Chapter 2: FAQs 139
sided difference and tends to be preferred for the delta
approximation, but when used for the time derivative
it can lead to instabilities in the numerical scheme.
(Here I am going to describe the explicit finite-difference
scheme, which is the easiest such scheme, but is one
which suffers from being unstable if the wrong time
discretization is used.)
The central difference for the gamma is
=∂^2 V
∂S^2≈V(S+δS,t)− 2 V(S,t)+V(S−δS,t)
δS^2.Slightly changing the notation so thatVikis the option
value approximation at theith asset step andkth time
step we can write
θ≈Vik−Vik−^1
δt, ≈Vik+ 1 −Vik− 1
2 δSand≈Vik+ 1 − 2 Vik+Vik− 1
δS^2.Finally, plugging the above, together withS=iδS,into
the Black–Scholes equation gives the following dis-
cretized version of the equation:
Vik−Vik−^1
δt+^12 σ^2 i^2 δS^2Vik+ 1 − 2 Vik+Vik− 1
δS^2+riδSVik+ 1 −Vik− 1
2 δS−rVik= 0.This can easily be rearranged to giveVik−^1 in terms
ofVik+ 1 ,VikandVik− 1 , as shown schematically in the
following figure.
In practice we know what the option value is as a func-
tion ofS, and hencei, at expiration. And this allows us