HullFund8eCh15ProblemSolutions

CHAPTER 15

Options on Stock Indices and Currencies

Practice Questions

Problem 15.8.

Show that the formula in equation (15.9) for a put option to sell one unit of currency A for currency B at strike price K gives the same value as equation (15.8) for a call option to buy K units of currency B for currency A at a strike price of 1K /.

A put option to sell one unit of currency A for K units of currency

B is worth

201()()B A r T r T Ke N d S e N d -----

where 2

1d = 2

2d = and A r and B r are the risk-free rates in currencies A and B, respectively. The value of the

option is measured in units of currency B. Defining 001S S *=/ and 1K K *=/ 2

1d **=

2

2d **= The put price is therefore

0012[()()B A r T r T S K S e N d K e N d --****-

where

2

12d d ***

=-= 2

2

1d d ***=-= This shows that put option is equivalent to 0KS call options to buy 1 unit of currency A for 1K / units of currency B. In this case the value of the option is measured in units of currency

A. To obtain the call option value in units of currency B (the same units as the value of the put option was measured in) we must divide by 0S . This proves the result.

Problem 15.9.

A foreign currency is currently worth $1.50. The domestic and foreign risk-free interest rates are 5% and 9%, respectively. Calculate a lower bound for the value of a six-month call option on the currency with a strike price of $1.40 if it is (a) European and (b) American.

Lower bound for European option is

0090500505015140069f r T rT S e Ke e e ---.?.-.?.-=.-.=. Lower bound for American option is

0010S K -=.

Problem 15.10.

Consider a stock index currently standing at 250. The dividend yield on the index is 4% per annum, and the risk-free rate is 6% per annum. A three-month European call option on the index with a strike price of 245 is currently worth $10. What is the value of a three-month put option on the index with a strike price of 245?

In this case 0250S =, 004q =., 006r =., 025T =., 245K =, and 10c =. Using put –call parity

0rT qT c Ke p S e --+=+

or

0rT qT p c Ke S e --=+-

Substituting:

02500602500410245250384p e e -.?.-.?.=+-=.

The put price is 3.84.

Problem 15.11.

An index currently stands at 696 and has a volatility of 30% per annum. The risk-free rate of interest is 7% per annum and the index provides a dividend yield of 4% per annum. Calculate the value of a three-month European put with an exercise price of 700.

In this case 0696S =, 700K =, 007r =., 03=.σ, 025T =. and 004q =.. The option can be valued using equation (15.5).

12100868000632

d d d ==.=-.=-.

and

12()04654()05252N d N d -=.,-=.

The value of the put, p , is given by:

0070250040257000525269604654406p e e -.?.-.?.=?.-?.=.

i.e., it is $40.6.

Problem 15.12.

Show that if C is the price of an American call with exercise price K and maturity T on a stock paying a dividend yield of q , and P is the price of an American put on the same stock with the same strike price and exercise date,

00qT rT S e K C P S Ke ---<-<-

where 0S is the stock price, r is the risk-free rate, and 0r >. (Hint: To obtain the first half of the inequality, consider possible values of:

Portfolio A; a European call option plus an amount K invested at the risk-free rate

Portfolio B: an American put option plus qT e - of stock with dividends being reinvested in the stock

To obtain the second half of the inequality, consider possible values of:

Portfolio C: an American call option plus an amount rT Ke - invested at the risk-free rate Portfolio D: a European put option plus one stock with dividends being reinvested in the stock)

Following the hint, we first consider

Portfolio A : A European call option plus an amount K invested at the risk-free rate Portfolio B : An American put option plus qT e - of stock with dividends being reinvested in the stock.

Portfolio A is worth c K + while portfolio B is worth 0qT P S e -+. If the put option is exercised at time (0)T ≤<ττ, portfolio B becomes:

()q T K S S e K ---+≤τττ

where S τ is the stock price at time τ. Portfolio A is worth

r c Ke K +≥τ

Hence, portfolio A is worth at least as much as portfolio B. If both portfolios are held to maturity (time T ), portfolio A is worth max(0)max()(1)rT

T rT T S K Ke S K K e -,+=,+-

Portfolio B is worth max()T S K ,. Hence portfolio A is worth more than portfolio B. Because portfolio A is worth at least as much as portfolio B in all circumstances

0qT P S e c K -+≤+

Because c C ≤:

0qT P S e C K -+≤+

or

0qT S e K C P --≤-

This proves the first part of the inequality.

For the second part consider:

Portfolio C : An American call option plus an amount rT Ke - invested at the risk-free rate Portfolio D : A European put option plus one stock with dividends being reinvested in the stock.

Portfolio C is worth rT C Ke -+ while portfolio D is worth 0p S +. If the call option is exercised at time (0)T ≤<ττ portfolio C becomes:

()r T S K Ke S ---+<τττ

while portfolio D is worth

()q t p S e S -+≥τττ

Hence portfolio D is worth more than portfolio C. If both portfolios are held to maturity (time T ), portfolio C is worth max()T S K , while portfolio D is worth )1(),max()0,max(-+=+-qT T T qT T T e S K S e S S K

Hence portfolio D is worth at least as much as portfolio C.

Since portfolio D is worth at least as much as portfolio C in all circumstances:

0rT C Ke p S -+≤+

Since p P ≤:

0rT C Ke P S -+≤+

or

0rT C P S Ke --≤-

This proves the second part of the inequality. Hence:

00qT rT S e K C P S Ke ---≤-≤-

Problem 15.13.

Show that a European call option on a currency has the same price as the corresponding European put option on the currency when the forward price equals the strike price.

This follows from put –call parity and the relationship between the forward price, 0F , and the spot price, 0S 0f r T rT c Ke p S e

--+=+ and ()00f r r T F S e -=

so that

0rT rT c Ke p F e --+=+

If 0K F = this reduces to c p =. The result that c p = when 0K F = is true for options

on all underlying assets, not just options on currencies. An at-the-money option is frequently defined as one where 0K F = (or c p =) rather than one where 0K S =.

Problem 15.14.

Would you expect the volatility of a stock index to be greater or less than the volatility of a typical stock? Explain your answer.

The volatility of a stock index can be expected to be less than the volatility of a typical stock. This is because some risk (i.e., return uncertainty) is diversified away when a portfolio of stocks is created. In capital asset pricing model terminology, there exists systematic and unsystematic risk in the returns from an individual stock. However, in a stock index, unsystematic risk has been diversified away and only the systematic risk contributes to volatility.

Problem 15.15.

Does the cost of portfolio insurance increase or decrease as the beta of a portfolio increases? Explain your answer.

The cost of portfolio insurance increases as the beta of the portfolio increases. This is because portfolio insurance involves the purchase of a put option on the portfolio. As beta increases, the volatility of the portfolio increases causing the cost of the put option to increase. When index options are used to provide portfolio insurance, both the number of options required and the strike price increase as beta increases.

Problem 15.16.

Suppose that a portfolio is worth $60 million and the S&P 500 is at 1200. If the value of the portfolio mirrors the value of the index, what options should be purchased to provide

protection against the value of the portfolio falling below $54 million in one year’s time?

If the value of the portfolio mirrors the value of the index, the index can be expected to have dropped by 10% when the value of the portfolio drops by 10%. Hence when the value of the portfolio drops to $54 million the value of the index can be expected to be 1080. This indicates that put options with an exercise price of 1080 should be purchased. The options should be on: 60000000500001200

$,,=, times the index. Each option contract is for $100 times the index. Hence 500 contracts should be purchased.

Problem 15.17.

Consider again the situation in Problem 15.16. Suppose that the portfolio has a beta of 2.0, the risk-free interest rate is 5% per annum, and the dividend yield on both the portfolio and the index is 3% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $54 million in one year’s time?

When the value of the portfolio falls to $54 million the holder of the portfolio makes a capital loss of 10%. After dividends are taken into account the loss is 7% during the year. This is 12% below the risk-free interest rate. According to the capital asset pricing model, the

expected excess return of the portfolio above the risk-free rate equals beta times the expected excess return of the market above the risk-free rate.

Therefore, when the portfolio provides a return 12% below the risk-free interest rate, the market’s expected return is 6% below the risk -free interest rate. As the index can be assumed to have a beta of 1.0, this is also the excess expected return (including dividends) from the index. The expected return from the index is therefore -1% per annum. Since the index provides a 3% per annum dividend yield, the expected movement in the index is -4%. Thus when the p ortfolio’s value is $54 million the expected value of the index is 0.96×1,200 = 1,152. Hence European put options should be purchased with an exercise price of 1,152. Their maturity date should be in one year.

The number of options required is twice the number required in Problem 15.16. This is because we wish to protect a portfolio which is twice as sensitive to changes in market

conditions as the portfolio in Problem 15.16. Hence options on $100,000 (or 1,000 contracts) should be purchased. To check that the answer is correct consider what happens when the value of the portfolio declines by 20% to $48 million. The return including dividends is -17%. This is 22% less than the risk-free interest rate. The index can be expected to provide a return (including dividends) which is 11% less than the risk-free interest rate, i.e. a return of -6%. The index can therefore be expected to drop by 9% to 1,092. The payoff from the put options is (1,152-1,092)×100,000 = $6 million. This is exactly what is required to restore the value of the portfolio to $54 million.

Problem 15.18.

An index currently stands at 1,500. European call and put options with a strike price of 1,400 and time to maturity of six months have market prices of 154.00 and 34.25, respectively. The six-month risk-free rate is 5%.What is the implied dividend yield?

The implied dividend yield is the value of q that satisfies the put –call parity equation. It is

the value of q that solves

0050505154140034251500q e e -.?.-.+=.+

This is 1.99%.

Problem 15.19.

A total return index tracks the return, including dividends, on a certain portfolio. Explain how you would value (a) forward contracts and (b) European options on the index.

A total return index behaves like a stock paying no dividends. In a risk-neutral world it can be expected to grow on average at the risk-free rate. Forward contracts and options on total return indices should be valued in the same way as forward contracts and options on non-dividend-paying stocks.

Problem 15.20.

What is the put –call parity relationship for European currency options

The put –call parity relationship for European currency options is

f r T rT c Ke p Se --+=+

To prove this result, the two portfolios to consider are:

Portfolio A : one call option plus one zero-coupon domestic bond which will be worth K at time T .

Portfolio B : one put option plus one foreign currency bond that will be worth one unit of the foreign currency at time T .

Both portfolios are worth max()T S K , at time T . They must therefore be worth the same today. The result follows.

Problem 15.21.

Can an option on the yen-euro exchange rate be created from two options, one on the dollar-euro exchange rate, and the other on the dollar-yen exchange rate? Explain your answer.

There is no way of doing this. A natural idea is to create an option to exchange K euros for one yen from an option to exchange Y dollars for 1 yen and an option to exchange K euros for Y dollars. The problem with this is that it assumes that either both options are exercised or that neither option is exercised. There are always some circumstances where the first option is in-the-money at expiration while the second is not and vice versa.

Problem 15.22.

Prove the results in equation (15.1), (15.2), and (15.3) using the portfolios indicated.

In portfolio A, the cash, if it is invested at the risk-free interest rate, will grow to K at time T . If T S K >, the call option is exercised at time T and portfolio A is worth T S . If T S K <, the call option expires worthless and the portfolio is worth K . Hence, at time T , portfolio A is worth

max ()T S K ,

Because of the reinvestment of dividends, portfolio B becomes one share at time T . It is, therefore, worth T S at this time. It follows that portfolio A is always worth as much as, and is sometimes worth more than, portfolio B at time T . In the absence of arbitrage opportunities,

this must also be true today. Hence,

0rT qT c Ke S e --+≥

or

0qT rT c S e Ke --≥- This proves equation (15.1)

In portfolio C, the reinvestment of dividends means that the portfolio is one put option plus one share at time T . If T S K <, the put option is exercised at time T and portfolio C is worth K . If T S K >, the put option expires worthless and the portfolio is worth T S . Hence, at time T , portfolio C is worth

max ()T S K ,

Portfolio D is worth K at time T . It follows that portfolio C is always worth as much as, and is sometimes worth more than, portfolio D at time T . In the absence of arbitrage opportunities, this must also be true today. Hence,

0qT rT p S e Ke --+≥

or

0rT qT p Ke S e --≥-

This proves equation (15.2)

Portfolios A and C are both worth max ()T S K , at time T . They must, therefore, be worth the same today, and the put –call parity result in equation (15.3) follows.

Further Questions

Problem 15.23.

The Dow Jones Industrial Average on January 12, 2007 was 12,556 and the price of the March 126 call was $2.25. Use the DerivaGem software to calculate the implied volatility of this option. Assume that the risk-free rate was 5.3% and the dividend yield was 3%. The option expires on March 20, 2007. Estimate the price of a March 126 put. What is the

volatility implied by the price you estimate for this option? (Note that options are on the Dow Jones index divided by 100.

Options on the DJIA are European. There are 47 trading days between January 12, 2007 and March 20, 2007. Setting the time to maturity equal to 47/252 = 0.1865, DerivaGem gives the implied volatility as 10.23%. (If instead we use calendar days the time to maturity is 67/365=0.1836 and the implied volatility is 10.33%.)

From put call parity (equation 15.3) the price of the put, p , (using trading time) is given by

0053018650030186522512612556e p e -.?.-.?..+=+.

so that 21512p =.. DerivaGem shows that the implied volatility is 10.23% (as for the call). (If calendar time is used the price of the put is 2.1597 and the implied volatility is 10.33% as for the call.)

A European call has the same implied volatility as a European put when both have the same strike price and time to maturity. This is formally proved in the appendix to Chapter 19.

Problem 15.24

A stock index currently stands at 300 and has a volatility of 20%. The risk-free interest rate is 8% and the dividend yield on the index is 3%. Use a three-step binomial tree to value a

six-month put option on the index with a strike price of 300 if it is (a) European and (b) American?

(a) The price is 14.39 as indicated by the tree in Figure S15.1.

(b) The price is 14.97 as indicated by the tree in Figure S15.2

At each node:

Upper value = Underlying Asset Price

Lower value = Option Price

Values in red are a result of early exercise.

Strike price = 300

Discount factor per step = 0.9868

Time step, dt = 0.1667 years, 60.83 days

Growth factor per step, a = 1.0084

0.00000.16670.33330.5000

Figure S15.1 Tree for valuing the European option in Problem 15.24

At each node:

Upper value = Underlying Asset Price

Lower value = Option Price

Values in red are a result of early exercise.

Strike price = 300

Discount factor per step = 0.9868

Time step, dt = 0.1667 years, 60.83 days

Growth factor per step, a = 1.0084

0.00000.16670.33330.5000

Figure S15.2 Tree for valuing the American option in Problem 15.24

Problem 15.25.

Suppose that the spot price of the Canadian dollar is U.S. $0.95 and that the Canadian dollar/U.S. dollar exchange rate has a volatility of 8% per annum. The risk-free rates of interest in Canada and the United States are 4% and 5% per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for U.S. $0.95 in nine months. Use put-call parity to calculate the price of a European put option to sell one Canadian

dollar for U.S. $0.95 in nine months. What is the price of a call option to buy U.S. $0.95 with one Canadian dollar in nine months?

In this case 0095S =., 095K =., 005r =., 004f r =., 008=.σ and 075T =.. The option can be valued using equation (15.8)

12101429000736

d d d ==.=-.=.

and

12()05568()05293N d N d =.,=.

The value of the call, c , is given by

c = 0.95e -0.04×0.75×0.5558?0.95e -0.05×0.75×0.5293 = 0.0290

i.e., it is 2.90 cents. From put –call parity

0f r T rT p S e c Ke --+=+

so that

005912004912002909509500221p e e -.?/-.?/=.+.-.=.

The option to buy US$0.95 with C$1.00 is the same as the same as an option to sell one

Canadian dollar for US$0.95. This means that it is a put option on the Canadian dollar and its price is US$0.0221.

Problem 15.26

The spot price of an index is 1,000 and the risk-free rate is 4%. The prices of three month European call and put options when the strike price is 950 are 78 and 26. Estimate (a) the dividend yield and (b) the implied volatility.

(a) From the formula at the end of Section 15.4

0299.01000

9502678ln 25.0125

.004.0=+--=?-e q

The dividend yield is 2.99%

(b) We can calculate the implied volatility using either the call or the put. The answer (given by DerivaGem) is 24.68% in both cases.

Problem 15.27

The USD/euro exchange rate is 1.3000. The exchange rate volatility is 15%. A US company will have to pay 1 million euros in three months. The euro and USD risk-free rates are 5% and 4%, respectively. The company decides to use a range forward contract with the lower strike equal to 1.2500.

a. What should the higher strike be to create a zero-cost contract?

b. What position in calls and puts should the company take?

c.Show that your answer to (a) does not depend on interest rates provided that the

interest rate differential between the two currencies, r –r f , remains the same.

(a) A put with a strike price of 1.25 is worth $0.019. By trial and error DerivaGem can be used to show that the strike price of a call that leads to a call having a price of $0.019 is

1.3477. This is the higher strike price to create a zero cost contract.

(b) The company should sell a put with strike price 1.25 and buy a call with strike price

1.3477. This ensures that the exchange rate it pays for the euros is between 1.2500 and

1.3477.

(c) If the interest rates change so that the spread between the dollar and euro interest rates remains the same, forward prices remain the same. From equations (15.10) and (15.11). changes to r have the same proportional effect on both c and p. If the relationship c = p holds for one value of r, it holds for all values of r . as a result the answer to (a) is unchanged when the spread between the two rates is held the same.

Problem 15.28

In Business Snapshot 15.1 what is the cost of a guarantee that the return on the fund will not be negative over the next 10 years?

In this case the guarantee is valued as a put option with S0 = 1000, K = 1000, r = 5%, q = 1%, = 15%, and T=10. The value of the guarantee is given by equation (15.5) as 38.46 or 3.8% of the value of the portfolio.

Problem 15.29

The one-year forward price of the Mexican peso is $0.0750 per MXN. The U.S. risk-free rate is 1.25%. The exchange rate volatility is 13%. What is the value of one-year European call and put options with a strike price of 0.0800.

Using equations (15.10) and (15.11) the values of the call and put are 0.0020 and 0.0069, respectively Note that we do not need the Mexican risk-free rate when we use forward prices for the valuation.