# Binomial Pricing Model d and u

In the binomial pricing model, why do the magnitude of the up factor $$(u)$$ and down factor $$(d)$$ have to be multiplicative inverses? I have read from multiple sources that the reason for this is that an up move followed by a down move $$(ud)$$ will have the same effect as a down move followed by an up move $$(du)$$, which simplifies calculations. However, doesn't this property still hold when $$u$$ and $$d$$ are not inverses? For example, $$1.1*.8 = .8*1.1$$

• It is just convenient that an up move followed by a down move is the same as no move. So up 10 times, down 9 times, in any order, is the same as up once. – Dimitri Vulis Jan 11 at 18:58
• To expand on @DimitriVulis s answer: this is the so-called recombining property of the tree. It has been put there deliberately so that the number of nodes increases only linearly in the number of steps instead of quadratically. You could easily drop the assumption (but you have to fill the degree of freedom) and come up with another tree - which will not be recombining, though. – Kermittfrog Jan 11 at 21:29

## 1 Answer

It does not need to be so always. You can always relax that assumption and come with the pricing by using the fundamental principles. As @Kermittfrog and @Dimitri Vulis commented it is just a matter of convenience for calculations and is called the recombining property. You can find an example in this link here which does not use this assumption to price the options.