Note 12: Finite difference approach to solve Heston model(1)

1) By Ito formula, it is easy to write down the PDE for
Heston model
2) The inmportant thing is the boundary condition
s->0 s-> infinity, u->0 u-> infinity
a) For European call option
we have s->0 U=0 u-> infinity U=S
u=0, we solve the PDE separatly which can be regarded a boundary
s-> infinite , Let us use U'=1(linear boundary condition)
b)For European call option
we have s->0 U=K u-> infinity U=0
u=0, max(K-S,0)
s-> infinite , Let us use U'=0(linear boundary condition)
c) we can have some other conditions
3) Numerical method for Heston model

Let us give basic introduction to the numerical procedure
1) Create Mesh
2) Define initial condition
3) update the boundary conditions for time level n+1 first at time level n
4) Propogate the state at time level n to time level n+1
5) Iterative Until the time is ended.