where f is the price of European stock options, S is the price of the underlying stock, sigma is the volatility of the stock price per year and r is the riskless interest rate.

For call option, the initial and boundary conditions are:

Last time, we have discussed the Crank-Nicolson scheme. It is unconditionally stable and has second order accuracy in both time and space. However, when the convection term is dominated, the spurious oscillation phenomenon will occurs, and thus making the method ineffective. In the rest part, we discuss two other schemes, which can avoid spurious oscillation.

**Implicit-Explicit method**

Implicit-Explicit (IMEX) methods are fully coupled methods that are designed to handle some terms implicitly and others explicitly. A simple example is obtained by combining forward Euler with backward Euler:

In this scheme, backward Euler is used for diffusion term and zero term while forward Euler is used for convection term.

It is not difficult to find that the diffusion term and zero term are unconditionally stable. As for the convection term where forward Euler is used, it would be unstable if central difference is applied. Therefore, upwind scheme is used instead. It shows that the stability condition for this scheme is:

where

rearrange the inequality, we get:

Thus, there is a restriction on the time step size. The matrix of the linear algebra system formed by using this scheme is strictly diagonal dominant. Thus, this method avoid the oscillations phenomenon occurred when using Crank-Nicolson scheme. But it is only conditionally stable and first order accuracy in both time and space.

**Exponentially fitted scheme**

In general, for convection-diffusion equation:

We have scheme:

where the fitting factor is

As what has been discussed above, Implicit Euler method is unconditionally stable. However, when the convection term is dominated, spurious oscillation will occur. The modified exponentially fitted scheme is designed to overcome this weakness.

Here we take an extreme case as example. Suppose the coefficient of convection term is positive and that of diffusion term is zero. Then the equation degenerates to a hyperbolic equation. In this case, the fitting factor becomes:

Inserting this result into the degenerated hyperbolic equation gives us the first-order scheme:

This is the so-called implicit upwind scheme which is stable and convergent. Like the IMEX method, the exponentially fitted scheme successfully avoid oscillation phenomenon at the cost of lowering the accuracy.

To summarize, the Crank-Nicolson scheme is very popular due to its higher order accuracy and non-conditional stability. However, it is notorious that this scheme will give rise to “spurious oscillation” when the convection term is dominated. To overcome this problem, we introduce ‘IMEX’ scheme and exponential fitted scheme. Both of these two methods eliminate oscillation phenomenon by scarifying the accuracy and stability. There is no perfect scheme here. But one could choose a suitable one for the certain situation.