Please answer (a) and (b). NOTE: Problem 1 content is below.

(Hedging Portfolio, 6 points) In an extension to Problem 1, we have two individual stocks, both begin with the price $1, and by the end of the period their prices are

S₁ = 1 + + X, S₂ = 1 + + Y

where X and Y are two random variables with mean zero, X has variance ², and

Y = X + Z, 1 < < 0

where Z is another random variable that is independent of X. The variance of Z is denoted by ².

(a)

We now construct a hedging portfolio with one share of S₁ and shares of S₂, such that 1 + = 0. Show that the return of the hedging portfolio is

(V₁ V₀)/V₀ = + Z /(1+ ) ,

and the variance of the return is less than ².

(b)

Assume = 5%, = 0.5, ₁ = 0.25, and ₂ = 0.2, compute the variance of the return of the hedging portfolio.

Problem 1 content below

In this exercise, we consider a stock price in a one-period model, with beginning price S₀, and end of the period price

S₁ = S₀(1 + + X)

where X is a random variable with mean zero and variance ², and > 0 is the expected growth of the stock. The return of the stock over this period of time is

R = (S₁ S₀₎/S₀ = + X

so the expected return (growth) is and the variance of return is ².

Suppose we have exactly the amount S₀ to invest, we can do one of the followings:

1. purchase one share of the stock;

2. borrow some money to invest in > 1 shares of the stock. This will be a leveraged portfolio, consisting of shares of the stock, and a loan with the principal Q = ( 1)S₀.

The return for the portfolio in case 2 is given by

( + X) ( 1)r

where r is the simple interest rate over this period of time.