Dec 9 / Exam Helper

# Derivative Market - Stock Index Problems Solutions

Problem 1
A stock index currently stands at \$350.00. The risk-free interest rate is 4.57% per annum (with continuous compounding) and the dividend yield on the index is 314% per annum.
What should the forward price for a 4.00-month contract on the index be?
Problem 2,
Part (a) 0.015)0 points (graded) The risk-free interest rates in Switzerland and the United States are, respectively, 1.57% and 2.14% per annum with continuous compounding. The spot price of the Swiss franc is \$1.05. The forward price for a contract deliverable in 2.00 months is also \$1.05.
Use the above to answer the following (a) & (b).
(a) What is the theoretical forward price?
Part (b) Which of the following is an arbitrage opportunity?
• a)     Borrow one unit of Swiss franc to invest in U.S_ dollars at risk-free interest rate for 2.00 months; enter forward contract to exchange \$1.05 x e2•14%x (2.00/12) U.S. dollars to Swiss francs at \$1.05 in 2.00 months.
• b)    Borrow 1.05 units of U.S. dollars to invest in Swiss francs at risk-free interest rate for 2.00 months; enter forward x contract to exchange \$1 x (2.0012} Swiss francs to U.S. dollars at \$1.05 in 2.00 months.
Problem 3

Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 2.57% per annum (continuously compounded). Consider the transactions that would allow you to make an arbitrage profit in March by trading in May and December forward contracts. Use the forward prices provided in the table below and assume you can lock in the settlement prices on either a long or short position.

Forward prices as of March 26

 Month Settlement Price Change Volume May 51.51 2.76 9,315 December 55.23 2.19 7,055
Table Notes: Crude oil contracts are for 1,000 barrels, quoted in \$ per barrel.

Consider the following strategy. In March, enter a long position in a May forward contract and a short position in a December forward contract. In May, borrow at the interest rate and buy oil from the forward contract. Store the crude oil till December. In December, sell oil to the forward contract and repay the loan.

What is the profit per barrel from the above strategy? Use negative numbers for losses.
Problem 4
We know how to derive forward prices for stocks in a frictionless market. In this problem, you are asked to explain how the presence of frictions such as bid-ask spreads or different interest rates for borrowing and lending imply no-arbitrage bounds on prices rather than a single no-arbitrage price.
Specifically, suppose that a trader faces bid-ask spreads posted by a dealer such that the stock and forward contract have bid and ask prices of Sb < S„ and Fb < Fa, respectively, and the interest rates for borrowing and lending are such that rb > ri.
Derive a no-arbitrage upper bound F+ for the forward bid price Fb with time-to-maturity T using no-arbitrage arguments. Assume that the stock pays no dividends, that there are no trading fees incurred at time T, and that the forward contract is settled by delivery of the stock.
Which of the following is correct?
a)   F+ = (Sa)erl*T
b)   F+ = (Sa)erb*T
c)   F+ = (Sb)erl*T
d)   F+ = (Sb)erb*T