Chapter 20: Volatility Smiles Flashcards
CURRENCY OPTIONS: Explain how the two below graphs are related/connected
First: consider a call option that’s deep OTM with at high strike price of K(2). This option will only be exercised/pays off if the underlying asset is higher than K(2). The bottom figure (20.2) shows that the probability of this is higher for the implied probability distribution than for the lognormal distribution. It is therefore expected that the implied distribution will give a relatively high price for the option. A relatively high price leads to a relatively high implied volatility and that is exactly what we observe in the top figure (20.1)
Second: consider a put that’s deep OTM with a low strike price of K(1). This option only pays off if the value of the underlying is below K(1). The probability for this is also higher for the implied probability distribution than for the lognormal distribution. We therefore expect the implied distribution to give a relatively high price and a relatively high volatility which again is what we observe in the top graph (20.1).
CURRENCY OPTIONS: Why do we distinct between the lognormal and implied probability distribution?
The volatility smile used by traders implies that they consider that the lognormal distribution understates the probability of extreme movements in the underlying. So that’s why the implied probability distribution is introduced.
CURRENCY OPTIONS: Why is the return of the underlying not lognormally distributed?
Two of the conditions for an asset price to have a lognormal distribution are:
1: The volatility of the asset is constant.
2: The price of the asset changes smoothly with no jumps.
In practice, neither of these conditions is satisfied for an exchange rate. The volatility of an exchange rate is far from constant, and exchange rates frequently exhibit jumps. It turns out that the effect of both a nonconstant volatility and jumps is that extreme outcomes become more likely.
CURRENCY OPTIONS: What is the relation between the volatility smile and time to maturity?
The impact of jumps and nonconstant volatility depends on the option maturity. As the maturity of the option is increased, the percentage impact of a nonconstant volatility on prices becomes more pronounced, but its percentage impact on implied volatility usually becomes less pronounced. The percentage impact of jumps on both prices and the implied volatility becomes less pronounced as the maturity of the option is increased. The result of all this is that the volatility smile becomes less pronounced as option maturity increases.
EQUITY OPTIONS: Explain the relation between the below two graphs
The volatility decreases at the strike price increases. The volatility used to price a low-strike-price option (i.e., a deep-out-of-the- money put or a deep-in-the-money call) is significantly higher than that used to price a high-strike-price option (i.e., a deep-in-the-money put or a deep-out-of-the-money call) - this gives rise to the volatility smirk.
In regard to figure (20.4) it can be seen that the implied distribution has a heavier left tail and a less heavy right tail than the lognormal distribution.
From figure (20.4) we see that a deep OTM call with a strike price of K(2) has a lower price when the implied distribution is used than when the lognormal distribution is used. This is because the option pays off only if the stock price proves to be above K(2), and the probability of this is lower for the implied probability distribution than for the lognormal distribution. Therefore, we expect the implied distribution to give a relatively low price for the option. A relatively low price leads to a relatively low implied volatility—and this is exactly what we observe in figure (20.3) for the option.
Consider next a deep-out-of-the-money put option with a strike price of K(1). This option pays off only if the stock price proves to be below K(1). Figure (20.4) shows that the probability of this is higher for the implied probability distribution than for the lognormal distribution. We therefore expect the implied distribution to give a relatively high price, and a relatively high implied volatility, for this option. Again, this is exactly what we observe in figure (20.3).
EQUITY OPTIONS: Explain why we observe the volatility smile
One possible explanation for the smile in equity options concerns leverage. As a company’s equity declines in value, the company’s leverage increases. This means that the equity becomes more risky and its volatility increases. As a company’s equity increases in value, leverage decreases. The equity then becomes less risky and its volatility decreases. This argument suggests that we can expect the volatility of a stock to be a decreasing function of the stock price and is consistent with figures (20.3) and (20.4).
Why does the volatility smile complicate the calculation of greek letters?
Assume that the relationship between the implied volatility and K/S for an option with a certain time to maturity remains the same. As the price of the underlying asset changes, the implied volatility of the option changes to reflect the option’s ‘‘moneyness’’ (i.e., the extent to which it is in or out of the money). The formulas for Greek letters given in chapter 19 are no longer correct.
Volatility is a decreasing function of K/S. This means that the implied volatility increases as the asset price increases. As a result, delta is higher than that given by the Black–Scholes–Merton assumptions.
What volatility smile is likely to be observed when both tails of the stock distribution are less heavy than those of the lognormal distribution?
See below
What volatility smile is likely to be observed when the right tail is heavier, and the left tail is less heavy, than that of the lognormal distribution?
We will see a volatility smile that resembles a smirk that starts of with high implied volatility with high strike price and a implied volatility that drops as the strike price decrease.
Imagine the below volatility smirk but spejlvendt