don't have the book but I think the argument for min variance hedge is that if you have a S invested in an asset and you want to "hedge" it with F. then the proportion of F that you need to hold is h. then the variance of the hedged portfolio is [$] \sigma_{\pi}^2 = \sigma_{s}^2 + 2h\rho\sigma_{s}\sigma_{f} + h^{2}\sigma_{f}^2 [$]Differentiating with respect to h then gives you the h that minimises the variance. [$] 2\sigma_\pi \frac {d\sigma_\pi}{dh}=2\rho\sigma_{s}\sigma_{f}+2h\sigma_{f}^2=0[$]and therefore [$] h = -\frac{\rho\sigma_s}{\sigma_f}[$]

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 8:25 am**

by **EdisonCruise**

Thank you daveangel. I thank your results are correct and I have no problem in calcuating h.However, after obtaining h, how to calculate the number of foward contract N one needs to hold?My result is N = h*QA*S/(QF*F)but John Hull's book suggestsN = h*QA/(QF)Where QA is Size of position being hedged (units) and QF is Size of one foward contract (units).You can see an online version of John Hull's book online with below link:

http://financetrain.com/minimum-variance-hedge-ratio/If h is money amount, I think N = h*QA*S/(QF*F) should be used to calculate the optimal foward contracts.

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 8:42 am**

by **daveangel**

QuoteOriginally posted by: EdisonCruiseThank you daveangel. I thank your results are correct and I have no problem in calcuating h.However, after obtaining h, how to calculate the number of foward contract N one needs to hold?My result is N = h*QA*S/(QF*F)but John Hull's book suggestsN = h*QA/(QF)Where QA is Size of position being hedged (units) and QF is Size of one foward contract (units).You can see an online version of John Hull's book online with below link:

http://financetrain.com/minimum-variance-hedge-ratio/If h is money amount, I think N = h*QA*S/(QF*F) should be used to calculate the optimal foward contracts.I agree with you. I think Hull has assumed that S=F and if you look at jet and heating oil prices they do seem to be approximately equal. however, I think the Hull example is not right. The airline would want to hedge the price risk of 200k gallons of jet fuel i.e 200,000*S

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 9:21 am**

by **EdisonCruise**

Thank you daveangel.Most people are directly using that formula on Hull's book for a long time. It is difficult to make a correction now.

### Some confusion on John Hull?s method to calculate Minimum Variance Hedge Ratio

Posted: **December 22nd, 2015, 9:31 am**

by **daveangel**

most people are going to work with notional amounts rather than quantity though