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 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?) (2 ½ marks)

b) What is the value (correct to 5 decimal places) of this portfolio in dollars in one year's time? (3/4 mark)

c) What is the value (correct to 5 decimal places) of this portfolio in dollars today? (3/4 mark)

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? (2 ½
marks)

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)? (1 mark)

(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. (2 ½ marks)

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. (2 marks)

(Note to
students: I want you to be explicit in your calculations, by explaining your reasoning. I do
NOT
want you to use the formulae from class (and, in any event, we did not cover this type of security explicitly in class so they may muddy the waters). Instead, I want you to apply the
principles
from class to answer this question. You may or will lose marks if you do not explain your reasoning.)

Question 6: (9
marks total)

Infinity inc is a U.S. company whose shares are listed on and freely traded on the New York stock exchange.

You are the Global Head of Equity Options trading at Goldman Sachs. You are approached by a hedge fund that today wants to buy a security that has the following features:

As long as the Infinity share price never trades at or above $125 per share, the hedge fund receives nothing. However, if the Infinity share price ever gets to trade at or above $125 per share, the hedge fund will receive at that time, from Goldman Sachs, a payment in cash equal to $15,000,000, after which the security expires.

The contract is
perpetual
(it means it has no fixed end date and the security continues
forever
until it expires (i.e., until the price level of $125 per share is reached)).

Infinity inc can potentially go bankrupt in which case the shares in Infinity inc become worthless (and assume that the shares would remain worthless forever).

Assume that, in the absence of bankruptcy, the shares of Infinity inc will trade forever (no takeovers or delistings, for example).

The share price of Infinity inc today is $75 per share.

Assume the absence of arbitrage throughout. Assume that you can freely trade Infinity inc shares (without transactions costs) and that you can borrow or lend at the risk-free interest-rate (forever, if needed).

Assume that Goldman Sachs can never go bankrupt and also assume that Goldman Sachs continues to exist for ever (so it can never renege on this contract).

We make two further simplifying assumptions (that we relax in parts b and c of the question):

Assumption 1./ Assume that the risk-free interest-rate is zero per cent (forever).

Assumption 2./ Assume that Infinity inc will never pay a dividend and will never do share buybacks.

a) The hedge fund wants to buy the above-mentioned security today. What price (in dollars) do you quote? How do you hedge this security so that Goldman Sachs has no risk? Carefully explain your strategy. (6 ½
marks)

b) How would your answer in part (a) change if, instead of assuming that the risk-free interest-rate is zero per cent forever (as in Assumption 1), I told you that the risk-free interest-rate today is 3.85% per annum (continuously compounded) but will fluctuate through time as the Fed changes its monetary policy? (2
marks)

c) How would your answer in part (a) change if, instead of assuming that Infinity inc will never pay a dividend (as in Assumption 2), I told you that Infinity Inc will pay a continuous dividend yield of 0.5% per annum (continuously-compounded) (forever)? We maintain Assumption 1 (so the risk-free interest-rate is zero (forever)). (½ mark)

(Note to
students: I assure you that the
principles
that you learnt in class apply to this question even if the security looks unfamiliar. Explain your reasoning in
words
– like I do in lectures -- as you go along.)

Question 7: (12
marks total)

Kangaroo inc is a U.S. company whose shares are listed on and freely traded on the New York stock exchange. Let be the price of Kangaroo inc shares in dollars, at time (measured in years). Time zero is today.

You have just been appointed by Barclays Capital to be their new Global Head of Options trading. It is your first day on the job and you are keen to show senior management at Barclays Capital that they made the right decision to hire you and to show to Wall Street that there is a new kid on the block!

You are telephoned by Sally, a fund manager, at Tiger Capital (a hedge fund). Sally tells you that Tiger Capital today wants to buy a security (called a SQUARED DIFFERENCE contract), linked to the share price of Kangaroo inc. The SQUARED DIFFERENCE security has the following features:

It has a maturity of one year.

At maturity, the SQUARED DIFFERENCE security pays an amount in dollars equal to the amount

. equation (*)

Here, and are, respectively, the Kangaroo inc share price one year from now and the share price today.

Assume that the risk-free interest rate is zero per cent and that Kangaroo inc shares pay no dividends. Assume that the share price, in dollars, today is

Using Excel, build a four-step binomial tree (this means each time step corresponds to three months). (Hint: It is the same idea as we did in classes and assignments but whereas, before, we had one, two or three steps, now you will have four binomial steps).

Assume the absence of arbitrage throughout and assume that there are no transactions costs.

a) Assume (to begin with) that the volatility of Kangaroo inc shares is 1%. Using your binomial tree, what is the price today of this SQUARED DIFFERENCE security? (4
marks)

b) Still assuming the volatility of Kangaroo inc shares is 1%, and using the binomial tree, what is the delta hedge at
each
step. To answer this, do a screen-shot (Control-C then Control V on a pc) of the delta hedges. Do you see a pattern in the delta hedges? What is it? (3
marks)

c) Now assume instead that the volatility of Kangaroo inc shares is 2%. What is the price today of this SQUARED DIFFERENCE security? (1
mark)

d) Now assume instead that the volatility of Kangaroo inc shares is 3%, then 4%, then 5%, then finally 10% (skip 6, 7, 8 and 9% - you will (hopefully) already see a pattern emerging). For each case, what is the price today of this SQUARED DIFFERENCE security? (Hint: If you do this in excel in an
efficient
manner, this can be done very rapidly). (2
marks)

Give your answers to parts (a), (c) and (d) (in dollars) by filling out the table below (replacing x.yyyyyyy) giving every answer to
7 decimal places
(you will see that the answers are quite small so that is why I am asking for 7 decimal places but this is no hassle - decimal places are “free” in excel since excel allows you to format up to 14 decimal places – Google this formatting feature if you have not seen it before):

Table of your results:

e) What is the pattern of prices? (A graph might be helpful here but is not obligatory). For example, could you guess (with a slight approximation – not to 7 decimal places! - by doing the calculations in your head) what the price would be if the volatility were, for example, to be 2.5% or 6%? How are you able to guess? In one or two
brief
sentences, what is the pattern? (2
marks)

a. The pattern of prices seem to increase with each 1% increase in volatility; thus, creating a closely linear graph. By doing the calculations in my head, I can quickly estimate that the prices for a volatility of 2.5% and 6% would be $0.042 and $0.24 respectively.

Hint: When you examine the payoff of this SQUARED DIFFERENCE security (i.e., in equation (*)), does the pattern of prices look intuitive? Why?