PPT 3 - 3.1 3. Optimal Hedge Ratios Using Hull (2011)...

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change in spot price over life of hedge change in futures price over life of hedge standard deviation of standard deviation of correlation between and S Δ F Δ S σ F ρ S Δ S Δ F Δ F Δ 9/28/2011 3. Optimal Hedge Ratios 3.1 # h # F A N N == of units of short (long) futures contract of units of asset sold (bought) at time 2 Hedge Ratio: Using Hull (2011) notation rather than Luenberger (1998)
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Cash flow: proceed at time t=T , e.g. from the sale and the short hedge: var( h ) SF Δ− Δ Minimum Variance Hedge Ratio 9/28/2011 3.2 0 00 0 0 0 () TA T F AT F AA F yS N F F N SN S S N F F N S N F N N S F h = + = + = + Δ Δ = + Δ Δ To minimize risk, we minimize the variance of y, which is equivalent to choosing h to minimize:
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Probability Notation • Expected value of X: E [X]= =probability weighted average of possible values of X. Variance and Standard Deviation of X: Covariance: 10/6/2011 3.3 X μ 22 2 2 () [ ( ) ] [ ] XX X X Var X E X E X SD X σ == = = cov( , ) [( )( )] XY XY E X Y μμ ρσ σ =
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2 22 2 2 var( h ) [( h ) ] [ 2h h ] )
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PPT 3 - 3.1 3. Optimal Hedge Ratios Using Hull (2011)...

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