136 Frequently Asked Questions In Quantitative Finance
What is the Finite-difference Method?
Short Answer
The finite-difference method is a way of approximating
differentialequations, in continuous variables, intodiffer-
enceequations, in discrete variables, so that they may
be solved numerically. It is a method particularly useful
when the problem has a small number of dimensions,
that is, independent variables.
Example
Many financial problems can be cast as partial dif-
ferential equations. Usually these cannot be solved
analytically and so they must be solved numerically.
Long Answer
Financial problems starting from stochastic differential
equations as models for quantities evolving randomly,
such as equity prices or interest rates, are using the
language of calculus. In calculus we refer to gradients,
rates of change, slopes, sensitivities. These mathemati-
cal ‘derivatives’ describe how fast a dependent variable,
such as an option value, changes as one of the indepen-
dent variables, such as an equity price, changes. These
sensitivities are technically defined as the ratio of the
infinitesimal change in the dependent variable to the
infinitesimal change in the independent. And we need
an infinite number of such infinitesimals to describe an
entire curve. However, when trying to calculate these
slopes numerically, on a computer, for example, we can-
not deal with infinites and infinitesimals, and have to
resort to approximations.
Technically, a definition of the delta of an option is
=
∂V
∂S
=lim
h→ 0
V(S+h,t)−V(S−h,t)
2 h
