Exam: CH 19 Volatility smiles

Question 1

What is a volatility smile?

Answer 1

• Plot of implied volatility of an option as a function of strike price

Question 2

What is the Put–Call Parity?

Answer 2

The relationship between the - price of a European call option and the
price of a European put option when they have the same strike price and maturity date

Question 3

Is the implied volatility of a European call different from the implied volatility of a European put?

Answer 3

• No the implied volatility of a european call and put are always the same.
• Approximately true for the American options too

Question 4

Similarities and differences of of lognormal and implied distribution?

Answer 4

-  Both distributions have the
 same mean and the same
 standard deviation
- But the implied distribution
 is Steeper & Fat tailed
- Why does this distribution
 hold?
-Consider deep out-of-the-
 money call option with high
 strike price K2
- Remember: Stock price would be in-between K1 and K2 (closer to mean)

Question 5

consider a deep out-of-the-money put option (or deep in-the-money call) with a low
strike price K1 since K1 < ST

At which point does the option pay off?

Answer 5

• The option pays off only if the exchange rate proves to be below K1.

Question 6

consider a deep out-of-the-money put option (or deep in-the-money call) with a low
strike price K1 since K1 < ST

Who has the higher probability of this happening?

Answer 6

• Implied distribution

Question 7

consider a deep out-of-the-money put option (or deep in-the-money call) with a low
strike price K1 since K1 < ST

What results will the implied probability give?

Answer 7

• Expect the implied distribution to give a relatively high price, and a relatively high implied
  volatility for this option too

Question 8

What does the smile used by traders show?

Answer 8

• shows that they believe that the lognormal distribution

understates the probability of extreme movements in exchange rates

Question 9

Real            Lognormal 
            World               model
> 1 SD     25.04              31.73
> 2 SD    5.27                4.55
> 3 SD     1.34                 0.27
> 4 SD     0.29                0.01
> 5 SD     0.08                0.00
> 6 SD     0.03                0.00

What does the following data show?

Answer 9

• We can see that there are much fatter tails in the real world
• Hence it is more likely to have a large movements.

Question 10

What two conditions need to hold for the lognormal distribution to hold?

Answer 10

• Volatility of the asset is constant

* Price of the asset changes smoothly with no jumps

Question 11

Do the 2 conditions for the lognormal distribution hold for an exchange rate?

Answer 11

• No Volatility of an exchange rate is not constant and exchange rates frequently jump.
• Both of these tend to increase the likelihood of extreme events

Question 12

What does the volatility smile for equity options look like?

Answer 12

• The volatility smile or
  volatility skew, has the
  form of a downward sloping
  parabola

Question 13

Volatility to price a ____ ____ ____ option (deep-out-
of-the-money put or deep-in-
the-money call) is
significantly ____ than that
used to price a ____ ____ _____ option (deep-in-the-money put or
deep-out-of-the-money call).

Answer 13

• Low strike price
• Higher
• High strike price

Question 14

Equity options:
Consider a deep-out-of-the-money call option with a strike price of K2 (High price)

Which distribution gives a lower price?

Answer 14

• This has a lower

price when the implied distribution is used than when the lognormal distribution is used

Question 15

Equity options:
Consider a deep-out-of-the-money call option with a strike price of K2 (High price)

When does this option pay off?

Answer 15

• This is because the option pays off only if the stock price is above K2, & the probability of this is lower for the implied probability distribution than for the lognormal distribution

Question 16

Equity options:
consider now a deep-out-of-the-money put option with strike price K1 (Low price)

When does the option pay off?

Answer 16

• The option pays off only when the stock price is below K1. The probability of this is
higher for implied probability distribution

Question 17

Equity options:
consider now a deep-out-of-the-money put option with strike price K1 (Low price)

What do we expect the implied distribution to give?

Answer 17

• Expect the implied distribution to give a relatively higher price, and a relatively high price
implies higher implied volatility.

Question 18

What are the 2 reasons for the equity volatility smile?

Answer 18

• Fear of a crash:
  Traders are concerned about the possibility of a crash, so they price the option accordingly
• Leverage:
  As a company’s equity declines in value, the equity becomes more risky
  and its volatility increases. As a company’s equity increases in value, the equity
  becomes less risky and its volatility decreases

Question 19

Volatility term structure:
Volatility tends to be an _____ function of maturity when ____ ____ ____ are historically low, since there is expectation that volatility will _____

Answer 19

• Increasing
• Short dated volatilities
• increase

Question 20

Volatility term structure:
Volatility tends to be a ______ function of maturity when _____ ______ _____ are historically high, since there is expectation that volatility will _____

Answer 20

• Decreasing
• Short dated volatilities
• Decrease

Question 21

What is a volatility surface?

Answer 21

• Volatility surfaces combine volatility smiles with the volatility term structure to tabulate the volatilities appropriate for pricing an option with any strike price and any maturity.

Question 22

How important is the pricing model if traders are prepared to use a different volatility for every option?

Answer 22

• We can think of Black-Scholes as an interpolation tool used by traders to unsure that an option is priced consistently with the market prices of other actively traded options
• If traders used a different model, then volatility surfaces and the shape of the smile
  would change, but the dollar prices found in the market should not change appreciably

Question 23

Single large jumps:
Consider that a pharmaceutical stock is currently at $50 and an announcement on a
pending lawsuit against it is expected in a few days. The news is expected to send the
stock either up by $8 or down by $8.

What should the probability distribution in one month consist of?

Answer 23

• The probability distribution of the stock price in one month should consist of a mixture of
  two lognormal distributions, the first corresponding to favorable news, the second to
  unfavorable news