Constructing a binomial tree decrease, something, only, increase

prav_Cfa7

Hey guys,

I have a question based on an example from Schweser's study notes. This is probably something really simple but I'm not sure why it doesn't make sense.

on page 247 of the first book an example is demonstrated with probability of a stock price going up as "P" and the price going down as "(1- p)", so only 2 possible outcomes since we're dealing with a binomial distribution. It assumes that the stock will either increase in value by 1% or decrease in value by 1% and states the up-move factor as 1.01, which makes perfect sense but provides the down-move factor as 1/1.01, which does not make sense. For the stock to go down 1% shouldn't the down-move factor be 0.99 and not 0.990099009=1/1.01?

Obviously if a stock goes down 1% today, and up 1% tomorrow, the stock will not be at it's initial trading price and would need to go higher by more than 1% on the second day to get back to that initial value. But using the down move factor of 1/1.01 would leave the stock value unchanged in such scenario.

A quick explanation would be very much appreciated.
Rina

chetan86

I think in the world of option pricing, all binomial trees are constructed with the recombinant property i.e. up moves and down moves offset each other.

So i guess it is a problem in framing the question (by stating that it will decrease by 1%).

Beatnik

Hi, Rina. The quick answer is that the up-factor and the down-factor can be whatever you want it to be. Let's say you are constructing a binomial tree to model the price of a bond. The expected future price of the bond is not equal to the current price. So, if the current price is S, p*U + (1-p)*D <> S. When constructing binomial trees, you are supposed to match the model inputs to the asset you are trying to model.

TLDR version: U and P can be any values, since zero drift is not always true.

kamara5

Thanks to both of you guys.

Ohai, what you said is completely logical and if the example stated the down-move factor explicitly as 1/1.01, the example would have made perfect sense but the description of the case starts out by indicating that the value can either increase by 1% or decrease by 1%. The very same description is again provided in the key concepts section of the reading, however, this time they state the U and D as 1.01 and 1/1.01 respectively instead of providing an absolute percentage for either side.

Thank you again.