Too large timesteps?
However, due to the explicitness of trinomial trees, there is a second source of instability in trinomials: When timesteps are too large compared to the discretization in the space (i.e. the short rate) dimension, the scheme becomes unstable. Assume we want to valuate a fixed income instrument under a one-factor Hull-White model with a time horizon of 30 yearsm and we would likt o have a grid resolution for the interest rates of 10 basis points. With a typical (absolute) volatility of, say, 1 percent (a reasonable guess ), this leads to a time step of the order of 1 day and that 10000 time steps are needed.
The grid then looks 50 times finer (in both directions) than the following plot.
|Only 200 of the necessary 10000 timesteps plotted here.|
No. There are much cleverer methods available.
For a more detailed description of the stability conditions, see section 4.5 of the Workout.