To tree or not to tree: trinomials

We have seen in a former blog article that binomial trees may lead to oscillating solutions and even to heding parameters of the wrong sign arising from these oscillations. Therefore, trinomial trees were invented and are still widely used for short rate models like Hull-White or Black-Karasinski.

As I am not a well-known tree-addict, the reader may expect some good reasons against trinomials.

The basic branching of trinomials looks like this

At each node there are three possible branches with probabilities p1, p2, p3 who should add up to 1. In contrast to the binomial model, the trinomial model is not a complete one leading to a unique price only from no-arbitrage arguments. Therefore, there is a certain arbitrariness to assign the probabilities to the different branches. The advantage of this arbitrariness is that it is fairly easy to obtain recombining trinomial trees.

When we think of a (mean-reverting) Hull-White model, the nodes and the probabilities are typically chosen in such a way that the expected value of the short rate (at the next time step) and its variance fit the analytic values available for these. If a node is too far away from the mean reverting level, this has the consequence that the one of the branches would be assigned with a negative probability leading necessarily to severe oscillations as the explicit scheme is not stable any more.

Therefore down-branching and up-branching are used for trinomials in mean reverting models.

So what?
Well, down-branching and up-branching cuts off certain parts of the calculation domain and therefore leads to wrong values even for the most easy instrument, a zero coupon bond. Yes, certainly you can fiddle around and stretch your calculation tree in such a way that you meet the discount factors again. But if you do this, you never ever can use analytic formulae for bonds, options or CMS rates any more. It would be much much easier if you had a consistent and stable numerical scheme which does not change your model.

And yes, there is one: Finite elements with proper treatment of convection (which is the reason for the stability problem) lead to much much better results.

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