138 Frequently Asked Questions In Quantitative Finance
ticated schemes these can vary. Our task will be to find
numerically an approximation to the option values at
each of the nodes on this grid.The classical option pricing differential equations are
written in terms of the option function,V(S,t), say, a
single derivative with respect to time,∂∂Vt, the option’s
theta, the first derivative with respect to the underlying,
∂V
∂S, the option’s delta, and the second derivative with
respect to the underlying,∂(^2) V
∂S^2 , the option’s gamma. I
am explicitly assuming we have an equity or exchange
rate as the underlying in these examples. In the world of
fixed income we might have similar equations but just
read interest rate,r, for underlying,S, and the ideas
carry over.
A simple discrete approximation to the partial derivative
for theta is
θ=
∂V
∂t
≈
V(S,t)−V(S,t−δt)
δt
whereδtis the time step between grid points. Similarly,
=
∂V
∂S
≈
V(S+δS,t)−V(S−δS,t)
2 δS
whereδSis the asset step between grid points. There
is a subtle difference between these two expressions.
Note how the time derivative has been discretized by
evaluating the functionVat the ‘current’Sandt,and
also one time step before. But the asset derivative uses
an approximation that straddles the pointS,usingS+
δSandS−δS. The first type of approximation is called a
one-sided difference, the second is a central difference.
The reasons for choosing one type of approximation
over another are to do with stability and accuracy.
The central difference is more accurate than a one-