# Binomial model in Björk's Arbitrage Theory in Continuous Time

I am having some trouble with variable $$Z$$ introduced in chapter $$2$$ in Björk's text. In the beginning, it is the random variable that attains $$u$$ resp. $$d$$ with probabilities $$p_{1}$$ and $$p_{2}$$, i.e., the actual probabilities of the stock going up or down.

Later on, however, it seems that he assumes that $$Z$$ attains $$u$$ resp. $$d$$ with probabilities related to the martingale measure $$\mathbb{Q}$$.

From what I can see, these do not have to be the same. In fact, $$u$$ and $$d$$ determine $$q_{1}$$ and $$q_{2}$$, but $$u$$ and $$d$$ are fixed from the beginning and, thus, can not be chosen freely.

Is this observation correct? If it is correct, can someone explain why this is reasonable?

The probability $$p_1$$ and $$p_2$$ are the probabilities of the two states under the P(physical) measure, and these same states have probabilities $$q_1$$ and $$q_2$$ under the Q(risk neutral) measure.
• I don't get it. We have a stock which can go up or down with some probabilities say $p_{1}$ and $p_{2}$ i.e it hits say 120 by $p_{1}$ and $0.8$ by $p_{2}$. But now we change these probabilities based the values $120$ and $80$ to some new porbabilites i.e the $q_{i}$, which the variable hit these values with. Is that right? The probabiltes which $Z$ hit $u$ or $d$ is changed to new probablites depending on $u$ and $d$? – user1 Sep 9 '19 at 12:48