Options, Futures, and Other Derivatives

John C. Hull

Chapter 26 Exotic options - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain the difference between a forward start option and a chooser option.

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Problem 2

Describe the payoff from a portfolio consisting of a floating lookback call and a floating lookback put with the same maturity.

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Consider a chooser option where the holder has the right to choose between a European call and a European put at any time during a 2 -year period. The maturity dates and strike prices for the calls and puts are the same regardless of when the choice is made. Is it ever optimal to make the choice before the end of the 2-year period? Explain your answer.

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Problem 4

Suppose that $c_1$ and $p_1$ are the prices of a European average price call and a European average price put with strike price $K$ and maturity $T, c_2$ and $p_2$ are the prices of a European average strike call and European average strike put with maturity $T$, and $c_3$ and $p_3$ are the prices of a regular European call and a regular European put with strike price $K$ and maturity $T$. Show that $c_1+c_2-c_3=p_1+p_2-p_3$.

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06:06

Problem 5

The text derives a decomposition of a particular type of chooser option into a call maturing at time $T_2$ and a put maturing at time $T_1$. Derive an alternative decomposition into a call maturing at time $T_1$ and a put maturing at time $T_2$.

Nicholas Barvinok

Numerade Educator

Problem 6

Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, $H$, is less than or equal to the strike price, $K$. The second applies to the situation where $H \geqslant K$. Show that the two formulas are the same when $H=K$.

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01:16

Problem 7

Explain why a down-and-out put is worth zero when the barrier is greater than the strike price.

Christopher Stanley

Numerade Educator

Problem 8

Suppose that the strike price of an American call option on a non-dividend-paying stock grows at rate $g$. Show that if $g$ is less than the risk-free rate, $r$, it is never optimal to exercise the call early.

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Problem 9

How can the value of a forward start put option on a non-dividend-paying stock be calculated if it is agreed that the strike price will be $10 \%$ greater than the stock price at the time the option starts?

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Problem 10

If a stock price follows geometric Brownian motion, what process does $A(t)$ follow where $A(t)$ is the arithmetic average stock price between time zero and time $t$ ?

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Problem 11

Explain why delta hedging is easier for Asian options than for regular options.

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Problem 12

Calculate the price of a 1-year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are $$\$ 1,520$$ and $$\$ 16$$, respectively; the risk-free interest rate is $10 \%$ per annum; the volatility of each commodity price is $20 \%$; and the correlation between the two prices is 0.7 . Ignore storage costs.

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04:10

Problem 13

Is a European down-and-out option on an asset worth the same as a European downand-out option on the asset's futures price for a futures contract maturing at the same time as the option?

Narayan Hari

Numerade Educator

Problem 14

Answer the following questions about compound options:
(a) What put-call parity relationship exists between the price of a European call on a call and a European put on a call? Show that the formulas given in the text satisfy the relationship.
(b) What put-call parity relationship exists between the price of a European call on a put and a European put on a put? Show that the formulas given in the text satisfy the relationship.

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Problem 15

Does a floating lookback call become more valuable or less valuable as we increase the frequency with which we observe the asset price in calculating the minimum?

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Problem 16

Does a down-and-out call become more valuable or less valuable as we increase the frequency with which we observe the asset price in determining whether the barrier has been crossed? What is the answer to the same question for a down-and-in call?

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Problem 17

Explain why a regular European call option is the sum of a down-and-out European call and a down-and-in European call. Is the same true for American call options?

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Problem 18

What is the value of a derivative that pays off $$\$ 100$$ in 6 months if an index is greater than 1,000 and zero otherwise? Assume that the current level of the index is 960 , the riskfree rate is $8 \%$ per annum, the dividend yield on the index is $3 \%$ per annum, and the volatility of the index is $20 \%$.

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Problem 19

In a 3-month down-and-out call option on silver futures the strike price is $$\$ 20$$ per ounce and the barrier is $$\$ 18$$. The current futures price is $$\$ 19$$, the risk-free interest rate is $5 \%$, and the volatility of silver futures is $40 \%$ per annum. Explain how the option works and calculate its value. What is the value of a regular call option on silver futures with the same terms? What is the value of a down-and-in call option on silver futures with the same terms?

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Problem 20

A new European-style floating lookback call option on a stock index has a maturity of 9 months. The current level of the index is 400 , the risk-free rate is $6 \%$ per annum, the dividend yield on the index is $4 \%$ per annum, and the volatility of the index is $20 \%$. Use DerivaGem to value the option.

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Problem 21

Estimate the value of a new 6-month European-style average price call option on a nondividend-paying stock. The initial stock price is $$\$ 30$$, the strike price is $$\$ 30$$, the risk-free interest rate is $5 \%$, and the stock price volatility is $30 \%$.

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Problem 22

Use DerivaGem to calculate the value of:
(a) A regular European call option on a non-dividend-paying stock where the stock price is $$\$ 50$$, the strike price is $$\$ 50$$, the risk-free rate is $5 \%$ per annum, the volatility is $30 \%$, and the time to maturity is one year
(b) A down-and-out European call which is as in (a) with the barrier at $$\$ 45$$
(c) A down-and-in European call which is as in (a) with the barrier at $$\$ 45$$.
Show that the option in (a) is worth the sum of the values of the options in (b) and (c).

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Problem 23

Explain adjustments that have to be made when $r=q$ for (a) the valuation formulas for floating lookback call options in Section 26.11 and (b) the formulas for $M_1$ and $M_2$ in Section 26.13 .

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Problem 24

Value the variance swap in Example 26.4 of Section 26.16 assuming that the implied volatilities for options with strike prices $800,850,900,950,1,000,1,050,1,100,1,150$, 1,200 are $20 \%, 20.5 \%, 21 \%, 21.5 \%, 22 \%, 22.5 \%, 23 \%, 23.5 \%, 24 \%$, respectively.

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03:10

Problem 25

Verify that the results in Section 26.2 for the value of a derivative that pays $Q$ when $S=H$ are consistent with those in Section 15.6.

Margaret Keymakh

Numerade Educator

Problem 26

What is the value in dollars of a derivative that pays off $£ 10,000$ in 1 year provided that the dollar/sterling exchange rate is greater than 1.5000 at that time? The current exchange rate is 1.4800 . The dollar and sterling interest rates are $4 \%$ and $8 \%$ per annum, respectively. The volatility of the exchange rate is $12 \%$ per annum.

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Problem 27

Consider an up-and-out barrier call option on a non-dividend-paying stock when the stock price is 50 , the strike price is 50 , the volatility is $30 \%$, the risk-free rate is $5 \%$, the time to maturity is 1 year, and the barrier at $$\$ 80$$. Use the DerivaGem software to value the option and graph the relationship between (a) the option price and the stock price, (b) the delta and the stock price, (c) the option price and the time to maturity, and (d) the option price and the volatility. Provide an intuitive explanation for the results you get. Show that the delta, gamma, theta, and vega for an up-and-out barrier call option can be either positive or negative.

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Problem 28

Sample Application $\mathrm{F}$ in the DerivaGem Application Builder Software considers the static options replication example in Section 26.17. It shows the way a hedge can be constructed using four options (as in Section 26.17) and two ways a hedge can be constructed using 16 options.
(a) Explain the difference between the two ways a hedge can be constructed using 16 options. Explain intuitively why the second method works better.
(b) Improve on the four-option hedge by changing Tmat for the third and fourth options.
(c) Check how well the 16-option portfolios match the delta, gamma, and vega of the barrier option.

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04:10

Problem 29

Consider a down-and-out call option on a foreign currency. The initial exchange rate is 0.90 , the time to maturity is 2 years, the strike price is 1.00 , the barrier is 0.80 , the domestic risk-free interest rate is $5 \%$, the foreign risk-free interest rate is $6 \%$, and the volatility is $25 \%$ per annum. Use DerivaGem to develop a static option replication strategy involving five options.

Narayan Hari

Numerade Educator

Problem 30

Suppose that a stock index is currently 900 . The dividend yield is $2 \%$, the risk-free rate is $5 \%$, and the volatility is $40 \%$. Use the results in Technical Note 27 on the author's website to calculate the value of a 1-year average price call where the strike price is 900 and the index level is observed at the end of each quarter for the purposes of the averaging. Compare this with the price calculated by DerivaGem for a 1-year average price option where the price is observed continuously. Provide an intuitive explanation for any differences between the prices.

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Problem 31

Use the DerivaGem Application Builder software to compare the effectiveness of daily delta hedging for (a) the option considered in Tables 19.2 and 19.3 and (b) an average price call with the same parameters. Use Sample Application C. For the average price option you will find it necessary to change the calculation of the option price in cell $\mathrm{C} 16$, the payoffs in cells H15 and H16, and the deltas (cells G46 to G186 and N46 to N186). Carry out 20 Monte Carlo simulation runs for each option by repeatedly pressing F9. On each run record the cost of writing and hedging the option, the volume of trading over the whole 20 weeks and the volume of trading between weeks 11 and 20. Comment on the results.

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Problem 32

In the DerivaGem Application Builder Software modify Sample Application D to test the effectiveness of delta and gamma hedging for a call on call compound option on a 100,000 units of a foreign currency where the exchange rate is 0.67 , the domestic risk-free rate is $5 \%$, the foreign risk-free rate is $6 \%$, the volatility is $12 \%$. The time to maturity of the first option is 20 weeks, and the strike price of the first option is 0.015 . The second option matures 40 weeks from today and has a strike price of 0.68 . Explain how you modified the cells. Comment on hedge effectiveness.

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Problem 33

Outperformance certificates (also called "sprint certificates," "accelerator certificates," or "speeders") are offered to investors by many European banks as a way of investing in a company's stock. The initial investment equals the stock price, $S_0$. If the stock price goes up between time 0 and time $T$, the investor gains $k$ times the increase at time $T$, where $k$ is a constant greater than 1.0. However, the stock price used to calculate the gain at time $T$ is capped at some maximum level $M$. If the stock price goes down, the investor's loss is equal to the decrease. The investor does not receive dividends.
(a) Show that an outperformance certificate is a package.
(b) Calculate using DerivaGem the value of a one-year outperformance certificate when the stock price is 50 euros, $k=1.5, M=70$ euros, the risk-free rate is $5 \%$, and the stock price volatility is $25 \%$. Dividends equal to 0.5 euros are expected in 2 months, 5 months, 8 months, and 11 months.

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01:32

Problem 34

Carry out the analysis in Example 26.4 of Section 26.16 to value the variance swap on the assumption that the life of the swap is 1 month rather than 3 months.

Manik Pulyani

Numerade Educator

Problem 35

What is the relationship between a regular call option, a binary call option, and a gap call option?

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Problem 36

Produce a formula for valuing a cliquet option where an amount $Q$ is invested to produce a payoff at the end of $n$ periods. The return earned each period is the greater of the return on an index (excluding dividends) and zero.

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