Chapter 20, Volatility smiles Video Solutions, Options, Futures, and Other Derivatives

Problem 1

What volatility smile is likely to be observed when:
(a) Both tails of the stock price distribution are less heavy than those of the lognormal distribution?
(b) The right tail is heavier, and the left tail is less heavy, than that of a lognormal distribution?

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Problem 2

What volatility smile is observed for equities?

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Problem 3

What volatility smile is likely to be caused by jumps in the underlying asset price? Is the pattern likely to be more pronounced for a 2-year option than for a 3-month option?

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Problem 4

A European call and put option have the same strike price and time to maturity. The call has an implied volatility of $30 \%$ and the put has an implied volatility of $25 \%$. What trades would you do?

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00:32

Problem 5

Explain carefully why a distribution with a heavier left tail and less heavy right tail than the lognormal distribution gives rise to a downward sloping volatility smile.

William Scherer

Numerade Educator

Problem 6

The market price of a European call is $$\$ 3.00$$ and its price given by Black-ScholesMerton model with a volatility of $30 \%$ is $$\$ 3.50$$. The price given by this Black-ScholesMerton model for a European put option with the same strike price and time to maturity is $$\$ 1.00$$. What should the market price of the put option be? Explain the reasons for your answer.

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01:09

Problem 7

Explain what is meant by "crashophobia."

Ajay Singhal

Numerade Educator

Problem 8

A stock price is currently $$\$ 20$$. Tomorrow, news is expected to be announced that will either increase the price by $$\$ 5$$ or decrease the price by $$\$ 5$$. What are the problems in using Black-Scholes-Merton to value 1-month options on the stock?

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Problem 9

What volatility smile is likely to be observed for 6-month options when the volatility is uncertain and positively correlated to the stock price?

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Problem 10

Explain the problems in testing a stock option pricing model empirically.

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00:48

Problem 11

Suppose that a central bank's policy is to allow an exchange rate to fluctuate between 0.97 and 1.03 . What pattern of implied volatilities for options on the exchange rate would you expect to see?

Majid Borumand

Numerade Educator

Problem 12

Option traders sometimes refer to deep-out-of-the-money options as being options on volatility. Why do you think they do this?

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Problem 13

A European call option on a certain stock has a strike price of $$\$ 30$$, a time to maturity of 1 year, and an implied volatility of $30 \%$. A European put option on the same stock has a strike price of $$\$ 30$$, a time to maturity of 1 year, and an implied volatility of $33 \%$. What is the arbitrage opportunity open to a trader? Does the arbitrage work only when the lognormal assumption underlying Black-Scholes-Merton holds? Explain carefully the reasons for your answer.

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Problem 14

Suppose that the result of a major lawsuit affecting a company is due to be announced tomorrow. The company's stock price is currently $$\$ 60$$. If the ruling is favorable to the company, the stock price is expected to jump to $$\$ 75$$. If it is unfavorable, the stock is expected to jump to $$\$ 50$$. What is the risk-neutral probability of a favorable ruling? Assume that the volatility of the company's stock will be $25 \%$ for 6 months after the ruling if the ruling is favorable and $40 \%$ if it is unfavorable. Use DerivaGem to calculate the relationship between implied volatility and strike price for 6-month European options on the company today. The company does not pay dividends. Assume that the 6-month risk-free rate is $6 \%$. Consider call options with strike prices of $$\$ 30$$, $$\$ 40$$, $$\$ 50$$ , $$\$60$$ ,$$\$ 70$$ , and $$\$ 80$$.

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Problem 15

An exchange rate is currently 0.8000 . The volatility of the exchange rate is quoted as $12 \%$ and interest rates in the two countries are the same. Using the lognormal assumption, estimate the probability that the exchange rate in 3 months will be (a) less than 0.7000 , (b) between 0.7000 and 0.7500 , (c) between 0.7500 and 0.8000 , (d) between 0.8000 and 0.8500 , (e) between 0.8500 and 0.9000 , and (f) greater than 0.9000 . Based on the volatility smile usually observed in the market for exchange rates, which of these estimates would you expect to be too low and which would you expect to be too high?

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Problem 16

A stock price is $$\$ 40$$. A 6-month European call option on the stock with a strike price of $$\$ 30$$ has an implied volatility of $35 \%$. A 6-month European call option on the stock with a strike price of $$\$ 50$$ has an implied volatility of $28 \%$. The 6-month risk-free rate is $5 \%$ and no dividends are expected. Explain why the two implied volatilities are different. Use DerivaGem to calculate the prices of the two options. Use put-call parity to calculate the prices of 6-month European put options with strike prices of $$\$ 30$$ and $$\$ 50$$. Use DerivaGem to calculate the implied volatilities of these two put options.

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Problem 17

"The Black-Scholes-Merton model is used by traders as an interpolation tool." Discuss this view.

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Problem 18

Using Table 20.2, calculate the implied volatility a trader would use for an 8-month option with $K / S_0=1.04$.

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00:54

Problem 19

A company's stock is selling for $$\$ 4$$. The company has no outstanding debt. Analysts consider the liquidation value of the company to be at least $$\$ 300,000$$ and there are 100,000 shares outstanding. What volatility smile would you expect to see?

Lily An

Numerade Educator

Problem 20

A company is currently awaiting the outcome of a major lawsuit. This is expected to be known within 1 month. The stock price is currently $$\$ 20$$. If the outcome is positive, the stock price is expected to be $$\$ 24$$ at the end of 1 month. If the outcome is negative, it is expected to be $$\$ 18$$ at this time. The 1 -month risk-free interest rate is $8 \%$ per annum.
(a) What is the risk-neutral probability of a positive outcome?
(b) What are the values of 1 -month call options with strike prices of $$\$ 19$$, $$\$ 20$$, $$\$ 21$$, $$\$ 22$$, and $$\$ 23$$ ?
(c) Use DerivaGem to calculate a volatility smile for 1-month call options.
(d) Verify that the same volatility smile is obtained for 1-month put options.

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Problem 21

A futures price is currently $$\$ 40$$. The risk-free interest rate is $5 \%$. Some news is expected tomorrow that will cause the volatility over the next 3 months to be either $10 \%$ or $30 \%$. There is a $60 \%$ chance of the first outcome and a $40 \%$ chance of the second outcome. Use DerivaGem to calculate a volatility smile for 3-month futures options.

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00:25

Problem 22

Data for a number of foreign currencies are provided on the author's website: http://www-2.rotman.utoronto.ca/ hull/data
Choose a currency and use the data to produce a table similar to Table 20.1.

Michelle Nguyen

Numerade Educator

00:47

Problem 23

Data for a number of stock indices are provided on the author's website: http://www-2.rotman.utoronto.ca/ hull/data
Choose an index and test whether a three-standard-deviation down movement happens more often than a three-standard-deviation up movement.

Sheryl Ezze

Numerade Educator

Problem 24

Consider a European call and a European put with the same strike price and time to maturity. Show that they change in value by the same amount when the volatility increases from a level $\sigma_1$ to a new level $\sigma_2$ within a short period of time. (Hint: Use put-call parity.)

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04:10

Problem 25

An exchange rate is currently 1.0 and the implied volatilities of 6-month European options with strike prices $0.7,0.8,0.9,1.0,1.1,1.2,1.3$ are $13 \%, 12 \%, 11 \%, 10 \%, 11 \%, 12 \%$, $13 \%$. The domestic and foreign risk-free rates are both $2.5 \%$. Calculate the implied probability distribution using an approach similar to that used for Example 20A.1 in the appendix to this chapter. Compare it with the implied distribution where all the implied volatilities are $11.5 \%$.

Narayan Hari

Numerade Educator

Problem 26

Using Table 20.2 , calculate the implied volatility a trader would use for an 11-month option with $K / S_0=0.98$.

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