The Setup

You are:

  • Long one share of stock A
  • Considering a short position of \( h \) shares in stock B

Let:

  • \( \mathrm{Var}(R_A) = \sigma_A^2 \)
  • \( \mathrm{Var}(R_B) = \sigma_B^2 \)
  • \( \mathrm{Corr}(R_A, R_B) = \rho \)

Objective

Choose \( h \) to minimize the variance of your net position:

\[ \mathrm{Var}(R_A - h R_B) \]

Step-by-Step Solution

We compute:

\[ \mathrm{Var}(R_A - h R_B) = \sigma_A^2 + h^2 \sigma_B^2 - 2h \rho \sigma_A \sigma_B \]

To minimize, differentiate with respect to \( h \):

\[ \frac{d}{dh} \mathrm{Var}(R_A - h R_B) = 2h \sigma_B^2 - 2 \rho \sigma_A \sigma_B \]

Set derivative to zero:

\[ 2h \sigma_B^2 = 2 \rho \sigma_A \sigma_B \Rightarrow h = \frac{\rho \sigma_A}{\sigma_B} \]

Final Answer

\[ \boxed{h = \frac{\rho \sigma_A}{\sigma_B}} \]

This is the minimum-variance hedge ratio, the amount of B to short to optimally hedge your position in A.

Reference