# Dimensionality

refers to the number of underlying independent variables. The vanilla option has two independent variables, *S* and *t,* and is thus two dimensional. The weakly path-dependent contracts have the same number of dimensions as their non-path-dependent cousins.

We can have two types of three-dimensional problem. The first type of problem that is three dimensional is the strongly path-dependent contract. Typically, the new independent variable is a measure of the path-dependent quantity on which the option is contingent. In this case, derivatives of the option value with respect to this new variable are only of the first order. Thus the new variable acts more like another time-like variable.

The second type of three-dimensional problem occurs when we have a second source of randomness, such as a second underlying asset. In the governing equation we see a second derivative of the option value with respect to each asset. We say that there is diffusion in two dimensions.

• Higher dimensions means longer computing time.

• The number of dimensions we have also tells us what kind of numerical method to use. High dimensions mean that we probably want to use Monte Carlo; low means finite difference.

# The order of an option

refers to options whose payoff, and hence value, is contingent on the value of *another* option. The obvious second-order options are compound options, for example, a call option giving the holder the right to buy a put option.

• When an option is second or higher order we have to solve for the first-order option, first. We thus have a layer cake, we must work on the lower levels and the results of those feed into the higher levels.

• This means that computationally we have to solve more than one problem to price our option.

# Embedded decisions

are when the holder or the writer has some control over the payoff. He may be able to exercise early, as in American options, or the issuer may be able to call the contract back for a specified price.

When a contract has embedded decisions you need an algorithm for deciding how that decision will be made. That algorithm amounts to assuming that the holder of the contract acts to *make the option value as high as possible for the delta-hedging writer.* The pricing algorithm then amounts to searching across all possible holder decision strategies for the one that maximizes the option value. That sounds hard, but approached correctly is actually remarkably straightforward, especially if you use the finite-difference method. The justification for seeking the strategy that maximizes the value is that the writer cannot afford to sell the option for anything less, otherwise he would be exposed to 'decision risk.' When the option writer or issuer is the one with the decision to make, then the value is based on seeking the strategy that minimizes the value.

• Decision features mean that we'd really like to price via finite differences.

• The code will contain a line in which we seek the best price, so watch out for > or < signs.

# 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.