Exam: CH 19 Volatility smiles Flashcards Preview

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Flashcards in Exam: CH 19 Volatility smiles Deck (23)
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1
Q

What is a volatility smile?

A
  • Plot of implied volatility of an option as a function of strike price
2
Q

What is the Put–Call Parity?

A

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

3
Q

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

A
  • No the implied volatility of a european call and put are always the same.
  • Approximately true for the American options too
4
Q

Similarities and differences of of lognormal and implied distribution?

A
-  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)
5
Q

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?

A
  • The option pays off only if the exchange rate proves to be below K1.
6
Q

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?

A
  • Implied distribution
7
Q

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?

A
  • Expect the implied distribution to give a relatively high price, and a relatively high implied
    volatility for this option too
8
Q

What does the smile used by traders show?

A
  • shows that they believe that the lognormal distribution

understates the probability of extreme movements in exchange rates

9
Q
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?

A
  • We can see that there are much fatter tails in the real world
  • Hence it is more likely to have a large movements.
10
Q

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

A
  • Volatility of the asset is constant

* Price of the asset changes smoothly with no jumps

11
Q

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

A
  • 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
12
Q

What does the volatility smile for equity options look like?

A
  • The volatility smile or
    volatility skew, has the
    form of a downward sloping
    parabola
13
Q

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).

A
  • Low strike price
  • Higher
  • High strike price
14
Q

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?

A
  • This has a lower

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

15
Q

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?

A
  • 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
16
Q

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

When does the option pay off?

A

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

17
Q

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?

A

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

18
Q

What are the 2 reasons for the equity volatility smile?

A
  • 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
19
Q

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

A
  • Increasing
  • Short dated volatilities
  • increase
20
Q

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

A
  • Decreasing
  • Short dated volatilities
  • Decrease
21
Q

What is a volatility surface?

A
  • 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.
22
Q

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

A
  • 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
23
Q

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?

A
  • 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