Question 1
Zero coupon yields (all yields are continuously compounded) are 3.00% for three months, 3.50% for six months, 3.60% for nine months and 3.80% for twelve months. NorthBank is contemplating 'buying' a one-year plain vanilla interest rate swap involving the quarterly exchange of fixed and floating interest rate payments on a notional principal of $200 million. Note: a swap buyer pays fixed, receives floating.
(a) Use the portfolio of bonds approach to calculate the 'fair' swap rate.
(b) Assume that NorthBank 'buys' the swap at the 'fair' swap rate determined in (a) above. However, within minutes of entering the swap, the zero coupon yield curve falls by 10 basis points. Calculate the value of the swap to NorthBank.
Question 2
A non-dividend paying stock price is currently $8.00. It is known that at the end of three months it will be either $4.00 or $11.00. The risk free rate of interest with continuous compounding is 25% p.a.. Calculate the value of a European put option with an exercise price of $10.00.
Verify that the value of the put option is the same under:
(a) the no arbitrage valuation method (i.e. a portfolio comprising a short or long position in delta shares and one option); and
(b) the risk-neutral valuation method (based on the probability of upward and downward stock price movements).
Question 3
A European put option on a dividend-paying stock is selling for $2.15. The underlying stock price is $21, the exercise price is $24, a dividend of $0.20 is expected in two months and the option expires in six months. The risk free rate is 6% p.a. continuously compounded (all maturities).
Required
Show how an arbitrageur can exploit this situation. You can assume the arbitrageur can borrow or lend at the risk free rate, incurs no transactions costs and can short-sell the stock if necessary. Ensure that the net cash flow at time 0 is positive.
Question 4
In December 2009 an options trader bought a March 2010 $40 put on Bayco stock for $2.50 and sold one June 2010 $40 put on Bayco stock for $3.30 (i.e., the exercise price for both options is $40).
Required
Draw a profit and loss diagram in Excel of the trader's portfolio at the expiration date of the March 2010 put option. The diagram should show the outcome for a range of stock prices between $30 and $50 in increments of $1. Ignore any transaction costs incurred to create the portfolio other than the initial cost of buying or selling the options.
You should use the Black Scholes model to price the June 2010 option that remains alive in March 2010. To value this option assume the remaining time to maturity of the option is 3 months, Bayco's annual volatility is 25% and the continuously compounded risk free rate is 6% p.a. The stock does not pay dividends.
Question 5
Assume that the current level of the S&P ASX 200 index is 4,500 points, the volatility of the index is 35% p.a., the continuous dividend yield on the index is 3.0% p.a. and the nine-month risk free rate (continuously compounded) is 5.0% p.a.
Required
(a) Use a five-period binomial option pricing model to value a nine-month American put option on the S&P ASX 200 index with an exercise price of 4,750 points. Show the binomial tree diagram.
(b) What is the value of the "right of early exercise"?