Question

An investor owns a stock. Daily change in stock price, ∆S, has
the standard deviation of 13. To hedge risks of the stock price, the investor considers
cross-hedging using one of the following futures contracts. The following table shows
each futures contract’s standard deviation σF of futures price change, ∆F, and the
correlation coefficient ρ between ∆S and ∆F.
Futures contract σF ρ
A 22 0.7
B 20 0.9
C 15 0.6
D 10 0.8

If the investor shorts h units of futures, the change in the portfolio value is ∆S −h∆F.

(a) Which futures contract results in the smallest variance, Var (∆S − h∆F)? (Assume
that for each futures, we use respective minimum-variance hedge ratio).
(b) What is the minimum variance if we use the futures contract found in question 20?

Answer 1

Minimum variance=standard deviation of spot^2+h^2*standard deviation of futures^2-2*h*standard deviation of spot*standard deviation of futures*correlation between spot and futures

h=correlation*standard deviation of spot/standard deviation of futures

Minimum variance=standard deviation of spot^2-(correlation*standard deviation of spot)^2=standard deviation of spot^2*(1-correlation^2)

A=13^2*(1-0.7^2)=86.1900

B=13^2*(1-0.9^2)=32.1100

C=13^2*(1-0.6^2)=108.1600

D=13^2*(1-0.8^2)=60.8400

1.
Contract B

2.
32.11