Construct a binomial tree to describe stock price movements

CFA Level I Quant
#1

Good day, colleagues!

I have one question about LOS 9g (Schweser): Construct a binomial tree to describe stock price movements:

“A binomial model can be applied to stock price movements. We just need to define the two possible outcomes and the probability that each outcome will occur. Consider a stock with current price S that will, over the next period, either increase in value by 1% or decrease in value by 1% (the only two possible outcomes). The probability of an up-move (the up transition probability, u) is p and the probability of a down-move (the down transition probability, d) is (1 - p). For our example, the up-move factor (U) is 1.01 and the down-move factor (D) is1 / 1.01. So there is a probability p that the stock price will move to S (1.01) over the next period and a probability (1 - p) that the stock price will move to S/1.01.”

I can`t understand, why the down-move factor (D) is 1 / 1.01, not 1 - 0,01=0,99?

#2

I remember reading somewhere that theres not that big of a difference so dont worry too much about it…

#3

d is often calculated as 1/u.

This means that u*d will equal the initial stock price, so an upward movement follwed by a downward movement (or vice versa) yields no change. Also, the binomial tree will work out nicely as time progesses.

Hope this helps

#4

Under this binomial tree model, the price scenarios are horizontal. That is, if your net price movement is X up moves, you are supposed to have the same price, regardless of how many up/down steps you took. This is only possible if d = 1/u.

There are other ways to construct binomial trees. However, this is just the one that CFA teaches. The horizontal prices are convenient to use, so lots of people do it like this.

#5

Ohai,

Wouldnt you need the same number of up/down movements to have a horizontal price shift?

I thought under this model a u*d (or d*u) move is what gives rise to a horizontal shift. This examples seems different because u and d are so close to 1.

I’m getting ready to take a derivatives test that covers the binomial model, so hopefully I’m not confused…

#6

I mean that u*u*d*u*d will give you the same result as u*d*u, since both have net movement of 1*u. We’re probably thinking of something similar.

#7

Ok, I was talking about a net movement of 0, you meant a net movement of u.

Sorry to beat a dead horse, but I haven’t officially started my CFA studying yet. Just wanted to make sure the way I learned it in class was consistent with the CFA stuff.