ch 20 Flashcards
What is the relationship between the implied volatility of a European call option and a European put option?
The implied volatility of a European call option is always the same as the implied volatility of a European put option when they have the same strike price and time to maturity.
What is put-call parity and how is it related to option pricing?
Put-call parity is a relationship between the prices of European call and put options with the same strike price and time to maturity. It provides a no-arbitrage argument for option pricing and holds true regardless of the underlying asset price distribution.
What does the concept of volatility smile refer to in option pricing?
The volatility smile refers to the relationship between implied volatility and strike price for a particular maturity in option pricing. It is the same for both European call and put options, and the same is true for the volatility surface, which is the implied volatility as a function of strike price and time to maturity.
What is the volatility smile for foreign currency options?
The implied volatility for foreign currency options is relatively low for at-the-money options and becomes progressively higher as the option moves into or out of the money.
What is the implied distribution for an asset price in the future?
The implied distribution is the risk-neutral probability distribution for an asset price in the future, determined from the volatility smile given by options maturing at that time. It has heavier tails than a lognormal distribution with the same mean and standard deviation.
How do the figures in this section show consistency between the implied probability distribution and implied volatility for deep out-of-the-money options?
For a deep out-of-the-money call option with a high strike price, the implied distribution shows a higher probability of the exchange rate being above the strike price, leading to a higher implied volatility and a relatively high option price, which is consistent with the volatility smile. The same is true for a deep out-of-the-money put option with a low strike price.
Traders’ assumption of probability distribution
Traders assume heavier left tail for equity prices and heavier right and left tails for exchange rates
Volatility smile and its use in nonlognormality
Volatility smile is used to relate implied volatility and strike price; often combined with volatility term structure to produce a volatility surface
Reasons for the Smile in Foreign Currency Options
- Exchange rates are not lognormally distributed because the volatility of an exchange rate is not constant, and exchange rates frequently exhibit jumps.
- Nonconstant volatility and jumps make extreme outcomes more likely.
- The impact of jumps and nonconstant volatility depends on the option maturity, with the volatility smile becoming less pronounced as option maturity increases.
Volatility Smile and Implied Probability Distribution for Equity Options
The volatility smile for equity options, also known as the volatility skew, shows that the volatility decreases as the strike price increases. The implied probability distribution for equity options has a heavier left tail and a less heavy right tail than a lognormal distribution with the same mean and standard deviation. There is a negative correlation between equity prices and volatility, which means that stock price declines are accompanied by increases in volatility, making even greater declines possible. Stock price increases are accompanied by decreases in volatility, making further stock price increases less likely.
Reasons for Negative Correlation between Equity Prices and Volatility
The negative correlation between equity prices and volatility can be explained by several factors. One is leverage, as equity prices move down (up), leverage increases (decreases) and as a result volatility increases (decreases). Another is the volatility feedback effect, where as volatility increases (decreases) due to external factors, investors require a higher (lower) return and as a result the stock price declines (increases). Crashophobia, or the fear of a market crash, may also play a role. This negative correlation means that stock price declines are accompanied by increases in volatility, making even greater declines possible, while stock price increases are accompanied by decreases in volatility, making further stock price increases less likely.
Reasons for the Smile in Foreign Currency Options
Exchange rates are not lognormally distributed because neither of the two conditions for an asset price to have a lognormal distribution are satisfied for an exchange rate.
The volatility of an exchange rate is far from constant, and exchange rates frequently exhibit jumps, which make extreme outcomes more likely.
The impact of jumps and nonconstant volatility depends on the option maturity, with the volatility smile becoming less pronounced as option maturity increases.
Prior to the crash of 1987, there was no marked volatility smile for equity options.
Before the 1987 market crash, there was no pronounced volatility smile for equity options.
Since 1987, traders have used a volatility skew to price equity options, with the volatility decreasing as the strike price increases.
The implied probability distribution for the volatility smile has a heavier left tail and a less heavy right tail than the lognormal distribution.
There is a negative correlation between equity prices and volatility, with stock price declines being accompanied by increases in volatility, making further declines possible, and stock price increases being accompanied by decreases in volatility, making further increases less likely.
