Exam: CH 19 Volatility smiles Flashcards
What is a volatility smile?
- Plot of implied volatility of an option as a function of strike price
What is the Put–Call Parity?
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
Is the implied volatility of a European call different from the implied volatility of a European put?
- No the implied volatility of a european call and put are always the same.
- Approximately true for the American options too
Similarities and differences of of lognormal and implied distribution?
- 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)
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?
- The option pays off only if the exchange rate proves to be below K1.
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?
- Implied distribution
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?
- Expect the implied distribution to give a relatively high price, and a relatively high implied
volatility for this option too
What does the smile used by traders show?
- shows that they believe that the lognormal distribution
understates the probability of extreme movements in exchange rates
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?
- We can see that there are much fatter tails in the real world
- Hence it is more likely to have a large movements.
What two conditions need to hold for the lognormal distribution to hold?
- Volatility of the asset is constant
* Price of the asset changes smoothly with no jumps
Do the 2 conditions for the lognormal distribution hold for an exchange rate?
- 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
What does the volatility smile for equity options look like?
- The volatility smile or
volatility skew, has the
form of a downward sloping
parabola
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).
- Low strike price
- Higher
- High strike price
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?
- This has a lower
price when the implied distribution is used than when the lognormal distribution is used
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?
- 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
Equity options:
consider now a deep-out-of-the-money put option with strike price K1 (Low price)
When does the option pay off?
• The option pays off only when the stock price is below K1. The probability of this is
higher for implied probability distribution
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?
• Expect the implied distribution to give a relatively higher price, and a relatively high price
implies higher implied volatility.
What are the 2 reasons for the equity volatility smile?
- 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
Volatility term structure:
Volatility tends to be an _____ function of maturity when ____ ____ ____ are historically low, since there is expectation that volatility will _____
- Increasing
- Short dated volatilities
- increase
Volatility term structure:
Volatility tends to be a ______ function of maturity when _____ ______ _____ are historically high, since there is expectation that volatility will _____
- Decreasing
- Short dated volatilities
- Decrease
What is a volatility surface?
- 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.
How important is the pricing model if traders are prepared to use a different volatility for every option?
- 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
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?
- 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
