General Motors has issued a zero-coupon bond. The zero-coupon bond has a price in the market today of 97.5 dollars. The zero-coupon bond pays 100 dollars in exactly one year from now. To be precise, General Motors promises to pay 100 dollars in one year's time. Of course, there is the possibility that General Motors may default on its promised payment. In the event of default, liquidators would be appointed to sell off General Motors's assets and then distribute the proceeds to bond-holders so they get some of their money back. Analysts estimate (and we accept their estimate as fact) that General Motors's assets will generate a payment of $60, per every $100 promised, in one year's time in the event of General Motors defaulting.

General Motors's bonds trade freely in the market and you can sell them short if you have to (without any extra costs). You can also borrow or lend risk-free at 2% per annum (continuously compounded). There are no transactions costs.

You are working as a trader at Morgan Stanley (a prestigious Wall Street Investment Bank). You get a phone call from a customer, Deborah. Deborah is a senior executive at a company which has significant exposure to the risk of General Motors defaulting and she is really worried about this. She needs a hedge and she doesn't mean the gardening type! She asks if, today, you will sell her a security to help her hedge her default risk. She wants to buy a security (let us call it a DIGITAL DEFAULT contract) from you today which pays nothing at all if General Motors does NOT default but will pay her 100 dollars exactly one year from now if General Motors does default (in essence, she wants to buy a kind of insurance contract) in the intervening time period.

The aim of this question is to establish at what price should you sell this DIGITAL DEFAULT contract to your customer, Deborah.

You assume the absence of arbitrage.

a) Set up a portfolio consisting of a short position in the DIGITAL DEFAULT contract and a position in the General Motors's bond which is risk-free to Morgan Stanley. I want you to be explicit about this portfolio (long? short? how many?)

b) What is the value (correct to 5 decimal places) of this portfolio in dollars in one year's time?

c) What is the value (correct to 5 decimal places) of this portfolio in dollars today?

d) In the absence of arbitrage, and using your answer to part (c), at what price (in dollars and correct to 4 decimal places) would you sell the DIGITAL DEFAULT contract to your customer, Deborah?

e) By using the risk-neutral valuation principle, check your calculation in part (d). What is the risk-neutral probability of General Motors defaulting (correct to 4 decimal places)?

(Continuation of question 5): Deborah agrees to buy the DIGITAL DEFAULT contract at the price you calculated in part d. You finish the telephone call with her and you are just about to do your hedge (i.e., setting up the risk-free portfolio in part a) when, before you can actually hedge using General Motors's bonds, the phone rings again! A voice at the other end of the phone line says “G’day, mate!” It is Hermione at Wallaby Capital, a bond investment fund based in Sydney Australia! Hermione tells you that talk of General Motors defaulting is over-done and she goes on to say that she and her colleagues at Wallaby Capital are quite bullish about General Motors. She is interested to buy four hundred (that is, in numbers, 400) call options on the General Motors zero-coupon bond (the same bond that was discussed at the start of the question). Each call option has a strike of 99.75 dollars and has a maturity of one year (the same maturity as the bond).

f) Continuing to assume the absence of arbitrage, what is the total option premium for these four hundred call options (in dollars and correct to 4 decimal places). Remember from above that the General Motors zero-coupon bond has a price in the market today of 97.5 dollars and the interest-rate is 2% per annum (continuously compounded). Explain your reasoning and show your working.

g) Hermione agrees to buy the four hundred call options at the price you calculated in part f and then you finish the phone call. As you put the phone down, you remember that, because of the call from Hermione, you forgot to do the hedge for the DIGITAL DEFAULT contract trade with Deborah! Oh dear! You now also need to do the hedge for the four hundred call options! What do you do! Tell me explicitly how you are going to hedge the DIGITAL DEFAULT contract AND the four hundred call options (together). I want you to explain carefully in words what are you doing here and why.