Binomial tree: calculating up move and down move

[sridhar]

Thought I'd share something that some may find as a nit...I came across a question like this:

"The current price of XYZ is $20. In each of the next two years you expect the stock price to either move up 20 percent or down 20 percent. The probability of an upward move is 0.65 and the probability of a downward move is 0.35. The risk-free rate is 5 percent. .................."

I draw your attention to the boldface. Tactical exam stuff:

Here if I take U = 1.2 and D = 0.8 -- then I get one set of calculations....

However if I take U = 1.2 and then treat D = 1 / U, then D is really 0.833. This seems six-of-one and half-dozen of other, but it really isn.t

If you take D = 0.833, then S-ud = 20, as we have seen in many examples.

If you take D = 0.8, then S-ud = 19.20. These differences can be material, no?

David -- I am coming to the conclusion that if a question is phrased as: stock moves up x% or down y%, then I should not calculate D from U, but calculate D directly from the language posed in the question at the top...

Your thoughts?

--sridhar

 

[David Harper CFA FRM]

sridhar,

Great example. I agree they are different, i'd even venture yours is the better approach (following Hull) and the "arithmetic" approach in the question is atypical except for interest rate trees. First, this question is specific, so as phrased I agree it deserves 1.2/0.8 as explicitly stated.

But if it weren't stated, it is better to use your approach (following Hull) and match up/down magnitudes to the volatility. The difference relates to geometric vs. arithmetic mean:

0.8 and 1.2 has an arithmetic mean = 0, but (ud) or (du) does not "get back" to 1.0: (0.8)(1.2) < 1.0
Whereas your numbers recombine to 1.0: u(1/u) = 1.0
(i.e., both trees recombine, but the question has arithmetic mean of 0 and geometric mean nonzero. Your numbers conversely have geometric mean of 0 and arithmetic mean nonzero)

So, unless told otherwise (as in this question) I agree that following Hull is the right thing to do. The only time it's not quite like these is with Tuckman's interest rate trees; e.g., 5% goes up to 5.5% or down to 4.5%. Because with interest rates we are using, in Linda Allen's terms, absolute changes in the variable.

David