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 who 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.

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

correctto4decimalplaces)wouldyouselltheDIGITALDEFAULTcontracttoyourcustomer?

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)?