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程序代写案例-A 2021

时间：2021-02-21

Problem Set 6

FIN 424, Spring A 2021

Robert Novy-Marx

This problem set is due before the start of class on Thuresday, February 25. You should

continue to work with your study group, but every one should be involved in answering

each question (this is really practice for the exams, which you will have to do solo, so

please do not just divide the problems up). Each group should hand in one solution. No

question is deliberately designed to trick or confuse you, or be overly complex, so assume

the obvious if you feel that sufficient detail is missing.

Solutions should be submitted via Blackboard before the start of class the day they are

due.

Please think of your product/deliverable as something you might give your employer!

• Please submit a single PDF document.

• Include your names!

Note: The first four (multi-part) problems are last year’s final (it was a three hour exam,

closed book, but with a “cheat-sheet” I provided). This year’s final will be very different

(on-line and multiple choice), but the problems here should give you a good idea of the

sorts of things I expect you to know. You should all work on every problem! The last two

questions are additional fixed-income exercises.

1

1. (30 points total) SHORT ANSWER QUESTIONS

(a) (7 points total) Suppose a stock is trading at 40, and over the next quarter will

either

• go up to 50, or

• go down to 30

$40

$50

$30

Three-month T-bills with $100 faces are trading at $96.

i. (4 points) What’s the market price of a three-month at-the-money call on

the stock?

ii. (3 points) If the stock’s expected return over the next quarter is 7%, what

is the stock volatility implied by the tree?

(b) (7 points total) Consider the following note “linked” to the common stock of

AAP (AAP is currently priced at S0 = $100, and pays no dividends):

• The note pays coupons of $5 in six month and one year;

• In one year (right after paying the second dividend) it converts to AAP

common stock, under the following formula:

1 shares of if ST < 100

100

ST

shares of if 100 < ST < 125

100

125

shares of if ST > 125.

i. (4 points) Draw the note’s payoff diagram, i.e., it’s payoff on the maturity

date, as a function the price of AAP in one year. (Blank payoff diagram

on next page)

ii. (5 points) What is today’s replicating portfolio for the note, using the un-

derlying, zero coupon bonds (possibly with more than one time-to-maturity),

and puts and/or calls on the underlying (possibly with more than one strike).

• That is, what portfolio of bonds, AAP stock, and options on AAP has

exactly the same cash flows as the note?

2

(c) (8 points total) Tell whether each of the following statements is true, false, or

ambiguous.

• In each case provide a short justification for your answer.

– Score based solely on justification!

– No points for unjustified (or wrongly justified) answers.

i. (4 points) A bear spread (i.e., a vertical put spread) written at low strikes

is more expensive than a bear spread written at higher strikes (assume the

same strike-spread, i.e., the difference between the high and low strikes in

each spread is the same).

ii. (4 points) The Black-Scholes formula tells us that the Gamma of a put is

equal to the Gamma of the call with the same strike and time-to-maturity.

In the real world, however, where we observe the volatility “smile,” put op-

tion prices fall slower as the underlying rises, which increases their Gam-

mas. As a result, a deep OTM put option has a slightly higher Gammas

than a calls with the same strike and maturity.

(d) (8 points total) ATM option intuition: Suppose a stock is currently trading at

100, and has a volatility of 50%.

i. (4 points) What’s the approximate price of the ATM option on the stock

with four days to maturity?

3

ii. (4 points) What’s the approximate Theta of this option?

• Hint: How much will its price fall over the next day, holding all else

equal?

2. (20 points total) Black-Scholes and its Extentions

(a) Suppose that a stock, Esco, with a volatility of 56%, is trading at 55. T-bills

with faces of $100 with three- and five-months to maturity are trading at $99.00

and $98.35, respectively.

i. (5 point) If Esco doesn’t pay and dividends, what inputs (S, K, T , r, and

σ) would you use in the Black-Scholes formula to get the price of the five-

month ATM European call on ESCO?

ii. (5 point) If ESCO announces that it will pay a $1 in one quarter, how does

your answer change?

(b) Suppose a stock that pays no dividends is currently trading at 67.03, and has a

volatility of 57.7%. Nine-month T-bills with $100 faces are trading at $97.53

(a continuously compounded rate of 3.33%).

i. (5 points) What is the Delta of the nine-month ATM call option on the

stock?

• Here I would like an explicit calculation, resulting in a number.

ii. (5 points) What is the risk-neutral probability that the stock price exceeds

$100 in nine-months?

• Again, please quantify your answer, providing an explicit number.

3. (20 points total) The Greeks:

Suppose you can trade a stock (S), and three-month European calls on the stock

struck at 250, 300, and 350. The stock pays no dividends, is trading at $312.50,

and has a volatility of 64%. Three-month zero-coupon bonds with faces of $100 are

trading at $99.50 (a continuously compounded yield of r = 2%). The Greeks of each

of the calls are given below:

K 250 300 350

∆ 0.809 0.619 0.429

Γ 2.725 ×10−3 3.810 ×10−3 3.926 ×10−3

Θ -58.04 -79.15 -80.68

ν 44.14 36.81 26.91

ρ 42.58 59.53 61.35

4

(a) (5 point) Use the Black-Scholes PDE (our pricing formula in terms of Greeks)

to calculate the price of the call struck at 300.

• That is, figure out the call price without using the Black-Scholes formula.

(b) (5 points) Suppose you own the call struck at 250. What positions in the call

struck at 350 and the underlying do you need to take to simultaneously gamma

and delta hedge your position?

(c) (5 points) Use put-call parity to calculate the Delta, the Gamma, and the Theta

of a three-month European put on the stock struck at 250.

• Remember, you can think about the forward as its replicating portfolio.

– And the Greeks of a portfolio are the sum of the Greeks of the portfo-

lio’s holdings.

(d) (5 point) Suppose you bought a butterfly on 1,000 shares of the stock (using the

same options provided in the start of the problem). If over the next five days

the underlying falls $30, what will be the approximate change in the value of

your position?

• You can assume interest rates and implied volatility are unchanged.

4. (30 points total) Fixed Income

Suppose you see the following prices for LIBOR-quality zero-coupon bonds ($100

faces):

Maturity (years) Price ($)

0.25 98.85

0.50 97.65

0.75 96.40

1.00 95.15

1.25 93.59

(a) (5 points) What’s the one year forward price of the three month rate?

• I.e., the rate you can lock in today for three month loans starting in one

year.

(b) (5 points) Price the floorlet that:

• Matures in one year.

• Provides the owner protection against the three-month rate falling below a

floor of 7.75%.

5

• On a notional of $100,000.

Assume that the volatility of the three-month rate is 30%.

(c) (5 points) What’s the one-year swap rate?

• For swaps making quarterly swap payments against three-month LIBOR.

(d) (5 points) What’s the market price of a one-year receiver swap, making quar-

terly payments against LIBOR on a notional of $1 million, with a contract fixed

rate of 5.5%?

(e) (5 points) What’s the three-month forward one-year swap rate?

• That is, at what fixed rate could you contract today for entering into a one

year swap (making quarterly payments) in three months.

(f) (5 points) Price the payer swaption that:

• Matures in three months,

• Gives the owner the right to enter into the pay-fixed side of a one-year swap

– Making quarterly payments,

– On a notional of $100,000,

• Is struck at a contract swap rate of 5.5%.

Assume that the volatility of the one-year swap rate is 20%.

• Note: If you were not able to answer part (c), feel free to make whatever

assumptions you need to in order to answer this one. Please clearly state

them and that they are your assumptions.

Additional fixed income problems

5. (Valuing Caps.)

Assume that the 6-month forward rate volatility is 18%, and you see the following

prices for 1-year and 18-month LIBOR-quality zero coupon bonds:

B0,1 = 95.784

B0,1.5 = 93.585.

(a) Use the Black ’76 model to value a one-year caplet on the 6-month T-Bill rate,

with a cap rate of 4.5%, written on a notional principle of $10,000.

(b) What is the value of the one-year floor on 6-month LIBOR? (Same 4.5% strike

(here the floor rate), also written on a notional of $10,000).

6

6. (Valuing Swaps and Swaption)

Suppose you observe the following term structure of interest rates (zero coupon bond

prices, per $100 dollars of face):

Maturity (years) Zero Coupon Bond Prices

0.5 98.5222

1 96.1169

1.5 92.1838

2 87.9913

2.5 84.1973

3 81.0959

(a) What is the three year swap-rate?

• Assume the swaps pay semi-annually, and quote the swap-rate accordingly

(i.e., the annual rate, semi-annually compounding rate).

• It’s a lot easier if you use a spreadsheet (or python).

(b) What’s the value of a three-year payer swap with a notional of $1,000,000 and

a fixed rate of 6% (annualized with semi-annual compounding) worth?

(c) Use Black’s model to value a 1-year swaption on a 2-year, pay-fixed swap struck

at 7.0% on a notional of $10,000.

• That is, how much should you pay for the right, but not the obligation,

to enter into a 2-year pay-fixed swap on $10,000 one year from now at a

swap-rate of 7%?

• Assume the forward rate volatility (i.e., the vol. of the 2-year swap rate) is

20%.

(d) What’s the value of the receiver swaption with the same contractual terms?

(e) What’s the value of a forward start receiver-swap, which makes semi-annual

payments based on a notional of $10,000 and a fixed rate of 7% for two years,

with a start date in one year? (Note: it then makes its first payment in 18

months, one six month period after it starts.)

(f) Which is bigger, the value of the forward-start payer-swap (i.e., the opposite of

your answer to part e), or the difference in value between the payer and receiver

swaptions (i.e., your answers to part c and d)?

7

学霸联盟

FIN 424, Spring A 2021

Robert Novy-Marx

This problem set is due before the start of class on Thuresday, February 25. You should

continue to work with your study group, but every one should be involved in answering

each question (this is really practice for the exams, which you will have to do solo, so

please do not just divide the problems up). Each group should hand in one solution. No

question is deliberately designed to trick or confuse you, or be overly complex, so assume

the obvious if you feel that sufficient detail is missing.

Solutions should be submitted via Blackboard before the start of class the day they are

due.

Please think of your product/deliverable as something you might give your employer!

• Please submit a single PDF document.

• Include your names!

Note: The first four (multi-part) problems are last year’s final (it was a three hour exam,

closed book, but with a “cheat-sheet” I provided). This year’s final will be very different

(on-line and multiple choice), but the problems here should give you a good idea of the

sorts of things I expect you to know. You should all work on every problem! The last two

questions are additional fixed-income exercises.

1

1. (30 points total) SHORT ANSWER QUESTIONS

(a) (7 points total) Suppose a stock is trading at 40, and over the next quarter will

either

• go up to 50, or

• go down to 30

$40

$50

$30

Three-month T-bills with $100 faces are trading at $96.

i. (4 points) What’s the market price of a three-month at-the-money call on

the stock?

ii. (3 points) If the stock’s expected return over the next quarter is 7%, what

is the stock volatility implied by the tree?

(b) (7 points total) Consider the following note “linked” to the common stock of

AAP (AAP is currently priced at S0 = $100, and pays no dividends):

• The note pays coupons of $5 in six month and one year;

• In one year (right after paying the second dividend) it converts to AAP

common stock, under the following formula:

1 shares of if ST < 100

100

ST

shares of if 100 < ST < 125

100

125

shares of if ST > 125.

i. (4 points) Draw the note’s payoff diagram, i.e., it’s payoff on the maturity

date, as a function the price of AAP in one year. (Blank payoff diagram

on next page)

ii. (5 points) What is today’s replicating portfolio for the note, using the un-

derlying, zero coupon bonds (possibly with more than one time-to-maturity),

and puts and/or calls on the underlying (possibly with more than one strike).

• That is, what portfolio of bonds, AAP stock, and options on AAP has

exactly the same cash flows as the note?

2

(c) (8 points total) Tell whether each of the following statements is true, false, or

ambiguous.

• In each case provide a short justification for your answer.

– Score based solely on justification!

– No points for unjustified (or wrongly justified) answers.

i. (4 points) A bear spread (i.e., a vertical put spread) written at low strikes

is more expensive than a bear spread written at higher strikes (assume the

same strike-spread, i.e., the difference between the high and low strikes in

each spread is the same).

ii. (4 points) The Black-Scholes formula tells us that the Gamma of a put is

equal to the Gamma of the call with the same strike and time-to-maturity.

In the real world, however, where we observe the volatility “smile,” put op-

tion prices fall slower as the underlying rises, which increases their Gam-

mas. As a result, a deep OTM put option has a slightly higher Gammas

than a calls with the same strike and maturity.

(d) (8 points total) ATM option intuition: Suppose a stock is currently trading at

100, and has a volatility of 50%.

i. (4 points) What’s the approximate price of the ATM option on the stock

with four days to maturity?

3

ii. (4 points) What’s the approximate Theta of this option?

• Hint: How much will its price fall over the next day, holding all else

equal?

2. (20 points total) Black-Scholes and its Extentions

(a) Suppose that a stock, Esco, with a volatility of 56%, is trading at 55. T-bills

with faces of $100 with three- and five-months to maturity are trading at $99.00

and $98.35, respectively.

i. (5 point) If Esco doesn’t pay and dividends, what inputs (S, K, T , r, and

σ) would you use in the Black-Scholes formula to get the price of the five-

month ATM European call on ESCO?

ii. (5 point) If ESCO announces that it will pay a $1 in one quarter, how does

your answer change?

(b) Suppose a stock that pays no dividends is currently trading at 67.03, and has a

volatility of 57.7%. Nine-month T-bills with $100 faces are trading at $97.53

(a continuously compounded rate of 3.33%).

i. (5 points) What is the Delta of the nine-month ATM call option on the

stock?

• Here I would like an explicit calculation, resulting in a number.

ii. (5 points) What is the risk-neutral probability that the stock price exceeds

$100 in nine-months?

• Again, please quantify your answer, providing an explicit number.

3. (20 points total) The Greeks:

Suppose you can trade a stock (S), and three-month European calls on the stock

struck at 250, 300, and 350. The stock pays no dividends, is trading at $312.50,

and has a volatility of 64%. Three-month zero-coupon bonds with faces of $100 are

trading at $99.50 (a continuously compounded yield of r = 2%). The Greeks of each

of the calls are given below:

K 250 300 350

∆ 0.809 0.619 0.429

Γ 2.725 ×10−3 3.810 ×10−3 3.926 ×10−3

Θ -58.04 -79.15 -80.68

ν 44.14 36.81 26.91

ρ 42.58 59.53 61.35

4

(a) (5 point) Use the Black-Scholes PDE (our pricing formula in terms of Greeks)

to calculate the price of the call struck at 300.

• That is, figure out the call price without using the Black-Scholes formula.

(b) (5 points) Suppose you own the call struck at 250. What positions in the call

struck at 350 and the underlying do you need to take to simultaneously gamma

and delta hedge your position?

(c) (5 points) Use put-call parity to calculate the Delta, the Gamma, and the Theta

of a three-month European put on the stock struck at 250.

• Remember, you can think about the forward as its replicating portfolio.

– And the Greeks of a portfolio are the sum of the Greeks of the portfo-

lio’s holdings.

(d) (5 point) Suppose you bought a butterfly on 1,000 shares of the stock (using the

same options provided in the start of the problem). If over the next five days

the underlying falls $30, what will be the approximate change in the value of

your position?

• You can assume interest rates and implied volatility are unchanged.

4. (30 points total) Fixed Income

Suppose you see the following prices for LIBOR-quality zero-coupon bonds ($100

faces):

Maturity (years) Price ($)

0.25 98.85

0.50 97.65

0.75 96.40

1.00 95.15

1.25 93.59

(a) (5 points) What’s the one year forward price of the three month rate?

• I.e., the rate you can lock in today for three month loans starting in one

year.

(b) (5 points) Price the floorlet that:

• Matures in one year.

• Provides the owner protection against the three-month rate falling below a

floor of 7.75%.

5

• On a notional of $100,000.

Assume that the volatility of the three-month rate is 30%.

(c) (5 points) What’s the one-year swap rate?

• For swaps making quarterly swap payments against three-month LIBOR.

(d) (5 points) What’s the market price of a one-year receiver swap, making quar-

terly payments against LIBOR on a notional of $1 million, with a contract fixed

rate of 5.5%?

(e) (5 points) What’s the three-month forward one-year swap rate?

• That is, at what fixed rate could you contract today for entering into a one

year swap (making quarterly payments) in three months.

(f) (5 points) Price the payer swaption that:

• Matures in three months,

• Gives the owner the right to enter into the pay-fixed side of a one-year swap

– Making quarterly payments,

– On a notional of $100,000,

• Is struck at a contract swap rate of 5.5%.

Assume that the volatility of the one-year swap rate is 20%.

• Note: If you were not able to answer part (c), feel free to make whatever

assumptions you need to in order to answer this one. Please clearly state

them and that they are your assumptions.

Additional fixed income problems

5. (Valuing Caps.)

Assume that the 6-month forward rate volatility is 18%, and you see the following

prices for 1-year and 18-month LIBOR-quality zero coupon bonds:

B0,1 = 95.784

B0,1.5 = 93.585.

(a) Use the Black ’76 model to value a one-year caplet on the 6-month T-Bill rate,

with a cap rate of 4.5%, written on a notional principle of $10,000.

(b) What is the value of the one-year floor on 6-month LIBOR? (Same 4.5% strike

(here the floor rate), also written on a notional of $10,000).

6

6. (Valuing Swaps and Swaption)

Suppose you observe the following term structure of interest rates (zero coupon bond

prices, per $100 dollars of face):

Maturity (years) Zero Coupon Bond Prices

0.5 98.5222

1 96.1169

1.5 92.1838

2 87.9913

2.5 84.1973

3 81.0959

(a) What is the three year swap-rate?

• Assume the swaps pay semi-annually, and quote the swap-rate accordingly

(i.e., the annual rate, semi-annually compounding rate).

• It’s a lot easier if you use a spreadsheet (or python).

(b) What’s the value of a three-year payer swap with a notional of $1,000,000 and

a fixed rate of 6% (annualized with semi-annual compounding) worth?

(c) Use Black’s model to value a 1-year swaption on a 2-year, pay-fixed swap struck

at 7.0% on a notional of $10,000.

• That is, how much should you pay for the right, but not the obligation,

to enter into a 2-year pay-fixed swap on $10,000 one year from now at a

swap-rate of 7%?

• Assume the forward rate volatility (i.e., the vol. of the 2-year swap rate) is

20%.

(d) What’s the value of the receiver swaption with the same contractual terms?

(e) What’s the value of a forward start receiver-swap, which makes semi-annual

payments based on a notional of $10,000 and a fixed rate of 7% for two years,

with a start date in one year? (Note: it then makes its first payment in 18

months, one six month period after it starts.)

(f) Which is bigger, the value of the forward-start payer-swap (i.e., the opposite of

your answer to part e), or the difference in value between the payer and receiver

swaptions (i.e., your answers to part c and d)?

7

学霸联盟

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