optiix

Assume that the current price of a stock is $100. A call option on that stock with an exercise price of $97 costs $7. A call option on the stock with the same expiration and an exercise price of $103 costs $3. Using these options what is the profit for a long bull spread if the stock price at expiration of the options is equal to $110?

A) -$2.

B) $2.

C) $6.

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The buyer of a bull spread buys the call with an exercise price below the current stock price and sells the call option with an exercise price above the stock price. Therefore, for a stock price of $110 at expiration of the options, he gets a payoff $13 from his long position and a payoff of -$7 from his short position for a net payoff of $6. The cost of the strategy is $4. Hence the profit is equal to $2.

optiix

Which of the following best explains put-call parity?

A) No arbitrage requires that using any three of the four instruments (stock, call, put, bond) the fourth can be synthetically replicated.

B) No arbitrage requires that only the underlying stock can be synthetically replicated using at the money call and put options and a zero coupon bond with a face value equal to the strike price of the options.

C) A stock can be replicated using any call option, put option and bond.

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A portfolio of the three instruments will have the identical profit and loss pattern as the fourth instrument and therefore the same value by no arbitrage. So the fourth security can be synthetically replicated using the remaining three.

optiix

In 30 days, a firm wishes to borrow $15 million for 90 days. The borrowing rate is LIBOR plus 250 basis points. The current LIBOR is 3.8%. The firm buys an interest-rate call that matures in 30 days with a notional principal of $15 million, 90 days in underlying, and a strike rate of 4%. The call premium is $4,000. What is the maximum effective annual rate the firm can anticipate paying?

A) 0.0687.

B) 0.0671.

C) 0.0603.

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First we compute the implied net amount to be borrowed after the cost of the call:

$ 14,995,979 = $15,000,000 − $4,000 × (1 + (0.038 + 0.025) × (30 / 360))

The most the firm will expect to pay is the rate associated with the strike rate: 4% plus the 250 basis-point spread equals 6.5%. This gives the nominal cost of the loan:

$243,750 = $15,000,000 × 0.065 (90 / 360)

The highest effective annual rate is:

0.0687 = ($15,243,750 / $14,995,979)(365/90) − 1

optiix

In 90 days, a firm wishes to borrow $10 million for 180 days. The borrowing rate is LIBOR plus 200 basis points. The current LIBOR is 4%. The firm buys an interest-rate call that matures in 90 days with a notional principal of $10 million, 180 days in underlying, and a strike rate of 4.1%. The call premium is $9,000. What is the effective annual rate of the loan if at expiration LIBOR = 4%?

A) 0.0619.

B) 0.0787.

C) 0.0637.

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The call option is out-of-the-money. The implied net amount to be borrowed after the cost of the call is:

$9,990,865 =$10,000,000 - $9,000 × (1 + (0.04+0.02) × (90/360))

For LIBOR = 0.04 at expiration, the dollar cost is:

$300,000 = $10,000,000 × 0.06 × (180/360)

The effective annual rate is:

0.0637 = ($10,300,000 / $9,990,865)(365/180) - 1

optiix

In 60 days, a bank plans to lend $10 million for 180 days. The lending rate is LIBOR plus 200 basis points. The current LIBOR is 4.5%. The bank buys an interest-rate put that matures in 60 days with a notional principal of $10 million, days in underlying of 180 days, and a strike rate of 4.3%. The put premium is $4,000. What is the effective annual rate of the loan if at expiration LIBOR = 4.1%?

A) 0.0648.

B) 0.0640.

C) 0.0619.

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The effective amount the bank parts with or “lends” at time of the loan is:

$10,004,043 = $10,000,000 + $4,000 × (1 + (0.045 + 0.02) × (60/360))

If LIBOR at maturity equals 4.1%, the payoff of the put would be:

payoff = ($10,000,000) × [max(0, 0.043 – 0.041) × (180/360)

payoff = $10,000

The dollar interest earned is:

$305,000=$10,000,000 × (0.041 + 0.02) × (180/360), and
EAR = [($10,000,000 + $10,000 +$305,000) / ($10,004,043)](365/180) - 1
EAR = 0.0640 or 6.40%

optiix

A firm purchases a collar with floor rate of 3% and a cap rate of 4.4%. The cap and floor have quarterly settlement and a notional principal of $10 million. The maximum outflow and inflow the buyer can expect on a given settlement is (assume equal settlement periods):

A) $110,000 and maximum inflow = $140,000.

B) $75,000 and maximum inflow = $140,000.

C) $75,000 and maximum inflow = infinite.

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Given the possible answers, this must be a collar consisting of a short floor and long cap. The firm’s maximum outflow would occur from the floor when the reference rate is zero: $10,000,000 × (0.03 − 0) / 4 = $75,000. Although interest rates cannot go to infinity, there is no upper limit on what the owner can expect from the cap. Thus “infinite” is the best answer.

optiix

Which of the following is equivalent to a pay-fixed interest rate swap?

A) Buying a cap and selling a floor.

B) Buying a cap and selling an interest rate collar.

C) Selling a cap and buying a floor.

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A pay-fixed interest rate swap has the same payoffs as a long position in the corresponding interest rate collar (with the strike rate equal to the swap fixed rate).

optiix

A firm purchases a one-year cap with a strike rate of 4%, a notional principal of $3 million, and semiannual settlement. The reference rate at the initiation of the cap is 5%, falls to 4.5% at the next settlement and then to 4% one year after the cap’s initiation. The total payoffs (without discounting) over the maturity of the swap would be:

A) $22,792.

B) $25,500.

C) $7,583.

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Since the number of days is not given for each period, approximate it with 182 in the first period and 183 in the second period. Remember that payments are made in arrears.

First payoff = $ 15,167 = $3,000,000 × max(0, 0.05 – 0.04) × (182/360).
Second payoff = $7,625 = $3,000,000 × max(0, 0.045 – 0.04) × (183/360)
Total = $22,792 = $7,625 + $ 15,167

optiix

A manager would delta hedge a position to:

A) earn extra “dividend” income on a given position.

B) earn the risk-free rate.

C) place a floor on the position while leaving the potential for upside risk.

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A delta hedged position should earn the risk-free rate. The position does not earn a “dividend” although it should increase in value gradually (at the risk-free rate). The upside potential is limited to the risk-free rate. The manager would have to constantly monitor and adjust the position to achieve the goal.

optiix

A short position in naked calls on an asset can be delta hedged by:

A) shorting the underlying asset.

B) buying the put.

C) buying the underlying asset.

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Delta hedging a naked call can be accomplished by owning the underlying asset in an amount that will make the value of the short-call/long-asset portfolio immune to changes in the price of the underlying asset.