Hull: Options, Futures, and Other Derivatives, Ninth Edition Chapter 20: Volatility Smiles Multiple Choie !est "an#: $uestions %ith &ns%ers
1. Which of of the follow following ing is true true of a volat volatility ility smile? smile? A. Implied Implied volatilit volatility y is on the horizontal horizontal axis axis and strike strike price is on the vertical axis B. istorical istorical volatil volatility ity is on the horizontal horizontal axis axis and strike strike price is on the vertical axis !. Implied Implied volatilit volatility y is on the vertical vertical axis and strike strike price price is on the horizontal axis ". istorical volatility volatility is on the vertical vertical axis and strike strike price is on the horizontal axis Answer# ! A volatility smile shows implied volatility $which is on the vertical axis% as a function of the strike price $which is on the horizontal axis%. &. Which Which of the the follo followi wing ng is true? true? A. 'olatility olatility smile smile for (uropean (uropean puts is the same same as for (uropean (uropean calls B. 'olatility olatility smile smile for (uropean (uropean puts is the same same as for American American puts !. 'olatility smile smile for (uropean (uropean calls calls is the same as for for American calls ". 'olatility smile smile for American American puts is the same as for American American calls Answer# A )ut call parity shows that the volatility smile for ( uropean puts should *e exactly the same as that for (uropean calls. +he volatility smile for American options is usually close to *ut not exactly the same as that for (uropean options. ,. Which of the follow following ing is true when the tails tails of a future future foreign foreign currency currency distri*ution are compared with those of a lognormal distri*ution with the same mean and standard deviation? A. +he left left tail tail and right tail are thinner B. +he left left tail is is thinner thinner and the the right tail tail is fatter fatter !. +he right right tail tail is thinner thinner and the the left left tail is fatter fatter ". Both Both tails tails are are fatt fatter er Answer# " Both tails of the th e foreign currency distri*ution are fatter. fatter. +his leads to a - shaped smile. /. Which of the the following following is true true when the tails tails of a future future stock stock price price distri*ution are compared with those of a lognormal distri*ution with the same mean and standard deviation? A. +he left left tail tail and right tail are thinner
B. +he left tail is thinner and the right tail is fatter !. +he right tail is thinner and the left tail is fatter ". Both tails are fatter Answer# ! +he left tail is fatter and the right tail is thinner. 0. Which of the following could cause the volatility smile typically seen for foreign currency options? A. !urrencies are traded in dierent countries at dierent times of the day B. !urrencies tend to have low volatilities !. +he activities of central *anks causes occasional 2umps in the exchange rate ". Interest rates may *e dierent in the two countries Answer# ! +he possi*ility of 2umps in the exchange rate makes extreme exchange rates more likely and is consistent with the volatility smile that is o*served. 3. Which of the following is true? A. +he volatility skew for e4uities is much more pronounced now than it was in 1560. B. +he volatility skew for e4uities has a positive gradient !. +he volatility skew for e4uities is consistent with the Black7choles 8erton model. ". +he volatility skew for e4uities is similar to that for foreign currencies. Answer# A +here was very little volatility skew or smile for e4uities prior to the crash of 1569 9. Why do traders use volatility smiles for pricing options? A. +o allow for nonlognormality of the pro*a*ility distri*ution of future asset price B. Because it is consistent with recent market moves !. As a tool to re:ect their views a*out extreme market moves ". Because extreme market moves are always more likely than Black 7choles8erton assumes Answer# A 'olatility smiles allow for the fact that the assumptions underlying the Black 7choles8erton model do not hold exactly. +he B78 assumptions imply a lognormally distri*uted future asset price.
6. What does the shape of the volatility smile reveal a*out put options on e4uity? A. ;ptions closetothemoney have the lowest implied volatility B. ;ptions deepinthemoney have a relatively high implied volatility !. ;ptions deepoutofthemoney have a relatively high implied volatility ". All of the a*ove Answer# ! +he volatility smile shows that lowstrikeprice options have high implied volatilities relative to atthemoney options. ighstrikeprice options have low implied volatilities relative to atthemoney options. ;ut ofthemoney put options have a low strike price. ence ! is correct. 5. What does the shape of the volatility smile reveal a*out call options on a currency? A. ;ptions closetothemoney have the lowest implied volatility B. ;ptions deepinthemoney have a relatively high implied volatility !. ;ptions deepoutofthemoney have a relatively high implied volatility ". All of the a*ove Answer# " " is correct. +he volatility smile for currency options is -shaped. 1<.Which of the following is =;+ true? A. A volatility surface provides more information than a single volatility smile B. A volatility surface is used to determine the implied volatility of an option that does not trade actively !. A volatility surface can *e determined from a single volatility smile using interpolation ". A volatility surface incorporates information a*out options with dierent maturity dates Answer# ! A volatility surface re4uires a knowledge of how implied volatilities vary with option maturity. ! is therefore not true. 11.A volatility surface is a ta*le showing the relationship *etween which of the following A. Implied volatility> time to maturity> and strike price B. Implied volatility> historical volatility> and time to maturity !. istorical volatility> strike price> and time to maturity ". =one of the a*ove
Answer# A A volatility surface is a lookup ta*le showing implied volatilities as a function of strike price and time to maturity. 1&.Which of the following causes a volatility smile that is a frown? A. +here is a small pro*a*ility of a large stock price decrease in one week B. +here is a small pro*a*ility of a large stock price increase in one week !. +he outcome of a lawsuit $roughly e4ual chance of *eing favora*le or unfavora*le% will create a large movement up or down in one week ". =one of the a*ove Answer# ! As shown at the end of the chapter the situation in ! leads to a @frown 1,.Which of the following could *e a result of @crashopho*ia? A. igh volatilities for inthemoney calls B. igh volatilities for inthemoney puts !. igh volatilities for atthemoney calls ". ow volatilities for atthemoney puts Answer# A !rashopho*ia is the word used to descri*e a possi*le @pho*ia that traders might have that the market will crash. It would lead to outofthemoney put options with low strike prices having high values and therefore high implied volatilities. Inthemoney call options have the same implied volatilities as the corresponding outofthemoney put options. ence they also have high implied volatilities and the correct answer is A. 1/.Which of the following is true for (uropean call and put options? A. If they have the same strike price> they have the same implied volatility B. If they have the same time to maturity> they have the same implied volatility !. If they have the same strike price and time to maturity> they have the same implied volatility ". =one of the a*ove Answer# ! If a (uropean call and put option have the same strike price and time to maturity putcall parity shows that they mu st have the same implied volatility. ;therwise there are ar*itrage opportunities. 10.Which of the following is true a*out daily exchange rate moves? A. Cour standard deviation daily moves in an exchange rate happen less
fre4uently than they would do if changes were normally distri*uted B. Cour standard deviation daily movements in an exchange rate happen more fre4uently than three standard deviation moves in the exchange rate !. +he fre4uency of six standard deviation daily movements in an exchange rate is a*out once every 1<< years ". =one of the a*ove Answer# " A> B> and ! are not true. arge daily exchange rate moves happen more fre4uently than they would if they were normally distri*uted. By deDnition a four standard deviation move occurs less fre4uently than a three standard deviation move. A six standard deviation move happens more fre4uency than once every 1<< years. +he ta*le in the text suggests the pro*a*ility is ,E1<><<<. Fiven that there are &0< *usiness days in a day this is a*out once every 1, years.
13.+he daily percentage change in an exchange rate is compared to a normal distri*ution with the same mean and standard deviation. Which of the following is true A. Both small and large exchange rate moves are more likely than with the normal distri*ution B. 7mall exchange rate moves are less likely and large exchange rate moves are more likely than with the normal distri*ution !. arge exchange rate moves are less likely and small exchange rate moves are more likely than with the normal distri*ution ". Both small and large exchange rate moves are less likely than with the normal distri*ution Answer# A Both large and small movements in the exchange rate happen more often than the normal distri*ution would predict. 19.Which of the following is true as time to maturity increases? A. +he volatility smile for currency options tends to *ecome more pronounced B. +he volatility smile for currency options tends to *ecome less pronounced !. +he volatility smile for currency options Drst *ecomes less pronounced and then *ecomes more pronounced ". +he volatility smile for currency options remains approximately the same Answer# B +he volatility smile tends to *ecome less pronounced as the time to maturity increases. $+his does not mean that its impact on prices *ecomes less
pronounced.%
16.If the volatility implied from an atthemoney put currency option were used to price other put options on the currency> which of the following would *e true? A. ;utofthe money and inthemoney prices would *e too high B. ;utofthe money and inthemoney prices would *e too low !. ;utofthemoney option prices would *e too high and inthemoney option prices would *e too low ". ;utofthemoney option prices would *e too low and inthemoney option prices would *e too high Answer# B +he volatility smile for currency options shows that atthemoney options have lower implied volatilities than out ofthemoney and inthemoney options. +his means that using atthe money implied volatilities for outof themoney or inthemoney options would lead to the volatility *eing too low so that there would *e underpricing. 15.If the volatility implied from an atthemoney put stock option were used to price other put options on the stock> which of the following would *e true? A. ;utofthe money and inthemoney prices would *e too high B. ;utofthe money and inthemoney prices would *e too low !. ;utofthemoney option prices would *e too high and inthemoney option prices would *e too low ". ;utofthemoney option prices would *e too low and inthemoney option prices would *e too high Answer# " +he volatility smile for e4uity options shows that atthemoney put options have lower implied volatilities than outofthemoney put options and a higher volatility than inthemoney put options. +his means that using atthemoney implied volatilities for outofthemoney put options would lead to underpricing and using atthemoney implied volatilities for inthemoney put options would lead to overpricing . ence the correct answer is ".
&<.+he implied volatilities for strike prices of 1.1 and 1.& when the time to maturity is 3 months are & what is the implied volatility for a strike price of 1.1& and a time to maturity of 1< months? A. 15.&/G B. 15.0&G !. &<./6G
". 15.53G Answer# B +his involves a twodimensional interpolation. We can either interpolate *etween strike prices Drst and then interpolate *etween maturities> or the other way round. 7uppose we interpolate *etween strike prices Drst. We get the implied volatility for a 3month option with a time to maturity of 3 months and a strike price of 1.1& as <.6H&<<.&H&&J&<./G. 7imilarly> we get the implied volatility for a 1 year option and a strike price of 1.1& as <.6H16.6<.&H&<.&J15.<6G. We next interpolate *etween maturities to get the volatility for a 1< month option with a strike price of 1.1& as <.,,H &<./<.33H15.<6J15.0&G. If you chose to interpolate *etween maturities Drst and then *etween strike prices you should have got the same answerK
