Derivatives FIN 249

There is a margin call if $1,000 is lost on the contract. This will happen if the price of wheat futures rises by 20 cents from 750 cents to 770 cents per bushel. $1,500 can be withdrawn if the futures price falls by 30 cents to 720 cents per bushel.

A company enters into a short futures contract to sell 5,000 bushels of wheat for 750 cents per bushel. The initial margin is $3,000 and the maintenance margin is $2,000. What price change would lead to a margin call? Under what circumstances could $1,500 be with-drawn from the margin account?

The formula for the number of contracts that should be shorted gives 88.9 Rounding to the nearest whole number, 89 contracts should be shorted. To reduce the beta to 0.6, half of this position, or a short position in 44 contracts, is required.

A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts on a stock index to hedge its risk. The index futures price is currently standing at 1080, and each contract is for delivery of $250 times the index. What is the hedge that minimizes risk? What should the company do if it wants to reduce the beta of the portfolio to 0.6?

When the expected return on the market is −30%, the expected return on a portfolio with a beta of 0.2 is 0.05 + 0.2 × (−0.30 − 0.05) = −0.02 or -2%. The actual return of -10% is worse than the expected return. The portfolio manager has done 8% worse than a simple strategy of forming a portfolio that is 20% invested in an equity index and 80% invested in risk-free investments.

A portfolio manager has maintained an actively managed portfolio with a beta of 0.2. During the last year, the risk-free rate was 5% and equities performed very badly providing a return of -30,. The portfolio manager produced a return of -10, and claims that in the circumstances it was a good performance. Discuss this claim.

There is a margin call if more than $1,500 is lost on one contract. This happens if the futures price of frozen orange juice falls by more than 10 cents to below 150 cents per pound. $2,000 can be withdrawn from the margin account if there is a gain on one contract of $1,000. This will happen if the futures price rises by 6.67 cents to 166.67 cents per pound.

A trader buys two July futures contracts on frozen orange juice concentrate. Each contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound, the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract. What price change would lead to a margin call? Under what circumstances could $2,000 be withdrawn from the margin account (aka have 2000 excess in the account)?

(a) The trader sells for 50 cents per pound something that is worth 48.20 cents per pound. Gain ($0.5000-$0.4820) * 50,000= $900 (b) The trader sells for 50 cents per pound something that is worth 51.30 cents per pound. Loss ($0.5130-$0.5000) * 50,000 = $650

A trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents per pound?

a) The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0084 per yen. The gain is 100M * .0006, or $60,000. b) The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0101 per yen. The loss is 100M * .0011, or $110,000.

A trader enters into a short forward contract on 100 million yen. The forward exchange rate is $0.0090 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is (a) $0.0084 per yen; (b) $0.0101 per yen?

(a) The minimum variance hedge ratio, h*, is 0.95×0.43/0.40=1.02125. (b) The hedger should take a short position. (c) The optimal number of contracts ignoring daily settlement is 1.02125×55,000/5,000=11.23 (or 11 when rounded to the nearest whole number)

A trader owns 55,000 units of a particular asset and decides to hedge the value of her position with futures contracts on another related asset. Each futures contract is on 5,000 units. The spot price of the asset that is owned is $28 and the standard deviation of the change in this price over the life of the hedge is estimated to be $0.43. The futures price of the related asset is $27 and the standard deviation of the change in this over the life of the hedge is $0.40. The coefficient of correlation between the spot price change and futures price change is 0.95. (a) What is the minimum variance hedge ratio? (b) Should the hedger take a long or short futures position? (c) What is the optimal number of futures contracts when adjustments for daily settlement are not considered?

a) The investor is obligated to sell pounds for 1.3000 when they are worth 1.2900. The gain is (1.3000−1.2900) ×100,000 = $1,000. (b) The investor is obligated to sell pounds for 1.3000 when they are worth 1.3200. The loss is (1.3200−1.3000)×100,000 = $2,000

An investor enters into a short forward contract to sell 100,000 British pounds for U.S. dollars at an exchange rate of 1.3000 USD per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.2900 and (b) 1.3200?

at time T at maturity roll risk

Basis risk is faced at this of rolling the hedge forward, everything else is

The open interest of a futures contract at a particular time is the total number of long positions outstanding. (Equivalently, it is the total number of short positions outstanding.) The trading volume during a certain period of time is the number of contracts traded during this period.

Distinguish between the terms open interest and trading volume.

Basis risk arises from the hedger's uncertainty as to the difference between the spot price and futures price at the expiration of the hedge.

Explain how basis risk arises in hedging.

A perfect hedge is one that completely eliminates the hedger's risk. A perfect hedge does not always lead to a better outcome than an imperfect hedge. It just leads to a more certain outcome. Consider a company that hedges its exposure to the price of an asset. Suppose the asset's price movements prove to be favorable to the company. A perfect hedge totally neutralizes the company's gain from these favorable price movements. An imperfect hedge, which only partially neutralizes the gains, might well give a better outcome.

Explain what is meant by a perfect hedge. Does a perfect hedge always lead to a better outcome than an imperfect hedge? Explain your answer.

Stack and roll involves creating a long-dated futures contract from a series of short-dated contracts.

Explain what the stack and roll hedging strategy involves.

To increase beta, a long position in index futures is required. To reduce beta, a short position in index futures is required

How can index futures be used to change the beta of a well-diversified portfolio?

They can hedge exposure to market moves using index futures. They will make money if the stocks they have picked outperform the market.

How might investors who consider themselves adept at stock picking use index futures?

It should short five contracts. It has basis risk. It is exposed to the difference between the October futures price and the spot price of light sweet crude at the time it closes out its position in September. It is also possibly exposed to the difference between the spot price of light sweet crude and the spot price of the type of oil it is selling.

It is now June. A company knows that it will sell 5,000 barrels of crude oil in September. It uses the October CME Group futures contract to hedge the price it will receive. Each contract is on 1,000 barrels of "light sweet crude." What position should it take? What price risks is it still exposed to after taking the position?

10M (F2-F1)

On July 1, 2021, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2022. On September 1, 2021, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2022. Describe the payoff from this strategy.

The open interest went down by 600. We can see this in two ways. First, 1,400 shorts closed out and there were 800 new shorts. Second, 1,200 longs closed out and there were 600 new longs.

On a particular day, there were 2,000 trades in a particular futures contract. This means that there were 2,000 buyers (going long) and 2,000 sellers (going short). Of the 2,000 buyers, 1,400 were closing out positions and 600 were entering into new positions. Of the 2,000 sellers, 1,200 were closing out positions and 800 were entering into new positions. What is the impact of the day's trading on open interest?

the spot price of underlying asset at time t (right now)

S(t) means

The excess of the spot over the futures at the time the hedge is closed out is $0.20 per ounce. If the trader is hedging the purchase of silver, the price paid is the futures price plus the basis. The trader therefore loses 60×5,000×$0.20=$60,000. If the trader is hedging the sales of silver, the price received is the futures price plus the basis. The trader therefore gains $60,000.

Sixty futures contracts are used to hedge an exposure to the price of silver. Each futures contract is on 5,000 ounces of silver. At the time the hedge is closed out, the basis is $0.20 per ounce. What is the effect of the basis on the hedger's financial position if (a) the trader is hedging the purchase of silver and (b) the trader is hedging the sale of silver?

The optimal hedge ratio is .642 This means that the size of the futures position should be 64.2% of the size of the company's exposure in a three-month hedge.

Suppose that the standard deviation of quarterly changes in the prices of a commodity is $0.65, the standard deviation of quarterly changes in a futures price on the commodity is $0.81, and the coefficient of correlation between the two changes is 0.8. What is the optimal hedge ratio for a 3-month contract? What does it mean?

30*e^(.05*.5)=30.76

Suppose that you enter into a 6-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 5% per annum. What is the forward price?

There will be a margin call when $1,000 has been lost from the margin account. This will occur when the price of silver increases by 1,000/5,000 = $0.20. The price of silver must therefore rise to $17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes out your position.

Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is $3,000. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call?

a. 6.4% b. 8.5% c. 14.8%

The expected return on the S&P 500 is 12% and the risk-free rate is 5%. What is the expected return on an investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4?

These options make the contract less attractive to the party with the long position and more attractive to the party with the short position. They therefore tend to reduce the futures price.

The party with a short position in a futures contract sometimes has options as to the precise asset that will be delivered, where delivery will take place, when delivery will take place, and so on. Do these options increase or decrease the futures price? Explain your reasoning

A minimum variance hedge leads to no hedging when the coefficient of correlation between the futures price changes and changes in the price of the asset being hedged is zero.

Under what circumstances does a minimum-variance hedge portfolio lead to no hedging at all?

The contract would not be a success. Parties with short positions would hold their contracts until delivery and then deliver the cheapest form of the asset. This might well be viewed by the party with the long position as garbage!

What do you think would happen if an exchange started trading a contract in which the quality of the underlying asset was incompletely specified?

Arbitrage involves carrying out two or more different trades to lock in a profit. In this case, traders can buy shares on the TSX and sell them on the NYSE to lock in a USD profit of 50-60/1.21=0.41 per share. As they do this the NYSE price will fall and the TSX price will rise so that the arbitrage opportunity disappears.

What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed mining company stock is $50 (USD) on the New York Stock Exchange and $60 (CAD) on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.21 CAD. Explain what is likely to happen to prices as traders take advantage of this opportunity.

The present value is Ae^-RT where T is the time until payment.

What is the formula for calculating the present value of an amount A when the interest rate is R and the compounding frequency is continuous?

A margin account is designed to protect the exchange against losses from defaults by traders.

What is the purpose of margin accounts?

A company that knows it will purchase a commodity in the future is able to lock in a price close to the futures price. This is likely to be particularly attractive when the futures price is less than the spot price.

When the futures prices of an asset is less than spot prices, long hedges are likely to be particularly attractive." Explain this statement.

loss for short gain for long

a backwardation market is when the short term price is greater than the long term price, and this is a (gain/loss) for short hedgers and a (gain/loss) for long hedgers

long futures h* = .85 (.045/.058) = .66 .66 * 2M = 1,320,000 1,320,000 / 42,000 = 31.43 ~long 31 contracts jet fuel price volatility will be reduced by P^2 = .85^2 = 72.25%

a company will need 2 M gallons of jet fuel in six months. the standard deviation of jet fuel prices over the next six months is calc as .045. the company will use heating oil futures (one contract is 42,000 gal) to hedge its need for jet fuel. the standard dev of heating oil futures price change over 6 months is .058. and the correlation between fuel prices and heating oil is .85. how would the oil company set up its hedge? how much of jet fuel price volatility will be reduced?

gain for short loss for long

a contango market is when the long term maturity price is greater than the short term maturity price, and this is a (gain/loss) for short hedgers and a (gain/loss) for long hedgers

constructed at zero cost at some specific future date it has no risk of a loss and positive value

an arbitrage opportunity is a portfolio with the following 3 properties

derivatives

any broad class of financial securities whose value is determined by the value of more basic underlying variables

spot price of asset to be hedged - futures price of contract used

basis is calculated as

can be quoted at anytime during its life

because futures are traded on exchanges, the futures price

130 pounds * 1.28 = $166.40 in London $170 in NY Buy in London and sell immediately in NY 170-166.4 = $3.6 immediate profit per share

consider a stock that is traded on both NYSE and the London stock exchange. supposed that the stock price is $170 in NY and 130 pounds in London at a time when exchange rate is $1.28 per British pound. describe the arbitrage opportunity.

offsetting trade

delivery almost never takes place in the case of futures contracts. most contracts are closed out before maturity, which involves entering into an

absence of arbitrage

derivative pricing is based on the

the forward price for a contract with maturity T, agreed upon at time t

f(t,T) means

e ^ rt e ^ -rt

for continuously compounded interest rate, this is the formula we use for FV, and this is the formula we use for PV

do offsetting trade close to maturity

if a contract expires there will be physical delivery required. so to avoid this, investors will

you want to hedge 60% of what you have in spot

if the optimal hedge is .6 that means

futures price = spot price basis risk

in a perfect hedge, that expiration (maturity) ________, and if this is not the case there is _________

a sale a buy

in an offsetting trade, a long position is closed out by _______ and a short position is closed out by _______

100,000+100,000 / 1000 april 17: short 200 oct 17 contracts sep 17: long 200 oct 17 contracts and short 200 march 18 contracts feb 18: long 200 march 18 contracts and short 200 aug 18 contracts june 18: long 100 aug 18 contracts july 18: long 100 aug 18 contracts short 100 jan 19 contracts dec 18: long 100 jan 19 contracts

in april 2017 a company realizes that it needs to sell 100,000 barrels of oil in June 2018 and another 100,000 barrels in dec 2018. it decides to hedge its risk with a hedge ratio of 1 using crude oil futures traded on NYMEX (one crude oil futures contract is for delivery of 1000 bbls). it wants to use contracts that are relatively liquid so that the hedge position can be liquidated if so desired. this means contracts with maturities less than 6 months. it therefore shorts october 2017 contracts. in september 2017 it rolls over to march 2018 contracts. in february 2018 it rolls over to august 2018 contracts. fill out the strategy for each date below: Apr 17 Sep 17 Feb 18 June 18 July 18 Dec 18

optimal hedge ratio=beta=1.5 short index because you have the shares # of contracts: 20,000 * 1100 * 1.5 / 2750 * 250 = 48 short 28 contracts

in jan 2018 you hold 20,000 alphabet shares, parents of company google, each worth $1,100. you think the market is going to be very volatile over the next 3 months, but you feel that google has a good chance of outperforming the market. so you decide to use the may futures contract on the S&P 500 index to hedge the market's return over the next three months. the beta of alphabet's stock is 1.5. suppose that the current futures price for the june contract on the S&P is 2750. each contract is for delivery for $250 times the index. how many futures contract should you use. and should you long or short?

spot-forward 4200-3990 210*5= loss of 1050

in jan, investor enters into a short forward contract to sell 5 bitcoins for 3990/BTC in april. in april the spot price for one bitcoin is $4200. what is the investors profit?

A good rule of thumb is to choose a futures contract that has a delivery month as close as possible to, but later than, the month containing the expiration of the hedge. The contracts that should be used are therefore (a) July (b) September (c) March

in the corn futures contract, the following delivery months are available: March, May, July, September, and December. State the contract that should be used for hedging when the expiration of the hedge is in a) June, b) July, and c) January

a. company should short 140 contracts (1.2-0.5) * 100,000,000 / 2000*250 b. long 60 contacts (1.5-1.2) * 100,000,000 / 2000*250

it is July 16. A company has a portfolio of stocks worth $100 million. The beta of the portfolio is 1.2. The company would like to use the December futures contract on a stock index to change the beta of the portfolio to 0.5 during the period July 16 to November 16. The index futures price is currently 2,000 and each contract is on $250 times the index. (a) What position should the company take? (b) Suppose that the company changes its mind and decides to increase the beta of the portfolio from 1.2 to 1.5. What position in futures contracts should it take?

they don't want to deal with physical delivery

people close out their orders before maturity because

S(T) - f(t,T)

profit for the long position of forward contract at time T (maturity) is

liquidity counterparty risk

standardizations of contracts eliminates flexibility, but improves ______, and minimizes __________

short 1 oz of gold in spot payoff at t 590 payoff at T : -S(T) long 1 oz of gold forward payoff at t 0 payoff at T: S(T)-610 (basis) proceeds of selling gold invested payoff at t: -590 payoff at T : 590 * e^.05 = 620.25 Net payoff payoff at t: 0 payoff at T: -S(T) + S(T) - 610 + 620.25 = 10.25 strategy will stop working when the price in forward market goes up by 10.25 to 620.25

suppose that the spot price of gold is US $590 and the quoted one year forward price is US$610. the one year US$ interest rate is 5% continuously compounded. is there an arbitrage opportunity? if so, the strategy will stop making money when ___________ transactions (payoff at t and T which is in one year): short 1 oz of gold in spot long1 oz of gold forward proceeds of selling gold invested (giving money to bank to invest) Net payoff

long 1 oz of gold in spot payoff at t is -590 payoff at T is S(T) short 1 oz of gold forward payoff at t is 0 payoff at T is 625-S(T) borrow $590 payoff at t is 590 payoff at T is 590*e^.05= -620.25 Net payoff payoff at t is 0 payoff at T is S(T)-S(T)+625-620.25=4.75 strategy will stop working when the price in forward market goes down by 4.75 to 620.25

suppose that the spot price of gold is US $590 and the quoted one year forward price is US$625. the one year US$ interest rate is 5% continuously compounded. is there an arbitrage opportunity? if so, the strategy will stop making money when ___________ transactions (payoff at t and T which is in one year): long 1 oz of gold in spot short 1 oz of gold forward borrow $590 Net payoff

1 year: $54 3 years: 62.99 1 month: 50.32 11 months: 53.66 make sure to put time in reference to one year

suppose you deposited $50 into a bank account at a continuously compounded interest rate of 7.7%, how much would you have after a year? three years? 1 month? 11 months?

spot price

the ________ is the price of the asset for immediate delivery

maturity date

the _________ is the time of the scheduled delivery

forward price

the __________ is the delivery price applicable to a contract today and may be different for contracts of different maturities.

forward price zero

the ____________________ is not the price of the forward contract itself. since no cash changes hands when a forward contract is set up, both parties enter the contract voluntarily and this implies that the value of the forward contract when it is initiated is ______

OTC

the forward contract is an ________ agreement between two parties

beta

the hedge ratio is also equal to

h* = P (correlation coefficient between spot and futures) * (sigma s / sigma f)

the optimal hedge percentage can be found with this equation

basis risk

this arises because of a mismatch between the expiration date of the futures and the actual selling date of the asset

basis risk

this arises because of the difference between the asset whose price is to be hedged and the asset underlying the futures

spot contract

this is an agreement to buy or sell an asset immediately

hedge ratio

this is how many futures you want to make up for the spot market

marking to market

this is rebalancing the margin acct everyday to reflect the gains and losses, instead of waiting until maturity to realize all the gains and losses like in a forward contract

initial margin

this is the amount of money deposited by each position in an acct to ensure that the trader is able to satisfy their obligation in the futures contract

hedging

this is when you are reducing exposure to adverse price movement

arbitraging

this is when you attempt to lock in a riskless profit by simultaneously entering into transactions in 2 or more markets

speculating

this is when you use derivatives to reflect a view on the future direction of the market

arbitrager

this person ensures that there is a "correct" price for derivatives that the actual price should not wander too far away from

the clearing house

this works as an intermediary between parties. from the moment the trade is made, the legal contract between both parties is canceled and the ________ takes the opposite position to each of the parties

hedging, speculating, arbitraging

three ways derivatives are used

long short

to _______ a forward means someone has entered in the forward contract while agreeing to buy asset in future. to ______a forward means someone has entered in the forward contract while agreeing to sell asset in future.

standardized on exchanges

unlike forwards, futures are _________ contracts traded _________

maintenance margin

values below which the balance of the margin acct must not fall, if the margin account falls below this, the trader must immediately increase it

buy

when you are long hedging you have assets you need to

sell

when you are short hedging you have assets you need to

up

when you long a forward contract, the price you pay in the future is the forward price. the more the spot price goes _______ the more $ you make

down

when you long a forward contract, the price you pay in the future is the forward price. the more the spot price goes _______ the more loss you will have

up

when you short a forward contract, the price you receive in the future is the forward price. the more the spot price goes _______ the less $ you make

down

when you short a forward contract, the price you receive in the future is the forward price. the more the spot price goes _______ the more $ you make

you have long term hedging strategy but short dated futures to hedge with

you roll the hedge forward when