Tutorial 1 Solution - Futures option PDF

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BFF3751 Derivatives 1 Tutorial 1 Answers Question 1 When we have an exposure to a particular asset (be it stocks, bonds, exchange rates, commodities, etc), movement in prices can be favourable or unfavourable. For most assets, it is extremely difficult to predict future movements. Exchange rate movements, for example, are pretty much a 50:50 chance of going up or down. The random walk hypothesis for stock prices implies a similar thing. Businesses operate in a very uncertain world. There are two main ways of approaching this exposure: •

Hedging: take the attitude that I am not prepared to gamble on correctly predicting price movements … not prepared to risk making a loss. Therefore, I prefer to remove all uncertainty by ‘locking-in’ a certain value (be it an exchange rate at which you transact, an interest rate at which you borrow, a price at which you buy or sell, etc).

•

Speculating: is the exact opposite to hedging. You don't lock-in; you basically take a gamble that you can correctly predict price movements. In doing so, you are fully exposed to price movements. If you guess right, you gain; if you guess wrong, you lose.

The majority of our focus in BFF3751 will be on using derivative securities to hedge various risk exposures. Of course, if you understand how derivatives work, you can easily use them to speculate. A few tutorial questions will do this (including Q2, Q3 and Q4 of Tutorial 1).

a) A business like Qantas buys a lot of jet fuel and is hugely exposed to the risk of fuel prices rising. If fuel prices rise, that hurts Qantas. If fuel prices fall, that’s good for Qantas. A long forward contract allows the purchase of oil at the pre-specified price. This gives Qantas some certainty over what fuel will cost by locking in a purchase price. Answer: hedging.

b) Cattle farmers worry about falling prices for live cattle. Often, they enter short futures contracts to lock in the sale price and protect against falling cattle prices. However, this farmer has decided not to do this – effectively, s/he is gambling that prices won’t fall. It might sound odd, but to do nothing in this case is in fact speculation. If cattle prices rise, that’s good for the farmer. If cattle prices fall, that bad for the farmer. Such a scenario is usually a clear indication that they are not hedged. Fifteen to twenty years ago, there were a number of high-profile cases where company directors were found to be negligent for ‘doing nothing’ about risk management. They refused to enter derivative positions to manage significant exchange-rate and interest-rate exposures.

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c) Entering long futures positions to buy electricity is obviously a hedging activity. Electricity prices can change very quickly due to extreme weather conditions, generator outages, etc. if the electiricty price rises, that hurts the retailer. If electricity price falls, that’s good for the retailer. Entering long electricity futures contracts locks in the price at which the wholesaler can purchase electricity and therefore protects against rising prices. A: hedging.

d) It makes sense that Cadbury has been using long futures contracts written on cocoa. Since they purchase large amounts of cocoa to make their products, they are exposed to rising prices of their key ingredient. However, the decision of senior execs to close out the hedge (by entering short cocoa futures) is essentially speculation. They are gambling that, when it comes time to purchase cocoa, prices will be lower than the delivery price written into the futures contract. If prices do fall, management look wise; however, if prices continue to rise, they look foolish. The fact that the final outcome is sensitive to what happens with the price of cocoa is a sure sign of a speculative move.

e) We saw in Lecture 1 that short forward/futures positions make money when the price of the underlying assets falls. Our fund manager will indeed make money on a short SPI200 position if the market drops. Of course, had the market risen, she would have lost lots of money with a short SPI200 position. Again, this is a sure sign of speculation.

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Question 2 We have seen in Lecture 1 that the short gold forward position profits when the price of the underlying commodity falls and makes a loss when the price of the underlying commodity rises. The lecture notes displayed the payoff diagram for a short forward position. It was a downward sloping line – the further the price rises (falls), the more the short forward position loses (profits). This question merely asks you to re-produce this payoff diagram calculating the profit/loss for a range of gold prices. Two key things to note about the calculations: •

The question is assuming that the size of the contract is 1,000 ounces of gold. That is, the forward contract is for the delivery of 1,000 ounces of gold. So all calculations below multiply by 1,000 ounces.

•

Our initial position was to enter a short forward contract. We will close this out in December by entering an offsetting long forward contract. The table gives a range of possible prices for gold in December 2020. The calculations implicitly assume that the forward price available to us when we go long to close out equals the December spot price for gold. It is a fact that, on the expiry date of the contract, the quoted forward price must equal the spot price of the underlying asset (i.e., gold). If this does not happen, there is a trivial arbitrage opportunity available (see Q6 of Tutorial 1).

Opening trade: Short forward to sell 1,000 ounces of gold in Dec-2020: 1,000 × 1975 =USD 1,975,000 Closing trade: Long forward to buy 1,000 ounces of gold in Dec-2020: 1,000 × 1,625 =USD 1,625,000 Profit = 1,975,000 – 1,625,000

= +350,000

Calculate profit or loss if Dec-2020 gold price is:

Profit or loss on short gold forward

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1400

1500

1625

1700

1800

2000

2200

2300

2400

+575k

+475k

+350k

+275k

+175k

-25k

-225k

-325k

-425k

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Profit Diagram for Short Forward Position

1.5

Our forward contract has a delivery price (F) = 1975 1

A short forward position profits when Dec-2020 gold price is below F = 1975

Profit or Loss

0.5

1000

1500

2000

2500

3000

0

Dec-2020 Gold Price (per ounce)

-0.5

A short forward position loses when Dec-2020 gold price is

-1

above F = 1975

-1.5

10 6

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Question 3 a) By entering a long futures contract, we have an obligation to buy the underlying asset at the contract price (F) on the specified delivery date. In this case, we are long ten FCOJ futures contracts. Futures contracts are highly standardised. Each FCOJ contract covers 15,000 pounds of orange juice solids. 1 Since we have ten contracts, we are obliged to purchase 150,000 pounds of orange juice in September 2020 at a fixed price of 200 cents per pound. Therefore, the purchase will cost us: 150,000 × $2 = $300,000. We went long because we are speculating that orange juice prices will rise. We hope they will be something much higher than 200 cents. For example, we would love the price to rise to say 300 cents. We have a contract that allows us to buy orange juice at 200c and we could then sell it for 300c. The danger is that the predicted price rise does not eventuate. If orange juice price falls to say 150c, we are obliged to buy OJ at 200c per pound and it will only be worth 150c. In other words, the danger with a long position is that prices will fall.

b) The FCOJ futures traded on the ICE can be physically delivered. That is, if we did not close out the contract before expiry, we would have to take delivery of 150,000 pounds of orange juice – not something a trader really wants to do! The vast majority of futures contracts are not physically delivered. Rather, the trader closes out by entering an equal and opposite position. In our case, we can close out any time we want between now and September 2020. The question says that, once price hits 230c, we close out and take the profit.

Opening trade: Long ten Sep-20 FCOJ contracts: Closing trade: Short ten Sep-20 FCOJ contracts: Profit

10 × 15,000 × 200 10 × 15,000 × 230

USD 300,000 USD 345,000 USD 45,000

Calculate profit or loss if Sep-20 FCOJ futures price is:

Profit or loss on ten long FCOJ futures

100

125

150

175

200

225

250

275

300

-150k

-112.5k

-75k

-37.5k

nil

+37.5k

+75k

+112.5k

+150k

1

I don’t really know what orange juice ‘solids’ are. And I don’t really need to. As a trader, I have no intention of ever physically taking delivery or receiving orange juice. Once the price moves in my favoured direction, I will simply close out the position and take the profit.

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10

Payoff Diagram for ten Long FCOJ Futures

5

3

Our futures contracts have a delivery price F = 200c

2

1

Long position profits when Sep-20 FCOJ price

Profit/Loss

is above F = 200c

0

Long position loses when Sep-20 FCOJ -1

price is below F = 200c

-2

-3

0

0.5

1

1.5

2

2.5

3

3.5

4

Sep-20 FCOJ Price (per pound)

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Question 4 The convention is to quote an exchange rate as AUD 1.00 buys X foreign dollars (this is referred to as an indirect quote). For example, AUD 1.00 = CHF 0.74. However, it simplifies the intuition if we view the foreign currency as just another commodity. Q: How much does it cost to buy one Swiss franc? A: AUD 1.3514.2 This is a direct quotation (e.g., CHF 1.00 = AUD 1.3514 . This practice also makes it much easier to understand the jargon of currencies ‘strengthening’ and ‘weakening’.

a)

If the CHF strengthens, it becomes more expensive to buy. Buying CHF 1.00 will cost more than AUD 1.3514. For example, the cost of one CHF might be AUD 1.40. In such a case, the quoted exchange rate after strengthening is thus 1/1.40 = 0.7143. So you can see why the indirect quoting convention is counter-intuitive. CHF quotes have gone from 0.7400 down to 0.7143, yet this is in fact a strengthening of the CHF. In contrast, if we use direct quotes, it is intuitive that the cost of a CHF going from 1.3514 up to 1.4000 is a strengthening.

b)

The way to make money on anything is to sell at a higher price than that at which you bought (i.e., buy low, sell high). If we are speculating that the CHF will strengthen, part (a) showed that this means the cost of the CHF will rise. So we want a long position in the CHF. Long positions make money when the price of the underlying asset rises. Even ignoring the availability of a forward contract, we could speculate by taking a long position in CHF (i.e., just buy some Swiss francs today at cost of AUD 1.3514). We hold the CHF, hope they strengthen, then sell them at the higher price (i.e., convert them back to AUD). However, the question wants us to speculate using the forward contract. Again, we would take a long forward contract on CHF at the six-month rate of 0.7450. Entering a long forward contract at 0.7450 gives us the right to buy Swiss francs in six months’ time at a fixed price of AUD 1.3423. To profit from this speculation, we not only need the CHF to strengthen, but to strengthen beyond the quoted forward rate of 0.7450. This happens in part (d) and we do in fact profit.

2

These answers will round the exchange rates to four decimal places. Students may get slightly different answers if they use different rounding. I suggest that 4 dps is sufficient. Using just 2 dps can lead to quite large rounding errors.

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c)

The spot exchange rate at 31 December maturity is 0.80. This represents a weakening of the CHF (thus, our calculation below must show a loss). The quoted forward rate for delivery on 31 Dec must also be 0.80 (else trivial arbitrage opportunity – see Tutorial 1 Question 6). So the ‘price’ of a CHF is 1.2500. We close out by shorting CHF 60,000 at this low price. (in July) (in Dec)

Enter long forward contract: CHF 60,000 × 1.3423 Close out by shorting forward contract CHF 60,000 × 1.2500 Loss

AUD 80,538 AUD 75,000 AUD (5,538)

Note: when we enter this long forward contract in July, no money changes hands. It is just a contract which has a notional value of $A80,538. Likewise, when we close out by shorting the contract, we do not receive the notional value of $A75,000. The only cashflow is us handing over $5,538.

d)

If the spot exchange rate on 31 December is 0.70, the CHF has strengthened considerably. This what we were hoping for when we entered a long forward on Swiss Francs. The price of a CHF is AUD 1.4286.

(in July) (in Dec)

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Enter long forward contract CHF 60,000 × 1.3423 Close out by shorting forward contract CHF 60,000 × 1.4286 Gain

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AUD 80,538 AUD 85,716 AUD 5,178

8

Question 5 nb: all of these answers round the exchange rate to 4 decimal places. If students carry more dps, their answer may differ just a little.

a) This part illustrates the danger of remaining exposed to exchange-rate movements (i.e., being unhedged). The AUD has weakened relative to the Euro between August and December. Back in August, one Euro cost AUD 1.4706. Now, one Euro costs AUD 2.0000. So the EUR10,000 needed for your holiday cost you a lot more. At the spot rate of 0.50, buying EUR 10,000 will cost AUD 20,000. Ouch!

b) If we purchase EUR 10,000 at today’s spot rate of 0.6800, it will cost us AUD 14,706. If the exchange rate did weaken to something like 0.50 in part a, we would be very glad we bought the Euros early. But this strategy isn’t ideal, in that we need to have access to AUD 14,706 today and thus miss out on the potential interest we could have earned.

c) By using a forward contract, we can lock in today the price at which we can buy Euros in December, thus removing any risk of exchange-rate movements. And doing this is better than (b) because we don’t need to have any cash now. Since we intend to buy Euros in December, we would enter a long forward contract written on Euros. As soon as we do this, we know for certain that, in December, the EUR 10,000 will cost us AUD 14,656. And another good thing about this long forward contract is that we don’t need any cash now. We merely need to have AUD 14,656 in December to meet our commitment under the forward contract.

d) To buy EUR 10,000 at today’s spot rate of 0.6800, we need AUD 14,706. So we can borrow AUD 14,706 from a local bank, convert them into Euros today at the spot exchange rate, all ready for our vacation. Four months later, we repay the loan balance of AUD 14,804 (=14,706 × exp(0.02 × 4/12)).

e) If we borrow AUD 14,560 today and use it to buy Euros at the spot rate of 0.6800, this produces EUR 9,901 (14,560 × 0.68). Next, we deposit these Euros into our European bank account to earn interest at 3% p.a. continuously compounded. In four months’ time, this account has grown to EUR 10,000 (9901 × exp(0.03 × 4/12)). So that’s our target spending money for the vacation! Back home, our loan of AUD 14,560 has grown to AUD 14,657 (14,560 × exp(0.02 × 4/12)). So this is the effective cost of the holiday. Note that the vacation cost in part e is identical to the vacation cost when a forward contract was utilised in part c (with $1 rounding difference). In Lecture 3, we will see the same logic being used to establish the fair price for currency forward contracts (i.e., factoring in the spot exchange rate and the interest rates in the two countries). This is how banks determine forward rates.

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Question 6 On the expiry date of forward contract, if the quoted forward price does not match the spot price on that day, there is an arbitrage opportunity. In other words, anyone who recognises the mis-pricing can implement some simple transactions to make a certain (or riskless) profit. For example, consider a forward contract on gold with delivery in October. Assume that one hour before the October contract expires, the spot price of gold is $1900/ounce and the forward is quoted at $1910/ounce. If this happens, the forward is overpriced. We could implement the following trades to capture the arbitrage profit: •

Take a short forward contract on the October-maturity gold. Why short? At $1910, the forward contract is overpriced. When something is overpriced, we would want to sell it. The short forward means I have a commitment to sell one ounce of gold (in one hour's time) and I will receive the delivery price of $1910.

•

I also immediately purchase the one ounce of gold on the spot market for $1900. This will enable me to meet my commitments.

•

In one hour's time, the forward contract matures. I physically deliver the gold to satisfy my commitment under the short forward. I receive the contracted forward price of $1910. And I have made $10 arbitrage profit.

And, of course, I wouldn't do this for just one ounce of gold. I'd do it for as many ounces as possible. There are plenty of smart arbitrageurs out there who would quickly jump on this mispriced forward contract and trade heavily until the forward contract was correctly priced.

Similarly, if the forward contract was quoted at $1895 very close to maturity, I'd do the opposite. Effectively, the forward contract is underpriced: •

Enter long forward contract on gold. Why long? At $1895, the forward is underpriced. When something is underpriced, we like to buy it. The long forward means I have a commitment to purchase gold in one hour's time at price $1895.

•

Short-sell one ounce of gold at the spot price of $1900. In simple terms, short selling means someone ‘loans’ you some gold, you sell it today on the spot market and pocket the $1900. Of course, at some point in the future, you must repay the gold to them.

•

In one hour, I honour my commitment under the forward contract to buy one ounce of gold for $1895. I use the gold to settle my short-sale debt.

Note that this question is concerned with the convergence of the spot and forward price exactly at expiry. At any time before expiry, we expect they will be different. More on this next week. Note also that the ability of arbitrageurs to implement these trades depends, in some cases, on the ability to short sell the underlying asset. This is not always possible for every asset.

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