solution problem of financial derivative

8/19/2019 Solution Problem of Financial Derivative

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FN426/452: Financial DerivativesSemester 1/2010Instructor: 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

FP

0,1 = 49.0002*e-0.06*1

= Bt 46.1467

DescriptionReceive 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 contractT T ?*e

rT

0.250 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?

FP

0,1 = 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 takethe 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.

TransactionCash 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. ใชขอมลราคาของ zero-coupon bonds ในตารางขางลางตอบคาถามขอ 1.1 และ 1.2

11.1 จงหาอัตราดอกเบ ยสาหรับ synthetic FRA loan ท มอาย 90 วัน โดยสัญญาเร มตนในวันท 90 และ

เตมคาตอบในตารางท กาหนดให หมายเหต: synthetic FRA loan คอ การซ อ zero-coupon bonds ท จะครบกาหนดในเวลา t+s บวกกับการ

ขาย zero-coupon bonds ท จะครบกาหนดในเวลา t

Days to MaturityZero-Coupon Bond

Price

90 0.99009

180 0.97943

270 0.96525

360 0.95238


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11.2 จงหาอัตราดอกเบ ยสาหรับ synthetic FRA loan ท มอาย 180 วัน โดยสัญญาเร มตนในวันท 180

11.3 หากคณเปนผ จัดการธนาคารและมค สัญญาคอลกคาท เปนผ ใหก (คณเปนผ ก ) คณซ อสัญญา FRA

เพ อลอคอัตราดอกเบ ยไวสาหรับเงนก จานวน 10 ลานดอลลาร โดยสัญญาเร มในวันท 270 และครบกาหนดใน

อก 90 วันหลังจากนั น คณจะ hedge สถานะของคณไดอยางไร

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

below:

12. ผ ก วางแผนก เงน 100 ลานดอลลาร โดยเร มตนในอก 60 วันขางหนา และครบกาหนดในอก 150 วันหลังจาก

นั น ขณะน Implied forward rate (อัตราดอกเบ ยของสัญญา FRA) สาหรับช วงระยะเวลา 150 วัน เทากับ

2.5% โดยท ดอกเบ ยท แทจรงในชวงท มการก ยมอาจจะเปล ยนแปลงเปน 2.2% หรอ 2.8%

12.1 หากในอก 60 วัน อัตราดอกเบ ยเปน 2.8% ผ ก จะตองจายเทาไรถาสัญญา FRA มการหักลาง

สถานะในวันท 60 และ ผ ก จะตองจายเทาไรถาสัญญา FRA มการหักลางสถานะในวันท 210

r0(t, t+s)

r0(90, 180)

r0(90, 270)

r0(90, 360)


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เน องจากผ ก ไดประโยชนหากดอกเบ ยเพ มข น ดังนั นหากมการหั กลางสถานะในวันท 60 ผ ก ไดรับเงน

หากมการหักลางสถานะในวันท 210 ผ ก ไดรับเงน

12.2 หากในอก 60 วัน อัตราดอกเบ ยเปน 2.2% ผ ก จะตองจายเทาไรถาสัญญา FRA มการหักลาง

สถานะในวันท 60 และ ผ ก จะตองจายเทาไรถาสัญญา FRA มการหักลางสถานะในวันท 210

ผ ก เสยประโยชนหากดอกเบ ยลดลง ดั งนั นหากมการหักลางสถานะในวันท 60 ผ ก ตองจายเงน = (0.022-0.025)/(1+0.022) * 100,000,000 = -$293,542.07

หากมการหักลางสถานะในวันท 210 ผ ก ตองจายเงน

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

13.

T-bill ท มวันครบกาหนดเทากบั 90 วัน และม face value เทากับ $1,000,000 อัตราดอกเบ ย (discountyield) ของ T-bill น เทากับ 8.75% จงหาราคาของ T-bill น

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 จงหาอัตราดอกเบ ย (discount yield) ของ T-bill น และหาก

คณซ อ T-bill futures ในราคาเทากับ price index นั นคอ 88.70 และไมนาน price index เพ มข นเปน 88.90 ถามวา ทานไดกาไรหรอขาดทน คดเปนมลคาเทาไร

DR = [Face - P]/[Face*(360/t)]

= (100-88.7)/(100*(360/90))

= 2.825% per 90 days or 11.3% per year

หากซ อมาราคา 88.70 และราคาปจจบันเทากั บ 88.90 จะไดกาไร (88.90-88.70)/88.70 = 0.2255%


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

เทากับ 93.00 จงหา implied interest rate ในชวงเดอนมถนายนถงเดอนกันยายน

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.

คณกระแสเงนสดจากคปองในแตละงวดและเงนตนดวยราคาของ zero-coupon bond ท เหมาะสม จะได

ราคาห นก เทากับ $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:

ใชขอมลในตารางขางลางตอบคาถามขอ 23-30

Quarter 1 2 3 4 5 6 7 8

Oil forward

price21 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-denominatedzero-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

กาหนดใหอัตราแลกเปล ยน ณ เวลาปจจบันเทากับ $/ € 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:

solution problem of financial derivative

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