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FN426/452: Financial Derivatives Semester 1/2010 Instructor: Satjaporn Tungsong

Solution to Problem Set #1

1. Suppose you want to sell a share of stock that has price Bt 100 at time 0. At time 0 you

agree to a price, which is paid either today or at time T. The share is delivered either at

time 0 or T. The interest rate is r. Fill in the following table:

2. A 50-Baht stock pays 1-Baht dividend every 3 months, with the first dividend coming

3 months from today. The continuously compounded risk-free rate is 6%.

a. What is the price of a prepaid forward contract that expires 1 year from today,

immediately after the fourth-quarter dividend?

 

F0,1 = 50*e0.06*1 -1 -1*e0.06*0.25 -1*e0.06*0.5 -1*e0.06*0.75 = Bt 49.0002

FP0,1 = 49.0002*e-0.06*1 = Bt 46.1467

Description Receive payment

at Time

Deliver Security

at Time

Payment

Received (Bt)

1. Outright purchase 0 0 100 at time 0

2. Fully leverage purchase T 0 100erT at T

3. Prepaid forward contract 0 T ?

4. Forward contract

T T ?*erT

0.25 0 0.5 0.75 1

1 1 1 1

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b. What is the price of a forward contract that expires at the same time?

F0,1 = 50*e0.06*1 -1 -1*e0.06*0.25 -1*e0.06*0.5 -1*e0.06*0.75 = Bt 49.0002

3. A 50-Baht stock pays an 8% continuous dividend. The continuously compounded

risk-free rate is 6%.

a. What is the price of a prepaid forward contract that expires 1 year from today?

FP0,1 = 50*e-0.08*1 = Bt 46.1558

b. What is the price of a forward contract that expires at the same time?

F0,T = S0 * e (r-δ)T

F0,1 = 50*e(0.06-0.08)*1 = Bt 49.0099

4. Suppose the stock price is BHT 35 and the continuously compounded interest rate is

5%.

a. What is the 6-month forward price, assuming dividends are zero?

Bt 35.886

b. If the 6-month forward price is BHT 35.50, what is the annualized continuous

dividend yield?

F0,T = S0 * e (r-δ)T

35.50 = 35* e (0.05-δ)*0.5

δ = 0.0216 = 2.16%

5. Suppose you are a market-maker in S&R index forward contracts. The S&R index

spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0.

a. What is the no-arbitrage forward price for delivery in 9 months?

1142.02

b. Suppose a customer wishes to enter a short index futures position. If you take

the opposite position, demonstrate how you would hedge your resulting long

position using the index and borrowing or lending.

Description Today In 9 months

Long forward, resulting from 0 ST - F0,T

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customer purchase

Sell short the index S0 - ST

Lend + S0, the proceeds from

short-selling

- S0 S0*erT

Total 0 S0*erT- F0,T

With the numbers given in the problem:

Description Today In 9 months

Long forward, resulting from

customer purchase

0 ST - 1,142.02

Sell short the index 1,100 - ST

Lend + S0, the proceeds from

short-selling

- 1,100 1,100*e0.5*0.75

=1,142.02

Total 0 0

We have perfect hedge.

c. Suppose a customer wishes to enter a long index futures position. If you take

the opposite position, demonstrate how you would hedge your resulting short

position using the index and borrowing or lending.

6. The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous

dividend yield on the index is 2%.

a. Suppose you observe a 6-month forward price of 1120. What arbitrage would

you undertake?

The forward price implied from cost-of-carry model is

F0,0.5 = S0*e(r-δ)T = 1,101.652

Thus, the forward in the market is too expensive. To make arbitrage

profit, you sell the forward at 1,120 and borrow S0*e-δT =1,100*e0.02*0.5 =

1,101.101 to buy a fraction of the index in the spot market.

Transaction Cash flows

Time 0 Time T

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Borrow S0*e-δT 1,101.101 -1,101.101* e(0.05-0.02)*0.5

= -1,117.75

Buy stock -1,101.101 ST

Short forward 0 1,120- ST

Total 0 2.25

b. Suppose you observe a 6-month forward price of 1100. What arbitrage would

you undertake?

7. Suppose the SET50 index futures price is currently 500. You wish to purchase 10

futures contracts on margin.

a. What is the notional value of your position?

= 500*1000*10 = Bt 5,000,000

b. Assuming a 10% initial margin, what is the value of the initial margin?

= 0.1* 5,000,000 = Bt 500,000

c. Suppose you earn a continuously compounded rate of 6% on your margin

balance, your position is marked to market weekly, and the maintenance

margin is 80% of the initial margin. What is the greatest SET50 index futures

price 1 week from today at which you receive a margin call?

You earn interest on Bt 500,000 in the first week

Your balance at the end of the first week is

500,000*e(0.06*7/52) + (gain or loss on futures price*1000*10)

= 504,054.81 + *1000*10*(S1-500)

You will receive a margin call if your balance falls below

0.8*500,000 = 400,000

Thus, 400,000 = 504,054.81 + *1000*10*(S1-500)

S1 = 489.59

8. Suppose the SET50 index is 800, and that the dividend yield is 0. You are an

arbitrageur with a continuously compounded borrowing rate of 5.5% and a

continuously compounded lending rate of 5%.

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a. Suppose there are no transaction fees, show that a cash-and-carry arbitrage is

not profitable if the forward price is less than 845.23

Suppose the forward contract will mature in 1 year. The theoretical

forward price is F0,T = S0*e rT

Highest possible forward price = 800*e0.055*1 = 845.2325

Lowest possible forward price = 800*e0.05*1 = 841.0169

Cash-and-carry arbitrage refers to an arbitrage transaction in which you

buy the underlying asset using the proceeds from the sale of forward (buy spot,

sell forward). Cash-and-carry is profitable when the actual forward price is

above the theoretical forward price. Therefore, the actual forward price has to

be higher than 845.2325 to make a cash-and-carry profit.

b. Suppose there are no transaction fees, show that a reverse cash-and-carry

arbitrage is not profitable if the forward price is greater than 841.02.

Reverse cash-and-carry arbitrage refers to an arbitrage transaction in

which you sell the underlying asset and use the proceeds to buy the forward (sell

spot, buy forward). Reverse cash-and-carry is profitable when the actual

forward price is below the theoretical forward price. Therefore, the actual

forward price has to be lower than 841.0169 to make a reverse cash-and-carry

profit.

9. (Bonus) Suppose the SET50 currently has a level of 875. The continuously

compounded return on a 1-year T-bill is 4.75%. You wish to hedge an $800,000

portfolio that has a beta of 1.0 and a correlation of 1.0 with the SET50.

a. What is a 1-year futures price for the SET50 assuming no dividends?

917.57

b. How many SET50 futures contract should you short to hedge your portfolio?

What return do you expect on the hedged portfolio?

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Short 3.65714 contracts to hedge your portfolio. The return you can

expect is the risk-free rate. Because if you perfectly hedge the position

and your portfolio is now a risk-less investment.

1. (Bonus) Synthetic Replication

Verify that going long a forward contract and lending the present value of the forward

price creates a payoff of one share of stock when:

a. The stock pays no dividends.

b. The stock pays discrete dividends

c. The stock pays continuous dividends.

Solution already given in class

11. ใชขอมูลราคาของ zero-coupon bonds ในตารางขางลางตอบคําถามขอ 1.1 และ 1.2 

 

 

 

 

 

 

 

 

 

 

11.1 จงหาอัตราดอกเบ้ียสําหรับ synthetic FRA loan ท่ีมีอายุ 90 วัน โดยสัญญาเร่ิมตนในวันท่ี 90 และ

เติมคําตอบในตารางท่ีกําหนดให 

หมายเหต:ุ synthetic FRA loan คือ การซ้ือ zero-coupon bonds ท่ีจะครบกําหนดในเวลา t+s บวกกับการ

ขาย zero-coupon bonds ท่ีจะครบกําหนดในเวลา t

 

Days to Maturity Zero-Coupon Bond

Price

90 0.99009

180 0.97943

270 0.96525

360 0.95238

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11.2 จงหาอัตราดอกเบ้ียสําหรับ synthetic FRA loan ท่ีมีอายุ 180 วัน โดยสัญญาเริ่มตนในวันท่ี 180 

 

 

11.3 หากคณุเปนผูจดัการธนาคารและมีคูสัญญาคอืลูกคาที่เปนผูใหกู (คุณเปนผูกู) คุณซ้ือสัญญา FRA 

เพื่อล็อคอัตราดอกเบ้ียไวสําหรับเงินกูจํานวน 10 ลานดอลลาร โดยสัญญาเร่ิมในวนัที่ 270 และครบกําหนดใน

อีก 90 วันหลังจากน้ัน คุณจะ hedge สถานะของคุณไดอยางไร 

 

To hedge, you go long on the FRA and buy/sell zero coupon bonds as shown below:

 

 

12. ผูกูวางแผนกูเงนิ 100 ลานดอลลาร โดยเร่ิมตนในอีก 60 วันขางหนา และครบกําหนดในอีก 150 วันหลังจาก

นั้น ขณะนี้ Implied forward rate (อัตราดอกเบ้ียของสัญญา FRA) สําหรับชวงระยะเวลา 150 วัน เทากับ

2.5% โดยท่ีดอกเบ้ียท่ีแทจริงในชวงท่ีมีการกูยืมอาจจะเปล่ืยนแปลงเปน 2.2% หรือ 2.8%

12.1 หากในอีก 60 วัน อัตราดอกเบ้ียเปน 2.8% ผูกูจะตองจายเทาไรถาสัญญา FRA มีการหักลาง

สถานะในวันท่ี 60 และ ผูกูจะตองจายเทาไรถาสัญญา FRA มีการหักลางสถานะในวันท่ี 210

r0(t, t+s)

r0(90, 180)

r0(90, 270)

r0(90, 360)

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เนื่องจากผูกูไดประโยชนหากดอกเบี้ยเพ่ิมขึ้น ดังนั้นหากมีการหักลางสถานะในวันที่ 60 ผูกูไดรับเงิน

หากมีการหักลางสถานะในวันที่ 210 ผูกูไดรับเงิน

12.2 หากในอีก 60 วัน อัตราดอกเบ้ียเปน 2.2% ผูกูจะตองจายเทาไรถาสัญญา FRA มีการหักลาง

สถานะในวันท่ี 60 และ ผูกูจะตองจายเทาไรถาสัญญา FRA มีการหักลางสถานะในวันท่ี 210

ผูกูเสียประโยชนหากดอกเบ้ียลดลง ดังนั้นหากมีการหักลางสถานะในวันที่ 60 ผูกูตองจายเงนิ = (0.022-0.025)/(1+0.022) * 100,000,000 = -$293,542.07

หากมีการหักลางสถานะในวันที่ 210 ผูกูตองจายเงิน

= (0.022-0.025) * 100,000,000 = -$300,000

13. T-bill ท่ีมีวันครบกําหนดเทากับ 90 วัน และมี face value เทากับ $1,000,000 อัตราดอกเบ้ีย (discount

yield) ของ T-bill นี้เทากับ 8.75% จงหาราคาของ T-bill นี ้

Price = Face*[1 - DR*(t/360)] = $1,000,000*(1-0.0875*(90/360)) = $978,125.00

14. หาก price index ของ T-bill เทากับ 88.70 จงหาอัตราดอกเบ้ีย (discount yield) ของ T-bill นี้ และหาก

คุณซือ้ T-bill futures ในราคาเทากับ price index นั่นคือ 88.70 และไมนาน price index เพิ่มขึ้นเปน

88.90 ถามวา ทานไดกําไรหรือขาดทุน คิดเปนมูลคาเทาไร

DR = [Face - P]/[Face*(360/t)] = (100-88.7)/(100*(360/90)) = 2.825% per 90 days or 11.3% per year

หากซ้ือมาราคา 88.70 และราคาปจจุบันเทากับ 88.90 จะไดกําไร (88.90-88.70)/88.70 = 0.2255%

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15. June T-bill futures มี index value เทากับ 92.80 และ September T-bill futures มี index value

เทากับ 93.00 จงหา implied interest rate ในชวงเดือนมิถุนายนถึงเดือนกันยายน

92.80/93 – 1 = -0.22%

16. Suppose you observe the following zero-coupon bond prices per $1 of maturity payment:

0.96154 (1-year), 0.91573 (2-year), 0.87630 (3-year), 0.87630 (4-year), 0.77611 (5-year). For each maturity year compute the zero-coupon bond yields (effective annual and continuously compounded), the par coupon rate, and the 1-year implied forward rate.

17. Using the information in question 16, find the price of a 5-year coupon bond that has a

par payment of $1,000.00 and annual coupon payments of $60.00. คูณกระแสเงินสดจากคูปองในแตละงวดและเงินตนดวยราคาของ zero-coupon bond ที่เหมาะสม จะได

ราคาหุนกูเทากับ $1,037.2528

18. Suppose that in order to hedge interest rate risk on your borrowing, you enter into an

FRA that will guarantee a 6% effective annual interest rate for 1 year on $500,000.00. On the day you borrow the $500,000.00, the actual interst rate is 5%. Determine the dollar settlement of the FRA:

18.1 If settlement occurs on the date the loan is initiated 18.2 If settlement occurs on the date the loan is repaid

Solution:

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19. What is the yield to maturity of the 10-year zero coupon bond with a face value of $100 and current price $69.20205?

P0 = Face*e-rT $69.20205 = 100*e(-r*10) -r = (1/10)ln(69.20205/100) r = -(1/10)ln(69.20205/100)

r = 3.6814%

20. Suppose that oil forward prices for 1 year, 2 years, and 3 years are $20, $21, and $22.

The 1-year effective annual interest rate is 6%, the 2-year interest rate is 6.5%, and the 3-year interest rate is 7%.

20.1 What is the 3-year swap price?

The present value of the cost per 3 barrels based on the forward price is:

The swap price per barrel is:

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20.2 What is the price of a 2-year swap beginning in one year? (That is, the first swap settlement will be in 2 years and the second in 3 years.) The present value of the cost per 2 barrels based on the forward price is:

The swap price per barrel is:

21. Consider the same 3-year oil swap in question 20. Suppose a dealer is paying the fixed price and receive floating. What position in oil forward contracts will hedge oil price risk in this position? Verify that the present value of the lock-in net ash flows is zero.

Solution:

22. Consider the same 3-year swap in question 20. Suppose you are a dealer who is paying

the fixed oil price and receive the floating price. Suppose that you enter into the swap and immediately thereafter all interest rates rise 50 basis points but oil forward prices are unchanged. What happens to the value of your swap position? What if interest rates fall 50 basis points? What hedging instrument would have protected you against interest rate risk in this position?

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Solution:

ใชขอมูลในตารางขางลางตอบคําถามขอ 23-30

Quarter 1 2 3 4 5 6 7 8

Oil forward price

21 21.1 20.8 20.5 20.2 20 19.9 19.8

Gas swap price

2.25 2.42 2.35 2.24 2.23 2.28 2.26 2.20

Zero-coupon bond price

.9852 .9701 .9546 .9388 .9231 .9075 .8919 .8763

Euro-denominated zero-coupon bond price

.9913 .9825 .9735 .9643 .9551 .9459 .9367 .9274

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Euro forward price ($/€)

.9056 .9115 .9178 .9244 .9312 .9381 .9452 .9524

กําหนดใหอัตราแลกเปล่ียน ณ เวลาปจจุบันเทากับ $/€ 0.9

23. Suppose the effective quarterly interest rate is 1.5%, what are the per-barrel swap prices

for 4-quarter and 8-quarter oil swaps? What is the total cost of prepaid 4- and 8-quarter swaps?

Solution:

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24. Construct the set of swap prices for oil for 1 through 8 quarters.  

25. What is the swap price of a 4-quarter oil swap with the first settlement occurring in the third quarter? Solution:

26. Using the zero-coupon bond prices and oil forward prices in the table provided above, what is the price of an 8-period swap for which two barrels of oil are delivered in even-numbered quarters and one barrel of oil in odd-numbered quarters?

Solution:

27. Using the zero-coupon bond prices and oil forward prices in the table provided above, what are the gas forward prices for each of the 8 quarters?

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Solution:

28. What is the fixed rate in a 5-quarter interest rate swap with the first settlement in quarter 2? Solution:

 

29. What is the fixed rate in a 4-quarter interest rate swap? What is the fixed rate in an 8-

quarter interest rate swap?

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Solution:

30. What are the euro-denominated fixed rates for 4- and 8-quarter swaps? Solution: