Non optimal hedge x Basis Risk
- Thread starter luccacerf
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Hi @luccacerf
I think you are asking about a relationship between optimal hedge and basis risk? (as non-optimal allows for virtually infinite possibilities). Beta = correlation * cross-volatility; that is β(f,s) = ρ(s,f) * σ(s)/σ(f) = h(*) is the optimal hedge ratio. So, if you know beta is 0.8, then you know β(f,s) = h(*) = 0.80 = ρ(s,f) * σ(s)/σ(f). In which case, the optimal hedge is 0.80 units of future per 1.0 unit of spot, and by "optimal hedge" we mean the hedge that minimizes the variance of the portfolio consisting of the spot plus the futures hedge. However, as there are many (virtually infinite) different combinations of correlation, spot volatility and futures volatility that imply 0.80 minimum variance hedge ratio, you cannot know the basis risk (which presumably refers to the variance of the basis) without additional information because σ(basis) = σ(s - h*f) = σ(s)^2 + h^2*σ(f)^2 - 2*h*σ(s)*σ(s)*ρ, so you really only know in the case of optimal hedge: σ(s - 0.8*f) = σ(s)^2 + 0.8^2*σ(f)^2 - 2*0.8*σ(s)*σ(s)*ρ, for which there are many solutions. But it's interesting to think about!
I think you are asking about a relationship between optimal hedge and basis risk? (as non-optimal allows for virtually infinite possibilities). Beta = correlation * cross-volatility; that is β(f,s) = ρ(s,f) * σ(s)/σ(f) = h(*) is the optimal hedge ratio. So, if you know beta is 0.8, then you know β(f,s) = h(*) = 0.80 = ρ(s,f) * σ(s)/σ(f). In which case, the optimal hedge is 0.80 units of future per 1.0 unit of spot, and by "optimal hedge" we mean the hedge that minimizes the variance of the portfolio consisting of the spot plus the futures hedge. However, as there are many (virtually infinite) different combinations of correlation, spot volatility and futures volatility that imply 0.80 minimum variance hedge ratio, you cannot know the basis risk (which presumably refers to the variance of the basis) without additional information because σ(basis) = σ(s - h*f) = σ(s)^2 + h^2*σ(f)^2 - 2*h*σ(s)*σ(s)*ρ, so you really only know in the case of optimal hedge: σ(s - 0.8*f) = σ(s)^2 + 0.8^2*σ(f)^2 - 2*0.8*σ(s)*σ(s)*ρ, for which there are many solutions. But it's interesting to think about!
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