Suppose I bought $100 worth of stock A and I want to hedge it by shorting stock B, they have correlation of rho and respective standard deviations. How do I know how much of Stock B to sell? that's the problem I am trying to solve.

More generally, suppose we own $\$X$ of Asset A and we wish to hedge this by buying $\$Y$ of Asset B, we know that the standard deviations of the Returns of A, and B are $\sigma_A$ and $\sigma_B$ respectively, and that the returns have a correlation of $\rho$. Using portfolio theory we want to minimize $VAR[w_aX+(1-w_A)Y]$ and we get the optimal weight of asset A in the portfolio as $w_A = \frac{\sigma_B^2-\rho\sigma_A\sigma_B}{\sigma_A+\sigma_B^2-2\rho\sigma_A\sigma_B}$

Furthermore, our total portfolio in dollar terms is $X+Y$, and the percentage of our portfolio in asset A is $\frac{X}{X+Y}$, and since we want to this to equal the weight neccessary to be the minimum variance portfolio we set $w_A=\frac{X}{X+Y}$ and solve for Y to find how much of asset B in dollar terms we need to short/purchase in order to acheive our desired portfolio.

Is this method valid for finding the optimal hedging strategy in terms of asset B?

  • $\begingroup$ where did you find the first formula for $w_A$ using $\sigma$s and $\rho$s, what does its numerator and denominator mean, and how does that fraction equate to $\frac{X}{X+Y}$? $\endgroup$ – develarist Dec 2 '19 at 19:53
  • $\begingroup$ @develarist thats simply the formula for finding the portfolio weight for the minimum variance portfolio. minimize $var(w_aA+w_bB)$ $\endgroup$ – Skrrrrrtttt Dec 2 '19 at 19:55
  • $\begingroup$ yes but how does that break into the numerator and denominator shown in the first formula, which you also restate to be equal to the second formula originally shown $\endgroup$ – develarist Dec 2 '19 at 20:02
  • $\begingroup$ I still don't understand what the significance of the numerator, and denominator are, thats just the formula that will tell you what percentage of portfolio should be invested in asset A, which is also equal to second formula? $\endgroup$ – Skrrrrrtttt Dec 2 '19 at 20:09
  • $\begingroup$ well that's what i mean, because your question is proposing $w_A = \frac{\sigma_B^2-\rho\sigma_A\sigma_B}{\sigma_A+\sigma_B^2-2\rho\sigma_A\sigma_B} = \frac{X}{X+Y}$ so i wanted to see what made you draw this connection $\endgroup$ – develarist Dec 2 '19 at 20:15

No, this approach to the mean-variance model should not be interpreted as an optimal hedging strategy in terms of asset $B$. Given $w_A$, solving for $Y$ simply provides the solution for the total dollar amount of the $B$ side of the portfolio that completes and satisfies the $w_A + w_B = 1$ constraint, if that constraint is in fact kept intact somewhere in the equations you provided.

The purpose of the mean-variance model is to compute optimal weights that sum to 1 that minimize portfolio variance, it's not meant for finding offseting hedge positions, especially not for a simple two-asset portfolio.

Moving up to $N>2$ asset portfolios, you might see optimized weights being very aggressive and completely offset by negative weights of the same magnitude in other assets, but this is due to those particular assets' collinearity and well-known misestimation error that the model is known for, and should not be interpreted as hedging since hedging is not an intended mechanism of mean-variance optimization, diversification is.

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