FINA 2322_1D              Problem Set (2)

Deadline for Submission: November 26, 2023 (Sunday), Noon-time 12:00 pm.

Submission Method: upload to course moodle

Submissions after the deadline above will NOT be accepted and students will get zero points for the exercise.

There are 7 questions and the total points of the problem set is 80.

Q. 1 (10 points)

Cathay Pacific plans to buy 20,000 barrels of jet fuel four months from now and it uses futures  contract on crude oil that expires six months from now to hedge against the risk of jet fuel price fluctuations.

The estimated correlation of the change in the spot price of jet fuel and crude oil futures is  p = 0.8,

the standard deviation of the change in crude oil futures prices is  σF   = 1.32  and the standard deviation of the change in spot jet fuel prices is  σs   = 1.10.

At time 0, the spot price of crude oil is $100 and the spot price of jet fuel is $110. The contract size of crude oil futures is 1,000 barrels.

The annual riskfree interest rate (cc) is 5%.

The storage cost of crude oil is assumed to occur on a continuous basis and it is estimated to be a constant fraction 2% of the spot price of crude oil.

(a) Based on equations (8) and (9) of Handout 5, how many crude oil futures contracts should Cathay Pacific buy or sell to achieve the optimal hedge?

You answer should specify whether Cathay Pacific should buy or sell the futures contracts and the number of contracts to be used (the number of contracts has to be rounded to the closest integer).

(b) Suppose that four months from now when Cathay Pacific closes its position on the future, the spot price of crude oil turns out to be $120 and the spot price of jet fuels turns out to be$123.33.

Compute the following values:

(i)        the cost that Cathay Pacific has to pay for the 20,000 barrels of jet fuel in the spot market (i.e., the unhedged position),

(ii)       the gains/losses from the futures contracts based on the optimal (rounded) number of contracts that you have solved in part (a),

(iii)      the effective cost that Cathay Pacific has to pay for the 20,000 barrels of jet fuel taking

into account of the gains/loss from the futures contracts (i.e., the hedged position).

(c) We consider five possible outcomes of the spot price of crude oil and jet fuel prices four months from now when Cathay Pacific closes its position on the futures.

In an excel file, compute the following values in each of the outcomes above:

(i) the cost that Cathay Pacific has to pay for the 20,000 barrels of jet fuel in the spot market (i.e., the unhedged position),

(ii)   the gains/losses from the futures contracts based on the optimal (rounded) number of contracts that you have solved in part (a),

(iii)  the effective cost that Cathay Pacific has to pay for the 20,000 barrels of jet fuel taking into account of the gains/loss from the futures contracts (i.e., the hedged position).

Does the hedging strategy appear to enable Cathay Pacific to hedge against the fluctuations of the jet fuel prices? Explain.

Remark:

Eventhough the answer in part (b) is done in the excel file in part (c), you have to write down your answers in part (b) on a separate piece of paper. It is because in the term test, there might be similar questions. However, we will not be able to ask you to solve the problem using excel file during the  test. Thus, you will have to compute the numbers using a calculator and write down your answers.   Thus, part (b) is for you to practice for similar questions in the term test. Yet part (c) allows you to   have more observations and is more interesting.

Q.2 (10 points) Suppose that you are holding a stock portfolio that has a current market value of

$1,000,000 and its beta is 1.25. You expect the general market will go up in the next few months

and you try to gain from the bull market by increasing the beta of your portfolio to 2.0 three months from now.

You will use the e-mini S&P 500 that expires four months from today to raise the beta of your portfolio.

The current spot price of the S&P 500 index is 4200.

The annual riskfree interest rate is 5% (cc) and the estimated annual dividend yield of the S&P 500 index portfolio is 1%.

(a) Specify clearly how many e-mini S&P 500 index futures you will buy or sell to raise the beta of your portfolio to 2.0 (the number of contracts has to be round to the closest integer).

(b) Three months from today when you close your position on the futures, we will consider five possible outcomes:

In an excel file, compute the following values in each of the outcomes above:

(i)  the gains/losses from the futures contracts based on the (rounded) number of contracts that you have stated in part (a),

(ii)       the value of your portfolio including the gains/losses from the futures in each of the outcomes three months from now.

(c) Loosely using the “returns” from outcome 1 to outcome 5 (just like what we have done in

class), does the beta of your portfolio appear to have increased to 2.0 as what you have hoped for? Explain.

Q. 3 (10 points)

Consider a call option and a put option that have the same underlying stock (which does not pay out any dividend during the life of the option), same exercise price (K) and time to expiration (T).

The annual riskfree interest rate is r (cc).

At the current moment (t = 0), the spot price of the underlying stock is  S0 , the call premium is C

and the put premium isP.

If P > C + e−rTK − S0 , identify an arbitrage strategy and use the payoff table to verify that the strategy you identify is indeed an arbitrage strategy.

Q.4 (10 points)

Consider the following trading strategy: buy a put option with an exercise price of $30 and buy a

call option with an exercise price of $40. P and Care the premium of the put and the call respectively.

(a) Draw (i) the time T state-contingent payoff, and (ii) the state-contingent profit of the strategy in a diagram.

(b) Present the cash flows of the strategy in a payoff table.

(c) Do you think the strategy is a neutral market trading strategy, bullish, bearish or volatility? Explain.

For questions that involve trading of two or more options, it is understood that the options have the same underlying stock (which does not pay out any dividend during the life of the option) and the same time to expiration (T) unless specified otherwise.

Q. 5 (10 points)

Consider the following trading strategy:

sell a call and a put at the sametime.

The two options have the same underlying stock (which does not pay out any dividend during the life of the option), expiration date (T) and exercise price (K). P and C are the premium of the put  and the call respectively.

Present the cash flows of the strategy in a payoff table, and draw the state-contingent payoff at time T and the profit in a diagram.

You may use K = $60 as a numerical example though not required.

How would you compare this strategy with a butterfly spread? Explain.

Q.6 (20 points in total, 5 points for each part)

Consider a European call option on a stock that does not pay out any dividend during the life of the option.

We use a one-step Binomial Model to solve for the premium of the call, C, at time 0.

The followings are the relevant information we need.

(i)        The option expires six months from now, i.e.,  T  = 1/2.

(ii)       The exercise price is K = $106.

(iii)      The current stock price is  S0   = $100.

(iv)      The annual riskfree interest rate is r = 5% (cc)

(v)       Six months from now when the option expires, there are two possible outcomes of the stock price. The stock price either goes up by a factor u or goes down by a factor d . That is, STcan take on two possible values only:  uS0      or  dS0 .

We assume that u = 1.1 and d = 0.9.

(a) Solve for the premium of the Call by arbitrage arguments with a synthetic portfolio that

generates the same state-contingent payoff of the call option at time T. Answer this question with the help of a payoff table.