  In this article, we investigate the behavior of long-term options. In many
cases, option prices follow an exponential decay (or growth) rate for further
maturity dates. We determine under what conditions option prices are
characterized by this property. To see this, we use the martingale extraction
method through which a pricing operator is transformed into a semigroup
operator, which is easier to address.
  We also explore notions of hedging long-term options. Hedging is an attempt
to reduce market risks, and we investigate the price sensitivities (Greeks)
with respect to such risks, which are typically repre- sented by variations in
the underlying process of an option. We combine the Malliavin calculus with the
martingale extraction method to analyze Greeks. We see that the ratios between
Greeks and the option price are expressed in a simple form in the long term.
