APPENDIX TO CHAPTER 6Regression Hedging and Principal Component Analysis

A6.1 REGRESSION HEDGES AND P&L VARIANCE

This section proves that i) the regression hedge minimizes the variance of the P&L of the hedged portfolio; and ii) the volatility of the regression‐hedged portfolio equals the DV01 of the position being hedged times the standard deviation of the regression residuals.

Begin with least‐squares estimation, which finds the parameters ModifyingAbove alpha With Ì‚ and ModifyingAbove beta With Ì‚ to minimize,

To solve this minimization, differentiate (A6.1) with respect to each of the parameters, set each result to zero, and obtain the following two equations,

(A6.2)minus 2 sigma-summation Underscript t Endscripts left-parenthesis normal upper Delta y Subscript t Baseline minus ModifyingAbove alpha With Ì‚ minus ModifyingAbove beta With Ì‚ normal upper Delta x Subscript t Baseline right-parenthesis equals 0
(A6.3)minus 2 sigma-summation Underscript t Endscripts left-parenthesis normal upper Delta y Subscript t Baseline minus ModifyingAbove alpha With Ì‚ minus ModifyingAbove beta With Ì‚ normal upper Delta x Subscript t Baseline right-parenthesis normal upper Delta x Subscript t Baseline equals 0

These equations can be solved to show that,

where and are the sample averages; and the standard deviations; the covariance; and the correlation. ...

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