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Wednesday, April 18, 2012

binomial tree - why identical diamonds

The standard CRR btree is always drawn with all straight lines, equally spaced vertically and equally spaced horizontally. Therefore you always see nothing but a strict pattern of identical diamonds. Let's zoom into this "geometry".

First, let's set the stage for the discussion. In this conceptual "world", a price (say IBM) can only be observed/sampled at periodic discrete moments, either once a second, or once a day, though the interval should be small relative to time to maturity. Price may change mid-interval, but we can't observe that. Further, during each interval, the price either moves up or down. It can remain unchanged only in a trinomial tree -- not popular in industry.

Why diamonds? Because of interlocking/recombinant. See http://bigblog.tanbin.com/2011/06/option-pricing-recombinant-binomial.html

Why equally spaced horizontally? Because the intervals are fixed and constant -- at each clock tick, the variable must either rise or fall, never stay flat like a trinomial.

Why equally spaced vertically? Because the y-axis is log(price). An interesting feature of the CRR btree. If after n-1 intervals you plot the n price values in a bar chart, they don't fit a straight line -- but try plotting log(price).

Why are all the diamonds identical? Because the nodes are equally spaced both vertically and horizontally.

I feel the regularity is a great simplification and helps us focus on the real issue -- the probability of an upswing at each node -- the transition probability function, which is individually determined at each node position.