Derivatives Final

Which of the following describes a short position in an option?

A position where an option has been sold A short position is a position where the option has been sold (the opposite to a long position).

The price of a stock is $64. A trader buys 1 put option contract on the stock with a strike price of $60 when the option price is $10. When does the trader make a profit?

The payoff must be more than the $10 paid for the option. The stock price must therefore be below $50.

The party with short position in a Treasury bond futures contract has a right to deliver at the closing price for a period of time after the close of trading. This is called

Wild card play Wild card play is the strategy to deliver to a short futures contract at the closing price after the close of trading.

The spot price of an investment asset that provides no income is $30 and the risk-free rate for all maturities (with continuous compounding) is 10%. What is the three-year forward price?

3-year forward rate = F = 30e0.10*3 = 40.50.

Which of the following are NOT true?

A hedge set up to value an option does not need to be changed The hedge set up to value an option needs to be changed as time passes. Both of the following are true: Risk-neutral valuation and no-arbitrage arguments give the same option prices Risk-neutral valuation involves assuming that the expected return is the risk-free rate

A short forward contract on an asset plus a long position in a European call option on the asset with a strike price equal to the forward price is equivalent to

A long position in a put option A short position on an asset plus a long position in a European call option is a long position in a put option.

Which of the following describes a protective put?

A long put option on a stock plus a long position in the stock A protective put consists of a long put plus the stock. The holder of the put owns the stock that might become deliverable.

A company enters into a short futures contract to sell 50,000 units of a commodity for 70 cents per unit. The initial margin is $4,000 and the maintenance margin is $3,000. What is the futures price per unit above which there will be a margin call?

A margin call is required when the loss is more than $1,000/50,000 = $0.02 per unit of the commodity. The price has to go higher than 0.70 + 0.02 = 0.72 cents.

Which of the following describes a long position in an option?

A position where an option has been purchased A long position is a position where an option has been purchased. It can be contrasted with a short position which is a position where an option has been sold.

Which of the following describes a covered call?

A short call option on a stock plus a long position in the stock A covered call consists of a short call plus a long position in the stock. Then the owner of the position has the stock ready to deliver if the other side exercises the call.

Which of the following could NOT be a delta-neutral portfolio?

A short position in call options plus a short position in the underlying stock Calls have a positive delta. Puts have a negative delta. A long stock position has a positive delta. A short stock position has a negative delta. B cannot be delta neutral (i.e., have a delta of zero) because both parts of the portfolio have a negative delta.

Which of the following is true for an interest rate swap?

A swap is usually worth close to zero when it is first negotiated A swap is worth close to zero at the beginning of its life. (It may not be worth exactly zero because of the impact of the market maker's bid-offer spread.) It is not true that each of the forward contracts underlying the swap are worth zero. (The sum of the value of the forward contracts is zero, but this does not mean that each one is worth zero.) The remaining floating payments on a swap are worth the notional principal immediately after a swap payment date, but this is not necessarily true for the remaining fixed payments.

Which of the following describes European options?

A. Exercisable only at maturity European options can be exercised only at maturity. This is in contrast to American options which can be exercised at any time. The term "European" has nothing to do with geographical location, currencies, or whether the option is a call or a put.

Which of the following describes an interest rate swap?

All of the above An interest rate swap is all of the above. It is one way to convert a liability from fixed to floating; it is a portfolio of forward rate agreements, and it is an agreement to exchange interest at a fixed rate for interest at a floating rate.

Clearing houses are

Always used in futures markets and sometimes used in OTC markets Clearing houses are always used by exchanges trading futures. Increasingly, OTC products are cleared through CCPs, which are a type of clearing house.

The bonds that can be delivered in a Treasury bond futures contract are

Assets that provide a known cash income A bond is an asset that provides a known cash income (the coupons)

Which of the following is true?

B. If all companies in an industry do not hedge, a company is liable increase its risk by hedging If all companies in a industry hedge, the price of the end product tends to reflect movements in relevant market variables. Attempting to hedge those movements can therefore increase risk.

The current price of a non-dividend-paying stock is $40. Over the next year it is expected to rise to $42 or fall to $37. An investor buys one-year put options with a strike price of $41. Which of the following is necessary to hedge the position?

Buy 0.8 shares for each option purchased The payoff from the put option is zero if there is an up movement and 4 if there is a down movement. Suppose that the investor buys one put option and buys shares. If there is an up movement the value of the portfolio is ×42. If there is a down movement it is worth ×37+4. These are equal when 37 +4=42 or =0.8. The investor should therefore buy 0.8 shares for each option purchased.

Which of the following can be used to create a long position in a European put option on a stock?

Buy a call on the stock and short the stock As payoff diagrams show a call on a stock combined with a short position in the stock gives a payoff similar to a put option. Alternatively we can use put-call parity, which shows that a call minus the stock equals the put minus the present value of the strike price.

Which of the following creates a bull spread?

Buy a low strike price call and sell a high strike price call A bull spread is created by buying a low strike call and selling a high strike call. Alternatively, it can be created by buying a low strike put and selling a high strike put.

Which of the following is true about a long forward contract

C. The contract becomes more valuable as the price of the asset rises A long forward contract is an agreement to buy the asset at a predetermined price. The contract becomes more attractive as the market price of the asset rises. The contract is only worth zero when the predetermined price in the forward contract equals the current forward price (as it usually does at the beginning of the contract).

The basis is defined as spot minus futures. A trader is hedging the sale of an asset with a short futures position. The basis increases unexpectedly. Which of the following is true?

C. The hedger's position improves. The price received by the trader is the futures price plus the basis. It follows that the trader's position improves when the basis increases.

When the stock price increases with all else remaining the same, which of the following is true?

Calls increase in value while puts decrease in value Stock price increases cause the values of calls to increase and the values of puts to decline.

Assume the Treasury bond futures price is 93.25. A bond that is deliverable to this futures contract has a price of 99.50 and a conversion factor of 1.0382. What is the cost of delivering this bond to the futures contract?

Cost = Quoted price - Futures price x Conversion Factor = 99.50 - 93.25 x 1.0382 = $2.69

Which of the following is a reason for hedging a portfolio with an index futures?

D. The investor believes the stocks in the portfolio will perform better than the market but is uncertain about the future performance of the market Index futures can be used to remove the impact of the performance of the overall market on the portfolio. If the market is expected to do well hedging against the performance of the market is not appropriate. Hedging cannot correct for a poorly diversified portfolio.

Which of the following is NOT true about call and put options:

D. The price of a call option increases as the strike price increases. A call option is the option to buy for the strike price. As the strike price increases this option becomes less attractive and is therefore less valuable. A, B, and C are true.

The modified duration of a bond portfolio worth $1 million is 6 years. By approximately how much does the value of the portfolio change if all yields increase by 5 basis points?

Decrease of $3,000 When yields increase bond prices decrease. The proportional decrease is the modified duration times the yield increase. In this case, it is 6×0.0005=0.003. The decrease is therefore 0.003×1,000,000 or $3,000.

A company invests $10,000 in a two-year zero-coupon bond and $5,000 in a five-year zero-coupon bond. What is the duration of the portfolio?

Duration = (10,000*2 + 5,000*5)/15,000 = 3.

Under liquidity preference theory, which of the following is always true?

Forward rates are higher than expected future spot rates. Liquidity preference theory argues that individuals like their borrowings to have a long maturity and their deposits to have a short maturity. To induce people to lend for long periods forward rates are raised relative to what expected future short rates would predict.

Given a choice between 5-year and 1-year instruments most people would choose 5-year instruments when borrowing and 1-year instruments when lending. Which of the following is a theory consistent with this observation?

Liquidity preference theory Borrowing 5 years and lending 1 year is consistent with liquidity preference theory.

A company has a $36 million portfolio with a beta of 1.2. The futures price for a contract on an index is 900. Futures contracts on $250 times the index can be traded. What trade is necessary to increase beta to 1.8?

Long 96 contracts To increase beta by 0.6 we need to go long 0.6×36,000,000/(900×250) or 96 contracts

The current price of a stock is $124. A trader sells 200 European put options on the stock with a strike price of $120. Expiration is 3 months. The option price is $5. The options are exercised when the stock price is $110. The trader's net profit or loss is

Loss of $1,000 The payoff that must be made on the options is 200×(120−110) or $2000. The amount received for the options is 5×200 or $1000. The net loss is therefore 2000−1000 or $1000.

You sell one December futures contracts when the futures price is $1,010 per unit. Each contract is on 100 units and the initial margin per contract that you provide is $2,000. The maintenance margin per contract is $1,500. During the next day the futures price rises to $1,012 per unit. What is the balance of your margin account at the end of the day?

Loss on the contract is $1,012 - $1,010 = $2 per unit. Total loss is $2 x 100 = $200. Margin the following day is $2,000 - $200 = $1,800.

Six-month call options with strike prices of $35 and $40 cost $6 and $4, respectively. What is the maximum gain when a bull spread is created by trading a total of 200 options?

Max profit when ST > 40: Profit = 100*[40 - 35 + (-6+4)] = 300.

Suppose that the standard deviation of monthly changes in the price of commodity A is $2. The standard deviation of monthly changes in a futures price for a contract on commodity B (which is similar to commodity A) is $3. The correlation between the futures price and the commodity price is 0.9. What hedge ratio should be used when hedging a one month exposure to the price of commodity A?

Minimum variance hedge ratio = Qs / Qf = 0.9*2/3 = 0.60

Which of the following is true about bonds?

None of the above When interest rates increase the impact of discounting is to make future cash flows worth less. Bond prices therefore decline. A is therefore wrong. As coupons increase a bond becomes more valuable because higher cash flows will be received. B is therefore wrong. When the coupon is higher than prevailing interest rates, longer maturity bonds are worth more than shorter maturity bonds. When it is less than prevailing interest rates, longer maturity bonds are worth less than shorter maturity bonds. C is therefore not true. The correct answer is therefore D.

A trader uses 3-month Eurodollar futures to lock in a rate on $10 million for six months. How many contracts are required?

One Eurodollar contract locks in $1 million for 3 months. To lock in $10 million for 6 months would require 10x 2 = 20 contracts.

Which of the following is true for American options?

Put-call parity provides an upper and lower bound for the difference between call and put prices Put-call parity provides both an upper and a lower bound for the difference between call and put prices.

Which of the following is NOT true?

Risk-neutral valuation provides prices that are only correct in a world where investors are risk-neutral Risk-neutral valuation produces a valuation that is correct in all situations not just those where investors are risk-neutral. The expected return on all investments is assumed to be the risk-free rate and the risk-free rate is used to discount expected payoffs.

An exchange rate is 1.1200 and the 3-month US and foreign risk-free interest rates are 2% and 0% (both expressed with continuous compounding). What is the 3-month forward rate?

The 3-month forward rate is 1.12e(.02-0)(0.25) = 1.1256.

Which of the following is the put-call parity result for a non-dividend-paying stock?

The European put price plus the stock price must equal the European call price plus the present value of the strike price The put-call parity result is c+Ke-rT=p+S0.

An interest rate is 5% per annum with continuous compounding. What is the equivalent rate with semiannual compounding?

The equivalent rate with semiannual compounding is 2×(e0.05/2−1) = 0.0506 or 5.06%.

A semi-annual pay interest rate swap where the fixed rate is 5.00% (with semi-annual compounding) has a remaining life of nine months. The six-month LIBOR rate observed three months ago was 4.85% with semi-annual compounding. Today's three and nine month LIBOR rates are 5.3% and 5.8% (continuously compounded) respectively. From this it can be calculated that the forward LIBOR rate for the period between three- and nine-months is 6.14% with semi-annual compounding. If the swap has a principal value of $15,000,000, what is closest to the value of the swap to the party receiving a fixed rate of interest?

The forward rates for the floating payment at time 9 months is 6.14%. The swap can be valued assuming that the fixed payments are 2.5% of principal at 3 months and 9 months and that the floating payments are 2.425% and 3.07% of the principal at 3 months and 9 months. The value of the swap to the party receiving fixed is therefore 15,000,000(0.025-0.02425)e-0.053×0.25+15,000,000(0.025-0.0307)e-0.058×0.75 = -$70,760

Which of the following is NOT an option open to the party with a short position in the Treasury bond futures contract?

The interest rate used in the calculation of the conversion factor A, B, and C describe options that the party with the short position has. D does not

A trader enters into a short position in 50 Eurodollar futures contract. What is the gain or loss of the trader when the futures price quote increases by 5 basis points?

The loss is 50 x 5 x $25 = $6,250.

A stock price (which pays no dividends) is $50 and the strike price of a two year European put option is $54. The risk-free rate is 3% (continuously compounded). Which of the following is a lower bound for the option such that there are arbitrage opportunities if the price is below the lower bound and no arbitrage opportunities if it is above the lower bound?

The lower bound in Ke-rT −S0 In this case it is 54e−0.03×2 - 50= $0.86.

Which of the following is true?

The optimal hedge ratio is the slope of the best fit line when the change in the spot price (on the y-axis) is regressed against the change in the futures price (on the x-axis). The optimal hedge ratio is the slope of the best fit line when the change in the spot price (on the y-axis) is regressed against the change in the futures price (on the x-axis).

When a six-month option is purchased

The price must be paid in full Only options lasting more than 9 months can be bought on margin.

Which of the following is NOT true about duration?

The prices of two bonds with the same duration change by the same percentage amount when interest rate moves up by 100 basis points D is only approximately true. A, B, and C are exactly true.

Which of the following describes the five-year swap rate?

The rate that can be earned over five years from a series of short-term loans to AA-rated companies By considering the effect of making a series of LIBOR loans to AA-rated companies and entering into a swap we see that the swap rate corresponds to the risk in a series of short-term loans.

In a binomial tree created to value an option on a stock, the expected return on stock is

The risk-free rate The expected return on the stock on the tree is the risk-free rate. This is an application of risk-neutral valuation.

Which of the following must post margin?

The seller of an option The seller of the option must post margin as a guarantee that the payoff on the option (if there is one) will be made. The buyer of the option usually pays for the option upfront and so no margin is required.

Which of the following is a definition of volatility?

The standard deviation of the return, measured with continuous compounding, in one year Volatility when multiplied by the square root of t is the standard deviation of the return in a short period of time of length t. It is also the standard deviation of the continuously compounded return in one year.

What should a trader do when the one-year forward price of an asset is too low? Assume that the asset provides no income.

The trader should short the asset, invest the proceeds of the short sale at the risk-free rate, enter into a long forward contract to buy the asset in one year If the forward price is too low relative to the spot price the trader should short the asset in the spot market and buy it in the forward market.

A short forward contract that was negotiated some time ago will expire in three months and has a delivery price of $40. The current forward price for three-month forward contract is $42. The three month risk-free interest rate (with continuous compounding) is 8%. What is the value of the short forward contract?

The value of a short forward contract is -(F - K)-rT = -(42-40)e-.08*0.25 = -1.96

A company enters into an interest rate swap where it is paying fixed and receiving LIBOR. When interest rates increase, which of the following is true?

The value of the swap to the company increases It is receiving the floating rate. When interest rates increase the floating rate can be expected to be higher and so the swap becomes more valuable. The answer is therefore A.

A trader has a portfolio worth $5 million that mirrors the performance of a stock index. The stock index is currently 1,250. Futures contract trade on the index with one contract being on 250 times the index. To remove market risk from the portfolio the trader should

To hedge the long stock position, sell $5,000,000/($1,250 x 250) = 16 contracts.

A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. The breakeven stock price above which the trader makes a profit is

Total cost of options = 2x$3 + 1x$4 = $10. Breakeven price on the call option = $30 + $10/2 = $35.

The risk-free yield curve is flat at 6% per annum compounded continuously. What is the value of an FRA where the holder receives LIBOR at the rate of 9% per annum (with semi-annual compounding) for a six-month period on a principal of $1,000 starting in two years? The forward LIBOR rate is 7% (with semiannual compounding).

VFRA = L(RK-RF)(T2-T1)e-R2T2 = 1,000(.09-.07)*0.5*e-.06*2.5 = 8.61

A company can invest funds for three years at LIBOR minus 25 basis points. The three-year swap rate is 3.25%. What fixed rate of interest can the company earn by using the swap?

When a company invests at LIBOR minus 0.25% and enters into a swap where it pays LIBOR and receives 3.25%, it earns (LIBOR - 0.25%) - LIBOR + 3.25% = 3.0% per annum.