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Monday, December 12, 2011

recombinant binomial tree, price relative and lognormal

I find the recombinant/interlock concept a nice simplification. I was told in practice, all pricing btrees are interlocked – much easier without loss of generality. wikipedia says

"The CRR method ensures that the tree is *recombinant*, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u) -- here the two paths merge or recombine."

Often (u,d) will 100% cancel each other. At each tiny step, either

newPrice = oldPrice * u (typically 1.0101010101010101) or
newPrice = oldPrice * d (typically 0.99)
u * d == 1

Note the standard definition of u and d [[CFA]] as __PriceRelative__ of day2closing/day1closing. (Useful in h-vol...)

For a simple example, u = 2, so our underlying price either doubles or halves at each step. Same result if it double-then-half, or half-then-double. Consistent with the lognormal model...

Warning: many texts use illustrations of u=1.01000000000, d=0.99, which violates lognormal assumption but still recombinant...

In binomial tree, the next value of u is kind of random. In other words, the multiply factor could change from 2.0 to 1.01 to 1.002... usually just above 1 if step is small.

More precisely, u is computed using the underlying volatility, σ, and the time duration of a step. In a complete model, the pdf of random variable "u" can be derived.

Q: in industry, are most binomial trees recombinant?
%%A: I think so.