# 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?

## 1 Answer

It essentially boils down to: same random variable, different probability measures. So when you set u and d, you fix the values that the random variable can take. Probability Measure does not change that- it only re-weights the probability in a way.

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.

In the textbooks, when the binomial model is introduced, u and d are assumed to be given, but these will need to be calibrated based on the market prices, just as one would calibrate the volatility of the geometric brownian process as in the Black scholes model.

• but then the variable has nothing to do with the price of the stock under this new measure. Why care about this new measure how is it meaningful? – user1 Sep 9 '19 at 12:25
• The current price is known right? It is just describing how the price changes, the more volatile the stock, the higher the u. You can view it as an approximation to the log normal that the Black Scholes assumes. – Magic is in the chain Sep 9 '19 at 12:40
• 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
• If you assume you know u and d, then it is relatively straightforward to determine the q’s through the hedging/replicating portfolio argument. There are three main things: 1) How to determine q’s based on hedging/replication argument, 2) how the p’s and q’s are related, this will lead to the very key concept- radon-nikodym derivative, and 3) how to calibrate/ determine the u and d. Eac – Magic is in the chain Sep 9 '19 at 13:27
• By the way, CRR original paper that I can see is available here static.stevereads.com/papers_to_read/… is very accessible and explains the concept really well. Shreve Stochastic Calculus for Finance I covers the details really well. – Magic is in the chain Sep 9 '19 at 13:36