Note 12: Finite difference approach to solve Heston model(2)

It is important for choosing method for the time propogating.

1. Explicit schemes are easily to implement, do not suffer from oscillation problems but are only conditionally stable.

2. Implicit schemes are also oscillation-free, unconditionally stable but only first-order accurate.

3. The Crank–Nicolson scheme is (theoretically) second-order accurate but it is well known that it produce spurious oscillations and spikes near the strike price, barriers and monitoring points.

4. Hybrid method. Such as in Rannacher (1984) it uses fully implicit
Euler for the first few time steps and Crank–Nicolson after that. The scheme is stable, firstorder accurate and is oscillation-free. And use Richardson extrapolation in combination with implicit Euler (Gourlay, 1980). The resulting
scheme is stable, second-order accurate and again oscillation free.

5. Function smooth.1) Projecting the initial condition onto a set of basis functions


6. One can combine smoothing and the Rannacher method to get a stable, second-order accurate and oscillation-free finite difference scheme