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Swaps and Properties of Options

This assignment is intended to provide some practice in calculations involving the use of swap contracts, and understanding the properties and uses of options.

1. Finding a Swap Rate from the Term Structure
Suppose that the term structure of interest rates for the next five years is as given in the chart and table below.

Use Excel to determine the swap rate for a five-year swap where payments are made every six months. In order to do this, you should follow the steps given below

1) Calculate the continuously compounded forward rates for each six-month period in the five-year term structure.

2) Convert the continuously compounded forward rates to semi-annually compounded rates, and use these to find the floating-rate cash flows to be received at the end of each six-month period.

3) Assume that the fixed-rate cash flow to be paid is a constant percentage of the face value of some notional amount, and assume that all these payments will remain the same throughout the tenor of the swap.

4) Calculate the net cash flow for each payment period, and discount these using the appropriate continuously compounded zero rate.

5) Use “Goal Seek” or “Solver” in Excel to set the initial value of the swap to zero, by changing the swap rate in your spreadsheet.

6) Print a copy of your Excel calculations and attach them to your homework submission.

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2. Probability of Default for a CDS (Problem 23.14)

[Use Excel to set up the example in Tables 23.2 to 23.5.] Verify that if the CDS spread for the example is 100 basis points then the probability of default in a year (conditional on no earlier default) must be 1.61%. What is the implied probability of default when the CDS spread is 100 basis points but the recovery rate is 20% instead of 40%? Finally, use your Excel model to estimate the credit default spread (in basis points) with the same general assumptions as for the example, but with a probability of default of 1.50%, a risk-free interest rate of 4.0% and an expected recovery rate of 35.0%.

[Print a copy of your Excel model and attach it to your homework submission.]

3. Dividends and Stock Splits

A trader has a put option contract to sell 100 shares of a stock for a strike price of $60. What is the effect on the terms of the contract of

(a) A $5 dividend being declared
(b) A $5 dividend being paid
(c) A 5-for-2 stock split
(d) A 5% stock dividend being paid.

4. Margin Requirements

A trader writes five naked put option contracts, with each contract being on 100 shares. The option price is $10, the time to maturity is six months, and the strike price is $64.

(a) What is the margin requirement if the stock price is $58?
(b) How would the answer to (a) change if the rules for index options applied?
(c) How would the answer to (a) change if the stock price were $70?
(d) How would the answer to (a) change if the trader is buying instead of selling the options?

5. Put-Call Parity on a Non-Dividend-Paying Stock

The prices of European call and put options on a non-dividend-paying stock with 12 months to maturity, a strike price of $120, and an expiration date in 12 months are $20 and $5, respectively. The current stock price is $130. What is the implied risk-free rate?

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