Up move and down move factors for binomial option valuation

So is there an accepted way to come up with the down move factor for binomial option valuation? The way it was done on the CFAI mock exam is a way I have never seen before.

If they tell me that a stock can drop 20% over two periods I would just take the current price and multiply by .8 twice. Absent information about how much the stock is going to drop I have seen people take the up move factor that is given and take the inverse. So a 15% up move would mean a 1/1.15 down move so that you are taking .87 by the price to get how much it would be on the way down.

The way they do it on the mock exam is to take the negative down move of 20% and basically say it’s the inverse of 1.2 for a .833 down move factor.

I guess that is the way I will do it on the exam; but that is not how they show you how to do it in Schweser. Perhaps Schweser is giving too simplistic of examples in their readings.

Honestly, all the questions i have seen provide both up and down %s.

The general way to do it is for the up factor and the down factor to be reciprocals of each other. If the up factor is, say, 1.20, then the down factor is 1/1.20 = 0.83; if the up factor is 1.15, then the down factor is 1/1.15 = 0.870, and so on.

This is done so that the the binomial tree is recombining (i.e., an up move followed by a down move gives the same price as a down move followed by an up move).

[2020 update: I’d forgotten that I’d written this seven years ago. In fact, the tree will be recombining whether the down factor is the reciprocal of the up factor or not. As best I can tell, the reason for having them be reciprocals of each other is so that after an up move followed by a down move (or vice versa), the price will be the same as it is today. Why anyone thinks that is important is beyond me; it isn’t.]

As a practical matter, they’re likely to give you the up and down factors.

I was stuck on that question last night for a long time. I was very confused with the way the mock explained this. I assumed it was an error by CFA.

Is there a mock exam errata? I looked for it but didn’t find anything on the CFA website.

I can’t believe you guys are doing mocks already

errata usually comes out soon.

What is the logic of taking reciprocal?

see S2000magician’s post

absent any additional info, there is no “standard” way of calculating the up and down factors. it was mentioned that the down factor may be a reciprocal of the up factor. that is not generally true.

ther are different definitions of binomial trees. see for example: Cox-Ross-Rubinstein Binomial Tree, Jarrow-Rudd Binomial Tree and Forward Tree, among just a few. They all have different formulas for calculating the up and down factors and will generally be different from one another, including the resulting option price. All of them, however, converges to the Black Scholes price in the limit.

Also, even though the up and down factors for different types of binomial trees are different, their ratio is the same*
ln (u / d) = 2 * sigma * sqrt(h)
where u is the up factor
d is the down factor
and h is the length of a single period

and of course, they all assume that interest rate and volatility of the stock does not change over time.

  • from Robert McDonald, Derivatives Markets, 3rd ed