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Options, Futures, and Other Derivatives

John C. Hull

Chapter 19

The Greek letters - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain how a stop-loss trading rule can be implemented for the writer of an out-of-themoney call option. Why does it provide a relatively poor hedge?

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Problem 2

What does it mean to assert that the delta of a call option is 0.7 ? How can a short position in 1,000 options be made delta neutral when the delta of each option is 0.7 ?

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Problem 3

Calculate the delta of an at-the-money six-month European call option on a nondividend-paying stock when the risk-free interest rate is $10 \%$ per annum and the stock price volatility is $25 \%$ per annum.

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Problem 4

What does it mean to assert that the theta of an option position is -0.1 when time is measured in years? If a trader feels that neither a stock price nor its implied volatility will change, what type of option position is appropriate?

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Problem 5

What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is highly negative and the delta is zero?

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01:26

Problem 6

"The procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position." Explain this statement.

Jennifer Stoner
Jennifer Stoner
Numerade Educator

Problem 7

Why did portfolio insurance not work well on October 19,1987 ?

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Problem 8

The Black-Scholes-Merton price of an out-of-the-money call option with an exercise price of $$\$ 40$$ is $$\$ 4$$. A trader who has written the option plans to use a stop-loss strategy. The trader's plan is to buy at $$\$ 40.10$$ and to sell at $$\$ 39.90$$. Estimate the expected number of times the stock will be bought or sold.

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Problem 9

Suppose that a stock price is currently $$\$ 20$$ and that a call option with an exercise price of $$\$ 25$$ is created synthetically using a continually changing position in the stock. Consider the following two scenarios: (a) Stock price increases steadily from $$\$ 20$$ to $$\$ 35$$ during the life of the option; (b) Stock price oscillates wildly, ending up at $$\$ 35 $$. Which scenario would make the synthetically created option more expensive? Explain your answer.

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Problem 10

What is the delta of a short position in 1,000 European call options on silver futures? The options mature in 8 months, and the futures contract underlying the option matures in 9 months. The current 9 -month futures price is $$\$ 8$$ per ounce, the exercise price of the options is $$\$ 8$$, the risk-free interest rate is $12 \%$ per annum, and the volatility of silver futures prices is $18 \%$ per annum.

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Problem 11

In Problem 19.10, what initial position in 9-month silver futures is necessary for delta hedging? If silver itself is used, what is the initial position? If 1 -year silver futures are used, what is the initial position? Assume no storage costs for silver.

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Problem 12

A company uses delta hedging to hedge a portfolio of long positions in put and call options on a currency. Which of the following would give the most favorable result?
(a) A virtually constant spot rate
(b) Wild movements in the spot rate Explain your answer.

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Problem 13

Repeat Problem 19.12 for a financial institution with a portfolio of short positions in put and call options on a currency.

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Problem 14

A financial institution has just sold 1,000 7-month European call options on the Japanese yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is 0.81 cent per yen, the risk-free interest rate in the United States is $8 \%$ per annum, the risk-free interest rate in Japan is $5 \%$ per annum, and the volatility of the yen is $15 \%$ per annum. Calculate the delta, gamma, vega, theta, and rho of the financial institution's position. Interpret each number.

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Problem 15

Under what circumstances is it possible to make a European option on a stock index both gamma neutral and vega neutral by adding a position in one other European option?

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05:04

Problem 16

A fund manager has a well-diversified portfolio that mirrors the performance of the S&P 500 and is worth $$\$ 360$$ million. The value of the S&P 500 is 1,200 , and the portfolio manager would like to buy insurance against a reduction of more than $5 \%$ in the value of the portfolio over the next 6 months. The risk-free interest rate is $6 \%$ per annum. The dividend yield on both the portfolio and the S\&P 500 is $3 \%$, and the volatility of the index is $30 \%$ per annum.
(a) If the fund manager buys traded European put options, how much would the insurance cost?
(b) Explain carefully alternative strategies open to the fund manager involving traded European call options, and show that they lead to the same result.
(c) If the fund manager decides to provide insurance by keeping part of the portfolio in risk-free securities, what should the initial position be?
(d) If the fund manager decides to provide insurance by using 9-month index futures, what should the initial position be?

James Kiss
James Kiss
Numerade Educator
01:08

Problem 17

Repeat Problem 19.16 on the assumption that the portfolio has a beta of 1.5 . Assume that the dividend yield on the portfolio is $4 \%$ per annum.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator

Problem 18

Show by substituting for the various terms in equation (19.4) that the equation is true for:
(a) A single European call option on a non-dividend-paying stock
(b) A single European put option on a non-dividend-paying stock
(c) Any portfolio of European put and call options on a non-dividend-paying stock.

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Problem 19

What is the equation corresponding to equation (19.4) for (a) a portfolio of derivatives on a currency and (b) a portfolio of derivatives on a futures price?

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Problem 20

Suppose that $$\$ 70$$ billion of equity assets are the subject of portfolio insurance schemes. Assume that the schemes are designed to provide insurance against the value of the assets declining by more than $5 \%$ within 1 year. Making whatever estimates you find necessary, use the DerivaGem software to calculate the value of the stock or futures contracts that the administrators of the portfolio insurance schemes will attempt to sell if the market falls by $23 \%$ in a single day.

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04:47

Problem 21

Does a forward contract on a stock index have the same delta as the corresponding futures contract? Explain your answer.

Manasvee Singh
Manasvee Singh
Numerade Educator

Problem 22

A bank's position in options on the dollar/euro exchange rate has a delta of 30,000 and a gamma of $-80,000$. Explain how these numbers can be interpreted. The exchange rate (dollars per euro) is 0.90 . What position would you take to make the position delta neutral? After a short period of time, the exchange rate moves to 0.93 . Estimate the new delta. What additional trade is necessary to keep the position delta neutral? Assuming the bank did set up a delta-neutral position originally, has it gained or lost money from the exchange-rate movement?

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Problem 23

Use the put-call parity relationship to derive, for a non-dividend-paying stock, the relationship between:
(a) The delta of a European call and the delta of a European put
(b) The gamma of a European call and the gamma of a European put
(c) The vega of a European call and the vega of a European put
(d) The theta of a European call and the theta of a European put.

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Problem 24

A financial institution has the following portfolio of over-the-counter options on sterling:
$$
\begin{array}{lrrcc}
\hline \text { Type } & \text { Position } & \begin{array}{c}
\text { Delta } \\
\text { of option }
\end{array} & \begin{array}{c}
\text { Gamma } \\
\text { of option }
\end{array} & \begin{array}{c}
\text { Vega } \\
\text { of option }
\end{array} \\
\hline \text { Call } & -1,000 & 0.50 & 2.2 & 1.8 \\
\text { Call } & -500 & 0.80 & 0.6 & 0.2 \\
\text { Put } & -2,000 & -0.40 & 1.3 & 0.7 \\
\text { Call } & -500 & 0.70 & 1.8 & 1.4 \\
\hline
\end{array}
$$
A traded option is available with a delta of 0.6 , a gamma of 1.5 , and a vega of 0.8 .
(a) What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?
(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral? Assume that all implied volatilities change by the same amount so that vegas can be aggregated.

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Problem 25

Consider again the situation in Problem 19.24. Suppose that a second traded option with a delta of 0.1 , a gamma of 0.5 , and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega neutral?

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Problem 26

Consider a 1-year European call option on a stock when the stock price is $$\$ 30$$, the strike price is $$\$ 30$$, the risk-free rate is $5 \%$, and the volatility is $25 \%$ per annum. Use the DerivaGem software to calculate the price, delta, gamma, vega, theta, and rho of the option. Verify that delta is correct by changing the stock price to $$\$ 30.1$$ and recomputing the option price. Verify that gamma is correct by recomputing the delta for the situation where the stock price is $$\$ 30.1$$. Carry out similar calculations to verify that vega, theta, and rho are correct. Use the DerivaGem Applications Builder functions to plot the option price, delta, gamma, vega, theta, and rho against the stock price for the stock option.

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Problem 27

A deposit instrument offered by a bank guarantees that investors will receive a return during a 6-month period that is the greater of (a) zero and (b) $40 \%$ of the return provided by a market index. An investor is planning to put $$\$ 100,000$$ in the instrument. Describe the payoff as an option on the index. Assuming that the risk-free rate of interest is $8 \%$ per annum, the dividend yield on the index is $3 \%$ per annum, and the volatility of the index is $25 \%$ per annum, is the product a good deal for the investor?

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Problem 28

The formula for the price $c$ of a European call futures option in terms of the futures price $F_0$ is given in Chapter 18 as
$$
c=e^{-r T}\left[F_0 N\left(d_1\right)-K N\left(d_2\right)\right]
$$
Where
$$
d_1=\frac{\ln \left(F_0 / K\right)+\sigma^2 T / 2}{\sigma \sqrt{T}} \quad \text { and } \quad d_2=d_1-\sigma \sqrt{T}
$$
and $K, r, T$, and $\sigma$ are the strike price, interest rate, time to maturity, and volatility, respectively.
(a) Prove that $F_0 N^{\prime}\left(d_1\right)=K N^{\prime}\left(d_2\right)$.
(b) Prove that the delta of the call price with respect to the futures price is $e^{-r T} N\left(d_1\right)$.
(c) Prove that the vega of the call price is $F_0 \sqrt{T} N^{\prime}\left(d_1\right) e^{-r T}$.
(d) Prove the formula for the rho of a call futures option given in Section 19.12.
The delta, gamma, theta, and vega of a call futures option are the same as those for a call option on a stock paying dividends at rate $q$, with $q$ replaced by $r$ and $S_0$ replaced by $F_0$. Explain why the same is not true of the rho of a call futures option.

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01:05

Problem 29

Use DerivaGem to check that equation (19.4) is satisfied for the option considered in Section 19.1. (Note: DerivaGem produces a value of theta "per calendar day." The theta in equation (19.4) is "per year.")

Raj Bala
Raj Bala
Numerade Educator

Problem 30

Use the DerivaGem Application Builder functions to reproduce Table 19.2. (In Table 19.2 the stock position is rounded to the nearest 100 shares.) Calculate the gamma and theta of the position each week. Calculate the change in the value of the portfolio each week and check whether equation (19.3) is approximately satisfied. (Note: DerivaGem produces a value of theta "per calendar day." The theta in equation (19.3) is "per year.")

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