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## Tuesday, March 13, 2012

### binomial trees -- flexible and adaptable

Binomial Tree is widely used in investment banks such as GS and Barcap. One of the most popular numeric methods for option pricing. I think the only other common pricing methodology is PDE, but less popular.
Simpler pricers use formula directly. So I'd say binomial tree is the #1 most practical and wide-spread pricing methodology for non-trivial instruments.

All binomial trees I have seen are interlocked. After 54 intervals, there are exactly 55 possible prices. See http://bigblog.tanbin.com/2011/06/option-pricing-recombinant-binomial.html

"Level 55" refers collectively to the 55 tree nodes after 54 intervals. The word "Level" should NOT be used for price-level. "Level" means height of the tree as it grows. Rotate the tree to put the root down to visualize the 55 "Levels". However, most btrees are drawn left to right because time (X-axis) grows rightward, and price values are naturally arranged along Y-axis.

The interlock/recombinant rule is simplifying, and all intervals are equal on a tree, but many other aspects of the tree are flexible/complex and need to be modeled. It's these flexibilities that allow the tree to be so versatile and adaptable.

- the probability of an upswing at each node is unique and probably independent. On the 5 nodes at Level 5, there are 5 distinct up-probabilities, and 6 on the next level.
- the magnitude of the upswing at each node is independent. Ditto the downswing. However CRR btree don't provide this flexibility.

Trinomial tree isn't popular (perhaps never used) in the finance industry.