## Tuesday, June 18, 2013

### The Finite Difference Theta Scheme Optimal Theta

The theta finite difference scheme is a common generalization of Crank-Nicolson. In finance, the book from Wilmott, a paper from A. Sepp, one from Andersen-Ratcliffe present it. Most of the time, it's just a convenient way to handle implicit $$\theta=1$$, explicit $$\theta=0$$ and Crank-Nicolson $$\theta=0.5$$ with the same algorithm.

Wilmott makes an interesting remark: one can choose a theta that will cancel out higher order terms in the local truncation error and therefore should lead to increased accuracy.
$$\theta = \frac{1}{2}- \frac{(\Delta x)^2}{12 b \Delta t}$$

where $$b$$ is the diffusion coefficient.

This leads to $$\theta < \frac{1}{2}$$, which means the scheme is not unconditionally stable anymore but needs to obey (see Morton & Mayers p 30):
$$b \frac{\Delta t}{(\Delta x)^2} \leq \frac{5}{6}$$.

and to ensure that $$\theta \geq 0$$:

$$b \frac{\Delta t}{(\Delta x)^2} \geq \frac{1}{6}$$

Crank-Nicolson has a similar requirement to ensure the absence of oscillations given non smooth initial value, but because it is unconditionality stable, the condition is actually much weaker if $$b$$ depends on $$x$$. Crank-Nicolson will be oscillation free if $$b(x_{j0}) \frac{\Delta t}{(\Delta x)^2} < 1$$ where $$j0$$ is the index of the discontinuity, while the theta scheme needs to be stable, that is $$\max(b) \frac{\Delta t}{(\Delta x)^2} \leq \frac{5}{6}$$
This is a much stricter condition if $$b$$ varies a lot, as it is the case for the arbitrage free SABR PDE. where $$\max(b) > 200 b_{j0}$$

The advantages of such a scheme are then not clear compared to a simpler explicit scheme (eventually predictor corrector), that will have a similar constraint on the ratio $$\frac{\Delta t}{(\Delta x)^2}$$.