Question

The market in which Inaho trades has three possibilities for investment:

• a risk-free asset with the continuously compounded, risk-free interest rate equal to r;

• a risky asset whose price is denoted by S(t),t ≥ 0 and whose dividend yield is δ_S;

• a risky asset whose price is denoted by Q(t),t ≥ 0 and whose dividend yield is δ_Q.

Initially, the market prices of assets S and Q are equal. Inaho opens a one-share short position in the asset S and uses the proceeds of the short sale to purchase a share of the asset Q. At time−T, Inaho sells the shares of asset Q she owns and closes the short sale of the asset S.

(i) What is the initial cost of this portfolio?

(ii) What is the profit of this portfolio?

(iii) What is the condition on the ratio of the final prices of assets S and Q for Inaho to break even?

Answer 1

(i) The initial cost of this portfolio is zero. This is because the proceeds from the short sale of one share of S are used to purchase one share of Q. Since the markets prices of S and Q are initially equal (S(t) = Q(t)), the net outlay or net cost of this portfolio is equal to zero.

(ii) The profit of this portfolio is calculated as (Profit from Long position in Q) + (Profit from short sale of S) + (Dividend Yield of Q) - (Dividend Yield of S)

Profit from the long position is calculated as price at time T minus price at time t

Profit from the short position is calculated as price at time t minus price at time T

Dividend yield of Q is a positive number and is earned as income on the portfolio. Dividend Yield of S is a negative number since this is a short sale. The dividend yield on S is an expense since Inaho as the borrower of the share must pay any dividend on the share to the lender of the share.

Profit on portfolio = S(t) - S(T) + Q(T) - Q(t) -δS+δQ.  

(iii) To break even, the ratio of final prices of assets S and Q is given by:

S(t) - S(T) - δS >= Q(T) - Q(t) +δQ

That is, the profit from short sale of S must be greater than equal to the profit from the long position in Q. The short sale of S results in an expense/loss - the dividend yield that must be paid to the lender of shares. The long position in Q has a positive dividend yield and is an income/profit for the portfolio.