There are at least two types of smile "dynamics." In the first, we would fix the time parameter t and consider options with longer and longer expirations, T. In the second case, we would keep T constant, but let time t pass and study how changes in various factors affect the volatility smile. In particular, we can observe if changes in St affect the smile when the moneyness Ki/St is kept constant.
We first keep t fixed and increase T. We consider two series of options that trade at the same time t. Both series have comparable strikes, but one series has a relatively longer maturity. How would the smiles implied by the two series of options with expirations, say, T\,T2, compare with each other?
The second question of interest is how the smile of the same option series moves over time as St changes. In particular, would the smile be a function of the ratio Ki/St only, or would it also depend on the level of St over and above the moneyness?
The answers to these questions change depending on which underlying asset is considered. This is because there is more than one explanation for the existence of the smile, and for different sectors, different explanations seem to prevail. Thus, before we analyze the smile dynamics and its properties any further, we need to discuss the major explanations advanced for the existence of the volatility smile.
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