Hull J.C. Risk management and Financial institutions

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2.17. (a) The trader buys a 180-day call option and takes a short position

in a 180-day forward contract. (b) The trader buys 90-day put options

and takes a long position in a 90-day forward contract.

2.18. It enters into a 5-year swap where it pays 6.51% and receives

LIBOR. Its investment is then at LIBOR minus 1.51%.

2.19. It enters into a 5-year swap where it receives 6.47% and pays

LIBOR. Its net cost of borrowing is LIBOR +0.53%.

2.20. It enters into a three-year swap where it receives LIBOR and pays

6.24%. Its net borrowing cost for the three years is then 7.24% per

annum.

2.21. Suppose that the weather is bad and the farmer's production is

lower than expected. Other farmers are likely to have been affected

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2.14. A long position in a four-month put option can provide insurance

against the exchange rate falling below the strike price. It ensures that the

foreign currency can be sold for at least the strike price.

2.15. The company could enter into a long forward contract to buy

1 million Canadian dollars in six months. This would have the effect of

locking in an exchange rate equal to the current forward exchange rate.

Alternatively the company could buy a call option giving it the right (but

not the obligation) to purchase 1 million Canadian dollars at a certain

exchange rate in six months. This would provide insurance against a

strong Canadian dollar in six months while still allowing the company

to benefit from a weak Canadian dollar at that time.

2.16. The payoff from an ICON is the payoff from (a) a regular bond,

(b) a short position in call options to buy 169,000 yen with an exercise

price of 1/169, (c) a long position in call options to buy 169,000 yen with

an exercise price of 1/84.5. This is demonstrated by the following table:

428 Answers to Problems and Questions

similarly. Corn production overall will be low and as a consequence the

price of corn will be relatively high. The farmer is likely to be overhedged

relative to actual production. The farmer's problems arising from the bad

harvest will be made worse by losses on the short futures position. This

problem emphasizes the importance of looking at the big picture when

hedging. The farmer is correct to question whether hedging price risk

while ignoring other risks is a good strategy.

2.22. It may well be true that there is just as much chance that the

company will lose as that it will gain. This means that the use of a futures

contract for speculation would be like betting on whether a coin comes up

heads or tails. But it might make sense for the airline to use futures for

hedging rather than speculation. The futures contract then has the effect

of reducing risks. It can be argued that an airline should not expose its

shareholders to risks associated with the future price of oil when there are

contracts available to hedge the risks.

2.23. The optimal hedge ratio is

The beef producer requires a long position in 200,000 x 0.6 = 120,000 lbs

of cattle. The beef producer should therefore take a long position in three

December contracts closing out the position on November 15.

2.24. Microsoft is choosing an option on a portfolio of assets instead of

the corresponding portfolio of options. The former is always less expen-

sive because there is the potential for an increase in the price of one asset

to be netted off against a decrease in the price of another. Compare (a) an

option with a strike price of $20 on a portfolio of two assets each worth

$10 and (b) a portfolio of two options with a strike price of $10, one on

each of assets. If both assets increase in price or both assets decrease in

price, the payoffs are the same. But if one decreases and the other

increases, the payoff from (a) is less than that from (b). Both the Asian

feature and the basket feature in Microsoft's options help to reduce the

cost of the options because of the possibility of gains and loss being

netted.

CHAPTER 3

3.1. The value of the portfolio decreases by $10,500.

3.2. The value of the portfolio increases by $400.

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3.3. In both cases it increases by 0.5 x 30 x 2

, or $60.

3.4. A delta of 0.7 means that, when the price of the stock increases by a

small amount, the price of the option increases by 70% of this amount.

Similarly, when the price of the stock decreases by a small amount, the

price of the option decreases by 70% of this amount. A short position in

1,000 options has a delta of —700 and can be made delta neutral with the

purchase of 700 shares.

3.5. A theta of — 100 per day means that if one day passes with no change

in either the stock price or its volatility, the value of the option position

declines by $100. If a trader feels that neither the stock price nor its

implied volatility will change, she should write an option with as high a

theta as possible. Relatively short-life at-the-money options have the

highest theta.

3.6. The gamma of an option position is the rate of change of the delta of

the position with respect to the asset price. For example, a gamma of 0.1

would indicate that, when the asset price increases by a certain small

amount, delta increases by 0.1 of this amount. When the gamma of an

option-writer's position is large and negative and the delta is zero, the

option writer will lose significant amounts of money if there is a large

movement (either an increase or a decrease) in the asset price.

3.7. To hedge an option position, it is necessary to create the opposite

option position synthetically. For example, to hedge a long position in a

put, it is necessary to create a short position in a put synthetically. It

follows that the procedure for creating an option position synthetically is

the reverse of the procedure for hedging the option position.

3.8. A long position in either a put or a call option has a positive gamma.

From Figure 15.8, when gamma is positive the hedger gains from a large

change in the stock price and loses from a small change in the stock price.

Hence the hedger will fare better in case (b). When the portfolio contains

short option position, the hedger will similarly fare better in (a).

3.9. The delta indicates that, when the value of the euro exchange rate

increases by $0.01, the value of the bank's position increases by

0.01 x 30,000 = $300. The gamma indicates that, when the euro exchange

rate increases by $0.01, the delta of the portfolio decreases by

0.01 x 80,000 = 800. For delta neutrality, 30,000 euros should be shorted.

When the exchange rate moves up to 0.93, we expect the delta of the

portfolio to decrease by (0.93 - 0.90) x 80,000 = 2,400, so that it becomes

27,600. To maintain delta neutrality, it is therefore necessary for the bank

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to unwind its short position 2,400 euros so that a net 27,600 have been

shorted. When a portfolio is delta neutral and has a negative gamma, a loss

is experienced when there is a large movement in the underlying asset price

We can conclude that the bank is likely to have lost money.

3.10. When used in the way described in the text, it does assume volatility

is constant. In theory, we could implement a static options replication

scheme where there are three dimensions: time, the stock price, and volatil-

ity. Prices are then matched on a surface in the three-dimensional space.

3.11. Ten regular options are likely to be needed. This is because there are

ten equations to be satisfied, one for each point on the boundary.

3.12. The payoff from an Asian option becomes more certain with the

passage of time. As a result, the amount of uncertainty that needs to be

hedged decreases with the passage of time.

3.13. Consider a portfolio of options dependent on a single market

variable. A single trade is all that is necessary to make the position delta

neutral, regardless of the size of the position.

3.14. The price, delta, gamma, vega, theta, and rho are 0.0217, -0.396,

5.415, 0.00203, -0.0000625, and -0.00119. Delta predicts that the option

price should decrease by approximately 0.000396 when the exchange rate

increases by 0.001. This is what we find. When the exchange rate is

increased to 0.751, the option price decreases to 0.0213.

CHAPTER 4

4.1. (a) 13.76% per annum, (b) 14.75% per annum.

4.2. (a) 10% per annum, (b) 9.76% per annum, (c) 9.57%> per annum,

(d) 9.53%o per annum.

4.3. The equivalent rate of interest with quarterly compounding is

12.18%o. The amount of interest paid each quarter is therefore

or $304.55.

4.4. The rate of interest is 14.91%) per annum.

4.5. The forward rates with continuous compounding for the 2nd, 3rd,

4th, and 5th years are 4.0%, 5.1%, 5.7%, and 5.7%, respectively.

4.6. The forward rates with continuous compounding for the 2nd, 3rd,

4th, 5th, and 6th quarters are 8.4%, 8.8%, 8.8%, 9.0%, and 9.2%,

respectively.

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4.7. When the term structure is upward sloping, c > a > b. When it is

downward sloping, b > a > c.

4.8. Suppose the bond has a face value of $100. Its price is obtained by

discounting the cash flows at 10.4%. The price is

If the 18-month zero rate is R, we must have

which gives R = 10.42%.

4.9. The bond pays $2 in 6, 12, 18, and 24 months, and $102 in

30 months. The cash price is

4.10. The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in

36 months. The bond yield is the value of y that solves

Using the Goal Seek tool in Excel, we get y = 0.06407, or 6.407%.

4.11. There are three reasons: (i) Treasury bills and Treasury bonds must

be purchased by financial institutions to fulfill a variety of regulatory

requirements. This increases demand for these Treasury instruments driv-

ing the price up and the yield down. (ii) The amount of capital a bank is

required to hold to support an investment in Treasury bills and bonds is

substantially smaller than the capital required to support a similar invest-

ment in other very-low-risk instruments. (iii) In the United States, Treas-

ury instruments are given a favorable tax treatment compared with most

other fixed-income investments because they are not taxed at the state level.

4.12. Duration provides information about the effect of a small parallel

shift in the yield curve on the value of a bond portfolio. The percentage

decrease in the value of the portfolio equals the duration of the portfolio

multiplied by the amount by which interest rates are increased in the

small parallel shift. Its limitation is that it applies only to parallel shifts in

the yield curve that are small.

4.13. (a) The bond's price is 86.80, (b) the bond's duration is 4.256 years,

bond's price increases by 0.74, (d) recomputing the bond's price with a

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yield of 10.8% gives a price of 87.54, which is approximately consistent

with (a) and (c).

4.14. (a) The bond's price is 88.91, (b) the bond's modified duration is

3.843 years, (c) the duration formula estimates that when the yield

decreases by 0.2% the bond's price increases by 0.68, (d) recomputing

the bond's price with a yield of 10.8% (annually compounded) gives a

price of 89.60 which is approximately consistent with (a) and (c).

4.15. The bond price is 104.80. The duration of the bond is 5.35. The

convexity is 30.60. The effect of a 1 % increase in the yield is estimated by

equation (4.14) as

104.80(-0.01 x 5.35 + 0.5 x 30.60 x 0.0001) = -5.44

The bond price actually changes to 99.36, which is consistent with the

estimate.

4.16. We can (a) perturb points on the yield curve (see Figure 4.4),

(b) perturb sections of the yield curve (see Figure 4.6), and (c) perturb

the market quotes used to create the yield curve.

4.17. The deltas are 10.7 and -190,1.

CHAPTER 5

5.1. 4.16%.

5.2. The standard deviation of the percentage price change in one day is

1.57%. The 95% confidence limits are from -3.09% to +3.09%.

5.3. Volatility is much higher when markets are open than when they are

closed. Traders therefore measure time in trading days rather than

calendar days when applying volatility.

5.4. Implied volatility is the volatility that leads to the option price

equaling the market price when Black-Scholes assumptions are used. It

is found by "trial and error". Because different options have different

implied volatilities, traders are not using the same assumptions as Black-

Scholes. (See Chapter 15 for a further discussion of this.)

5.5. The approach in Section 5.3 gives 0.547% per day. The simplified

approach in equation 5.4 gives 0.530%) per day.

5.6. (a) 0.25%, (b) 0.0625%.

5.7. The variance rate estimated at the end of day n equals times the

variance rate estimated at the end of day n — 1 plus 1 — times the

squared return on day n.

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433

5.8. GARCH(1, 1) adapts the EWMA model by giving some weight to a

long-run average variance rate. Whereas the EWMA has no mean rever-

sion, GARCH(1,1) is consistent with a mean-reverting variance rate

model.

5.9. In this case, = 0.015 and = 0.5/30 = 0.01667, so that equa-

tion (19.7) gives

= 0.94 x 0.015

+ 0.06 x 0.01667

= 0.0002281

The volatility estimate on day n is therefore = 0.015103, or

1.5103%.

5.10. Reducing from 0.95 to 0.85 means that more weight is put on

recent observations of and less weight is given to older observations.

Volatilities calculated with — 0.85 will react more quickly to new

information and will "bounce around" much more than volatilities

calculated with

— 0.95.

5.11. With the usual notation, = 20/1040 = 0.01923, so that

= 0.000002 + 0.06 x 0.01923

+ 0.92 x 0.01

= 0.0001162

This gives = 0.01078. The new volatility estimate is therefore 1.078%

per day.

5.12. The proportional daily change is -0.005/1.5000 = -0.003333. The

current daily variance estimate is 0.006 = 0.000036. The new daily

variance estimate is

0.9 x 0.000036 + 0.1 x 0.003333

= 0.000033511

The new daily volatility is the square root of this. It is 0.00579, or 0.579%.

5.13. The weight given to the long-run average variance rate is

and the long-run average variance rate is Increasing

increases the long-run average variance rate; increasing increases the

weight given to the most recent data item, reduces the weight given to the

long-run average variance rate, and increases the level of the long-run

average variance rate. Increasing increases the weight given to the

previous variance estimate, reduces the weight given to the long-run

average variance rate, and increases the level of the long-run average

variance rate.

5.14. The long-run average variance rate is or

0.000004/0.03 = 0.0001333. The long-run average volatility is

or 1.155%. The equation describing the way the variance

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rate reverts to its long-run average is

In this case,

If the current volatility is 20% per year, = 0.0126. The

expected variance rate in 20 days is

0.0001333 + 0.97

(0.0126

- 0.0001333) = 0.0001471

The expected volatility in 20 days is therefore =0.0121, or

1.21% per day.

5.15. The FTSE expressed in dollars is XY where X is the FTSE

expressed in sterling and Y is the exchange rate (value of one pound in

dollars). Define as the proportional change in X on day i and as the

proportional change in Y on day i. The proportional change in XY is

approximately The standard deviation of is 0.018 and the

standard deviation of is 0.009. The correlation between the two is

0.4. The variance of is therefore

0.018

+ 0.009

+ 2 x 0.018 x 0.009 x 0.4 = 0.0005346

so that the volatility of is 0.0231, or 2.31 %. This is the volatility of

the FTSE expressed in dollars. Note that it is greater than the volatility of

the FTSE expressed in sterling. This is the impact of the positive correla-

tion. When the FTSE increases, the value of sterling measured in dollars

also tends to increase. This creates an even bigger increase in the value of

FTSE measured in dollars. Similarly for a decrease in the FTSE.

5.16. In this case, V

= 0.00015 and the expected variance rate in 30 days

is 0.000123. The volatility is 1.11% per day.

5.17. In equation (5.15), V

=0.0001, a = 0.0202, T = 20, and V(0) =

0.000169, so that the volatility is 19.88%.

CHAPTER 6

6.1. You need the standard deviations of the two variables.

6.2. Loosely speaking correlation measures the extent of linear depen-

dence. It does not measure other types of dependence. When y = x

there

is perfect dependence between x and y. However E(xy) = E(x

). This is

zero for a symmetrical distribution such as the normal showing that the

coefficient of correlation between x and y is zero.

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6.3. In a factor model the correlation between two variables arises

entirely because of their correlation with one or more other variables.

The latter are known as factors. A factor model reduces the number of

estimates that have to be made when correlations between large numbers

of variables are being produced.

6.4. A Positive-semidefinite matrix is a matrix that satisfies equation (6.4)

for all vectors If a correlation matrix is not positive semidennite, the

correlations are internally inconsistent.

6.5. (a) The volatilities and correlation imply that the current estimate of

the Covariance is 0.25 x 0.016 x 0.025 = 0.0001. (b) If the prices of the

assets at close of trading are $20.5 and $40.5, the proportional changes

are 0.5/20 = 0.025 and 0.5/40 = 0.0125. The new Covariance estimate is

0.95 x 0.0001 + 0.05 x 0.025 x 0.0125 = 0.0001106

The new variance estimate for asset A is

0.95 x 0.016

+ 0.05 x 0.025

= 0.00027445

so that the new volatility is 0.0166. The new variance estimate for asset B is

0.95 x 0.025

+ 0.05 x 0.0125

= 0.000601562

so that the new volatility is 0.0245. The new correlation estimate is

6.6. Using the notation in the text, = 0.01 and =0.012 and

the most recent estimate of the Covariance between the asset returns is

= 0.01 x 0.012 x 0.50 = 0.00006

The variable = 1/30 = 0.03333 and the variable = 1/50 = 0.02.

The new estimate of the Covariance, cov

, is

0.000001 + 0.04 x 0.03333 x 0.02 + 0.94 x 0.00006 = 0.0000841

The new estimate of the variance of the first asset, is

0.000003 + 0.04 x 0.03333

+ 0.94 x 0.01

= 0.0001414

so that = 0.01189, or 1.189%. The new estimate of the

variance of the second asset, is

0.000003 + 0.04 x 0.02

+ 0.94 x 0.012

= 0.0001544

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so that , = 0.01242, or 1.242%. The new estimate of the

correlation between the assets is therefore

0.0000841/(0.01189 x 0.01242) = 0.569

6.7. Continuing with the notation in Problem 5.15, define as the pro-

portional change in the value of the S&P 500 on day i. The Covariance

between and is 0.7 x 0.018 x 0.016 = 0.0002016. The Covariance

between and is 0.3 x 0.009 x 0.016 = 0.0000432. The Covariance

between and equals the Covariance between and plus the

Covariance between and It is

0.0002016 + 0.0000432 = 0.0002448

The correlation between and is

Note that the volatility of the S&P 500 drops out in this calculation.

6.8.

0.25

0.5

0.75

0.25

0.095

0.163

0.216

0.5

0.163

0.298

0.413

0.75

0.216

0.413

0.595

6.9. Suppose x

, x

, and x

are random samples from three independent

normal distributions. Random samples with the required correlation

structure are where

where

This means that

6.10. Tail dependence is the tendency for extreme values for two or more

variables to occur together. The choice of copula affects tail dependence.