310 Frequently Asked Questions In Quantitative Finance
Examples
Accrual is a generic term applied to contracts in which an
amount gradually builds up until it is paid off in a lump sum.
An example would be an accrual range note in which for
every day that some underlying is within a specified range
a specified amount is accrued, to eventually be paid off in a
lump sum on a specified day. As long as there are no decision
features in the contract then the accrual is easily dealt with
by Monte Carlo simulation. If one wants to take a partial
differential approach to modelling then an extra state variable
will often be required to keep track of how much money has
been accrued.
American option is one where the holder has the right to
exercise at any time before expiration and receive the payoff.
Many contracts have such early exercise American features.
Mathematically, early exercise is the same as conversion of a
convertible bond. These contracts are priced assuming that
the holder exercises so as to give the contract its highest
value. Therefore a comparison must be made between the
value of the option assuming you don’t exercise and what
you would get if you immediately exercised. This makes finite
differences a much more natural numerical method for pricing
such contracts than Monte Carlo.
Asian option is an option whose payoff depends on the aver-
age value of the underlying during some period of the option’s
life. The average can be defined in many ways, as an arith-
metic or geometric mean, for example, and can use a large
set of data points in a continuously sampled Asian or only
a smaller set, in the discretely sampled Asian. In an Asian
tail the averaging only occurs over a short period before
option expiration. There are closed-form formulæ for some of
the simpler Asian options based on geometric averages, and
approximations for others. Otherwise they can be priced using
Monte Carlo methods, or sometimes by finite differences.
Because the average of an asset price path is less volatile
than the asset path itself these options can be cheaper than
their equivalent vanillas, but this will obviously depend on the
nature of the payoff. These contracts are very common in the
commodity markets because users of commodities tend to be
